diff --git a/thys/Padic_Field/Cring_Multivariable_Poly.thy b/thys/Padic_Field/Cring_Multivariable_Poly.thy
--- a/thys/Padic_Field/Cring_Multivariable_Poly.thy
+++ b/thys/Padic_Field/Cring_Multivariable_Poly.thy
@@ -1,8548 +1,8548 @@
 theory Cring_Multivariable_Poly
-imports  "HOL-Algebra.Indexed_Polynomials" Padic_Ints.Cring_Poly 
-begin 
+imports  "HOL-Algebra.Indexed_Polynomials" Padic_Ints.Cring_Poly
+begin
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 section\<open>Multivariable Polynomials Over a Commutative Ring\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 text\<open>
-  This theory extends the content of \texttt{HOL-Algebra.Indexed\_Polynomials}, mainly in the 
-  context of a commutative base ring. In particular, the ring of polynomials over an arbitrary 
+  This theory extends the content of \texttt{HOL-Algebra.Indexed\_Polynomials}, mainly in the
+  context of a commutative base ring. In particular, the ring of polynomials over an arbitrary
   variable set is explicitly witnessed as a ring itself, which is commutative if the base is
-  commutative, and a domain if the base ring is a domain. A universal property for polynomial 
-  evaluation is proved, which allows us to embed polynomial rings in a ring of functions over the 
+  commutative, and a domain if the base ring is a domain. A universal property for polynomial
+  evaluation is proved, which allows us to embed polynomial rings in a ring of functions over the
   base ring, as well as construe multivariable polynomials as univariate polynomials over a small
   base polynomial ring.
 \<close>
 
 type_synonym 'a monomial = "'a multiset"
 type_synonym ('b,'a) mvar_poly = "'a multiset \<Rightarrow> 'b"
 type_synonym ('a,'b) ring_hom = "'a \<Rightarrow> 'b"
 type_synonym 'a u_poly = "nat \<Rightarrow> 'a"
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsection\<open>Lemmas about multisets\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 text\<open>
-  Since multisets function as monomials in this formalization, we'll need some simple lemmas 
+  Since multisets function as monomials in this formalization, we'll need some simple lemmas
   about multisets in order to define degree functions and certain lemmas about factorizations.
   Those are provided in this section.
 \<close>
 
-lemma count_size:                       
+lemma count_size:
   assumes "size m \<le> K"
   shows "count m i \<le> K"
-  using assms 
+  using assms
   by (metis count_le_replicate_mset_subset_eq dual_order.trans order_refl size_mset_mono size_replicate_mset)
 
 text\<open>The following defines the set of monomials with nonzero coefficients for a given polynomial:\<close>
 
 definition monomials_of :: "('a,'c) ring_scheme \<Rightarrow> ('a, 'b) mvar_poly \<Rightarrow> ('b monomial) set" where
 "monomials_of R P = {m. P m \<noteq> \<zero>\<^bsub>R\<^esub>}"
 
-context ring 
-begin 
+context ring
+begin
 
 lemma monomials_of_index_free:
   assumes "index_free P i"
   assumes "m \<in> monomials_of R P"
   shows "count m i = 0"
-  using assms  
-  unfolding monomials_of_def index_free_def 
+  using assms
+  unfolding monomials_of_def index_free_def
   by (meson count_inI mem_Collect_eq)
 
 lemma index_freeI:
   assumes "\<And>m. m \<in> monomials_of R P \<Longrightarrow> count m i = 0"
   shows "index_free P i"
   unfolding index_free_def
 proof safe
   fix m
   assume "i \<in># m"
   then have "count m i \<noteq> 0"
     by simp
   then have "m \<notin> monomials_of R P"
-    using assms 
+    using assms
     by meson
   then show "P m = \<zero>"
-    unfolding monomials_of_def 
+    unfolding monomials_of_def
     by blast
 qed
 
 text\<open>\texttt{monomials\_of} R is subadditive\<close>
 
 lemma monomials_of_add:
 "monomials_of R (P \<Oplus> Q) \<subseteq> (monomials_of R P) \<union> (monomials_of R Q)"
 proof
   fix x
   assume "x \<in> monomials_of R (P \<Oplus> Q) "
   then have "P x \<oplus> Q x \<noteq>\<zero>"
     by (simp add: indexed_padd_def monomials_of_def)
   then have "P x \<noteq>\<zero> \<or> Q x \<noteq> \<zero>"
     by auto
-  then show "x \<in> (monomials_of R P) \<union> (monomials_of R Q)" 
+  then show "x \<in> (monomials_of R P) \<union> (monomials_of R Q)"
     by (simp add: monomials_of_def)
 qed
 
 lemma monomials_of_add_finite:
   assumes "finite (monomials_of R P)"
   assumes "finite (monomials_of R Q)"
   shows "finite (monomials_of R (P \<Oplus> Q))"
   by (meson assms(1) assms(2) finite_Un finite_subset monomials_of_add)
 
 lemma monomials_ofE:
   assumes "m \<in> monomials_of R p"
   shows "p m \<noteq> \<zero>"
-  using assms 
-  unfolding monomials_of_def 
-  by simp 
+  using assms
+  unfolding monomials_of_def
+  by simp
 
 lemma complement_of_monomials_of:
   assumes "m \<notin> monomials_of R P"
   shows "P m = \<zero>"
-  using assms  
+  using assms
   unfolding monomials_of_def
-  by blast 
+  by blast
 
 text\<open>Multiplication by an indexed variable corresponds to adding that index to each monomial:\<close>
 
 lemma monomials_of_p_mult:
 "monomials_of R (P \<Otimes> i) = {m. \<exists> n \<in> (monomials_of R P). m = add_mset i n}"
 proof
   show "monomials_of R (P \<Otimes> i) \<subseteq> {m. \<exists>n\<in>monomials_of R P. m = add_mset i n}"
     proof
       fix m
       assume A: "m \<in> monomials_of R (P \<Otimes> i)"
       show "m \<in> {m. \<exists>n\<in>monomials_of R P. m = add_mset i n}"
       proof-
         have "(P \<Otimes> i) m \<noteq> \<zero>"
           by (simp add: A monomials_ofE)
         then have "P (m - {# i #}) \<noteq>\<zero>"
-          unfolding indexed_pmult_def 
+          unfolding indexed_pmult_def
           by presburger
         then have 0: "(m - {# i #}) \<in> monomials_of R P"
           by (meson complement_of_monomials_of)
         have 1: " m = add_mset i  (m - {# i #})"
           by (metis \<open>(P \<Otimes> i) m \<noteq> \<zero>\<close> add_mset_remove_trivial_If indexed_pmult_def)
         then show ?thesis using 0 1
           by blast
       qed
     qed
   show "{m. \<exists>n\<in>monomials_of R P. m = add_mset i n} \<subseteq> monomials_of R (P \<Otimes> i)"
-    unfolding monomials_of_def indexed_pmult_def 
+    unfolding monomials_of_def indexed_pmult_def
     by auto
 qed
 
 lemma monomials_of_p_mult':
 "monomials_of R (p \<Otimes> i) = add_mset i ` (monomials_of R p)"
   using  monomials_of_p_mult
   by (simp add: monomials_of_p_mult image_def)
 
 lemma monomials_of_p_mult_finite:
   assumes "finite (monomials_of R P)"
   shows "finite (monomials_of R (P \<Otimes> i))"
   using assms monomials_of_p_mult'[of P i]
   by simp
 
 text\<open>Monomials of a constant either consist of the empty multiset, or nothing:\<close>
 
-lemma monomials_of_const: 
+lemma monomials_of_const:
 "(monomials_of R (indexed_const k)) = (if (k = \<zero>) then {} else {{#}})"
-  unfolding monomials_of_def indexed_const_def 
+  unfolding monomials_of_def indexed_const_def
   by simp
 
-lemma monomials_of_const_finite: 
+lemma monomials_of_const_finite:
 "finite (monomials_of R (indexed_const k))"
   by (simp add: monomials_of_const)
-  
+
 text\<open>A polynomial always has finitely many monomials:\<close>
 lemma monomials_finite:
   assumes "P \<in> indexed_pset K I"
   shows "finite (monomials_of R P)"
-  using assms 
+  using assms
   apply(induction P)
   using monomials_of_const_finite apply blast
     using monomials_of_add_finite apply blast
       by (simp add: monomials_of_p_mult_finite)
 end
 
 (**************************************************************************************************)
 (**************************************************************************************************)
     subsection\<open>Turning monomials into polynomials\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
- 
+
 text\<open>Constructor for turning a monomial into a polynomial\<close>
 
 definition mset_to_IP :: "('a, 'b) ring_scheme \<Rightarrow> 'c monomial \<Rightarrow> ('a,'c) mvar_poly" where
-"mset_to_IP R m n= (if (n = m) then \<one>\<^bsub>R\<^esub> else \<zero>\<^bsub>R\<^esub>)" 
+"mset_to_IP R m n= (if (n = m) then \<one>\<^bsub>R\<^esub> else \<zero>\<^bsub>R\<^esub>)"
 
 definition var_to_IP :: "('a, 'b) ring_scheme \<Rightarrow> 'c \<Rightarrow> ('a,'c) mvar_poly" ("pvar") where
 "var_to_IP R i = mset_to_IP R {#i#}"
 
 context ring
 begin
 
 lemma mset_to_IP_simp[simp]:
 "mset_to_IP R m m = \<one>"
   by (simp add: mset_to_IP_def)
 
 lemma mset_to_IP_simp'[simp]:
   assumes "n \<noteq>m"
   shows "mset_to_IP R m n = \<zero>"
   by (simp add: assms mset_to_IP_def)
 
 lemma(in cring) monomials_of_mset_to_IP:
   assumes "\<one> \<noteq>\<zero>"
   shows "monomials_of R (mset_to_IP R m) = {m}"
-  unfolding monomials_of_def mset_to_IP_def 
+  unfolding monomials_of_def mset_to_IP_def
 proof
   show "{ma. (if ma = m then \<one> else \<zero>) \<noteq> \<zero>} \<subseteq> {m}"
     by auto
   show "{m} \<subseteq> {ma. (if ma = m then \<one> else \<zero>) \<noteq> \<zero>}"
     using assms by auto
 qed
 
-end 
+end
 
 text\<open>The set of monomials of a fixed \<open>P\<close> which include a given variable:\<close>
 
 definition monomials_with :: "('a, 'b) ring_scheme \<Rightarrow> 'c \<Rightarrow> ('a,'c) mvar_poly \<Rightarrow> ('c monomial) set" where
 "monomials_with R i P = {m. m \<in> monomials_of R P \<and> i \<in># m}"
 
 context ring
 begin
 
 lemma monomials_withE:
   assumes "m \<in> monomials_with R i P"
   shows "i \<in># m"
         "m \<in> monomials_of R P"
-  using assms unfolding monomials_with_def  
+  using assms unfolding monomials_with_def
   apply blast
-     using assms unfolding monomials_with_def  
+     using assms unfolding monomials_with_def
      by blast
 
 lemma monomials_withI:
   assumes "i \<in># m"
   assumes "m \<in> monomials_of R P"
   shows "m \<in> monomials_with R i P"
-  using assms 
-  unfolding monomials_with_def 
-  by blast 
-
-end 
+  using assms
+  unfolding monomials_with_def
+  by blast
+
+end
 
 text\<open>Restricting a polynomial to be zero outside of a given monomial set:\<close>
 
-definition restrict_poly_to_monom_set :: 
+definition restrict_poly_to_monom_set ::
   "('a, 'b) ring_scheme \<Rightarrow> ('a,'c) mvar_poly \<Rightarrow> ('c monomial) set \<Rightarrow>('a,'c) mvar_poly"  where
 "restrict_poly_to_monom_set R P m_set m = (if m \<in> m_set then P m else \<zero>\<^bsub>R\<^esub>)"
 
 context ring
 begin
 
 lemma restrict_poly_to_monom_set_coeff:
   assumes "carrier_coeff P"
   shows "carrier_coeff (restrict_poly_to_monom_set R P Ms)"
   by (metis assms carrier_coeff_def restrict_poly_to_monom_set_def zero_closed)
 
 lemma restrict_poly_to_monom_set_coeff':
   assumes "P \<in> indexed_pset K I"
   assumes "I \<noteq> {}"
   assumes "\<And>m. P m \<in> S"
   assumes "\<zero> \<in> S"
   shows "\<And>m.  (restrict_poly_to_monom_set R P m_set m) \<in> S"
-  using assms 
+  using assms
   unfolding restrict_poly_to_monom_set_def
   by simp
 
 lemma restrict_poly_to_monom_set_monoms:
   assumes "P \<in> indexed_pset I K"
   assumes "m_set \<subseteq> monomials_of R P"
   shows "monomials_of R (restrict_poly_to_monom_set R P m_set) = m_set"
-  using assms 
+  using assms
   unfolding monomials_of_def restrict_poly_to_monom_set_def
   by (simp add: subset_iff)
 end
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsection\<open>Degree Functions\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 (**************************************************************************************************)
 
   (**************************************************************************************************)
   (**************************************************************************************************)
   subsubsection\<open>Total Degree Function\<close>
   (**************************************************************************************************)
   (**************************************************************************************************)
 
 
 lemma multi_set_size_count:
   fixes m :: "'c monomial"
   shows "size m \<ge> count m i"
   by (metis count_le_replicate_mset_subset_eq order_refl size_mset_mono size_replicate_mset)
 
 text\<open>Total degree function\<close>
 
 definition total_degree :: "('a, 'b) ring_scheme \<Rightarrow> ('a, 'c) mvar_poly \<Rightarrow> nat" where
 "total_degree R P = (if (monomials_of R P = {}) then 0 else Max (size ` (monomials_of R P)))"
 
 context ring
 begin
 
 lemma total_degree_ineq:
   assumes "m \<in> monomials_of R P"
   assumes "finite (monomials_of R P)"
   shows "total_degree R P \<ge> size m"
-  unfolding total_degree_def using assms   
+  unfolding total_degree_def using assms
   by force
 
 lemma total_degree_in_monomials_of:
   assumes "monomials_of R P \<noteq> {}"
   assumes "finite (monomials_of R P)"
   obtains m where "m \<in> monomials_of R P \<and> size m = total_degree R P"
-    using assms Max_in[of "(size ` monomials_of R P)"] unfolding total_degree_def  
+    using assms Max_in[of "(size ` monomials_of R P)"] unfolding total_degree_def
     by (metis (mono_tags, lifting) empty_is_image finite_imageI image_iff)
 
 lemma total_degreeI:
   assumes "finite (monomials_of R P)"
   assumes "\<exists>m. m \<in> monomials_of R P \<and> size m = n"
   assumes "\<And>m. m \<in> monomials_of R P \<Longrightarrow> size m \<le> n"
   shows "n = total_degree R P"
 proof(cases "monomials_of R P = {}")
   case True
-  then show ?thesis 
-    using assms by blast  
+  then show ?thesis
+    using assms by blast
 next
   case False
   obtain m where m_def: "m \<in> monomials_of R P \<and> size m = n"
-    using assms by blast 
+    using assms by blast
   have 0: "n \<in> (size ` (monomials_of R P))"
     using m_def
     by blast
   have "\<And>k. k \<in> (size ` (monomials_of R P)) \<Longrightarrow> k \<le> n"
-    using assms 
+    using assms
     by blast
   then have 1: "\<And>m.  m \<in> (monomials_of R P) \<Longrightarrow> size m\<le> n"
     by blast
   obtain m' where m'_def: "m' \<in> monomials_of R P \<and> size m' = total_degree R P"
     using assms total_degree_in_monomials_of
     by blast
   then have 2: "size m' \<le> n"
-    using 1 
+    using 1
     by blast
   have 3: "n \<le>size m'"
     using assms m'_def total_degree_ineq by auto
-  show ?thesis using 2 3 
+  show ?thesis using 2 3
     using dual_order.antisym m'_def by blast
-qed  
-end 
+qed
+end
 
   (**************************************************************************************************)
   (**************************************************************************************************)
   subsubsection\<open>Degree in One Variable\<close>
   (**************************************************************************************************)
   (**************************************************************************************************)
 
 definition degree_in_var ::
   "('a, 'b) ring_scheme \<Rightarrow> ('a, 'c) mvar_poly \<Rightarrow> 'c \<Rightarrow> nat" where
 "degree_in_var R P i =  (if (monomials_of R P = {}) then 0 else Max ((\<lambda>m. count m i) ` (monomials_of R P)))"
 
 context ring
 begin
 
 lemma degree_in_var_ineq:
   assumes "m \<in> monomials_of R P"
   assumes "finite (monomials_of R P)"
   shows "degree_in_var R P i \<ge> count m i"
-  unfolding degree_in_var_def using assms   
+  unfolding degree_in_var_def using assms
   by force
 
 lemma degree_in_var_in_monomials_of:
   assumes "monomials_of R P \<noteq> {}"
   assumes "finite (monomials_of R P)"
   obtains m where "m \<in> monomials_of R P \<and> count m i = degree_in_var R P i"
   using assms Max_in[of "((\<lambda>m. count m i) ` (monomials_of R P))"] unfolding degree_in_var_def
   by (metis (no_types, lifting) empty_is_image finite_imageI image_iff)
 
 lemma degree_in_varI:
   assumes "finite (monomials_of R P)"
   assumes "\<exists>m. m \<in> monomials_of R P \<and> count m i = n"
   assumes "\<And>c. c \<in> monomials_of R P \<Longrightarrow> count c i \<le> n"
   shows "n = degree_in_var R P i"
 proof-
   obtain l where l_def: "l \<in> monomials_of R P \<and> count l i = degree_in_var R P i"
     by (metis assms(1) assms(2) degree_in_var_in_monomials_of equals0D)
   have "degree_in_var R P i \<le> n"
-    using assms(3) l_def 
+    using assms(3) l_def
     by force
-  then show ?thesis 
-    using assms(1) assms(2) dual_order.antisym degree_in_var_ineq 
-    by fastforce  
+  then show ?thesis
+    using assms(1) assms(2) dual_order.antisym degree_in_var_ineq
+    by fastforce
 qed
 
 text\<open>Total degree bounds degree in a single variable:\<close>
 
 lemma total_degree_degree_in_var:
   assumes "finite (monomials_of R P)"
   shows "total_degree R P \<ge> degree_in_var R P i"
 proof(cases " (monomials_of R P) = {}")
   case True
-  then show ?thesis 
-    unfolding total_degree_def degree_in_var_def 
+  then show ?thesis
+    unfolding total_degree_def degree_in_var_def
     by simp
 next
   case False
   then   obtain m1 where m1_def: "m1 \<in> monomials_of R P \<and> count m1 i = degree_in_var R P i"
     by (meson assms degree_in_var_in_monomials_of)
   have "size m1 \<ge>count m1 i"
     by (simp add: multi_set_size_count)
-  then show ?thesis 
+  then show ?thesis
     by (simp add: assms local.ring_axioms m1_def order.trans ring.total_degree_ineq)
 qed
-end 
+end
 
 text\<open>The set of monomials of maximal degree in variable \<open>i\<close>:\<close>
 
-definition max_degree_monoms_in_var :: 
+definition max_degree_monoms_in_var ::
   "('a, 'b) ring_scheme \<Rightarrow> ('a, 'c) mvar_poly \<Rightarrow> 'c \<Rightarrow> ('c monomial) set" where
 "max_degree_monoms_in_var R P i = {m. m \<in> monomials_of R P \<and> count m i = degree_in_var R P i}"
 
 context ring
 begin
 
 lemma max_degree_monoms_in_var_memI:
   assumes "m \<in> monomials_of R P"
   assumes "count m i = degree_in_var R P i"
   shows "m \<in> max_degree_monoms_in_var R P i"
-  using assms unfolding max_degree_monoms_in_var_def 
+  using assms unfolding max_degree_monoms_in_var_def
   by blast
 
 lemma max_degree_monoms_in_var_memE:
   assumes "m \<in> max_degree_monoms_in_var R P i"
   shows   "m \<in> monomials_of R P"
           "count m i = degree_in_var R P i"
-  using assms unfolding max_degree_monoms_in_var_def 
+  using assms unfolding max_degree_monoms_in_var_def
   apply blast
-    using assms unfolding max_degree_monoms_in_var_def 
+    using assms unfolding max_degree_monoms_in_var_def
     by blast
-end 
-
-text\<open>The set of monomials of \<open>P\<close> of fixed degree in variable \<open>i\<close>:\<close> 
+end
+
+text\<open>The set of monomials of \<open>P\<close> of fixed degree in variable \<open>i\<close>:\<close>
 
 definition fixed_degree_in_var ::
   "('a, 'b) ring_scheme \<Rightarrow> ('a, 'c) mvar_poly \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> ('c monomial) set" where
 "fixed_degree_in_var R P i n = {m. m \<in> monomials_of R P \<and> count m i = n}"
 
-context ring 
+context ring
 begin
 
 lemma fixed_degree_in_var_subset:
 "fixed_degree_in_var R P i n \<subseteq> monomials_of R P"
-  unfolding fixed_degree_in_var_def 
+  unfolding fixed_degree_in_var_def
   by blast
 
 lemma fixed_degree_in_var_max_degree_monoms_in_var:
 "max_degree_monoms_in_var R P i = fixed_degree_in_var R P i (degree_in_var R P i)"
-  unfolding max_degree_monoms_in_var_def fixed_degree_in_var_def 
+  unfolding max_degree_monoms_in_var_def fixed_degree_in_var_def
   by auto
 
 lemma fixed_degree_in_varI:
   assumes "m \<in> monomials_of R P"
   assumes "count m i = n"
   shows "m \<in> fixed_degree_in_var R P i n"
-  unfolding fixed_degree_in_var_def 
-  using assms 
-  by blast 
+  unfolding fixed_degree_in_var_def
+  using assms
+  by blast
 
 lemma fixed_degree_in_varE:
   assumes "m \<in> fixed_degree_in_var R P i n"
   shows "m \<in> monomials_of R P"
           "count m i = n"
   apply (meson assms fixed_degree_in_var_subset in_mono)
     using assms fixed_degree_in_var_def by force
 
 definition restrict_to_var_deg ::
  "('a,'c) mvar_poly \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'c monomial \<Rightarrow> 'a"  where
 "restrict_to_var_deg P i n = restrict_poly_to_monom_set R P (fixed_degree_in_var R P i n)"
 
 lemma restrict_to_var_deg_var_deg:
   assumes "finite (monomials_of R P)"
   assumes "Q = restrict_to_var_deg P i n"
   assumes "monomials_of R Q \<noteq> {}"
   shows "n = degree_in_var R Q i"
   apply(rule degree_in_varI)
-    apply (metis assms(1) assms(2) fixed_degree_in_varE(1) monomials_ofE restrict_poly_to_monom_set_def 
+    apply (metis assms(1) assms(2) fixed_degree_in_varE(1) monomials_ofE restrict_poly_to_monom_set_def
       restrict_to_var_deg_def rev_finite_subset subsetI)
       apply (metis (full_types) assms(2) assms(3) equals0I fixed_degree_in_varE(2)
       monomials_ofE restrict_poly_to_monom_set_def restrict_to_var_deg_def)
         by (metis assms(2) eq_iff fixed_degree_in_varE(2) monomials_ofE restrict_poly_to_monom_set_def restrict_to_var_deg_def)
 
 lemma index_free_degree_in_var[simp]:
   assumes "index_free P i"
   shows "degree_in_var R P i = 0"
 proof(cases "monomials_of R P = {}")
   case True
-  then show ?thesis 
-    using assms 
-    unfolding degree_in_var_def 
+  then show ?thesis
+    using assms
+    unfolding degree_in_var_def
     by simp
 next
   case False
   then have 0: "degree_in_var R P i =  Max ((\<lambda>m. count m i) ` (monomials_of R P))"
-    unfolding degree_in_var_def 
-    by simp 
+    unfolding degree_in_var_def
+    by simp
   have"((\<lambda>m. count m i) ` (monomials_of R P)) = {0}"
   proof
     show "(\<lambda>m. count m i) ` monomials_of R P \<subseteq> {0}"
-      using False assms monomials_of_index_free[of P i] 
+      using False assms monomials_of_index_free[of P i]
       by auto
-    show "{0} \<subseteq> (\<lambda>m. count m i) ` monomials_of R P" 
-      using False  \<open>(\<lambda>m. count m i) ` monomials_of R P \<subseteq> {0}\<close> 
+    show "{0} \<subseteq> (\<lambda>m. count m i) ` monomials_of R P"
+      using False  \<open>(\<lambda>m. count m i) ` monomials_of R P \<subseteq> {0}\<close>
       by auto
   qed
-  then show ?thesis using 0 
+  then show ?thesis using 0
     by simp
 qed
 
 lemma degree_in_var_index_free:
   assumes "degree_in_var R P i = 0"
   assumes "finite (monomials_of R P)"
-  shows "index_free P i" 
+  shows "index_free P i"
   apply(rule index_freeI)
   by (metis assms(1) assms(2) degree_in_var_ineq le_zero_eq)
 
-end 
+end
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsection\<open>Constructing the Multiplication Operation on the Ring of Indexed Polynomials\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
   (**********************************************************************)
   (**********************************************************************)
   subsubsection\<open>The Set of Factors of a Fixed Monomial\<close>
   (**********************************************************************)
   (**********************************************************************)
 
   text\<open>The following function maps a monomial to the set of monomials which divide it:\<close>
 
 definition mset_factors :: "'c monomial \<Rightarrow> ('c monomial) set" where
 "mset_factors m = {n. n \<subseteq># m}"
 
-context ring 
+context ring
 begin
 
 lemma monom_divides_factors:
 "n \<in> (mset_factors m)\<longleftrightarrow> n \<subseteq># m"
-  unfolding mset_factors_def by auto 
+  unfolding mset_factors_def by auto
 
 lemma mset_factors_mono:
   assumes "n \<subseteq># m"
   shows "mset_factors n \<subseteq> mset_factors m"
-  unfolding mset_factors_def 
+  unfolding mset_factors_def
   by (simp add: Collect_mono_iff assms subset_mset.order_trans)
 
 lemma mset_factors_size_bound:
   assumes "n \<in> mset_factors m"
   shows "size n \<le> size m"
-  using assms 
-  unfolding mset_factors_def 
+  using assms
+  unfolding mset_factors_def
   by (simp add: size_mset_mono)
 
 lemma sets_to_inds_finite:
   assumes "finite I"
   shows "finite S \<Longrightarrow> finite (Pi\<^sub>E S (\<lambda>_. I))"
-  using assms 
+  using assms
   by (simp add: finite_PiE)
 
 end
 
     (**********************************************************************************************)
     (**********************************************************************************************)
     subsubsection\<open>Finiteness of the Factor Set of a Monomial\<close>
     (**********************************************************************************************)
     (**********************************************************************************************)
 
 text\<open>
   This section shows that any monomial m has only finitely many factors. This is done by mapping
-  the set of factors injectively into a finite extensional function set. Explicitly, a monomial is 
+  the set of factors injectively into a finite extensional function set. Explicitly, a monomial is
   just mapped to its count function, restricted to the set of indices where the count is nonzero.
 \<close>
 
 definition mset_factors_to_fun ::
   "('a,'b) ring_scheme \<Rightarrow> 'c monomial \<Rightarrow> 'c monomial \<Rightarrow> ('c \<Rightarrow> nat)" where
-"mset_factors_to_fun R m n = (if (n \<in> mset_factors m) then 
+"mset_factors_to_fun R m n = (if (n \<in> mset_factors m) then
                                            (restrict  (count n) (set_mset m)) else undefined)"
-context ring 
+context ring
 begin
 
 lemma mset_factors_to_fun_prop:
   assumes "size m = n"
   shows "mset_factors_to_fun R m \<in> (mset_factors m) \<rightarrow> (Pi\<^sub>E (set_mset m) (\<lambda>_. {0.. n}))"
 proof
-  fix x 
+  fix x
   assume A: "x \<in> (mset_factors m)"
-  show "mset_factors_to_fun R m x \<in> (set_mset m) \<rightarrow>\<^sub>E {0..n} "    
+  show "mset_factors_to_fun R m x \<in> (set_mset m) \<rightarrow>\<^sub>E {0..n} "
   proof
     show "\<And>xa. xa \<in># m \<Longrightarrow> mset_factors_to_fun R m x xa \<in> {0..n}"
       proof-
         fix y
         assume Ay: "y \<in># m"
         show P0: "mset_factors_to_fun R m x y \<in> {0..n}"
         proof-
           have "mset_factors_to_fun R m x = restrict (count x) (set_mset m)"
             using A  unfolding mset_factors_to_fun_def
             by simp
           then have "mset_factors_to_fun R m x y  = count x y"
-            using Ay 
-            by simp        
+            using Ay
+            by simp
           then show ?thesis
-            using A INTEG.R.mset_factors_size_bound assms count_size by fastforce             
+            using A INTEG.R.mset_factors_size_bound assms count_size by fastforce
         qed
       qed
     fix z
     assume Ay: "z \<notin>#m"
     show "mset_factors_to_fun R m x z = undefined"
-      using A Ay unfolding mset_factors_to_fun_def  
+      using A Ay unfolding mset_factors_to_fun_def
       by simp
-  qed 
+  qed
 qed
 
 lemma mset_factors_to_fun_inj:
   shows "inj_on (mset_factors_to_fun R m) (mset_factors m) "
 proof
   fix x y
   assume A: "x \<in> mset_factors m" "y \<in> mset_factors m"
   show "mset_factors_to_fun R m x = mset_factors_to_fun R m y \<Longrightarrow> x = y"
   proof-
     assume A0:  "mset_factors_to_fun R m x = mset_factors_to_fun R m y"
     show " x = y"
     proof-
       have "\<And>i. count x i = count y i"
       proof- fix i
         show "count x i = count y i"
         proof(cases "i \<in># m")
           case True
-          then show ?thesis using A0 A unfolding mset_factors_to_fun_def  
+          then show ?thesis using A0 A unfolding mset_factors_to_fun_def
             by (metis restrict_def)
         next
           case False
-          then show ?thesis 
+          then show ?thesis
              using A0 A unfolding mset_factors_to_fun_def
              by (metis monom_divides_factors count_inI mset_subset_eqD)
         qed
       qed
-      then show ?thesis 
+      then show ?thesis
         using multiset_eqI by blast
     qed
   qed
 qed
 
 lemma finite_target:
 "finite (Pi\<^sub>E (set_mset m) (\<lambda>_. {0..(n::nat)}))"
 proof-
   have 0: "finite (set_mset m)"
-    by simp 
+    by simp
   have 1: "finite ({0..n})"
     by simp
-  then show ?thesis using 0 
+  then show ?thesis using 0
     by (simp add: finite_PiE)
 qed
 
 text\<open>A multiset has only finitely many factors:\<close>
 
 lemma mset_factors_finite[simp]:
 "finite (mset_factors m)"
 proof-
   have 0: "inj_on (mset_factors_to_fun R m) (mset_factors m) "
     by (simp add: mset_factors_to_fun_inj)
   have 1: "(mset_factors_to_fun R m) \<in> (mset_factors m) \<rightarrow> (Pi\<^sub>E (set_mset m) (\<lambda>_. {0 .. (size m)}))"
     by (metis mset_factors_to_fun_prop)
   have 2: "finite (Pi\<^sub>E (set_mset m) (\<lambda>_. {0 .. (size m)}))"
     by (simp add: finite_target)
   have 3: "((mset_factors_to_fun R m) ` (mset_factors m)) \<subseteq> (Pi\<^sub>E (set_mset m) (\<lambda>_. {0 .. (size m)}))"
-    using 1 2 
+    using 1 2
     by blast
   then have "finite ((mset_factors_to_fun R m) ` (mset_factors m))"
-    using 2 finite_subset by auto    
+    using 2 finite_subset by auto
   then show ?thesis using 0 1 2
-    finite_imageD[of "mset_factors_to_fun R m" "mset_factors m" ] 
+    finite_imageD[of "mset_factors_to_fun R m" "mset_factors m" ]
     by blast
 qed
 
 end
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsubsection\<open>Definition of Indexed Polynomial Multiplication.\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
-context ring 
+context ring
 begin
 
 text\<open>Monomial division:\<close>
 
 lemma monom_divide:
   assumes "n \<in> mset_factors m"
   shows "(THE k. n + k = m ) = m - n "
   apply(rule the_equality)
-  using assms unfolding mset_factors_def  
+  using assms unfolding mset_factors_def
   apply simp
-    using assms unfolding mset_factors_def  
+    using assms unfolding mset_factors_def
     by auto
 
 text\<open>A monomial is a factor of itself:\<close>
 
 lemma m_factor[simp]:
 "m \<in> mset_factors m"
   using local.ring_axioms ring.monom_divides_factors by blast
 
-end 
+end
 
 text\<open>The definition of polynomial multiplication:\<close>
 
 definition P_ring_mult :: "('a, 'b) ring_scheme \<Rightarrow> ('a,'c) mvar_poly \<Rightarrow> ('a,'c) mvar_poly \<Rightarrow> 'c monomial \<Rightarrow> 'a"
   where
 "P_ring_mult R P Q m = (finsum R (\<lambda>x. (P x) \<otimes>\<^bsub>R\<^esub> (Q (m - x))) (mset_factors m))"
 
 abbreviation(in ring) P_ring_mult_in_ring (infixl "\<Otimes>\<^sub>p" 65)where
 "P_ring_mult_in_ring \<equiv>  P_ring_mult R"
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsubsection\<open>Distributivity Laws for Polynomial Multiplication\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
-context ring 
-begin 
+context ring
+begin
 
 lemma P_ring_rdistr:
   assumes "carrier_coeff a"
   assumes "carrier_coeff b"
   assumes "carrier_coeff c"
   shows "a \<Otimes>\<^sub>p (b \<Oplus> c) = (a \<Otimes>\<^sub>p b)\<Oplus> (a \<Otimes>\<^sub>p c)"
 proof
   fix m
   show "(a \<Otimes>\<^sub>p (b \<Oplus> c)) m = (a \<Otimes>\<^sub>p b \<Oplus> (a \<Otimes>\<^sub>p c)) m"
   proof-
-    have RHS: "(a \<Otimes>\<^sub>p b \<Oplus> (a \<Otimes>\<^sub>p c)) m = 
+    have RHS: "(a \<Otimes>\<^sub>p b \<Oplus> (a \<Otimes>\<^sub>p c)) m =
                   (\<Oplus>x\<in>mset_factors m. a x \<otimes> b (m - x)) \<oplus> (\<Oplus>x\<in>mset_factors m. a x \<otimes> c (m - x))"
-      unfolding indexed_padd_def P_ring_mult_def by auto 
-    have LHS: "(a \<Otimes>\<^sub>p (b \<Oplus> c)) m= 
+      unfolding indexed_padd_def P_ring_mult_def by auto
+    have LHS: "(a \<Otimes>\<^sub>p (b \<Oplus> c)) m=
                   ( \<Oplus>x\<in>mset_factors m. a x \<otimes> b (m - x) \<oplus> a x \<otimes> c (m - x))"
-      unfolding indexed_padd_def  P_ring_mult_def 
+      unfolding indexed_padd_def  P_ring_mult_def
       by (meson assms(1) assms(2) assms(3) local.ring_axioms r_distr ring.carrier_coeff_def)
-    have RHS': "(a \<Otimes>\<^sub>p b \<Oplus> (a \<Otimes>\<^sub>p c)) m = 
+    have RHS': "(a \<Otimes>\<^sub>p b \<Oplus> (a \<Otimes>\<^sub>p c)) m =
                   (\<Oplus>x\<in>mset_factors m. a x \<otimes> b (m - x) \<oplus> a x \<otimes> c (m - x))"
     proof-
       have 0: "(\<lambda>x. a x \<otimes> b (m - x)) \<in> mset_factors m \<rightarrow> carrier R"
       proof
         fix x assume "x \<in> mset_factors m"
         then show "a x \<otimes> b (m - x) \<in> carrier R"
-          using assms carrier_coeffE 
+          using assms carrier_coeffE
           by blast
       qed
       have 1: "(\<lambda>x. a x \<otimes> c (m - x)) \<in> mset_factors m \<rightarrow> carrier R"
       proof
         fix x assume "x \<in> mset_factors m"
         then show "a x \<otimes> c (m - x) \<in> carrier R"
-          using assms carrier_coeffE 
-          by blast 
+          using assms carrier_coeffE
+          by blast
       qed
-      then show ?thesis 
+      then show ?thesis
         using 0 1 RHS assms finsum_addf[of "\<lambda>x. a x \<otimes> b (m - x)" "mset_factors m"
                                                     "\<lambda>x. a x \<otimes> c (m - x)"]
-        by metis  
+        by metis
     qed
-    show ?thesis using LHS RHS' by auto 
+    show ?thesis using LHS RHS' by auto
   qed
 qed
 
 lemma P_ring_ldistr:
   assumes "carrier_coeff a"
   assumes "carrier_coeff b"
   assumes "carrier_coeff c"
   shows "  (b \<Oplus> c) \<Otimes>\<^sub>p  a = (b \<Otimes>\<^sub>p a)\<Oplus> (c \<Otimes>\<^sub>p a)"
 proof
   fix m
   show "((b \<Oplus> c) \<Otimes>\<^sub>p  a) m = ((b \<Otimes>\<^sub>p a)\<Oplus> (c \<Otimes>\<^sub>p a)) m"
   proof-
-    have RHS: "((b \<Otimes>\<^sub>p a)\<Oplus> (c \<Otimes>\<^sub>p a)) m = 
+    have RHS: "((b \<Otimes>\<^sub>p a)\<Oplus> (c \<Otimes>\<^sub>p a)) m =
                   (\<Oplus>x\<in>mset_factors m. b x \<otimes> a (m - x)) \<oplus> (\<Oplus>x\<in>mset_factors m. c x \<otimes> a (m - x))"
-      unfolding indexed_padd_def P_ring_mult_def by auto 
-    have LHS: "((b \<Oplus> c) \<Otimes>\<^sub>p  a)  m= 
+      unfolding indexed_padd_def P_ring_mult_def by auto
+    have LHS: "((b \<Oplus> c) \<Otimes>\<^sub>p  a)  m=
                   ( \<Oplus>x\<in>mset_factors m. b x \<otimes> a (m - x) \<oplus> c x \<otimes> a (m - x))"
-      unfolding indexed_padd_def  P_ring_mult_def 
+      unfolding indexed_padd_def  P_ring_mult_def
       by (meson assms(1) assms(2) assms(3) l_distr local.ring_axioms ring.carrier_coeff_def)
-    have RHS': "((b \<Otimes>\<^sub>p a)\<Oplus> (c \<Otimes>\<^sub>p a)) m = 
+    have RHS': "((b \<Otimes>\<^sub>p a)\<Oplus> (c \<Otimes>\<^sub>p a)) m =
                   (\<Oplus>x\<in>mset_factors m. b x \<otimes> a (m - x) \<oplus> c x \<otimes> a (m - x))"
     proof-
       have 0: "(\<lambda>x. b x \<otimes> a (m - x)) \<in> mset_factors m \<rightarrow> carrier R"
       proof
         fix x assume "x \<in> mset_factors m"
         then show "b x \<otimes> a (m - x) \<in> carrier R"
-          using assms carrier_coeffE 
+          using assms carrier_coeffE
           by blast
       qed
       have 1: "(\<lambda>x. c x \<otimes> a (m - x)) \<in> mset_factors m \<rightarrow> carrier R"
       proof
         fix x assume "x \<in> mset_factors m"
         then show "c x \<otimes> a (m - x) \<in> carrier R"
-          using assms carrier_coeffE 
-          by blast 
+          using assms carrier_coeffE
+          by blast
       qed
-      then show ?thesis 
+      then show ?thesis
         using 0 1 RHS assms finsum_addf[of "\<lambda>x. b x \<otimes> a (m - x)" "mset_factors m"
                                                     "\<lambda>x. c x \<otimes> a (m - x)"]
-        by metis  
+        by metis
     qed
-    show ?thesis using LHS RHS' by auto 
-  qed
-qed
-end 
+    show ?thesis using LHS RHS' by auto
+  qed
+qed
+end
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsubsection\<open>Multiplication Commutes with \texttt{indexed\_pmult}\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 context ring
 begin
 
 text\<open>This lemma shows how we can write the factors of a monomial $m$ times a variable $i$ in terms
       of the factors of m:\<close>
 
 lemma mset_factors_add_mset:
 "mset_factors (add_mset i m) = mset_factors m \<union> add_mset i ` (mset_factors m)"
 proof
   show "mset_factors (add_mset i m) \<subseteq> mset_factors m \<union> add_mset i ` mset_factors m"
   proof fix x
     assume A: "x \<in> mset_factors (add_mset i m)"
-    show "x \<in> mset_factors m \<union> add_mset i ` mset_factors m" 
+    show "x \<in> mset_factors m \<union> add_mset i ` mset_factors m"
     proof(cases "i \<in># x")
       case True
       then have "x - {#i#} \<subseteq># m"
         using A INTEG.R.monom_divides_factors subset_eq_diff_conv by force
-      then show ?thesis 
+      then show ?thesis
         by (metis INTEG.R.monom_divides_factors True UnI2 add_mset_remove_trivial image_iff multi_member_split)
     next
       case False
       have 0: "x \<subseteq># add_mset i m"
         using A INTEG.R.monom_divides_factors by blast
-      hence "x \<subseteq>#  m" 
-        using mset_subset_eqI[of x m] 0 False 
-        by (metis count_add_mset count_greater_zero_iff count_inI less_Suc_eq_le 
+      hence "x \<subseteq>#  m"
+        using mset_subset_eqI[of x m] 0 False
+        by (metis count_add_mset count_greater_zero_iff count_inI less_Suc_eq_le
             subseteq_mset_def union_single_eq_member)
-      thus ?thesis 
+      thus ?thesis
         using INTEG.R.monom_divides_factors by blast
     qed
   qed
   show "mset_factors m \<union> add_mset i ` mset_factors m \<subseteq> mset_factors (add_mset i m)"
   proof fix x assume A: "x \<in> mset_factors m \<union> add_mset i ` mset_factors m"
     have "x \<subseteq># add_mset i m"
     proof(cases "x \<in> mset_factors m")
       case True
       then have "x \<subseteq># m"
         by (simp add: INTEG.R.monom_divides_factors)
       hence "(\<And>a. count x a \<le> count (add_mset i m) a)"
-        using mset_subset_eq_count[of x m] count_add_mset[of i m] 
+        using mset_subset_eq_count[of x m] count_add_mset[of i m]
               nat_le_linear not_less_eq_eq by fastforce
-      thus ?thesis 
-        using mset_subset_eqI by blast          
+      thus ?thesis
+        using mset_subset_eqI by blast
     next
       case False
       then obtain n where n_def: "n \<in> mset_factors m \<and> x = add_mset i n"
         using A by blast
       then have "x \<subseteq># add_mset i m"
         by (simp add: INTEG.R.monom_divides_factors)
-      then show ?thesis 
+      then show ?thesis
         by simp
     qed
     thus "x \<in> mset_factors (add_mset i m)"
       by (simp add: INTEG.R.monom_divides_factors)
   qed
 qed
 
-end 
+end
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsubsection\<open>Associativity of Polynomial Multiplication.\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 context ring
 begin
 
 lemma finsum_eq:
   assumes "f \<in> S \<rightarrow> carrier R"
   assumes "g \<in> S \<rightarrow> carrier R"
   assumes "(\<lambda> x \<in> S. f x) = (\<lambda> x \<in> S. g x)"
   shows " finsum R f S = finsum R g S"
         by (metis assms(1) assms(3) finsum_cong' restrict_apply')
-        
+
 lemma finsum_eq_induct:
   assumes "f \<in> S \<rightarrow> carrier R"
   assumes "g \<in> T \<rightarrow> carrier R"
   assumes "finite S"
   assumes "finite T"
   assumes "bij_betw h S T"
   assumes "\<And>s. s \<in> S \<Longrightarrow> f s = g (h s)"
   shows "finite U \<Longrightarrow> U \<subseteq> S \<Longrightarrow> finsum R f U = finsum R g (h ` U)"
-  apply(induct rule: finite_induct) 
+  apply(induct rule: finite_induct)
   apply (simp; fail)
 proof-
   fix x
   fix F :: "'c set"
   assume F_fin: "finite F"
   assume x_not_in: "x \<notin> F"
   assume IH: "(F \<subseteq> S \<Longrightarrow> finsum R f F = finsum R g (h ` F)) "
   show "insert x F \<subseteq> S \<Longrightarrow> finsum R f (insert x F) = finsum R g (h ` insert x F)"
   proof-
     assume A: "insert x F \<subseteq> S"
     then have F_sub: "F \<subseteq>S"
       by simp
     have x_in: "x \<in> S"
       using A by blast
     have fx_in: "f x \<in>  carrier R"
       using x_in assms(1)
-      by auto 
+      by auto
     have ghx_fx_eq:
       "f x = g (h x)"
-      using x_in assms 
+      using x_in assms
       by blast
     have I: "finsum R f F = finsum R g (h ` F)"
       using F_sub IH by blast
     have f_fact: "f \<in> F \<rightarrow> carrier R "
-      using assms(1) F_sub 
+      using assms(1) F_sub
       by blast
     have "finsum R f (insert x F) = (f x) \<otimes>\<^bsub>add_monoid R\<^esub> (finsum R f F) "
-      using F_fin  x_not_in comm_monoid.finprod_insert[of "(add_monoid R)" F x f ] 
+      using F_fin  x_not_in comm_monoid.finprod_insert[of "(add_monoid R)" F x f ]
       unfolding finsum_def
-      using f_fact fx_in local.add.comm_monoid_axioms 
+      using f_fact fx_in local.add.comm_monoid_axioms
       by auto
     have "finsum R g (h ` insert x F) = finsum R g (insert (h x) (h ` F))"
       by simp
     then have "finsum R g (h ` insert x F) = (g (h x)) \<otimes>\<^bsub>add_monoid R\<^esub> finsum R g (h ` F)"
     proof-
        have 0: "finite (h ` F)"
          by (simp add: F_fin)
        have 1: "h x \<notin> h ` F"
        proof-
          have 10: "bij_betw h F (h` F)"
-           using assms(5) F_sub bij_betw_subset 
+           using assms(5) F_sub bij_betw_subset
            by blast
-         show ?thesis 
+         show ?thesis
          proof
            assume "h x \<in> h ` F "
            then  obtain s where s_def: "s \<in> F \<and> h s = h x"
-             using 10 
+             using 10
              by auto
            have "s \<in> S"
              using F_sub s_def by blast
            then have "s = x"
              using x_in assms(5)
                s_def
              unfolding bij_betw_def inj_on_def
              by blast
            then show False using x_not_in
              using s_def by blast
          qed
        qed
        have 2: "g \<in> h ` F \<rightarrow> carrier (add_monoid R)"
        proof
          fix y
          assume Ay: "y \<in> h ` F"
          then obtain s where s_def: "y = h s \<and> s \<in> F"
            by blast
          then have s_S: "s \<in> S"
            using F_sub by blast
          have "h ` F \<subseteq> T"
-           using F_sub assms(5) 
-           unfolding bij_betw_def  
+           using F_sub assms(5)
+           unfolding bij_betw_def
            by blast
          then have "g y \<in> carrier R"
            using Ay assms(2)
            by blast
          then show "g y \<in> carrier (add_monoid R)"
            by simp
        qed
        have "g (h x) \<in> carrier (add_monoid R)"
-         using fx_in ghx_fx_eq 
+         using fx_in ghx_fx_eq
          by auto
-       then show ?thesis 
-         using 0 1 2 comm_monoid.finprod_insert[of "(add_monoid R)" "(h ` F)"  "h x" g ] 
-         unfolding finsum_def 
+       then show ?thesis
+         using 0 1 2 comm_monoid.finprod_insert[of "(add_monoid R)" "(h ` F)"  "h x" g ]
+         unfolding finsum_def
          by (simp add: local.add.comm_monoid_axioms)
     qed
-    then have "finsum R g (h ` insert x F) = f x  \<otimes>\<^bsub>add_monoid R\<^esub> (finsum R f F)"    
+    then have "finsum R g (h ` insert x F) = f x  \<otimes>\<^bsub>add_monoid R\<^esub> (finsum R f F)"
       using I ghx_fx_eq by auto
-    then show ?thesis 
+    then show ?thesis
       by (simp add: \<open>finsum R f (insert x F) = f x \<otimes>\<^bsub>add_monoid R\<^esub> finsum R f F\<close>)
   qed
 qed
 
 lemma finsum_bij_eq:
   assumes "f \<in> S \<rightarrow> carrier R"
   assumes "g \<in> T \<rightarrow> carrier R"
   assumes "finite S"
   assumes "bij_betw h S T"
   assumes "\<And>s. s \<in> S \<Longrightarrow> f s = g (h s)"
   shows "finsum R f S = finsum R g T"
 proof-
   have 0: "finite T"
-    using assms bij_betw_finite 
+    using assms bij_betw_finite
     by blast
   have 1: "(\<And>s. s \<in> S \<Longrightarrow> f s = g (h s)) "
-    using assms 
+    using assms
     by blast
   have "(\<And>s. s \<in> S \<Longrightarrow> f s = g (h s)) \<Longrightarrow> finite S \<Longrightarrow> S \<subseteq> S \<Longrightarrow> finsum R f S = finsum R g (h ` S)"
-    using 0 assms finsum_eq_induct[of f S g T h S ] 
+    using 0 assms finsum_eq_induct[of f S g T h S ]
   by blast
   then have "finsum R f S = finsum R g (h ` S)"
     using assms(5) assms(3)
     by blast
-  then show ?thesis 
-    using assms(5) assms(4) bij_betw_imp_surj_on 
+  then show ?thesis
+    using assms(5) assms(4) bij_betw_imp_surj_on
     by blast
 qed
 
 lemma monom_comp:
   assumes "x \<subseteq># m"
   assumes "y \<subseteq># m - x"
   shows "x \<subseteq># m - y"
-using assms 
-  by (metis add_diff_cancel_left' subset_eq_diff_conv 
+using assms
+  by (metis add_diff_cancel_left' subset_eq_diff_conv
       subset_mset.le_diff_conv2 subset_mset.order_refl subset_mset.order_trans)
 
 lemma monom_comp':
   assumes "x \<subseteq># m"
   assumes "y = m - x"
   shows "x = m - y"
-  using assms 
+  using assms
   by (metis add_diff_cancel_right' subset_mset.add_diff_inverse)
 
 text \<open>
   This lemma turns iterated sums into sums over a product set. The first lemma is only a technical
-  phrasing of \texttt{double\_finsum'} to facilitate an inductive proof, and likely can and should 
+  phrasing of \texttt{double\_finsum'} to facilitate an inductive proof, and likely can and should
   be simplified.
 \<close>
 
 lemma double_finsum_induct:
   assumes "finite S"
   assumes "\<And>s. s \<in> S \<Longrightarrow> finite (F s)"
   assumes "P = (\<lambda>S. {(s, t). s \<in> S \<and>  t \<in> (F s)})"
   assumes "\<And>s y. s \<in> S \<Longrightarrow> y \<in> (F s) \<Longrightarrow> g s y \<in> carrier R"
-  shows "finite T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> finsum R (\<lambda>s. finsum R (g s) (F s)) T = 
-         finsum R (\<lambda> c. g (fst c) (snd c)) (P  T)" 
-  apply(induct rule: finite_induct) 
+  shows "finite T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> finsum R (\<lambda>s. finsum R (g s) (F s)) T =
+         finsum R (\<lambda> c. g (fst c) (snd c)) (P  T)"
+  apply(induct rule: finite_induct)
   apply (simp add: assms(3); fail)
 proof-
   fix x
   fix T :: "'c set"
   assume AT: "finite T"
   assume x_notin: "x \<notin> T"
   assume IH: "(T \<subseteq> S \<Longrightarrow> (\<Oplus>s\<in>T. finsum R (g s) (F s)) = (\<Oplus>c\<in>P T. g (fst c) (snd c)))"
   assume A: "insert x T \<subseteq>S"
   show " (\<Oplus>s\<in>insert x T. finsum R (g s) (F s)) = (\<Oplus>c\<in>P (insert x T). g (fst c) (snd c))"
   proof-
     have 0: "(\<lambda>s. finsum R (g s) (F s)) \<in> T \<rightarrow> carrier R"
     proof
       fix v
       assume A0: "v \<in> T"
       then have A1: "v \<in> S"
         using A by blast
       then show "finsum R (g v) (F v) \<in> carrier R"
       proof-
         have 00: "finite (F v)"
-          using assms A 
-          using \<open>v \<in> T\<close> by blast   
+          using assms A
+          using \<open>v \<in> T\<close> by blast
         have "g v \<in> F v \<rightarrow> carrier R"
         proof
           fix l assume "l \<in> F v"
           then show "g v l \<in> carrier R"
-            using A1 assms(4)[of v l] 
+            using A1 assms(4)[of v l]
             by blast
         qed
-        then show ?thesis 
+        then show ?thesis
           using finsum_closed by blast
       qed
     qed
     have 1: "finsum R (g x) (F x) \<in> carrier R"
     proof-
       have "x \<in> S"
         using A by blast
-      then show ?thesis using assms(4) finsum_closed[of "(g x)" "F x"] 
+      then show ?thesis using assms(4) finsum_closed[of "(g x)" "F x"]
         by blast
     qed
     have 2: " (\<Oplus>s\<in>insert x T. finsum R (g s) (F s)) = finsum R (g x) (F x) \<oplus> (\<Oplus>s\<in> T. finsum R (g s) (F s))"
-      using 0 1 finsum_insert[of T x "(\<lambda>s. finsum R (g s) (F s))"] AT x_notin  
+      using 0 1 finsum_insert[of T x "(\<lambda>s. finsum R (g s) (F s))"] AT x_notin
       by blast
     have 3: "P (insert x T) = P {x} \<union> P T"
     proof
       show "P (insert x T) \<subseteq> P {x} \<union> P T"
       proof
         fix y
         assume "y \<in> P (insert x T)"
         then show "y \<in> P {x} \<union> P T"
-          using assms 
+          using assms
           by blast
       qed
       show "P {x} \<union> P T \<subseteq> P (insert x T)"
       proof
         fix y
         assume Ay: "y \<in> P {x} \<union> P T"
         show "y  \<in> P (insert x T)"
         proof(cases "y \<in> P {x}")
           case True
           then have "y \<in> ({(s, t). s \<in> {x} \<and> t \<in> F s})"
-            using assms(3) by auto 
+            using assms(3) by auto
           then have "fst y = x \<and> snd y \<in> F x"
             using Product_Type.Collect_case_prodD by auto
-          then show ?thesis using assms(3) 
+          then show ?thesis using assms(3)
             using fst_def by auto
         next
           case False
           then have "y \<in> P T"
-            using Ay 
+            using Ay
             by blast
           then have "y \<in> ({(s, t). s \<in> T \<and> t \<in> F s})"
-            using assms(3) by blast 
+            using assms(3) by blast
           then obtain s t where st_def: "y = (s, t) \<and>s \<in> T \<and> t \<in> F s"
-            by blast 
+            by blast
           then have "y = (s, t) \<and>s \<in> (insert x T) \<and> t \<in> F s"
-            by blast 
-          then show ?thesis using assms(3) 
+            by blast
+          then show ?thesis using assms(3)
             by simp
         qed
       qed
     qed
     have 4: "P {x} \<inter> P T = {}"
     proof
       show "P {x} \<inter> P T \<subseteq> {}"
       proof
         fix y
         assume B: "y \<in> P {x} \<inter> P T"
         then have "fst y = x"
         proof-
           have "y \<in> {(s, t). s \<in> {x} \<and> t \<in> F s}"
             using assms(3) B by auto
           then obtain s t where "y = (s, t) \<and> s \<in> {x} \<and> t \<in> F s"
-            by blast 
-          then show ?thesis 
+            by blast
+          then show ?thesis
             by auto
         qed
         have "fst y \<in> T"
         proof-
           have "y \<in> {(s, t). s \<in> T \<and> t \<in> F s}"
             using assms(3) B by auto
           then obtain s t where "y = (s, t) \<and> s \<in> T \<and> t \<in> F s"
-            by blast 
-          then show ?thesis 
+            by blast
+          then show ?thesis
             by simp
         qed
-        then have False 
-          using x_notin 
+        then have False
+          using x_notin
           by (simp add: \<open>fst y = x\<close>)
-        then show "y \<in> {}" 
+        then show "y \<in> {}"
           by simp
       qed
-      show "{} \<subseteq> P {x} \<inter> P T" 
+      show "{} \<subseteq> P {x} \<inter> P T"
         by simp
     qed
     have 5: "(\<lambda>c. g (fst c) (snd c)) \<in> P {x} \<rightarrow> carrier R"
     proof
       fix y
       assume X0: "y \<in> P {x}"
       obtain s t where st_def: "y = (s, t) \<and> (s, t) \<in> P {x}"
         by (metis X0 old.prod.exhaust)
       then have st: "s = x \<and> t \<in> F x"
         using assms(3) by blast
       then have "g (fst y) (snd y) = g x t \<and>t \<in> F x"
         by (simp add: st_def)
       then show "g (fst y) (snd y) \<in> carrier R"
-        using assms(4)[of x t] A 
+        using assms(4)[of x t] A
         by simp
     qed
     have 6: "(\<lambda>c. g (fst c) (snd c)) \<in> P T \<rightarrow> carrier R"
     proof
       fix y
       assume X0: "y \<in> P T"
       obtain s t where st_def: "y = (s, t) \<and> (s, t) \<in> P T"
         by (metis X0 old.prod.exhaust)
       then have st: "s \<in> T \<and> t \<in> F s"
         using assms(3) by blast
       then have "g (fst y) (snd y) = g s t \<and>t \<in> F s"
         by (simp add: st_def)
       then show "g (fst y) (snd y) \<in> carrier R"
-        using assms(4)[of s t] A st 
+        using assms(4)[of s t] A st
         by auto
     qed
     have 07: "\<And>x. x \<in> S \<Longrightarrow>  finite (P {x})"
     proof-
       fix x
       assume "x \<in> S"
       have "bij_betw snd (P {x}) (F x)"
         unfolding bij_betw_def
       proof
         show "inj_on snd (P {x})"
         proof
           fix a b
           assume Aa: "a \<in> P {x}"
           assume Ab: "b \<in> P {x}"
           have 0: "fst a = x"
           proof-
             have" a \<in> {(s, t). s \<in> {x} \<and> t \<in> F s} "
               using Aa  assms(3)
-              by blast 
+              by blast
             then obtain s t where st_def: "a = (s, t) \<and> s \<in> {x} \<and> t \<in> F s"
-              by blast 
-            then show ?thesis 
+              by blast
+            then show ?thesis
               by auto
           qed
           have 1: "fst b = x"
           proof-
             have" b \<in> {(s, t). s \<in> {x} \<and> t \<in> F s} "
               using Ab  assms(3)
-              by blast 
+              by blast
             then obtain s t where st_def: "b = (s, t) \<and> s \<in> {x} \<and> t \<in> F s"
-              by blast 
-            then show ?thesis 
+              by blast
+            then show ?thesis
               by auto
           qed
-          show "snd a = snd b \<Longrightarrow> a = b" 
-            using 0 1 
+          show "snd a = snd b \<Longrightarrow> a = b"
+            using 0 1
             by (simp add: prod_eqI)
         qed
         show "snd ` P {x} = F x"
         proof
           show "snd ` P {x} \<subseteq> F x"
           proof
             fix y
             assume 0: "y \<in> snd ` P {x}"
             then obtain q where q_def: " q \<in> P {x} \<and> y = snd q"
-              by blast 
+              by blast
             then obtain s t where st: "q = (s, t) \<and> s \<in> {x} \<and> t \<in> F s"
-              using assms(3) by blast 
-            then have  1: "s = x" 
-              by blast 
+              using assms(3) by blast
+            then have  1: "s = x"
+              by blast
             have 2: "snd q = t"
               by (simp add: st)
-            then show "y \<in> F x" 
+            then show "y \<in> F x"
               using q_def st by blast
           qed
           show "F x \<subseteq> snd ` P {x}"
           proof
             fix y
             assume "y \<in> F x"
             then have C: "(x, y) \<in> P {x}"
-              using assms(3) 
+              using assms(3)
               by simp
             then have "y = snd (x, y)"
               by auto
             then show "y \<in> snd ` P {x}"
-              using C  
+              using C
               by blast
           qed
         qed
       qed
       then show "finite (P {x})"
-        using assms(2)[of x] bij_betw_finite 
-        \<open>x \<in> S\<close> 
-        by blast    
+        using assms(2)[of x] bij_betw_finite
+        \<open>x \<in> S\<close>
+        by blast
     qed
     have 7: "finite (P {x})"
-      using 07 A 
+      using 07 A
       by blast
     have 8: "finite (P T)"
     proof-
       have "\<And>D. finite D \<Longrightarrow> D \<subseteq> S \<Longrightarrow> finite (P D)"
       proof-
         fix D
         show "finite D \<Longrightarrow> D \<subseteq> S  \<Longrightarrow> finite (P D)"
-          apply(induct rule: finite_induct) 
+          apply(induct rule: finite_induct)
         proof-
-          have "P {} = {}" using assms(3) 
+          have "P {} = {}" using assms(3)
             by blast
           then show "finite (P {})"
-            by auto 
+            by auto
           show "\<And>x F. finite F \<Longrightarrow> x \<notin> F \<Longrightarrow> (F \<subseteq> S \<Longrightarrow> finite (P F)) \<Longrightarrow> insert x F \<subseteq> S \<Longrightarrow> finite (P (insert x F))"
           proof-
             fix x
             fix Q :: "'c set "
             assume fin: "finite Q"
             assume notin: "x \<notin> Q"
             assume fin_pf: "(Q \<subseteq> S \<Longrightarrow> finite (P Q))"
             assume I: "insert x Q \<subseteq> S "
             show "finite (P (insert x Q))"
             proof-
               have x_in: "x \<in> S"
                 using I by blast
               have 0: "(P (insert x Q)) \<subseteq> (P Q) \<union> P {x}"
               proof
                 fix y
                 assume "y \<in> P (insert x Q)"
                 obtain s t where st: "y = (s, t) \<and> s \<in> (insert x Q) \<and> t \<in> F s"
-                  using assms(3) x_in  \<open>y \<in> P (insert x Q)\<close> 
+                  using assms(3) x_in  \<open>y \<in> P (insert x Q)\<close>
                   by blast
                 show "y \<in> (P Q) \<union> P {x}"
                 proof(cases "s \<in> Q")
                   case True
                   then have "y \<in> P Q"
                     using st assms(3)
                     by simp
-                  then show ?thesis using st assms(3) 
+                  then show ?thesis using st assms(3)
                     by blast
                 next
                   case False
                   then have "s = x"
                     using st by blast
                   then have  "s \<in>{x} \<and> t \<in> F s"
-                    using st by blast 
+                    using st by blast
                   then have "y \<in> P {x}"
-                    using st assms(3) 
-                    by auto 
-                  then show ?thesis by auto                      
+                    using st assms(3)
+                    by auto
+                  then show ?thesis by auto
                 qed
               qed
               have 1: "finite  (P Q)"
                 using I fin_pf by blast
               have "finite (P {x})"
                 using "07" x_in by blast
-              then show ?thesis using 0 1 
+              then show ?thesis using 0 1
                 using finite_subset by auto
             qed
           qed
         qed
-      qed   
-      then show ?thesis 
+      qed
+      then show ?thesis
         using A AT by blast
     qed
-    have 9: "(\<Oplus>q\<in>P (insert x T). g (fst q) (snd q)) = 
+    have 9: "(\<Oplus>q\<in>P (insert x T). g (fst q) (snd q)) =
           (\<Oplus>q\<in>P T. g (fst q) (snd q)) \<oplus> (\<Oplus>q\<in>P {x}. g (fst q) (snd q))"
-      using 8 7 6 5 4 3 finsum_Un_disjoint[of "P {x}" "P T" "\<lambda>q. g (fst q) (snd q)"] 
+      using 8 7 6 5 4 3 finsum_Un_disjoint[of "P {x}" "P T" "\<lambda>q. g (fst q) (snd q)"]
       by (simp add: "7" \<open>\<lbrakk>finite (P {x}); finite (P T); P {x} \<inter> P T = {}; (\<lambda>p. g (fst p) (snd p))
            \<in> P {x} \<rightarrow> carrier R; (\<lambda>p. g (fst p) (snd p)) \<in> P T \<rightarrow> carrier R\<rbrakk> \<Longrightarrow>
            (\<Oplus>p\<in>P {x} \<union> P T. g (fst p) (snd p)) = (\<Oplus>p\<in>P {x}. g (fst p) (snd p)) \<oplus>
-             (\<Oplus>p\<in>P T. g (fst p) (snd p))\<close> add.m_comm)      
-    have 10: "(\<Oplus>p\<in>P (insert x T). g (fst p) (snd p)) 
+             (\<Oplus>p\<in>P T. g (fst p) (snd p))\<close> add.m_comm)
+    have 10: "(\<Oplus>p\<in>P (insert x T). g (fst p) (snd p))
         =  (\<Oplus>s\<in>T. finsum R (g s) (F s))\<oplus> (\<Oplus>p\<in>P {x}. g (fst p) (snd p))"
-      using IH 9 A 
+      using IH 9 A
       by auto
     have 11: "(\<Oplus>p\<in>P {x}. g (fst p) (snd p)) = finsum R (g x) (F x)"
     proof-
       obtain h :: "('c \<times> 'd) \<Rightarrow> 'd" where h_def: "h = snd"
-        by simp 
+        by simp
       have 110: "bij_betw h (P {x}) (F x)"
         unfolding bij_betw_def
       proof
         show "inj_on h (P {x})"
           unfolding inj_on_def
         proof
           fix q
           assume Ap: "q \<in>  P {x}"
           show " \<forall>y\<in>P {x}. h q = h y \<longrightarrow> q = y"
           proof
             fix y
             assume q_def: "y \<in> P {x}"
             then have C0: "fst y = x"
-               using assms(3) 
+               using assms(3)
                by (simp add: case_prod_beta)
             have C1: "fst q = x"
               using assms(3) Ap
                by (simp add: case_prod_beta)
-             show  "h q = h y \<longrightarrow> q = y" 
-               using C0 C1 h_def 
+             show  "h q = h y \<longrightarrow> q = y"
+               using C0 C1 h_def
                by (simp add: prod_eq_iff)
           qed
         qed
         show "h ` P {x} = F x"
         proof
           show "h ` P {x} \<subseteq> F x"
           proof
             fix y
             assume "y \<in> h ` P {x}"
             then obtain d where d_def: "y = h d \<and> d \<in> P {x}"
-              by blast 
+              by blast
             then obtain s t where st_def: "d = (s, t)"
               by (meson surj_pair)
             then have "s = x \<and> t \<in> F x"
-              using assms(3) d_def 
+              using assms(3) d_def
               by blast
             then  show "y \<in> F x"
-              using assms(3) 
+              using assms(3)
               by (simp add: d_def h_def st_def)
           qed
           show "F x \<subseteq> h ` P {x}"
           proof
             fix y
             assume E0:  "y \<in> F x"
             have E1: "y = h (x , y)"
               by (simp add: h_def)
             have "(x, y) \<in> P {x}"
-              using E0 assms(3) by blast  
+              using E0 assms(3) by blast
             then show "y \<in> h ` P {x} "
-              using E1 assms(3) by blast 
+              using E1 assms(3) by blast
           qed
         qed
       qed
       have 111: "g x \<in> F x \<rightarrow> carrier R "
       proof
         fix y
         assume "y \<in> F x"
-        then show "g x y \<in> carrier R" 
-          using assms A 
+        then show "g x y \<in> carrier R"
+          using assms A
           by blast
       qed
       have 112: "(\<And>s. s \<in> P {x} \<Longrightarrow> g (fst s) (snd s) = g x (h s))"
       proof-
         fix s
         assume "s \<in> P{x}"
         then have "s \<in> {(s, t). s = x \<and> t \<in> F s}"
-          using assms(3) by blast 
+          using assms(3) by blast
         then obtain t where "s = (x, t) \<and> t \<in> F x"
-          using assms(3) 
+          using assms(3)
           by blast
         then show "g (fst s) (snd s) = g x (h s)"
           by (simp add: h_def)
       qed
 
-        
- 
+
+
       show ?thesis using 5 7  110 111 112 finsum_bij_eq[of "\<lambda>p. g (fst p) (snd p)" "P {x}" "g x" "F x" h ]
-        by auto 
+        by auto
     qed
-    have 12: "(\<Oplus>p\<in>P (insert x T). g (fst p) (snd p)) 
+    have 12: "(\<Oplus>p\<in>P (insert x T). g (fst p) (snd p))
         =  (\<Oplus>s\<in>T. finsum R (g s) (F s))\<oplus> finsum R (g x) (F x)"
-      using 10 11 by auto 
-    then show ?thesis using 2 
+      using 10 11 by auto
+    then show ?thesis using 2
       by (simp add: "2" "0" "1" add.m_comm)
   qed
 qed
 
 lemma double_finsum:
   assumes "finite S"
   assumes "\<And>s. s \<in> S \<Longrightarrow> finite (F s)"
   assumes "P = {(s, t). s \<in> S \<and>  t \<in> (F s)}"
   assumes "\<And>s y. s \<in> S \<Longrightarrow> y \<in> (F s) \<Longrightarrow> g s y \<in> carrier R"
-  shows "finsum R (\<lambda>s. finsum R (g s) (F s)) S = 
-         finsum R (\<lambda> p. g (fst p) (snd p)) P" 
+  shows "finsum R (\<lambda>s. finsum R (g s) (F s)) S =
+         finsum R (\<lambda> p. g (fst p) (snd p)) P"
 proof-
   obtain P' where P'_def: "P' =  (\<lambda>S. {(s, t). s \<in> S \<and>  t \<in> (F s)})"
     by simp
-  have "finsum R (\<lambda>s. finsum R (g s) (F s)) S = 
+  have "finsum R (\<lambda>s. finsum R (g s) (F s)) S =
          finsum R (\<lambda> p. g (fst p) (snd p)) (P' S)"
     using double_finsum_induct[of S F P' g S]
-          assms P'_def 
+          assms P'_def
     by blast
-  then show ?thesis 
-    using P'_def assms 
+  then show ?thesis
+    using P'_def assms
     by blast
 qed
 
-end 
+end
 
 text\<open>
-  The product index set for the double sums in the coefficients of the 
+  The product index set for the double sums in the coefficients of the
   $((a \otimes_p b) \otimes_p c)$. It is the set of pairs $(x,y)$ of monomials, such that
   $x$ is a factor of monomial $m$, and $y$ is a factor of monomial $x$.
 \<close>
 
 definition right_cuts :: "'c monomial \<Rightarrow> ('c monomial \<times> 'c monomial) set" where
 "right_cuts m = {(x, y). x \<subseteq># m \<and> y \<subseteq># x}"
 
 context ring
 begin
 
 lemma right_cuts_alt_def:
 "right_cuts m = {(x, y). x \<in> mset_factors m \<and> y \<in> mset_factors x}"
   unfolding mset_factors_def right_cuts_def
   by simp
 
 lemma right_cuts_finite:
 "finite (right_cuts m)"
 proof-
   have "finite (mset_factors m \<times> mset_factors m)"
     using mset_factors_finite
     by blast
   have "right_cuts m \<subseteq> (mset_factors m \<times> mset_factors m)"
   proof
     fix p
     assume p_def: "p \<in> right_cuts m"
     obtain x y where xy: "p = (x , y) \<and> x \<subseteq># m \<and> y \<subseteq>#x"
-      using p_def unfolding right_cuts_def 
+      using p_def unfolding right_cuts_def
       by blast
     then have "x \<in> mset_factors m \<and> y \<in> mset_factors m"
-      using monom_divides_factors 
-      by auto    
+      using monom_divides_factors
+      by auto
     then show "p \<in> (mset_factors m \<times> mset_factors m)"
       by (simp add: xy)
   qed
-  then show ?thesis 
+  then show ?thesis
     using \<open>finite (mset_factors m \<times> mset_factors m)\<close> finite_subset
     by blast
 qed
-    
-lemma assoc_aux0:  
+
+lemma assoc_aux0:
   assumes "carrier_coeff a"
   assumes "carrier_coeff b"
   assumes "carrier_coeff c"
   assumes "g = (\<lambda>x y.  a x \<otimes> (b y \<otimes> c (m - x - y)))"
-  shows "\<And>s y. s \<in> mset_factors m \<Longrightarrow> y \<in> mset_factors (m - x) 
+  shows "\<And>s y. s \<in> mset_factors m \<Longrightarrow> y \<in> mset_factors (m - x)
             \<Longrightarrow> g s y \<in> carrier R"
   using assms carrier_coeffE by blast
-  
-lemma assoc_aux1:  
+
+lemma assoc_aux1:
   assumes "carrier_coeff a"
   assumes "carrier_coeff b"
   assumes "carrier_coeff c"
   assumes "g = (\<lambda>x y.  (a y  \<otimes> b (x - y)) \<otimes> c (m - x))"
   shows "\<And>s y. s \<in> mset_factors m \<Longrightarrow> y \<in> mset_factors x \<Longrightarrow> g s y \<in> carrier R"
   using assms carrier_coeffE by blast
 end
 
 text\<open>
-  The product index set for the double sums in the coefficients of the 
+  The product index set for the double sums in the coefficients of the
   $(a \otimes_p (b \otimes_p c))$. It is the set of pairs $(x,y)$ such that $x$ is a factor
   of $m$ and $y$ is a factor of $m/x$.
 \<close>
 
 definition left_cuts :: "'c monomial \<Rightarrow> ('c monomial \<times> 'c monomial) set" where
 "left_cuts m = {(x, y). x \<subseteq>#m \<and> y \<subseteq># (m - x)}"
 
 context ring
 begin
 
 lemma left_cuts_alt_def:
 "left_cuts m = {(x, y). x \<in> mset_factors m \<and> y \<in> mset_factors (m - x)}"
   unfolding mset_factors_def left_cuts_def
   by simp
 
 text\<open>This lemma witnesses the bijection between left and right cuts for the proof of associativity:\<close>
 
 lemma left_right_cuts_bij:
 "bij_betw (\<lambda> (x,y). (x + y, x))  (left_cuts m) (right_cuts m)"
   unfolding bij_betw_def right_cuts_def left_cuts_def
-proof 
+proof
   show "inj_on (\<lambda>(x, y). (x + y, x)) {(x, y). x \<subseteq># m \<and> y \<subseteq># m - x}"
-    unfolding inj_on_def 
-    by auto 
+    unfolding inj_on_def
+    by auto
   show "(\<lambda>(x, y). (x + y, x)) ` {(x, y). x \<subseteq># m \<and> y \<subseteq># m - x} = {(x, y). x \<subseteq># m \<and> y \<subseteq># x}"
   proof
     show "(\<lambda>(x, y). (x + y, x)) ` {(x, y). x \<subseteq># m \<and> y \<subseteq># m - x} \<subseteq> {(x, y). x \<subseteq># m \<and> y \<subseteq># x}"
     proof
       fix p
       assume "p  \<in> (\<lambda>(x, y). (x + y, x)) ` {(x, y). x \<subseteq># m \<and> y \<subseteq># m - x}"
       then obtain a b where ab: "a \<subseteq># m \<and> b \<subseteq># m - a \<and> p = (\<lambda>(x, y). (x + y, x)) (a, b)"
         by blast
       then have "p = (a + b, a)"
         by simp
       then show "p \<in> {(x, y). x \<subseteq># m \<and> y \<subseteq># x}"
-        using ab 
+        using ab
         by (metis (no_types, lifting) case_prodI mem_Collect_eq mset_subset_eq_add_left
             subset_mset.le_diff_conv2 union_commute)
     qed
     show "{(x, y). x \<subseteq># m \<and> y \<subseteq># x} \<subseteq> (\<lambda>(x, y). (x + y, x)) ` {(x, y). x \<subseteq># m \<and> y \<subseteq># m - x}"
     proof
       fix p
       assume p_def: "p \<in> {(x, y). x \<subseteq># m \<and> y \<subseteq># x}"
       then obtain a b where ab: "p = (a, b) \<and> a \<subseteq># m \<and> b \<subseteq># a"
-        by blast 
+        by blast
       then have "p = (\<lambda>(x, y). (x + y, x)) (b, a - b)"
-        using ab 
+        using ab
         by simp
       then show " p \<in> (\<lambda>(x, y). (x + y, x)) ` {(x, y). x \<subseteq># m \<and> y \<subseteq># m - x}"
         by (metis (mono_tags, lifting) ab case_prodI image_eqI mem_Collect_eq
             subset_mset.diff_add subset_mset.dual_order.trans subset_mset.le_diff_conv2)
     qed
   qed
 qed
 
 lemma left_cuts_sum:
   assumes "carrier_coeff a"
   assumes "carrier_coeff b"
   assumes "carrier_coeff c"
   shows "(a \<Otimes>\<^sub>p (b \<Otimes>\<^sub>p c)) m = (\<Oplus>p \<in> left_cuts m. a (fst p) \<otimes> (b (snd p) \<otimes> c (m - (fst p) - (snd p))))"
 proof-
   have U: "(a \<Otimes>\<^sub>p (b \<Otimes>\<^sub>p c)) m = (\<Oplus>x\<in>mset_factors m. (\<Oplus>xa\<in>mset_factors (m - x). a x \<otimes> (b xa \<otimes> c (m - x - xa))))"
     unfolding P_ring_mult_def
   proof-
     obtain f where f_def:
           "f = (\<lambda>x . a x \<otimes> (\<Oplus>xa\<in>mset_factors (m - x). b xa \<otimes> c (m - x - xa)))"
-      by simp 
+      by simp
     obtain g where g_def:
       "g =  (\<lambda>x . \<Oplus>xa\<in>mset_factors (m - x). a x \<otimes> (b xa \<otimes> c (m - x - xa)))"
-      by simp 
+      by simp
     have 0: "restrict f (mset_factors m) = restrict g (mset_factors m)"
     proof
       fix x
       show "restrict f (mset_factors m) x = restrict g (mset_factors m) x"
       proof(cases "x \<in> mset_factors m")
-        case True 
+        case True
          have T0: "restrict f (mset_factors m) x = a x \<otimes> (\<Oplus>xa\<in>mset_factors (m - x). b xa \<otimes> c (m - x - xa))"
-           using f_def True by auto 
+           using f_def True by auto
          have T1: "restrict g (mset_factors m) x = (\<Oplus>xa\<in>mset_factors (m - x). a x \<otimes> (b xa \<otimes> c (m - x - xa)))"
-           using True g_def by auto 
+           using True g_def by auto
          show "restrict f (mset_factors m) x = restrict g (mset_factors m) x"
            using assms finsum_rdistr[of "mset_factors (m - x)" "a x" "\<lambda> xa. b xa \<otimes> c (m - x - xa)"]
            by (metis (mono_tags, lifting) mset_factors_finite Pi_I T0 T1 carrier_coeffE m_closed)
       next
         case False
-        then have "restrict f (mset_factors m)  x = undefined" using f_def 
+        then have "restrict f (mset_factors m)  x = undefined" using f_def
           by (simp add: restrict_def)
-        have  "restrict g (mset_factors m) x = undefined" using g_def False 
+        have  "restrict g (mset_factors m) x = undefined" using g_def False
           using restrict_def by auto
-        then show ?thesis using f_def g_def 
+        then show ?thesis using f_def g_def
           using \<open>restrict f (mset_factors m) x = undefined\<close>
           by auto
-           
+
       qed
     qed
     have 1: "f \<in> mset_factors m \<rightarrow> carrier R"
-      using f_def assms 
+      using f_def assms
       by (simp add: carrier_coeffE)
     have 2: "g \<in> mset_factors m \<rightarrow> carrier R"
       using g_def assms
       by (simp add: carrier_coeffE)
     show "(\<Oplus>x\<in>mset_factors m. a x \<otimes> (\<Oplus>xa\<in>mset_factors (m - x). b xa \<otimes> c (m - x - xa))) =
     (\<Oplus>x\<in>mset_factors m. \<Oplus>xa\<in>mset_factors (m - x). a x \<otimes> (b xa \<otimes> c (m - x - xa)))"
       using f_def g_def finsum_eq[of "f"
                 "mset_factors m" "g"] 0 1 2
-      by blast 
+      by blast
   qed
   have 0: "(\<And>s. s \<in> mset_factors m \<Longrightarrow> finite (mset_factors (m - s)))"
     by simp
   have 1: "finite (mset_factors m)"
-    by simp 
+    by simp
   have 2: "(\<And>s y. s \<in> mset_factors m \<Longrightarrow> y \<in> mset_factors (m - s) \<Longrightarrow> a s \<otimes> (b y \<otimes> c (m - s - y)) \<in> carrier R)"
     using assms assoc_aux0[of a b c ]
     by blast
     have "(\<Oplus>x\<in>mset_factors m. (\<Oplus>xa\<in>mset_factors (m - x). a x \<otimes> (b xa \<otimes> c (m - x - xa)))) =
-          (\<Oplus>p \<in> left_cuts m. a (fst p) \<otimes> (b (snd p) \<otimes> c (m - (fst p) - (snd p)))) " 
+          (\<Oplus>p \<in> left_cuts m. a (fst p) \<otimes> (b (snd p) \<otimes> c (m - (fst p) - (snd p)))) "
     using assms left_cuts_alt_def[of m] 0 1 2
           double_finsum[of "mset_factors m" "\<lambda>x. mset_factors (m - x)" "left_cuts m" "(\<lambda>x y.  a x \<otimes> (b y \<otimes> c (m - x - y)))"]
     by blast
-  then show ?thesis using U 
+  then show ?thesis using U
     by auto
 qed
 
 lemma right_cuts_sum:
   assumes "carrier_coeff a"
   assumes "carrier_coeff b"
   assumes "carrier_coeff c"
   shows "(a \<Otimes>\<^sub>p b \<Otimes>\<^sub>p c) m = (\<Oplus>p \<in> right_cuts m. a (snd  p) \<otimes> (b ((fst p) -(snd p)) \<otimes> c (m - (fst p))))"
 proof-
   have 0: "finite (mset_factors m)"
-    by simp 
+    by simp
   have 1: "(\<And>s. s \<in> mset_factors m \<Longrightarrow> finite (mset_factors s))"
-    by auto 
+    by auto
   have 2: "right_cuts m = {(s, t). s \<in> mset_factors m \<and> t \<in> mset_factors s}"
-    unfolding right_cuts_def 
+    unfolding right_cuts_def
     by (simp add: monom_divides_factors)
   have 3: "(\<And>s y. s \<in> mset_factors m \<Longrightarrow> y \<in> mset_factors s \<Longrightarrow> a y \<otimes> b (s - y) \<otimes> c (m - s) \<in> carrier R)"
-    using assoc_aux1 assms(1) assms(2) assms(3) 
+    using assoc_aux1 assms(1) assms(2) assms(3)
     by blast
   have 4: "(\<Oplus>s\<in>mset_factors m. (\<Oplus>y\<in>mset_factors s. a y \<otimes> b (s - y) \<otimes> c (m - s))) =
     (\<Oplus>p\<in>right_cuts m. a (snd p) \<otimes> b (fst p - snd p) \<otimes> c (m - fst p))"
-    using 0 1 2 3 
-      double_finsum[of "mset_factors m" _ "right_cuts m"  
+    using 0 1 2 3
+      double_finsum[of "mset_factors m" _ "right_cuts m"
         "(\<lambda>x y.  (a y  \<otimes> b (x - y)) \<otimes> c (m - x))"]
-    by auto 
-  have 5: "(\<Oplus>x\<in>mset_factors m. (\<Oplus>xa\<in>mset_factors x. a xa \<otimes> b (x - xa)) \<otimes> c (m - x)) = 
+    by auto
+  have 5: "(\<Oplus>x\<in>mset_factors m. (\<Oplus>xa\<in>mset_factors x. a xa \<otimes> b (x - xa)) \<otimes> c (m - x)) =
            (\<Oplus>x\<in>mset_factors m. \<Oplus>xa\<in>mset_factors x. a xa \<otimes> b (x - xa) \<otimes> c (m - x))"
   proof-
     obtain f where f_def: "f =( \<lambda>x. (\<Oplus>xa\<in>mset_factors x. a xa \<otimes> b (x - xa)) \<otimes> c (m - x))"
       by simp
     obtain g where g_def: "g = (\<lambda>x.  \<Oplus>xa\<in>mset_factors x. a xa \<otimes> b (x - xa) \<otimes> c (m - x))"
       by simp
     have 50: "\<And>s. s \<in> (mset_factors m) \<Longrightarrow> f s = g s"
     proof-
       fix x
       assume As:  "x \<in> mset_factors m"
       show "f x = g x"
       proof-
         have f_eq: "f x = (\<Oplus>xa\<in>mset_factors x. a xa \<otimes> b (x - xa)) \<otimes> c (m - x)"
-          using f_def 
-          by auto 
+          using f_def
+          by auto
         have g_eq: "g x =  (\<Oplus>xa\<in>mset_factors x. a xa \<otimes> b (x - xa) \<otimes> c (m - x))"
-          using g_def  
-          by auto 
+          using g_def
+          by auto
         have f_eq': "f x =  (\<Oplus>xa\<in>mset_factors x. (a xa \<otimes> b (x - xa)) \<otimes> c (m - x))"
-          using f_eq finsum_ldistr[of "mset_factors x" "c (m - x)" "\<lambda>xa. (a xa \<otimes> b (x - xa))" ] assms 
+          using f_eq finsum_ldistr[of "mset_factors x" "c (m - x)" "\<lambda>xa. (a xa \<otimes> b (x - xa))" ] assms
           by (simp add: \<open>\<lbrakk>finite (mset_factors x); c (m - x) \<in> carrier R; (\<lambda>xa. a xa \<otimes> b (x - xa)) \<in> mset_factors x \<rightarrow> carrier R\<rbrakk> \<Longrightarrow> (\<Oplus>i\<in>mset_factors x. a i \<otimes> b (x - i)) \<otimes> c (m - x) = (\<Oplus>i\<in>mset_factors x. a i \<otimes> b (x - i) \<otimes> c (m - x))\<close> Pi_I carrier_coeffE)
         then show "f x = g x"
           by (simp add: g_eq)
       qed
     qed
     have 51: "f \<in> mset_factors m \<rightarrow> carrier R"
     proof
       fix x
       assume "x \<in> mset_factors m"
       have "(\<Oplus>xa\<in>mset_factors x. a xa \<otimes> b (x - xa)) \<otimes> c (m - x )\<in> carrier R"
-        using assms carrier_coeffE finsum_closed[of "\<lambda>xa.  a xa \<otimes> b (x - xa)" "mset_factors x"]  
+        using assms carrier_coeffE finsum_closed[of "\<lambda>xa.  a xa \<otimes> b (x - xa)" "mset_factors x"]
         by blast
-      then show "f x \<in> carrier R" using assms f_def by auto 
+      then show "f x \<in> carrier R" using assms f_def by auto
     qed
     have 52: "g \<in> mset_factors m \<rightarrow> carrier R"
     proof
       fix x
       assume "x \<in> mset_factors m"
       show "g x \<in> carrier R"
         using assms finsum_closed[of "\<lambda>xa. a xa \<otimes> b (x - xa) \<otimes> c (m - x)" "mset_factors x"] g_def
         by (metis (no_types, lifting) "50" "51" Pi_iff \<open>x \<in> mset_factors m\<close>)
     qed
-      
-    show ?thesis 
-      using 50 51 52 finsum_eq[of f "(mset_factors m) " g] 
+
+    show ?thesis
+      using 50 51 52 finsum_eq[of f "(mset_factors m) " g]
       by (metis (mono_tags, lifting) f_def finsum_cong' g_def)
   qed
   then have  6: "(\<Oplus>x\<in>mset_factors m. (\<Oplus>xa\<in>mset_factors x. a xa \<otimes> b (x - xa)) \<otimes> c (m - x)) =
     (\<Oplus>p\<in>right_cuts m. a (snd p) \<otimes> b (fst p - snd p) \<otimes> c (m - fst p))"
     by (simp add: "4")
-  have 7: "(\<Oplus>p\<in>right_cuts m. a (snd p) \<otimes> b (fst p - snd p) \<otimes> c (m - fst p)) 
+  have 7: "(\<Oplus>p\<in>right_cuts m. a (snd p) \<otimes> b (fst p - snd p) \<otimes> c (m - fst p))
       = (\<Oplus>p\<in>right_cuts m. a (snd p) \<otimes> (b (fst p - snd p) \<otimes> c (m - fst p)))"
-    using assms 
+    using assms
     by (meson local.ring_axioms m_assoc ring.carrier_coeff_def)
-  then show ?thesis 
-    using assms 5 6  
-    unfolding P_ring_mult_def 
+  then show ?thesis
+    using assms 5 6
+    unfolding P_ring_mult_def
     by simp
-qed  
+qed
 
 text\<open>The Associativity of Polynomial Multiplication:\<close>
 
 lemma P_ring_mult_assoc:
   assumes "carrier_coeff a"
   assumes "carrier_coeff b"
   assumes "carrier_coeff c"
   shows "a \<Otimes>\<^sub>p (b \<Otimes>\<^sub>p c) = (a \<Otimes>\<^sub>p b) \<Otimes>\<^sub>p c"
-proof  
+proof
   fix m
   show "(a \<Otimes>\<^sub>p (b \<Otimes>\<^sub>p c)) m = (a \<Otimes>\<^sub>p b \<Otimes>\<^sub>p c) m"
   proof-
     obtain f where f_def: "f = (\<lambda>p.  a (snd  p) \<otimes> (b ((fst p) -(snd p)) \<otimes> c (m - (fst p))))"
       by simp
     obtain g where g_def: "g = (\<lambda>p. a (fst p) \<otimes> (b (snd p) \<otimes> c (m - (fst p) - (snd p))))"
       by simp
     have f_dom: "f \<in> right_cuts m \<rightarrow> carrier R"
-      using assms f_def unfolding right_cuts_def 
+      using assms f_def unfolding right_cuts_def
       by (simp add: carrier_coeffE)
     have g_dom: "g \<in> left_cuts m \<rightarrow> carrier R"
-      using assms g_def unfolding left_cuts_def 
+      using assms g_def unfolding left_cuts_def
       by (simp add: carrier_coeffE)
     have 0: "finite (right_cuts m)"
       by (simp add: right_cuts_finite)
     have 1: "bij_betw (\<lambda> (x,y). (x + y, x))  (left_cuts m) (right_cuts m)"
       by (simp add: left_right_cuts_bij)
     have 2: "(\<And>s. s \<in> left_cuts m \<Longrightarrow> g s = f (case s of (x, y) \<Rightarrow> (x + y, x)))"
     proof-
       fix s
       assume "s \<in> left_cuts m"
       then obtain x y where xy: "s = (x, y) \<and> x \<subseteq># m \<and> y \<subseteq># m - x"
-        using  left_cuts_def 
+        using  left_cuts_def
         by blast
       then have g_eq: "g s = a x \<otimes> (b y \<otimes> c (m - x - y))"
-        using g_def fst_conv 
+        using g_def fst_conv
         by auto
       have f_eq: "f (case s of (x, y) \<Rightarrow> (x + y, x)) = f (x + y, x)"
         by (simp add: xy)
       then have f_eq': "f (case s of (x, y) \<Rightarrow> (x + y, x)) = a x \<otimes> (b ((x + y) - x) \<otimes> c (m - (x + y)))"
-        using f_def 
+        using f_def
         by simp
       then show "g s = f (case s of (x, y) \<Rightarrow> (x + y, x))"
         by (simp add: g_eq)
     qed
     have 3: "finsum R g (left_cuts m) = finsum R f (right_cuts m)"
     using 0 1 2 finsum_bij_eq[of g "left_cuts m" f "right_cuts m" "\<lambda> (x,y). (x + y, x)" ]
       using bij_betw_finite f_dom g_dom by blast
     then show ?thesis using assms right_cuts_sum left_cuts_sum
       by (metis (mono_tags, lifting) f_def f_dom finsum_cong' g_def g_dom)
   qed
 qed
 
 end
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsubsection\<open>Commutativity of Polynomial Multiplication\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
-context ring 
+context ring
 begin
 
 lemma mset_factors_bij:
 "bij_betw (\<lambda>x. m - x) (mset_factors m) (mset_factors m)"
   apply(rule bij_betwI')
   apply (metis monom_comp' monom_divides_factors)
    apply (simp add: monom_divides_factors)
     by (meson monom_comp' monom_divides_factors diff_subset_eq_self)
 
 lemma(in cring) P_ring_mult_comm:
   assumes "carrier_coeff a"
   assumes "carrier_coeff b"
   shows "a \<Otimes>\<^sub>p b  = b \<Otimes>\<^sub>p a"
 proof
   fix m
   show "(a \<Otimes>\<^sub>p b) m = (b \<Otimes>\<^sub>p a) m"
   unfolding P_ring_mult_def
-  apply (rule finsum_bij_eq[of "\<lambda> x. a x \<otimes> b (m - x)" "mset_factors m" 
+  apply (rule finsum_bij_eq[of "\<lambda> x. a x \<otimes> b (m - x)" "mset_factors m"
                           "\<lambda>x. b x \<otimes> a (m - x)" "mset_factors m" "\<lambda>x. m - x"])
   proof
     show "\<And>x. x \<in> mset_factors m \<Longrightarrow> a x \<otimes> b (m - x) \<in> carrier R"
     proof-
       fix x
       assume "x \<in> mset_factors m"
       show "a x \<otimes> b (m - x) \<in> carrier R"
-        using assms carrier_coeffE 
+        using assms carrier_coeffE
         by blast
     qed
     show "(\<lambda>x. b x \<otimes> a (m - x)) \<in> mset_factors m \<rightarrow> carrier R"
     proof
       fix x
       assume "x \<in> mset_factors m"
       show "b x \<otimes> a (m - x) \<in> carrier R"
-        using assms  carrier_coeffE 
-        by blast 
+        using assms  carrier_coeffE
+        by blast
     qed
-    show "finite (mset_factors m)" 
-      by simp 
+    show "finite (mset_factors m)"
+      by simp
     show "bij_betw ((-) m) (mset_factors m) (mset_factors m)"
       by (simp add: mset_factors_bij)
     show "\<And>s. s \<in> mset_factors m \<Longrightarrow> a s \<otimes> b (m - s) = b (m - s) \<otimes> a (m - (m - s))"
     proof-
       fix s
       assume "s \<in> mset_factors m"
       then show "a s \<otimes> b (m - s) = b (m - s) \<otimes> a (m - (m - s))"
-        using assms carrier_coeffE 
+        using assms carrier_coeffE
         by (metis local.ring_axioms m_comm ring.monom_comp' ring.monom_divides_factors)
     qed
   qed
-qed 
+qed
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsubsection\<open>Closure properties for multiplication\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 text\<open>Building monomials from polynomials:\<close>
 
-lemma indexed_const_P_mult_eq: 
+lemma indexed_const_P_mult_eq:
   assumes "a \<in> carrier R"
   assumes "b \<in> carrier R"
   shows "(indexed_const a) \<Otimes>\<^sub>p  (indexed_const b) = indexed_const (a \<otimes> b)"
 proof-
   have 0: "monomials_of R (indexed_const a) = (if (a = \<zero>) then {} else {{#}})"
     unfolding monomials_of_def indexed_const_def
-    by auto 
+    by auto
   have 1: "monomials_of R (indexed_const b) = (if (b = \<zero>) then {} else {{#}})"
     unfolding monomials_of_def indexed_const_def
-    by auto 
-  show ?thesis 
-    unfolding P_ring_mult_def 
+    by auto
+  show ?thesis
+    unfolding P_ring_mult_def
   proof
     fix m:: "'c monomial"
     show "(\<Oplus>x\<in>mset_factors m. indexed_const a x \<otimes> indexed_const b (m - x)) = indexed_const (a \<otimes> b) m "
     proof(cases "m = {#}")
       case True
       then have T0: "mset_factors m = {{#}}"
-        unfolding mset_factors_def 
-        by simp        
+        unfolding mset_factors_def
+        by simp
       have "(\<Oplus>x\<in>mset_factors m. indexed_const a x \<otimes> indexed_const b (m - x)) =
                    indexed_const a {#} \<otimes> indexed_const b ({#} - {#}) "
       proof-
         have 0: "(\<lambda>x. indexed_const a x \<otimes> indexed_const b (m - x)) \<in> {} \<rightarrow> carrier R"
           by blast
         have 1: "indexed_const a {#} \<otimes> indexed_const b (m - {#}) \<in> carrier R"
           by (simp add: assms(1) assms(2) indexed_const_def)
         have 2: "(\<Oplus>x\<in>{{#}}. indexed_const a x \<otimes> indexed_const b (m - x)) =
                  indexed_const a {#} \<otimes> indexed_const b (m - {#}) \<oplus> (\<Oplus>x\<in>{}. indexed_const a x \<otimes> indexed_const b (m - x))"
-        using True T0 0 1  finsum_insert[of "{}" "{#}" "\<lambda>x. indexed_const a x \<otimes> indexed_const b (m - x)"  ] 
+        using True T0 0 1  finsum_insert[of "{}" "{#}" "\<lambda>x. indexed_const a x \<otimes> indexed_const b (m - x)"  ]
             by (simp add: indexed_const_def)
-        then show ?thesis 
-          using True T0 0 1  finsum_insert[of "{}" "{#}" "\<lambda>x. indexed_const a x \<otimes> indexed_const b (m - x)"  ] 
+        then show ?thesis
+          using True T0 0 1  finsum_insert[of "{}" "{#}" "\<lambda>x. indexed_const a x \<otimes> indexed_const b (m - x)"  ]
               finsum_empty[of "\<lambda>x. indexed_const a x \<otimes> indexed_const b (m - x)"]
           by (simp add: \<open>\<lbrakk>finite {}; {#} \<notin> {}; (\<lambda>x. indexed_const a x \<otimes> indexed_const b (m - x)) \<in> {} \<rightarrow> carrier R; indexed_const a {#} \<otimes> indexed_const b (m - {#}) \<in> carrier R\<rbrakk> \<Longrightarrow> (\<Oplus>x\<in>{{#}}. indexed_const a x \<otimes> indexed_const b (m - x)) = indexed_const a {#} \<otimes> indexed_const b (m - {#}) \<oplus> (\<Oplus>x\<in>{}. indexed_const a x \<otimes> indexed_const b (m - x))\<close> indexed_const_def)
       qed
       then show ?thesis
         by (simp add: True indexed_const_def)
     next
       case False
-      then show ?thesis using 0 1 
-        by (simp add: add.finprod_one_eqI assms(1) assms(2) 
+      then show ?thesis using 0 1
+        by (simp add: add.finprod_one_eqI assms(1) assms(2)
             indexed_const_def )
     qed
   qed
 qed
 
-lemma indexed_const_P_multE: 
+lemma indexed_const_P_multE:
   assumes "P \<in> indexed_pset I (carrier R)"
   assumes "c \<in> carrier R"
   shows "(P \<Otimes>\<^sub>p (indexed_const c)) m =  (P m) \<otimes> c"
   unfolding P_ring_mult_def
 proof-
   have 3: "m \<notin> mset_factors m - {m}"
     by simp
   have 4: "finite ((mset_factors m) - {m})"
     by simp
   have 5: "P m \<otimes> indexed_const c (m - m) \<in> carrier R "
-  by (metis assms(1) assms(2) cancel_comm_monoid_add_class.diff_cancel 
-      carrier_coeffE indexed_const_def indexed_pset_in_carrier local.ring_axioms 
+  by (metis assms(1) assms(2) cancel_comm_monoid_add_class.diff_cancel
+      carrier_coeffE indexed_const_def indexed_pset_in_carrier local.ring_axioms
       ring.ring_simprules(5) subsetI)
   have 0: "(\<lambda>x. P x \<otimes> indexed_const c (m - x)) \<in> mset_factors m - {m} \<rightarrow> carrier R"
   proof
     fix x
     assume "x \<in> mset_factors m - {m}"
     then show "P x \<otimes> indexed_const c (m - x) \<in> carrier R "
-      using assms 
+      using assms
       by (meson carrier_coeffE indexed_const_in_carrier indexed_pset_in_carrier m_closed subsetI)
   qed
   have 1: "P m \<otimes> indexed_const c ( m - m) =  (P m \<otimes> c)"
     by (simp add: indexed_const_def)
   have 2: "\<And>x. x \<in> mset_factors m \<Longrightarrow>  P x \<otimes> indexed_const c (m - x)  = (if x = m then (P m \<otimes> c) else \<zero>)"
   proof-
     fix x
     assume "x \<in> mset_factors m "
     then have "indexed_const c (m - x) = (if x = m then c else \<zero>)"
-      unfolding indexed_const_def 
-      by (metis cancel_comm_monoid_add_class.diff_cancel 
+      unfolding indexed_const_def
+      by (metis cancel_comm_monoid_add_class.diff_cancel
           diff_zero monom_comp' monom_divides_factors)
     then show  "P x \<otimes> indexed_const c (m - x)  = (if x = m then (P m \<otimes> c) else \<zero>)"
       by (metis assms(1) carrier_coeffE indexed_pset_in_carrier
             local.semiring_axioms semiring.r_null set_eq_subset)
   qed
   then have "(\<Oplus>x\<in>(mset_factors m) - {m}. P x \<otimes> indexed_const c (m - x)) = \<zero>"
-    using assms 
+    using assms
     by (metis (no_types, lifting) DiffD1 DiffD2 add.finprod_one_eqI singletonI)
   then show "(\<Oplus>x\<in>(mset_factors m). P x \<otimes> indexed_const c (m - x)) = P m \<otimes> c"
-    using assms 0 1 3 4 5 finsum_insert[of "(mset_factors m) - {m}" 
-                                    m "\<lambda>x. P x \<otimes> indexed_const c (m - x) "] 
+    using assms 0 1 3 4 5 finsum_insert[of "(mset_factors m) - {m}"
+                                    m "\<lambda>x. P x \<otimes> indexed_const c (m - x) "]
     by (metis (no_types, lifting) m_factor add.l_cancel_one
               insert_Diff_single insert_absorb zero_closed)
 qed
 
 lemma indexed_const_var_mult:
   assumes "P \<in> indexed_pset I (carrier R)"
   assumes "c \<in> carrier R"
   assumes "i \<in> I"
   shows "P \<Otimes> i \<Otimes>\<^sub>p indexed_const c = (P \<Otimes>\<^sub>p (indexed_const c)) \<Otimes> i "
 proof
   fix m
   show "(P \<Otimes> i \<Otimes>\<^sub>p indexed_const c) m = (P \<Otimes>\<^sub>p indexed_const c \<Otimes> i) m"
   proof(cases "i \<in># m")
     case True
     then have T0: "(P \<Otimes> i \<Otimes>\<^sub>p indexed_const c) m  = (P \<Otimes> i) m \<otimes> c"
       using assms indexed_const_P_multE[of "P \<Otimes> i" I c m]
       by (simp add: indexed_pset.indexed_pmult)
-    then show ?thesis using assms indexed_const_P_multE[of P I c m]  
-      unfolding  indexed_pmult_def 
-      using True indexed_const_P_multE by fastforce      
+    then show ?thesis using assms indexed_const_P_multE[of P I c m]
+      unfolding  indexed_pmult_def
+      using True indexed_const_P_multE by fastforce
   next
     case False
         then have T0: "(P \<Otimes> i \<Otimes>\<^sub>p indexed_const c) m  = (P \<Otimes> i) m \<otimes> c"
       using assms indexed_const_P_multE[of "P \<Otimes> i" I c m]
       by (simp add: indexed_pset.indexed_pmult)
-    then show ?thesis using assms indexed_const_P_multE[of P I c m]  
-      unfolding  indexed_pmult_def 
+    then show ?thesis using assms indexed_const_P_multE[of P I c m]
+      unfolding  indexed_pmult_def
       using False by auto
   qed
-qed    
-    
+qed
+
 lemma indexed_const_P_mult_closed:
   assumes "a \<in> indexed_pset I (carrier R)"
-  assumes "c \<in> carrier R" 
+  assumes "c \<in> carrier R"
   shows "a \<Otimes>\<^sub>p (indexed_const c) \<in> indexed_pset I (carrier R)"
   apply(rule indexed_pset.induct[of a I "(carrier R)" ])
      apply (simp add: assms(1); fail)
 proof-
   show "\<And>k. (k \<in> carrier R) \<Longrightarrow> ((indexed_const k) \<Otimes>\<^sub>p (indexed_const c)) \<in> ((carrier R) [\<X>\<^bsub>I\<^esub>])"
-    using assms 
+    using assms
     by (metis indexed_const_P_mult_eq indexed_pset.simps m_closed)
   show "\<And>P Q. P \<in> ((carrier R) [\<X>\<^bsub>I\<^esub>]) \<Longrightarrow>
            P \<Otimes>\<^sub>p indexed_const c \<in> ((carrier R) [\<X>\<^bsub>I\<^esub>]) \<Longrightarrow>
            Q \<in> ((carrier R) [\<X>\<^bsub>I\<^esub>]) \<Longrightarrow> Q \<Otimes>\<^sub>p indexed_const c \<in> ((carrier R) [\<X>\<^bsub>I\<^esub>]) \<Longrightarrow> P \<Oplus> Q \<Otimes>\<^sub>p indexed_const c \<in> ((carrier R) [\<X>\<^bsub>I\<^esub>])"
-    using P_ring_ldistr 
+    using P_ring_ldistr
     by (metis assms(2) carrier_coeff_def indexed_const_in_carrier indexed_pset.indexed_padd indexed_pset_in_carrier subsetI)
   show "\<And>P i. P \<in> ((carrier R) [\<X>\<^bsub>I\<^esub>]) \<Longrightarrow> P \<Otimes>\<^sub>p indexed_const c \<in> ((carrier R) [\<X>\<^bsub>I\<^esub>]) \<Longrightarrow> i \<in> I \<Longrightarrow> P \<Otimes> i \<Otimes>\<^sub>p indexed_const c \<in> ((carrier R) [\<X>\<^bsub>I\<^esub>])"
   proof-
     fix P i
     assume A0: "P \<in> ((carrier R) [\<X>\<^bsub>I\<^esub>])"
     assume A1: " (P \<Otimes>\<^sub>p indexed_const c) \<in> ((carrier R) [\<X>\<^bsub>I\<^esub>])"
     assume A2: "i \<in> I"
     show "P \<Otimes> i \<Otimes>\<^sub>p indexed_const c \<in> (carrier R [\<X>\<^bsub>I\<^esub>])"
-      using assms  A0 A1 A2 indexed_const_var_mult local.ring_axioms ring.indexed_pset.simps 
+      using assms  A0 A1 A2 indexed_const_var_mult local.ring_axioms ring.indexed_pset.simps
       by (metis (no_types, opaque_lifting))
   qed
 qed
 
-lemma monom_add_mset: 
+lemma monom_add_mset:
 "mset_to_IP R (add_mset i m) = mset_to_IP R m \<Otimes> i"
-  unfolding indexed_pmult_def 
+  unfolding indexed_pmult_def
   by (metis (no_types, opaque_lifting) add_mset_diff_bothsides diff_empty mset_to_IP_def multi_member_split union_single_eq_member)
 
 text\<open>Monomials are closed under multiplication:\<close>
 
 lemma monom_mult:
 "mset_to_IP R m \<Otimes>\<^sub>p mset_to_IP R n = mset_to_IP R (m + n)"
 proof-
   have "(\<lambda>ma. \<Oplus>x\<in>mset_factors ma. (if x = m then \<one> else \<zero>) \<otimes> (if ma - x = n then \<one> else \<zero>))
        = (\<lambda>ma. (if ma = n + m then \<one> else \<zero>))"
   proof
     fix ma
     show "(\<Oplus>x\<in>mset_factors ma. (if x = m then \<one> else \<zero>) \<otimes> (if ma - x = n then \<one> else \<zero>)) = (if ma = n + m then \<one> else \<zero>)"
-    
+
     proof-
-      have 0: "\<And>x. x \<in> mset_factors ma \<Longrightarrow> 
-            (\<lambda> x. (if x = m then \<one> else \<zero>) \<otimes> (if ma - x = n then \<one> else \<zero>)) x = 
+      have 0: "\<And>x. x \<in> mset_factors ma \<Longrightarrow>
+            (\<lambda> x. (if x = m then \<one> else \<zero>) \<otimes> (if ma - x = n then \<one> else \<zero>)) x =
                   (if (x = m) then (if ma - x = n then \<one> else \<zero>) else \<zero>) "
         by simp
       then have 1: "(\<Oplus>x\<in>((mset_factors ma) - {m}). (if x = m then \<one> else \<zero>) \<otimes> (if ma - x = n then \<one> else \<zero>)) =
             \<zero>"
         by (metis (no_types, lifting) add.finprod_one_eqI mem_Collect_eq set_diff_eq singletonI)
       have 2: "finite (mset_factors ma - {m}) "
         by simp
       have 3: "m \<notin> mset_factors ma - {m}"
         by simp
       have 4: "(\<lambda>x. (if x = m then \<one> else \<zero>) \<otimes> (if ma - x = n then \<one> else \<zero>)) \<in> mset_factors ma - {m} \<rightarrow> carrier R"
       proof
         fix x
         assume " x \<in> mset_factors ma - {m}"
         show "(if x = m then \<one> else \<zero>) \<otimes> (if ma - x = n then \<one> else \<zero>) \<in> carrier R "
           by auto
       qed
       show ?thesis
       proof(cases "m \<in> (mset_factors ma)")
         case True
         have T0: " (\<Oplus>x\<in>insert m (mset_factors ma - {m}). (if x = m then \<one> else \<zero>) \<otimes> (if ma - x = n then \<one> else \<zero>)) =
       (if m = m then \<one> else \<zero>) \<otimes> (if ma - m = n then \<one> else \<zero>) \<oplus>
       (\<Oplus>x\<in>mset_factors ma - {m}. (if x = m then \<one> else \<zero>) \<otimes> (if ma - x = n then \<one> else \<zero>))"
-          using True 1 2 3 4 finsum_insert[of "((mset_factors ma) - {m})" m 
-                                   "\<lambda>x. (if x = m then \<one> else \<zero>) \<otimes> (if ma - x = n then \<one> else \<zero>)"] 
+          using True 1 2 3 4 finsum_insert[of "((mset_factors ma) - {m})" m
+                                   "\<lambda>x. (if x = m then \<one> else \<zero>) \<otimes> (if ma - x = n then \<one> else \<zero>)"]
           using m_closed one_closed zero_closed by presburger
         have T1: "(mset_factors ma) = insert m (mset_factors ma - {m})"
-          using True 
+          using True
           by blast
         then have  "(\<Oplus>x\<in>(mset_factors ma). (if x = m then \<one> else \<zero>) \<otimes> (if ma - x = n then \<one> else \<zero>))
                       = (if m = m then \<one> else \<zero>) \<otimes> (if ma - m = n then \<one> else \<zero>)"
           using T0 T1 1 add.l_cancel_one insert_Diff_single insert_absorb2 l_one mk_disjoint_insert one_closed zero_closed
           by presburger
         then have "(\<Oplus>x\<in>(mset_factors ma). (if x = m then \<one> else \<zero>) \<otimes> (if ma - x = n then \<one> else \<zero>))
                       = (if ma - m = n then \<one> else \<zero>)"
           by simp
-        then show ?thesis 
+        then show ?thesis
           by (metis (no_types, lifting) monom_divides_factors True add.commute add_diff_cancel_right' subset_mset.add_diff_inverse)
       next
         case False
-        then show ?thesis 
+        then show ?thesis
           by (metis (no_types, lifting) "0" monom_divides_factors add.finprod_one_eqI mset_subset_eq_add_right)
       qed
     qed
   qed
-  then show ?thesis 
-  unfolding mset_to_IP_def P_ring_mult_def   
+  then show ?thesis
+  unfolding mset_to_IP_def P_ring_mult_def
   by (simp add: union_commute)
 qed
 
 lemma poly_index_mult:
   assumes "a \<in> indexed_pset I (carrier R)"
   assumes "i \<in> I"
   shows "a \<Otimes> i = a \<Otimes>\<^sub>p mset_to_IP R {#i#}"
 proof
   fix m
   show "(a \<Otimes> i) m = (a \<Otimes>\<^sub>p mset_to_IP R {#i#}) m"
   proof(cases "i \<in># m")
     case True
     have T0: "(a \<Otimes> i) m = a (m - {#i#})"
       by (simp add: True indexed_pmult_def)
     have T1: "(a \<Otimes>\<^sub>p mset_to_IP R {#i#}) m = a (m - {#i#})"
     proof-
-      have T10: "(a \<Otimes>\<^sub>p mset_to_IP R {#i#}) m 
+      have T10: "(a \<Otimes>\<^sub>p mset_to_IP R {#i#}) m
                 = (\<Oplus>x\<in>mset_factors m. a x \<otimes> mset_to_IP R {#i#} (m - x)) "
-        unfolding P_ring_mult_def 
-        by auto 
-      have T11: "\<And>x. x \<in> mset_factors m \<Longrightarrow> 
+        unfolding P_ring_mult_def
+        by auto
+      have T11: "\<And>x. x \<in> mset_factors m \<Longrightarrow>
                   mset_to_IP R {#i#} (m - x) = (if x = (m - {#i#}) then \<one> else \<zero>)"
-        unfolding mset_to_IP_def 
+        unfolding mset_to_IP_def
         by (metis Multiset.diff_cancel Multiset.diff_right_commute True diff_single_eq_union
             diff_single_trivial diff_zero monom_comp' monom_divides_factors multi_drop_mem_not_eq)
-      have T12: "\<And>x. x \<in> mset_factors m \<Longrightarrow> 
+      have T12: "\<And>x. x \<in> mset_factors m \<Longrightarrow>
               a x \<otimes> mset_to_IP R {#i#} (m - x) = (if  x = (m - {#i#}) then a (m - {#i#}) else \<zero>)"
       proof-
         fix x
         assume "x \<in> mset_factors m"
         then show " a x \<otimes> mset_to_IP R {#i#} (m - x) = (if x = m - {#i#} then a (m - {#i#}) else \<zero>)"
           apply(cases "x = m - {#i#}")
            apply (metis T0 T11 \<open>x \<in> mset_factors m\<close> assms(1) carrier_coeffE empty_subsetI
               ideal_is_subalgebra indexed_pset_in_carrier oneideal r_one subalgebra_in_carrier)
-          by (metis T11 \<open>x \<in> mset_factors m\<close> assms(1) carrier_coeffE carrier_is_subalgebra 
+          by (metis T11 \<open>x \<in> mset_factors m\<close> assms(1) carrier_coeffE carrier_is_subalgebra
               empty_subsetI  indexed_pset_in_carrier  r_null  subalgebra_in_carrier)
       qed
       have  T13: "(\<Oplus>x\<in>(mset_factors m) - {m - {#i#}}. a x \<otimes> mset_to_IP R {#i#} (m - x)) = \<zero>"
-        using T12 
+        using T12
         by (metis (no_types, lifting) DiffE add.finprod_one_eqI singletonI)
       have T14: "finite (mset_factors m - {m - {#i#}})"
         using mset_factors_finite by blast
       have T15: "m - {#i#} \<notin> mset_factors m - {m - {#i#}}"
         by simp
       have T16: "  (\<lambda>x. a x \<otimes> mset_to_IP R {#i#} (m - x)) \<in> mset_factors m - {m - {#i#}} \<rightarrow> carrier R"
       proof
         fix x
         assume "x \<in> mset_factors m - {m - {#i#}}"
         then show " a x \<otimes> mset_to_IP R {#i#} (m - x) \<in> carrier R"
           using assms T12 by auto
       qed
       have "m - (m - {#i#}) = {#i#}"
         by (metis monom_comp' True single_subset_iff)
       then have T17: " a (m - {#i#}) \<otimes> mset_to_IP R {#i#} (m - (m - {#i#})) \<in> carrier R"
-        unfolding mset_to_IP_def apply auto using assms  
+        unfolding mset_to_IP_def apply auto using assms
         by (meson carrier_coeffE carrier_is_subalgebra empty_subsetI indexed_pset_in_carrier m_closed one_closed subalgebra_in_carrier)
       have T18:"(\<Oplus>x\<in>insert (m - {#i#}) (mset_factors m - {m - {#i#}}). a x \<otimes> mset_to_IP R {#i#} (m - x)) =
-      a (m - {#i#}) \<otimes> mset_to_IP R {#i#} (m - (m - {#i#})) \<oplus> (\<Oplus>x\<in>mset_factors m - {m - {#i#}}. a x \<otimes> mset_to_IP R {#i#} (m - x))" 
+      a (m - {#i#}) \<otimes> mset_to_IP R {#i#} (m - (m - {#i#})) \<oplus> (\<Oplus>x\<in>mset_factors m - {m - {#i#}}. a x \<otimes> mset_to_IP R {#i#} (m - x))"
         using T12 T13 T14 T15 T16 T17 unfolding P_ring_mult_def
-        using finsum_insert[of "mset_factors m - {m - {#i#}}" "m - {#i#}" 
-            "\<lambda>x. a x \<otimes> mset_to_IP R {#i#} (m - x)"] 
+        using finsum_insert[of "mset_factors m - {m - {#i#}}" "m - {#i#}"
+            "\<lambda>x. a x \<otimes> mset_to_IP R {#i#} (m - x)"]
         by blast
       have T19: "a (m - {#i#}) \<otimes> mset_to_IP R {#i#} (m - (m - {#i#})) = a (m - {#i#}) "
       proof-
         have " (m - (m - {#i#})) = {#i#}"
-          using True 
+          using True
           by (metis monom_comp' single_subset_iff)
         then have "mset_to_IP R {#i#} (m - (m - {#i#})) = \<one>"
           by (metis mset_to_IP_def)
         have "a (m - {#i#}) \<in> carrier R"
-          using assms 
-          by (meson carrier_coeffE carrier_is_subalgebra exp_base_closed 
+          using assms
+          by (meson carrier_coeffE carrier_is_subalgebra exp_base_closed
               indexed_pset_in_carrier one_closed subalgebra_in_carrier)
-        then show ?thesis using assms 
+        then show ?thesis using assms
           using \<open>mset_to_IP R {#i#} (m - (m - {#i#})) = \<one>\<close> by auto
       qed
       have T20: "(\<Oplus>x\<in>insert (m - {#i#}) ((mset_factors m) - {m - {#i#}}). a x \<otimes> mset_to_IP R {#i#} (m - x)) =
-       a (m - {#i#}) \<oplus> (\<Oplus>x\<in>mset_factors m - {m - {#i#}}. a x \<otimes> mset_to_IP R {#i#} (m - x))" 
+       a (m - {#i#}) \<oplus> (\<Oplus>x\<in>mset_factors m - {m - {#i#}}. a x \<otimes> mset_to_IP R {#i#} (m - x))"
         using T18 T19 by auto
       have T21: "insert (m - {#i#}) ((mset_factors m) - {m - {#i#}}) = mset_factors m"
       proof-
         have "(m - {#i#}) \<in> mset_factors m"
           by (simp add: monom_divides_factors)
-        then show ?thesis 
+        then show ?thesis
           by blast
       qed
-      then show ?thesis using T13 T20 True unfolding P_ring_mult_def  
+      then show ?thesis using T13 T20 True unfolding P_ring_mult_def
         using T17 T19 by auto
     qed
-    then show ?thesis 
+    then show ?thesis
       by (simp add: T0)
   next
     case False
     have F0: "(a \<Otimes> i) m = \<zero> "
       by (simp add: False indexed_pmult_def)
     have F1: "  (a \<Otimes>\<^sub>p mset_to_IP R {#i#}) m = \<zero>"
-      unfolding P_ring_mult_def 
+      unfolding P_ring_mult_def
     proof-
       have "\<And>x. x \<in> mset_factors m \<Longrightarrow>  a x \<otimes> mset_to_IP R {#i#} (m - x) = \<zero>"
       proof-
         fix x
         assume A: "x \<in> mset_factors m"
-        have B: "m - x \<noteq> {#i#}" using False A 
+        have B: "m - x \<noteq> {#i#}" using False A
           by (metis diff_subset_eq_self single_subset_iff)
         then show "a x \<otimes> mset_to_IP R {#i#} (m - x) = \<zero>"
-          using assms False unfolding mset_to_IP_def  
+          using assms False unfolding mset_to_IP_def
           using  carrier_coeffE indexed_pset_in_carrier by fastforce
       qed
       then show "(\<Oplus>x\<in>mset_factors m. a x \<otimes> mset_to_IP R {#i#} (m - x)) = \<zero>"
         by (simp add: add.finprod_one_eqI)
     qed
-    then show ?thesis 
-      by (simp add: F0)  
+    then show ?thesis
+      by (simp add: F0)
   qed
 qed
 
 lemma mset_to_IP_mult_closed:
   assumes "a \<in> indexed_pset I (carrier R)"
   shows "set_mset m \<subseteq> I \<Longrightarrow> a \<Otimes>\<^sub>p mset_to_IP R m \<in> indexed_pset I (carrier R)"
 proof(induction m)
   case empty
   then have "mset_to_IP R {#} = indexed_const \<one>"
     unfolding mset_to_IP_def indexed_const_def by auto
-  then show ?case 
+  then show ?case
     by (simp add: \<open>mset_to_IP R {#} = indexed_const \<one>\<close> assms indexed_const_P_mult_closed)
 next
   case (add x m)
-  fix x 
+  fix x
   fix m :: "'c monomial"
   assume A: "set_mset (add_mset x m) \<subseteq> I"
   assume B: "set_mset m \<subseteq> I \<Longrightarrow> a \<Otimes>\<^sub>p mset_to_IP R m \<in> (carrier R [\<X>\<^bsub>I\<^esub>])"
   show " a \<Otimes>\<^sub>p mset_to_IP R (add_mset x m) \<in> (carrier R [\<X>\<^bsub>I\<^esub>])"
   proof-
     have 0: "x \<in> I"
       using A by auto
     have 1: "set_mset m \<subseteq> I"
       using A by auto
     have 2: " a \<Otimes>\<^sub>p mset_to_IP R m \<in> (carrier R [\<X>\<^bsub>I\<^esub>])"
       using "1" B by blast
-    have 3: " a \<Otimes>\<^sub>p mset_to_IP R (add_mset x m) = (a \<Otimes>\<^sub>p mset_to_IP R m) \<Otimes> x" 
+    have 3: " a \<Otimes>\<^sub>p mset_to_IP R (add_mset x m) = (a \<Otimes>\<^sub>p mset_to_IP R m) \<Otimes> x"
     proof-
-      have 30: " a \<Otimes>\<^sub>p mset_to_IP R (add_mset x m) = a \<Otimes>\<^sub>p (mset_to_IP R m \<Otimes> x)" 
+      have 30: " a \<Otimes>\<^sub>p mset_to_IP R (add_mset x m) = a \<Otimes>\<^sub>p (mset_to_IP R m \<Otimes> x)"
         using monom_add_mset[of x m]
         by auto
       have 31: "(a \<Otimes>\<^sub>p mset_to_IP R m) \<Otimes> x = a \<Otimes>\<^sub>p (mset_to_IP R m \<Otimes> x)"
       proof
         fix y
         show "(a \<Otimes>\<^sub>p mset_to_IP R m \<Otimes> x) y = (a \<Otimes>\<^sub>p (mset_to_IP R m \<Otimes> x)) y "
         proof-
           have 310: "(a \<Otimes>\<^sub>p mset_to_IP R m \<Otimes> x) y = (a \<Otimes>\<^sub>p mset_to_IP R m \<Otimes>\<^sub>p mset_to_IP R {#x#}) y"
-            using poly_index_mult "0" "2" 
+            using poly_index_mult "0" "2"
             by fastforce
           have 311: " (mset_to_IP R m \<Otimes> x) = mset_to_IP R m \<Otimes>\<^sub>p mset_to_IP R {#x#}"
-            using  "0" "2" 
+            using  "0" "2"
             by (metis add_mset_add_single monom_add_mset monom_mult)
           have 312: "carrier_coeff a "
             using assms indexed_pset_in_carrier by blast
           have 313: "carrier_coeff (mset_to_IP R m)"
             by (simp add: carrier_coeff_def mset_to_IP_def)
-          have 314: "carrier_coeff (mset_to_IP R {#x#})" 
+          have 314: "carrier_coeff (mset_to_IP R {#x#})"
             by (metis carrier_coeff_def mset_to_IP_def one_closed zero_closed)
-          show ?thesis 
-            using 310 311 312 313 314 
-                  P_ring_mult_assoc[of a "mset_to_IP R m" "mset_to_IP R {#x#}"]  
-            by simp 
+          show ?thesis
+            using 310 311 312 313 314
+                  P_ring_mult_assoc[of a "mset_to_IP R m" "mset_to_IP R {#x#}"]
+            by simp
         qed
       qed
-      then show ?thesis 
-        by (simp add: monom_add_mset)        
+      then show ?thesis
+        by (simp add: monom_add_mset)
     qed
-    then show ?thesis 
+    then show ?thesis
       by (simp add: "0" "1" B indexed_pset.indexed_pmult)
   qed
 qed
 
 text\<open>
   A predicate which identifies when the variables used in a given polynomial $P$ are only
    drawn from a fixed variable set $I$.
 \<close>
 abbreviation monoms_in where
 "monoms_in I P \<equiv> (\<forall>m \<in> monomials_of R P. set_mset m \<subseteq> I)"
 
 lemma monoms_of_const:
 "monomials_of R (indexed_const k) = (if k = \<zero> then {} else {{#}})"
   unfolding indexed_const_def monomials_of_def
   by auto
 
 lemma const_monoms_in:
 "monoms_in I (indexed_const k)"
   unfolding indexed_const_def monomials_of_def
   using  count_empty count_eq_zero_iff monomials_ofE subsetI
   by simp
 
 lemma mset_to_IP_indices:
   shows  "P \<in> indexed_pset I (carrier R) \<Longrightarrow> monoms_in I P"
   apply(erule indexed_pset.induct[of])
-   apply (simp add: const_monoms_in; fail) 
+   apply (simp add: const_monoms_in; fail)
 proof-
   show "\<And>P Q. P \<in> (carrier R [\<X>\<^bsub>I\<^esub>]) \<Longrightarrow>
            \<forall>m\<in>monomials_of R P. set_mset m \<subseteq> I \<Longrightarrow>
            Q \<in> (carrier R [\<X>\<^bsub>I\<^esub>]) \<Longrightarrow> \<forall>m\<in>monomials_of R Q. set_mset m \<subseteq> I \<Longrightarrow> \<forall>m\<in>monomials_of R (P \<Oplus> Q). set_mset m \<subseteq> I"
   proof
     fix P Q
     fix m
     assume A: "P \<in> (carrier R [\<X>\<^bsub>I\<^esub>])" "\<forall>m\<in>monomials_of R P. set_mset m \<subseteq> I"
                 "Q \<in> (carrier R [\<X>\<^bsub>I\<^esub>])" "\<forall>m\<in>monomials_of R Q. set_mset m \<subseteq> I"
                   " m \<in> monomials_of R (P \<Oplus> Q)"
     show "set_mset m \<subseteq> I"
     proof-
       have "m \<in> monomials_of R P \<or> m \<in> monomials_of R Q"
-        using A using monomials_of_add[of P Q] 
+        using A using monomials_of_add[of P Q]
         by blast
-      then show ?thesis using A 
+      then show ?thesis using A
         by blast
     qed
   qed
   show "\<And>P i. P \<in> (carrier R [\<X>\<^bsub>I\<^esub>]) \<Longrightarrow> \<forall>m\<in>monomials_of R P. set_mset m \<subseteq> I \<Longrightarrow> i \<in> I \<Longrightarrow> \<forall>m\<in>monomials_of R (P \<Otimes> i). set_mset m \<subseteq> I"
   proof
-    fix P 
-    fix i 
+    fix P
+    fix i
     fix m
     assume A: "P \<in> (carrier R [\<X>\<^bsub>I\<^esub>])" "\<forall>m\<in>monomials_of R P. set_mset m \<subseteq> I" "i \<in> I" "m \<in> monomials_of R (P \<Otimes> i)"
     obtain n where "n \<in> monomials_of R P \<and> m = add_mset i n"
-      using A 
+      using A
       by (metis image_iff monomials_of_p_mult')
-    then show "set_mset m \<subseteq> I" 
-      using A 
+    then show "set_mset m \<subseteq> I"
+      using A
       by auto
   qed
 qed
 
 lemma mset_to_IP_indices':
   assumes "P \<in> indexed_pset I (carrier R)"
   assumes "m \<in> monomials_of R P"
   shows "set_mset m \<subseteq> I"
   using assms(1) assms(2) mset_to_IP_indices by blast
 
-lemma one_mset_to_IP: 
+lemma one_mset_to_IP:
  "mset_to_IP R {#} = indexed_const \<one>"
   unfolding mset_to_IP_def indexed_const_def
   by blast
 
 lemma mset_to_IP_closed:
   shows "set_mset m \<subseteq> I \<Longrightarrow>mset_to_IP R m \<in> indexed_pset I (carrier R) "
   apply(induction m)
   apply (simp add: indexed_pset.indexed_const one_mset_to_IP)
   by (metis (no_types, lifting) add_mset_commute indexed_pset.simps
       monom_add_mset mset_add subset_iff union_single_eq_member)
 
 lemma mset_to_IP_closed':
   assumes "P \<in> indexed_pset I (carrier R)"
   assumes "m \<in> monomials_of R P"
   shows "mset_to_IP R m \<in> indexed_pset I (carrier R)"
   by (meson assms(1) assms(2) mset_to_IP_closed mset_to_IP_indices')
 
 text\<open>This lemma expresses closure under multiplcation for indexed polynomials.\<close>
 
 lemma  P_ring_mult_closed:
   assumes "carrier_coeff P"
   assumes "carrier_coeff Q"
   shows  "carrier_coeff (P_ring_mult R P Q)"
   unfolding carrier_coeff_def
 proof
   fix m
   have "(\<lambda>x. P x \<otimes> Q (m - x)) \<in> mset_factors m \<rightarrow> carrier R"
   proof
     fix x
     assume "x \<in> mset_factors m"
     then show "P x \<otimes> Q (m - x) \<in> carrier R"
       using assms  carrier_coeffE
       by blast
   qed
   then show "(P \<Otimes>\<^sub>p Q) m \<in> carrier R"
     using assms finsum_closed[of "(\<lambda>x. (P x) \<otimes> (Q (m - x)))" "mset_factors m"]
     unfolding carrier_coeff_def  P_ring_mult_def
     by blast
 qed
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsection\<open>Multivariable Polynomial Induction\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 lemma mpoly_induct:
   assumes "\<And>Q. Q \<in> indexed_pset I (carrier R) \<and> card (monomials_of R Q) = 0 \<Longrightarrow> P Q"
   assumes "\<And>n. (\<And>Q. Q \<in> indexed_pset I (carrier R) \<and> card (monomials_of R Q) \<le>  n \<Longrightarrow> P Q)
              \<Longrightarrow> (\<And>Q. Q \<in> indexed_pset I (carrier R) \<and> card (monomials_of R Q) \<le> (Suc n) \<Longrightarrow> P Q)"
   assumes "Q \<in> indexed_pset I (carrier R)"
   shows "P Q"
 proof-
   have "\<And>m. \<And>Q. Q \<in> indexed_pset I (carrier R) \<and> card(monomials_of R Q) \<le> m \<Longrightarrow> P Q"
   proof-
     fix m
     show "\<And>Q. Q \<in> indexed_pset I (carrier R) \<and> card(monomials_of R Q) \<le> m \<Longrightarrow> P Q"
       apply(induction m)
       using assms(1)
        apply blast
-      using assms 
+      using assms
       by blast
   qed
-  then show ?thesis 
+  then show ?thesis
     using assms(3) by blast
 qed
 
 lemma monomials_of_card_zero:
   assumes "Q \<in> indexed_pset I (carrier R) \<and> card (monomials_of R Q) = 0"
   shows "Q = indexed_const \<zero>"
 proof-
   have 0: "carrier_coeff  Q"
-    using assms  indexed_pset_in_carrier 
+    using assms  indexed_pset_in_carrier
     by blast
   have "\<And>m. Q m = \<zero>"
     unfolding monomials_of_def
-    using assms monomials_finite 
+    using assms monomials_finite
     by (metis card_0_eq complement_of_monomials_of empty_iff)
-  then show  ?thesis 
-  using 0 assms unfolding indexed_const_def 
+  then show  ?thesis
+  using 0 assms unfolding indexed_const_def
   by auto
 qed
 
 text\<open>Polynomial induction on the number of monomials with nonzero coefficient:\<close>
 
 lemma mpoly_induct':
   assumes "P (indexed_const \<zero>)"
   assumes "\<And>n. (\<And>Q. Q \<in> indexed_pset I (carrier R) \<and> card (monomials_of R Q) \<le>  n \<Longrightarrow> P Q)
              \<Longrightarrow> (\<And>Q. Q \<in> indexed_pset I (carrier R) \<and> card (monomials_of R Q) = (Suc n) \<Longrightarrow> P Q)"
   assumes "Q \<in> indexed_pset I (carrier R)"
   shows "P Q"
   apply(rule mpoly_induct)
   using assms(1) monomials_of_card_zero apply blast
 proof-
   show "\<And>n Q. (\<And>Q. Q \<in> (carrier R [\<X>\<^bsub>I\<^esub>]) \<and> card (monomials_of R Q) \<le> n \<Longrightarrow> P Q) \<Longrightarrow> Q \<in> (carrier R [\<X>\<^bsub>I\<^esub>]) \<and> card (monomials_of R Q) \<le> Suc n \<Longrightarrow> P Q"
   proof-
     fix n
     fix Q
     assume A0: "(\<And>Q. Q \<in> (carrier R [\<X>\<^bsub>I\<^esub>]) \<and> card (monomials_of R Q) \<le> n \<Longrightarrow> P Q)"
     assume A1: "Q \<in> (carrier R [\<X>\<^bsub>I\<^esub>]) \<and> card (monomials_of R Q) \<le> Suc n"
     show "P Q"
       apply(cases  "card (monomials_of R Q) = Suc n")
-      using assms A0 A1 
+      using assms A0 A1
        apply blast
-      using assms A0 A1 
+      using assms A0 A1
       using le_SucE by blast
   qed
   show "Q \<in> (carrier R [\<X>\<^bsub>I\<^esub>])"
-    using assms by auto 
+    using assms by auto
 qed
 
 lemma monomial_poly_split:
   assumes "P \<in> indexed_pset I (carrier R)"
   assumes "m \<in> monomials_of R P"
   shows "(restrict_poly_to_monom_set R P ((monomials_of R P) - {m})) \<Oplus> ((mset_to_IP R m) \<Otimes>\<^sub>p (indexed_const (P m))) = P"
 proof fix x
   show "(restrict_poly_to_monom_set R P (monomials_of R P - {m}) \<Oplus> (mset_to_IP R m \<Otimes>\<^sub>p indexed_const (P m))) x = P x"
   proof(cases "x = m")
     case True
     have T0: "(restrict_poly_to_monom_set R P (monomials_of R P - {m})) x = \<zero>"
-      unfolding restrict_poly_to_monom_set_def 
+      unfolding restrict_poly_to_monom_set_def
       by (simp add: True)
     have T1: "(mset_to_IP R m \<Otimes>\<^sub>p indexed_const (P m)) x = P x"
       using assms True indexed_const_P_multE[of "mset_to_IP R m" I "P m" m]
-              mset_to_IP_simp 
-      by (metis Idl_subset_ideal' carrier_coeffE genideal_one indexed_pset_in_carrier 
+              mset_to_IP_simp
+      by (metis Idl_subset_ideal' carrier_coeffE genideal_one indexed_pset_in_carrier
           l_one mset_to_IP_closed' one_closed)
-    then show ?thesis 
-      using T0 True 
-      unfolding indexed_padd_def 
+    then show ?thesis
+      using T0 True
+      unfolding indexed_padd_def
       using assms(1) carrier_coeffE indexed_pset_in_carrier l_zero
-      by fastforce    
+      by fastforce
   next
     case False
     then have F0: "(restrict_poly_to_monom_set R P (monomials_of R P - {m})) x = P x"
     proof(cases "x \<in> monomials_of R P")
       case True
-      then show ?thesis 
+      then show ?thesis
         by (simp add: False restrict_poly_to_monom_set_def)
     next
       case False
-      then show ?thesis 
-        by (simp add: complement_of_monomials_of restrict_poly_to_monom_set_def)              
+      then show ?thesis
+        by (simp add: complement_of_monomials_of restrict_poly_to_monom_set_def)
     qed
     have F1: "mset_to_IP R m \<in> (carrier R [\<X>\<^bsub>I\<^esub>])"
       using assms(1) assms(2) mset_to_IP_closed' by blast
     have F2: "P m \<in> carrier R"
       using assms(1) carrier_coeffE indexed_pset_in_carrier by blast
     have F3: "(mset_to_IP R m \<Otimes>\<^sub>p indexed_const (P m)) x = \<zero>"
-      using False F1 F2 assms indexed_const_P_multE[of "mset_to_IP R m" I "P m" x] 
+      using False F1 F2 assms indexed_const_P_multE[of "mset_to_IP R m" I "P m" x]
       by (simp add: F1 \<open>m \<in> monomials_of R P\<close>)
-    then show ?thesis using F0 unfolding indexed_padd_def  
+    then show ?thesis using F0 unfolding indexed_padd_def
       using assms(1) carrier_coeffE indexed_pset_in_carrier
       by fastforce
   qed
 qed
 
 lemma restrict_not_in_monoms:
   assumes "a \<notin> monomials_of R P"
   shows "restrict_poly_to_monom_set R P A = restrict_poly_to_monom_set R P (insert a A)"
 proof
   fix x
   show " restrict_poly_to_monom_set R P A x = restrict_poly_to_monom_set R P (insert a A) x "
-    unfolding restrict_poly_to_monom_set_def using assms unfolding monomials_of_def 
+    unfolding restrict_poly_to_monom_set_def using assms unfolding monomials_of_def
   by simp
 qed
 
 lemma restriction_closed':
   assumes "P \<in> indexed_pset I (carrier R)"
   assumes "finite ms"
   shows "(restrict_poly_to_monom_set R P ms) \<in> indexed_pset I (carrier R)"
   apply(rule finite.induct[of ms])
   apply (simp add: assms(2); fail)
 proof-
   show "restrict_poly_to_monom_set R P {} \<in> (carrier R [\<X>\<^bsub>I\<^esub>])"
   proof-
     have "restrict_poly_to_monom_set R P {} = indexed_const \<zero>"
       unfolding restrict_poly_to_monom_set_def indexed_const_def by auto
-    then show ?thesis 
+    then show ?thesis
       by (simp add: indexed_pset.indexed_const)
   qed
   show "\<And>A a. finite A \<Longrightarrow> restrict_poly_to_monom_set R P A \<in> (carrier R [\<X>\<^bsub>I\<^esub>]) \<Longrightarrow> restrict_poly_to_monom_set R P (insert a A) \<in> (carrier R [\<X>\<^bsub>I\<^esub>])"
   proof-
     fix A
-    fix a 
+    fix a
     assume A: "restrict_poly_to_monom_set R P A \<in> (carrier R [\<X>\<^bsub>I\<^esub>])"
     show "restrict_poly_to_monom_set R P (insert a A) \<in> (carrier R [\<X>\<^bsub>I\<^esub>])"
     proof(cases "a \<in> monomials_of R P")
       case True
       then have T0: "mset_to_IP R a \<in> (carrier R [\<X>\<^bsub>I\<^esub>])"
         using assms(1) mset_to_IP_closed' by blast
       then have T1: "(mset_to_IP R a \<Otimes>\<^sub>p indexed_const (P a)) \<in>  (carrier R [\<X>\<^bsub>I\<^esub>])"
-        by (meson assms(1) carrier_coeffE indexed_pset_in_carrier 
+        by (meson assms(1) carrier_coeffE indexed_pset_in_carrier
       local.ring_axioms ring.indexed_const_P_mult_closed subset_refl)
 
       show ?thesis
       proof(cases "a \<in> A")
         case True
-        then show ?thesis 
-          by (simp add: A insert_absorb)        
+        then show ?thesis
+          by (simp add: A insert_absorb)
       next
         case False
         have "restrict_poly_to_monom_set R P (insert a A) = restrict_poly_to_monom_set R P A \<Oplus>  (mset_to_IP R a \<Otimes>\<^sub>p indexed_const (P a))"
         proof
           fix x
           show "restrict_poly_to_monom_set R P (insert a A) x = (restrict_poly_to_monom_set R P A \<Oplus> (mset_to_IP R a \<Otimes>\<^sub>p indexed_const (P a))) x"
           proof(cases "x = a")
             case True
             then have FT0: "(if x \<in> insert a A then P x else \<zero>) = P x"
-              by simp 
+              by simp
             have FT1: "(if x \<in> A then P x else \<zero>) = \<zero>"
               by (simp add: False True)
             have FT2: "P a \<in> carrier R"
               using assms(1) carrier_coeffE indexed_pset_in_carrier by blast
             have FT3: "(mset_to_IP R a \<Otimes>\<^sub>p indexed_const (P a)) x = P x"
               using T0 FT2 True indexed_const_P_multE[of "mset_to_IP R a" I "P a" x] mset_to_IP_simp[of a]
               by simp
-            then show ?thesis           
-              using False 
+            then show ?thesis
+              using False
               unfolding restrict_poly_to_monom_set_def indexed_padd_def
-              using FT0 FT1 FT3  assms  
+              using FT0 FT1 FT3  assms
             by (simp add: FT2 True)
           next
             case F: False
             then have FT0: "(if x \<in> insert a A then P x else \<zero>) = (if x \<in> A then P x else \<zero>)"
-              by simp 
+              by simp
             have FT2: "P a \<in> carrier R"
               using assms(1) carrier_coeffE indexed_pset_in_carrier by blast
             have FT3: "(mset_to_IP R a \<Otimes>\<^sub>p indexed_const (P a)) x = \<zero>"
               using T0 FT2 True indexed_const_P_multE[of "mset_to_IP R a" I "P a" x] mset_to_IP_simp[of a]
               by (simp add: F mset_to_IP_def)
-            show ?thesis             
+            show ?thesis
               unfolding restrict_poly_to_monom_set_def indexed_padd_def
             proof-
 
               show "(if x \<in> insert a A then P x else \<zero>) = (if x \<in> A then P x else \<zero>) \<oplus> (mset_to_IP R a \<Otimes>\<^sub>p indexed_const (P a)) x"
                 apply(cases "x \<in> A")
-                using FT0 FT2 FT3 F False assms  carrier_coeffE indexed_pset_in_carrier local.ring_axioms 
+                using FT0 FT2 FT3 F False assms  carrier_coeffE indexed_pset_in_carrier local.ring_axioms
                 apply fastforce
-                    using FT0 FT2 FT3 F False assms 
+                    using FT0 FT2 FT3 F False assms
                     by simp
                 qed
           qed
         qed
-        then show ?thesis 
-          using A T1 indexed_pset.simps by blast      
+        then show ?thesis
+          using A T1 indexed_pset.simps by blast
       qed
     next
       case False
-      then show ?thesis 
-        using A restrict_not_in_monoms by fastforce    
+      then show ?thesis
+        using A restrict_not_in_monoms by fastforce
     qed
   qed
 qed
 
 lemma restriction_restrict:
 "restrict_poly_to_monom_set R P ms = restrict_poly_to_monom_set R P (ms \<inter> monomials_of R P)"
 proof
   fix x
   show "restrict_poly_to_monom_set R P ms x = restrict_poly_to_monom_set R P (ms \<inter> monomials_of R P) x"
-    unfolding restrict_poly_to_monom_set_def monomials_of_def 
+    unfolding restrict_poly_to_monom_set_def monomials_of_def
   by simp
 qed
 
 lemma restriction_closed:
   assumes "P \<in> indexed_pset I (carrier R)"
   assumes "Q = restrict_poly_to_monom_set R P ms"
   shows "Q \<in> indexed_pset I (carrier R)"
 proof-
   have "Q = restrict_poly_to_monom_set R P (ms \<inter> monomials_of R P)"
-    using assms restriction_restrict 
+    using assms restriction_restrict
     by blast
-  then show ?thesis 
+  then show ?thesis
     using assms restriction_closed'[of P I "(ms \<inter> monomials_of R P)"]
-    using monomials_finite 
+    using monomials_finite
     by blast
 qed
 
 lemma monomial_split_card:
   assumes "P \<in> indexed_pset I (carrier R)"
   assumes "m \<in> monomials_of R P"
-  shows "card (monomials_of R (restrict_poly_to_monom_set R P ((monomials_of R P) - {m})))= 
+  shows "card (monomials_of R (restrict_poly_to_monom_set R P ((monomials_of R P) - {m})))=
         card (monomials_of R P) -1"
 proof-
-  have 0: "(monomials_of R (restrict_poly_to_monom_set R P ((monomials_of R P) - {m}))) = 
+  have 0: "(monomials_of R (restrict_poly_to_monom_set R P ((monomials_of R P) - {m}))) =
         (monomials_of R P) - {m}"
-    using assms(1) 
-    by (meson Diff_subset restrict_poly_to_monom_set_monoms)    
-  then have 1:  "card (monomials_of R (restrict_poly_to_monom_set R P ((monomials_of R P) - {m}))) = 
+    using assms(1)
+    by (meson Diff_subset restrict_poly_to_monom_set_monoms)
+  then have 1:  "card (monomials_of R (restrict_poly_to_monom_set R P ((monomials_of R P) - {m}))) =
               card ((monomials_of R P) - {m})"
-    by auto 
+    by auto
   have " card ((monomials_of R P) - {m}) =  card (monomials_of R P) - 1"
-    using assms(2) 
+    using assms(2)
   using assms(1) monomials_finite by fastforce
-    then show ?thesis 
+    then show ?thesis
     by (simp add: "1")
 qed
 
 lemma P_ring_mult_closed':
   assumes "a \<in> indexed_pset I (carrier R)"
   assumes "b \<in> indexed_pset I (carrier R)"
   shows "a \<Otimes>\<^sub>p b \<in> indexed_pset I (carrier R)"
   apply(rule mpoly_induct'[of "\<lambda>b. a \<Otimes>\<^sub>p b \<in> indexed_pset I (carrier R)" I b])
   using assms(1) indexed_const_P_mult_closed apply blast
 proof-
   show "b \<in> (carrier R [\<X>\<^bsub>I\<^esub>])"
-    using assms by auto 
+    using assms by auto
   show "\<And>n Q. (\<And>Q. Q \<in>  (carrier R [\<X>\<^bsub>I\<^esub>]) \<and> card (monomials_of R Q) \<le> n \<Longrightarrow> a \<Otimes>\<^sub>p Q \<in>  (carrier R [\<X>\<^bsub>I\<^esub>])) \<Longrightarrow>
            Q \<in>  (carrier R [\<X>\<^bsub>I\<^esub>]) \<and> card (monomials_of R Q) = Suc n \<Longrightarrow> a \<Otimes>\<^sub>p Q \<in>  (carrier R [\<X>\<^bsub>I\<^esub>])"
   proof-
-    fix n 
+    fix n
     fix Q
     assume A0: "(\<And>Q. Q \<in>  (carrier R [\<X>\<^bsub>I\<^esub>]) \<and> card (monomials_of R Q) \<le> n \<Longrightarrow> a \<Otimes>\<^sub>p Q \<in>  (carrier R [\<X>\<^bsub>I\<^esub>]))"
     assume A1: "Q \<in>  (carrier R [\<X>\<^bsub>I\<^esub>]) \<and> card (monomials_of R Q) = Suc n"
     obtain m where m_def: "m \<in> monomials_of R Q"
-      using A1 
+      using A1
       by fastforce
     obtain P where P_def: "P = (restrict_poly_to_monom_set R Q ((monomials_of R Q) - {m}))"
       by simp
     have "P \<in> (carrier R [\<X>\<^bsub>I\<^esub>])"
-      using P_def A1 restriction_closed 
+      using P_def A1 restriction_closed
       by blast
     have 0: "a \<Otimes>\<^sub>p P \<in>  (carrier R [\<X>\<^bsub>I\<^esub>])"
-      using A0 P_def 
+      using A0 P_def
       by (metis A1 One_nat_def \<open>P \<in> (carrier R [\<X>\<^bsub>I\<^esub>])\<close> diff_Suc_Suc m_def
           minus_nat.diff_0 monomial_split_card nat_le_linear)
     have 1: "a \<Otimes>\<^sub>p (mset_to_IP R m) \<in> (carrier R [\<X>\<^bsub>I\<^esub>])"
-      using assms mset_to_IP_mult_closed[of a I m] A1 m_def mset_to_IP_indices 
+      using assms mset_to_IP_mult_closed[of a I m] A1 m_def mset_to_IP_indices
       by blast
     have 2: "a \<Otimes>\<^sub>p (mset_to_IP R m \<Otimes>\<^sub>p indexed_const (Q m)) \<in> (carrier R [\<X>\<^bsub>I\<^esub>])"
       using P_ring_mult_assoc[of a "mset_to_IP R m" " indexed_const (Q m)"]
-        1 
+        1
       by (metis A1 assms(1) carrier_coeffE indexed_pset.indexed_const
-          indexed_pset_in_carrier local.ring_axioms m_def mset_to_IP_closed' 
+          indexed_pset_in_carrier local.ring_axioms m_def mset_to_IP_closed'
           ring.indexed_const_P_mult_closed set_eq_subset)
     have 3: "(P \<Oplus> (mset_to_IP R m \<Otimes>\<^sub>p indexed_const (Q m))) = Q"
       using P_def monomial_poly_split[of Q I m]
       using A1 m_def by blast
     have 4: "(a \<Otimes>\<^sub>p P ) \<Oplus> (a \<Otimes>\<^sub>p (mset_to_IP R m \<Otimes>\<^sub>p indexed_const (Q m))) = a \<Otimes>\<^sub>p Q"
-      using assms 2 3 P_ring_rdistr[of a P "(mset_to_IP R m \<Otimes>\<^sub>p indexed_const (Q m))"] 
-      by (metis A1 Idl_subset_ideal' P_ring_mult_closed \<open>P \<in> (carrier R [\<X>\<^bsub>I\<^esub>])\<close> 
-          carrier_coeffE indexed_pset.indexed_const indexed_pset_in_carrier m_def 
+      using assms 2 3 P_ring_rdistr[of a P "(mset_to_IP R m \<Otimes>\<^sub>p indexed_const (Q m))"]
+      by (metis A1 Idl_subset_ideal' P_ring_mult_closed \<open>P \<in> (carrier R [\<X>\<^bsub>I\<^esub>])\<close>
+          carrier_coeffE indexed_pset.indexed_const indexed_pset_in_carrier m_def
           mset_to_IP_closed' onepideal principalideal.generate)
     then show " a \<Otimes>\<^sub>p Q \<in>  (carrier R [\<X>\<^bsub>I\<^esub>])"
       by (metis "0" "2" indexed_pset.indexed_padd)
   qed
 qed
 
-end 
+end
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsection\<open>Subtraction of Polynomials\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 definition P_ring_uminus :: "('a,'b) ring_scheme \<Rightarrow> ('a,'c) mvar_poly \<Rightarrow> ('a,'c) mvar_poly" where
 "P_ring_uminus R P = (\<lambda>m. \<ominus>\<^bsub>R\<^esub> (P m))"
 
 context ring
 
 begin
 
 
 lemma P_ring_uminus_eq:
   assumes "a \<in> indexed_pset I (carrier R)"
   shows "P_ring_uminus R a  = a \<Otimes>\<^sub>p (indexed_const (\<ominus> \<one>))"
 proof
     fix x
     have "(a \<Otimes>\<^sub>p indexed_const (\<ominus> \<one>)) x = a x \<otimes> \<ominus> \<one>"
       using indexed_const_P_multE[of a I "\<ominus> \<one>" x] assms
       by blast
     then show "P_ring_uminus R a x = (a \<Otimes>\<^sub>p indexed_const (\<ominus> \<one>)) x"
-      unfolding P_ring_uminus_def 
-      using assms 
+      unfolding P_ring_uminus_def
+      using assms
       by (metis Idl_subset_ideal' carrier_coeffE indexed_pset_in_carrier
         one_closed onepideal principalideal.generate r_minus r_one)
 qed
 
 lemma P_ring_uminus_closed:
   assumes "a \<in> indexed_pset I (carrier R)"
   shows "P_ring_uminus R a \<in> indexed_pset I (carrier R)"
   using assms P_ring_uminus_eq
   by (metis add.l_inv_ex indexed_const_P_mult_closed minus_equality one_closed)
 
 lemma P_ring_uminus_add:
   assumes "a \<in> indexed_pset I (carrier R)"
   shows "P_ring_uminus R a \<Oplus> a = indexed_const \<zero>"
 proof
   fix x
   show "(P_ring_uminus R a \<Oplus> a) x = indexed_const \<zero> x"
-    using assms 
+    using assms
     unfolding P_ring_uminus_def indexed_const_def indexed_padd_def
-    by (meson carrier_coeffE indexed_pset_in_carrier 
+    by (meson carrier_coeffE indexed_pset_in_carrier
         local.ring_axioms ring.ring_simprules(9) set_eq_subset)
 qed
 
 text\<open>multiplication by 1\<close>
 
 lemma one_mult_left:
   assumes "a \<in> indexed_pset I (carrier R)"
   shows "(indexed_const \<one>) \<Otimes>\<^sub>p a = a"
 proof
   fix m
   show "(indexed_const \<one> \<Otimes>\<^sub>p a) m = a m "
     unfolding indexed_const_def P_ring_mult_def
   proof-
     have 0: "(\<Oplus>x\<in>((mset_factors m) - {{#}}). (if x = {#} then \<one> else \<zero>) \<otimes> a (m - x)) = \<zero>"
     proof-
       have "\<And>x. x\<in>((mset_factors m) - {{#}}) \<Longrightarrow>  (if x = {#} then \<one> else \<zero>) \<otimes> a (m - x) = \<zero> \<otimes> a (m - x)"
         by simp
       then have "\<And>x. x\<in>((mset_factors m) - {{#}}) \<Longrightarrow>  (if x = {#} then \<one> else \<zero>) \<otimes> a (m - x) = \<zero>"
-        using assms 
+        using assms
         by (metis carrier_coeffE  indexed_pset_in_carrier local.ring_axioms ring.ring_simprules(24) set_eq_subset)
-      then show ?thesis 
+      then show ?thesis
         by (meson add.finprod_one_eqI)
     qed
     have 1: "(\<lambda>x. (if x = {#} then \<one> else \<zero>) \<otimes> a (m - x)) {#} = a m"
-      by (metis (full_types) assms carrier_coeffE diff_zero 
+      by (metis (full_types) assms carrier_coeffE diff_zero
           indexed_pset_in_carrier local.ring_axioms ring.ring_simprules(12) set_eq_subset)
-    have 2: "(\<Oplus>x\<in>insert {#} (mset_factors m - {{#}}). (if x = {#} then \<one> else \<zero>) \<otimes> a (m - x)) = 
+    have 2: "(\<Oplus>x\<in>insert {#} (mset_factors m - {{#}}). (if x = {#} then \<one> else \<zero>) \<otimes> a (m - x)) =
               (\<lambda>x. (if x = {#} then \<one> else \<zero>) \<otimes> a (m - x)) {#} \<oplus>
              (\<Oplus>x\<in>((mset_factors m) - {{#}}). (if x = {#} then \<one> else \<zero>) \<otimes> a (m - x))"
     proof-
-      have 20:"finite (mset_factors m - {{#}})" 
+      have 20:"finite (mset_factors m - {{#}})"
         by simp
       have 21: " {#} \<notin> mset_factors m - {{#}}"
         by blast
       have 22: " (\<lambda>x. (if x = {#} then \<one> else \<zero>) \<otimes> a (m - x)) \<in> mset_factors m - {{#}} \<rightarrow> carrier R"
       proof
         fix x
         assume A: "x \<in> mset_factors m - {{#}}"
         show "(if x = {#} then \<one> else \<zero>) \<otimes> a (m - x) \<in> carrier R"
           apply(cases "x = {#}")
-          using A 
+          using A
           apply blast
-              using assms carrier_coeffE indexed_pset_in_carrier  local.ring_axioms 
+              using assms carrier_coeffE indexed_pset_in_carrier  local.ring_axioms
               one_closed ring.ring_simprules(5) set_eq_subset zero_closed
               by (metis (mono_tags, opaque_lifting))
       qed
       have 23: "(if {#} = {#} then \<one> else \<zero>) \<otimes> a (m - {#}) \<in> carrier R"
         by (metis "1" assms carrier_coeffE indexed_pset_in_carrier set_eq_subset)
-      show ?thesis 
+      show ?thesis
         using 20 21 22 23 finsum_insert[of  "((mset_factors m) - {{#}})" "{#}"
             " (\<lambda>x. (if x = {#} then \<one> else \<zero>) \<otimes> a (m - x))"]
         by blast
     qed
     have 3: "insert {#} (mset_factors m - {{#}}) = mset_factors m"
     proof-
       have "{#} \<in> mset_factors m"
-        using monom_divides_factors subset_mset.zero_le 
+        using monom_divides_factors subset_mset.zero_le
         by blast
-      then show ?thesis 
+      then show ?thesis
         by blast
     qed
     show "(\<Oplus>x\<in>mset_factors m. (if x = {#} then \<one> else \<zero>) \<otimes> a (m - x)) = a m"
-      using 0 1 2 3 assms 
-      by (metis (no_types, lifting) Idl_subset_ideal' carrier_coeffE 
-          genideal_one indexed_pset_in_carrier one_closed r_zero) 
-  qed
-qed
-
-end 
+      using 0 1 2 3 assms
+      by (metis (no_types, lifting) Idl_subset_ideal' carrier_coeffE
+          genideal_one indexed_pset_in_carrier one_closed r_zero)
+  qed
+qed
+
+end
 
 (**************************************************************************************************)
 (**************************************************************************************************)
                      subsection\<open>The Carrier of the Ring of Indexed Polynomials\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 
 abbreviation(input) Pring_set where
 "Pring_set R I \<equiv> ring.indexed_pset R I (carrier R) "
 
 context ring
 
 begin
 
 lemma Pring_set_zero:
   assumes "f \<in> Pring_set R I"
-  assumes "\<not> set_mset m \<subseteq>  I" 
+  assumes "\<not> set_mset m \<subseteq>  I"
   shows " f m = \<zero>\<^bsub>R\<^esub>"
 proof-
   have "\<not> set_mset m \<subseteq>  I \<Longrightarrow> f m = \<zero>\<^bsub>R\<^esub>"
   apply(induction m)
   apply simp
   by (meson assms complement_of_monomials_of mset_to_IP_indices')
-  then show ?thesis 
+  then show ?thesis
     using assms(2) by blast
 qed
 
 lemma(in ring) Pring_cfs_closed:
   assumes "P \<in> Pring_set R I"
   shows "P m \<in> carrier R"
-  using assms carrier_coeffE indexed_pset_in_carrier 
+  using assms carrier_coeffE indexed_pset_in_carrier
   by blast
 
 lemma indexed_pset_mono_aux:
   assumes "P \<in> indexed_pset I S"
   shows "S \<subseteq> T \<Longrightarrow> P \<in> indexed_pset I T"
-  using assms 
+  using assms
   apply(induction P)
   using indexed_pset.indexed_const apply blast
   using indexed_pset.indexed_padd apply blast
   by (simp add: indexed_pset.indexed_pmult)
 
 lemma indexed_pset_mono:
   assumes "S \<subseteq> T"
   shows "indexed_pset I S \<subseteq> indexed_pset I T"
-  using assms indexed_pset_mono_aux 
+  using assms indexed_pset_mono_aux
   by blast
 
 end
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsection\<open>Scalar Multiplication\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 definition poly_scalar_mult :: "('a, 'b) ring_scheme \<Rightarrow> 'a \<Rightarrow> ('a,'c) mvar_poly \<Rightarrow> ('a,'c) mvar_poly" where
 "poly_scalar_mult R c P = (\<lambda> m. c \<otimes>\<^bsub>R\<^esub> P m)"
 
 lemma(in cring) poly_scalar_mult_eq:
   assumes "c \<in> carrier R"
   shows "P \<in> Pring_set R (I :: 'c set) \<Longrightarrow> poly_scalar_mult R c P = indexed_const c \<Otimes>\<^sub>p P"
 proof(erule indexed_pset.induct)
   show "\<And>k. k \<in> carrier R \<Longrightarrow> poly_scalar_mult R c (indexed_const k) = indexed_const c \<Otimes>\<^sub>p indexed_const k"
   proof-
-    fix k 
+    fix k
     assume A0: "k \<in> carrier R"
     show "poly_scalar_mult R c (indexed_const k)  = (indexed_const c \<Otimes>\<^sub>p indexed_const k) "
-      unfolding poly_scalar_mult_def 
+      unfolding poly_scalar_mult_def
     proof
       show "\<And>m. c \<otimes> indexed_const k m = (indexed_const c \<Otimes>\<^sub>p indexed_const k) m"
           using indexed_const_P_mult_eq
           by (metis A0  assms indexed_const_P_mult_eq indexed_const_def local.semiring_axioms semiring.r_null)
     qed
-  qed   
+  qed
   show "\<And>P Q. P \<in> Pring_set R I \<Longrightarrow>
            poly_scalar_mult R c P = indexed_const c \<Otimes>\<^sub>p P \<Longrightarrow>
            Q \<in> Pring_set R I \<Longrightarrow> poly_scalar_mult R c Q = indexed_const c \<Otimes>\<^sub>p Q \<Longrightarrow> poly_scalar_mult R c (P \<Oplus> Q) = indexed_const c \<Otimes>\<^sub>p (P \<Oplus> Q)"
   proof
     fix P Q :: "'c monomial \<Rightarrow> 'a"
     fix x :: "'c monomial"
     assume A0: "P \<in> Pring_set R I"
     assume A1: "poly_scalar_mult R c P = indexed_const c \<Otimes>\<^sub>p P"
     assume A2: "Q \<in> Pring_set R I"
     assume A3: "poly_scalar_mult R c Q = indexed_const c \<Otimes>\<^sub>p Q"
     show "poly_scalar_mult R c (P \<Oplus> Q) x = (indexed_const c \<Otimes>\<^sub>p (P \<Oplus> Q)) x"
     proof-
       have P0: "poly_scalar_mult R c (P \<Oplus> Q) = (poly_scalar_mult R c P) \<Oplus> (poly_scalar_mult R c Q)"
         unfolding poly_scalar_mult_def
       proof
         fix m
         show "c \<otimes> (P \<Oplus> Q) m = ((\<lambda>m. c \<otimes> P m) \<Oplus> (\<lambda>m. c \<otimes> Q m)) m"
         proof-
           have LHS: "c \<otimes> (P \<Oplus> Q) m = c \<otimes> ( P m \<oplus> Q m) "
             by (simp add: indexed_padd_def)
           then have LHS1: "c \<otimes> (P \<Oplus> Q) m = (c \<otimes>  P m) \<oplus> (c \<otimes>Q m) "
           proof-
             have 0: "carrier_coeff P"
-              using A0 indexed_pset_in_carrier 
+              using A0 indexed_pset_in_carrier
               by blast
             have 1: "P m \<in> carrier R"
-              using 0 carrier_coeffE by blast 
+              using 0 carrier_coeffE by blast
             have 2: "carrier_coeff Q"
-              using A2 indexed_pset_in_carrier 
+              using A2 indexed_pset_in_carrier
               by blast
-            have 3: "Q m \<in> carrier R" using 2 
-              using carrier_coeff_def 
+            have 3: "Q m \<in> carrier R" using 2
+              using carrier_coeff_def
               by blast
-            show ?thesis using 1 3 assms 
+            show ?thesis using 1 3 assms
               by (simp add: LHS r_distr)
           qed
-          then show ?thesis 
+          then show ?thesis
             by (simp add: indexed_padd_def)
         qed
       qed
       have P1: "poly_scalar_mult R c (P \<Oplus> Q) = (indexed_const c \<Otimes>\<^sub>p P) \<Oplus> (indexed_const c \<Otimes>\<^sub>p Q)"
-        using P0 A1 A3 
+        using P0 A1 A3
         by simp
       have P2: "indexed_const c \<in> Pring_set R I"
         using assms indexed_pset.indexed_const by blast
-      show ?thesis 
-        using P2 A0 A2 P1 
+      show ?thesis
+        using P2 A0 A2 P1
         by (metis P_ring_rdistr indexed_pset_in_carrier set_eq_subset)
     qed
-  qed      
+  qed
   show "\<And>P i. P \<in> Pring_set R I \<Longrightarrow>
            poly_scalar_mult R c P = indexed_const c \<Otimes>\<^sub>p P \<Longrightarrow> i \<in> I \<Longrightarrow> poly_scalar_mult R c (P \<Otimes> i) = indexed_const c \<Otimes>\<^sub>p (P \<Otimes> i)"
   proof-
-    fix P 
+    fix P
     fix i
     assume A0:  "P \<in> Pring_set R I"
     assume A1: "poly_scalar_mult R c P = indexed_const c \<Otimes>\<^sub>p P"
     assume A2: "i \<in> I"
     show "poly_scalar_mult R c (P \<Otimes> i) = indexed_const c \<Otimes>\<^sub>p (P \<Otimes> i)"
     proof
       fix x
       show "poly_scalar_mult R c (P \<Otimes> i) x = (indexed_const c \<Otimes>\<^sub>p (P \<Otimes> i)) x"
       proof-
         have RHS: "(indexed_const c \<Otimes>\<^sub>p (P \<Otimes> i)) = (indexed_const c \<Otimes>\<^sub>p P) \<Otimes> i"
         proof-
           have B0: "P \<Otimes> i = P \<Otimes>\<^sub>p (mset_to_IP R {#i#})"
             by (meson A0 A2 poly_index_mult)
           have B1: " (indexed_const c \<Otimes>\<^sub>p P) \<Otimes> i = (indexed_const c \<Otimes>\<^sub>p P) \<Otimes>\<^sub>p (mset_to_IP R {#i#})"
             by (meson A0 A2 P_ring_mult_closed' assms indexed_pset.simps poly_index_mult)
           have B2: "(indexed_const c \<Otimes>\<^sub>p (P \<Otimes> i)) = indexed_const c \<Otimes>\<^sub>p P \<Otimes>\<^sub>p (mset_to_IP R {#i#})"
             by (metis A0 A2 B1 assms cring.P_ring_mult_comm indexed_const_var_mult
                 indexed_pmult_in_carrier indexed_pset.indexed_const indexed_pset_in_carrier
                 is_cring set_eq_subset)
-          show ?thesis using A0 A1 A2 B2 B1 assms 
+          show ?thesis using A0 A1 A2 B2 B1 assms
           by (simp add: A0)
         then have RHS': "(indexed_const c \<Otimes>\<^sub>p (P \<Otimes> i)) = (poly_scalar_mult R c P) \<Otimes> i"
-          using A0 A1 A2 assms 
+          using A0 A1 A2 assms
           by simp
         qed
         show ?thesis apply(cases "i \<in># x")
-          unfolding poly_scalar_mult_def 
+          unfolding poly_scalar_mult_def
            apply (metis A1 RHS poly_scalar_mult_def indexed_pmult_def )
           by (metis RHS assms indexed_pmult_def r_null)
       qed
     qed
   qed
 qed
 
 lemma(in cring) poly_scalar_mult_const:
   assumes "c \<in> carrier R"
   assumes "k \<in> carrier R"
   shows "poly_scalar_mult R k (indexed_const c) = indexed_const (k \<otimes> c)"
-  using assms poly_scalar_mult_eq 
+  using assms poly_scalar_mult_eq
   by (simp add: poly_scalar_mult_eq indexed_const_P_mult_eq indexed_pset.indexed_const)
- 
+
 lemma(in cring) poly_scalar_mult_closed:
   assumes "c \<in> carrier R"
-  assumes "P \<in> Pring_set R I" 
+  assumes "P \<in> Pring_set R I"
   shows "poly_scalar_mult R c P \<in> Pring_set R I"
-  using assms poly_scalar_mult_eq 
+  using assms poly_scalar_mult_eq
   by (metis P_ring_mult_closed' indexed_pset.indexed_const)
 
 lemma(in cring) poly_scalar_mult_zero:
   assumes "P \<in> Pring_set R I"
   shows "poly_scalar_mult R \<zero> P = indexed_const \<zero>"
 proof
-  fix x 
+  fix x
   show "poly_scalar_mult R \<zero> P x = indexed_const \<zero> x"
-    unfolding poly_scalar_mult_def 
-    using assms 
-    by (metis Pring_cfs_closed indexed_zero_def l_null)    
+    unfolding poly_scalar_mult_def
+    using assms
+    by (metis Pring_cfs_closed indexed_zero_def l_null)
 qed
 
 lemma(in cring) poly_scalar_mult_one:
   assumes "P \<in> Pring_set R I"
   shows "poly_scalar_mult R \<one> P = P"
 proof
   fix x show "poly_scalar_mult R \<one> P x = P x"
-    using assms 
+    using assms
     by (metis one_closed one_mult_left poly_scalar_mult_eq)
-qed   
+qed
 
 lemma(in cring) times_poly_scalar_mult:
   assumes "P \<in> Pring_set R I"
   assumes "Q \<in> Pring_set R I"
   assumes "k \<in> carrier R"
   shows "P \<Otimes>\<^sub>p (poly_scalar_mult R k Q) = poly_scalar_mult R k (P \<Otimes>\<^sub>p Q)"
 proof-
   have "P \<Otimes>\<^sub>p (poly_scalar_mult R k Q) = P \<Otimes>\<^sub>p (indexed_const k) \<Otimes>\<^sub>p Q"
-  by (metis assms(1) assms(2) assms(3)  indexed_pset_mono_aux 
-      local.ring_axioms poly_scalar_mult_eq ring.P_ring_mult_assoc ring.indexed_pset.intros(1) 
+  by (metis assms(1) assms(2) assms(3)  indexed_pset_mono_aux
+      local.ring_axioms poly_scalar_mult_eq ring.P_ring_mult_assoc ring.indexed_pset.intros(1)
       ring.indexed_pset_in_carrier subset_refl)
   then have  "P \<Otimes>\<^sub>p (poly_scalar_mult R k Q) =  (indexed_const k) \<Otimes>\<^sub>p P \<Otimes>\<^sub>p Q"
   by (metis P_ring_mult_comm assms(1) assms(3) local.ring_axioms ring.indexed_pset.indexed_const ring.indexed_pset_in_carrier subset_refl)
-  then show ?thesis 
-    by (metis (no_types, opaque_lifting) P_ring_mult_closed' Pring_cfs_closed assms(1) assms(2) assms(3) 
+  then show ?thesis
+    by (metis (no_types, opaque_lifting) P_ring_mult_closed' Pring_cfs_closed assms(1) assms(2) assms(3)
         carrier_coeff_def  indexed_pset.indexed_const  local.ring_axioms poly_scalar_mult_eq ring.P_ring_mult_assoc)
 qed
 
 lemma(in cring) poly_scalar_mult_times:
   assumes "P \<in> Pring_set R I"
   assumes "Q \<in> Pring_set R I"
   assumes "k \<in> carrier R"
   shows " poly_scalar_mult R k (Q \<Otimes>\<^sub>p P) = (poly_scalar_mult R k Q) \<Otimes>\<^sub>p P"
-  using assms times_poly_scalar_mult 
-  by (metis (no_types, opaque_lifting) P_ring_mult_comm  cring.P_ring_mult_comm 
+  using assms times_poly_scalar_mult
+  by (metis (no_types, opaque_lifting) P_ring_mult_comm  cring.P_ring_mult_comm
       cring.poly_scalar_mult_closed is_cring local.ring_axioms ring.indexed_pset_in_carrier subset_refl)
 
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsection\<open>Defining the Ring of Indexed Polynomials\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 definition Pring :: "('b, 'e) ring_scheme  \<Rightarrow> 'a set  \<Rightarrow> ('b, ('b,'a) mvar_poly) module" where
 
 "Pring R I \<equiv> \<lparr>  carrier = Pring_set R I,
                 Group.monoid.mult = P_ring_mult R ,
                 one = ring.indexed_const R \<one>\<^bsub>R\<^esub>,
                 zero = ring.indexed_const R  \<zero>\<^bsub>R\<^esub>,
                 add = ring.indexed_padd R,
                 smult = poly_scalar_mult R\<rparr>"
 
 context ring
 
 begin
 
 lemma Pring_car:
 "carrier (Pring R I) = Pring_set R I"
   unfolding Pring_def
-  by auto 
+  by auto
 
 text\<open>Definitions of the operations and constants:\<close>
 
 lemma Pring_mult:
 "a \<otimes>\<^bsub>Pring R I\<^esub> b = a \<Otimes>\<^sub>p b"
-  unfolding Pring_def 
-  by simp 
+  unfolding Pring_def
+  by simp
 
 lemma Pring_add:
 "a \<oplus>\<^bsub>Pring R I\<^esub> b = a \<Oplus> b"
-  unfolding Pring_def 
-  by simp 
+  unfolding Pring_def
+  by simp
 
 lemma Pring_zero:
 "\<zero>\<^bsub>Pring R I\<^esub> = indexed_const \<zero>"
-  unfolding Pring_def by simp 
+  unfolding Pring_def by simp
 
 lemma Pring_one:
 "\<one>\<^bsub>Pring R I\<^esub> = indexed_const \<one>"
-  unfolding Pring_def by simp 
+  unfolding Pring_def by simp
 
 lemma Pring_smult:
 "(\<odot>\<^bsub>Pring R I\<^esub>) = (poly_scalar_mult R)"
-  unfolding Pring_def by simp 
+  unfolding Pring_def by simp
 
 lemma Pring_carrier_coeff:
   assumes "a \<in> carrier (Pring R I)"
   shows "carrier_coeff a"
   using assms indexed_pset_in_carrier[of "(carrier R)" a I] Pring_car
   by blast
 
 lemma Pring_carrier_coeff'[simp]:
   assumes "a \<in> carrier (Pring R I)"
   shows "a m \<in> carrier R"
-  using Pring_car[of I] assms carrier_coeffE indexed_pset_in_carrier 
+  using Pring_car[of I] assms carrier_coeffE indexed_pset_in_carrier
   by blast
 
 lemma Pring_add_closed:
   assumes "a \<in> carrier (Pring R I)"
   assumes "b \<in> carrier (Pring R I)"
   shows "a \<oplus>\<^bsub>Pring R I\<^esub> b \<in> carrier (Pring R I)"
   using assms  Pring_def[of R I]
   by (simp add: Pring_def[of R I] indexed_pset.indexed_padd)
 
 lemma Pring_mult_closed:
   assumes "a \<in> carrier (Pring R I)"
   assumes "b \<in> carrier (Pring R I)"
   shows "a \<otimes>\<^bsub>Pring R I\<^esub> b \<in> carrier (Pring R I)"
   using assms  P_ring_mult_closed'[of a I b] Pring_car[of I] Pring_mult[of I a b]
   by (simp add: \<open>a \<otimes>\<^bsub>Pring R I\<^esub> b = a \<Otimes>\<^sub>p b\<close> \<open>carrier (Pring R I) = Pring_set R I\<close>)
 
 lemma Pring_one_closed:
 "\<one>\<^bsub>Pring R I\<^esub> \<in> carrier (Pring R I)"
 proof-
   have "indexed_const \<one> \<in> carrier (Pring R I)"
     using Pring_car indexed_pset.simps by blast
-  then show ?thesis 
-    unfolding Pring_def by auto 
+  then show ?thesis
+    unfolding Pring_def by auto
 qed
 
 lemma Pring_zero_closed:
 "\<zero>\<^bsub>Pring R I\<^esub> \<in> carrier (Pring R I)"
 proof-
   have "indexed_const \<zero> \<in> carrier (Pring R I)"
     using Pring_car indexed_pset.simps by blast
-  then show ?thesis 
-    unfolding Pring_def by auto 
+  then show ?thesis
+    unfolding Pring_def by auto
 qed
 
 lemma Pring_var_closed:
   assumes "i \<in> I"
   shows "var_to_IP R i \<in> carrier (Pring R I)"
-unfolding var_to_IP_def 
-  by (metis Pring_car Pring_one_closed assms indexed_pset.indexed_pmult 
+unfolding var_to_IP_def
+  by (metis Pring_car Pring_one_closed assms indexed_pset.indexed_pmult
       local.ring_axioms monom_add_mset one_mset_to_IP ring.Pring_one)
 
 text\<open>Properties of addition:\<close>
 
 lemma Pring_add_assoc:
   assumes "a \<in> carrier (Pring R I)"
   assumes "b \<in> carrier (Pring R I)"
   assumes "c \<in> carrier (Pring R I)"
   shows "a \<oplus>\<^bsub>Pring R I\<^esub> (b \<oplus>\<^bsub>Pring R I\<^esub> c) = (a \<oplus>\<^bsub>Pring R I\<^esub> b) \<oplus>\<^bsub>Pring R I\<^esub> c"
 proof-
   have "a \<Oplus> (b \<Oplus> c) = (a \<Oplus>  b) \<Oplus>  c"
   proof
-    fix x 
+    fix x
     have "carrier_coeff a" "carrier_coeff b" "carrier_coeff c"
       using assms Pring_car[of I] Pring_carrier_coeff   apply blast
         using assms Pring_car[of I] Pring_carrier_coeff   apply blast
           using assms Pring_car[of I] Pring_carrier_coeff   by blast
     then have "a x \<in> carrier R" "b x \<in> carrier R" "c x \<in> carrier R"
       using carrier_coeffE apply blast
         using \<open>carrier_coeff b\<close> carrier_coeffE apply blast
           using \<open>carrier_coeff c\<close> carrier_coeffE by blast
     show "(a \<Oplus> (b \<Oplus> c)) x = (a \<Oplus> b \<Oplus> c) x"
-      unfolding indexed_padd_def 
-      using assms 
+      unfolding indexed_padd_def
+      using assms
       by (simp add: \<open>a x \<in> carrier R\<close> \<open>b x \<in> carrier R\<close> \<open>c x \<in> carrier R\<close> add.m_assoc)
   qed
   then show ?thesis
-    using assms  
+    using assms
   by (simp add: Pring_add)
 qed
 
 lemma Pring_add_comm:
   assumes "a \<in> carrier (Pring R I)"
   assumes "b \<in> carrier (Pring R I)"
   shows "a \<oplus>\<^bsub>Pring R I\<^esub> b = b \<oplus>\<^bsub>Pring R I\<^esub> a"
 proof-
   have "a \<Oplus> b = b \<Oplus> a"
-  proof fix x 
+  proof fix x
     show "(a \<Oplus> b) x = (b \<Oplus> a) x"
-      using assms 
-      by (metis abelian_monoid.a_comm abelian_monoid_axioms 
+      using assms
+      by (metis abelian_monoid.a_comm abelian_monoid_axioms
           indexed_padd_def local.ring_axioms ring.Pring_carrier_coeff')
   qed
-  then show ?thesis 
+  then show ?thesis
     by (simp add: Pring_add)
 qed
 
 lemma Pring_add_zero:
   assumes "a \<in> carrier (Pring R I)"
   shows "a \<oplus>\<^bsub>Pring R I\<^esub> \<zero>\<^bsub>Pring R I\<^esub> = a"
         "\<zero>\<^bsub>Pring R I\<^esub> \<oplus>\<^bsub>Pring R I\<^esub> a  = a"
-  using assms Pring_zero Pring_add 
+  using assms Pring_zero Pring_add
    apply (metis Pring_carrier_coeff indexed_padd_zero(1))
-  using assms Pring_zero Pring_add 
+  using assms Pring_zero Pring_add
    by (metis Pring_carrier_coeff indexed_padd_zero(2))
 
 text\<open>Properties of multiplication\<close>
 
 lemma Pring_mult_assoc:
   assumes "a \<in> carrier (Pring R I)"
   assumes "b \<in> carrier (Pring R I)"
   assumes "c \<in> carrier (Pring R I)"
   shows "a \<otimes>\<^bsub>Pring R I\<^esub> (b \<otimes>\<^bsub>Pring R I\<^esub> c) = (a \<otimes>\<^bsub>Pring R I\<^esub> b) \<otimes>\<^bsub>Pring R I\<^esub> c"
   using assms P_ring_mult_assoc[of a b c]
   by (metis Pring_carrier_coeff Pring_mult)
 
 lemma Pring_mult_comm:
   assumes "cring R"
   assumes "a \<in> carrier (Pring R I)"
   assumes "b \<in> carrier (Pring R I)"
   shows "a \<otimes>\<^bsub>Pring R I\<^esub> b = b \<otimes>\<^bsub>Pring R I\<^esub> a"
   using assms Pring_carrier_coeff[of a I] Pring_carrier_coeff[of b I]
         Pring_mult[of I b a]  Pring_mult[of I a b] cring.P_ring_mult_comm[of R a b]
-  by metis 
+  by metis
 
 lemma Pring_mult_one:
   assumes "a \<in> carrier (Pring R I)"
   shows "a \<otimes>\<^bsub>Pring R I\<^esub> \<one>\<^bsub>Pring R I\<^esub> = a"
 proof
   fix x
   show "(a \<otimes>\<^bsub>Pring R I\<^esub> \<one>\<^bsub>Pring R I\<^esub>) x = a x "
   proof-
     have 0: "(a \<otimes>\<^bsub>Pring R I\<^esub> \<one>\<^bsub>Pring R I\<^esub>) x = (a \<Otimes>\<^sub>p indexed_const \<one>) x"
         using assms Pring_mult[of I a "\<one>\<^bsub>Pring R I\<^esub>" ] Pring_one[of I]
-        by metis 
+        by metis
     have 1: "\<one> \<in> carrier R"
       by simp
     have 2: "(a \<otimes>\<^bsub>Pring R I\<^esub> \<one>\<^bsub>Pring R I\<^esub>) x =  a x \<otimes> \<one>"
       using 0 1 indexed_const_P_multE[of a I \<one> x]
         assms Pring_car[of I]
-      by metis  
-    then show ?thesis 
+      by metis
+    then show ?thesis
       using assms by auto
   qed
 qed
 
 lemma Pring_mult_one':
   assumes "a \<in> carrier (Pring R I)"
   shows "\<one>\<^bsub>Pring R I\<^esub>  \<otimes>\<^bsub>Pring R I\<^esub> a = a"
   using one_mult_left[of a I]
         assms Pring_one Pring_mult
   by (simp add: Pring_mult Pring_one Pring_car)
 
 text\<open>Distributive laws\<close>
 
 lemma Pring_mult_rdistr:
   assumes "a \<in> carrier (Pring R I)"
   assumes "b \<in> carrier (Pring R I)"
   assumes "c \<in> carrier (Pring R I)"
   shows "a \<otimes>\<^bsub>Pring R I\<^esub> (b \<oplus>\<^bsub>Pring R I\<^esub> c) = (a \<otimes>\<^bsub>Pring R I\<^esub> b) \<oplus>\<^bsub>Pring R I\<^esub> (a \<otimes>\<^bsub>Pring R I\<^esub> c)"
 proof-
   have "a \<Otimes>\<^sub>p (b \<Oplus> c) = a \<Otimes>\<^sub>p b \<Oplus> (a \<Otimes>\<^sub>p c)"
-  using P_ring_rdistr[of a b c] 
-        assms Pring_carrier_coeff 
+  using P_ring_rdistr[of a b c]
+        assms Pring_carrier_coeff
   by metis
   then have "a \<Otimes>\<^sub>p (b \<oplus>\<^bsub>Pring R I\<^esub> c) = a \<Otimes>\<^sub>p b \<oplus>\<^bsub>Pring R I\<^esub> (a \<Otimes>\<^sub>p c)"
     using       Pring_add[of I b c] Pring_add[of I "a \<Otimes>\<^sub>p b" "a \<Otimes>\<^sub>p c"]
-    by auto 
+    by auto
   then have "a \<otimes>\<^bsub>Pring R I\<^esub> (b \<oplus>\<^bsub>Pring R I\<^esub> c) = (a \<Otimes>\<^sub>p b) \<oplus>\<^bsub>Pring R I\<^esub> (a \<Otimes>\<^sub>p c)"
-    using  Pring_mult[of I a "(b \<oplus>\<^bsub>Pring R I\<^esub> c)"] 
-    by auto 
+    using  Pring_mult[of I a "(b \<oplus>\<^bsub>Pring R I\<^esub> c)"]
+    by auto
   then have "a \<otimes>\<^bsub>Pring R I\<^esub> (b \<oplus>\<^bsub>Pring R I\<^esub> c) = (a \<otimes>\<^bsub>Pring R I\<^esub> b) \<oplus>\<^bsub>Pring R I\<^esub> (a \<Otimes>\<^sub>p c)"
-    using Pring_mult[of I a b] by metis 
-  then show ?thesis 
-    using Pring_mult[of I a c] by metis 
-qed
-      
+    using Pring_mult[of I a b] by metis
+  then show ?thesis
+    using Pring_mult[of I a c] by metis
+qed
+
 lemma Pring_mult_ldistr:
   assumes "a \<in> carrier (Pring R I)"
   assumes "b \<in> carrier (Pring R I)"
   assumes "c \<in> carrier (Pring R I)"
   shows "(b \<oplus>\<^bsub>Pring R I\<^esub> c) \<otimes>\<^bsub>Pring R I\<^esub> a  = (b \<otimes>\<^bsub>Pring R I\<^esub> a) \<oplus>\<^bsub>Pring R I\<^esub> (c \<otimes>\<^bsub>Pring R I\<^esub> a)"
 proof-
   have "(b \<Oplus> c) \<Otimes>\<^sub>p a = b \<Otimes>\<^sub>p a \<Oplus> (c \<Otimes>\<^sub>p a)"
-  using P_ring_ldistr[of a b c] 
-        assms Pring_carrier_coeff 
+  using P_ring_ldistr[of a b c]
+        assms Pring_carrier_coeff
   by metis
   then have " (b \<oplus>\<^bsub>Pring R I\<^esub> c)  \<Otimes>\<^sub>p a = b \<Otimes>\<^sub>p a \<oplus>\<^bsub>Pring R I\<^esub> (c \<Otimes>\<^sub>p a)"
     using       Pring_add[of I b c] Pring_add[of I "b \<Otimes>\<^sub>p a" "c \<Otimes>\<^sub>p a"]
-    by auto 
+    by auto
   then have " (b \<oplus>\<^bsub>Pring R I\<^esub> c) \<otimes>\<^bsub>Pring R I\<^esub> a = (b \<Otimes>\<^sub>p a) \<oplus>\<^bsub>Pring R I\<^esub> (c \<Otimes>\<^sub>p a)"
-    using  Pring_mult[of I "(b \<oplus>\<^bsub>Pring R I\<^esub> c)" a] 
-    by auto 
+    using  Pring_mult[of I "(b \<oplus>\<^bsub>Pring R I\<^esub> c)" a]
+    by auto
   then have "(b \<oplus>\<^bsub>Pring R I\<^esub> c) \<otimes>\<^bsub>Pring R I\<^esub> a = (b \<otimes>\<^bsub>Pring R I\<^esub> a) \<oplus>\<^bsub>Pring R I\<^esub> (c \<Otimes>\<^sub>p a)"
-    using Pring_mult[of I b a] by metis 
-  then show ?thesis 
-    using Pring_mult[of I c a] by metis 
+    using Pring_mult[of I b a] by metis
+  then show ?thesis
+    using Pring_mult[of I c a] by metis
 qed
 
 text\<open>Properties of subtraction:\<close>
 
 lemma Pring_uminus:
   assumes "a \<in> carrier (Pring R I)"
   shows "P_ring_uminus R a \<in> carrier (Pring R I)"
-  using P_ring_uminus_closed[of a I] Pring_car[of I] assms 
-  by metis 
+  using P_ring_uminus_closed[of a I] Pring_car[of I] assms
+  by metis
 
 lemma Pring_subtract:
   assumes "a \<in> carrier (Pring R I)"
   shows "P_ring_uminus R a \<oplus>\<^bsub>Pring R I\<^esub> a = \<zero>\<^bsub>Pring R I\<^esub>"
         "a \<oplus>\<^bsub>Pring R I\<^esub> P_ring_uminus R a = \<zero>\<^bsub>Pring R I\<^esub>"
   using assms Pring_add[of I "P_ring_uminus R a " a] Pring_zero[of I]
   apply  (simp add: Pring_car local.ring_axioms ring.P_ring_uminus_add)
     using assms Pring_add[of I "P_ring_uminus R a " a] Pring_zero[of I]
     by (metis P_ring_uminus_add P_ring_uminus_closed Pring_add_comm Pring_car)
 
 text\<open>Pring R I is a ring\<close>
 
 lemma Pring_is_abelian_group:
   shows "abelian_group (Pring R I)"
   apply(rule abelian_groupI)
   apply (simp add: Pring_add_closed)
-    apply (simp add: local.ring_axioms ring.Pring_zero_closed)    
+    apply (simp add: local.ring_axioms ring.Pring_zero_closed)
       apply (simp add: Pring_add_assoc)
         apply (simp add: Pring_add_comm)
-          apply (simp add: local.ring_axioms ring.Pring_add_zero(2))      
-  using Pring_subtract(1) Pring_uminus 
+          apply (simp add: local.ring_axioms ring.Pring_add_zero(2))
+  using Pring_subtract(1) Pring_uminus
   by blast
 
 lemma Pring_is_monoid:
 "Group.monoid (Pring R I)"
   apply(rule monoidI)
   using Pring_mult_closed apply blast
-    apply (simp add: Pring_one_closed)   
+    apply (simp add: Pring_one_closed)
       apply (metis Pring_mult_assoc)
-        using Pring_mult_one' 
+        using Pring_mult_one'
         apply blast
   using Pring_mult_one by blast
 
 lemma Pring_is_ring:
   shows "ring (Pring R I)"
   apply(rule ringI)
   apply (simp add: Pring_is_abelian_group)
     apply (simp add: Pring_is_monoid)
       apply (simp add: Pring_mult_ldistr)
   by (simp add: Pring_mult_rdistr)
 
 lemma Pring_is_cring:
   assumes "cring R"
   shows "cring (Pring R I)"
   apply(rule cringI)
   apply (simp add: Pring_is_abelian_group)
-   apply (simp add: Pring_is_monoid assms local.ring_axioms 
+   apply (simp add: Pring_is_monoid assms local.ring_axioms
           monoid.monoid_comm_monoidI ring.Pring_mult_comm)
      by (simp add: Pring_mult_ldistr)
 
 lemma Pring_a_inv:
   assumes "P \<in> carrier (Pring R I)"
   shows "\<ominus>\<^bsub>Pring R I\<^esub> P = P_ring_uminus R P"
 proof-
   have 0: "P_ring_uminus R P \<in> carrier (Pring R I)"
-    using Pring_uminus assms 
+    using Pring_uminus assms
     by blast
   have 1: "P_ring_uminus R P \<oplus>\<^bsub>Pring R I\<^esub> P = \<zero>\<^bsub>Pring R I\<^esub>"
-    using Pring_subtract(1) assms 
+    using Pring_subtract(1) assms
     by blast
   show ?thesis using 0 1 assms Pring_is_ring
   by (simp add: Pring_car Pring_is_abelian_group abelian_group.minus_equality)
 qed
 
 
 end
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsection\<open>Defining the R-Algebra of Indexed Polynomials\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 context cring
 begin
 
 lemma Pring_smult_cfs:
   assumes "a \<in> carrier R"
   assumes "P \<in> carrier (Pring R I)"
   shows "(a \<odot>\<^bsub>Pring R I\<^esub> P) m = a \<otimes> (P m)"
-  using assms Pring_smult 
-  by (simp add: Pring_smult poly_scalar_mult_def)  
+  using assms Pring_smult
+  by (simp add: Pring_smult poly_scalar_mult_def)
 
 lemma Pring_smult_closed:
       "\<And> a x. [| a \<in> carrier R; x \<in> carrier (Pring R I) |] ==> a \<odot>\<^bsub>(Pring R I)\<^esub> x \<in> carrier (Pring R I)"
   by (simp add: Pring_car Pring_smult poly_scalar_mult_closed)
-  
+
 lemma Pring_smult_l_distr:
       "!!a b x. [| a \<in> carrier R; b \<in> carrier R; x \<in> carrier (Pring R I) |] ==>
       (a \<oplus> b) \<odot>\<^bsub>(Pring R I)\<^esub> x = (a \<odot>\<^bsub>(Pring R I)\<^esub> x) \<oplus>\<^bsub>(Pring R I)\<^esub> (b \<odot>\<^bsub>(Pring R I)\<^esub> x)"
 proof- fix a b x assume A: "a \<in> carrier R" "b \<in> carrier R" "x \<in> carrier (Pring R I)"
   show "(a \<oplus> b) \<odot>\<^bsub>Pring R I\<^esub> x = a \<odot>\<^bsub>Pring R I\<^esub> x \<oplus>\<^bsub>Pring R I\<^esub> b \<odot>\<^bsub>Pring R I\<^esub> x"
   proof fix m
     have "(a \<oplus> b) \<otimes> x m = (a \<otimes> (x m)) \<oplus> (b \<otimes> (x m))"
       by (meson A(1) A(2) A(3) Pring_carrier_coeff' local.semiring_axioms semiring.l_distr)
     thus "((a \<oplus> b) \<odot>\<^bsub>Pring R I\<^esub> x) m = (a \<odot>\<^bsub>Pring R I\<^esub> x \<oplus>\<^bsub>Pring R I\<^esub> b \<odot>\<^bsub>Pring R I\<^esub> x) m"
-      using Pring_smult_cfs[of "a \<oplus> b" x I m] 
-            Pring_smult_cfs[of "a" x I m] 
+      using Pring_smult_cfs[of "a \<oplus> b" x I m]
+            Pring_smult_cfs[of "a" x I m]
             Pring_smult_cfs[of "b" x I m]
             Pring_smult_closed[of a x I]
             Pring_smult_closed[of b x I]
-            Pring_add  A  
+            Pring_add  A
       by (simp add: \<open>(a \<oplus> b) \<otimes> x m = a \<otimes> x m \<oplus> b \<otimes> x m\<close> Pring_def indexed_padd_def poly_scalar_mult_def)
-  qed   
+  qed
 qed
 
 lemma Pring_smult_r_distr:
       "!!a x y. [| a \<in> carrier R; x \<in> carrier (Pring R I); y \<in> carrier (Pring R I) |] ==>
       a \<odot>\<^bsub>(Pring R I)\<^esub> (x \<oplus>\<^bsub>(Pring R I)\<^esub> y) = (a \<odot>\<^bsub>(Pring R I)\<^esub> x) \<oplus>\<^bsub>(Pring R I)\<^esub> (a \<odot>\<^bsub>(Pring R I)\<^esub> y)"
 proof fix a x y m assume A: "a \<in> carrier R" "x \<in> carrier (Pring R I)" "y \<in> carrier (Pring R I)"
   show "(a \<odot>\<^bsub>Pring R I\<^esub> (x \<oplus>\<^bsub>Pring R I\<^esub> y)) m =
        (a \<odot>\<^bsub>Pring R I\<^esub> x \<oplus>\<^bsub>Pring R I\<^esub> a \<odot>\<^bsub>Pring R I\<^esub> y) m"
-    using Pring_smult_cfs[of a "x \<oplus>\<^bsub>Pring R I\<^esub> y" I m] 
-            Pring_smult_cfs[of a x I m] 
+    using Pring_smult_cfs[of a "x \<oplus>\<^bsub>Pring R I\<^esub> y" I m]
+            Pring_smult_cfs[of a x I m]
             Pring_smult_cfs[of a y I m]
             Pring_smult_closed[of a x I]
             Pring_smult_closed[of a y I]
-            Pring_add  A  
+            Pring_add  A
   by (metis (no_types, lifting) Pring_add_closed Pring_carrier_coeff' indexed_padd_def l_distr m_comm)
 qed
 
 lemma Pring_smult_assoc1:
       "!!a b x. [| a \<in> carrier R; b \<in> carrier R; x \<in> carrier (Pring R I) |] ==>
       (a \<otimes> b) \<odot>\<^bsub>Pring R I\<^esub> x = a \<odot>\<^bsub>Pring R I\<^esub> (b \<odot>\<^bsub>Pring R I\<^esub> x)"
 proof fix a b x m  assume A: "a \<in> carrier R" "b \<in> carrier R" "x \<in> carrier (Pring R I)"
   show " (a \<otimes> b \<odot>\<^bsub>Pring R I\<^esub> x) m = (a \<odot>\<^bsub>Pring R I\<^esub> (b \<odot>\<^bsub>Pring R I\<^esub> x)) m"
-      using Pring_smult_cfs[of "a \<otimes> b" x I m] 
-            Pring_smult_cfs[of a "b \<odot>\<^bsub>Pring R I\<^esub> x" I m] 
+      using Pring_smult_cfs[of "a \<otimes> b" x I m]
+            Pring_smult_cfs[of a "b \<odot>\<^bsub>Pring R I\<^esub> x" I m]
             Pring_smult_cfs[of "b" x I m]
             Pring_smult_closed[of a "b \<odot>\<^bsub>Pring R I\<^esub> x" I]
             Pring_smult_closed[of b x I]
             A(1) A(2) A(3) m_assoc m_closed by auto
 qed
 
 lemma Pring_smult_one:
       "!!x. x \<in> carrier (Pring R I) ==> (one R) \<odot>\<^bsub>Pring R I\<^esub> x = x"
   by (simp add: Pring_car Pring_smult poly_scalar_mult_one)
-  
+
 
 lemma Pring_smult_assoc2:
       "!!a x y. [| a \<in> carrier R; x \<in> carrier (Pring R I); y \<in> carrier (Pring R I) |] ==>
       (a \<odot>\<^bsub>Pring R I\<^esub> x) \<otimes>\<^bsub>Pring R I\<^esub> y = a \<odot>\<^bsub>Pring R I\<^esub> (x \<otimes>\<^bsub>Pring R I\<^esub> y)"
   by (simp add: Pring_def poly_scalar_mult_times)
 
 lemma Pring_algebra:
 "algebra R (Pring R I)"
   apply(rule algebraI)
          apply (simp add: is_cring)
         apply (simp add: Pring_is_cring is_cring)
        apply (simp add: Pring_smult_closed)
       apply (simp add: Pring_smult_l_distr)
      apply (simp add: Pring_smult_r_distr)
     apply (simp add: Pring_smult_assoc1)
    apply (simp add: Pring_smult_one)
   by (simp add: Pring_smult_assoc2)
 
 
-end 
+end
 
 
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsection\<open>Evaluation of Polynomials and Subring Structure\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 text\<open>
   In this section the aim is to define the evaluation of a polynomial over its base ring. We define
-  both total evaluation of a polynomial at all variables, and partial evaluation at only a subset 
-  of variables. The basic input for evaluation is a variable assignment function mapping variables 
-  to ring elements. Once we can evaluate a polynomial $P$ in variables $I$ over a ring $R$ at an 
-  assignment $f: I \to R$, this can be generalized to evaluation of $P$ in some other ring $S$, 
-  given a variable assignment $f: I \to S$ and a ring homomorphism $\phi: R \to S$. We chose to 
-  define this by simply applying $\phi$ to the coefficients of $P$, and then using the first 
-  evaluation function over $S$. This could also have been done the other way around: define 
-  general polynomial evaluation over any ring, given a ring hom $\phi$, and then defining 
-  evaluation over the base ring $R$ as the specialization of this function to the case there 
+  both total evaluation of a polynomial at all variables, and partial evaluation at only a subset
+  of variables. The basic input for evaluation is a variable assignment function mapping variables
+  to ring elements. Once we can evaluate a polynomial $P$ in variables $I$ over a ring $R$ at an
+  assignment $f: I \to R$, this can be generalized to evaluation of $P$ in some other ring $S$,
+  given a variable assignment $f: I \to S$ and a ring homomorphism $\phi: R \to S$. We chose to
+  define this by simply applying $\phi$ to the coefficients of $P$, and then using the first
+  evaluation function over $S$. This could also have been done the other way around: define
+  general polynomial evaluation over any ring, given a ring hom $\phi$, and then defining
+  evaluation over the base ring $R$ as the specialization of this function to the case there
   $\phi = \mathit{id}_R$.\<close>
 
-definition remove_monom :: 
+definition remove_monom ::
 "('a,'b) ring_scheme \<Rightarrow> 'c monomial \<Rightarrow> ('a, 'c) mvar_poly \<Rightarrow> ('a, 'c) mvar_poly" where
 "remove_monom R m P = ring.indexed_padd R P (poly_scalar_mult R (\<ominus>\<^bsub>R\<^esub> P m) (mset_to_IP R m))"
- 
-context cring 
+
+context cring
 begin
 
 lemma remove_monom_alt_def:
-  assumes "P \<in> Pring_set R I" 
+  assumes "P \<in> Pring_set R I"
   shows "remove_monom R m P n = (if n = m then \<zero> else P n)"
-  unfolding remove_monom_def 
+  unfolding remove_monom_def
   apply(cases "n = m")
-  using assms 
+  using assms
    apply (metis Pring_cfs_closed cring.cring_simprules(3) poly_scalar_mult_def
           indexed_padd_def is_cring mset_to_IP_simp r_neg r_one)
-   using assms 
+   using assms
    by (metis Pring_cfs_closed add.l_cancel_one cring.cring_simprules(3)
        poly_scalar_mult_def indexed_padd_def is_cring mset_to_IP_simp' r_null zero_closed)
 
 lemma remove_monom_zero:
   assumes "m \<notin> monomials_of R P"
   assumes "P \<in> Pring_set R I"
   shows "remove_monom R m P = P"
 proof
   fix x
   show "remove_monom R m P x = P x "
     apply(cases "x \<in> monomials_of R P")
-    using assms unfolding monomials_of_def remove_monom_def 
+    using assms unfolding monomials_of_def remove_monom_def
   apply (metis cring.remove_monom_alt_def is_cring remove_monom_def)
-    using assms unfolding monomials_of_def remove_monom_def 
-    by (metis assms(1) cring.remove_monom_alt_def is_cring local.ring_axioms 
+    using assms unfolding monomials_of_def remove_monom_def
+    by (metis assms(1) cring.remove_monom_alt_def is_cring local.ring_axioms
         remove_monom_def ring.complement_of_monomials_of)
 qed
-  
+
 lemma remove_monom_closed:
-  assumes "P \<in> Pring_set R I" 
+  assumes "P \<in> Pring_set R I"
   shows "remove_monom R m P \<in> Pring_set R I"
- 
+
   apply(cases "m \<in> monomials_of R P")
   using assms poly_scalar_mult_closed[of "(\<ominus> P m)" "(mset_to_IP R m)" I] mset_to_IP_closed[of m I]
-  unfolding remove_monom_def 
+  unfolding remove_monom_def
   apply (meson Pring_cfs_closed add.inv_closed indexed_pset.indexed_padd mset_to_IP_closed')
   by (metis assms remove_monom_def remove_monom_zero)
 
 lemma remove_monom_monomials:
   assumes "P \<in> Pring_set R I"
   shows "monomials_of R (remove_monom R m P) = monomials_of R P - {m}"
 proof
   show "monomials_of R (remove_monom R m P) \<subseteq> monomials_of R P - {m}"
     using assms remove_monom_alt_def[of P I m]
-    unfolding monomials_of_def 
+    unfolding monomials_of_def
     by auto
   show "monomials_of R P - {m} \<subseteq> monomials_of R (remove_monom R m P)"
     using assms remove_monom_alt_def[of P I m]
-    unfolding monomials_of_def 
+    unfolding monomials_of_def
     by auto
 qed
 
 text\<open>The additive decomposition of a polynomial by a monomial\<close>
 
 lemma remove_monom_eq:
   assumes "P \<in> Pring_set R I"
   shows "P = (remove_monom R a P) \<Oplus> poly_scalar_mult R  (P a) (mset_to_IP R a)"
   unfolding remove_monom_def poly_scalar_mult_def
 proof
   fix x
   show "P x = (P \<Oplus> (\<lambda>m. \<ominus> P a \<otimes> mset_to_IP R a m) \<Oplus> (\<lambda>m. P a \<otimes> mset_to_IP R a m)) x"
   apply(cases "x = a")
      apply (metis Pring_cfs_closed assms l_minus l_zero local.ring_axioms mset_to_IP_simp one_closed r_neg r_one ring.indexed_padd_def)
   proof-
     assume A: "x \<noteq> a"
     show "P x = (P \<Oplus> (\<lambda>m. \<ominus> P a \<otimes> mset_to_IP R a m) \<Oplus> (\<lambda>m. P a \<otimes> mset_to_IP R a m)) x"
       using assms A
-      unfolding mset_to_IP_def indexed_padd_def  
-      using Pring_cfs_closed by fastforce     
+      unfolding mset_to_IP_def indexed_padd_def
+      using Pring_cfs_closed by fastforce
   qed
 qed
 
 lemma remove_monom_restrict_poly_to_monom_set:
   assumes "P \<in> Pring_set R I"
   assumes "monomials_of R P = insert a M"
   assumes "a \<notin> M"
   shows "(remove_monom R a P) = restrict_poly_to_monom_set R P M"
 proof
   fix m
   show "remove_monom R a P m= restrict_poly_to_monom_set R P M m"
     apply(cases "m = a")
     using assms apply (metis remove_monom_alt_def restrict_poly_to_monom_set_def)
-    using assms 
+    using assms
     by (metis complement_of_monomials_of insert_iff remove_monom_alt_def restrict_poly_to_monom_set_def)
 qed
 
 end
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsubsection\<open>Nesting of Polynomial Rings According to Nesting of Index Sets\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 lemma(in ring) Pring_carrier_subset:
   assumes "J \<subseteq> I"
   shows "(Pring_set R J) \<subseteq> (Pring_set R I)"
 proof
   fix P
   show "P \<in> Pring_set R J \<Longrightarrow>P \<in> Pring_set R I"
     apply(erule indexed_pset.induct)
     using indexed_pset.indexed_const apply blast
       using indexed_pset.indexed_padd apply blast
         by (meson assms indexed_pset.indexed_pmult subsetD)
 qed
 
 lemma(in cring) Pring_set_restrict_induct:
   shows "finite S \<Longrightarrow> \<forall>P. monomials_of R P = S \<and>  P \<in> Pring_set R I \<and>  (\<forall> m \<in> monomials_of R P. set_mset m \<subseteq> J) \<longrightarrow>  P \<in> Pring_set R J"
 proof(erule finite.induct)
   show "\<forall>P. monomials_of R P = {} \<and> P \<in> Pring_set R I \<and> (\<forall>m\<in>monomials_of R P. set_mset m \<subseteq> J) \<longrightarrow> P \<in> Pring_set R J"
   proof
     fix P
     show "monomials_of R P = {} \<and> P \<in> Pring_set R I \<and> (\<forall>m\<in>monomials_of R P. set_mset m \<subseteq> J) \<longrightarrow> P \<in> Pring_set R J"
     proof
       assume A0: "monomials_of R P = {} \<and> P \<in> Pring_set R I \<and> (\<forall>m\<in>monomials_of R P. set_mset m \<subseteq> J)"
-      show "P \<in> Pring_set R J " 
+      show "P \<in> Pring_set R J "
         by (metis A0 card_eq_0_iff indexed_pset.indexed_const monomials_of_card_zero zero_closed)
     qed
   qed
   show "\<And>A a. finite A \<Longrightarrow>
            \<forall>P. monomials_of R P = A \<and> P \<in> Pring_set R I \<and> (\<forall>m\<in>monomials_of R P. set_mset m \<subseteq> J) \<longrightarrow> P \<in> Pring_set R J \<Longrightarrow>
            \<forall>P. monomials_of R P = insert a A \<and> P \<in> Pring_set R I \<and> (\<forall>m\<in>monomials_of R P. set_mset m \<subseteq> J) \<longrightarrow> P \<in> Pring_set R J"
   proof
     fix A :: "('c monomial) set" fix a fix P
     assume A1: "finite A"
     assume A2: "\<forall>P. monomials_of R P = A \<and> P \<in> Pring_set R I \<and> (\<forall>m\<in>monomials_of R P. set_mset m \<subseteq> J) \<longrightarrow> P \<in> Pring_set R J"
     show "monomials_of R P = insert a A \<and> P \<in> Pring_set R I \<and> (\<forall>m\<in>monomials_of R P. set_mset m \<subseteq> J) \<longrightarrow> P \<in> Pring_set R J "
     proof
       assume A3: "monomials_of R P = insert a A \<and> P \<in> Pring_set R I \<and> (\<forall>m\<in>monomials_of R P. set_mset m \<subseteq> J)"
       show "P \<in> Pring_set R  J"
         apply(cases "a \<in>   A")
          apply (metis A2 A3 insert_absorb)
       proof-
         assume N: "a \<notin> A"
         obtain Q where Q_def: "Q = remove_monom R a P"
           by simp
         have Q0: "monomials_of R Q = A"
         proof-
           have 0:  "monomials_of R P = insert a A"
             by (simp add: A3)
           have 1: "P \<in> Pring_set R I"
             using A3 by blast
           have 2: "monomials_of R (remove_monom R a P) = monomials_of R P - {a}"
             using A3 remove_monom_monomials by blast
           then show ?thesis
-            using Q_def 0 
+            using Q_def 0
             by (simp add: N)
         qed
         have Q1: "Q \<in> Pring_set R I"
           using A3 Q_def remove_monom_closed by blast
         have Q2: "(\<forall>m\<in>monomials_of R Q. set_mset m \<subseteq> J)"
-          using Q0 A3 
+          using Q0 A3
           by blast
         have "Q \<in> Pring_set R J"
           using A2 Q0 Q1 Q2 by blast
         then show "P \<in> Pring_set R J"
         proof-
           have "P = Q \<Oplus> poly_scalar_mult R  (P a) (mset_to_IP R a)"
             using Q_def remove_monom_eq
             using A3 by blast
-          then show ?thesis 
+          then show ?thesis
             by (metis A3 Pring_cfs_closed \<open>Q \<in> Pring_set R J\<close> indexed_pset.indexed_padd insertCI mset_to_IP_closed poly_scalar_mult_closed)
         qed
       qed
     qed
   qed
 qed
 
 lemma(in cring) Pring_set_restrict:
   assumes "P \<in> Pring_set R I"
   assumes "(\<And>m.  m \<in> monomials_of R P \<Longrightarrow>  set_mset m \<subseteq> J)"
   shows " P \<in> Pring_set R J"
   using assms Pring_set_restrict_induct[of "monomials_of R P"]
   by (metis monomials_finite)
-             
+
 lemma(in ring) Pring_mult_eq:
   fixes I:: "'c set"
   fixes J:: "'c set"
   shows "(\<otimes>\<^bsub>Pring R I\<^esub>) = (\<otimes>\<^bsub>Pring R J\<^esub>)"
   by (simp add: Pring_def)
-  
+
 lemma(in ring) Pring_add_eq:
   fixes I:: "'c set"
   fixes J:: "'c set"
   shows "(\<oplus>\<^bsub>Pring R I\<^esub>) = (\<oplus>\<^bsub>Pring R J\<^esub>)"
-  using Pring_def 
+  using Pring_def
   by (simp add: Pring_def)
 
 lemma(in ring) Pring_one_eq:
   fixes I:: "'c set"
   fixes J:: "'c set"
   shows "(\<one>\<^bsub>Pring R I\<^esub>) = (\<one>\<^bsub>Pring R J\<^esub>)"
-  using Pring_def 
+  using Pring_def
   by (simp add: Pring_def)
 
 lemma(in ring) Pring_zero_eq:
   fixes I:: "'c set"
   fixes J:: "'c set"
   shows "(\<zero>\<^bsub>Pring R I\<^esub>) = (\<zero>\<^bsub>Pring R J\<^esub>)"
-  using Pring_def 
+  using Pring_def
   by (simp add: Pring_def)
 
 lemma(in ring) index_subset_Pring_subring:
   assumes "J \<subseteq> I"
   shows "subring (carrier (Pring R J)) (Pring R I)"
   apply(rule ring.subringI)
   apply (simp add: Pring_is_ring; fail)
-    using assms 
+    using assms
    apply (simp add: Pring_car Pring_carrier_subset; fail)
-   using Pring_def 
+   using Pring_def
          apply (simp add: Pring_def indexed_pset.indexed_const; fail)
     proof-
       show "\<And>h. h \<in> carrier (Pring R J) \<Longrightarrow> \<ominus>\<^bsub>Pring R I\<^esub> h \<in> carrier (Pring R J)"
       proof-
         fix h
         assume A: "h \<in> carrier (Pring R J)"
         then have A0: "\<ominus>\<^bsub>Pring R J\<^esub> h = P_ring_uminus R h"
-          using Pring_a_inv[of h J] 
-          by auto 
+          using Pring_a_inv[of h J]
+          by auto
         have A1: "\<ominus>\<^bsub>Pring R I\<^esub> h = P_ring_uminus R h"
-          using assms A Pring_carrier_subset[of J I] Pring_a_inv[of h I] Pring_car 
+          using assms A Pring_carrier_subset[of J I] Pring_a_inv[of h I] Pring_car
           by blast
         show "\<ominus>\<^bsub>Pring R I\<^esub> h \<in> carrier (Pring R J)"
-          using A0 A1 A 
-          by (simp add: Pring_uminus)          
+          using A0 A1 A
+          by (simp add: Pring_uminus)
       qed
       show " \<And>h1 h2. h1 \<in> carrier (Pring R J) \<Longrightarrow> h2 \<in> carrier (Pring R J) \<Longrightarrow> h1 \<otimes>\<^bsub>Pring R I\<^esub> h2 \<in> carrier (Pring R J)"
-        using assms Pring_mult_eq 
+        using assms Pring_mult_eq
         by (metis Pring_mult_closed)
       show " \<And>h1 h2. h1 \<in> carrier (Pring R J) \<Longrightarrow> h2 \<in> carrier (Pring R J) \<Longrightarrow> h1 \<oplus>\<^bsub>Pring R I\<^esub> h2 \<in> carrier (Pring R J)"
-        using assms Pring_add_eq 
+        using assms Pring_add_eq
         by (metis Pring_add_closed)
     qed
 
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsubsection\<open>Inclusion Maps\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 definition Pring_inc :: "('a, 'c) mvar_poly \<Rightarrow> ('a, 'c) mvar_poly" where
 "Pring_inc \<equiv> (\<lambda>P. P)"
 
 lemma(in ring) Princ_inc_is_ring_hom:
   assumes "J \<subseteq> I"
   shows "ring_hom_ring (Pring R J) (Pring R I) Pring_inc"
   unfolding Pring_inc_def
   apply(rule ring_hom_ringI)
   apply (simp add: Pring_is_ring)
     using Pring_is_ring apply blast
-       using index_subset_Pring_subring[of I J] assms index_subset_Pring_subring subringE(1) 
+       using index_subset_Pring_subring[of I J] assms index_subset_Pring_subring subringE(1)
        apply blast
           using Pring_mult_eq[of I J]
           apply (simp add: Pring_mult)
              using Pring_add_eq[of I J]
              apply (simp add: Pring_add)
-             using Pring_one_eq 
+             using Pring_one_eq
   by (simp add: Pring_one_eq)
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsubsection\<open>Restricting a Multiset to a Subset of Variables\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 definition restrict_to_indices :: "'c monomial \<Rightarrow> 'c set \<Rightarrow> 'c monomial" where
 "restrict_to_indices m S = filter_mset (\<lambda>i. i \<in> S) m"
 
 lemma restrict_to_indicesE:
   assumes "i \<in># restrict_to_indices m S"
   shows "i \<in> S"
-  using assms 
-  unfolding restrict_to_indices_def 
+  using assms
+  unfolding restrict_to_indices_def
   by simp
-  
+
 lemma restrict_to_indicesI[simp]:
   assumes "i \<in># m"
   assumes "i \<in> S"
   shows "i \<in># restrict_to_indices m S"
-  using assms 
-  unfolding restrict_to_indices_def 
+  using assms
+  unfolding restrict_to_indices_def
   by simp
 
 lemma restrict_to_indices_not_in[simp]:
   assumes "i \<in># m"
   assumes "i \<notin> S"
   shows "i \<notin># restrict_to_indices m S"
-  using assms 
-  unfolding restrict_to_indices_def 
+  using assms
+  unfolding restrict_to_indices_def
   by (meson count_eq_zero_iff count_filter_mset)
 
 lemma restrict_to_indices_submultiset[simp]:
 "restrict_to_indices m S \<subseteq># m"
-  unfolding restrict_to_indices_def 
-  using multiset_filter_subset 
+  unfolding restrict_to_indices_def
+  using multiset_filter_subset
   by blast
 
 lemma restrict_to_indices_add_element:
-"restrict_to_indices (add_mset x m) S = (if x \<in> S then (add_mset x (restrict_to_indices m S) ) 
+"restrict_to_indices (add_mset x m) S = (if x \<in> S then (add_mset x (restrict_to_indices m S) )
                                             else (restrict_to_indices m S) )"
-  unfolding restrict_to_indices_def 
+  unfolding restrict_to_indices_def
   by (metis filter_mset_add_mset)
 
-lemma restrict_to_indices_count[simp]: 
+lemma restrict_to_indices_count[simp]:
 "count (restrict_to_indices m S) i = (if (i \<in> S) then (count m i) else 0)"
-  unfolding restrict_to_indices_def 
+  unfolding restrict_to_indices_def
   by (metis count_filter_mset)
 
 lemma restrict_to_indices_subset:
 "restrict_to_indices m S = restrict_to_indices m (set_mset m \<inter> S)"
 proof(induction m)
   case empty
-  then show ?case unfolding restrict_to_indices_def 
+  then show ?case unfolding restrict_to_indices_def
     by (metis filter_empty_mset)
 next
   case (add x m)
   assume IH: "restrict_to_indices m S = restrict_to_indices m (set_mset m \<inter> S)"
   show "restrict_to_indices (add_mset x m) S = restrict_to_indices (add_mset x m) (set_mset (add_mset x m) \<inter> S)"
   proof-
-    have "\<And>i. count (restrict_to_indices (add_mset x m) S) i  = 
+    have "\<And>i. count (restrict_to_indices (add_mset x m) S) i  =
               count (restrict_to_indices (add_mset x m) (set_mset (add_mset x m) \<inter> S)) i  "
     proof-
       fix i
       show "count (restrict_to_indices (add_mset x m) S) i  = count (restrict_to_indices (add_mset x m) (set_mset (add_mset x m) \<inter> S)) i"
         apply(cases "i \<in> S")
-        using restrict_to_indices_count 
+        using restrict_to_indices_count
         apply (metis IntI count_inI)
         by (metis restrict_to_indices_count Int_iff)
     qed
-    then show ?thesis 
+    then show ?thesis
       using multiset_eq_iff by blast
   qed
 qed
 
 text\<open>\texttt{Restrict\_to\_indices} only depends on the intersection
      of the index set with the set of indices in m:\<close>
 
 lemma restrict_to_indices_subset':
   assumes "(set_mset m) \<inter> S = (set_mset m) \<inter> S'"
   shows "restrict_to_indices m S = restrict_to_indices m S'"
   by (metis restrict_to_indices_subset assms)
 
 lemma mset_add_plus:
   assumes "m = n + k"
   shows "add_mset x m = (add_mset x n) + k"
-  using assms 
+  using assms
   by simp
 
 text\<open>Restricting to \<open>S\<close> and the complement of \<open>S\<close> partitions \<open>m\<close>:\<close>
 
 lemma restrict_to_indices_decomp:
   "m = (restrict_to_indices m S) + (restrict_to_indices m ((set_mset m) - S))"
   apply(induction m)
   apply (metis add.right_neutral empty_Diff restrict_to_indices_submultiset set_mset_empty subset_mset.le_zero_eq)
 proof-
   fix x
   fix m
   assume A: "m = restrict_to_indices m S + restrict_to_indices m (set_mset m - S)"
   show "add_mset x m = restrict_to_indices (add_mset x m) S + restrict_to_indices (add_mset x m) (set_mset (add_mset x m) - S)"
   proof(cases "x \<in> S")
     case True
     then have T0: "restrict_to_indices (add_mset x m) S = (add_mset x (restrict_to_indices m S) ) "
       by (simp add: True restrict_to_indices_add_element)
-    have T1: "restrict_to_indices (add_mset x m) (set_mset (add_mset x m) - S) = 
+    have T1: "restrict_to_indices (add_mset x m) (set_mset (add_mset x m) - S) =
             restrict_to_indices m (set_mset (add_mset x m) - S)"
       using True by (metis DiffD2 restrict_to_indices_add_element)
     have T2: "restrict_to_indices (add_mset x m) S +  restrict_to_indices (add_mset x m) (set_mset (add_mset x m) - S)
             = (add_mset x (restrict_to_indices m S) )  + restrict_to_indices m (set_mset (add_mset x m) - S)"
-      using T0 T1 
+      using T0 T1
       by presburger
     have T3: " add_mset x m = add_mset x (restrict_to_indices m S) + restrict_to_indices m (set_mset m - S)"
-      using T2 A using mset_add_plus[of m "restrict_to_indices m S" " restrict_to_indices m (set_mset m - S)" x] 
+      using T2 A using mset_add_plus[of m "restrict_to_indices m S" " restrict_to_indices m (set_mset m - S)" x]
       by blast
     have T4: "set_mset m \<inter> (set_mset (add_mset x m) - S) = set_mset m \<inter> (set_mset m - S)"
     proof
       show "set_mset m \<inter> (set_mset (add_mset x m) - S) \<subseteq> set_mset m \<inter> (set_mset m - S)"
         by blast
       show "set_mset m \<inter> (set_mset m - S) \<subseteq> set_mset m \<inter> (set_mset (add_mset x m) - S)"
         by (simp add: True)
     qed
     have T5: "restrict_to_indices m (set_mset (add_mset x m) - S) = restrict_to_indices m (set_mset m - S)"
       using T4 restrict_to_indices_subset'[of m "(set_mset (add_mset x m) - S)" " (set_mset m - S)" ]
       by blast
-    then show ?thesis using T4 
+    then show ?thesis using T4
       by (metis T0 T1 T3)
   next
     case False
     then have F0: "restrict_to_indices (add_mset x m) S = (restrict_to_indices m S) "
       by (simp add: False restrict_to_indices_add_element)
-    have F1: "restrict_to_indices (add_mset x m) (set_mset (add_mset x m) - S) = 
+    have F1: "restrict_to_indices (add_mset x m) (set_mset (add_mset x m) - S) =
             (add_mset x (restrict_to_indices m (set_mset (add_mset x m) - S)))"
-      using False 
+      using False
       by (meson DiffI restrict_to_indices_add_element union_single_eq_member)
-      
+
     have F2: "restrict_to_indices (add_mset x m) S +  restrict_to_indices (add_mset x m) (set_mset (add_mset x m) - S)
             = (restrict_to_indices m S) + (add_mset x (restrict_to_indices m (set_mset (add_mset x m) - S)))"
       using F0 F1
       by presburger
     have F3: " add_mset x m =  (restrict_to_indices m S) +  (add_mset x (restrict_to_indices m (set_mset m - S)))"
-      using F2 A  mset_add_plus[of m "restrict_to_indices m (set_mset m - S)" "restrict_to_indices m S" x] 
+      using F2 A  mset_add_plus[of m "restrict_to_indices m (set_mset m - S)" "restrict_to_indices m S" x]
       by (metis union_mset_add_mset_right)
     have F4: " add_mset x m =  restrict_to_indices (add_mset x m) S+  (add_mset x (restrict_to_indices m (set_mset m - S)))"
       using F0 F3 by auto
     have F5: "add_mset x (restrict_to_indices m (set_mset m - S)) = restrict_to_indices (add_mset x m) (set_mset (add_mset x m) - S) "
     proof(cases "x \<in> set_mset m")
       case True
-      then show ?thesis 
+      then show ?thesis
         by (metis F1 add_mset_remove_trivial more_than_one_mset_mset_diff)
     next
       case F: False
       have "set_mset m \<inter> (set_mset (add_mset x m) - S) = set_mset m \<inter>(set_mset m - S)"
       proof
         show "set_mset m \<inter> (set_mset (add_mset x m) - S) \<subseteq> set_mset m \<inter> (set_mset m - S)"
-          using F False 
+          using F False
           by blast
         show "set_mset m \<inter> (set_mset m - S) \<subseteq> set_mset m \<inter> (set_mset (add_mset x m) - S)"
-          using F False 
-          by (metis Diff_mono mset_add_plus Int_mono add_cancel_right_left  
+          using F False
+          by (metis Diff_mono mset_add_plus Int_mono add_cancel_right_left
               set_eq_subset subsetI subset_iff union_commute union_iff)
       qed
-      then show ?thesis 
+      then show ?thesis
         by (metis F1 restrict_to_indices_subset')
     qed
-    then show ?thesis 
-      using False F4 
+    then show ?thesis
+      using False F4
       by presburger
   qed
 qed
 
 definition remove_indices :: "'c monomial \<Rightarrow> 'c set \<Rightarrow> 'c monomial" where
 "remove_indices m S = (restrict_to_indices m (set_mset m - S))"
 
 lemma remove_indices_decomp:
-"m = (restrict_to_indices m S) + (remove_indices m S)" 
-  unfolding remove_indices_def 
-  using restrict_to_indices_decomp 
+"m = (restrict_to_indices m S) + (remove_indices m S)"
+  unfolding remove_indices_def
+  using restrict_to_indices_decomp
   by blast
 
 lemma remove_indices_indices[simp]:
   assumes "set_mset m \<subseteq> I"
   shows "set_mset (remove_indices m S) \<subseteq> I - S"
-  unfolding remove_indices_def using assms 
+  unfolding remove_indices_def using assms
   by (meson Diff_iff restrict_to_indicesE subsetD subsetI)
- 
+
 subsubsection\<open>Total evaluation of a monomial\<close>
 
 text\<open>
   We define total evaluation of a monomial first, and then define the partial evaluation of a
   monomial in terms of this.
 \<close>
 
 abbreviation(input) closed_fun where
 "closed_fun R g \<equiv> g \<in> UNIV \<rightarrow> carrier R"
 
 definition monom_eval :: "('a, 'b) ring_scheme \<Rightarrow> 'c monomial \<Rightarrow> ('c \<Rightarrow> 'a) \<Rightarrow> 'a"  where
 "monom_eval R (m:: 'c monomial) g = fold_mset (\<lambda> x . \<lambda> y. if y \<in> carrier R then (g x) \<otimes>\<^bsub>R\<^esub> y else \<zero>\<^bsub>R\<^esub>) \<one>\<^bsub>R\<^esub> m"
 
-context cring 
+context cring
 begin
 
 lemma closed_fun_simp:
   assumes "closed_fun R g"
   shows "g n \<in> carrier R"
-  using assms 
-  by blast 
+  using assms
+  by blast
 
 lemma closed_funI:
   assumes "\<And>x. g x \<in> carrier R"
   shows "closed_fun R g"
   by (meson Pi_I assms)
 
 
 text\<open>The following are necessary technical lemmas to prove properties of about folds over multisets:\<close>
 
 lemma monom_eval_comp_fun:
   fixes g:: "'c \<Rightarrow> 'a"
   assumes "closed_fun R g"
   shows "comp_fun_commute (\<lambda> x . \<lambda>y. if y \<in> carrier R then  (g x) \<otimes> y else \<zero>)"
-  unfolding comp_fun_commute_def 
+  unfolding comp_fun_commute_def
 proof-
   have "\<And> x y. (\<lambda>ya. if ya \<in> carrier R then g y \<otimes> ya else \<zero>) \<circ> (\<lambda>y. if y \<in> carrier R then g x \<otimes> y else \<zero>) =
           (\<lambda>y. if y \<in> carrier R then g x \<otimes> y else \<zero>) \<circ> (\<lambda>ya. if ya \<in> carrier R then g y \<otimes> ya else \<zero>)"
   proof
     fix x y a
     show "((\<lambda>ya. if ya \<in> carrier R then g y \<otimes> ya else \<zero>) \<circ> (\<lambda>y. if y \<in> carrier R then g x \<otimes> y else \<zero>)) a =
        ((\<lambda>y. if y \<in> carrier R then g x \<otimes> y else \<zero>) \<circ> (\<lambda>ya. if ya \<in> carrier R then g y \<otimes> ya else \<zero>)) a"
     proof(cases "a \<in> carrier R")
       case True
-      then show ?thesis 
+      then show ?thesis
       proof-
-        have LHS: "((\<lambda>ya. if ya \<in> carrier R then g y \<otimes> ya else \<zero>) \<circ> (\<lambda>y. if y \<in> carrier R then g x \<otimes> y else \<zero>)) a = 
+        have LHS: "((\<lambda>ya. if ya \<in> carrier R then g y \<otimes> ya else \<zero>) \<circ> (\<lambda>y. if y \<in> carrier R then g x \<otimes> y else \<zero>)) a =
                     ((\<lambda>ya. if ya \<in> carrier R then g y \<otimes> ya else \<zero>)  (g x \<otimes> a))"
-          using True assms(1)  m_closed m_lcomm 
-          unfolding o_def  
+          using True assms(1)  m_closed m_lcomm
+          unfolding o_def
           by presburger
 
       then have LHS': "((\<lambda>ya. if ya \<in> carrier R then g y \<otimes> ya else \<zero>) \<circ> (\<lambda>y. if y \<in> carrier R then g x \<otimes> y else \<zero>)) a =   g y \<otimes> (g x \<otimes> a) "
-        using True assms m_closed 
-        by (meson PiE UNIV_I)        
+        using True assms m_closed
+        by (meson PiE UNIV_I)
       then show ?thesis
-        unfolding o_def 
-        using True assms m_closed m_lcomm 
-        by (smt PiE UNIV_I)       
+        unfolding o_def
+        using True assms m_closed m_lcomm
+        by (smt PiE UNIV_I)
       qed
     next
       case False
-      then show ?thesis 
-        unfolding o_def 
+      then show ?thesis
+        unfolding o_def
         using assms r_null closed_fun_simp
-        by smt        
+        by smt
     qed
   qed
   then show " \<forall>y x. (\<lambda>ya. if ya \<in> carrier R then g y \<otimes> ya else \<zero>) \<circ> (\<lambda>y. if y \<in> carrier R then g x \<otimes> y else \<zero>) =
           (\<lambda>y. if y \<in> carrier R then g x \<otimes> y else \<zero>) \<circ> (\<lambda>ya. if ya \<in> carrier R then g y \<otimes> ya else \<zero>)"
-    by blast 
+    by blast
 qed
 
 lemma monom_eval_car:
   assumes "closed_fun R g"
-  shows "monom_eval R (m:: 'c monomial) g \<in> carrier R" 
+  shows "monom_eval R (m:: 'c monomial) g \<in> carrier R"
 proof(induction m)
 case empty
-  then show ?case 
+  then show ?case
     unfolding monom_eval_def
-    by (metis fold_mset_empty one_closed) 
+    by (metis fold_mset_empty one_closed)
 next
   case (add x m)
   fix x m
   assume A: "monom_eval R m g \<in> carrier R"
   obtain f where f_def: "f = (\<lambda> x . \<lambda>y. if y \<in> carrier R then (g x) \<otimes> y else \<zero>)"
-    by blast  
+    by blast
   have 0: "comp_fun_commute f"
-    using assms monom_eval_comp_fun[of g] f_def 
-    by blast        
+    using assms monom_eval_comp_fun[of g] f_def
+    by blast
   have 1: "\<And>m. monom_eval R m g = fold_mset f \<one> m"
-    using f_def monom_eval_def 
-    by blast 
+    using f_def monom_eval_def
+    by blast
   have 2: "monom_eval R (add_mset x m) g = fold_mset f \<one> (add_mset x m)"
-    using 1 by blast 
-  have 3: "g x \<in> carrier R" 
-    using assms  by blast 
+    using 1 by blast
+  have 3: "g x \<in> carrier R"
+    using assms  by blast
   then show "monom_eval R (add_mset x m) g \<in> carrier R"
     using assms 0 1 2 3
     by (metis A comp_fun_commute.fold_mset_add_mset f_def m_closed)
 qed
 
 text\<open>Formula for recursive (total) evaluation of a monomial:\<close>
 lemma monom_eval_add:
   assumes "closed_fun R g"
   shows "monom_eval R (add_mset x M) g = (g x) \<otimes> (monom_eval R M g)"
 proof-
   obtain f where f_def: "f = (\<lambda> x . \<lambda>y. if y \<in> carrier R then (g x) \<otimes> y else \<zero>)"
-    by blast  
+    by blast
   have 0: "comp_fun_commute f"
     using assms monom_eval_comp_fun f_def
-    by blast 
+    by blast
   have 1: "\<And>m. monom_eval R m g = fold_mset f \<one> m"
-    using f_def monom_eval_def 
-    by blast 
+    using f_def monom_eval_def
+    by blast
   have 2: "monom_eval R (add_mset x M) g = fold_mset f \<one> (add_mset x M)"
-    using 1 by blast 
-  have 3: "g x \<in> carrier R" 
-    using assms  by blast 
+    using 1 by blast
+  have 3: "g x \<in> carrier R"
+    using assms  by blast
   have 4: "(g x) \<otimes> (monom_eval R M g) = f x (monom_eval R M g)"
-    using f_def 3  
-    by (meson assms monom_eval_car)    
-  then show ?thesis 
+    using f_def 3
+    by (meson assms monom_eval_car)
+  then show ?thesis
     by (metis "0" "1" comp_fun_commute.fold_mset_add_mset)
 qed
 
 end
 
 text\<open>
   This function maps a polynomial $P$ to the set of monomials in $P$ which, after evaluating all
   variables in the set $S$ to values in the ring $R$, reduce to the monomial $n$.
 \<close>
 
-definition monomials_reducing_to :: 
+definition monomials_reducing_to ::
 "('a,'b) ring_scheme \<Rightarrow> 'c monomial \<Rightarrow> ('a,'c) mvar_poly \<Rightarrow> 'c set \<Rightarrow> ('c monomial) set" where
 "monomials_reducing_to R n P S = {m \<in> monomials_of R P. remove_indices m S = n}"
 
 lemma monomials_reducing_to_subset[simp]:
  "monomials_reducing_to R n P s \<subseteq> monomials_of R P"
-  unfolding monomials_reducing_to_def 
+  unfolding monomials_reducing_to_def
   by blast
 
 context cring
 begin
 
 lemma monomials_reducing_to_finite:
   assumes "P \<in> Pring_set R I"
   shows "finite (monomials_reducing_to R n P s)"
   by (meson assms monomials_finite monomials_reducing_to_subset rev_finite_subset)
 
 lemma monomials_reducing_to_disjoint:
   assumes "n1 \<noteq> n2"
   shows "monomials_reducing_to R n1 P S \<inter> monomials_reducing_to R n2 P S = {}"
-  unfolding monomials_reducing_to_def 
-  using assms 
+  unfolding monomials_reducing_to_def
+  using assms
   by blast
 
 lemma monomials_reducing_to_submset:
 assumes "n \<subset># m"
 shows "n \<notin> monomials_reducing_to R m P S"
 proof(rule ccontr)
   assume C: "\<not> n \<notin> monomials_reducing_to R m P S "
   then have "n \<in>  monomials_reducing_to R m P S "
-    by blast 
+    by blast
   then have "remove_indices n S = m"
-    unfolding monomials_reducing_to_def 
+    unfolding monomials_reducing_to_def
     by blast
-  then show False 
+  then show False
     by (metis (full_types) remove_indices_def restrict_to_indices_submultiset assms subset_mset.less_asym' subset_mset.less_irrefl subset_mset_def)
 qed
 
 end
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsubsection\<open>Partial Evaluation of a Polynomial\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 text\<open>
   This function takes as input a set $S$ of variables, an evaluation function $g$, and a polynomial
   to evaluate $P$. The output is a polynomial which is the result of evaluating the variables from
   the set $S$ which occur in $P$, according to the evaluation function $g$.
 \<close>
 
-definition poly_eval :: 
+definition poly_eval ::
   "('a,'b) ring_scheme \<Rightarrow> 'c set \<Rightarrow> ('c \<Rightarrow> 'a) \<Rightarrow> ('a, 'c) mvar_poly \<Rightarrow> ('a, 'c) mvar_poly"where
 "poly_eval R S g P m = (finsum R (\<lambda>n. monom_eval R (restrict_to_indices n S) g  \<otimes>\<^bsub>R\<^esub> (P n)) (monomials_reducing_to R m P S))"
 
 context cring
 begin
 
 lemma finsum_singleton:
   assumes "S = {s}"
   assumes "f s \<in> carrier R"
   shows "finsum R f S = f s"
 proof-
   have "finsum R f S = finsum R f (insert s {})"
-    using assms(1) 
+    using assms(1)
     by blast
-  then show ?thesis using finsum_insert[of "{}" s f] assms 
+  then show ?thesis using finsum_insert[of "{}" s f] assms
     by (metis Pi_I empty_iff finite.emptyI finsum_empty r_zero)
 qed
 
 lemma poly_eval_constant:
   assumes "k \<in> carrier R"
   shows "poly_eval R S g (indexed_const k) = (indexed_const k)"
 proof
   have S: "monomials_of R (indexed_const k) \<subseteq> {{#}}"
-    unfolding indexed_const_def monomials_of_def 
-    by (metis (mono_tags, lifting) mem_Collect_eq singletonI subset_iff)    
+    unfolding indexed_const_def monomials_of_def
+    by (metis (mono_tags, lifting) mem_Collect_eq singletonI subset_iff)
   fix x
   show "poly_eval R S g (indexed_const k) x = indexed_const k x "
   proof(cases "x = {#}")
     case True
     have "(monomials_reducing_to R x (indexed_const k) S) \<subseteq> {{#}}"
-      using S monomials_reducing_to_subset 
+      using S monomials_reducing_to_subset
       by blast
-    then show ?thesis 
+    then show ?thesis
     proof(cases "k = \<zero>")
       case True
       then have "(monomials_reducing_to R x (indexed_const k) S) = {}"
-        by (metis S \<open>monomials_reducing_to R x (indexed_const k) S \<subseteq> {{#}}\<close> 
+        by (metis S \<open>monomials_reducing_to R x (indexed_const k) S \<subseteq> {{#}}\<close>
             monomials_reducing_to_subset monoms_of_const subset_antisym subset_singletonD)
-      then show ?thesis 
-        unfolding poly_eval_def 
-        by (metis True finsum_empty indexed_const_def)      
+      then show ?thesis
+        unfolding poly_eval_def
+        by (metis True finsum_empty indexed_const_def)
     next
       case False
       then have "monomials_of R (indexed_const k) = {{#}}"
         by (meson monoms_of_const)
       have "remove_indices {#} S = {#}"
         using remove_indices_decomp by blast
       then have "{#} \<in> monomials_reducing_to R x (indexed_const k) S"
-        using True False unfolding monomials_reducing_to_def 
-              \<open>monomials_of R (indexed_const k) = {{#}}\<close> 
+        using True False unfolding monomials_reducing_to_def
+              \<open>monomials_of R (indexed_const k) = {{#}}\<close>
         by blast
       then have 0: "monomials_reducing_to R x (indexed_const k) S = {{#}}"
-        using \<open>monomials_reducing_to R x (indexed_const k) S \<subseteq> {{#}}\<close> 
+        using \<open>monomials_reducing_to R x (indexed_const k) S \<subseteq> {{#}}\<close>
         by blast
       have 1: "restrict_to_indices {#} S = {#}"
-        using restrict_to_indices_submultiset remove_indices_decomp by blast 
+        using restrict_to_indices_submultiset remove_indices_decomp by blast
       have 2: "monom_eval R (restrict_to_indices {#} S) g = \<one>"
-        unfolding monom_eval_def 
-        using 1 
-        by (metis fold_mset_empty)   
+        unfolding monom_eval_def
+        using 1
+        by (metis fold_mset_empty)
       have 3: "poly_eval R S g (indexed_const k)  x =
                 (\<Oplus>n\<in>{{#}}. monom_eval R (restrict_to_indices n S) g \<otimes> indexed_const k n)"
-        unfolding poly_eval_def 
-        using 0 
-        by presburger           
-      have 4: "(\<Oplus>n\<in>{{#}}. monom_eval R (restrict_to_indices n S) g \<otimes> indexed_const k n) = monom_eval R (restrict_to_indices {#} S) g \<otimes> indexed_const k {#}" 
+        unfolding poly_eval_def
+        using 0
+        by presburger
+      have 4: "(\<Oplus>n\<in>{{#}}. monom_eval R (restrict_to_indices n S) g \<otimes> indexed_const k n) = monom_eval R (restrict_to_indices {#} S) g \<otimes> indexed_const k {#}"
         using finsum_singleton[of "{{#}}" "{#}" "\<lambda>n. monom_eval R (restrict_to_indices n S) g \<otimes> indexed_const k n" ]
         by (metis "2" assms indexed_const_def l_one)
-      then show ?thesis unfolding poly_eval_def 
-        using 0 1 2 
+      then show ?thesis unfolding poly_eval_def
+        using 0 1 2
         by (metis True assms indexed_const_def l_one)
     qed
   next
     case False
     then have F0: "(indexed_const k) x = \<zero>"
       by (meson indexed_const_def)
     have "(monomials_reducing_to R x (indexed_const k) S) = {}"
-      unfolding monomials_reducing_to_def 
+      unfolding monomials_reducing_to_def
     proof(rule ccontr)
       assume A: "{m \<in> monomials_of R (indexed_const k). remove_indices m S = x} \<noteq> {}"
       then obtain m where m_def: "m \<in> monomials_of R (indexed_const k) \<and> remove_indices m S = x"
         by blast
-        then show False using A F0 
-          by (metis False S empty_eq_union empty_iff  
+        then show False using A F0
+          by (metis False S empty_eq_union empty_iff
               remove_indices_decomp singletonD subset_singletonD)
     qed
-    then show ?thesis 
-      unfolding poly_eval_def 
+    then show ?thesis
+      unfolding poly_eval_def
       by (metis False finsum_empty indexed_const_def)
   qed
 qed
 
 lemma finsum_partition:
   assumes "finite S"
   assumes "f \<in> S \<rightarrow> carrier R"
   assumes "T \<subseteq> S"
   shows "finsum  R f S  = finsum R f T  \<oplus> finsum  R f (S - T) "
 proof-
   have "\<And>U. finite U \<Longrightarrow> U \<subseteq> S \<longrightarrow> finsum  R f S  = finsum R f U  \<oplus> finsum  R f (S - U) "
   proof-
-    fix U 
+    fix U
     show "finite U \<Longrightarrow> U \<subseteq> S \<longrightarrow> finsum R f S = finsum R f U \<oplus> finsum R f (S - U) "
     apply(erule finite.induct)
        apply (metis Diff_empty assms(2) finsum_closed finsum_empty l_zero)
     proof
       fix A :: "'c set" fix a
       assume A0: "finite A"
       show "A \<subseteq> S \<longrightarrow> finsum R f S = finsum R f A \<oplus> finsum R f (S - A) \<Longrightarrow> insert a A \<subseteq> S \<Longrightarrow> finsum R f S = finsum R f (insert a A) \<oplus> finsum R f (S - insert a A)"
 
       proof-
         assume A1: "A \<subseteq> S \<longrightarrow> finsum R f S = finsum R f A \<oplus> finsum R f (S - A)"
         assume A2: "insert a A \<subseteq> S"
         show "finsum R f S = finsum R f (insert a A) \<oplus> finsum R f (S - insert a A)"
           apply(cases "a \<in> A")
            apply (metis A1 A2 insert_absorb)
         proof-
           assume A3: "a \<notin> A"
           have A4: "f a \<in> carrier R"
             by (metis A2 Pi_iff assms(2) insert_subset)
           have A5: " finsum R f (insert a A) = f a \<oplus> finsum R f A"
 
-            using A0 A1 A2 finsum_insert[of A a f] assms  A3 
+            using A0 A1 A2 finsum_insert[of A a f] assms  A3
             by blast
           have A6: "a \<in> S"
             using A2 by blast
           have A7: "finsum R f S \<in> carrier R"
             using assms(2) finsum_closed by blast
           have A8: " finsum R f (S - A)\<in> carrier R"
             using Diff_subset[of S A] Pi_iff assms(2) finsum_closed[of f] subsetD[of _ S ]
             by (meson Pi_anti_mono in_mono)
           have A9: "finsum R f A \<in> carrier R"
             by (meson A2 Pi_anti_mono assms(2) finsum_closed insert_subset subsetD)
           have A10: "finsum R f A =  finsum R f S \<ominus> finsum R f (S - A)"
-            using A7 A8 A9 
+            using A7 A8 A9
             by (metis A1 A2 add.inv_solve_right' insert_subset minus_eq)
           have A11: " finsum R f (insert a A) = f a \<oplus> (finsum R f S \<ominus> finsum R f (S - A))"
             using A5 A6 A1 A2 assms  A10
             by presburger
           then have A12: " finsum R f (insert a A) =finsum R f S \<oplus>( f a  \<ominus> finsum R f (S - A))"
             using A4 A7 A8 add.inv_closed add.m_lcomm minus_eq by presburger
           have A13: "finsum R f (insert a A) \<in> carrier R"
-            using A4 A5 A9 add.m_closed 
+            using A4 A5 A9 add.m_closed
             by presburger
           have A14: " finsum R f (S - insert a A) \<in> carrier R"
             by (meson Diff_subset Pi_anti_mono assms(2) finsum_closed in_mono)
           have A15: "finsum R f S = finsum R f (insert a A) \<ominus> ( f a  \<ominus> finsum R f (S - A)) "
             by (metis A12 A13 A4 A7 A8 add.inv_solve_right' minus_closed minus_eq)
           have A16: "finsum R f S = finsum R f (insert a A) \<oplus> finsum R f (S - A) \<ominus> f a"
             using  A1 A2 A4 A5 A8 A9 add.inv_closed add.m_assoc add.m_comm insert_subset minus_closed minus_eq r_neg1
-            unfolding a_minus_def 
+            unfolding a_minus_def
             by (metis add.m_closed)
           have A16: "finsum R f S = finsum R f (insert a A) \<oplus> (finsum R f (S - A) \<ominus> f a)"
-            unfolding a_minus_def 
+            unfolding a_minus_def
             using A13 A16 A4 A8 add.inv_closed add.m_assoc minus_eq by presburger
           have A17: "(finsum R f (S - A) \<ominus> f a) = finsum R f (S - insert a A) "
           proof-
             have A170: "(S - A) = insert a (S - insert a A)"
               using A3 A6 by blast
             have A171: "a \<notin> S - insert a A"
               by blast
             then have "finsum R f (S - A)  =(f a) \<oplus> finsum R f (S - insert a A) "
-              using A170 finsum_insert[of "(S - insert a A)" a f] 
-              by (metis A4 Diff_subset Pi_anti_mono assms(1) assms(2)  
+              using A170 finsum_insert[of "(S - insert a A)" a f]
+              by (metis A4 Diff_subset Pi_anti_mono assms(1) assms(2)
                   rev_finite_subset subsetD )
-            then show ?thesis 
-              by (metis A1 A13 A14 A16 A2 A4 A5 A8 A9 add.l_cancel 
+            then show ?thesis
+              by (metis A1 A13 A14 A16 A2 A4 A5 A8 A9 add.l_cancel
                   add.m_assoc add.m_comm insert_subset minus_closed)
           qed
-          then show "finsum R f S = finsum R f (insert a A) \<oplus> finsum R f (S - insert a A) "  
+          then show "finsum R f S = finsum R f (insert a A) \<oplus> finsum R f (S - insert a A) "
             using A16
             by presburger
         qed
       qed
     qed
   qed
-  then show ?thesis 
+  then show ?thesis
     by (meson assms(1) assms(3) rev_finite_subset)
 qed
 
 lemma finsum_eq_parition:
   assumes "finite S"
   assumes "f \<in> S \<rightarrow> carrier R"
   assumes "T \<subseteq> S"
   assumes "\<And>x. x \<in> S - T \<Longrightarrow> f x = \<zero>"
   shows "finsum  R f S  = finsum R f T"
-  using assms 
+  using assms
   by (metis add.finprod_mono_neutral_cong_right)
 
 lemma poly_eval_scalar_mult:
   assumes "k \<in> carrier R"
   assumes "closed_fun R g"
   assumes "P \<in> Pring_set R I"
-  shows "poly_eval R  S g (poly_scalar_mult R k P)= 
+  shows "poly_eval R  S g (poly_scalar_mult R k P)=
          (poly_scalar_mult R k (poly_eval R S g P))"
 proof
   fix m
   show "poly_eval R S g (poly_scalar_mult R k P) m = poly_scalar_mult R k (poly_eval R S g  P) m"
-    unfolding poly_eval_def poly_scalar_mult_def 
+    unfolding poly_eval_def poly_scalar_mult_def
   proof-
-    have 0: "(\<Oplus>n\<in>monomials_reducing_to R m P S. monom_eval R (restrict_to_indices n S) g \<otimes> (k \<otimes> P n)) = 
+    have 0: "(\<Oplus>n\<in>monomials_reducing_to R m P S. monom_eval R (restrict_to_indices n S) g \<otimes> (k \<otimes> P n)) =
               (\<Oplus>n\<in>monomials_reducing_to R m (\<lambda>m. k \<otimes> P m) S. monom_eval R (restrict_to_indices n S) g \<otimes> (k \<otimes> P n)) "
     proof-
       have 00: "monomials_reducing_to R m (\<lambda>m. k \<otimes> P m) S \<subseteq> monomials_reducing_to R m P S"
       proof
         fix x show " x \<in> monomials_reducing_to R m (\<lambda>m. k \<otimes> P m) S \<Longrightarrow> x \<in> monomials_reducing_to R m P S"
-        unfolding monomials_reducing_to_def 
+        unfolding monomials_reducing_to_def
         using assms  assms(1) monomials_ofE complement_of_monomials_of  r_null[of k]
         by (metis (no_types, lifting) mem_Collect_eq)
        have 01: "(\<lambda>n. monom_eval R (restrict_to_indices n S) g \<otimes> (k \<otimes> P n)) \<in> monomials_reducing_to R m P S \<rightarrow> carrier R"
-        by (smt Pi_I Pring_cfs_closed assms(1) assms(2) assms(3) closed_fun_simp m_closed monom_eval_car)        
+        by (smt Pi_I Pring_cfs_closed assms(1) assms(2) assms(3) closed_fun_simp m_closed monom_eval_car)
        have 02: "finite (monomials_reducing_to R m P S)"
-        using assms(3) monomials_reducing_to_finite 
+        using assms(3) monomials_reducing_to_finite
         by blast
        have 03: "  (\<And>x. x \<in> monomials_reducing_to R m P S - monomials_reducing_to R m (\<lambda>m. k \<otimes> P m) S \<Longrightarrow>
           monom_eval R (restrict_to_indices x S) g \<otimes> (k \<otimes> P x) = \<zero>)"
        proof-
         fix x
         assume A: "x \<in> monomials_reducing_to R m P S - monomials_reducing_to R m (\<lambda>m. k \<otimes> P m) S "
         have "x \<notin> monomials_of R ((\<lambda>m. k \<otimes> P m))"
         proof
           assume "x \<in> monomials_of R (\<lambda>m. k \<otimes> P m)"
           then have "x \<in>  monomials_reducing_to R m (\<lambda>m. k \<otimes> P m) S"
             using A
-            unfolding monomials_reducing_to_def 
+            unfolding monomials_reducing_to_def
             by blast
-          then show False 
+          then show False
             using A by blast
         qed
         then show "monom_eval R (restrict_to_indices x S) g \<otimes> (k \<otimes> P x) = \<zero>"
-          by (metis assms(2) complement_of_monomials_of monom_eval_car r_null)          
+          by (metis assms(2) complement_of_monomials_of monom_eval_car r_null)
        qed
       qed
       have 01: " (\<And>x. x \<in> monomials_reducing_to R m P S - monomials_reducing_to R m (\<lambda>m. k \<otimes> P m) S \<Longrightarrow> monom_eval R (restrict_to_indices x S) g \<otimes> (k \<otimes> P x) = \<zero>)"
       proof- fix x assume A: "x \<in> monomials_reducing_to R m P S - monomials_reducing_to R m (\<lambda>m. k \<otimes> P m) S"
         hence "x \<notin> monomials_reducing_to R m (\<lambda>m. k \<otimes> P m) S"
         by blast
-         hence "(k \<otimes> P x) = \<zero>" unfolding monomials_reducing_to_def 
-        by (metis (no_types, lifting) A DiffD1 complement_of_monomials_of mem_Collect_eq monomials_reducing_to_def)      
+         hence "(k \<otimes> P x) = \<zero>" unfolding monomials_reducing_to_def
+        by (metis (no_types, lifting) A DiffD1 complement_of_monomials_of mem_Collect_eq monomials_reducing_to_def)
         thus "monom_eval R (restrict_to_indices x S) g \<otimes> (k \<otimes> P x) = \<zero>"
         using monom_eval_car[of g]  assms(2) r_null by presburger
       qed
       have 02: "finite (monomials_reducing_to R m P S)"
         using assms(3) monomials_reducing_to_finite by blast
       have 04: "(\<lambda>n. monom_eval R (restrict_to_indices n S) g \<otimes> (k \<otimes> P n)) \<in> monomials_reducing_to R m P S \<rightarrow> carrier R"
-        by (smt Pi_I assms(1) assms(2) assms(3) closed_fun_simp cring.axioms(1) is_cring m_closed monom_eval_car ring.Pring_cfs_closed)      
+        by (smt Pi_I assms(1) assms(2) assms(3) closed_fun_simp cring.axioms(1) is_cring m_closed monom_eval_car ring.Pring_cfs_closed)
       show ?thesis
         using 00 01 02 04
-            finsum_eq_parition[of "monomials_reducing_to R m P S" 
-                                "(\<lambda>n. monom_eval R (restrict_to_indices n S) g \<otimes> (k \<otimes> P n))" 
-                                "monomials_reducing_to R m (\<lambda>m. k \<otimes> P m) S"]    
+            finsum_eq_parition[of "monomials_reducing_to R m P S"
+                                "(\<lambda>n. monom_eval R (restrict_to_indices n S) g \<otimes> (k \<otimes> P n))"
+                                "monomials_reducing_to R m (\<lambda>m. k \<otimes> P m) S"]
         by blast
     qed
     then have 1: " (\<Oplus>n\<in>monomials_reducing_to R m (\<lambda>m. k \<otimes> P m) S. monom_eval R (restrict_to_indices n S) g \<otimes> (k \<otimes> P n))
        = (\<Oplus>n\<in>monomials_reducing_to R m P S. k \<otimes> (monom_eval R (restrict_to_indices n S) g \<otimes> (P n)))      "
     proof-
       have "\<And>n. monom_eval R (restrict_to_indices n S) g \<otimes> (k \<otimes> P n) = k \<otimes> (monom_eval R (restrict_to_indices n S) g \<otimes> (P n))"
         by (metis Pring_cfs_closed assms(1) assms(2) assms(3) m_lcomm monom_eval_car)
-      then show ?thesis 
-        using 0 
+      then show ?thesis
+        using 0
         by presburger
     qed
     show " (\<Oplus>n\<in>monomials_reducing_to R m (\<lambda>m. k \<otimes> P m) S. monom_eval R (restrict_to_indices n S) g \<otimes> (k \<otimes> P n))
-       = k \<otimes>(\<Oplus>n\<in>monomials_reducing_to R m P S.  monom_eval R (restrict_to_indices n S) g \<otimes> (P n))      "   
+       = k \<otimes>(\<Oplus>n\<in>monomials_reducing_to R m P S.  monom_eval R (restrict_to_indices n S) g \<otimes> (P n))      "
     proof-
       have "(\<lambda>n. monom_eval R (restrict_to_indices n S) g \<otimes> (P n)) \<in> (monomials_reducing_to R m P S) \<rightarrow> carrier R"
         by (meson Pi_I Pring_cfs_closed assms(2) assms(3) m_closed monom_eval_car)
       then have "k \<otimes> (\<Oplus>i\<in>monomials_reducing_to R m P S. monom_eval R (restrict_to_indices i S) g \<otimes> P i) =
       (\<Oplus>i\<in>monomials_reducing_to R m P S. k \<otimes> (monom_eval R (restrict_to_indices i S) g \<otimes> P i))"
-        using finsum_rdistr[of "monomials_reducing_to R m P S" 
-                                  k 
-                                  "(\<lambda>n. monom_eval R (restrict_to_indices n S) g \<otimes> (P n))"] 
+        using finsum_rdistr[of "monomials_reducing_to R m P S"
+                                  k
+                                  "(\<lambda>n. monom_eval R (restrict_to_indices n S) g \<otimes> (P n))"]
               assms  monomials_reducing_to_finite by blast
       then show ?thesis
-        using 1 
+        using 1
         by presburger
     qed
   qed
 qed
 
 lemma poly_eval_monomial:
   assumes "closed_fun R g"
   assumes "\<one> \<noteq>\<zero>"
   shows "poly_eval R  S g (mset_to_IP R m)
            = poly_scalar_mult R (monom_eval R (restrict_to_indices m S) g)
                               (mset_to_IP R (remove_indices m S))"
 proof
   have 0: "monomials_of R (mset_to_IP R m) = {m}"
-    using assms  monomials_of_mset_to_IP 
+    using assms  monomials_of_mset_to_IP
     by blast
 
   fix x
   show "poly_eval R  S g (mset_to_IP R m) x =
          poly_scalar_mult R (monom_eval R (restrict_to_indices m S) g)
                            (mset_to_IP R (remove_indices m S)) x "
   proof(cases "x = (remove_indices m S)")
     case True
     have "monomials_reducing_to R x (mset_to_IP R m) S = {m}"
-      unfolding monomials_reducing_to_def 
+      unfolding monomials_reducing_to_def
       using True 0
       by auto
     then have "poly_eval R  S g (mset_to_IP R m) x = monom_eval R (restrict_to_indices m S) g \<otimes> (mset_to_IP R m m)"
-      unfolding poly_eval_def 
-      by (metis (mono_tags, lifting) assms(1) finsum_singleton monom_eval_car mset_to_IP_simp r_one)      
-    then show ?thesis 
-      by (metis True mset_to_IP_simp poly_scalar_mult_def)    
+      unfolding poly_eval_def
+      by (metis (mono_tags, lifting) assms(1) finsum_singleton monom_eval_car mset_to_IP_simp r_one)
+    then show ?thesis
+      by (metis True mset_to_IP_simp poly_scalar_mult_def)
   next
     case False
     then have "monomials_reducing_to R x (mset_to_IP R m) S = {}"
-      unfolding monomials_reducing_to_def 
-      using 0 
+      unfolding monomials_reducing_to_def
+      using 0
       by auto
     then have "poly_eval R S g (mset_to_IP R m) x = \<zero>"
-      unfolding poly_eval_def 
+      unfolding poly_eval_def
       by (metis finsum_empty)
-    then show ?thesis 
-      using False 
+    then show ?thesis
+      using False
       by (metis assms(1) monom_eval_car mset_to_IP_simp' poly_scalar_mult_def r_null)
   qed
 qed
 
 
 lemma(in cring) poly_eval_monomial_closed:
   assumes "closed_fun R g"
   assumes "\<one> \<noteq>\<zero>"
   assumes "set_mset m \<subseteq> I"
   shows "poly_eval R S g (mset_to_IP R m) \<in> Pring_set R (I - S)"
 proof-
   have "(mset_to_IP R (remove_indices m S)) \<in> Pring_set R (I - S)"
     using assms  mset_to_IP_closed[of "(remove_indices m S)" "I - S"]
     by (metis remove_indices_indices)
   then show ?thesis
     using assms poly_eval_monomial[of g S m  ]
           poly_scalar_mult_closed[of "(monom_eval R (restrict_to_indices m S) g)"
-                                        "(mset_to_IP R (remove_indices m S))"] 
+                                        "(mset_to_IP R (remove_indices m S))"]
     by (metis monom_eval_car)
 qed
 
 lemma poly_scalar_mult_iter:
   assumes "\<one> \<noteq>\<zero>"
   assumes "P \<in> Pring_set R I"
   assumes "k \<in> carrier R"
   assumes "n \<in> carrier R"
   shows "poly_scalar_mult R k (poly_scalar_mult R n P) = poly_scalar_mult R (k \<otimes> n) P"
-  using assms 
-  unfolding poly_scalar_mult_def 
+  using assms
+  unfolding poly_scalar_mult_def
   by (metis Pring_cfs_closed m_assoc)
 
 lemma poly_scalar_mult_comm:
   assumes "\<one> \<noteq>\<zero>"
   assumes "P \<in> Pring_set R I"
   assumes "a \<in> carrier R"
   assumes "b \<in> carrier R"
   shows "poly_scalar_mult R a (poly_scalar_mult R b P) = poly_scalar_mult R b (poly_scalar_mult R a P)"
   using assms  poly_scalar_mult_iter m_comm[of a b]
   by metis
 
 lemma poly_eval_monomial_term:
   assumes "closed_fun R g"
   assumes "\<one> \<noteq>\<zero>"
   assumes "set_mset m \<subseteq> I"
   assumes "k \<in> carrier R"
   shows "poly_eval R  S g (poly_scalar_mult R k (mset_to_IP R m)) = poly_scalar_mult R (k\<otimes>(monom_eval R (restrict_to_indices m S) g))
                               (mset_to_IP R (remove_indices m S))"
 proof-
   have 0: "poly_eval R  S g (poly_scalar_mult R k (mset_to_IP R m)) =
           poly_scalar_mult R k (poly_eval R S g (mset_to_IP R m))"
-    using assms poly_eval_scalar_mult[of k g "mset_to_IP R m" I S] mset_to_IP_closed 
+    using assms poly_eval_scalar_mult[of k g "mset_to_IP R m" I S] mset_to_IP_closed
     by blast
   have 1: "poly_eval R  S g (poly_scalar_mult R k (mset_to_IP R m)) =
            poly_scalar_mult R k (poly_scalar_mult R (monom_eval R (restrict_to_indices m S) g)
                               (mset_to_IP R (remove_indices m S))) "
-    using 0 assms 
+    using 0 assms
     by (metis poly_eval_monomial)
   have 2: "mset_to_IP R (remove_indices m S) \<in> Pring_set R I"
-    using assms mset_to_IP_closed 
+    using assms mset_to_IP_closed
     by (metis Diff_subset remove_indices_def restrict_to_indicesE subset_iff)
   have 3: "monom_eval R (restrict_to_indices m S) g \<in> carrier R"
     using assms monom_eval_car by blast
-  show ?thesis 
+  show ?thesis
     using 1 2 3 assms   poly_scalar_mult_iter[of  "mset_to_IP R (remove_indices m S)" I k "(monom_eval R (restrict_to_indices m S) g)"]
   by presburger
 qed
 
 lemma poly_eval_monomial_term_closed:
   assumes "closed_fun R g"
   assumes "\<one> \<noteq>\<zero>"
   assumes "set_mset m \<subseteq> I"
   assumes "k \<in> carrier R"
   shows "poly_eval R S g (poly_scalar_mult R k (mset_to_IP R m)) \<in> Pring_set R (I - S)"
 proof-
   have "(mset_to_IP R (remove_indices m S)) \<in> Pring_set R (I - S)"
-    using assms 
+    using assms
     by (meson remove_indices_indices mset_to_IP_closed)
-  then show ?thesis 
+  then show ?thesis
     using assms poly_eval_monomial_term remove_indices_indices[of m I S]
     by (metis cring.cring_simprules(5) cring.monom_eval_car is_cring poly_scalar_mult_closed)
 qed
 
 lemma finsum_split:
   assumes "finite S"
   assumes "f \<in> S \<rightarrow> carrier R"
   assumes "g \<in> S \<rightarrow> carrier R"
   assumes "k \<in> carrier  R"
   assumes "c \<in> S"
   assumes "\<And>s. s \<in> S \<and> s \<noteq> c \<Longrightarrow> f s = g s"
   assumes "g c = f c \<oplus> k"
   shows "finsum R g S = k \<oplus> finsum R f S"
 proof-
   have 0: "finsum R f S = f c \<oplus> finsum R f (S - {c})"
   proof-
     have "f \<in> S - {c} \<rightarrow> carrier R"
       by (metis Pi_split_insert_domain assms(2) assms(5) insert_Diff)
-    then show ?thesis  
-    using assms finsum_insert[of "S - {c}" c f] 
+    then show ?thesis
+    using assms finsum_insert[of "S - {c}" c f]
     by (metis DiffD2 Pi_iff finite_Diff insert_Diff singletonI)
   qed
   have 1: "finsum R g S = g c \<oplus> finsum R g (S - {c})"
   proof-
     have "g \<in> S - {c} \<rightarrow> carrier R"
       by (metis Pi_split_insert_domain  assms(3) assms(5) insert_Diff)
-    then show ?thesis  
-    using assms finsum_insert[of "S - {c}" c g] 
+    then show ?thesis
+    using assms finsum_insert[of "S - {c}" c g]
     by (metis DiffD2 Pi_iff finite_Diff insert_Diff singletonI)
   qed
   have "finsum R f (S - {c}) = finsum R g (S - {c})"
-    using assms Diff_iff Pi_split_insert_domain finsum_cong'[of "S- {c}" "S - {c}" g f] 
-      insert_Diff singletonI 
+    using assms Diff_iff Pi_split_insert_domain finsum_cong'[of "S- {c}" "S - {c}" g f]
+      insert_Diff singletonI
     by blast
   then have "finsum R g S = f c \<oplus> k \<oplus> finsum R g (S - {c})"
-    using assms  \<open>finsum R g S = g c \<oplus> finsum R g (S - {c})\<close> 
+    using assms  \<open>finsum R g S = g c \<oplus> finsum R g (S - {c})\<close>
     by presburger
   then have 1: "finsum R g S = f c \<oplus> k \<oplus> finsum R f (S - {c})"
     using \<open>finsum R f (S - {c}) = finsum R g (S - {c})\<close> by presburger
   have "finsum R g S = k \<oplus> ( f c \<oplus> finsum R f (S - {c}))"
-  proof- 
+  proof-
     have "f c \<in> carrier R"
       by (metis PiE assms(2) assms(5))
     have "finsum R f (S - {c}) \<in> carrier R"
       by (metis Pi_split_insert_domain assms(2) assms(5) finsum_closed insert_Diff)
-    then show ?thesis using assms(4) 1 
-      using \<open>f c \<in> carrier R\<close> add.m_assoc add.m_lcomm 
+    then show ?thesis using assms(4) 1
+      using \<open>f c \<in> carrier R\<close> add.m_assoc add.m_lcomm
       by presburger
   qed
-  then show ?thesis 
-    using "0" 
+  then show ?thesis
+    using "0"
     by presburger
-qed 
+qed
 
 lemma poly_monom_induction:
 assumes "P (indexed_const \<zero>)"
 assumes "\<And>m k. set_mset m \<subseteq> I \<and> k \<in> carrier R \<Longrightarrow> P (poly_scalar_mult R k (mset_to_IP R m))"
 assumes "\<And>Q m k. Q \<in> Pring_set R I \<and> (P Q) \<and> set_mset m \<subseteq> I \<and> k \<in> carrier R\<Longrightarrow> P (Q \<Oplus> (poly_scalar_mult R k (mset_to_IP R m)))"
 shows "\<And>Q. Q \<in> Pring_set R I \<Longrightarrow> P Q"
 proof-
   have 0: "\<And>Ms. finite Ms \<Longrightarrow> (\<forall> Q \<in> Pring_set R I. monomials_of R Q = Ms \<longrightarrow> P Q)"
   proof-
     fix Ms
     show " finite Ms \<Longrightarrow> (\<forall> Q \<in> Pring_set R I. monomials_of R Q = Ms \<longrightarrow> P Q)"
     proof(erule finite.induct)
       show "\<forall>Q\<in>Pring_set R I. monomials_of R Q = {} \<longrightarrow> P Q"
       proof
         fix Q
         assume "Q \<in> Pring_set R I"
         show "monomials_of R Q = {} \<longrightarrow> P Q "
-          using assms 
+          using assms
           by (metis \<open>Q \<in> Pring_set R I\<close> card_0_eq monomials_finite monomials_of_card_zero)
       qed
-      show "\<And>A a. finite A \<Longrightarrow> \<forall>Q\<in>Pring_set R I. monomials_of R Q = A \<longrightarrow> P Q \<Longrightarrow> 
+      show "\<And>A a. finite A \<Longrightarrow> \<forall>Q\<in>Pring_set R I. monomials_of R Q = A \<longrightarrow> P Q \<Longrightarrow>
                     \<forall>Q\<in>Pring_set R I. monomials_of R Q = insert a A \<longrightarrow> P Q"
       proof
         fix A :: "'c monomial set" fix  a fix  Q
         assume A0: "finite A"
         assume A1: "\<forall>Q\<in>Pring_set R I. monomials_of R Q = A \<longrightarrow> P Q"
         assume A2: " Q \<in> Pring_set R I "
         show "monomials_of R Q = insert a A \<longrightarrow> P Q"
         proof
           assume A3: "monomials_of R Q = insert a A"
           show "P Q"
             apply(cases "a \<in> A")
              apply (metis A1 A2 A3 insert_absorb)
           proof-
             assume A4: "a \<notin> A"
             show "P Q"
             proof-
               have A5: "set_mset a \<subseteq> I"
                 by (metis A2 A3 insert_iff mset_to_IP_indices)
               have A6: "set_mset a \<subseteq> I \<and> Q a \<in> carrier R"
                 using A2 A5 Pring_cfs_closed by blast
               obtain Q' where Q'_def: "Q' = remove_monom R a Q"
                 by simp
               then have "Q = Q' \<Oplus> poly_scalar_mult R (Q a) (mset_to_IP R a)"
                 using A2 cring.remove_monom_eq is_cring by blast
               then show ?thesis using A6 assms
-                by (metis A1 A2 A3 A4 Diff_empty Diff_insert0 Q'_def insert_Diff1 
+                by (metis A1 A2 A3 A4 Diff_empty Diff_insert0 Q'_def insert_Diff1
                     remove_monom_closed remove_monom_monomials singletonI)
             qed
           qed
         qed
       qed
     qed
   qed
   show "\<And>Q. Q \<in> Pring_set R I \<Longrightarrow> P Q"
   proof-
     fix Q
     assume "Q \<in> Pring_set R I"
     show "P Q"
-      using 0[of "monomials_of R Q"]  \<open>Q \<in> Pring_set R I\<close> monomials_finite 
+      using 0[of "monomials_of R Q"]  \<open>Q \<in> Pring_set R I\<close> monomials_finite
       by blast
   qed
 qed
 
 lemma Pring_car_induct:
   assumes "q \<in> carrier (Pring R I)"
   assumes "P \<zero>\<^bsub>Pring R I\<^esub>"
   assumes "\<And>m k. set_mset m \<subseteq> I \<and> k \<in> carrier R \<Longrightarrow> P (k \<odot>\<^bsub>Pring R I\<^esub>(mset_to_IP R m))"
   assumes "\<And>Q m k. Q \<in> carrier (Pring R I) \<and> (P Q) \<and> set_mset m \<subseteq> I \<and> k \<in> carrier R\<Longrightarrow>
              P (Q \<Oplus> (k \<odot>\<^bsub>Pring R I\<^esub> (mset_to_IP R m)))"
   shows "P q"
   using poly_monom_induction[of P I q] assms Pring_smult[of I] Pring_car[of I] Pring_zero
-  by metis 
+  by metis
 
 lemma poly_monom_induction2:
 assumes "P (indexed_const \<zero>)"
 assumes "\<And>m k. set_mset m \<subseteq> I \<and> k \<in> carrier R \<Longrightarrow> P (poly_scalar_mult R k (mset_to_IP R m))"
 assumes "\<And>Q m k. Q \<in> Pring_set R I \<and> (P Q) \<and> set_mset m \<subseteq> I \<and> k \<in> carrier R \<Longrightarrow> P (Q \<Oplus> (poly_scalar_mult R k (mset_to_IP R m)))"
 assumes "Q \<in>  Pring_set R I"
 shows "P Q"
   using assms(1) assms(2) assms(3) assms(4) poly_monom_induction by blast
 
 lemma poly_monom_induction3:
 assumes "Q \<in>  Pring_set R I"
 assumes "P (indexed_const \<zero>)"
 assumes "\<And>m k. set_mset m \<subseteq> I \<and> k \<in> carrier R \<Longrightarrow> P (poly_scalar_mult R k (mset_to_IP R m))"
 assumes "\<And>p q. p \<in> Pring_set R I \<Longrightarrow> (P p) \<Longrightarrow> q \<in> Pring_set R I \<Longrightarrow> (P q) \<Longrightarrow>  P (p \<Oplus> q)"
 shows "P Q"
   apply(rule poly_monom_induction2[of _  I])
   using assms(2) apply blast
   using assms(3) apply blast
   apply (meson assms(3) assms(4) mset_to_IP_closed poly_scalar_mult_closed)
   using assms(1) by blast
 
 lemma Pring_car_induct':
 assumes "Q \<in> carrier (Pring R I)"
 assumes "P \<zero>\<^bsub>Pring R I\<^esub>"
 assumes "\<And>m k. set_mset m \<subseteq> I \<and> k \<in> carrier R \<Longrightarrow> P (k \<odot>\<^bsub>Pring R I\<^esub> mset_to_IP R m)"
 assumes "\<And>p q. p \<in> carrier (Pring R I) \<Longrightarrow> (P p) \<Longrightarrow> q \<in> carrier (Pring R I) \<Longrightarrow> (P q) \<Longrightarrow>  P (p \<oplus>\<^bsub>Pring R I\<^esub> q)"
 shows "P Q"
-  using poly_monom_induction3[of Q I P] assms Pring_smult Pring_add Pring_zero Pring_car 
-  by metis 
+  using poly_monom_induction3[of Q I P] assms Pring_smult Pring_add Pring_zero Pring_car
+  by metis
 
 lemma poly_eval_mono:
   assumes "P \<in> Pring_set R I"
   assumes "closed_fun R g"
   assumes "finite F"
   assumes "monomials_reducing_to R m P S \<subseteq> F"
   assumes "\<And>n. n \<in> F \<Longrightarrow> remove_indices n S = m"
   shows "poly_eval R S g P m = (\<Oplus>n\<in> F. monom_eval R (restrict_to_indices n S) g \<otimes> P n)"
 proof-
-  have 0: "(\<Oplus>n\<in> F. monom_eval R (restrict_to_indices n S) g \<otimes> P n)= 
+  have 0: "(\<Oplus>n\<in> F. monom_eval R (restrict_to_indices n S) g \<otimes> P n)=
           (\<Oplus>n\<in> F - (monomials_reducing_to R m P S). monom_eval R (restrict_to_indices n S) g \<otimes> P n) \<oplus> poly_eval R S g P m"
   proof-
     have 00: "  (\<lambda>n. monom_eval R (restrict_to_indices n S) g \<otimes> P n) \<in> F \<rightarrow> carrier R"
       by (meson Pi_I Pring_cfs_closed assms(1) assms(2) m_closed monom_eval_car)
     have 01: "monomials_reducing_to R m P S \<subseteq> F"
       by (simp add: assms(4))
     have 02: "(\<Oplus>n\<in>F. monom_eval R (restrict_to_indices n S) g \<otimes> P n) =
     (\<Oplus>n\<in>monomials_reducing_to R m P S. monom_eval R (restrict_to_indices n S) g \<otimes> P n) \<oplus>
     (\<Oplus>n\<in>F - monomials_reducing_to R m P S. monom_eval R (restrict_to_indices n S) g \<otimes> P n)"
       using "00" "01" assms(3) finsum_partition by blast
     have 03: " (\<Oplus>n\<in>F - monomials_reducing_to R m P S. monom_eval R (restrict_to_indices n S) g \<otimes> P n)\<in> carrier R"
       by (metis (mono_tags, lifting) "00" DiffD1 PiE Pi_I finsum_closed)
     have "    (\<Oplus>n\<in>monomials_reducing_to R m P S. monom_eval R (restrict_to_indices n S) g \<otimes> P n) \<in> carrier R"
     proof -
       have "(\<lambda>m. monom_eval R (restrict_to_indices m S) g \<otimes> P m) \<in> monomials_reducing_to R m P S \<rightarrow> carrier R"
         using "00" "01" by blast
       then show ?thesis
         using finsum_closed by blast
     qed
-    then show ?thesis 
-      unfolding poly_eval_def 
+    then show ?thesis
+      unfolding poly_eval_def
       using  00 01 02 03  add.m_comm
       by presburger
   qed
   have "(\<Oplus>n\<in> F - (monomials_reducing_to R m P S). monom_eval R (restrict_to_indices n S) g \<otimes> P n) = \<zero>"
   proof-
     have "\<And>n. n \<in> F - (monomials_reducing_to R m P S) \<Longrightarrow>  monom_eval R (restrict_to_indices n S) g \<otimes> P n = \<zero>"
     proof-
       fix n
       assume A: "n \<in>  F - (monomials_reducing_to R m P S)"
       have "n \<notin> monomials_of R P"
       proof
         assume "n \<in> monomials_of R P"
-        then have "n \<in> (monomials_reducing_to R m P S)" 
+        then have "n \<in> (monomials_reducing_to R m P S)"
           unfolding monomials_reducing_to_def
           using assms(5) A
           by blast
-        then show False using A by blast 
+        then show False using A by blast
       qed
       then show " monom_eval R (restrict_to_indices n S) g \<otimes> P n = \<zero>"
         by (metis assms(2) monom_eval_car complement_of_monomials_of r_null)
     qed
-    then show ?thesis 
+    then show ?thesis
       by (meson add.finprod_one_eqI)
   qed
   then have 1: "(\<Oplus>n\<in>F. monom_eval R (restrict_to_indices n S) g \<otimes> P n) =
     \<zero> \<oplus> poly_eval R S g P m"
-    using 0  Pi_I Pring_cfs_closed add.r_cancel_one assms(1) assms(2) 
+    using 0  Pi_I Pring_cfs_closed add.r_cancel_one assms(1) assms(2)
     by presburger
   have " (\<lambda>n. monom_eval R (restrict_to_indices n S) g \<otimes> P n) \<in> monomials_reducing_to R m P S \<rightarrow> carrier R"
     by (meson Pi_I Pring_cfs_closed assms(1) assms(2) m_closed monom_eval_car)
   hence "poly_eval R S g P m \<in> carrier R"
     using assms
-    unfolding poly_eval_def 
-    using finsum_closed[of "\<lambda> n. monom_eval R (restrict_to_indices n S) g \<otimes> P n" 
+    unfolding poly_eval_def
+    using finsum_closed[of "\<lambda> n. monom_eval R (restrict_to_indices n S) g \<otimes> P n"
                           "monomials_reducing_to R m P S"]
-    by (meson Pi_I Pring_cfs_closed m_closed monom_eval_car)    
+    by (meson Pi_I Pring_cfs_closed m_closed monom_eval_car)
   then have "(\<Oplus>n\<in>F. monom_eval R (restrict_to_indices n S) g \<otimes> P n) = poly_eval R S g P m"
-    using "1" add.r_cancel_one zero_closed 
+    using "1" add.r_cancel_one zero_closed
     by presburger
-  then show ?thesis 
+  then show ?thesis
     by presburger
 qed
 
 lemma finsum_group:
   assumes "\<And>n. f n \<in> carrier R"
   assumes "\<And>n. g n \<in>  carrier R"
   shows "finite S \<Longrightarrow> finsum R f S \<oplus> finsum R g S = finsum R (\<lambda>n. f n \<oplus> g n) S "
   apply(erule finite.induct)
   apply (metis finsum_empty r_zero zero_closed)
 proof-
   fix A :: "'c set "
   fix a
   assume A0: "finite A"
   assume A1: "finsum R f A \<oplus> finsum R g A = (\<Oplus>n\<in>A. f n \<oplus> g n)"
   show "finsum R f (insert a A) \<oplus> finsum R g (insert a A) = (\<Oplus>n\<in>insert a A. f n \<oplus> g n)"
   proof(cases "a \<in> A")
     case True
-    then show ?thesis 
-      by (metis A1 insert_absorb)   
+    then show ?thesis
+      by (metis A1 insert_absorb)
   next
     case False
-    have LHS: "finsum R f (insert a A) \<oplus> finsum R g (insert a A) = 
+    have LHS: "finsum R f (insert a A) \<oplus> finsum R g (insert a A) =
                   (f a \<oplus> finsum R f A) \<oplus> (g a \<oplus> finsum R g A)"
       using assms  finsum_insert[of A a f] finsum_insert[of A a g]
       by (metis A0 False Pi_I)
     have F0: "(\<lambda>n. f n \<oplus> g n) \<in> A \<rightarrow> carrier R"
-      using assms 
+      using assms
       by blast
     have F1: "(f a \<oplus> g a) \<in> carrier  R"
-      using assms 
+      using assms
       by blast
     have RHS: " (\<Oplus>n\<in>insert a A. f n \<oplus> g n) = (f a \<oplus> g a) \<oplus> (\<Oplus>n\<in>A. f n \<oplus> g n)"
-      using F0 F1 assms finsum_insert[of A a " (\<lambda>n. f n \<oplus> g n)"] False A0 
+      using F0 F1 assms finsum_insert[of A a " (\<lambda>n. f n \<oplus> g n)"] False A0
       by blast
     have F2: " f a \<oplus> finsum R f A \<oplus> (g a \<oplus> finsum R g A) = (f a \<oplus> g a) \<oplus> (finsum R f A \<oplus>  finsum R g A)"
     proof-
       have F20: "f a \<in> carrier R"
         using assms(1) by blast
       have F21: "g a \<in> carrier R"
         using assms(2) by blast
       have F22: "finsum R f A \<in> carrier R"
         by (metis Pi_iff assms(1) finsum_closed)
       have F23: "finsum R g A \<in> carrier R"
         by (metis Pi_I assms(2) finsum_closed)
       show ?thesis using F21 F20 F22 F23
         using add.m_assoc add.m_closed add.m_lcomm
         by presburger
     qed
     show ?thesis
-      using RHS LHS assms A1 F2 
+      using RHS LHS assms A1 F2
       by presburger
   qed
 qed
 
 lemma poly_eval_add:
   assumes "P \<in> Pring_set R I"
   assumes "Q \<in> Pring_set R I"
   assumes "closed_fun R g"
   shows "poly_eval R S g (P \<Oplus> Q)   = poly_eval R S g P  \<Oplus> poly_eval R S g Q "
 proof
   fix m
   show "poly_eval R S g (P \<Oplus> Q)  m = (poly_eval R S g P \<Oplus> poly_eval R S g Q) m"
   proof-
-    obtain F where F_def: "F = monomials_reducing_to R m (P \<Oplus> Q) S \<union> monomials_reducing_to R m P S \<union> 
+    obtain F where F_def: "F = monomials_reducing_to R m (P \<Oplus> Q) S \<union> monomials_reducing_to R m P S \<union>
                               monomials_reducing_to R m Q S"
-      by simp 
+      by simp
     have 0: "finite F"
     proof-
       have 00: "finite (monomials_reducing_to R m (P \<Oplus> Q) S)"
-        using assms 
+        using assms
         by (meson finite_subset monomials_finite monomials_of_add_finite monomials_reducing_to_subset)
       have 01: "finite (monomials_reducing_to R m P S)"
         using assms(1) monomials_reducing_to_finite by blast
       have 02: "finite (monomials_reducing_to R m Q S)"
         using assms(2) monomials_reducing_to_finite by blast
-      show ?thesis  
-        using F_def 00 01 02 
+      show ?thesis
+        using F_def 00 01 02
         by blast
     qed
     have 1: "\<And>n. n \<in> F \<Longrightarrow> remove_indices n S = m"
     proof-
       fix n
       assume A: "n \<in> F"
       show "remove_indices n S = m"
-        using F_def 
-        unfolding monomials_reducing_to_def 
-        using A 
+        using F_def
+        unfolding monomials_reducing_to_def
+        using A
         by blast
     qed
     have 2: "poly_eval R S g (P \<Oplus> Q)  m  = (\<Oplus>n\<in> F. monom_eval R (restrict_to_indices n S) g \<otimes> (P \<Oplus> Q) n)"
       using assms 0 1 poly_eval_mono[of "P \<Oplus> Q" I g F m] F_def indexed_pset.indexed_padd
       by blast
     have 3: "poly_eval R S g P m  = (\<Oplus>n\<in> F. monom_eval R (restrict_to_indices n S) g \<otimes> P n)"
       using assms 0 1 poly_eval_mono[of "P" I g F m] F_def indexed_pset.indexed_padd
-      by blast  
+      by blast
     have 4: "poly_eval R S g Q m  = (\<Oplus>n\<in> F. monom_eval R (restrict_to_indices n S) g \<otimes> Q n)"
       using assms 0 1 poly_eval_mono[of "Q" I g F m] F_def indexed_pset.indexed_padd
-      by blast  
+      by blast
     have 5: "poly_eval R S g P m \<oplus> poly_eval R S g Q m = (\<Oplus>n\<in> F. monom_eval R (restrict_to_indices n S) g \<otimes> P n
                                                           \<oplus> monom_eval R (restrict_to_indices n S) g \<otimes> Q n)"
     proof-
       have 50: "(\<lambda>n. monom_eval R (restrict_to_indices n S) g \<otimes> P n) \<in> F \<rightarrow> carrier R"
         by (meson Pi_I Pring_cfs_closed assms(1) assms(3) m_closed monom_eval_car)
       have 51: "(\<lambda>n. monom_eval R (restrict_to_indices n S) g \<otimes> Q n) \<in> F \<rightarrow> carrier R"
         by (meson Pi_I Pring_cfs_closed assms(2) assms(3) m_closed monom_eval_car)
       then show ?thesis
         using 0 2 3 50 finsum_group[of "(\<lambda>n. monom_eval R (restrict_to_indices n S) g \<otimes> P n)"
-                                       "(\<lambda>n. monom_eval R (restrict_to_indices n S) g \<otimes> Q n)" F] 
+                                       "(\<lambda>n. monom_eval R (restrict_to_indices n S) g \<otimes> Q n)" F]
         by (metis (mono_tags, lifting) "4" Pring_cfs_closed assms(1) assms(2) assms(3) m_closed monom_eval_car)
     qed
     have 6: "poly_eval R S g P m \<oplus> poly_eval R S g Q m
            = (\<Oplus>n\<in> F. monom_eval R (restrict_to_indices n S) g \<otimes>( P n \<oplus>  Q n))"
     proof-
       have 0 : "(\<lambda>n. monom_eval R (restrict_to_indices n S) g \<otimes> P n \<oplus> monom_eval R (restrict_to_indices n S) g \<otimes> Q n) \<in> F \<rightarrow> carrier R"
         apply(rule Pi_I)
         by (meson Pring_cfs_closed add.m_closed assms(1) assms(2) assms(3) m_closed monom_eval_car)
       have 1: "(\<lambda>n. monom_eval R (restrict_to_indices n S) g \<otimes> (P n \<oplus> Q n)) \<in> F \<rightarrow> carrier R"
         apply(rule Pi_I)
         by (meson Pring_cfs_closed add.m_closed assms(1) assms(2) assms(3) m_closed monom_eval_car)
       have "\<And>n. n \<in> F \<Longrightarrow>  monom_eval R (restrict_to_indices n S) g \<otimes>( P n \<oplus>  Q n) = monom_eval R (restrict_to_indices n S) g \<otimes> P n
                                                           \<oplus> monom_eval R (restrict_to_indices n S) g \<otimes> Q n"
         using assms Pring_cfs_closed cring.monom_eval_car is_cring r_distr
-        by metis 
-      then have "(\<lambda>x\<in>F. monom_eval R (restrict_to_indices x S) g \<otimes> P x \<oplus> monom_eval R (restrict_to_indices x S) g \<otimes> Q x) 
+        by metis
+      then have "(\<lambda>x\<in>F. monom_eval R (restrict_to_indices x S) g \<otimes> P x \<oplus> monom_eval R (restrict_to_indices x S) g \<otimes> Q x)
                 = (\<lambda>x\<in>F. monom_eval R (restrict_to_indices x S) g \<otimes> (P x \<oplus> Q x))"
         by (metis (no_types, lifting) restrict_ext)
       then show ?thesis
         using 5 finsum_eq[of "(\<lambda>n. monom_eval R (restrict_to_indices n S) g \<otimes> P n  \<oplus> monom_eval R (restrict_to_indices n S) g \<otimes> Q n)"
-                            F "(\<lambda>n. monom_eval R (restrict_to_indices n S) g \<otimes>( P n \<oplus>  Q n))" ] 0 1 
+                            F "(\<lambda>n. monom_eval R (restrict_to_indices n S) g \<otimes>( P n \<oplus>  Q n))" ] 0 1
         by presburger
-    qed                  
+    qed
     have 7: "monomials_reducing_to R m (P \<Oplus> Q) S \<subseteq> F"
-      using F_def 
+      using F_def
       by blast
     have 8: "poly_eval R S g (P \<Oplus> Q)  m  = (\<Oplus>n\<in>F. monom_eval R (restrict_to_indices n S) g \<otimes> (P \<Oplus> Q) n)"
       using 7 0 1 "2"
       by blast
     obtain f where f_def: "f = (\<lambda>n. monom_eval R (restrict_to_indices n S) g \<otimes> (P \<Oplus> Q) n)"
-      by blast 
+      by blast
     obtain h where h_def: "h = (\<lambda>n.  monom_eval R (restrict_to_indices n S) g \<otimes>( P n \<oplus>  Q n))"
-      by blast 
+      by blast
     have 9: "f \<in> F \<rightarrow> carrier R"
-      using f_def 
+      using f_def
       by (metis (mono_tags, lifting) Pi_I Pring_cfs_closed add.m_closed
           assms(1) assms(2) assms(3) indexed_padd_def m_closed monom_eval_car)
     have 10: "h \<in> F \<rightarrow> carrier R"
-      using h_def 
+      using h_def
       by (metis (mono_tags, lifting) "9" Pi_cong f_def indexed_padd_def)
     have 11: "restrict f F = restrict h F"
-      using f_def h_def 
-      by (metis indexed_padd_def)      
+      using f_def h_def
+      by (metis indexed_padd_def)
     have "finsum R f F = finsum R h F"
-      using 9 10 11 finsum_eq[of f F h]  
+      using 9 10 11 finsum_eq[of f F h]
       by blast
-    then show ?thesis 
-      using f_def h_def 
+    then show ?thesis
+      using f_def h_def
       by (metis (no_types, lifting) "6" "8"  indexed_padd_def)
   qed
-qed   
+qed
 
 lemma poly_eval_Pring_add:
   assumes "P \<in> carrier (Pring R I)"
   assumes "Q \<in> carrier (Pring R I)"
   assumes "closed_fun R g"
   shows "poly_eval R S g (P \<oplus>\<^bsub>Pring R I\<^esub> Q)   = poly_eval R S g P \<oplus>\<^bsub>Pring R I\<^esub> poly_eval R S g Q "
-  using assms poly_eval_add[of P I Q g S]  
+  using assms poly_eval_add[of P I Q g S]
   by (metis Pring_add Pring_car)
 
 text\<open>Closure of partial evaluation maps:\<close>
 lemma(in cring) poly_eval_closed:
   assumes "closed_fun R g"
   assumes "P \<in> Pring_set R I"
   shows "poly_eval R S g P \<in> Pring_set R (I - S)"
 proof-
   obtain Pr where Pr_def[simp]: "Pr = (\<lambda>Q. poly_eval R S g Q \<in> Pring_set R (I - S))"
     by blast
   have "Pr P"
     apply(rule poly_monom_induction2[of _ I _ ])
      apply (metis Pr_def indexed_pset.indexed_const poly_eval_constant zero_closed)
-       apply (metis Pr_def assms(1) indexed_pset.indexed_const mset_to_IP_closed 
+       apply (metis Pr_def assms(1) indexed_pset.indexed_const mset_to_IP_closed
         poly_eval_constant poly_eval_monomial_term_closed poly_scalar_mult_zero r_null r_one)
   proof-
-    show "P \<in> Pring_set R I" using assms by blast 
+    show "P \<in> Pring_set R I" using assms by blast
     show "\<And>Q m k. Q \<in> Pring_set R I \<and> Pr Q \<and> set_mset m \<subseteq> I \<and> k \<in> carrier R\<Longrightarrow> Pr (Q \<Oplus> poly_scalar_mult R k (mset_to_IP R m))"
     proof-
-      fix Q 
-      fix m 
+      fix Q
+      fix m
       fix k
       assume A: "Q \<in> Pring_set R I \<and> Pr Q \<and> set_mset m \<subseteq> I\<and> k \<in> carrier R"
       then have 0: "poly_eval R S g Q \<in> Pring_set R (I - S)"
         using Pr_def by blast
       have 1: "poly_scalar_mult R k (mset_to_IP R m) \<in> Pring_set R I"
-        using  A mset_to_IP_closed poly_scalar_mult_closed 
+        using  A mset_to_IP_closed poly_scalar_mult_closed
         by blast
-      have  "poly_eval R  S g (Q \<Oplus> poly_scalar_mult R k (mset_to_IP R m)) = 
+      have  "poly_eval R  S g (Q \<Oplus> poly_scalar_mult R k (mset_to_IP R m)) =
      poly_eval R S g Q \<Oplus> poly_eval R S g (poly_scalar_mult R k (mset_to_IP R m))   "
-        using assms poly_eval_add[of Q I " (poly_scalar_mult R k (mset_to_IP R m))" g]  "1" A 
+        using assms poly_eval_add[of Q I " (poly_scalar_mult R k (mset_to_IP R m))" g]  "1" A
         by blast
       then show "Pr (Q \<Oplus> poly_scalar_mult R k (mset_to_IP R m))"
-        using Pr_def 
+        using Pr_def
         by (metis "1" A assms(1) indexed_pset.indexed_padd poly_eval_monomial_term_closed poly_scalar_mult_one poly_scalar_mult_zero)
     qed
   qed
-  then show ?thesis 
-    using Pr_def 
+  then show ?thesis
+    using Pr_def
     by blast
 qed
 
 lemma poly_scalar_mult_indexed_pmult:
   assumes "P \<in> Pring_set R I"
   assumes "k \<in> carrier R"
   shows " poly_scalar_mult R k (P \<Otimes> i) = (poly_scalar_mult R k P) \<Otimes> i"
 proof-
   have 0: "mset_to_IP R {#i#} \<in> Pring_set R (I \<union> {i})"
     by (metis Un_upper2 mset_to_IP_closed set_mset_add_mset_insert set_mset_empty)
   have 1: "P \<in> Pring_set R (I \<union> {i})"
     using Pring_carrier_subset Un_upper1 assms(1) by blast
-  show ?thesis 
+  show ?thesis
     using 0 1  poly_scalar_mult_times[of "mset_to_IP R {#i#}" "I \<union> {i}" P  k]
-        poly_index_mult[of P "I \<union> {i}" i] assms 
+        poly_index_mult[of P "I \<union> {i}" i] assms
     by (metis Un_upper2 insert_subset poly_index_mult poly_scalar_mult_closed)
 qed
 
 lemma remove_indices_add_mset:
   assumes "i \<notin> S"
   shows "remove_indices (add_mset i m) S = add_mset i (remove_indices m S)"
   apply(induction m)
   apply (smt assms empty_eq_union remove_indices_decomp restrict_to_indicesE single_is_union union_single_eq_member)
   by (metis assms multi_union_self_other_eq remove_indices_decomp restrict_to_indices_add_element union_mset_add_mset_right)
 
 lemma poly_eval_monom_insert:
   assumes "closed_fun R g"
   assumes "\<one> \<noteq>\<zero>"
   assumes "i \<in> S"
   shows "poly_eval R  S g (mset_to_IP R (add_mset i m))
            = poly_scalar_mult R (g i)
                     (poly_eval R  S g (mset_to_IP R m))"
 proof-
-  have 0: "poly_eval R  S g (mset_to_IP R (add_mset i m)) = 
+  have 0: "poly_eval R  S g (mset_to_IP R (add_mset i m)) =
                 poly_scalar_mult R (monom_eval R (restrict_to_indices (add_mset i m) S) g)
                               (mset_to_IP R (remove_indices (add_mset i m) S))"
     using assms(1) assms(2) poly_eval_monomial by blast
   have 1: "(mset_to_IP R (remove_indices (add_mset i m) S)) =
                            (mset_to_IP R (remove_indices m S))"
-    using assms 
+    using assms
     by (metis (full_types) Diff_iff insert_Diff1 remove_indices_def restrict_to_indices_add_element set_mset_add_mset_insert)
-  have 2: "(monom_eval R (restrict_to_indices (add_mset i m) S) g) = 
+  have 2: "(monom_eval R (restrict_to_indices (add_mset i m) S) g) =
             (g i) \<otimes> ((monom_eval R (restrict_to_indices m S) g))"
-    using assms 
+    using assms
     by (metis monom_eval_add restrict_to_indices_add_element)
-  have 3: "poly_eval R  S g (mset_to_IP R (add_mset i m)) = 
+  have 3: "poly_eval R  S g (mset_to_IP R (add_mset i m)) =
                 poly_scalar_mult R ((g i) \<otimes> ((monom_eval R (restrict_to_indices m S) g)))
                               (mset_to_IP R (remove_indices m S))"
-    using 0 1 2 assms 
+    using 0 1 2 assms
     by presburger
-  hence "poly_eval R  S g (mset_to_IP R (add_mset i m)) = 
-                poly_scalar_mult R (g i) 
+  hence "poly_eval R  S g (mset_to_IP R (add_mset i m)) =
+                poly_scalar_mult R (g i)
                   (poly_scalar_mult R ((monom_eval R (restrict_to_indices m S) g))
                               (mset_to_IP R (remove_indices m S)))"
     using assms poly_scalar_mult_iter[of "(mset_to_IP R (remove_indices m S))"
                                           UNIV "g i" "monom_eval R (restrict_to_indices m S) g"]
           mset_to_IP_closed[of "remove_indices m S" UNIV]
     by (metis PiE UNIV_I monom_eval_car subsetI)
-  thus ?thesis using assms 
+  thus ?thesis using assms
     by (metis poly_eval_monomial)
-qed    
+qed
 
 lemma poly_eval_monom_insert':
   assumes "closed_fun R g"
   assumes "\<one> \<noteq>\<zero>"
   assumes "i \<notin> S"
   shows "poly_eval R  S g (mset_to_IP R (add_mset i m))
            =  (poly_eval R  S g (mset_to_IP R m)) \<Otimes> i"
 proof-
-  have 0: "poly_eval R  S g (mset_to_IP R (add_mset i m)) = 
+  have 0: "poly_eval R  S g (mset_to_IP R (add_mset i m)) =
                 poly_scalar_mult R (monom_eval R (restrict_to_indices (add_mset i m) S) g)
                               (mset_to_IP R (remove_indices (add_mset i m) S))"
     using assms(1) assms(2) poly_eval_monomial by blast
-  hence "poly_eval R  S g (mset_to_IP R (add_mset i m)) = 
+  hence "poly_eval R  S g (mset_to_IP R (add_mset i m)) =
                 poly_scalar_mult R (monom_eval R (restrict_to_indices m S) g)
                               (mset_to_IP R (remove_indices (add_mset i m) S))"
     by (metis assms(3) restrict_to_indices_add_element)
-  hence "poly_eval R  S g (mset_to_IP R (add_mset i m)) = 
+  hence "poly_eval R  S g (mset_to_IP R (add_mset i m)) =
                 poly_scalar_mult R (monom_eval R (restrict_to_indices m S) g)
                               (mset_to_IP R (add_mset i (remove_indices m S)))"
     by (metis assms(3) remove_indices_add_mset)
-  hence "poly_eval R  S g (mset_to_IP R (add_mset i m)) = 
+  hence "poly_eval R  S g (mset_to_IP R (add_mset i m)) =
                 poly_scalar_mult R (monom_eval R (restrict_to_indices m S) g)
                               (mset_to_IP R (remove_indices m S) \<Otimes> i)"
     by (metis monom_add_mset)
-  hence "poly_eval R  S g (mset_to_IP R (add_mset i m)) = 
+  hence "poly_eval R  S g (mset_to_IP R (add_mset i m)) =
                 (poly_scalar_mult R (monom_eval R (restrict_to_indices m S) g)
                               (mset_to_IP R (remove_indices m S))) \<Otimes> i"
-    by (metis assms(1) cring.monom_eval_car is_cring local.ring_axioms 
+    by (metis assms(1) cring.monom_eval_car is_cring local.ring_axioms
         poly_scalar_mult_indexed_pmult ring.mset_to_IP_closed set_eq_subset)
-  thus ?thesis 
-    by (metis assms(1) assms(2) poly_eval_monomial)  
-qed    
+  thus ?thesis
+    by (metis assms(1) assms(2) poly_eval_monomial)
+qed
 
 lemma poly_eval_indexed_pmult_monomial:
   assumes "closed_fun R g"
   assumes "k \<in> carrier R"
   assumes "i \<in> S"
   assumes "\<one> \<noteq> \<zero>"
   shows "poly_eval R S g (poly_scalar_mult R k (mset_to_IP R m) \<Otimes> i) =
           poly_scalar_mult R (g i) (poly_eval R S g (poly_scalar_mult R k (mset_to_IP R m)))"
 proof-
-  have 0: "poly_scalar_mult R k (mset_to_IP R m) \<Otimes> i = 
+  have 0: "poly_scalar_mult R k (mset_to_IP R m) \<Otimes> i =
               poly_scalar_mult R k (mset_to_IP R (add_mset i m ))"
-    using monom_add_mset[of i m] poly_scalar_mult_indexed_pmult 
+    using monom_add_mset[of i m] poly_scalar_mult_indexed_pmult
     by (metis (no_types, opaque_lifting) assms(2) local.ring_axioms  ring.mset_to_IP_closed subsetD subset_refl)
-  hence 1: "poly_eval R S g (poly_scalar_mult R k (mset_to_IP R m) \<Otimes> i)  = 
+  hence 1: "poly_eval R S g (poly_scalar_mult R k (mset_to_IP R m) \<Otimes> i)  =
              poly_scalar_mult R k (poly_eval R S g (mset_to_IP R (add_mset i m )))"
     by (metis assms(1) assms(2) cring.poly_eval_scalar_mult is_cring local.ring_axioms ring.mset_to_IP_closed subsetI)
   have 2: "poly_scalar_mult R (g i) (poly_eval R S g (poly_scalar_mult R k (mset_to_IP R m)))
-            = poly_scalar_mult R (g i) 
+            = poly_scalar_mult R (g i)
               (poly_scalar_mult R k  (poly_eval R S g (mset_to_IP R m))) "
-    by (smt assms(1) assms(2) local.ring_axioms poly_eval_scalar_mult ring.mset_to_IP_closed subsetD subset_refl)    
+    by (smt assms(1) assms(2) local.ring_axioms poly_eval_scalar_mult ring.mset_to_IP_closed subsetD subset_refl)
   hence 3: "poly_scalar_mult R (g i) (poly_eval R S g (poly_scalar_mult R k (mset_to_IP R m)))
-            = poly_scalar_mult R k 
+            = poly_scalar_mult R k
               (poly_scalar_mult R (g i) (poly_eval R S g (mset_to_IP R m))) "
-    using assms poly_scalar_mult_comm[of "(poly_eval R S g (mset_to_IP R m))" UNIV] 
+    using assms poly_scalar_mult_comm[of "(poly_eval R S g (mset_to_IP R m))" UNIV]
           poly_eval_closed[of g "(mset_to_IP R m)" UNIV]
-    by (metis UNIV_I closed_fun_simp cring.poly_scalar_mult_comm 
+    by (metis UNIV_I closed_fun_simp cring.poly_scalar_mult_comm
         is_cring local.ring_axioms ring.mset_to_IP_closed subsetI)
   have 4: "(poly_eval R S g (mset_to_IP R (add_mset i m))) =  (poly_scalar_mult R (g i) (poly_eval R S g (mset_to_IP R m)))"
     using assms poly_eval_monom_insert[of g i S m]
     by blast
   thus ?thesis
     using 1 3 poly_scalar_mult_comm assms
-    by presburger   
+    by presburger
 qed
 
 lemma poly_eval_indexed_pmult_monomial':
   assumes "closed_fun R g"
   assumes "k \<in> carrier R"
   assumes "i \<notin> S"
   assumes "\<one> \<noteq> \<zero>"
   shows "poly_eval R S g (poly_scalar_mult R k (mset_to_IP R m) \<Otimes> i) =
          (poly_eval R S g (poly_scalar_mult R k (mset_to_IP R m))) \<Otimes> i"
 proof-
   have "poly_eval R S g (poly_scalar_mult R k (mset_to_IP R m) \<Otimes> i) =
       poly_scalar_mult R k (poly_eval R S g ( (mset_to_IP R m) \<Otimes> i))"
-    using poly_eval_scalar_mult[of k g] 
-    by (metis UNIV_I assms(1) assms(2) local.ring_axioms poly_scalar_mult_indexed_pmult ring.monom_add_mset ring.mset_to_IP_closed subsetI)   
+    using poly_eval_scalar_mult[of k g]
+    by (metis UNIV_I assms(1) assms(2) local.ring_axioms poly_scalar_mult_indexed_pmult ring.monom_add_mset ring.mset_to_IP_closed subsetI)
   hence "poly_eval R S g (poly_scalar_mult R k (mset_to_IP R m) \<Otimes> i) =
       poly_scalar_mult R k (poly_eval R S g (mset_to_IP R (add_mset i m)))"
     by (simp add: monom_add_mset)
   hence "poly_eval R S g (poly_scalar_mult R k (mset_to_IP R m) \<Otimes> i) =
       (poly_scalar_mult R k (poly_eval R S g (mset_to_IP R m))) \<Otimes> i"
-    using poly_eval_monom_insert'[of g i S m] 
+    using poly_eval_monom_insert'[of g i S m]
     by (smt assms(1) assms(2) assms(3) assms(4) poly_eval_monomial_closed poly_scalar_mult_indexed_pmult subsetD subset_refl)
-  thus ?thesis using assms poly_eval_scalar_mult[of k g _ UNIV S] 
-    by (metis UNIV_I mset_to_IP_closed subsetI)    
-qed
-    
+  thus ?thesis using assms poly_eval_scalar_mult[of k g _ UNIV S]
+    by (metis UNIV_I mset_to_IP_closed subsetI)
+qed
+
 lemma indexed_pmult_add:
   assumes "p \<in> Pring_set R I"
   assumes "q \<in> Pring_set R I"
   shows "p \<Oplus> q \<Otimes> i = (p \<Otimes> i) \<Oplus> (q \<Otimes> i)"
-  using assms poly_index_mult[of _ "I \<union> {i}"] 
+  using assms poly_index_mult[of _ "I \<union> {i}"]
   by (smt Pring_carrier_subset Set.basic_monos(1) Un_upper1 Un_upper2 cring.axioms(1)
-      insert_subset is_cring mk_disjoint_insert mset_to_IP_closed ring.P_ring_ldistr 
-      ring.indexed_pset.intros(2) ring.indexed_pset_in_carrier set_mset_add_mset_insert 
+      insert_subset is_cring mk_disjoint_insert mset_to_IP_closed ring.P_ring_ldistr
+      ring.indexed_pset.intros(2) ring.indexed_pset_in_carrier set_mset_add_mset_insert
       set_mset_empty)
 
 lemma poly_eval_indexed_pmult:
   assumes "P \<in> Pring_set R I"
   assumes "closed_fun R g"
   shows "poly_eval R S g (P \<Otimes> i)   = (if i \<in> S then poly_scalar_mult R (g i) (poly_eval R S g P) else (poly_eval R S g P)\<Otimes> i )"
 proof(cases "i \<in> S")
   case True
   have "poly_eval R S g (P \<Otimes> i) = poly_scalar_mult R (g i) (poly_eval R S g P)"
   apply(rule poly_monom_induction3[of P I])
     using assms apply blast
-      apply (metis PiE UNIV_I assms(2) indexed_pmult_zero poly_eval_constant poly_scalar_mult_const r_null zero_closed)      
+      apply (metis PiE UNIV_I assms(2) indexed_pmult_zero poly_eval_constant poly_scalar_mult_const r_null zero_closed)
         apply (meson True assms(2) poly_eval_indexed_pmult_monomial)
           apply (smt PiE True UNIV_I assms(2) genideal_one genideal_zero indexed_pmult_zero mset_to_IP_closed poly_eval_constant poly_eval_indexed_pmult_monomial poly_scalar_mult_one poly_scalar_mult_zero singletonD)
-  proof- fix p q assume A: 
+  proof- fix p q assume A:
           "p \<in> Pring_set R I"
           "poly_eval R S g (p \<Otimes> i) = poly_scalar_mult R (g i) (poly_eval R S g p)"
           "q \<in> Pring_set R I"
           "poly_eval R S g (q \<Otimes> i) = poly_scalar_mult R (g i) (poly_eval R S g q)"
     have "poly_eval R S g (p \<Oplus> q \<Otimes> i) = poly_eval R S g (p \<Otimes> i) \<Oplus> poly_eval R S g (q \<Otimes> i)"
-      using assms poly_eval_add[of "p \<Otimes> i" "I \<union> {i}" "q \<Otimes> i" g S] 
+      using assms poly_eval_add[of "p \<Otimes> i" "I \<union> {i}" "q \<Otimes> i" g S]
             indexed_pmult_add[of p  I q i]
       by (smt A(1) A(3) Pring_carrier_subset Un_insert_right Un_upper1 indexed_pset.indexed_pmult insert_iff insert_subset mk_disjoint_insert)
-    hence "poly_eval R S g (p \<Oplus> q \<Otimes> i) = poly_scalar_mult R (g i) (poly_eval R S g p) \<Oplus> 
+    hence "poly_eval R S g (p \<Oplus> q \<Otimes> i) = poly_scalar_mult R (g i) (poly_eval R S g p) \<Oplus>
                                            poly_scalar_mult R (g i) (poly_eval R S g q)"
-      using A 
+      using A
       by presburger
-    hence "poly_eval R S g (p \<Oplus> q \<Otimes> i) = poly_scalar_mult R (g i) 
-                                            ((poly_eval R S g p) \<Oplus> (poly_eval R S g q))"      
-      using Pring_smult poly_eval_closed[of g] A Pring_add Pring_car  \<open>poly_eval R S g (p \<Oplus> q \<Otimes> i) = poly_eval R S g (p \<Otimes> i) \<Oplus> poly_eval R S g (q \<Otimes> i)\<close> assms(1) assms(2) closed_fun_simp 
-            Pring_smult_r_distr[of "g i" "poly_eval R S g p" _ "poly_eval R S g q"] Pring_add Pring_car 
+    hence "poly_eval R S g (p \<Oplus> q \<Otimes> i) = poly_scalar_mult R (g i)
+                                            ((poly_eval R S g p) \<Oplus> (poly_eval R S g q))"
+      using Pring_smult poly_eval_closed[of g] A Pring_add Pring_car  \<open>poly_eval R S g (p \<Oplus> q \<Otimes> i) = poly_eval R S g (p \<Otimes> i) \<Oplus> poly_eval R S g (q \<Otimes> i)\<close> assms(1) assms(2) closed_fun_simp
+            Pring_smult_r_distr[of "g i" "poly_eval R S g p" _ "poly_eval R S g q"] Pring_add Pring_car
       by metis
     thus "poly_eval R S g (p \<Oplus> q \<Otimes> i) = poly_scalar_mult R (g i) (poly_eval R S g (p \<Oplus> q))"
       by (metis A(1) A(3) assms(2) poly_eval_add)
-  qed             
-  then show ?thesis 
-    using True by presburger 
+  qed
+  then show ?thesis
+    using True by presburger
 next
   case False
   have "poly_eval R S g (P \<Otimes> i) = (poly_eval R S g P) \<Otimes> i"
   apply(rule poly_monom_induction3[of P I])
        apply (simp add: assms(1))
         apply (metis indexed_pmult_zero poly_eval_constant zero_closed)
-          apply (metis False assms(2) indexed_pmult_zero inv_unique l_null  mset_to_IP_closed 
-           one_closed one_mset_to_IP poly_eval_constant  poly_eval_indexed_pmult_monomial' 
+          apply (metis False assms(2) indexed_pmult_zero inv_unique l_null  mset_to_IP_closed
+           one_closed one_mset_to_IP poly_eval_constant  poly_eval_indexed_pmult_monomial'
            poly_scalar_mult_zero  zero_closed)
-  proof-fix p q assume A: 
+  proof-fix p q assume A:
           "p \<in> Pring_set R I"
           "poly_eval R S g (p \<Otimes> i) = poly_eval R S g p \<Otimes> i"
           "q \<in> Pring_set R I"
           "poly_eval R S g (q \<Otimes> i) = poly_eval R S g q \<Otimes> i"
     have "poly_eval R S g (p \<Oplus> q \<Otimes> i) = poly_eval R S g (p \<Otimes> i) \<Oplus> poly_eval R S g (q \<Otimes> i)"
-      using assms poly_eval_add[of "p \<Otimes> i" "I \<union> {i}" "q \<Otimes> i" g S] 
+      using assms poly_eval_add[of "p \<Otimes> i" "I \<union> {i}" "q \<Otimes> i" g S]
             indexed_pmult_add[of p  I q i]
       by (smt A(1) A(3) Pring_carrier_subset Un_insert_right Un_upper1 indexed_pset.indexed_pmult insert_iff insert_subset mk_disjoint_insert)
     thus "poly_eval R S g (p \<Oplus> q \<Otimes> i) = poly_eval R S g (p \<Oplus> q) \<Otimes> i"
       by (metis A(1) A(2) A(3) A(4) assms(2) indexed_pmult_add poly_eval_add poly_eval_closed)
-  qed             
-  then show ?thesis 
-    by (simp add: False) 
+  qed
+  then show ?thesis
+    by (simp add: False)
 qed
 
 lemma poly_eval_index:
   assumes "\<one> \<noteq>\<zero>"
   assumes "closed_fun R g"
   shows "poly_eval R S g (mset_to_IP R {#i#})= (if i \<in> S then (indexed_const (g i)) else mset_to_IP R {#i#})"
 proof-
   have 0: "poly_eval R S g (mset_to_IP R {#i#})= poly_scalar_mult R (monom_eval R (restrict_to_indices {#i#} S) g)
                               (mset_to_IP R (remove_indices {#i#} S))"
     using poly_eval_monomial[of g S "{#i#}" ] assms(1) assms(2) by blast
   show ?thesis proof(cases "i \<in> S")
     case True
     then have T0: "(restrict_to_indices {#i#} S) =  {#i#}"
       by (metis restrict_to_indices_add_element restrict_to_indices_submultiset add_mset_subseteq_single_iff)
     then have T1: "(monom_eval R (restrict_to_indices {#i#} S) g) = (monom_eval R {#i#} g)"
       by presburger
     then have "(monom_eval R (restrict_to_indices {#i#} S) g)  = g i \<otimes>  monom_eval R {#} g"
-      using assms monom_eval_add[of g i "{#}"] 
+      using assms monom_eval_add[of g i "{#}"]
       by presburger
     then have T2: "(monom_eval R (restrict_to_indices {#i#} S) g)  = g i"
-      unfolding monom_eval_def  
+      unfolding monom_eval_def
       using T0
-      by (metis PiE UNIV_I assms(2) fold_mset_empty r_one)      
+      by (metis PiE UNIV_I assms(2) fold_mset_empty r_one)
     have T3: "(remove_indices {#i#} S) = {#}"
-      by (metis Diff_iff remove_indices_indices restrict_to_indicesE 
+      by (metis Diff_iff remove_indices_indices restrict_to_indicesE
           T0 multiset_cases subset_iff union_single_eq_member)
     then have T4: " (mset_to_IP R (remove_indices {#i#} S)) = indexed_const \<one>"
       by (metis one_mset_to_IP)
     have T5: "poly_eval R S g (mset_to_IP R {#i#})= poly_scalar_mult R (g i) (indexed_const \<one>)"
-      using "0" T2 T4 
+      using "0" T2 T4
       by presburger
-    then show ?thesis using True  poly_scalar_mult_const 
-      by (metis T2 assms(2) monom_eval_car one_closed r_one)      
+    then show ?thesis using True  poly_scalar_mult_const
+      by (metis T2 assms(2) monom_eval_car one_closed r_one)
   next
     case False
     have F0: "(restrict_to_indices {#i#} S) = {#}"
-      using False restrict_to_indices_def 
+      using False restrict_to_indices_def
       by (metis restrict_to_indices_add_element filter_empty_mset)
     have F1: "(monom_eval R (restrict_to_indices {#i#} S) g) = \<one>"
-      using F0 
-      unfolding monom_eval_def 
+      using F0
+      unfolding monom_eval_def
       by (metis fold_mset_empty)
     have F2: "(remove_indices {#i#} S) = {#i#}"
-      using False 
+      using False
       by (metis Diff_iff remove_indices_def restrict_to_indices_add_element
           restrict_to_indices_def filter_empty_mset set_mset_single singletonI)
     have F3: "(mset_to_IP R (remove_indices {#i#} S)) = mset_to_IP R {#i#}"
       by (simp add: F2)
     have F4: " poly_eval R S g (mset_to_IP R {#i#})= poly_scalar_mult R \<one> (mset_to_IP R {#i#})"
-      using "0" F1 F3 
+      using "0" F1 F3
       by presburger
-    show ?thesis using False 
+    show ?thesis using False
       by (metis F4 mset_to_IP_closed poly_scalar_mult_one subset_iff)
   qed
-qed  
-    
+qed
+
 lemma poly_eval_indexed_pmult':
   assumes "P \<in> Pring_set R I"
   assumes "closed_fun R g"
   assumes "i \<in> I"
   shows "poly_eval R  S g (P \<Otimes>\<^sub>p  (mset_to_IP R {#i#}))  =  poly_eval R S g P  \<Otimes>\<^sub>p  poly_eval R S g (mset_to_IP R {#i#})"
 proof(cases "i \<in> S")
   case True
   have "(P \<Otimes>\<^sub>p mset_to_IP R {#i#})=(P \<Otimes> i) "
-    using assms poly_index_mult 
+    using assms poly_index_mult
     by metis
   then have 0: " poly_eval R  S g (P \<Otimes>\<^sub>p mset_to_IP R {#i#}) = poly_scalar_mult R (g i) (poly_eval R S g P)"
-    using True assms poly_eval_indexed_pmult[of P I g S i] 
-    by presburger  
+    using True assms poly_eval_indexed_pmult[of P I g S i]
+    by presburger
   have 1: "poly_eval R S g (mset_to_IP R {#i#}) = indexed_const (g i)"
-    using assms True 
-    by (smt PiE UNIV_I genideal_one genideal_zero indexed_pmult_zero monom_add_mset one_mset_to_IP poly_eval_constant poly_eval_index singletonD)    
+    using assms True
+    by (smt PiE UNIV_I genideal_one genideal_zero indexed_pmult_zero monom_add_mset one_mset_to_IP poly_eval_constant poly_eval_index singletonD)
   then have "poly_eval R S g P  \<Otimes>\<^sub>p  poly_eval R S g (mset_to_IP R {#i#})=  poly_eval R S g P\<Otimes>\<^sub>p indexed_const (g i) "
     by presburger
   then have "poly_eval R S g P  \<Otimes>\<^sub>p  poly_eval R S g (mset_to_IP R {#i#})= indexed_const (g i) \<Otimes>\<^sub>p  poly_eval R S g P"
-    using assms P_ring_mult_comm[of "indexed_const (g i)" "poly_eval R S g P"] 
-    unfolding carrier_coeff_def 
-    by (metis "1" Pring_cfs_closed cring.closed_fun_simp indexed_pset.indexed_const is_cring poly_eval_closed)    
+    using assms P_ring_mult_comm[of "indexed_const (g i)" "poly_eval R S g P"]
+    unfolding carrier_coeff_def
+    by (metis "1" Pring_cfs_closed cring.closed_fun_simp indexed_pset.indexed_const is_cring poly_eval_closed)
   then have "poly_eval R S g P  \<Otimes>\<^sub>p  poly_eval R S g (mset_to_IP R {#i#})= poly_scalar_mult R (g i)  (poly_eval R S g P)"
-    using assms 
-    by (metis "0" "1" \<open>P \<Otimes>\<^sub>p mset_to_IP R {#i#} = P \<Otimes> i\<close> cring.closed_fun_simp is_cring poly_eval_closed poly_scalar_mult_eq)    
-  then show ?thesis using 0 
+    using assms
+    by (metis "0" "1" \<open>P \<Otimes>\<^sub>p mset_to_IP R {#i#} = P \<Otimes> i\<close> cring.closed_fun_simp is_cring poly_eval_closed poly_scalar_mult_eq)
+  then show ?thesis using 0
     by presburger
 next
   case False
   then have 0: "poly_eval R  S g (P \<Otimes>\<^sub>p  (mset_to_IP R {#i#})) = (poly_eval R S g P)\<Otimes> i "
-    using assms 
+    using assms
     by (metis poly_eval_indexed_pmult poly_index_mult)
   have 1: "poly_eval R S g (mset_to_IP R {#i#})= mset_to_IP R {#i#}"
-    using False 
+    using False
     by (metis assms(2) indexed_pmult_zero monom_add_mset one_closed one_mset_to_IP poly_eval_constant poly_eval_index)
-  then show ?thesis 
+  then show ?thesis
     using 0 False assms poly_eval_index[of g]
     by (metis UNIV_I cring.Pring_set_restrict is_cring local.ring_axioms poly_eval_closed ring.poly_index_mult subsetI)
 qed
 
 lemma poly_eval_monom_mult:
   assumes "P \<in> Pring_set R I"
   assumes "closed_fun R g"
   shows "poly_eval R  S g (P  \<Otimes>\<^sub>p  (mset_to_IP R m))  = poly_eval R S g P  \<Otimes>\<^sub>p  poly_eval R S g (mset_to_IP R m) "
 proof(induct m)
   case empty
   have 0:  "mset_to_IP R {#} = indexed_const \<one>"
     using one_mset_to_IP by blast
   then have 1: "(P \<Otimes>\<^sub>p mset_to_IP R {#}) = P"
-    using assms 
-    by (metis P_ring_mult_comm Pring_cfs_closed carrier_coeff_def 
+    using assms
+    by (metis P_ring_mult_comm Pring_cfs_closed carrier_coeff_def
         mset_to_IP_simp mset_to_IP_simp' one_closed one_mult_left zero_closed)
   have 2: "poly_eval R S g  (mset_to_IP R {#}) = indexed_const \<one>"
     by (metis "0" one_closed poly_eval_constant)
-  show ?case using 0 1 2 
-    by (metis assms(1) assms(2) cring.P_ring_mult_comm indexed_pset.indexed_const 
-        is_cring local.ring_axioms one_closed one_mult_left poly_eval_closed  
+  show ?case using 0 1 2
+    by (metis assms(1) assms(2) cring.P_ring_mult_comm indexed_pset.indexed_const
+        is_cring local.ring_axioms one_closed one_mult_left poly_eval_closed
         ring.indexed_pset_in_carrier set_eq_subset)
 next
   case (add x m)
-  fix x 
+  fix x
   fix m
   assume A: "poly_eval R  S g (P \<Otimes>\<^sub>p mset_to_IP R m) = poly_eval R S g P \<Otimes>\<^sub>p poly_eval R S g (mset_to_IP R m)"
   show "poly_eval R  S g (P \<Otimes>\<^sub>p mset_to_IP R (add_mset x m))= poly_eval R S g P \<Otimes>\<^sub>p poly_eval R  S g (mset_to_IP R (add_mset x m))"
   proof-
     obtain J where J_def: "J = I \<union> set_mset m \<union>{x}"
-      by blast 
+      by blast
     have I0: "P \<in> Pring_set R J"
-      using J_def assms 
+      using J_def assms
       by (meson Pring_carrier_subset Un_upper1 subsetD)
     have I1: "set_mset m \<subseteq> J"
       using J_def by blast
     have I2: "x \<in> J"
       using J_def by blast
     have "mset_to_IP R (add_mset x m)= (mset_to_IP R m) \<Otimes> x"
       by (simp add: monom_add_mset)
     then have "(P \<Otimes>\<^sub>p mset_to_IP R (add_mset x m)) = P \<Otimes>\<^sub>p((mset_to_IP R m) \<Otimes> x)"
       by simp
     then have "(P \<Otimes>\<^sub>p mset_to_IP R (add_mset x m)) = P \<Otimes>\<^sub>p((mset_to_IP R m) \<Otimes>\<^sub>p (mset_to_IP R {#x#}))"
       by (metis add_mset_add_single monom_mult)
     then have I3: "(P \<Otimes>\<^sub>p mset_to_IP R (add_mset x m)) = (P \<Otimes>\<^sub>p(mset_to_IP R m)) \<Otimes>\<^sub>p (mset_to_IP R {#x#})"
       by (metis I0 P_ring_mult_assoc indexed_pset_in_carrier mset_to_IP_closed set_eq_subset)
-    have "poly_eval R  S g (P \<Otimes>\<^sub>p mset_to_IP R m \<Otimes>\<^sub>p mset_to_IP R {#x#}) = 
+    have "poly_eval R  S g (P \<Otimes>\<^sub>p mset_to_IP R m \<Otimes>\<^sub>p mset_to_IP R {#x#}) =
             poly_eval R  S g (P \<Otimes>\<^sub>p mset_to_IP R m) \<Otimes>\<^sub>p poly_eval R S g  (mset_to_IP R {#x#})"
-      using poly_eval_indexed_pmult'[of "(P \<Otimes>\<^sub>p(mset_to_IP R m))" J g x] 
-            I0 I1 I2 assms(2) assms(1) mset_to_IP_mult_closed 
+      using poly_eval_indexed_pmult'[of "(P \<Otimes>\<^sub>p(mset_to_IP R m))" J g x]
+            I0 I1 I2 assms(2) assms(1) mset_to_IP_mult_closed
       by blast
-    then have "poly_eval R S g (P \<Otimes>\<^sub>p mset_to_IP R (add_mset x m)) =   
+    then have "poly_eval R S g (P \<Otimes>\<^sub>p mset_to_IP R (add_mset x m)) =
           poly_eval R  S g (P \<Otimes>\<^sub>p(mset_to_IP R m)) \<Otimes>\<^sub>p  poly_eval R  S g (mset_to_IP R {#x#}) "
-      using I3 
+      using I3
       by simp
-    then have "poly_eval R  S g (P \<Otimes>\<^sub>p mset_to_IP R (add_mset x m)) = 
+    then have "poly_eval R  S g (P \<Otimes>\<^sub>p mset_to_IP R (add_mset x m)) =
          poly_eval R S g P \<Otimes>\<^sub>p poly_eval R S g (mset_to_IP R m)  \<Otimes>\<^sub>p  poly_eval R S g (mset_to_IP R {#x#}) "
       by (simp add: A)
-    then have "poly_eval R S g  (P \<Otimes>\<^sub>p mset_to_IP R (add_mset x m)) = 
+    then have "poly_eval R S g  (P \<Otimes>\<^sub>p mset_to_IP R (add_mset x m)) =
          poly_eval R S g P \<Otimes>\<^sub>p (poly_eval R S g (mset_to_IP R m)  \<Otimes>\<^sub>p  poly_eval R  S g (mset_to_IP R {#x#}))"
       using P_ring_mult_assoc[of "poly_eval R S g P " "poly_eval R S g (mset_to_IP R m)"  " poly_eval R  S g (mset_to_IP R {#x#})"]
       by (metis assms(1) assms(2)  indexed_pset_in_carrier mset_to_IP_closed poly_eval_closed set_eq_subset)
-    then have "poly_eval R  S g (P \<Otimes>\<^sub>p mset_to_IP R (add_mset x m)) = 
+    then have "poly_eval R  S g (P \<Otimes>\<^sub>p mset_to_IP R (add_mset x m)) =
          poly_eval R S g P \<Otimes>\<^sub>p (poly_eval R  S g ((mset_to_IP R m) \<Otimes>\<^sub>p (mset_to_IP R {#x#})))"
       by (metis I1 I2 assms(2)  mset_to_IP_closed poly_eval_indexed_pmult')
-    then show ?thesis 
+    then show ?thesis
       by (metis add_mset_add_single monom_mult)
   qed
 qed
 
 abbreviation mon_term ("Mt") where
 "Mt k m \<equiv> poly_scalar_mult R k (mset_to_IP R m)"
 
 lemma poly_eval_monom_term_mult:
   assumes "P \<in> Pring_set R I"
   assumes "closed_fun R g"
   assumes "k \<in> carrier R"
   shows "poly_eval R S g (P \<Otimes>\<^sub>p (Mt k m))   = poly_eval R S g P  \<Otimes>\<^sub>p  poly_eval R S g (Mt k m) "
 proof-
   obtain J where J_def: "J = I \<union> (set_mset m)"
-    by blast 
+    by blast
   have J0: "P \<in> Pring_set R J"
     using J_def Pring_carrier_subset Un_upper1 assms(1) by blast
   have J1: "mset_to_IP R m  \<in> Pring_set R J"
     by (metis J_def Un_upper2 mset_to_IP_closed)
   have 0: "(P  \<Otimes>\<^sub>p  (Mt k m)) = poly_scalar_mult R k (P  \<Otimes>\<^sub>p  mset_to_IP R m)"
     using times_poly_scalar_mult[of P J "mset_to_IP R m" k ] J0 J1 assms(3)
     by blast
   have 1: "poly_eval R S g (P \<Otimes>\<^sub>p (Mt k m)) = poly_scalar_mult R k (poly_eval R  S g (P  \<Otimes>\<^sub>p  mset_to_IP R m))"
     by (metis "0" J0 J1 P_ring_mult_closed' assms(2) assms(3) cring.poly_eval_scalar_mult is_cring)
   have 2: "poly_eval R S g (P \<Otimes>\<^sub>p (Mt k m)) = poly_scalar_mult R k ((poly_eval R S g P)  \<Otimes>\<^sub>p (poly_eval R S g (mset_to_IP R m)))"
     by (metis "1" assms(1) assms(2)  poly_eval_monom_mult)
   have 3: "poly_eval R S g (P \<Otimes>\<^sub>p (Mt k m)) = (poly_eval R S g P)  \<Otimes>\<^sub>p poly_scalar_mult R k  (poly_eval R S g (mset_to_IP R m))"
     by (metis "0" "2" J0 J1 assms(2) assms(3)  poly_eval_closed times_poly_scalar_mult)
-  then show ?thesis 
+  then show ?thesis
     by (metis J1 assms(3) assms(2) poly_eval_scalar_mult)
 qed
 
 lemma poly_eval_mult:
   assumes "P \<in> Pring_set R I"
   assumes "Q \<in> Pring_set R I"
   assumes "closed_fun R g"
   shows "poly_eval R S g (P \<Otimes>\<^sub>p Q)  = poly_eval R S g P  \<Otimes>\<^sub>p  poly_eval R S g Q "
 proof-
   obtain Pr where Pr_def: "Pr = (\<lambda>Q. poly_eval R S g (P \<Otimes>\<^sub>p Q)   = poly_eval R S g P  \<Otimes>\<^sub>p  poly_eval R S g Q )"
-    by blast 
+    by blast
   have "Pr Q"
   proof(rule poly_monom_induction2[of _ I])
     show "Q \<in> Pring_set R I"
       by (simp add: assms(2))
     show "Pr (indexed_const \<zero>)"
     proof-
       have 0: "(P \<Otimes>\<^sub>p indexed_const \<zero>) = indexed_const \<zero>"
         by (metis assms(1) cring.P_ring_mult_comm indexed_pset.indexed_const
-            indexed_pset_in_carrier is_cring poly_scalar_mult_eq poly_scalar_mult_zero 
+            indexed_pset_in_carrier is_cring poly_scalar_mult_eq poly_scalar_mult_zero
             set_eq_subset zero_closed)
       have 1: "poly_eval R  S g (indexed_const \<zero>) = indexed_const \<zero>"
         using poly_eval_constant by blast
       have 2: "poly_eval R S g P  \<Otimes>\<^sub>p poly_eval R  S g (indexed_const \<zero>) = indexed_const \<zero>"
-        using 1 
+        using 1
         by (metis "0" assms(1) assms(3) local.ring_axioms one_closed poly_eval_monom_term_mult
-            poly_scalar_mult_const r_one ring.one_mset_to_IP zero_closed)        
+            poly_scalar_mult_const r_one ring.one_mset_to_IP zero_closed)
       have 3: "poly_eval R S g (P \<Otimes>\<^sub>p indexed_const \<zero>) = indexed_const \<zero>"
         using "0" "1" by presburger
       show ?thesis
-        using 2 3 Pr_def 
+        using 2 3 Pr_def
         by presburger
     qed
     show "\<And>m k. set_mset m \<subseteq> I \<and> k \<in> carrier R \<Longrightarrow> Pr (poly_scalar_mult R k (mset_to_IP R m))"
       using Pr_def assms(1) assms(3)  poly_eval_monom_term_mult by blast
     show " \<And>Q m k. Q \<in> Pring_set R I \<and> Pr Q \<and> set_mset m \<subseteq> I \<and> k \<in> carrier R \<Longrightarrow> Pr (Q \<Oplus> Mt k m)"
     proof-
       fix Q
       fix m
       fix k
       assume A: "Q \<in> Pring_set R I \<and> Pr Q \<and> set_mset m \<subseteq> I \<and> k \<in> carrier R"
       show "Pr (Q \<Oplus> Mt k m)"
       proof-
         have 0: "poly_eval R S g (P \<Otimes>\<^sub>p Q)   = poly_eval R S g P  \<Otimes>\<^sub>p  poly_eval R S g Q "
           using A Pr_def by blast
         have 1: "poly_eval R S g (P \<Otimes>\<^sub>p  (Mt k m))   = poly_eval R S g P  \<Otimes>\<^sub>p  poly_eval R S g (Mt k m)"
-          using A assms(1) assms(3)  poly_eval_monom_term_mult 
+          using A assms(1) assms(3)  poly_eval_monom_term_mult
           by blast
         have 2: "P \<Otimes>\<^sub>p  (Q \<Oplus> Mt k m) =(P \<Otimes>\<^sub>p  Q)\<Oplus> (P \<Otimes>\<^sub>p  (Mt k m)) "
-          by (meson A assms(1) local.ring_axioms mset_to_IP_closed poly_scalar_mult_closed 
+          by (meson A assms(1) local.ring_axioms mset_to_IP_closed poly_scalar_mult_closed
               ring.P_ring_rdistr ring.indexed_pset_in_carrier set_eq_subset)
         have 3: "poly_eval R S g (P \<Otimes>\<^sub>p  (Q \<Oplus> Mt k m))= poly_eval R S g (P \<Otimes>\<^sub>p Q)  \<Oplus> poly_eval R S g (P \<Otimes>\<^sub>p  (Mt k m))     "
-          by (metis "2" A P_ring_mult_closed' assms(1) assms(3) 
+          by (metis "2" A P_ring_mult_closed' assms(1) assms(3)
               mset_to_IP_closed  poly_eval_add poly_scalar_mult_closed)
-        have 4: "poly_eval R S g (P \<Otimes>\<^sub>p  (Q \<Oplus> Mt k m))= 
+        have 4: "poly_eval R S g (P \<Otimes>\<^sub>p  (Q \<Oplus> Mt k m))=
         (poly_eval R S g P  \<Otimes>\<^sub>p  poly_eval R S g Q) \<Oplus> ( poly_eval R S g P  \<Otimes>\<^sub>p  poly_eval R S g (Mt k m) )"
           by (simp add: "0" "1" "3")
         have 5: " poly_eval R S g P \<Otimes>\<^sub>p (poly_eval R S g Q \<Oplus> poly_eval R S g (Mt k m)) = poly_eval R S g P \<Otimes>\<^sub>p poly_eval R S g Q \<Oplus> (poly_eval R S g P \<Otimes>\<^sub>p poly_eval R S g (Mt k m))"
           using ring.P_ring_rdistr[of R "(poly_eval R S g P)" "(poly_eval R S g Q)" "( poly_eval R S g (Mt k m) )"]
-          by (meson A assms(1) assms(3) indexed_pset_in_carrier local.ring_axioms 
+          by (meson A assms(1) assms(3) indexed_pset_in_carrier local.ring_axioms
               poly_eval_closed poly_scalar_mult_closed ring.mset_to_IP_closed subset_refl)
-        have 6: "poly_eval R S g (P \<Otimes>\<^sub>p  (Q \<Oplus> Mt k m))= 
+        have 6: "poly_eval R S g (P \<Otimes>\<^sub>p  (Q \<Oplus> Mt k m))=
         (poly_eval R S g P)  \<Otimes>\<^sub>p  ((poly_eval R S g Q) \<Oplus> ( poly_eval R S g (Mt k m) ))"
-          using 4 5 
+          using 4 5
           by simp
-        have 7: "poly_eval R S g (P \<Otimes>\<^sub>p  (Q \<Oplus> Mt k m))= 
+        have 7: "poly_eval R S g (P \<Otimes>\<^sub>p  (Q \<Oplus> Mt k m))=
         (poly_eval R S g P)  \<Otimes>\<^sub>p  (poly_eval R  S g (Q\<Oplus>(Mt k m)))"
-          using 6 poly_eval_add[of "(poly_eval R S g Q)" I "(Mt k m)" g ]  
+          using 6 poly_eval_add[of "(poly_eval R S g Q)" I "(Mt k m)" g ]
           by (metis A assms(3) mset_to_IP_closed poly_eval_add poly_scalar_mult_closed)
-        show ?thesis using 7 
+        show ?thesis using 7
           using Pr_def by blast
       qed
     qed
   qed
-  then show ?thesis 
+  then show ?thesis
     using Pr_def by blast
 qed
 
 lemma poly_eval_Pring_mult:
   assumes "P \<in> Pring_set R I"
   assumes "Q \<in> Pring_set R I"
   assumes "closed_fun R g"
   shows "poly_eval R S g (P \<otimes>\<^bsub>Pring R I\<^esub> Q)  = poly_eval R S g P \<otimes>\<^bsub>Pring R I\<^esub> poly_eval R S g Q "
   by (metis Pring_mult assms(1) assms(2) assms(3) poly_eval_mult)
 
 lemma poly_eval_smult:
   assumes "P \<in> Pring_set R I"
   assumes "a \<in> carrier R"
   assumes "closed_fun R g"
   shows "poly_eval R S g (a \<odot>\<^bsub>Pring R I\<^esub> P)  =a \<odot>\<^bsub>Pring R I\<^esub> poly_eval R S g P"
-  using poly_eval_scalar_mult assms 
+  using poly_eval_scalar_mult assms
   by (metis Pring_smult)
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsubsection\<open>Partial Evaluation is a Homomorphism\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 lemma poly_eval_ring_hom:
   assumes "I \<subseteq> J"
   assumes "closed_fun R g"
   assumes "J - S \<subseteq> I"
   shows "ring_hom_ring (Pring R J) (Pring R I) (poly_eval R S g)"
   apply(rule ring_hom_ringI)
   apply (simp add: Pring_is_ring)
   apply (simp add: Pring_is_ring)
-  apply (metis (full_types) Pring_car Pring_carrier_subset assms(2) assms(3) poly_eval_closed subset_iff)  
+  apply (metis (full_types) Pring_car Pring_carrier_subset assms(2) assms(3) poly_eval_closed subset_iff)
   apply (metis Pring_car assms(2) local.ring_axioms poly_eval_mult ring.Pring_mult)
   apply (metis Pring_add Pring_car assms(2) poly_eval_add)
   by (metis Pring_one one_closed poly_eval_constant)
 
 text\<open>\texttt{poly\_eval} R at the zero function is an inverse to the inclusion of polynomial rings\<close>
 
 lemma poly_eval_zero_function:
   assumes "g = (\<lambda>n. \<zero>)"
   assumes "J - S = I"
-  shows "P \<in> Pring_set R I \<Longrightarrow> poly_eval R S g P = P" 
+  shows "P \<in> Pring_set R I \<Longrightarrow> poly_eval R S g P = P"
   apply(erule indexed_pset.induct)
   using poly_eval_constant apply blast
-  using  assms poly_eval_add[of _ I _ g S]  zero_closed  
-  apply (metis Pi_I)  
+  using  assms poly_eval_add[of _ I _ g S]  zero_closed
+  apply (metis Pi_I)
   using Diff_iff assms(1) assms(2) Pi_I[of UNIV g ]
-      poly_eval_indexed_pmult set_eq_subset subsetD zero_closed 
+      poly_eval_indexed_pmult set_eq_subset subsetD zero_closed
   by smt
 
 lemma poly_eval_eval_function_eq:
   assumes "closed_fun R g"
   assumes "closed_fun R g'"
   assumes "restrict g S = restrict g' S"
   assumes "P \<in> Pring_set R I"
   shows "poly_eval R S g P = poly_eval R S g' P"
   apply(rule indexed_pset.induct[of P I "carrier R"])
      apply (simp add: assms(4))
     apply (metis poly_eval_constant)
    apply (metis assms(1) assms(2) poly_eval_add)
 proof- fix P i assume A: "P \<in> Pring_set R I" "poly_eval R S g P = poly_eval R S g' P" "i \<in> I "
   show "poly_eval R S g (P \<Otimes> i) = poly_eval R S g' (P \<Otimes> i)"
   proof(cases "i \<in> S")
     case True
     then have "g i = g' i"
-      using assms 
-      unfolding  restrict_def  
+      using assms
+      unfolding  restrict_def
       by meson
-    then show ?thesis     
-      using assms A poly_eval_indexed_pmult[of P I g S i] poly_eval_indexed_pmult[of P I g' S i] 
+    then show ?thesis
+      using assms A poly_eval_indexed_pmult[of P I g S i] poly_eval_indexed_pmult[of P I g' S i]
       by presburger
   next
     case False
-    then show ?thesis 
-      using assms A poly_eval_indexed_pmult[of P I g S i] poly_eval_indexed_pmult[of P I g' S i] 
+    then show ?thesis
+      using assms A poly_eval_indexed_pmult[of P I g S i] poly_eval_indexed_pmult[of P I g' S i]
       by presburger
   qed
-qed   
+qed
 
 lemma poly_eval_eval_set_eq:
   assumes "closed_fun R g"
   assumes "S \<inter> I = S' \<inter> I"
   assumes "P \<in> Pring_set R I"
   assumes "\<one> \<noteq>\<zero>"
   shows "poly_eval R S g P = poly_eval R S' g P"
   apply(rule indexed_pset.induct[of P I "carrier R"])
      apply (simp add: assms(3))
   apply (metis poly_eval_constant)
    apply (metis assms(1) poly_eval_add)
 proof- fix P i
   assume A: " P \<in> Pring_set R I" "poly_eval R S g P = poly_eval R S' g P" "i \<in> I "
   show "poly_eval R S g (P \<Otimes> i) = poly_eval R S' g (P \<Otimes> i)"
-  using assms poly_eval_index[of g _ i] A 
+  using assms poly_eval_index[of g _ i] A
   by (metis Diff_Diff_Int Diff_iff poly_eval_indexed_pmult)
 qed
 
 lemma poly_eval_trivial:
   assumes "closed_fun R g"
   assumes "P \<in> Pring_set R (I - S)"
   shows "poly_eval R S g P = P"
   apply(rule indexed_pset.induct[of P "I - S" "carrier R"])
   apply (simp add: assms(2))
   using poly_eval_constant apply blast
   apply (metis assms(1) poly_eval_add)
   by (metis Diff_iff assms(1) poly_eval_indexed_pmult)
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsubsection\<open>Total Evaluation of a Polynomial\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 lemma zero_fun_closed:
 "closed_fun R (\<lambda>n. \<zero>)"
   by blast
 
 lemma deg_zero_cf_eval:
   shows "P \<in> Pring_set R I \<Longrightarrow> poly_eval R I (\<lambda>n. \<zero>) P = indexed_const (P {#})"
   apply(erule indexed_pset.induct)
   apply (metis indexed_const_def poly_eval_constant)
 proof-
   show " \<And>P Q. P \<in> Pring_set R I \<Longrightarrow>
            poly_eval R I (\<lambda>n. \<zero>) P = indexed_const (P {#}) \<Longrightarrow>
            Q \<in> Pring_set R I \<Longrightarrow> poly_eval R I (\<lambda>n. \<zero>) Q = indexed_const (Q {#}) \<Longrightarrow> poly_eval R I (\<lambda>n. \<zero>) (P \<Oplus> Q) = indexed_const ((P \<Oplus> Q) {#})"
   proof-
-    fix P 
+    fix P
     fix Q
     assume A: "P \<in> Pring_set R I"
     assume B: " poly_eval R I (\<lambda>n. \<zero>) P = indexed_const (P {#})"
     assume C: "Q \<in> Pring_set R I "
     assume D: " poly_eval R I (\<lambda>n. \<zero>) Q = indexed_const (Q {#})"
     have "indexed_const ((P \<Oplus> Q) {#}) = indexed_const (P {#})  \<Oplus> indexed_const (Q {#})"
       by (metis indexed_padd_const indexed_padd_def)
     thus "poly_eval R I (\<lambda>n. \<zero>) (P \<Oplus> Q) = indexed_const ((P \<Oplus> Q) {#})"
-      using A B C D poly_eval_add[of P I Q "\<lambda>n. \<zero>" I] 
+      using A B C D poly_eval_add[of P I Q "\<lambda>n. \<zero>" I]
       by (smt zero_fun_closed)
   qed
   show "\<And>P i. P \<in> Pring_set R I \<Longrightarrow> poly_eval R I (\<lambda>n. \<zero>) P = indexed_const (P {#}) \<Longrightarrow> i \<in> I \<Longrightarrow> poly_eval R I (\<lambda>n. \<zero>) (P \<Otimes> i) = indexed_const ((P \<Otimes> i) {#})"
   proof-
     fix P
     fix i
     assume A: "P \<in> Pring_set R I" "poly_eval R I (\<lambda>n. \<zero>) P = indexed_const (P {#})" "i \<in> I"
     show "poly_eval R I (\<lambda>n. \<zero>) (P \<Otimes> i) = indexed_const ((P \<Otimes> i) {#})"
     proof-
       have "poly_eval R I (\<lambda>n. \<zero>) (P \<Otimes> i) = indexed_const \<zero>"
-        using A(1) A(3) cring.poly_eval_constant is_cring poly_eval_indexed_pmult 
+        using A(1) A(3) cring.poly_eval_constant is_cring poly_eval_indexed_pmult
               poly_eval_scalar_mult poly_scalar_mult_zero zero_closed
         by (metis Pi_I)
       have "(P \<Otimes> i) {#} = \<zero>"
-        using indexed_pmult_def 
+        using indexed_pmult_def
         by (metis empty_iff set_mset_empty)
-      then show ?thesis 
+      then show ?thesis
         using \<open>poly_eval R I (\<lambda>n. \<zero>) (P \<Otimes> i) = indexed_const \<zero>\<close>
         by presburger
     qed
   qed
 qed
 
 lemma deg_zero_cf_mult:
   assumes "P \<in> Pring_set R I"
   assumes "Q \<in> Pring_set R I"
   shows " (P \<Otimes>\<^sub>p Q) {#} = P {#} \<otimes> Q {#}"
 proof-
   have "poly_eval R I (\<lambda>n. \<zero>) (P \<Otimes>\<^sub>p Q) = poly_eval R I (\<lambda>n. \<zero>) P \<Otimes>\<^sub>p  poly_eval R I (\<lambda>n. \<zero>) Q"
-    using zero_fun_closed assms  
+    using zero_fun_closed assms
     by (metis poly_eval_mult)
   then have 0: "indexed_const ((P \<Otimes>\<^sub>p Q) {#}) = (indexed_const (P {#})) \<Otimes>\<^sub>p (indexed_const (Q {#}))"
     by (metis P_ring_mult_closed' Pring_cfs_closed assms(1) assms(2) deg_zero_cf_eval indexed_const_P_mult_eq indexed_const_def)
   have "indexed_const ((P \<Otimes>\<^sub>p Q) {#}) = (indexed_const ((P {#}) \<otimes>(Q {#})))"
     apply(rule ccontr)
-    using 0 
+    using 0
     by (metis Pring_cfs_closed assms(1) assms(2) indexed_const_P_mult_eq)
-  then show ?thesis 
+  then show ?thesis
   proof -
     show ?thesis
-      by (metis (no_types) \<open>indexed_const ((P \<Otimes>\<^sub>p Q) {#}) = indexed_const (P {#} \<otimes> Q {#})\<close> 
+      by (metis (no_types) \<open>indexed_const ((P \<Otimes>\<^sub>p Q) {#}) = indexed_const (P {#} \<otimes> Q {#})\<close>
         local.ring_axioms ring.indexed_const_def)
   qed
 qed
 
 definition deg_zero_cf :: "('a, 'c) mvar_poly \<Rightarrow> 'a" where
 "deg_zero_cf P = P {#}"
 
 lemma deg_zero_cf_ring_hom:
   shows "ring_hom_ring (Pring R I) R (deg_zero_cf)"
   apply(rule ring_hom_ringI)
   using Pring_is_ring apply blast
   apply (simp add: local.ring_axioms)
   apply (metis deg_zero_cf_def Pring_car Pring_cfs_closed)
   apply (metis deg_zero_cf_def Pring_car Pring_mult deg_zero_cf_mult)
   apply (metis deg_zero_cf_def Pring_add indexed_padd_def)
   by (metis deg_zero_cf_def Pring_one indexed_const_def)
 
 end
 
 definition eval_in_ring ::
  "('a,'b) ring_scheme \<Rightarrow> 'c set \<Rightarrow> ('c \<Rightarrow> 'a) \<Rightarrow> ('a, 'c) mvar_poly \<Rightarrow> 'a" where
 "eval_in_ring R S g P = (poly_eval R S g P) {#}"
 
 definition total_eval ::
 "('a,'b) ring_scheme \<Rightarrow> ('c \<Rightarrow> 'a) \<Rightarrow> ('a, 'c) mvar_poly \<Rightarrow> 'a"  where
 "total_eval R g P = eval_in_ring R UNIV g P"
 
 context cring
 begin
 
 lemma eval_in_ring_ring_hom:
   assumes "closed_fun R g"
   shows "ring_hom_ring (Pring R I) R (eval_in_ring R S g)"
-  unfolding eval_in_ring_def 
+  unfolding eval_in_ring_def
   apply(rule ring_hom_ringI)
   apply (simp add: Pring_is_ring)
   apply (simp add: local.ring_axioms)
-    using Pring_car Pring_carrier_coeff' assms poly_eval_closed 
-     apply (metis )     
-        using Pring_mult[of I] poly_eval_mult[of _ I _ g S] poly_eval_closed[of g _ I S] deg_zero_cf_mult[of _ I] assms 
+    using Pring_car Pring_carrier_coeff' assms poly_eval_closed
+     apply (metis )
+        using Pring_mult[of I] poly_eval_mult[of _ I _ g S] poly_eval_closed[of g _ I S] deg_zero_cf_mult[of _ I] assms
           Pring_car[of I]
         apply (metis deg_zero_cf_mult)
-        
+
           using Pring_add[of I ] poly_eval_add[of _ I _ g S] Pring_car[of I] assms indexed_padd_def
           apply metis
-            unfolding deg_zero_cf_def 
-            by (metis Pring_one indexed_const_def one_closed poly_eval_constant)        
+            unfolding deg_zero_cf_def
+            by (metis Pring_one indexed_const_def one_closed poly_eval_constant)
 
 lemma eval_in_ring_smult:
   assumes "P \<in> carrier (Pring R I)"
   assumes "a \<in> carrier R"
   assumes "closed_fun R g"
   shows "eval_in_ring R S g (a \<odot>\<^bsub>Pring R I\<^esub> P) = a \<otimes> eval_in_ring R S g P "
-  using assms unfolding eval_in_ring_def  
+  using assms unfolding eval_in_ring_def
   by (smt Pring_car Pring_smult poly_eval_scalar_mult poly_scalar_mult_def)
 
 
 lemma total_eval_ring_hom:
   assumes "closed_fun R g"
   shows "ring_hom_ring (Pring R I) R (total_eval R g)"
-  using assms unfolding total_eval_def 
+  using assms unfolding total_eval_def
   using eval_in_ring_ring_hom by blast
 
 lemma total_eval_smult:
   assumes "P \<in> carrier (Pring R I)"
   assumes "a \<in> carrier R"
   assumes "closed_fun R g"
   shows "total_eval R g (a \<odot>\<^bsub>Pring R I\<^esub> P) = a \<otimes> total_eval R g P"
-  using assms unfolding total_eval_def 
+  using assms unfolding total_eval_def
   using eval_in_ring_smult by blast
 
 lemma total_eval_const:
   assumes "k \<in> carrier R"
   shows "total_eval R g (indexed_const k) = k"
-  unfolding total_eval_def eval_in_ring_def 
-  using assms 
+  unfolding total_eval_def eval_in_ring_def
+  using assms
   by (metis indexed_const_def poly_eval_constant)
 
 lemma total_eval_var:
   assumes "closed_fun R g"
   shows "(total_eval R g (mset_to_IP R {#i#})) = g i"
-  unfolding total_eval_def eval_in_ring_def 
-  using UNIV_I assms indexed_const_def indexed_pmult_zero monom_add_mset one_closed 
+  unfolding total_eval_def eval_in_ring_def
+  using UNIV_I assms indexed_const_def indexed_pmult_zero monom_add_mset one_closed
       one_mset_to_IP one_zeroD poly_eval_constant poly_eval_index  singletonD
   by (smt PiE iso_tuple_UNIV_I monom_eval_add monom_eval_car mset_to_IP_simp poly_scalar_mult_const r_one)
 
 lemma total_eval_indexed_pmult:
   assumes "P \<in> carrier (Pring R I)"
   assumes "i \<in> I"
   assumes "closed_fun R g"
   shows "total_eval R g (P \<Otimes> i) = total_eval R g P \<otimes>\<^bsub>R\<^esub> g i"
 proof-
   have "P \<Otimes> i = P \<Otimes>\<^sub>p mset_to_IP R ((add_mset i) {#})"
-    using assms poly_index_mult[of P I i] Pring_car 
+    using assms poly_index_mult[of P I i] Pring_car
     by blast
   then have 0: "(P \<Otimes> i) =  P \<otimes>\<^bsub>Pring R I\<^esub> (mset_to_IP R {#i#})"
     by (simp add: Pring_mult)
   then have "total_eval R g (P \<Otimes> i) = (total_eval R g P) \<otimes>\<^bsub>R\<^esub> (total_eval R g (mset_to_IP R {#i#}))"
   proof-
     have "(mset_to_IP R {#i#}) \<in> carrier (Pring R I)"
       by (metis Pring_car assms(2) mset_to_IP_closed set_mset_single singletonD subset_iff)
-    then show ?thesis 
-      using assms total_eval_ring_hom[of g I] ring_hom_mult 
+    then show ?thesis
+      using assms total_eval_ring_hom[of g I] ring_hom_mult
       by (metis "0" ring_hom_ring.homh)
   qed
-  then show ?thesis 
+  then show ?thesis
     by (metis assms(3) cring.total_eval_var is_cring)
 qed
 
 lemma total_eval_mult:
   assumes "P \<in> carrier (Pring R I)"
   assumes "Q \<in> carrier (Pring R I)"
   assumes "closed_fun R g"
   shows "total_eval R g (P \<otimes>\<^bsub>Pring R I\<^esub> Q) = (total_eval R g P) \<otimes>\<^bsub>R\<^esub>(total_eval R g Q) "
   by (metis assms(1) assms(2) assms(3) ring_hom_mult ring_hom_ring.homh total_eval_ring_hom)
 
 lemma total_eval_add:
   assumes "P \<in> carrier (Pring R I)"
   assumes "Q \<in> carrier (Pring R I)"
   assumes "closed_fun R g"
   shows "total_eval R g (P \<oplus>\<^bsub>Pring R I\<^esub> Q) = (total_eval R g P) \<oplus>\<^bsub>R\<^esub>(total_eval R g Q) "
   by (metis assms(1) assms(2) assms(3) ring_hom_add ring_hom_ring.homh total_eval_ring_hom)
 
 lemma total_eval_one:
   assumes "closed_fun R g"
   shows "total_eval R g \<one>\<^bsub>Pring R I\<^esub> = \<one>"
   by (metis Pring_one one_closed total_eval_const)
 
 lemma total_eval_zero:
   assumes "closed_fun R g"
   shows "total_eval R g \<zero>\<^bsub>Pring R I\<^esub> = \<zero>"
   by (metis Pring_zero total_eval_const zero_closed)
 
 lemma total_eval_closed:
   assumes "P \<in> carrier (Pring R I)"
   assumes "closed_fun R g"
   shows "total_eval R g P \<in> carrier R"
   using assms total_eval_ring_hom[of g]
   by (metis ring_hom_closed ring_hom_ring.homh)
 
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsection\<open>Constructing Homomorphisms from Indexed Polynomial Rings and a Universal Property\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 text\<open>The inclusion of \<open>R\<close> into its polynomial ring\<close>
 
 lemma indexed_const_ring_hom:
 "ring_hom_ring R (Pring R I) (indexed_const)"
   apply(rule ring_hom_ringI)
   apply (simp add: local.ring_axioms)
   apply (simp add: Pring_is_ring)
   using Pring_car indexed_pset.indexed_const apply blast
   apply (metis Pring_mult indexed_const_P_mult_eq)
   apply (metis indexed_padd_const local.ring_axioms ring.Pring_add)
   by (metis Pring_one)
 
 lemma indexed_const_inj_on:
 "inj_on (indexed_const) (carrier R)"
   by (metis cring.total_eval_const inj_onI is_cring)
 
 end
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsubsection\<open>Mapping $R[x] \to S[x]$ along a homomorphism $R \to S$\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 definition ring_hom_to_IP_ring_hom ::
 "('a, 'e) ring_hom \<Rightarrow> ('a, 'c) mvar_poly \<Rightarrow> 'c monomial \<Rightarrow> 'e"  where
 "ring_hom_to_IP_ring_hom \<phi> P m = \<phi> (P m)"
 
 context cring
 begin
 
 lemma ring_hom_to_IP_ring_hom_one:
   assumes "cring S"
   assumes "ring_hom_ring R S \<phi>"
   shows "ring_hom_to_IP_ring_hom \<phi> \<one>\<^bsub>Pring R I\<^esub> = \<one>\<^bsub>Pring S I\<^esub>"
-  unfolding ring_hom_to_IP_ring_hom_def 
+  unfolding ring_hom_to_IP_ring_hom_def
 proof
   fix m
   show " \<phi> (\<one>\<^bsub>Pring R I\<^esub> m) = \<one>\<^bsub>Pring S I\<^esub> m"
   proof(cases "m = {#}")
     case True
     then have "(\<one>\<^bsub>Pring R I\<^esub> m) = \<one>\<^bsub>R\<^esub>"
-      by (metis Pring_one indexed_const_def)      
+      by (metis Pring_one indexed_const_def)
     then have "\<phi> (\<one>\<^bsub>Pring R I\<^esub> m) = \<one>\<^bsub>S\<^esub>"
-      using assms 
-      unfolding ring_hom_ring_def 
+      using assms
+      unfolding ring_hom_ring_def
       by (metis assms(2) ring_hom_one ring_hom_ring.homh)
-    then show ?thesis using assms cring True ring.indexed_const_def ring_hom_ring_def
+    then show ?thesis using assms True ring.indexed_const_def ring_hom_ring_def
       by (metis ring.Pring_one)
   next
     case False
       then have "(\<one>\<^bsub>Pring R I\<^esub> m) = \<zero>\<^bsub>R\<^esub>"
         by (metis indexed_const_def local.ring_axioms ring.Pring_one)
-        
+
     then have "\<phi> (\<one>\<^bsub>Pring R I\<^esub> m) = \<zero>\<^bsub>S\<^esub>"
-      using assms 
+      using assms
       unfolding ring_hom_ring_def cring_def
-      by (metis assms(2) ring_hom_zero ring_hom_ring.homh)     
+      by (metis assms(2) ring_hom_zero ring_hom_ring.homh)
     then show ?thesis using assms False ring.indexed_const_def ring_hom_ring_def
       by (metis ring.Pring_one)
   qed
 qed
 
 lemma ring_hom_to_IP_ring_hom_constant:
   assumes "cring S"
   assumes "ring_hom_ring R S \<phi>"
   assumes "a \<in> carrier R"
   shows "ring_hom_to_IP_ring_hom \<phi> ((indexed_const a):: 'c monomial \<Rightarrow> 'a) = ring.indexed_const S (\<phi> a)"
-  unfolding ring_hom_to_IP_ring_hom_def indexed_const_def 
+  unfolding ring_hom_to_IP_ring_hom_def indexed_const_def
 proof
-  fix m:: "'c monomial"  
+  fix m:: "'c monomial"
   show "\<phi> (if m = {#} then a else \<zero>) = ring.indexed_const S (\<phi> a) m"
     apply(cases "m = {#}")
   apply (simp add: assms(1) cring.axioms(1) ring.indexed_const_def)
   proof-
     assume "m \<noteq> {#} "
     then have "\<phi> (if m = {#} then a else \<zero>) = \<zero>\<^bsub>S\<^esub>"
-      by (metis (full_types) assms(1) assms(2) cring_def local.ring_axioms 
+      by (metis (full_types) assms(1) assms(2) cring_def local.ring_axioms
           ring_hom_ring.homh ring_hom_zero)
     then show " \<phi> (if m = {#} then a else \<zero>) = ring.indexed_const S (\<phi> a) m"
-      using assms  
-      by (metis \<open>m \<noteq> {#}\<close> cring.axioms(1) ring.indexed_const_def)      
+      using assms
+      by (metis \<open>m \<noteq> {#}\<close> cring.axioms(1) ring.indexed_const_def)
   qed
 qed
 
 lemma ring_hom_to_IP_ring_hom_add:
   assumes "cring S"
   assumes "ring_hom_ring R S \<phi>"
   assumes "P \<in> carrier (Pring R I)"
   assumes "Q \<in> carrier (Pring R I)"
-  shows "ring_hom_to_IP_ring_hom \<phi> (P \<oplus>\<^bsub>Pring R I\<^esub> Q) = 
+  shows "ring_hom_to_IP_ring_hom \<phi> (P \<oplus>\<^bsub>Pring R I\<^esub> Q) =
        (ring_hom_to_IP_ring_hom \<phi> P) \<oplus>\<^bsub>Pring S I\<^esub> (ring_hom_to_IP_ring_hom \<phi> Q)"
-  unfolding ring_hom_to_IP_ring_hom_def 
+  unfolding ring_hom_to_IP_ring_hom_def
 proof
   fix m
   show " \<phi> ((P \<oplus>\<^bsub>Pring R I\<^esub> Q) m) = ((\<lambda>m. \<phi> (P m)) \<oplus>\<^bsub>Pring S I\<^esub> (\<lambda>m. \<phi> (Q m))) m"
   proof-
     have RHS: "((\<lambda>m. \<phi> (P m)) \<oplus>\<^bsub>Pring S I\<^esub> (\<lambda>m. \<phi> (Q m))) m = \<phi> (P m) \<oplus>\<^bsub>S\<^esub> \<phi> (Q m)"
-      using ring.Pring_add[of  S I _ _ ] assms 
+      using ring.Pring_add[of  S I _ _ ] assms
       by (metis cring.axioms(1) ring.indexed_padd_def)
     have LHS: "\<phi> ((P \<oplus>\<^bsub>Pring R I\<^esub> Q) m) = \<phi> ((P m)\<oplus>\<^bsub>R\<^esub> (Q m))"
       by (metis Pring_add indexed_padd_def)
-    then show ?thesis 
-      using assms unfolding ring_hom_ring_def 
-      using  ring_hom_add[of \<phi> R S "P m" "Q m"] 
+    then show ?thesis
+      using assms unfolding ring_hom_ring_def
+      using  ring_hom_add[of \<phi> R S "P m" "Q m"]
       by (metis Pring_carrier_coeff' RHS assms(2) ring_hom_ring.homh)
   qed
 qed
 
 lemma ring_hom_to_IP_ring_hom_closed:
   assumes "cring S"
   assumes "ring_hom_ring R S \<phi>"
   assumes "P \<in> carrier (Pring R I)"
   shows "ring_hom_to_IP_ring_hom \<phi> P \<in> carrier (Pring S I)"
   apply(rule indexed_pset.induct[of P I "carrier R"])
   using Pring_car assms(3) apply blast
 proof-
   show "\<And>k. k \<in> carrier R \<Longrightarrow> ring_hom_to_IP_ring_hom \<phi> (indexed_const k) \<in> carrier (Pring S I)"
   proof-
     fix k
     show " k \<in> carrier R \<Longrightarrow> ring_hom_to_IP_ring_hom \<phi> (indexed_const k) \<in> carrier (Pring S I)"
     proof-
       assume A: "k \<in> carrier R"
       have "(\<phi> k) \<in> carrier S"
         by (meson A assms(2) ring_hom_closed ring_hom_ring.homh)
       then have 0: "ring.indexed_const S (\<phi> k) \<in> carrier (Pring S I)"
         by (metis assms(1) cring.axioms(1) ring.Pring_car ring.indexed_pset.indexed_const)
-      then show ?thesis  
-        using  assms(2) 
+      then show ?thesis
+        using  assms(2)
         by (simp add: "0" A ring_hom_to_IP_ring_hom_constant[of S \<phi> k] assms(1))
     qed
   qed
   show "\<And>P Q. P \<in> Pring_set R I \<Longrightarrow>
            ring_hom_to_IP_ring_hom \<phi> P \<in> carrier (Pring S I) \<Longrightarrow>
            Q \<in> Pring_set R I \<Longrightarrow>
            ring_hom_to_IP_ring_hom \<phi> Q \<in> carrier (Pring S I) \<Longrightarrow> ring_hom_to_IP_ring_hom \<phi> (P \<Oplus> Q) \<in> carrier (Pring S I)"
     using ring_hom_to_IP_ring_hom_add[of S \<phi> _ I] Pring_car[of I] ring.Pring_car[of S I]
     by (metis Pring_add assms(1) assms(2) cring.axioms(1) ring.Pring_add_closed )
   show "\<And>P i. P \<in> Pring_set R I \<Longrightarrow>
            ring_hom_to_IP_ring_hom \<phi> P \<in> carrier (Pring S I) \<Longrightarrow> i \<in> I \<Longrightarrow> ring_hom_to_IP_ring_hom \<phi> (P \<Otimes> i) \<in> carrier (Pring S I)"
   proof-
-    fix P 
+    fix P
     fix i
     assume A: "P \<in> Pring_set R I " "ring_hom_to_IP_ring_hom \<phi> P \<in> carrier (Pring S I) " "i \<in> I"
     have 0: "(\<lambda>m. \<phi> ((P \<Otimes> i) m)) = ring.indexed_pmult S (ring_hom_to_IP_ring_hom \<phi> P) i"
     proof
       fix m
       show "\<phi> ((P \<Otimes> i) m) = ring.indexed_pmult S (ring_hom_to_IP_ring_hom \<phi> P) i m"
       proof(cases "i \<in># m")
         case True
         then have LHS: "\<phi> ((P \<Otimes> i) m) = \<phi>  (P (m - {#i#}))"
           by (metis indexed_pmult_def)
-        then show ?thesis 
-          using True 
+        then show ?thesis
+          using True
           by (metis assms(1) cring.axioms(1) ring.indexed_pmult_def ring_hom_to_IP_ring_hom_def)
       next
         case False
         then have "\<phi> ((P \<Otimes> i) m) = \<phi> \<zero>\<^bsub>R\<^esub>"
           by (metis indexed_pmult_def)
         then have LHS: "\<phi> ((P \<Otimes> i) m) = \<zero>\<^bsub>S\<^esub>"
-          by (metis assms(1) assms(2) cring.axioms(1) local.ring_axioms 
+          by (metis assms(1) assms(2) cring.axioms(1) local.ring_axioms
               ring_hom_ring.homh ring_hom_zero)
-        then show ?thesis 
-          using False assms ring.indexed_pmult_def 
+        then show ?thesis
+          using False assms ring.indexed_pmult_def
           by (metis cring.axioms(1))
       qed
     qed
     then show "ring_hom_to_IP_ring_hom \<phi> (P \<Otimes> i) \<in> carrier (Pring S I)"
-      using assms 
-      unfolding ring_hom_to_IP_ring_hom_def 
+      using assms
+      unfolding ring_hom_to_IP_ring_hom_def
       by (metis "0" A(2) A(3) cring.axioms(1) ring.Pring_car ring.indexed_pset.simps)
-  qed  
+  qed
 qed
 
 lemma ring_hom_to_IP_ring_hom_monom:
   assumes "cring S"
   assumes "ring_hom_ring R S \<phi>"
   shows "ring_hom_to_IP_ring_hom \<phi> (mset_to_IP R m) = mset_to_IP S m"
 proof
-  fix x 
+  fix x
   show "ring_hom_to_IP_ring_hom \<phi> (mset_to_IP R m) x = mset_to_IP S m x"
-    unfolding ring_hom_to_IP_ring_hom_def mset_to_IP_def apply( cases "x = m" ) 
+    unfolding ring_hom_to_IP_ring_hom_def mset_to_IP_def apply( cases "x = m" )
      apply (metis (mono_tags, lifting) assms(2) ring_hom_one ring_hom_ring.homh)
-    by (metis (full_types) assms(1) assms(2) cring.axioms(1) local.ring_axioms 
+    by (metis (full_types) assms(1) assms(2) cring.axioms(1) local.ring_axioms
         ring_hom_ring.homh ring_hom_zero)
 qed
 
 lemma Pring_morphism:
   assumes "cring S"
   assumes "\<phi> \<in> (carrier (Pring R I)) \<rightarrow> (carrier S)"
   assumes "\<phi> \<one>\<^bsub>Pring R I\<^esub> = \<one>\<^bsub>S\<^esub>"
   assumes "\<phi> \<zero>\<^bsub>Pring R I\<^esub> = \<zero>\<^bsub>S\<^esub>"
-  assumes "\<And>P Q. P \<in> carrier (Pring R I) \<Longrightarrow> Q \<in> carrier (Pring R I) \<Longrightarrow> 
+  assumes "\<And>P Q. P \<in> carrier (Pring R I) \<Longrightarrow> Q \<in> carrier (Pring R I) \<Longrightarrow>
           \<phi> (P \<oplus>\<^bsub>Pring R I\<^esub> Q) = (\<phi> P) \<oplus>\<^bsub>S\<^esub> (\<phi> Q)"
   assumes "\<And> i . \<And> P. i \<in> I \<Longrightarrow> P \<in> carrier (Pring R I) \<Longrightarrow>  \<phi> (P \<Otimes> i) =  (\<phi> P) \<otimes>\<^bsub>S\<^esub> (\<phi> (mset_to_IP R {#i#}))"
-  assumes "\<And>k Q. k \<in> carrier R \<Longrightarrow> Q \<in> carrier (Pring R I) \<Longrightarrow>  \<phi> (poly_scalar_mult R k Q) = 
+  assumes "\<And>k Q. k \<in> carrier R \<Longrightarrow> Q \<in> carrier (Pring R I) \<Longrightarrow>  \<phi> (poly_scalar_mult R k Q) =
                                      (\<phi> (indexed_const k)) \<otimes>\<^bsub>S\<^esub> (\<phi> Q)"
   shows "ring_hom_ring (Pring R I) S \<phi>"
   apply(rule ring_hom_ringI)
   apply (simp add: Pring_is_ring; fail)
   apply (simp add: assms(1) cring.axioms(1); fail)
-  using assms(2) apply blast  
+  using assms(2) apply blast
 proof-
 
   show "\<And>x y. x \<in> carrier (Pring R I) \<Longrightarrow> y \<in> carrier (Pring R I) \<Longrightarrow> \<phi> (x \<otimes>\<^bsub>Pring R I\<^esub> y) = \<phi> x \<otimes>\<^bsub>S\<^esub> \<phi> y"
   proof-
     fix P Q
     assume A0: "P \<in> carrier (Pring R I)"
     assume A1: "Q \<in> carrier (Pring R I)"
     show "\<phi> (P \<otimes>\<^bsub>Pring R I\<^esub> Q) = \<phi> P \<otimes>\<^bsub>S\<^esub> \<phi> Q"
     proof(rule mpoly_induct'[of ])
       show "\<phi> (P \<otimes>\<^bsub>Pring R I\<^esub> indexed_const \<zero>) = \<phi> P \<otimes>\<^bsub>S\<^esub> \<phi> (indexed_const \<zero>)"
       proof-
         have 0: "(P \<otimes>\<^bsub>Pring R I\<^esub> indexed_const \<zero>) = \<zero>\<^bsub>Pring R I\<^esub>"
-          using assms 
+          using assms
           by (metis A0 Pring_is_cring Pring_zero cring.cring_simprules(27) is_cring)
         have 1: " \<phi> P \<otimes>\<^bsub>S\<^esub> \<phi> (indexed_const \<zero>) = \<zero>\<^bsub>S\<^esub>"
         proof-
           have "\<phi> P \<in> carrier S"
             using assms(2) A0
             by blast
-          then show ?thesis 
-            using assms(4) Pring_zero[of I] 
+          then show ?thesis
+            using assms(4) Pring_zero[of I]
             by (metis assms(1) cring.cring_simprules(27))
         qed
-        then show ?thesis using 0 1 
+        then show ?thesis using 0 1
           using assms(4) by presburger
-      qed 
+      qed
       show "\<And>n Q. (\<And>Q. Q \<in> Pring_set R I \<and> card (monomials_of R Q) \<le> n \<Longrightarrow> \<phi> (P \<otimes>\<^bsub>Pring R I\<^esub> Q) = \<phi> P \<otimes>\<^bsub>S\<^esub> \<phi> Q) \<Longrightarrow>
            Q \<in> Pring_set R I \<and> card (monomials_of R Q) = Suc n \<Longrightarrow> \<phi> (P \<otimes>\<^bsub>Pring R I\<^esub> Q) = \<phi> P \<otimes>\<^bsub>S\<^esub> \<phi> Q"
       proof- fix n fix Q
         assume IH: " (\<And>Q. Q \<in> Pring_set R I \<and> card (monomials_of R Q) \<le> n \<Longrightarrow> \<phi> (P \<otimes>\<^bsub>Pring R I\<^esub> Q) = \<phi> P \<otimes>\<^bsub>S\<^esub> \<phi> Q)"
         assume A: "Q \<in> Pring_set R I \<and> card (monomials_of R Q) = Suc n"
         show "\<phi> (P \<otimes>\<^bsub>Pring R I\<^esub> Q) = \<phi> P \<otimes>\<^bsub>S\<^esub> \<phi> Q"
         proof-
           obtain m M where m_M_def: "monomials_of R Q = insert m M \<and> m \<notin> M"
             using A
             by (metis card_0_eq ex_in_conv finite.emptyI mk_disjoint_insert nat.distinct(1))
           have "Q = (restrict_poly_to_monom_set R Q M) \<oplus>\<^bsub>Pring R I\<^esub> (poly_scalar_mult R (Q m) (mset_to_IP R m))"
             by (metis A Pring_add m_M_def remove_monom_eq remove_monom_restrict_poly_to_monom_set)
           obtain Q' where Q'_def: "Q' = (restrict_poly_to_monom_set R Q M)"
             by simp
           have Q'_fact: " Q' \<in> Pring_set R I \<and> card (monomials_of R Q') \<le> n"
             proof-
               have 0: "Q' \<in> Pring_set R I"
-                using Q'_def  A restriction_closed 
+                using Q'_def  A restriction_closed
                 by blast
-              have "monomials_of R Q' = M" 
-                using A m_M_def Q'_def 
+              have "monomials_of R Q' = M"
+                using A m_M_def Q'_def
                 by (metis restrict_poly_to_monom_set_monoms subset_insertI)
               then have "card (monomials_of R Q') = n" using A m_M_def
                 by (metis "0" card_insert_disjoint diff_Suc_1 monomials_finite)
-              then show ?thesis 
+              then show ?thesis
                 by (simp add: "0")
-            qed 
+            qed
           have 0:"(P \<otimes>\<^bsub>Pring R I\<^esub> Q) = (P \<otimes>\<^bsub>Pring R I\<^esub> (Q' \<oplus>\<^bsub>Pring R I\<^esub> (poly_scalar_mult R (Q m) (mset_to_IP R m))))"
             using Q'_def \<open>Q = restrict_poly_to_monom_set R Q M \<oplus>\<^bsub>Pring R I\<^esub> Mt (Q m) m\<close> by presburger
           have 1: "(P \<otimes>\<^bsub>Pring R I\<^esub> Q) = (P \<otimes>\<^bsub>Pring R I\<^esub> Q') \<oplus>\<^bsub>Pring R I\<^esub> (P \<otimes>\<^bsub>Pring R I\<^esub> (poly_scalar_mult R (Q m) (mset_to_IP R m)))"
           proof-
             have 10: "P \<in> carrier (Pring R I)"
               by (simp add: A0)
             have 11: "Q \<in> carrier (Pring R I)"
               by (simp add: A Pring_car)
             have 12: "Q' \<in> carrier (Pring R I)"
               using A Pring_car Q'_def restriction_closed by blast
             have 13: "(poly_scalar_mult R (Q m) (mset_to_IP R m)) \<in> carrier (Pring R I)"
               by (metis A Pring_car Pring_carrier_coeff' insert_iff m_M_def mset_to_IP_closed' poly_scalar_mult_closed)
-            then show ?thesis 
-              using 0 10 11 12 13 
+            then show ?thesis
+              using 0 10 11 12 13
               by (metis Pring_mult_rdistr)
           qed
           then have "\<phi> (P \<otimes>\<^bsub>Pring R I\<^esub> Q) = (\<phi> (P \<otimes>\<^bsub>Pring R I\<^esub> Q')) \<oplus>\<^bsub>S\<^esub> \<phi> (P \<otimes>\<^bsub>Pring R I\<^esub> (poly_scalar_mult R (Q m) (mset_to_IP R m)))"
             by (metis A A0 Pring_car Pring_mult_closed Pring_cfs_closed Q'_def assms(5) insert_iff
-                local.ring_axioms m_M_def poly_scalar_mult_closed ring.mset_to_IP_closed' 
+                local.ring_axioms m_M_def poly_scalar_mult_closed ring.mset_to_IP_closed'
                 ring.restriction_closed)
           then have "\<phi> (P \<otimes>\<^bsub>Pring R I\<^esub> Q) = (\<phi> P) \<otimes>\<^bsub>S\<^esub> (\<phi> Q') \<oplus>\<^bsub>S\<^esub> \<phi> (P \<otimes>\<^bsub>Pring R I\<^esub> (poly_scalar_mult R (Q m) (mset_to_IP R m)))"
             using IH[of Q'] Q'_fact
             by presburger
           then have 2: "\<phi> (P \<otimes>\<^bsub>Pring R I\<^esub> Q) = (\<phi> P) \<otimes>\<^bsub>S\<^esub> (\<phi> Q') \<oplus>\<^bsub>S\<^esub> \<phi> (poly_scalar_mult R (Q m) (P \<otimes>\<^bsub>Pring R I\<^esub>  (mset_to_IP R m)))"
             by (metis A A0 Pring_car Pring_mult Pring_cfs_closed insert_iff m_M_def mset_to_IP_closed' times_poly_scalar_mult)
-          have 3: "\<And>k. k \<in> carrier R \<Longrightarrow> set_mset m \<subseteq> I \<Longrightarrow> \<phi> (poly_scalar_mult R k (P \<otimes>\<^bsub>Pring R I\<^esub>  (mset_to_IP R m))) = 
+          have 3: "\<And>k. k \<in> carrier R \<Longrightarrow> set_mset m \<subseteq> I \<Longrightarrow> \<phi> (poly_scalar_mult R k (P \<otimes>\<^bsub>Pring R I\<^esub>  (mset_to_IP R m))) =
                     \<phi> (indexed_const k) \<otimes>\<^bsub>S\<^esub> (\<phi> P) \<otimes>\<^bsub>S\<^esub> \<phi> (mset_to_IP R m)"
           proof(induction m)
             case empty
             assume A: "set_mset {#} \<subseteq> I"
             then have E0: "(P \<otimes>\<^bsub>Pring R I\<^esub> mset_to_IP R {#}) = P"
               by (metis A0 Pring_mult_one Pring_one one_mset_to_IP)
             have E1: "k \<in> carrier R"
-              using A Pring_cfs_closed empty.prems(1) 
-              by linarith              
+              using A Pring_cfs_closed empty.prems(1)
+              by linarith
             have E2: "\<phi> (mset_to_IP R {#}) = \<one>\<^bsub>S\<^esub>"
               by (metis Pring_one assms(3) one_mset_to_IP)
             have E3: " \<phi> (poly_scalar_mult R k (P \<otimes>\<^bsub>Pring R I\<^esub> mset_to_IP R {#})) =
                        \<phi> (indexed_const k) \<otimes>\<^bsub>S\<^esub> \<phi> (P \<otimes>\<^bsub>Pring R I\<^esub> mset_to_IP R {#})"
-              using E1 E2 E0 assms(7)[of "k" "P \<otimes>\<^bsub>Pring R I\<^esub> mset_to_IP R {#}"]  A0 
-              by (simp add: A0 E1)            
+              using E1 E2 E0 assms(7)[of "k" "P \<otimes>\<^bsub>Pring R I\<^esub> mset_to_IP R {#}"]  A0
+              by (simp add: A0 E1)
             have E4: " \<phi> (P \<otimes>\<^bsub>Pring R I\<^esub> mset_to_IP R {#}) =  (\<phi> P) "
-              using E0 
+              using E0
               by auto
             have E5: " \<phi> (poly_scalar_mult R k (P \<otimes>\<^bsub>Pring R I\<^esub> mset_to_IP R {#})) =
                        \<phi> (indexed_const k) \<otimes>\<^bsub>S\<^esub> \<phi> P"
               using E0 E3 by presburger
             have E6: " \<phi> (indexed_const k) \<otimes>\<^bsub>S\<^esub> \<phi> P =  \<phi> (indexed_const k) \<otimes>\<^bsub>S\<^esub> \<phi> P \<otimes>\<^bsub>S\<^esub> \<one>\<^bsub>S\<^esub>"
             proof-
               have 0: "\<phi> P \<in> carrier S"
                 using assms A0
                 by blast
               have "indexed_const k \<in> carrier (Pring R I)"
-                using assms(2) E1 Pring_car indexed_pset.indexed_const 
+                using assms(2) E1 Pring_car indexed_pset.indexed_const
                 by blast
               then have 1: "\<phi> (indexed_const k) \<in> carrier S"
                 using assms(2)
-                by blast 
-              show ?thesis 
+                by blast
+              show ?thesis
                 using assms(1) 0 1
-                by (metis cring.cring_simprules(12) cring.cring_simprules(14) 
+                by (metis cring.cring_simprules(12) cring.cring_simprules(14)
                     cring.cring_simprules(5) cring.cring_simprules(6))
             qed
-            then show ?case 
-              using E0 E2 E3 by presburger           
+            then show ?case
+              using E0 E2 E3 by presburger
           next
             case (add x m)
             assume AA: "\<And>k. k \<in> carrier R \<Longrightarrow> set_mset m \<subseteq> I \<Longrightarrow> \<phi> (poly_scalar_mult R k (P \<otimes>\<^bsub>Pring R I\<^esub> mset_to_IP R m)) = \<phi> (indexed_const k) \<otimes>\<^bsub>S\<^esub> \<phi> P \<otimes>\<^bsub>S\<^esub> \<phi> (mset_to_IP R m)"
             assume P0: "set_mset (add_mset x m) \<subseteq> I"
             then have IA: "set_mset m \<subseteq>I"
               by (metis insert_subset set_mset_add_mset_insert)
-            have IH: "\<phi> (poly_scalar_mult R k (P \<otimes>\<^bsub>Pring R I\<^esub> mset_to_IP R m)) = 
+            have IH: "\<phi> (poly_scalar_mult R k (P \<otimes>\<^bsub>Pring R I\<^esub> mset_to_IP R m)) =
                         \<phi> (indexed_const k) \<otimes>\<^bsub>S\<^esub> \<phi> P \<otimes>\<^bsub>S\<^esub> \<phi> (mset_to_IP R m)"
               using AA IA add.prems(1)
-              by blast             
+              by blast
             then have x_mem: "x \<in> I"
               by (meson P0 subsetD union_single_eq_member)
             have 0: " mset_to_IP R (add_mset x m) =  mset_to_IP R m \<Otimes> x"
               by (simp add: monom_add_mset)
             then have 1: "P \<otimes>\<^bsub>Pring R I\<^esub> mset_to_IP R (add_mset x m) = P \<otimes>\<^bsub>Pring R I\<^esub> (mset_to_IP R m \<Otimes> x)"
               by simp
             have 2: "P \<otimes>\<^bsub>Pring R I\<^esub> mset_to_IP R (add_mset x m) =( P \<otimes>\<^bsub>Pring R I\<^esub> (mset_to_IP R m)) \<Otimes> x"
             proof-
               have "mset_to_IP R (add_mset x m) = (mset_to_IP R m) \<Otimes> x"
                 using "0" by blast
               then have "P \<otimes>\<^bsub>Pring R I\<^esub> mset_to_IP R (add_mset x m) = P \<otimes>\<^bsub>Pring R I\<^esub> ((mset_to_IP R m) \<Otimes> x)"
-                by simp 
+                by simp
               then have "P \<otimes>\<^bsub>Pring R I\<^esub> mset_to_IP R (add_mset x m) = P \<otimes>\<^bsub>Pring R I\<^esub> ((mset_to_IP R m) \<otimes>\<^bsub>Pring R I\<^esub> mset_to_IP R {#x#})"
                 by (metis IA Pring_mult mset_to_IP_closed poly_index_mult x_mem)
               then have "P \<otimes>\<^bsub>Pring R I\<^esub> mset_to_IP R (add_mset x m) = (P \<otimes>\<^bsub>Pring R I\<^esub> (mset_to_IP R m)) \<otimes>\<^bsub>Pring R I\<^esub> mset_to_IP R {#x#}"
-                by (metis A0 IA Pring_car Pring_mult_assoc Pring_one Pring_one_closed 
+                by (metis A0 IA Pring_car Pring_mult_assoc Pring_one Pring_one_closed
                     indexed_pset.indexed_pmult monom_add_mset mset_to_IP_closed one_mset_to_IP x_mem)
-              then show ?thesis 
+              then show ?thesis
                 by (metis A0 IA Pring_car Pring_mult Pring_mult_closed mset_to_IP_closed poly_index_mult x_mem)
             qed
             have 3: "poly_scalar_mult R k (P \<otimes>\<^bsub>Pring R I\<^esub> mset_to_IP R (add_mset x m)) =
                       poly_scalar_mult R k ( P \<otimes>\<^bsub>Pring R I\<^esub> (mset_to_IP R m) \<Otimes> x)"
               using "2" by presburger
             have 4: "poly_scalar_mult R k (P \<otimes>\<^bsub>Pring R I\<^esub> mset_to_IP R (add_mset x m)) =
                       poly_scalar_mult R k ( P \<otimes>\<^bsub>Pring R I\<^esub> (mset_to_IP R m)) \<Otimes> x"
             proof-
               have "poly_scalar_mult R k (P \<otimes>\<^bsub>Pring R I\<^esub> mset_to_IP R (add_mset x m)) =
                       poly_scalar_mult R k ( P \<otimes>\<^bsub>Pring R I\<^esub> (mset_to_IP R m) \<otimes>\<^bsub>Pring R I\<^esub> mset_to_IP R {#x#})"
                 by (metis "2" A0 IA Pring_car Pring_mult Pring_mult_closed mset_to_IP_closed poly_index_mult x_mem)
               have "poly_scalar_mult R k (P \<otimes>\<^bsub>Pring R I\<^esub> mset_to_IP R (add_mset x m)) =
                       (poly_scalar_mult R k ( P \<otimes>\<^bsub>Pring R I\<^esub> mset_to_IP R m)) \<otimes>\<^bsub>Pring R I\<^esub> mset_to_IP R {#x#}"
-                by (metis A0 IA Pring_car Pring_mult Pring_mult_closed Pring_one Pring_one_closed 
-                    \<open>poly_scalar_mult R k (P \<otimes>\<^bsub>Pring R I\<^esub> mset_to_IP R (add_mset x m)) = poly_scalar_mult R k (P \<otimes>\<^bsub>Pring R I\<^esub> mset_to_IP R m \<otimes>\<^bsub>Pring R I\<^esub> mset_to_IP R {#x#})\<close> 
-                    add.prems(1) indexed_pset.indexed_pmult monom_add_mset mset_to_IP_closed 
-                    one_mset_to_IP poly_scalar_mult_times x_mem)               
-              then show ?thesis 
-                by (metis A0 IA Pring_car Pring_mult Pring_mult_closed add.prems(1) 
-                    cring.poly_scalar_mult_closed is_cring mset_to_IP_closed 
-                    poly_index_mult x_mem)                
+                by (metis A0 IA Pring_car Pring_mult Pring_mult_closed Pring_one Pring_one_closed
+                    \<open>poly_scalar_mult R k (P \<otimes>\<^bsub>Pring R I\<^esub> mset_to_IP R (add_mset x m)) = poly_scalar_mult R k (P \<otimes>\<^bsub>Pring R I\<^esub> mset_to_IP R m \<otimes>\<^bsub>Pring R I\<^esub> mset_to_IP R {#x#})\<close>
+                    add.prems(1) indexed_pset.indexed_pmult monom_add_mset mset_to_IP_closed
+                    one_mset_to_IP poly_scalar_mult_times x_mem)
+              then show ?thesis
+                by (metis A0 IA Pring_car Pring_mult Pring_mult_closed add.prems(1)
+                    cring.poly_scalar_mult_closed is_cring mset_to_IP_closed
+                    poly_index_mult x_mem)
             qed
             have 5: "\<phi> (poly_scalar_mult R k (P \<otimes>\<^bsub>Pring R I\<^esub> mset_to_IP R (add_mset x m))) =
                       \<phi> (poly_scalar_mult R k ( P \<otimes>\<^bsub>Pring R I\<^esub> (mset_to_IP R m)) \<Otimes> x)"
-              using 4 by metis 
+              using 4 by metis
             have 6: "\<phi> (poly_scalar_mult R k (P \<otimes>\<^bsub>Pring R I\<^esub> mset_to_IP R (add_mset x m))) =
                       \<phi> (poly_scalar_mult R k ( P \<otimes>\<^bsub>Pring R I\<^esub> (mset_to_IP R m))) \<otimes>\<^bsub>S\<^esub> \<phi> (mset_to_IP  R {#x#})"
-              using assms  
-              by (metis "5" A0 IA Pring_car Pring_mult_closed add.prems(1) 
-                  cring.poly_scalar_mult_closed is_cring mset_to_IP_closed x_mem)             
+              using assms
+              by (metis "5" A0 IA Pring_car Pring_mult_closed add.prems(1)
+                  cring.poly_scalar_mult_closed is_cring mset_to_IP_closed x_mem)
             have 7: "\<phi> (poly_scalar_mult R k (P \<otimes>\<^bsub>Pring R I\<^esub> mset_to_IP R (add_mset x m))) =
                        \<phi> (indexed_const k) \<otimes>\<^bsub>S\<^esub> \<phi> P \<otimes>\<^bsub>S\<^esub> \<phi> (mset_to_IP R m) \<otimes>\<^bsub>S\<^esub> \<phi> (mset_to_IP  R {#x#})"
-              using assms 6 IH 
-              by presburger              
+              using assms 6 IH
+              by presburger
             have 8: " \<phi> (mset_to_IP R m) \<otimes>\<^bsub>S\<^esub> \<phi> (mset_to_IP  R {#x#}) = \<phi> (mset_to_IP R (add_mset x m))"
             proof-
-              have "\<phi> (mset_to_IP R m) \<otimes>\<^bsub>S\<^esub> \<phi> (mset_to_IP  R {#x#}) = 
+              have "\<phi> (mset_to_IP R m) \<otimes>\<^bsub>S\<^esub> \<phi> (mset_to_IP  R {#x#}) =
                     \<phi> ((mset_to_IP R m) \<otimes>\<^bsub>Pring R I\<^esub> (mset_to_IP  R {#x#}))"
                 by (metis IA Pring_car Pring_mult assms(6) mset_to_IP_closed poly_index_mult x_mem)
-              then show ?thesis 
+              then show ?thesis
                 by (metis "0" IA Pring_car assms(6) mset_to_IP_closed x_mem)
             qed
             have 9: "\<phi> (poly_scalar_mult R k (P \<otimes>\<^bsub>Pring R I\<^esub> mset_to_IP R (add_mset x m))) =
              \<phi> (indexed_const k) \<otimes>\<^bsub>S\<^esub> \<phi> P \<otimes>\<^bsub>S\<^esub> ( \<phi> (mset_to_IP R m) \<otimes>\<^bsub>S\<^esub> \<phi> (mset_to_IP  R {#x#}))"
             proof-
               have 0: "\<phi> (indexed_const k) \<in> carrier S"
               proof-
                 have "(indexed_const k) \<in> carrier (Pring R I)"
                   using A Pring_car Pring_cfs_closed indexed_pset.indexed_const add.prems(1)
-                  by blast                  
+                  by blast
                 then show ?thesis
-                  using assms(2) 
-                  by blast   
+                  using assms(2)
+                  by blast
               qed
-              have 1: "\<phi> P \<in> carrier S" 
-                using A0 assms(2) 
-                by blast  
+              have 1: "\<phi> P \<in> carrier S"
+                using A0 assms(2)
+                by blast
               have 2: "\<phi> (mset_to_IP R m) \<in> carrier S"
               proof-
                 have "(mset_to_IP R m) \<in> carrier (Pring R I)"
                   using IA Pring_car mset_to_IP_closed by blast
-                then show ?thesis using assms(2) 
-                  by blast 
+                then show ?thesis using assms(2)
+                  by blast
               qed
               have 3: " \<phi> (mset_to_IP  R {#x#}) \<in> carrier S"
               proof-
                 have "(mset_to_IP  R {#x#}) \<in> carrier (Pring R I)"
-                  by (metis Pring_car Pring_one Pring_one_closed indexed_pset.indexed_pmult 
+                  by (metis Pring_car Pring_one Pring_one_closed indexed_pset.indexed_pmult
                       monom_add_mset one_mset_to_IP x_mem)
-                then show ?thesis 
-                  using assms(2) 
-                  by blast 
+                then show ?thesis
+                  using assms(2)
+                  by blast
               qed
               have "\<phi> (poly_scalar_mult R k (P \<otimes>\<^bsub>Pring R I\<^esub> mset_to_IP R (add_mset x m))) =
                   \<phi> (indexed_const k) \<otimes>\<^bsub>S\<^esub> (\<phi> P \<otimes>\<^bsub>S\<^esub> \<phi> (mset_to_IP R m) \<otimes>\<^bsub>S\<^esub> \<phi> (mset_to_IP  R {#x#}))"
-                using 1 2 3 7 
-                by (metis "0" \<open>\<phi> (mset_to_IP R m) \<in> carrier S\<close> assms(1) 
+                using 1 2 3 7
+                by (metis "0" \<open>\<phi> (mset_to_IP R m) \<in> carrier S\<close> assms(1)
                     cring.cring_simprules(11) cring.cring_simprules(5))
-              then show ?thesis 
-                by (metis "0" "1" "2" "3" "8" Pring_mult assms(1) cring.cring_simprules(11) 
+              then show ?thesis
+                by (metis "0" "1" "2" "3" "8" Pring_mult assms(1) cring.cring_simprules(11)
                     cring.cring_simprules(5))
             qed
             have 10: "\<phi> (poly_scalar_mult R k (P \<otimes>\<^bsub>Pring R I\<^esub> mset_to_IP R (add_mset x m))) =
-             \<phi> (indexed_const k) \<otimes>\<^bsub>S\<^esub> \<phi> P \<otimes>\<^bsub>S\<^esub> \<phi> (mset_to_IP R (add_mset x m))"              
-              using 8 9 
+             \<phi> (indexed_const k) \<otimes>\<^bsub>S\<^esub> \<phi> P \<otimes>\<^bsub>S\<^esub> \<phi> (mset_to_IP R (add_mset x m))"
+              using 8 9
               by presburger
-            then show ?case 
+            then show ?case
               by blast
           qed
-          have 4: "\<phi> (poly_scalar_mult R (Q m) (P \<otimes>\<^bsub>Pring R I\<^esub>  (mset_to_IP R m))) = 
-                    \<phi> (indexed_const (Q m)) \<otimes>\<^bsub>S\<^esub> (\<phi> P) \<otimes>\<^bsub>S\<^esub> \<phi> (mset_to_IP R m)"  
-            using 3 
+          have 4: "\<phi> (poly_scalar_mult R (Q m) (P \<otimes>\<^bsub>Pring R I\<^esub>  (mset_to_IP R m))) =
+                    \<phi> (indexed_const (Q m)) \<otimes>\<^bsub>S\<^esub> (\<phi> P) \<otimes>\<^bsub>S\<^esub> \<phi> (mset_to_IP R m)"
+            using 3
             by (metis A Pring_mult insert_iff local.ring_axioms m_M_def mset_to_IP_indices
                 ring.Pring_cfs_closed)
-          have 5: "\<phi> (P \<otimes>\<^bsub>Pring R I\<^esub> Q) = (\<phi> P) \<otimes>\<^bsub>S\<^esub> (\<phi> Q') \<oplus>\<^bsub>S\<^esub> \<phi> (indexed_const (Q m)) \<otimes>\<^bsub>S\<^esub> (\<phi> P) \<otimes>\<^bsub>S\<^esub> \<phi> (mset_to_IP R m)"  
-            using 2 4 
+          have 5: "\<phi> (P \<otimes>\<^bsub>Pring R I\<^esub> Q) = (\<phi> P) \<otimes>\<^bsub>S\<^esub> (\<phi> Q') \<oplus>\<^bsub>S\<^esub> \<phi> (indexed_const (Q m)) \<otimes>\<^bsub>S\<^esub> (\<phi> P) \<otimes>\<^bsub>S\<^esub> \<phi> (mset_to_IP R m)"
+            using 2 4
             by presburger
-          have " \<phi> (indexed_const (Q m)) \<otimes>\<^bsub>S\<^esub> (\<phi> P) \<otimes>\<^bsub>S\<^esub> \<phi> (mset_to_IP R m) = 
-                    (\<phi> P) \<otimes>\<^bsub>S\<^esub> \<phi> (indexed_const (Q m)) \<otimes>\<^bsub>S\<^esub> \<phi> (mset_to_IP R m)"  
+          have " \<phi> (indexed_const (Q m)) \<otimes>\<^bsub>S\<^esub> (\<phi> P) \<otimes>\<^bsub>S\<^esub> \<phi> (mset_to_IP R m) =
+                    (\<phi> P) \<otimes>\<^bsub>S\<^esub> \<phi> (indexed_const (Q m)) \<otimes>\<^bsub>S\<^esub> \<phi> (mset_to_IP R m)"
           proof-
             have 0: "(\<phi> P)  \<in> carrier S"
-              using A0 assms(2) 
-              by blast 
+              using A0 assms(2)
+              by blast
             have 1: "\<phi> (indexed_const (Q m)) \<in> carrier S"
             proof-
               have "indexed_const (Q m ) \<in> carrier (Pring R I)"
                 using A Pring_car Pring_cfs_closed indexed_pset.indexed_const by blast
-              then show ?thesis 
-                using assms(2) 
-                by blast 
+              then show ?thesis
+                using assms(2)
+                by blast
             qed
-            have 2: "\<phi> (mset_to_IP R m) \<in> carrier S" 
+            have 2: "\<phi> (mset_to_IP R m) \<in> carrier S"
             proof-
               have "(mset_to_IP R m) \<in> carrier (Pring R I)"
                 by (metis A Pring_car insert_iff m_M_def mset_to_IP_closed')
-              then show ?thesis using assms(2) 
-                by blast 
+              then show ?thesis using assms(2)
+                by blast
             qed
-            show ?thesis using assms(1) 0 1 2   
+            show ?thesis using assms(1) 0 1 2
               by (metis cring.cring_simprules(14))
           qed
-          then  have 6: "\<phi> (P \<otimes>\<^bsub>Pring R I\<^esub> Q) = (\<phi> P) \<otimes>\<^bsub>S\<^esub> (\<phi> Q') \<oplus>\<^bsub>S\<^esub>  (\<phi> P) \<otimes>\<^bsub>S\<^esub> \<phi> (indexed_const (Q m)) \<otimes>\<^bsub>S\<^esub> \<phi> (mset_to_IP R m)"  
-            using 5 
+          then  have 6: "\<phi> (P \<otimes>\<^bsub>Pring R I\<^esub> Q) = (\<phi> P) \<otimes>\<^bsub>S\<^esub> (\<phi> Q') \<oplus>\<^bsub>S\<^esub>  (\<phi> P) \<otimes>\<^bsub>S\<^esub> \<phi> (indexed_const (Q m)) \<otimes>\<^bsub>S\<^esub> \<phi> (mset_to_IP R m)"
+            using 5
             by presburger
           then have 7: "\<phi> (P \<otimes>\<^bsub>Pring R I\<^esub> Q) = (\<phi> P) \<otimes>\<^bsub>S\<^esub> ((\<phi> Q') \<oplus>\<^bsub>S\<^esub>  \<phi> (indexed_const (Q m)) \<otimes>\<^bsub>S\<^esub> \<phi> (mset_to_IP R m))"
           proof-
             have 0: "(\<phi> P)  \<in> carrier S"
-              using A0 assms(2) 
-              by blast 
+              using A0 assms(2)
+              by blast
             have 1: "\<phi> (indexed_const (Q m)) \<in> carrier S"
             proof-
               have "indexed_const (Q m ) \<in> carrier (Pring R I)"
                 using A Pring_car Pring_cfs_closed indexed_pset.indexed_const by blast
-              then show ?thesis 
-                using assms(2) 
-                by blast 
+              then show ?thesis
+                using assms(2)
+                by blast
             qed
-            have 2: "\<phi> (mset_to_IP R m) \<in> carrier S" 
+            have 2: "\<phi> (mset_to_IP R m) \<in> carrier S"
             proof-
               have "(mset_to_IP R m) \<in> carrier (Pring R I)"
                 by (metis A Pring_car insert_iff m_M_def mset_to_IP_closed')
-              then show ?thesis using assms(2) 
-                by blast 
+              then show ?thesis using assms(2)
+                by blast
             qed
             have 3: "\<phi> Q' \<in> carrier S"
             proof-
               have "Q' \<in> carrier (Pring R I)"
-                using Q'_fact Pring_car 
+                using Q'_fact Pring_car
                 by blast
-              then show ?thesis 
-                using assms(2) 
-                by blast 
-            qed  
-            show ?thesis using 0 1 2 3 6 
+              then show ?thesis
+                using assms(2)
+                by blast
+            qed
+            show ?thesis using 0 1 2 3 6
               by (metis assms(1) cring.cring_simprules(11) cring.cring_simprules(25) cring.cring_simprules(5))
           qed
-          then show ?thesis 
-            by (metis A Pring_car Pring_cfs_closed Q'_def Q'_fact 
+          then show ?thesis
+            by (metis A Pring_car Pring_cfs_closed Q'_def Q'_fact
                 \<open>Q = restrict_poly_to_monom_set R Q M \<oplus>\<^bsub>Pring R I\<^esub> Mt (Q m) m\<close> assms(5) assms(7)
                 cring.poly_scalar_mult_closed insert_iff is_cring m_M_def mset_to_IP_closed')
         qed
       qed
-      show "Q \<in> Pring_set R I " 
-        using A1 Pring_car 
+      show "Q \<in> Pring_set R I "
+        using A1 Pring_car
         by blast
     qed
   qed
   show "\<And>x y. x \<in> carrier (Pring R I) \<Longrightarrow> y \<in> carrier (Pring R I) \<Longrightarrow> \<phi> (x \<oplus>\<^bsub>Pring R I\<^esub> y) = \<phi> x \<oplus>\<^bsub>S\<^esub> \<phi> y"
-    using assms(5) 
+    using assms(5)
     by blast
   show "\<phi> \<one>\<^bsub>Pring R I\<^esub> = \<one>\<^bsub>S\<^esub>"
     by (simp add: assms(3))
 qed
 
 lemma(in cring) indexed_const_Pring_mult:
   assumes "k \<in> carrier R"
   assumes "P \<in> carrier (Pring R I)"
   shows "(indexed_const k \<otimes>\<^bsub>Pring R I\<^esub> P) m = k \<otimes>\<^bsub>R\<^esub> (P m)"
         "(P \<otimes>\<^bsub>Pring R I\<^esub> indexed_const k) m = k \<otimes>\<^bsub>R\<^esub> (P m)"
   apply (metis Pring_car Pring_mult assms(1) assms(2) poly_scalar_mult_def poly_scalar_mult_eq)
   by (metis Pring_car Pring_carrier_coeff' Pring_mult assms(1) assms(2) indexed_const_P_multE m_comm)
 
 lemma(in cring) ring_hom_to_IP_ring_hom_is_hom:
   assumes "cring S"
   assumes "ring_hom_ring R S \<phi>"
   shows "ring_hom_ring (Pring R I) (Pring S I) (ring_hom_to_IP_ring_hom \<phi>)"
 proof(rule Pring_morphism)
   show 0: "cring (Pring S I)"
     by (simp add: assms(1) cring.axioms(1) ring.Pring_is_cring)
   show 1: "ring_hom_to_IP_ring_hom \<phi> \<in> carrier (Pring R I) \<rightarrow> carrier (Pring S I)"
     by (meson Pi_I assms(1) assms(2) ring_hom_to_IP_ring_hom_closed)
   show 2: "ring_hom_to_IP_ring_hom \<phi> \<one>\<^bsub>Pring R I\<^esub> = \<one>\<^bsub>Pring S I\<^esub>"
     by (simp add: assms(1) assms(2) ring_hom_to_IP_ring_hom_one)
   show 3: "ring_hom_to_IP_ring_hom \<phi> \<zero>\<^bsub>Pring R I\<^esub> = \<zero>\<^bsub>Pring S I\<^esub>"
   proof-
     have "\<And>m. \<phi> (\<zero>\<^bsub>Pring R I\<^esub> m) = \<zero>\<^bsub>S\<^esub>"
       using assms
-      by (metis Pring_carrier_coeff' Pring_zero Pring_zero_closed cring.axioms(1) 
-          cring.cring_simprules(26) indexed_const_Pring_mult(2) is_cring ring.Pring_is_cring 
+      by (metis Pring_carrier_coeff' Pring_zero Pring_zero_closed cring.axioms(1)
+          cring.cring_simprules(26) indexed_const_Pring_mult(2) is_cring ring.Pring_is_cring
           ring_hom_ring.homh ring_hom_zero zero_closed)
     then have "\<And>m. \<phi> (\<zero>\<^bsub>Pring R I\<^esub> m) = \<zero>\<^bsub>Pring S I\<^esub> m"
       by (metis assms(1) cring.axioms(1) ring.Pring_zero ring.indexed_const_def)
     then show ?thesis unfolding ring_hom_to_IP_ring_hom_def
-      by blast 
-  qed    
+      by blast
+  qed
   show 4: " \<And>P Q. P \<in> carrier (Pring R I) \<Longrightarrow>
            Q \<in> carrier (Pring R I) \<Longrightarrow> ring_hom_to_IP_ring_hom \<phi> (P \<oplus>\<^bsub>Pring R I\<^esub> Q) = ring_hom_to_IP_ring_hom \<phi> P \<oplus>\<^bsub>Pring S I\<^esub> ring_hom_to_IP_ring_hom \<phi> Q"
     using assms(1) assms(2) ring_hom_to_IP_ring_hom_add by blast
   show 5: " \<And>i P. i \<in> I \<Longrightarrow>
            P \<in> carrier (Pring R I) \<Longrightarrow>
            ring_hom_to_IP_ring_hom \<phi> (P \<Otimes> i) = ring_hom_to_IP_ring_hom \<phi> P \<otimes>\<^bsub>Pring S I\<^esub> ring_hom_to_IP_ring_hom \<phi> (mset_to_IP R {#i#})"
   proof-
     fix i P
     assume i0: "i \<in> I"
     assume P0: "P \<in> carrier (Pring R I)"
     show  "ring_hom_to_IP_ring_hom \<phi> (P \<Otimes> i) = ring_hom_to_IP_ring_hom \<phi> P \<otimes>\<^bsub>Pring S I\<^esub> ring_hom_to_IP_ring_hom \<phi> (mset_to_IP R {#i#})"
-      unfolding ring_hom_to_IP_ring_hom_def 
+      unfolding ring_hom_to_IP_ring_hom_def
     proof
       fix m
       show " \<phi> ((P \<Otimes> i) m) = ((\<lambda>m. \<phi> (P m)) \<otimes>\<^bsub>Pring S I\<^esub> (\<lambda>m. \<phi> (mset_to_IP R {#i#} m))) m"
       proof(cases "i \<in># m")
         case True
         have "(\<lambda>m. \<phi> (mset_to_IP R {#i#} m)) = mset_to_IP S {#i#}"
         proof
           fix m
           show "\<phi> (mset_to_IP R {#i#} m) = mset_to_IP S {#i#} m"
             apply(cases "{#i#} = m")
-             apply (metis (mono_tags, lifting) assms(1) assms(2) cring.axioms(1) local.ring_axioms 
-                mset_to_IP_def one_mset_to_IP ring.Pring_one ring.Pring_one ring.one_mset_to_IP 
+             apply (metis (mono_tags, lifting) assms(1) assms(2) cring.axioms(1) local.ring_axioms
+                mset_to_IP_def one_mset_to_IP ring.Pring_one ring.Pring_one ring.one_mset_to_IP
                 ring_hom_to_IP_ring_hom_def ring_hom_to_IP_ring_hom_one)
           proof-
             assume "{#i#} \<noteq> m"
             then have LHS: "(mset_to_IP R {#i#} m) = \<zero>\<^bsub>R\<^esub>"
               by (metis mset_to_IP_simp')
             have  RHS :"mset_to_IP S {#i#} m = \<zero>\<^bsub>S\<^esub>"
               by (metis \<open>{#i#} \<noteq> m\<close> mset_to_IP_def)
             have "\<phi> \<zero>\<^bsub>R\<^esub> = \<zero>\<^bsub>S\<^esub>"
-              using assms(1) assms(2) cring.axioms(1) local.ring_axioms 
+              using assms(1) assms(2) cring.axioms(1) local.ring_axioms
                 ring_hom_ring.homh ring_hom_zero by blast
-            then show ?thesis 
-              using  LHS RHS 
+            then show ?thesis
+              using  LHS RHS
               by presburger
           qed
         qed
-        then have RHS: "((\<lambda>m. \<phi> (P m)) \<otimes>\<^bsub>Pring S I\<^esub> (\<lambda>m. \<phi> (mset_to_IP R {#i#} m))) m = 
+        then have RHS: "((\<lambda>m. \<phi> (P m)) \<otimes>\<^bsub>Pring S I\<^esub> (\<lambda>m. \<phi> (mset_to_IP R {#i#} m))) m =
               ((\<lambda>m. \<phi> (P m)) \<otimes>\<^bsub>Pring S I\<^esub> mset_to_IP S {#i#}) m"
           by presburger
-        then have RHS': "((\<lambda>m. \<phi> (P m)) \<otimes>\<^bsub>Pring S I\<^esub> (\<lambda>m. \<phi> (mset_to_IP R {#i#} m))) m = 
+        then have RHS': "((\<lambda>m. \<phi> (P m)) \<otimes>\<^bsub>Pring S I\<^esub> (\<lambda>m. \<phi> (mset_to_IP R {#i#} m))) m =
               (ring.indexed_pmult S (\<lambda>m. \<phi> (P m)) i) m"
         proof-
           have 0: "(\<lambda>m. \<phi> (P m)) \<in> Pring_set S I"
-            using ring_hom_to_IP_ring_hom_closed[of S \<phi> P I] ring.Pring_car[of S I] assms 
-            unfolding ring_hom_to_IP_ring_hom_def 
+            using ring_hom_to_IP_ring_hom_closed[of S \<phi> P I] ring.Pring_car[of S I] assms
+            unfolding ring_hom_to_IP_ring_hom_def
             using P0 cring.axioms(1) by blast
           show ?thesis using ring.poly_index_mult[of S "(\<lambda>m. \<phi> (P m))" I i]
-            by (metis "0" \<open>(\<lambda>m. \<phi> (mset_to_IP R {#i#} m)) = mset_to_IP S {#i#}\<close> 
+            by (metis "0" \<open>(\<lambda>m. \<phi> (mset_to_IP R {#i#} m)) = mset_to_IP S {#i#}\<close>
                 assms(1) cring.axioms(1) i0 ring.Pring_mult)
         qed
-        then have RHS'': "((\<lambda>m. \<phi> (P m)) \<otimes>\<^bsub>Pring S I\<^esub> (\<lambda>m. \<phi> (mset_to_IP R {#i#} m))) m = 
-            (\<lambda>m. \<phi> (P m)) (m - {#i#})" using ring.indexed_pmult_def[of S "(\<lambda>m. \<phi> (P m))" i] True   
+        then have RHS'': "((\<lambda>m. \<phi> (P m)) \<otimes>\<^bsub>Pring S I\<^esub> (\<lambda>m. \<phi> (mset_to_IP R {#i#} m))) m =
+            (\<lambda>m. \<phi> (P m)) (m - {#i#})" using ring.indexed_pmult_def[of S "(\<lambda>m. \<phi> (P m))" i] True
           by (metis assms(1) cring.axioms(1))
-        then show ?thesis 
+        then show ?thesis
           by (metis True indexed_pmult_def)
       next
         case False
         have LHS: "((P \<Otimes> i) m) = \<zero>\<^bsub>R\<^esub>"
-          using False 
+          using False
           by (meson indexed_pmult_def)
         have "(\<lambda>m. \<phi> (mset_to_IP R {#i#} m)) = mset_to_IP S {#i#}"
         proof
           fix m
           show "\<phi> (mset_to_IP R {#i#} m) = mset_to_IP S {#i#} m"
             apply(cases "{#i#} = m")
-             apply (metis (mono_tags, lifting) assms(1) assms(2) cring.axioms(1) local.ring_axioms 
-                mset_to_IP_def one_mset_to_IP ring.Pring_one ring.Pring_one ring.one_mset_to_IP 
+             apply (metis (mono_tags, lifting) assms(1) assms(2) cring.axioms(1) local.ring_axioms
+                mset_to_IP_def one_mset_to_IP ring.Pring_one ring.Pring_one ring.one_mset_to_IP
                 ring_hom_to_IP_ring_hom_def ring_hom_to_IP_ring_hom_one)
           proof-
             assume "{#i#} \<noteq> m"
             then have LHS: "(mset_to_IP R {#i#} m) = \<zero>\<^bsub>R\<^esub>"
               by (metis mset_to_IP_simp')
             have  RHS :"mset_to_IP S {#i#} m = \<zero>\<^bsub>S\<^esub>"
               by (metis \<open>{#i#} \<noteq> m\<close> mset_to_IP_def)
             have "\<phi> \<zero>\<^bsub>R\<^esub> = \<zero>\<^bsub>S\<^esub>"
-              using assms(1) assms(2) cring.axioms(1) local.ring_axioms 
+              using assms(1) assms(2) cring.axioms(1) local.ring_axioms
                 ring_hom_ring.homh ring_hom_zero by blast
-            then show ?thesis 
-              using  LHS RHS 
+            then show ?thesis
+              using  LHS RHS
               by presburger
           qed
         qed
-        then have RHS: "((\<lambda>m. \<phi> (P m)) \<otimes>\<^bsub>Pring S I\<^esub> (\<lambda>m. \<phi> (mset_to_IP R {#i#} m))) m = 
+        then have RHS: "((\<lambda>m. \<phi> (P m)) \<otimes>\<^bsub>Pring S I\<^esub> (\<lambda>m. \<phi> (mset_to_IP R {#i#} m))) m =
               ((\<lambda>m. \<phi> (P m)) \<otimes>\<^bsub>Pring S I\<^esub> mset_to_IP S {#i#}) m"
           by presburger
-        then have RHS': "((\<lambda>m. \<phi> (P m)) \<otimes>\<^bsub>Pring S I\<^esub> (\<lambda>m. \<phi> (mset_to_IP R {#i#} m))) m = 
+        then have RHS': "((\<lambda>m. \<phi> (P m)) \<otimes>\<^bsub>Pring S I\<^esub> (\<lambda>m. \<phi> (mset_to_IP R {#i#} m))) m =
               (ring.indexed_pmult S (\<lambda>m. \<phi> (P m)) i) m"
         proof-
           have 0: "(\<lambda>m. \<phi> (P m)) \<in> Pring_set S I"
-            using ring_hom_to_IP_ring_hom_closed[of S \<phi> P I] ring.Pring_car[of S I] assms 
-            unfolding ring_hom_to_IP_ring_hom_def 
+            using ring_hom_to_IP_ring_hom_closed[of S \<phi> P I] ring.Pring_car[of S I] assms
+            unfolding ring_hom_to_IP_ring_hom_def
             using P0 cring.axioms(1) by blast
           show ?thesis using ring.poly_index_mult[of S "(\<lambda>m. \<phi> (P m))" I i]
-            by (metis "0" \<open>(\<lambda>m. \<phi> (mset_to_IP R {#i#} m)) = mset_to_IP S {#i#}\<close> 
+            by (metis "0" \<open>(\<lambda>m. \<phi> (mset_to_IP R {#i#} m)) = mset_to_IP S {#i#}\<close>
                 assms(1) cring.axioms(1) i0 ring.Pring_mult)
         qed
-        then have RHS'': "((\<lambda>m. \<phi> (P m)) \<otimes>\<^bsub>Pring S I\<^esub> (\<lambda>m. \<phi> (mset_to_IP R {#i#} m))) m = 
+        then have RHS'': "((\<lambda>m. \<phi> (P m)) \<otimes>\<^bsub>Pring S I\<^esub> (\<lambda>m. \<phi> (mset_to_IP R {#i#} m))) m =
               \<zero>\<^bsub>S\<^esub>"
-          using False 
+          using False
           by (metis assms(1) cring.axioms(1) ring.indexed_pmult_def)
-        then show ?thesis 
-          using LHS False assms ring_hom_zero[of \<phi> R S]  
+        then show ?thesis
+          using LHS False assms ring_hom_zero[of \<phi> R S]
           by (metis cring.axioms(1) local.ring_axioms ring_hom_ring.homh)
       qed
     qed
   qed
   show "\<And>k Q. k \<in> carrier R \<Longrightarrow>
            Q \<in> carrier (Pring R I) \<Longrightarrow>
            ring_hom_to_IP_ring_hom \<phi> (poly_scalar_mult R k Q) = ring_hom_to_IP_ring_hom \<phi> (indexed_const k) \<otimes>\<^bsub>Pring S I\<^esub> ring_hom_to_IP_ring_hom \<phi> Q"
   proof-
-    fix k Q 
+    fix k Q
     assume A0: "k \<in> carrier R"
     assume A1: " Q \<in> carrier (Pring R I)"
-    show "ring_hom_to_IP_ring_hom \<phi> (poly_scalar_mult R k Q) = 
+    show "ring_hom_to_IP_ring_hom \<phi> (poly_scalar_mult R k Q) =
           ring_hom_to_IP_ring_hom \<phi> (indexed_const k) \<otimes>\<^bsub>Pring S I\<^esub> ring_hom_to_IP_ring_hom \<phi> Q"
     proof
       fix x
       show "ring_hom_to_IP_ring_hom \<phi> (poly_scalar_mult R k Q) x = (ring_hom_to_IP_ring_hom \<phi> (indexed_const k) \<otimes>\<^bsub>Pring S I\<^esub> ring_hom_to_IP_ring_hom \<phi> Q) x"
       proof-
         have LHS: "ring_hom_to_IP_ring_hom \<phi> (poly_scalar_mult R k Q) x = \<phi> (k \<otimes>\<^bsub>R\<^esub> Q x)"
           by (metis poly_scalar_mult_def ring_hom_to_IP_ring_hom_def)
         then have LHS': "ring_hom_to_IP_ring_hom \<phi> (poly_scalar_mult R k Q) x = (\<phi> k) \<otimes>\<^bsub>S\<^esub> \<phi> (Q x)"
         proof-
           have "Q x \<in> carrier  R"
             using A1 Pring_car local.ring_axioms ring.Pring_cfs_closed by blast
-          then show ?thesis 
+          then show ?thesis
             using LHS assms ring_hom_mult[of \<phi> R S k "Q x"]
             by (metis A0 ring_hom_ring.homh)
         qed
-        have RHS: "(ring_hom_to_IP_ring_hom \<phi> (indexed_const k) \<otimes>\<^bsub>Pring S I\<^esub> ring_hom_to_IP_ring_hom \<phi> Q) x = 
-               (\<phi> k) \<otimes>\<^bsub>S\<^esub> (\<phi> (Q x))" 
+        have RHS: "(ring_hom_to_IP_ring_hom \<phi> (indexed_const k) \<otimes>\<^bsub>Pring S I\<^esub> ring_hom_to_IP_ring_hom \<phi> Q) x =
+               (\<phi> k) \<otimes>\<^bsub>S\<^esub> (\<phi> (Q x))"
         proof-
           have 0: "ring_hom_to_IP_ring_hom \<phi> (indexed_const k)= ring.indexed_const S (\<phi> k)"
             by (simp add: A0 assms(1) assms(2) ring_hom_to_IP_ring_hom_constant)
           have 1: "(\<phi> k) \<in> carrier S"
             by (meson A0 assms(2) ring_hom_closed ring_hom_ring.homh)
           have 2: "ring_hom_to_IP_ring_hom \<phi> Q \<in> carrier (Pring S I)"
-            using A1 assms ring_hom_to_IP_ring_hom_closed 
-            by blast 
+            using A1 assms ring_hom_to_IP_ring_hom_closed
+            by blast
           have 3: "(ring.indexed_const S (\<phi> k) \<otimes>\<^bsub>Pring S I\<^esub> ring_hom_to_IP_ring_hom \<phi> Q) x = (\<phi> k) \<otimes>\<^bsub>S\<^esub> (\<phi> (Q x))"
-            using assms(1) 1 2 
-                 cring.indexed_const_Pring_mult(1)[of S "\<phi> k" "ring_hom_to_IP_ring_hom \<phi> Q" I x] 
-                 ring_hom_to_IP_ring_hom_def[of \<phi> Q x] 
-            by presburger                 
-          then show ?thesis 
+            using assms(1) 1 2
+                 cring.indexed_const_Pring_mult(1)[of S "\<phi> k" "ring_hom_to_IP_ring_hom \<phi> Q" I x]
+                 ring_hom_to_IP_ring_hom_def[of \<phi> Q x]
+            by presburger
+          then show ?thesis
             by (metis "0")
         qed
-        then show ?thesis 
-          using LHS' 
-          by metis  
+        then show ?thesis
+          using LHS'
+          by metis
       qed
     qed
   qed
 qed
 
 lemma ring_hom_to_IP_ring_hom_smult:
   assumes "cring S"
   assumes "ring_hom_ring R S \<phi>"
   assumes "P \<in> carrier (Pring R I)"
   assumes "a \<in> carrier R"
   shows "ring_hom_to_IP_ring_hom \<phi> (a \<odot>\<^bsub>Pring R I\<^esub>P) =
                  \<phi> a \<odot>\<^bsub>Pring S I\<^esub> (ring_hom_to_IP_ring_hom \<phi> P)"
 proof fix m
   have 0: "\<phi> ((a \<odot>\<^bsub>Pring R I\<^esub> P) m) = \<phi> (a \<otimes> (P m))"
-    using assms 
+    using assms
     by (metis Pring_smult_cfs)
-  hence 1: "\<phi> ((a \<odot>\<^bsub>Pring R I\<^esub> P) m) = \<phi> a \<otimes>\<^bsub>S\<^esub> \<phi> (P m)"  
+  hence 1: "\<phi> ((a \<odot>\<^bsub>Pring R I\<^esub> P) m) = \<phi> a \<otimes>\<^bsub>S\<^esub> \<phi> (P m)"
     using assms ring_hom_mult[of \<phi> R S]
     by (metis Pring_carrier_coeff' ring_hom_ring.homh)
   have 2: "(\<phi> a \<odot>\<^bsub>Pring S I\<^esub> (\<lambda>m. \<phi> (P m))) m = \<phi> a \<otimes>\<^bsub>S\<^esub> \<phi> (P m)"
-    using assms ring.Pring_smult[of S I] 
-    unfolding poly_scalar_mult_def cring_def 
+    using assms ring.Pring_smult[of S I]
+    unfolding poly_scalar_mult_def cring_def
     by presburger
   show "ring_hom_to_IP_ring_hom \<phi> (a \<odot>\<^bsub>Pring R I\<^esub> P) m =
-         (\<phi> a \<odot>\<^bsub>Pring S I\<^esub> ring_hom_to_IP_ring_hom \<phi> P) m" 
-    unfolding ring_hom_to_IP_ring_hom_def using assms 1 2 
+         (\<phi> a \<odot>\<^bsub>Pring S I\<^esub> ring_hom_to_IP_ring_hom \<phi> P) m"
+    unfolding ring_hom_to_IP_ring_hom_def using assms 1 2
     by presburger
-qed 
+qed
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsubsection\<open>A Universal Property for Indexed Polynomial Rings\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 lemma Pring_universal_prop_0:
   assumes a_cring: "cring S"
   assumes index_map: "closed_fun S g"
   assumes ring_hom: "ring_hom_ring R S \<phi>"
   assumes "\<psi> = (total_eval S g) \<circ> (ring_hom_to_IP_ring_hom \<phi>)"
   shows "(ring_hom_ring (Pring R I) S \<psi>)"
         "(\<forall>i \<in> I. \<psi> (mset_to_IP R {#i#}) = g i)"
         "(\<forall>a \<in> carrier R. \<psi> (indexed_const a) = \<phi> a)"
-        "\<forall> \<rho>. (ring_hom_ring (Pring R I) S \<rho>) \<and> 
-          (\<forall>i \<in> I. \<rho> (mset_to_IP R {#i#}) = g i) \<and> 
-          (\<forall>a \<in> carrier R. \<rho> (indexed_const a) = \<phi> a) \<longrightarrow> 
+        "\<forall> \<rho>. (ring_hom_ring (Pring R I) S \<rho>) \<and>
+          (\<forall>i \<in> I. \<rho> (mset_to_IP R {#i#}) = g i) \<and>
+          (\<forall>a \<in> carrier R. \<rho> (indexed_const a) = \<phi> a) \<longrightarrow>
           (\<forall>x \<in> carrier (Pring R I). \<rho> x = \<psi> x)"
 proof-
   have 0: " (ring_hom_to_IP_ring_hom \<phi>) \<in> ring_hom (Pring R I) (Pring S I)"
-    using a_cring ring_hom ring_hom_ring.homh ring_hom_to_IP_ring_hom_is_hom 
+    using a_cring ring_hom ring_hom_ring.homh ring_hom_to_IP_ring_hom_is_hom
     by blast
-  have 1: "(total_eval S g) \<in> ring_hom (Pring S I) S " 
-    using a_cring cring.total_eval_ring_hom index_map ring_hom_ring.homh 
+  have 1: "(total_eval S g) \<in> ring_hom (Pring S I) S "
+    using a_cring cring.total_eval_ring_hom index_map ring_hom_ring.homh
     by blast
   show P0: "ring_hom_ring (Pring R I) S \<psi> "
-    using ring_hom_trans 0 1 Pring_is_ring a_cring assms(4) cring.axioms(1) ring_hom_ringI2 
+    using ring_hom_trans 0 1 Pring_is_ring a_cring assms(4) cring.axioms(1) ring_hom_ringI2
     by blast
   show P1: "\<forall>i\<in>I. \<psi> (mset_to_IP R {#i#}) = g i"
   proof
     fix i assume Pi: "i \<in> I"
     show "\<psi> (mset_to_IP R {#i#}) = g i"
     proof-
       have 0: "\<psi> (mset_to_IP R {#i#}) = (total_eval S g) ( (ring_hom_to_IP_ring_hom \<phi>) (mset_to_IP R {#i#}))"
         by (simp add: assms(4))
       have "( (ring_hom_to_IP_ring_hom \<phi>) (mset_to_IP R {#i#})) = (mset_to_IP S {#i#})"
         by (simp add: a_cring ring_hom ring_hom_to_IP_ring_hom_monom)
       then have "\<psi> (mset_to_IP R {#i#}) = (total_eval S g) (mset_to_IP S {#i#})"
         by (simp add: 0)
-      then show ?thesis 
+      then show ?thesis
         by (simp add: a_cring cring.total_eval_var index_map)
     qed
   qed
   show P2: "\<forall>a\<in>carrier R. \<psi> (indexed_const a) = \<phi> a"
   proof
     fix a assume A: "a \<in> carrier R"
     show "\<psi> (indexed_const a) = \<phi> a"
     proof-
       have 0: "ring_hom_to_IP_ring_hom \<phi> (indexed_const a) = ring.indexed_const S (\<phi> a)"
         by (simp add: A a_cring ring_hom ring_hom_to_IP_ring_hom_constant)
       have 1: "total_eval S g (ring.indexed_const S (\<phi> a)) = \<phi> a"
         by (meson A a_cring cring.total_eval_const ring_hom ring_hom_closed ring_hom_ring.homh)
-      show ?thesis 
-        using assms 0 1 
+      show ?thesis
+        using assms 0 1
         by (simp add: "0" index_map)
     qed
   qed
-  show "\<forall> \<rho>. (ring_hom_ring (Pring R I) S \<rho>) \<and> 
-          (\<forall>i \<in> I. \<rho> (mset_to_IP R {#i#}) = g i) \<and> 
-          (\<forall>a \<in> carrier R. \<rho> (indexed_const a) = \<phi> a) \<longrightarrow> 
+  show "\<forall> \<rho>. (ring_hom_ring (Pring R I) S \<rho>) \<and>
+          (\<forall>i \<in> I. \<rho> (mset_to_IP R {#i#}) = g i) \<and>
+          (\<forall>a \<in> carrier R. \<rho> (indexed_const a) = \<phi> a) \<longrightarrow>
           (\<forall>x \<in> carrier (Pring R I). \<rho> x = \<psi> x)"
   proof
     fix \<rho>
     show "ring_hom_ring (Pring R I) S \<rho> \<and> (\<forall>i\<in>I. \<rho> (mset_to_IP R {#i#}) = g i) \<and> (\<forall>a\<in>carrier R. \<rho> (indexed_const a) = \<phi> a) \<longrightarrow>
          (\<forall>x\<in>carrier (Pring R I). \<rho> x = \<psi> x)"
     proof
-        assume A: "(ring_hom_ring (Pring R I) S \<rho>) \<and> 
-          (\<forall>i \<in> I. \<rho> (mset_to_IP R {#i#}) = g i) \<and> 
+        assume A: "(ring_hom_ring (Pring R I) S \<rho>) \<and>
+          (\<forall>i \<in> I. \<rho> (mset_to_IP R {#i#}) = g i) \<and>
           (\<forall>a \<in> carrier R. \<rho> (indexed_const a) = \<phi> a)"
         show "(\<forall>x \<in> carrier (Pring R I). \<rho> x = \<psi> x)"
         proof
           fix x assume B: "x \<in> carrier (Pring R I)"
           show "\<rho> x = \<psi> x"
             apply(rule indexed_pset.induct[of x I "carrier R"])
             using B Pring_car apply blast
-              apply (metis A P2) 
+              apply (metis A P2)
           proof-
             show "\<And>P Q. P \<in> Pring_set R I \<Longrightarrow> \<rho> P = \<psi> P \<Longrightarrow> Q \<in> Pring_set R I \<Longrightarrow> \<rho> Q = \<psi> Q \<Longrightarrow> \<rho> (P \<Oplus> Q) = \<psi> (P \<Oplus> Q)"
             proof-
               fix P Q
               assume A0: "P \<in> Pring_set R I " "\<rho> P = \<psi> P" " Q \<in> Pring_set R I" " \<rho> Q = \<psi> Q "
               show "\<rho> (P \<Oplus> Q) = \<psi> (P \<Oplus> Q)"
                 using A A0 ring_hom_add[of \<psi> "Pring R I" S P Q] ring_hom_add[of \<rho> "Pring R I" S P Q] P0
                 by (metis Pring_add Pring_car ring_hom_ring.homh)
             qed
             show "\<And>P i. P \<in> Pring_set R I \<Longrightarrow> \<rho> P = \<psi> P \<Longrightarrow> i \<in> I \<Longrightarrow> \<rho> (P \<Otimes> i) = \<psi> (P \<Otimes> i)"
             proof-
               fix P i
               assume A0: " P \<in> Pring_set R I" " \<rho> P = \<psi> P" "i \<in> I"
               show "\<rho> (P \<Otimes> i) = \<psi> (P \<Otimes> i)"
               proof-
                 have "\<rho> (P \<otimes>\<^bsub>Pring R I\<^esub> (mset_to_IP R {#i#})) = \<psi> (P \<otimes>\<^bsub>Pring R I\<^esub> (mset_to_IP R {#i#}))"
-                  using A A0 ring_hom_mult[of \<psi> "Pring R I" S P "(mset_to_IP R {#i#})"] 
+                  using A A0 ring_hom_mult[of \<psi> "Pring R I" S P "(mset_to_IP R {#i#})"]
                         ring_hom_mult[of \<rho> "Pring R I" S P "(mset_to_IP R {#i#})"] P0
-                  by (metis P1 Pring_car Pring_one Pring_one_closed indexed_pset.indexed_pmult 
+                  by (metis P1 Pring_car Pring_one Pring_one_closed indexed_pset.indexed_pmult
                       monom_add_mset one_mset_to_IP ring_hom_ring.homh)
-                then show ?thesis 
+                then show ?thesis
                   by (metis A0(1) A0(3) Pring_mult poly_index_mult)
               qed
             qed
           qed
         qed
     qed
   qed
-qed    
-
-end 
+qed
+
+end
 
 definition close_fun :: "'c set \<Rightarrow> ('e, 'f) ring_scheme \<Rightarrow> ('c \<Rightarrow> 'e) \<Rightarrow> ('c \<Rightarrow> 'e)" where
 "close_fun I S g = (\<lambda>i. (if i \<in> I then g i else  \<zero>\<^bsub>S\<^esub>))"
 
 context cring
 begin
 
 lemma close_funE:
   assumes "cring S"
   assumes "g \<in> I \<rightarrow> carrier S"
   shows "closed_fun S (close_fun I S g)"
   apply(rule cring.closed_funI)
   apply (simp add: assms(1))
   by (metis close_fun_def PiE assms(1) assms(2) cring.cring_simprules(2))
 
 end
 
 definition indexed_poly_induced_morphism ::
  "'c set \<Rightarrow> ('e, 'f) ring_scheme \<Rightarrow> ('a, 'e) ring_hom \<Rightarrow> ('c \<Rightarrow> 'e) \<Rightarrow> (('a,'c) mvar_poly, 'e) ring_hom" where
 "indexed_poly_induced_morphism I S \<phi> g = (total_eval S (close_fun I S g)) \<circ> (ring_hom_to_IP_ring_hom \<phi>)"
 
-context cring 
+context cring
 begin
 
 lemma Pring_universal_prop:
   assumes a_cring: "cring S"
   assumes index_map: "g \<in> I \<rightarrow> carrier S"
   assumes ring_hom: "ring_hom_ring R S \<phi>"
   assumes "\<psi> = indexed_poly_induced_morphism I S \<phi> g"
   shows "(ring_hom_ring (Pring R I) S \<psi>)"
         "(\<forall>i \<in> I. \<psi> (mset_to_IP R {#i#}) = g i)"
         "(\<forall>a \<in> carrier R. \<psi> (indexed_const a) = \<phi> a)"
-        "\<forall> \<rho>. (ring_hom_ring (Pring R I) S \<rho>) \<and> 
-          (\<forall>i \<in> I. \<rho> (mset_to_IP R {#i#}) = g i) \<and> 
-          (\<forall>a \<in> carrier R. \<rho> (indexed_const a) = \<phi> a) \<longrightarrow> 
+        "\<forall> \<rho>. (ring_hom_ring (Pring R I) S \<rho>) \<and>
+          (\<forall>i \<in> I. \<rho> (mset_to_IP R {#i#}) = g i) \<and>
+          (\<forall>a \<in> carrier R. \<rho> (indexed_const a) = \<phi> a) \<longrightarrow>
           (\<forall>x \<in> carrier (Pring R I). \<rho> x = \<psi> x)"
 proof-
   obtain g' where g'_def: "g' =  (close_fun I S g)"
-    by simp 
+    by simp
   have 0: "closed_fun S g'"
-    using close_funE a_cring g'_def index_map 
+    using close_funE a_cring g'_def index_map
     by blast
-  show "(ring_hom_ring (Pring R I) S \<psi>)" using assms 0 
-    using cring.Pring_universal_prop_0(1) indexed_poly_induced_morphism_def g'_def is_cring 
+  show "(ring_hom_ring (Pring R I) S \<psi>)" using assms 0
+    using cring.Pring_universal_prop_0(1) indexed_poly_induced_morphism_def g'_def is_cring
     by blast
   show "(\<forall>i \<in> I. \<psi> (mset_to_IP R {#i#}) = g i)"
-  proof- 
+  proof-
     have "(\<forall>i \<in> I. \<psi> (mset_to_IP R {#i#}) = g' i)"
       apply(intro Pring_universal_prop_0[of S _ \<phi>] assms)
-      unfolding assms indexed_poly_induced_morphism_def g'_def 
+      unfolding assms indexed_poly_induced_morphism_def g'_def
       using assms 0 g'_def apply fastforce
       by auto
-    thus ?thesis unfolding g'_def using assms 
+    thus ?thesis unfolding g'_def using assms
       by (simp add: close_fun_def)
   qed
   show "\<forall>a\<in>carrier R. \<psi> (indexed_const a) = \<phi> a"
-    using 0  indexed_poly_induced_morphism_def Pring_universal_prop_0(3) a_cring assms(4) g'_def ring_hom 
+    using 0  indexed_poly_induced_morphism_def Pring_universal_prop_0(3) a_cring assms(4) g'_def ring_hom
     by blast
   show "\<forall>\<rho>. ring_hom_ring (Pring R I) S \<rho> \<and> (\<forall>i\<in>I. \<rho> (mset_to_IP R {#i#}) = g i) \<and> (\<forall>a\<in>carrier R. \<rho> (indexed_const a) = \<phi> a) \<longrightarrow>
         (\<forall>x\<in>carrier (Pring R I). \<rho> x = \<psi> x)"
   proof-
     have "\<forall>\<rho>. ring_hom_ring (Pring R I) S \<rho> \<and> (\<forall>i\<in>I. \<rho> (mset_to_IP R {#i#}) = g' i) \<and> (\<forall>a\<in>carrier R. \<rho> (indexed_const a) = \<phi> a) \<longrightarrow>
         (\<forall>x\<in>carrier (Pring R I). \<rho> x = \<psi> x)"
       using assms 0 Pring_universal_prop_0(4)  g'_def
-      unfolding indexed_poly_induced_morphism_def 
+      unfolding indexed_poly_induced_morphism_def
       by blast
-    then show ?thesis 
-      using g'_def 
-      unfolding close_fun_def 
+    then show ?thesis
+      using g'_def
+      unfolding close_fun_def
       by meson
   qed
 qed
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsection\<open>Mapping Mulitvariate Polynomials over a Single Variable to Univariate Polynomials\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 text\<open>Constructor for multisets which have one distinct element\<close>
 
 definition nat_to_mset :: "'c \<Rightarrow> nat \<Rightarrow> 'c monomial" where
 "nat_to_mset i n = Abs_multiset (\<lambda>j. if (j = i) then n else 0)"
 
 lemma nat_to_msetE: "count (nat_to_mset i n) i = n"
   unfolding nat_to_mset_def by simp
 
 lemma nat_to_msetE':
   assumes "j \<noteq> i"
   shows "count (nat_to_mset i n) j = 0"
   unfolding nat_to_mset_def using assms by simp
 
 lemma nat_to_mset_add: "nat_to_mset i (n + m) = (nat_to_mset i n) + (nat_to_mset i m)"
   apply(rule multiset_eqI)
   by (metis add.right_neutral nat_to_msetE nat_to_msetE' count_union)
 
-lemma nat_to_mset_inj: 
+lemma nat_to_mset_inj:
   assumes "n \<noteq> m"
   shows "(nat_to_mset i n) \<noteq> (nat_to_mset i m)"
-  using assms 
+  using assms
   by (metis nat_to_msetE)
 
 lemma nat_to_mset_zero: "nat_to_mset i 0 = {#}"
   by (metis add.right_neutral add_cancel_right_right nat_to_mset_add)
 
 lemma nat_to_mset_Suc: "nat_to_mset i (Suc n) = add_mset i (nat_to_mset i n)"
   using nat_to_mset_add[of i n 1]
   by (simp add: multiset_eqI nat_to_msetE nat_to_msetE')
 
 lemma nat_to_mset_Pring_singleton:
   assumes "cring R"
   assumes "P \<in> carrier (Pring R {i})"
   assumes "m \<in> monomials_of R P"
   shows "m = nat_to_mset i (count m i)"
 proof-
   have "\<And> j. count m j = count (nat_to_mset i (count m i)) j"
   proof-
     fix j
     show "count m j = count (nat_to_mset i (count m i)) j"
       apply(cases "j = i")
        apply (simp add: nat_to_msetE; fail)
     proof-
       assume A: "j \<noteq>i"
       have P0: "set_mset m \<subseteq> {i}"
-        using assms 
+        using assms
         by (metis cring.axioms(1) ring.Pring_car ring.mset_to_IP_indices)
       then have "count m j = 0"
-        using A assms 
+        using A assms
         by (metis count_inI empty_iff singletonD subset_singletonD)
       then show " count m j = count (nat_to_mset i (count m i)) j"
         by (simp add: A nat_to_msetE')
     qed
   qed
-  then show ?thesis 
+  then show ?thesis
     using multiset_eqI by blast
 qed
 
 definition IP_to_UP :: "'d \<Rightarrow> ('e, 'd) mvar_poly \<Rightarrow> 'e u_poly" where
 "IP_to_UP i P  = (\<lambda> (n::nat). P (nat_to_mset i n))"
 
 lemma IP_to_UP_closed:
   assumes "cring R"
   assumes "P \<in> carrier (Pring R {i::'c})"
   shows "IP_to_UP i P \<in> carrier (UP R)"
 proof-
   have "IP_to_UP i P \<in> up R"
     apply(rule mem_upI)
-    using assms 
+    using assms
      apply (metis cring_def IP_to_UP_def ring.Pring_carrier_coeff')
        unfolding bound_def IP_to_UP_def
        apply(rule ccontr)
      proof-
        assume A: "\<nexists>n. \<forall>m>n. P (nat_to_mset i m) = \<zero>\<^bsub>R\<^esub>"
        then have 0: "\<forall>n. \<exists>m> n. (nat_to_mset i m) \<in> monomials_of R P"
          by (meson assms(1) cring_def ring.complement_of_monomials_of)
        have "\<not> finite {m. (nat_to_mset i m) \<in> monomials_of R P }"
        proof
          assume "finite {m. nat_to_mset i m \<in> monomials_of R P}"
          then have "\<exists>n. \<forall>m>n.  m \<notin> {m. nat_to_mset i m \<in> monomials_of R P}"
            by (meson finite_nat_set_iff_bounded nat_less_le order.strict_trans)
          then have "\<exists>n. \<forall>m>n.  nat_to_mset i m \<notin> monomials_of R P"
            by (simp add: monomials_of_def)
-         then show False using 0 
+         then show False using 0
            by blast
        qed
        then obtain S where S_def: "infinite (S::nat set) \<and> (\<forall>m \<in> S. (nat_to_mset i m) \<in> monomials_of R P)"
          by blast
        have "inj_on (nat_to_mset i) S"
          using inj_onI[of S "nat_to_mset i"]
          by (meson nat_to_mset_inj)
        then have 1: "infinite (nat_to_mset i ` S)"
-         using S_def finite_imageD 
+         using S_def finite_imageD
          by blast
        have 2: "(nat_to_mset i ` S) \<subseteq> (monomials_of R P)"
-         using S_def 
-         by blast         
+         using S_def
+         by blast
        then have "infinite (monomials_of R P)"
          using S_def "1" finite_subset
          by blast
-       then show False using assms 
+       then show False using assms
          by (metis Pring_def cring_def partial_object.select_convs(1) ring.monomials_finite)
      qed
-     then show ?thesis 
+     then show ?thesis
        by (metis (no_types, lifting) UP_def partial_object.select_convs(1))
 qed
 
 lemma IP_to_UP_var:
   shows "IP_to_UP i (mset_to_IP R {#i#}) = X_poly R"
 proof
   have UP: "UP_cring R"
-    by (simp add: UP_cring_def cring_axioms)   
+    by (simp add: UP_cring_def cring_axioms)
   fix x
   show "IP_to_UP i (mset_to_IP R {#i#}) x = X_poly R x"
   proof(cases "x = 1")
     case True
     then have RHS: "X_poly R x = \<one>\<^bsub>R\<^esub>"
       unfolding X_poly_def using UP UP_ring.cfs_monom[of R]
-      unfolding UP_ring_def 
+      unfolding UP_ring_def
       using local.ring_axioms one_closed by presburger
     have LHS: "IP_to_UP i (mset_to_IP R ((add_mset i) {#})) x
         = mset_to_IP R ((add_mset i) {#}) (nat_to_mset i x)"
-      unfolding IP_to_UP_def 
+      unfolding IP_to_UP_def
       by blast
     have "(nat_to_mset i x) = {#i#}"
     proof-
       have "\<And>n. count (nat_to_mset i x) n = count {#i#} n"
         by (simp add: True nat_to_msetE nat_to_msetE')
-      then show ?thesis 
-        using multi_count_eq 
+      then show ?thesis
+        using multi_count_eq
         by blast
     qed
-    then have LHS: "IP_to_UP i (mset_to_IP R ((add_mset i) {#})) x 
+    then have LHS: "IP_to_UP i (mset_to_IP R ((add_mset i) {#})) x
        =  mset_to_IP R ((add_mset i) {#}) {#i#}"
-      using LHS by presburger  
-    then show ?thesis 
-      unfolding deg_zero_cf_def      
-      by (metis RHS mset_to_IP_simp)            
+      using LHS by presburger
+    then show ?thesis
+      unfolding deg_zero_cf_def
+      by (metis RHS mset_to_IP_simp)
   next
     case False
     then have RHS: "X_poly R x = \<zero>\<^bsub>R\<^esub>"
-      unfolding X_poly_def 
+      unfolding X_poly_def
       using UP UP_ring.cfs_monom[of R]
-      unfolding UP_ring_def 
-      using local.ring_axioms one_closed by presburger   
+      unfolding UP_ring_def
+      using local.ring_axioms one_closed by presburger
     have LHS: "IP_to_UP i (mset_to_IP R ((add_mset i) {#})) x
         = mset_to_IP R ((add_mset i) {#}) (nat_to_mset i x)"
-      unfolding IP_to_UP_def 
-      by blast      
-    then show ?thesis 
-      unfolding deg_zero_cf_def  
-      by (metis False RHS count_single mset_to_IP_def nat_to_msetE) 
+      unfolding IP_to_UP_def
+      by blast
+    then show ?thesis
+      unfolding deg_zero_cf_def
+      by (metis False RHS count_single mset_to_IP_def nat_to_msetE)
   qed
 qed
 
 end
 
 context UP_cring
 begin
 
 lemma IP_to_UP_monom:
   shows "IP_to_UP i (mset_to_IP R (nat_to_mset i n)) = ((X_poly R)[^]\<^bsub>UP R\<^esub>n) "
 proof
-  fix x 
+  fix x
   show "IP_to_UP i (mset_to_IP R (nat_to_mset i n)) x = (X_poly R [^]\<^bsub>UP R\<^esub> n) x"
   proof(cases "x = n")
     case True
-    have RHS: "(X_poly R [^]\<^bsub>UP R\<^esub> n) x = \<one>\<^bsub>R\<^esub>" 
-      unfolding X_poly_def 
-      by (metis P.nat_pow_closed P.nat_pow_eone P_def R.one_closed True UP_cring.X_closed 
+    have RHS: "(X_poly R [^]\<^bsub>UP R\<^esub> n) x = \<one>\<^bsub>R\<^esub>"
+      unfolding X_poly_def
+      by (metis P.nat_pow_closed P.nat_pow_eone P_def R.one_closed True UP_cring.X_closed
           UP_cring.monom_coeff UP_one_closed UP_r_one deg_one is_UP_cring monom_one monom_rep_X_pow
           to_poly_inverse to_poly_mult_simp(2))
     have LHS:  "IP_to_UP i (mset_to_IP R (nat_to_mset i n)) x = \<one>\<^bsub>R\<^esub>"
       by (metis R.mset_to_IP_simp True IP_to_UP_def)
     then show ?thesis
-      using RHS by presburger            
+      using RHS by presburger
   next
     case False
     have 0: "\<And>x y::nat. nat_to_mset x = nat_to_mset y \<Longrightarrow> x = y"
     proof-
-      fix a b::nat assume A:  "nat_to_mset a = nat_to_mset b" 
-      then show "a = b" unfolding nat_to_mset_def  
+      fix a b::nat assume A:  "nat_to_mset a = nat_to_mset b"
+      then show "a = b" unfolding nat_to_mset_def
         by (metis A nat_to_msetE nat_to_msetE' zero_neq_one)
     qed
     have 1: "IP_to_UP i (mset_to_IP R (nat_to_mset i n)) x  = (if nat_to_mset i x = nat_to_mset i n then \<one> else \<zero>) "
-      unfolding IP_to_UP_def mset_to_IP_def 
-      by blast 
+      unfolding IP_to_UP_def mset_to_IP_def
+      by blast
     hence 2: "IP_to_UP i (mset_to_IP R (nat_to_mset i n)) x  = (if x = n then \<one> else \<zero>) "
-      using 0 
+      using 0
       by (meson False nat_to_mset_inj)
     have 3: "(X_poly R [^]\<^bsub>UP R\<^esub> n) x = \<zero>"
-      unfolding X_poly_def using False 
-      by (smt ctrm_degree P.nat_pow_closed P.nat_pow_eone P.r_null P_def R.one_closed 
-          UP_cring.ltrm_of_X UP_cring.ltrm_rep_X_pow UP_cring.X_closed UP_cring.monom_coeff 
+      unfolding X_poly_def using False
+      by (smt ctrm_degree P.nat_pow_closed P.nat_pow_eone P.r_null P_def R.one_closed
+          UP_cring.ltrm_of_X UP_cring.ltrm_rep_X_pow UP_cring.X_closed UP_cring.monom_coeff
           UP_r_one UP_zero_closed X_mult_cf cfs_closed cfs_monom deg_nzero_nzero is_UP_cring
           monom_closed monom_one to_poly_inverse to_poly_mult_simp(2))
-    thus ?thesis using 2 1 
+    thus ?thesis using 2 1
       using False by presburger
   qed
 qed
 
 lemma IP_to_UP_one:
  "IP_to_UP i \<one>\<^bsub>Pring R {i}\<^esub> = \<one>\<^bsub>UP R\<^esub>"
 proof
   fix x
   show "IP_to_UP i \<one>\<^bsub>Pring R {i}\<^esub> x = \<one>\<^bsub>UP R\<^esub> x"
   proof(cases "x = 0")
     case True
     have RHS: "\<one>\<^bsub>UP R\<^esub> x = \<one>\<^bsub>R\<^esub>"
       using P_def True cfs_one by presburger
     have "\<one>\<^bsub>Pring R {i}\<^esub> = (\<lambda> m. if m = {#} then \<one>\<^bsub>R\<^esub> else \<zero>\<^bsub>R\<^esub>)"
       by (metis R.Pring_one R.indexed_const_def)
     then have "IP_to_UP i \<one>\<^bsub>Pring R {i}\<^esub> = IP_to_UP i (\<lambda> m. if m = {#} then \<one>\<^bsub>R\<^esub> else \<zero>\<^bsub>R\<^esub>)"
-      by presburger  
+      by presburger
     then have LHS: "IP_to_UP i \<one>\<^bsub>Pring R {i}\<^esub> x = \<one>\<^bsub>R\<^esub>"
       by (smt True count_empty IP_to_UP_def multi_count_eq nat_to_msetE nat_to_msetE')
-    then show ?thesis 
-      using RHS by presburger      
+    then show ?thesis
+      using RHS by presburger
   next
     case False
     have RHS: "\<one>\<^bsub>UP R\<^esub> x = \<zero>\<^bsub>R\<^esub>"
-      by (smt False UP_def monoid.simps(2))      
-    show ?thesis 
-      using False count_empty 
-            nat_to_msetE 
+      by (smt False UP_def monoid.simps(2))
+    show ?thesis
+      using False count_empty
+            nat_to_msetE
             ring.indexed_const_def
       unfolding IP_to_UP_def
       by (metis R.Pring_one R.ring_axioms RHS)
   qed
 qed
 
 lemma IP_to_UP_zero:
  "IP_to_UP i \<zero>\<^bsub>Pring R {i}\<^esub> = \<zero>\<^bsub>UP R\<^esub>"
 proof
   fix x
   show "IP_to_UP i \<zero>\<^bsub>Pring R {i}\<^esub> x = \<zero>\<^bsub>UP R\<^esub> x"
-    unfolding IP_to_UP_def using R.Pring_zero 
+    unfolding IP_to_UP_def using R.Pring_zero
     by (metis P_def R.indexed_zero_def cfs_zero)
 qed
 
 lemma IP_to_UP_add:
   assumes " x \<in> carrier (Pring R {i})"
   assumes " y \<in> carrier (Pring R {i})"
   shows " IP_to_UP i (x \<oplus>\<^bsub>Pring R {i}\<^esub> y) =
           IP_to_UP i x \<oplus>\<^bsub>UP R\<^esub> IP_to_UP i y"
 proof
   fix n
   have LHS: "IP_to_UP i (x \<oplus>\<^bsub>Pring R {i}\<^esub> y) n = (x \<oplus>\<^bsub>Pring R {i}\<^esub> y) (nat_to_mset i n)"
-    by (meson IP_to_UP_def)   
+    by (meson IP_to_UP_def)
   then have LHS: "IP_to_UP i (x \<oplus>\<^bsub>Pring R {i}\<^esub> y) n = x (nat_to_mset i n) \<oplus>\<^bsub>R\<^esub> y (nat_to_mset i n)"
     using assms  unfolding IP_to_UP_def
-    by (metis R.Pring_add R.indexed_padd_def)  
+    by (metis R.Pring_add R.indexed_padd_def)
   have RHS: "(IP_to_UP i x \<oplus>\<^bsub>UP R\<^esub> IP_to_UP i y) n =
    (IP_to_UP i x) n \<oplus>\<^bsub>R\<^esub> (IP_to_UP i y) n"
     using assms UP_ring.cfs_add IP_to_UP_closed
-    by (simp add: UP_ring.cfs_add R_cring cring.IP_to_UP_closed is_UP_ring)    
+    by (simp add: UP_ring.cfs_add R_cring cring.IP_to_UP_closed is_UP_ring)
   then   show "IP_to_UP i (x \<oplus>\<^bsub>Pring R {i}\<^esub> y) n = (IP_to_UP i x \<oplus>\<^bsub>UP R\<^esub> IP_to_UP i y) n"
-    using assms 
+    using assms
     by (metis LHS IP_to_UP_def)
 qed
 
 lemma IP_to_UP_indexed_const:
   assumes "k \<in> carrier R"
   shows "IP_to_UP i (ring.indexed_const R k) = to_polynomial R k"
 proof
   fix x
   show "IP_to_UP i (ring.indexed_const R k) x = to_polynomial R k x"
   proof(cases "x = 0")
     case True
     have LHS: "IP_to_UP i (ring.indexed_const R k) x = k"
-      using True unfolding IP_to_UP_def   
-      by (metis R.indexed_const_def nat_to_mset_zero)  
-    then show ?thesis 
-      using assms 
-      unfolding to_polynomial_def 
-      using True to_polynomial_def 
-      by (metis UP_ring.cfs_monom is_UP_ring)      
+      using True unfolding IP_to_UP_def
+      by (metis R.indexed_const_def nat_to_mset_zero)
+    then show ?thesis
+      using assms
+      unfolding to_polynomial_def
+      using True to_polynomial_def
+      by (metis UP_ring.cfs_monom is_UP_ring)
   next
     case False
     have LHS: "IP_to_UP i (ring.indexed_const R k) x = \<zero>\<^bsub>R\<^esub>"
-      using False unfolding IP_to_UP_def   
+      using False unfolding IP_to_UP_def
       by (metis R.indexed_const_def nat_to_mset_inj nat_to_mset_zero)
     then show ?thesis
-      using assms 
-      unfolding to_polynomial_def 
+      using assms
+      unfolding to_polynomial_def
       using False UP_cring.intro UP_cring.monom_coeff UP_cring.monom_rep_X_pow
       using P_def cfs_monom by presburger
   qed
 qed
 
 lemma IP_to_UP_indexed_pmult:
   assumes "p \<in> carrier (Pring R {i})"
   shows "IP_to_UP i (ring.indexed_pmult R p i) = (IP_to_UP i p) \<otimes>\<^bsub>UP R\<^esub> (X_poly R)"
 proof
   fix n
   have 0: "IP_to_UP i p \<in> carrier (UP R)"
-    by (simp add: R_cring assms cring.IP_to_UP_closed)     
+    by (simp add: R_cring assms cring.IP_to_UP_closed)
   show "IP_to_UP i (ring.indexed_pmult R p i) n = (IP_to_UP i p \<otimes>\<^bsub>UP R\<^esub> X_poly R) n"
   proof(cases "n = 0")
     case True
     then have RHS: "(IP_to_UP i p \<otimes>\<^bsub>UP R\<^esub> X_poly R) n = \<zero>\<^bsub>R\<^esub>"
       by (metis (no_types, lifting) "0" lcf_closed One_nat_def P.r_null P_def R.r_null
-          UP_cring.ltrm_of_X UP_cring.cfs_monom_mult UP_cring.cfs_monom_mult_l UP_zero_closed 
-          X_closed cfs_times_X deg_leE deg_nzero_nzero is_UP_cring lessI neq0_conv plus_1_eq_Suc to_poly_inverse)      
+          UP_cring.ltrm_of_X UP_cring.cfs_monom_mult UP_cring.cfs_monom_mult_l UP_zero_closed
+          X_closed cfs_times_X deg_leE deg_nzero_nzero is_UP_cring lessI neq0_conv plus_1_eq_Suc to_poly_inverse)
     have LHS: "IP_to_UP i (ring.indexed_pmult R p i) n = ring.indexed_pmult R p i (nat_to_mset i n)"
-      unfolding IP_to_UP_def 
-      by blast     
-    then have LHS': "IP_to_UP i (ring.indexed_pmult R p i) n = 
+      unfolding IP_to_UP_def
+      by blast
+    then have LHS': "IP_to_UP i (ring.indexed_pmult R p i) n =
                   (p \<otimes>\<^bsub>Pring R {i}\<^esub> (mset_to_IP R {#i#})) (nat_to_mset i n)"
       using  assms(1) ring.Pring_car ring.Pring_mult
-                ring.poly_index_mult singletonI 
+                ring.poly_index_mult singletonI
       by (metis R.ring_axioms)
-  then have LHS': "IP_to_UP i (ring.indexed_pmult R p i) n = 
+  then have LHS': "IP_to_UP i (ring.indexed_pmult R p i) n =
                   (p \<otimes>\<^bsub>Pring R {i}\<^esub> (mset_to_IP R {#i#})) {#}"
-    using True 
-    by (metis nat_to_mset_zero)          
+    using True
+    by (metis nat_to_mset_zero)
   then show ?thesis using RHS  LHS True assms(1) nat_to_mset_zero ring.indexed_pmult_def
     by (metis R.ring_axioms empty_iff set_mset_empty)
   next
     case False
     then have RHS: "(IP_to_UP i p \<otimes>\<^bsub>UP R\<^esub> X_poly R) n = (IP_to_UP i p ) (n -1)"
-      using "0" Suc_diff_1 Suc_eq_plus1 
+      using "0" Suc_diff_1 Suc_eq_plus1
            assms(1) bot_nat_def IP_to_UP_def nat_neq_iff not_less0
-      by (metis (no_types, lifting) P_def UP_cring X_closed cfs_times_X cring.cring_simprules(14))  
+      by (metis (no_types, lifting) P_def UP_cring X_closed cfs_times_X cring.cring_simprules(14))
     have LHS: "IP_to_UP i (ring.indexed_pmult R p i) n = ring.indexed_pmult R p i (nat_to_mset i n)"
-      unfolding IP_to_UP_def 
-      by blast      
-    then have LHS': "IP_to_UP i (ring.indexed_pmult R p i) n = 
+      unfolding IP_to_UP_def
+      by blast
+    then have LHS': "IP_to_UP i (ring.indexed_pmult R p i) n =
                   (p \<otimes>\<^bsub>Pring R {i}\<^esub> (mset_to_IP R {#i#})) (nat_to_mset i n)"
       using assms(1) ring.Pring_car ring.Pring_mult
-      ring.poly_index_mult singletonI 
+      ring.poly_index_mult singletonI
       by (metis R.ring_axioms)
-    then have LHS'': "IP_to_UP i (ring.indexed_pmult R p i) n = 
+    then have LHS'': "IP_to_UP i (ring.indexed_pmult R p i) n =
                   (p \<otimes>\<^bsub>Pring R {i}\<^esub> (mset_to_IP R {#i#})) (add_mset i (nat_to_mset i (n-1))) "
       by (metis False Suc_diff_1 nat_to_mset_Suc neq0_conv)
-    then show ?thesis using RHS unfolding IP_to_UP_def  
-      by (metis (no_types, lifting) False R.indexed_pmult_def Suc_diff_1 add_mset_remove_trivial add_mset_remove_trivial_If multi_self_add_other_not_self nat_to_mset_Suc neq0_conv)    
+    then show ?thesis using RHS unfolding IP_to_UP_def
+      by (metis (no_types, lifting) False R.indexed_pmult_def Suc_diff_1 add_mset_remove_trivial add_mset_remove_trivial_If multi_self_add_other_not_self nat_to_mset_Suc neq0_conv)
   qed
 qed
 
 lemma IP_to_UP_ring_hom:
   shows "ring_hom_ring (Pring R {i}) (UP R) (IP_to_UP i)"
   apply(rule cring.Pring_morphism)
-      apply (simp add: R_cring; fail) 
+      apply (simp add: R_cring; fail)
   using P_def UP_cring apply blast
       apply (simp add: R.IP_to_UP_closed R_cring; fail)
       apply (meson IP_to_UP_one)
             apply (meson IP_to_UP_zero)
     apply (meson IP_to_UP_add)
   apply (metis R.IP_to_UP_var IP_to_UP_indexed_pmult singletonD)
 proof-
   fix k Q
   assume A0: "k \<in> carrier R"
   assume A1: "Q \<in> carrier (Pring R {i})"
   show "IP_to_UP i (poly_scalar_mult R k Q) =
            IP_to_UP i (ring.indexed_const R k) \<otimes>\<^bsub>UP R\<^esub> IP_to_UP i Q"
-    unfolding poly_scalar_mult_def  
+    unfolding poly_scalar_mult_def
   proof
     fix x
     show "IP_to_UP i (\<lambda>m. k \<otimes>\<^bsub>R\<^esub> Q m) x =
          (IP_to_UP i (ring.indexed_const R k) \<otimes>\<^bsub>UP R\<^esub> IP_to_UP i Q) x"
     proof-
       have LHS: "IP_to_UP i (\<lambda>m. k \<otimes>\<^bsub>R\<^esub> Q m) x = k \<otimes>\<^bsub>R\<^esub> Q (nat_to_mset i x)"
-        unfolding IP_to_UP_def 
-        by blast       
-      have RHS: "(IP_to_UP i (ring.indexed_const R k) \<otimes>\<^bsub>UP R\<^esub> IP_to_UP i Q) x = 
+        unfolding IP_to_UP_def
+        by blast
+      have RHS: "(IP_to_UP i (ring.indexed_const R k) \<otimes>\<^bsub>UP R\<^esub> IP_to_UP i Q) x =
                 (to_polynomial R k \<otimes>\<^bsub>UP R\<^esub> IP_to_UP i Q) x"
-        by (metis A0 IP_to_UP_indexed_const)        
-      have RHS': "(IP_to_UP i (ring.indexed_const R k) \<otimes>\<^bsub>UP R\<^esub> IP_to_UP i Q) x = 
+        by (metis A0 IP_to_UP_indexed_const)
+      have RHS': "(IP_to_UP i (ring.indexed_const R k) \<otimes>\<^bsub>UP R\<^esub> IP_to_UP i Q) x =
                 k \<otimes>\<^bsub>R\<^esub> ((IP_to_UP i Q) x)"
       proof-
         have 0: "deg R (to_polynomial R k) = 0"
           using A0 degree_to_poly by blast
         have 1: "(IP_to_UP i Q) \<in> carrier (UP R)"
-          using IP_to_UP_closed  unfolding P_def 
-          by (simp add: A1 R.IP_to_UP_closed R_cring) 
-        then show ?thesis 
+          using IP_to_UP_closed  unfolding P_def
+          by (simp add: A1 R.IP_to_UP_closed R_cring)
+        then show ?thesis
         proof -
           have "UP_cring R \<and> IP_to_UP i Q \<in> carrier (UP R)"
             using "1" is_UP_cring by blast
           then show ?thesis
-            by (metis A0 UP_cring.to_poly_mult_simp(1) UP_ring.UP_mult_closed UP_ring.coeff_simp UP_ring.coeff_smult UP_ring.monom_closed IP_to_UP_indexed_const is_UP_ring to_polynomial_def)      
+            by (metis A0 UP_cring.to_poly_mult_simp(1) UP_ring.UP_mult_closed UP_ring.coeff_simp UP_ring.coeff_smult UP_ring.monom_closed IP_to_UP_indexed_const is_UP_ring to_polynomial_def)
         qed
-      qed    
-      then show ?thesis 
-        by (metis IP_to_UP_def)       
+      qed
+      then show ?thesis
+        by (metis IP_to_UP_def)
     qed
   qed
 qed
-      
+
 lemma IP_to_UP_ring_hom_inj:
   shows "inj_on  (IP_to_UP i) (carrier (Pring R {i}))"
 proof
   fix x y
   assume A: "x \<in> carrier (Pring R {i})" "y \<in> carrier (Pring R {i}) "
   assume B: "IP_to_UP i x = IP_to_UP i y"
   show "x = y"
   proof
-    fix a 
+    fix a
     show "x a = y a"
     proof(cases "set_mset a \<subseteq> {i}")
       case True
       then obtain n where "a = (nat_to_mset i n)"
-        by (metis count_eq_zero_iff insert_subset multiset_eqI nat_to_msetE nat_to_msetE' 
+        by (metis count_eq_zero_iff insert_subset multiset_eqI nat_to_msetE nat_to_msetE'
             set_eq_subset singletonD singleton_insert_inj_eq' subset_insertI subset_refl)
       then have LHS: "x a = IP_to_UP i x n"
-        by (metis IP_to_UP_def)    
-      then show ?thesis  
+        by (metis IP_to_UP_def)
+      then show ?thesis
         by (metis B \<open>a = nat_to_mset i n\<close> IP_to_UP_def)
     next
       case False
-      then show ?thesis 
-        using ring.Pring_set_zero[of R y "{i}" a] ring.Pring_set_zero[of R x "{i}" a] A 
-        by (metis R.Pring_car R.ring_axioms)  
+      then show ?thesis
+        using ring.Pring_set_zero[of R y "{i}" a] ring.Pring_set_zero[of R x "{i}" a] A
+        by (metis R.Pring_car R.ring_axioms)
     qed
   qed
 qed
 
 lemma IP_to_UP_scalar_mult:
   assumes "a \<in> carrier R"
   assumes "p \<in> carrier (Pring R {i})"
   shows "(IP_to_UP i (a \<odot>\<^bsub>Pring R {i}\<^esub> p)) = a\<odot>\<^bsub>UP R\<^esub> (IP_to_UP i p)"
   apply(rule ring.indexed_pset.induct[of R p "{i}" "carrier R"])
   apply (simp add: R.ring_axioms; fail)
   using R.Pring_car assms(2) apply blast
   apply (metis IP_to_UP_indexed_const P_def R.m_closed R.poly_scalar_mult_const R.ring_axioms assms(1) ring.Pring_smult to_poly_closed to_poly_mult to_poly_mult_simp(1))
 proof-
   show "\<And>P Q. P \<in> Pring_set R {i} \<Longrightarrow>
            IP_to_UP i (a \<odot>\<^bsub>Pring R {i}\<^esub> P) = a \<odot>\<^bsub>UP R\<^esub> IP_to_UP i P \<Longrightarrow>
            Q \<in> Pring_set R {i} \<Longrightarrow> IP_to_UP i (a \<odot>\<^bsub>Pring R {i}\<^esub> Q) = a \<odot>\<^bsub>UP R\<^esub> IP_to_UP i Q \<Longrightarrow> IP_to_UP i (a \<odot>\<^bsub>Pring R {i}\<^esub> (P \<Oplus> Q)) = a \<odot>\<^bsub>UP R\<^esub> IP_to_UP i (P \<Oplus> Q)"
   proof-
     fix p Q
     assume A0: "p \<in> Pring_set R {i}"
               "IP_to_UP i (a \<odot>\<^bsub>Pring R {i}\<^esub> p) = a \<odot>\<^bsub>UP R\<^esub> IP_to_UP i p"
               " Q \<in> Pring_set R {i}"
               "IP_to_UP i (a \<odot>\<^bsub>Pring R {i}\<^esub> Q) = a \<odot>\<^bsub>UP R\<^esub> IP_to_UP i Q"
-    show "IP_to_UP i (a \<odot>\<^bsub>Pring R {i}\<^esub> (p \<Oplus> Q)) = a \<odot>\<^bsub>UP R\<^esub> IP_to_UP i (p \<Oplus> Q)"      
+    show "IP_to_UP i (a \<odot>\<^bsub>Pring R {i}\<^esub> (p \<Oplus> Q)) = a \<odot>\<^bsub>UP R\<^esub> IP_to_UP i (p \<Oplus> Q)"
     proof-
       have "(a \<odot>\<^bsub>Pring R {i}\<^esub> (p \<Oplus> Q)) = a \<odot>\<^bsub>Pring R {i}\<^esub>  p \<oplus>\<^bsub>Pring R {i}\<^esub> a \<odot>\<^bsub>Pring R {i}\<^esub>  Q"
         by (metis A0(1) A0(3) R.Pring_add R.Pring_car R.Pring_smult_r_distr assms(1))
-      then show ?thesis using A0 
-        by (metis IP_to_UP_add P_def R.IP_to_UP_closed R.Pring_add R.Pring_car R.Pring_smult_closed R_cring UP_smult_r_distr assms(1)) 
-      qed           
+      then show ?thesis using A0
+        by (metis IP_to_UP_add P_def R.IP_to_UP_closed R.Pring_add R.Pring_car R.Pring_smult_closed R_cring UP_smult_r_distr assms(1))
+      qed
   qed
   show " \<And>P ia. P \<in> Pring_set R {i} \<Longrightarrow> IP_to_UP i (a \<odot>\<^bsub>Pring R {i}\<^esub> P) = a \<odot>\<^bsub>UP R\<^esub> IP_to_UP i P \<Longrightarrow> ia \<in> {i} \<Longrightarrow> IP_to_UP i (a \<odot>\<^bsub>Pring R {i}\<^esub> (P \<Otimes> ia)) = a \<odot>\<^bsub>UP R\<^esub> IP_to_UP i (P \<Otimes> ia)"
   proof
     fix P j x
     assume A0: "P \<in> Pring_set R {i}"
     assume A1: "IP_to_UP i (a \<odot>\<^bsub>Pring R {i}\<^esub> P) = a \<odot>\<^bsub>UP R\<^esub> IP_to_UP i P"
     assume A2: "j \<in> {i}"
     then have A3: "j = i"
       by blast
     have "IP_to_UP i (ring.indexed_pmult R P j) \<in> carrier (UP R)"
       by (simp add: A0 A3 R.Pring_car R.indexed_pset.indexed_pmult R_cring cring.IP_to_UP_closed)
     then have "(a \<odot>\<^bsub>UP R\<^esub> (\<lambda>n. ring.indexed_pmult R P j (nat_to_mset i n))) x = a \<otimes>\<^bsub>R\<^esub>((\<lambda>n. ring.indexed_pmult R P j (nat_to_mset i n))) x"
-      using A0 A1 A3 assms unfolding IP_to_UP_def 
+      using A0 A1 A3 assms unfolding IP_to_UP_def
       using P_def cfs_smult by blast
     then show " IP_to_UP i (a \<odot>\<^bsub>Pring R {i}\<^esub> (P \<Otimes> j)) x = (a \<odot>\<^bsub>UP R\<^esub> IP_to_UP i (P \<Otimes> j)) x"
-      by (metis A0 A2 IP_to_UP_def P_def R.Pring_car R.Pring_smult_cfs R.indexed_pset.indexed_pmult \<open>IP_to_UP i (P \<Otimes> j) \<in> carrier (UP R)\<close> assms(1) cfs_smult)            
+      by (metis A0 A2 IP_to_UP_def P_def R.Pring_car R.Pring_smult_cfs R.indexed_pset.indexed_pmult \<open>IP_to_UP i (P \<Otimes> j) \<in> carrier (UP R)\<close> assms(1) cfs_smult)
   qed
 qed
 
 end
 
 text\<open>Evaluation of indexed polynomials commutes with evaluation of univariate polynomials:\<close>
 
 lemma pvar_closed:
   assumes "cring R"
   assumes "i \<in> I"
   shows "(pvar R i) \<in> carrier (Pring R I)"
   by (meson assms(1) assms(2) cring.axioms(1) ring.Pring_var_closed)
 
 context UP_cring
 begin
 
 lemma pvar_mult:
   assumes "i \<in> I"
   assumes "j \<in>  I"
   shows "(pvar R i) \<otimes>\<^bsub>Pring R I\<^esub> (pvar R j) = mset_to_IP R {#i, j#}"
 proof-
   have "{#i#} + {#j#}  = {# i, j#}"
     by auto
-  then show ?thesis 
-    unfolding var_to_IP_def 
-    by (metis R.Pring_mult R.monom_mult)  
+  then show ?thesis
+    unfolding var_to_IP_def
+    by (metis R.Pring_mult R.monom_mult)
 qed
 
 lemma pvar_pow:
 
   assumes "i \<in> I"
   shows "(pvar R i)[^]\<^bsub>Pring R I\<^esub>(n::nat) = mset_to_IP R (nat_to_mset i n)"
   apply(induction n)
   apply (metis Group.nat_pow_0 R.one_mset_to_IP R.ring_axioms nat_to_mset_zero ring.Pring_one)
 proof-
   fix n
   assume IH: "pvar R i [^]\<^bsub>Pring R I\<^esub> n = mset_to_IP R (nat_to_mset i n)"
   show "pvar R i [^]\<^bsub>Pring R I\<^esub> Suc n = mset_to_IP R (nat_to_mset i (Suc n)) "
   proof-
     have "mset_to_IP R (nat_to_mset i (Suc n)) = mset_to_IP R (nat_to_mset i n) \<otimes>\<^bsub>Pring R I\<^esub> pvar R i"
-      using R.monom_mult[of "nat_to_mset i n" "nat_to_mset i 1"] 
-      by (metis One_nat_def R.Pring_mult Suc_eq_plus1 nat_to_mset_Suc nat_to_mset_add nat_to_mset_zero var_to_IP_def)   
-    then show ?thesis 
-      using IH 
+      using R.monom_mult[of "nat_to_mset i n" "nat_to_mset i 1"]
+      by (metis One_nat_def R.Pring_mult Suc_eq_plus1 nat_to_mset_Suc nat_to_mset_add nat_to_mset_zero var_to_IP_def)
+    then show ?thesis
+      using IH
       by simp
   qed
 qed
 
 lemma IP_to_UP_poly_eval:
   assumes "p \<in> Pring_set R {i}"
   assumes "closed_fun R g"
   shows "total_eval R g p = to_function R (IP_to_UP i p) (g i)"
   apply(rule R.indexed_pset.induct[of p "{i}" "carrier R" ])
   apply (simp add: assms(1); fail)
 proof-
   show "\<And>k. k \<in> carrier R \<Longrightarrow> total_eval R g (R.indexed_const k) = to_function R (IP_to_UP i (R.indexed_const k)) (g i)"
   proof-
     fix k
     assume A: "k \<in> carrier R"
     have P0: "total_eval R g (ring.indexed_const R k) = k"
-      unfolding total_eval_def eval_in_ring_def 
+      unfolding total_eval_def eval_in_ring_def
       using cring.poly_eval_constant[of R k UNIV g]
-      by (metis A R.indexed_const_def R_cring)      
+      by (metis A R.indexed_const_def R_cring)
     have P1: "(IP_to_UP i (ring.indexed_const R k)) = to_polynomial R k"
-      by (meson A IP_to_UP_indexed_const)     
-    have P2: "to_function R (IP_to_UP i (ring.indexed_const R k)) (g i) = 
+      by (meson A IP_to_UP_indexed_const)
+    have P2: "to_function R (IP_to_UP i (ring.indexed_const R k)) (g i) =
               to_function R (to_polynomial R k) (g i)"
-      using P1 by presburger      
+      using P1 by presburger
     have P3: "to_function R (to_polynomial R k) (g i) = k"
-      using A assms(2) to_fun_to_poly[of k "g i"] unfolding to_fun_def by blast           
+      using A assms(2) to_fun_to_poly[of k "g i"] unfolding to_fun_def by blast
     then show "total_eval R g (R.indexed_const k) = to_function R (IP_to_UP i (R.indexed_const k)) (g i)"
-      using P0 P2 by presburger      
+      using P0 P2 by presburger
   qed
   show "\<And>P Q. P \<in> Pring_set R {i} \<Longrightarrow>
            total_eval R g P = to_function R  (IP_to_UP i P) (g i) \<Longrightarrow>
            Q \<in> Pring_set R {i} \<Longrightarrow>
            total_eval R g Q = to_function R  (IP_to_UP i Q) (g i) \<Longrightarrow> total_eval R g (P \<Oplus> Q) = to_function R (IP_to_UP i (P \<Oplus> Q)) (g i)"
   proof-
     fix p Q
     assume A0: "p \<in> Pring_set R {i}"
     assume A1: " total_eval R g p = to_function R (IP_to_UP i p) (g i)"
     assume A2: " Q \<in> Pring_set R {i}"
     assume A3: "total_eval R g Q = to_function R (IP_to_UP i Q) (g i)"
     have "total_eval R g (R.indexed_padd p Q) = (total_eval R g p) \<oplus>\<^bsub>R\<^esub> (total_eval R g Q)"
-      using R.total_eval_add[of p "{i}" Q g] A0 A1 
-      by (metis A2 R.Pring_add R.Pring_car assms(2))  
+      using R.total_eval_add[of p "{i}" Q g] A0 A1
+      by (metis A2 R.Pring_add R.Pring_car assms(2))
     then
     have 0: "total_eval R g (p \<Oplus> Q) = total_eval R g p \<oplus> total_eval R g Q "
       by blast
     have 1: "IP_to_UP i (p \<Oplus> Q) = IP_to_UP i p \<oplus>\<^bsub>UP R\<^esub> IP_to_UP i Q"
-      using A0 A1 A3 assms A2 R.ring_axioms R_cring IP_to_UP_add 
+      using A0 A1 A3 assms A2 R.ring_axioms R_cring IP_to_UP_add
       by (metis R.Pring_add R.Pring_car)
     have "g i \<in> carrier R"
-      using assms by blast 
+      using assms by blast
     hence 2: "to_function R (IP_to_UP i (p \<Oplus> Q)) (g i) = to_function R (IP_to_UP i p) (g i) \<oplus> to_function R (IP_to_UP i Q) (g i)"
-      using A0 A1 A3 assms A2 R.ring_axioms R_cring to_fun_plus[of "IP_to_UP i p" "IP_to_UP i Q" "g i"] 
+      using A0 A1 A3 assms A2 R.ring_axioms R_cring to_fun_plus[of "IP_to_UP i p" "IP_to_UP i Q" "g i"]
             IP_to_UP_closed is_UP_cring UP_cring.to_fun_def
-            to_fun_def 0 1 
-      unfolding to_fun_def P_def 
-      by (smt (z3) P_def R.IP_to_UP_closed R.Pring_car to_fun_plus)       
+            to_fun_def 0 1
+      unfolding to_fun_def P_def
+      by (smt (z3) P_def R.IP_to_UP_closed R.Pring_car to_fun_plus)
     show "total_eval R g (R.indexed_padd p Q) = to_function R (IP_to_UP i (ring.indexed_padd R p Q)) (g i) "
       using A0 A1 A3 assms A2 R.ring_axioms R_cring is_UP_cring to_fun_def 0 1 2
-      unfolding to_fun_def   by metis 
+      unfolding to_fun_def   by metis
   qed
   show "\<And>P ia.
        P \<in> Pring_set R {i} \<Longrightarrow>
        total_eval R g P = to_function R (IP_to_UP i P) (g i) \<Longrightarrow> ia \<in> {i} \<Longrightarrow> total_eval R g (P \<Otimes> ia) = to_function R (IP_to_UP i (P \<Otimes> ia)) (g i)"
   proof-
     fix P
     fix j
     assume A0: "P \<in> Pring_set R {i}"
     assume A1: "total_eval R g P = to_function R (IP_to_UP i P) (g i)"
     assume A2: "j \<in> {i}"
     then have A3: "j = i"
       by blast
     show "total_eval R g (P \<Otimes> j) = to_function R (IP_to_UP i (P \<Otimes> j)) (g i)"
     proof-
       have LHS: "total_eval R g (P \<Otimes> j) = (total_eval R g P) \<otimes>\<^bsub>R\<^esub> (g i)"
-        using assms A0 A3 
-        by (metis R.Pring_car R_cring cring.total_eval_indexed_pmult insertI1)   
+        using assms A0 A3
+        by (metis R.Pring_car R_cring cring.total_eval_indexed_pmult insertI1)
       have RHS: "IP_to_UP i (P \<Otimes> j)=  IP_to_UP i P  \<otimes>\<^bsub> UP R\<^esub> X_poly R"
         by (metis A0 A3 IP_to_UP_indexed_pmult R.Pring_car)
       have "g i \<in> carrier R"
-        using assms by blast 
-      then show ?thesis 
+        using assms by blast
+      then show ?thesis
           using A0 A1 A3 X_closed to_fun_X[of "g i"] to_fun_mult[of "IP_to_UP i P" "X_poly R" "g i"] LHS RHS
-              assms  cring.axioms(1) domain.axioms(1) 
+              assms  cring.axioms(1) domain.axioms(1)
               IP_to_UP_indexed_pmult IP_to_UP_closed
               Pring_car unfolding to_fun_def P_def
-          by (smt (z3) P.m_comm P_def R.m_comm R_cring cring.IP_to_UP_closed ring.Pring_car to_fun_closed to_fun_def)                                                                                                                 
+          by (smt (z3) P.m_comm P_def R.m_comm R_cring cring.IP_to_UP_closed ring.Pring_car to_fun_closed to_fun_def)
       qed
    qed
 qed
 end
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsection\<open>Mapping Univariate Polynomials to Multivariate Polynomials over a Singleton Variable Set\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 definition UP_to_IP :: "('a,'b) ring_scheme \<Rightarrow> 'c \<Rightarrow> 'a u_poly \<Rightarrow> ('a, 'c) mvar_poly" where
 "UP_to_IP R i P = (\<lambda> m. if (set_mset m) \<subseteq>  {i} then P (count m i) else \<zero>\<^bsub>R\<^esub>)"
 
 context UP_cring
 begin
 
 lemma UP_to_IP_inv:
   assumes "p \<in> Pring_set R {i}"
   shows "UP_to_IP R i (IP_to_UP i p) = p"
 proof
   fix x
   show "UP_to_IP R i (IP_to_UP i p) x = p x"
   proof(cases "(set_mset x) = {i}")
     case True
     have "{a. 0 < (\<lambda>j. if j = i then count x i else 0) a} = {i}"
       by (smt Collect_cong True count_eq_zero_iff neq0_conv singletonI singleton_conv)
   then have "finite {j. (if j = i then count x i else 0) \<noteq> 0}"
-      by auto     
+      by auto
     have "(\<lambda>j. if j = i then count x i else 0) = count x"
     proof
       fix j
       show "(if j = i then count x i else 0) = count x j "
         apply(cases "j = i")
-        using True 
-        apply (simp; fail)     
-        using True 
+        using True
+        apply (simp; fail)
+        using True
         by (metis count_inI singletonD)
     qed
     then have "(Abs_multiset (\<lambda>j. if j = i then count x i else 0)) = x"
-      using count_inverse  
+      using count_inverse
       by simp
-    then show ?thesis 
-      unfolding UP_to_IP_def IP_to_UP_def nat_to_mset_def   
-      by (metis True set_eq_subset)          
+    then show ?thesis
+      unfolding UP_to_IP_def IP_to_UP_def nat_to_mset_def
+      by (metis True set_eq_subset)
   next
     case False
-    then show ?thesis 
+    then show ?thesis
       apply(cases "x = {#}")
        apply (metis count_empty empty_subsetI IP_to_UP_def nat_to_mset_zero set_mset_empty UP_to_IP_def)
-     unfolding UP_to_IP_def IP_to_UP_def nat_to_mset_def   
-     using False  assms 
-     by (metis R.Pring_set_zero set_mset_eq_empty_iff subset_singletonD)     
+     unfolding UP_to_IP_def IP_to_UP_def nat_to_mset_def
+     using False  assms
+     by (metis R.Pring_set_zero set_mset_eq_empty_iff subset_singletonD)
   qed
 qed
 
 lemma UP_to_IP_const:
   assumes "a \<in> carrier R"
   shows "UP_to_IP R i (to_polynomial R a) = ring.indexed_const R a"
 proof
   fix x
   show "UP_to_IP R i (to_polynomial R a) x = ring.indexed_const R a x"
     apply(cases "x = {#}")
-    unfolding UP_to_IP_def 
+    unfolding UP_to_IP_def
     apply (metis R.indexed_const_def UP_ring.cfs_monom assms count_eq_zero_iff insert_absorb insert_not_empty is_UP_ring set_mset_empty subset_insert subset_refl to_polynomial_def)
   by (metis R.indexed_const_def UP_ring.cfs_monom assms count_eq_zero_iff is_UP_ring set_mset_eq_empty_iff subset_empty subset_insert to_polynomial_def)
-qed      
+qed
 
 lemma UP_to_IP_add:
   assumes "p \<in> carrier (UP R)"
   assumes "Q \<in> carrier (UP R)"
-  shows "UP_to_IP R i (p \<oplus>\<^bsub>UP R\<^esub> Q) = 
+  shows "UP_to_IP R i (p \<oplus>\<^bsub>UP R\<^esub> Q) =
          UP_to_IP R i p \<oplus>\<^bsub>Pring R {i}\<^esub> UP_to_IP R i Q"
 proof
   fix x
   show "UP_to_IP R i (p \<oplus>\<^bsub>UP R\<^esub> Q) x = (UP_to_IP R i p \<oplus>\<^bsub>Pring R {i}\<^esub> UP_to_IP R i Q) x"
   proof(cases "set_mset x \<subseteq> {i}")
     case True
-    have "(UP_to_IP R i p \<oplus>\<^bsub>Pring R {i}\<^esub> UP_to_IP R i Q) x = 
+    have "(UP_to_IP R i p \<oplus>\<^bsub>Pring R {i}\<^esub> UP_to_IP R i Q) x =
           (UP_to_IP R i p) x \<oplus>\<^bsub>R\<^esub> (UP_to_IP R i Q) x"
-      using True assms 
-      by (metis R.Pring_add R.indexed_padd_def)        
+      using True assms
+      by (metis R.Pring_add R.indexed_padd_def)
     then show ?thesis     using assms True
-      unfolding UP_to_IP_def UP_def 
-      by (smt partial_object.select_convs(1) restrict_def ring_record_simps(12))    
+      unfolding UP_to_IP_def UP_def
+      by (smt partial_object.select_convs(1) restrict_def ring_record_simps(12))
   next
     case False
-    have "(UP_to_IP R i p \<oplus>\<^bsub>Pring R {i}\<^esub> UP_to_IP R i Q) x = 
+    have "(UP_to_IP R i p \<oplus>\<^bsub>Pring R {i}\<^esub> UP_to_IP R i Q) x =
           (UP_to_IP R i p) x \<oplus>\<^bsub>R\<^esub> (UP_to_IP R i Q) x"
-            using False assms 
-            by (metis R.Pring_add R.indexed_padd_def) 
-    then show ?thesis using False assms  
-    unfolding UP_to_IP_def UP_def 
+            using False assms
+            by (metis R.Pring_add R.indexed_padd_def)
+    then show ?thesis using False assms
+    unfolding UP_to_IP_def UP_def
     using R.l_zero R.zero_closed by presburger
   qed
 qed
 
 lemma UP_to_IP_var:
   shows "UP_to_IP R i (X_poly R) = pvar R i"
 proof
   have 0: "(count {#i#} i) = 1"
     by simp
   have 1: "set_mset {#i#} \<subseteq> {i}"
     by simp
   have 2: "pvar R i {#i#} = \<one>"
     by (metis R.mset_to_IP_simp var_to_IP_def)
   fix x
   show "UP_to_IP R i (X_poly R) x = pvar R i x"
     apply(cases "x = {#i#}")
     using X_poly_def[of R] cfs_monom[of \<one> 1 "count x i"] 0 1 2
     unfolding UP_to_IP_def P_def
     using R.one_closed apply presburger
   proof-
     assume A: "x \<noteq>{#i#}"
     then show "(if set_mset x \<subseteq> {i} then X_poly R (count x i) else \<zero>\<^bsub>R\<^esub>) = pvar R i x"
     proof(cases "set_mset x \<subseteq> {i}")
       case True
       have "count x i \<noteq> 1" using True A
-        by (metis One_nat_def count_empty count_inI count_single empty_iff multiset_eqI 
+        by (metis One_nat_def count_empty count_inI count_single empty_iff multiset_eqI
             set_mset_add_mset_insert set_mset_empty set_mset_eq_empty_iff singletonD singletonI subset_singletonD)
       then have 0: "X_poly R (count x i) = \<zero>\<^bsub>R\<^esub>"
         using A UP_cring.X_closed UP_cring.degree_X UP_cring.intro True
-        unfolding X_poly_def 
-        using P_def R.one_closed \<open>\<one> \<in> carrier R \<Longrightarrow> up_ring.monom P \<one> 1 (count x i) = (if 1 = count x i then \<one> else \<zero>)\<close> by presburger        
+        unfolding X_poly_def
+        using P_def R.one_closed \<open>\<one> \<in> carrier R \<Longrightarrow> up_ring.monom P \<one> 1 (count x i) = (if 1 = count x i then \<one> else \<zero>)\<close> by presburger
       have "pvar R i x = \<zero>\<^bsub>R\<^esub>"
         using A var_to_IP_def
-        by (metis R.mset_to_IP_simp')        
-      then show ?thesis 
+        by (metis R.mset_to_IP_simp')
+      then show ?thesis
         using A "0" by presburger
     next
       case False
-      have "pvar R i x = \<zero>\<^bsub>R\<^esub>" using A var_to_IP_def False 
-        by (metis "1" R.Pring_set_zero R.mset_to_IP_closed)      
-      then show ?thesis     
-        unfolding UP_to_IP_def 
-        using False by presburger       
+      have "pvar R i x = \<zero>\<^bsub>R\<^esub>" using A var_to_IP_def False
+        by (metis "1" R.Pring_set_zero R.mset_to_IP_closed)
+      then show ?thesis
+        unfolding UP_to_IP_def
+        using False by presburger
     qed
   qed
 qed
 
 lemma UP_to_IP_var_pow:
   shows "UP_to_IP R i ((X_poly R)[^]\<^bsub>UP R\<^esub> (n::nat)) = (pvar R i)[^]\<^bsub>Pring R {i}\<^esub>n"
 proof
   fix x
   show "UP_to_IP R i (X_poly R [^]\<^bsub>UP R\<^esub> n) x = (pvar R i [^]\<^bsub>Pring R {i}\<^esub> n) x "
   proof(cases "set_mset x \<subseteq> {i}")
     case True
-    show ?thesis 
+    show ?thesis
     proof(cases "count x i = n")
       case T: True
       then have 0: "x = nat_to_mset i n"
-        using True 
-        by (metis count_inI emptyE insert_iff multiset_eqI nat_to_msetE 
+        using True
+        by (metis count_inI emptyE insert_iff multiset_eqI nat_to_msetE
             nat_to_msetE' subsetD)
       have 1: "x = nat_to_mset i (count x i)"
         using "0" T by auto
       then have LHS: "(pvar R i [^]\<^bsub>Pring R {i}\<^esub> n) x = \<one>\<^bsub>R\<^esub>"
-        using T True  0 1  
-        by (metis R.mset_to_IP_simp insertI1 pvar_pow)   
+        using T True  0 1
+        by (metis R.mset_to_IP_simp insertI1 pvar_pow)
       have 2: "UP_to_IP R i (X_poly R [^]\<^bsub>UP R\<^esub> n) x = (up_ring.monom (UP R) \<one> n) (count x i)"
-        unfolding UP_to_IP_def X_poly_def using True 
-        by (metis ctrm_degree P.nat_pow_closed P.nat_pow_eone P_def R.one_closed UP_cring.monom_coeff 
-            UP_one_closed UP_r_one X_closed is_UP_cring monom_one monom_rep_X_pow to_poly_inverse 
+        unfolding UP_to_IP_def X_poly_def using True
+        by (metis ctrm_degree P.nat_pow_closed P.nat_pow_eone P_def R.one_closed UP_cring.monom_coeff
+            UP_one_closed UP_r_one X_closed is_UP_cring monom_one monom_rep_X_pow to_poly_inverse
             to_poly_mult_simp(2))
       then show ?thesis
-        using True T LHS P_def R.one_closed cfs_monom 
-        by presburger                                      
+        using True T LHS P_def R.one_closed cfs_monom
+        by presburger
     next
       case False
       have "(pvar R i [^]\<^bsub>Pring R {i}\<^esub> n) = mset_to_IP R (nat_to_mset i n)"
         by (simp add: pvar_pow)
       hence 0: "(pvar R i [^]\<^bsub>Pring R {i}\<^esub> n) x = \<zero>"
-        by (metis False R.mset_to_IP_simp' nat_to_msetE)  
+        by (metis False R.mset_to_IP_simp' nat_to_msetE)
       have 1: "UP_to_IP R i (X_poly R [^]\<^bsub>UP R\<^esub> n) x = (up_ring.monom (UP R) \<one> n) (count x i)"
-        unfolding UP_to_IP_def X_poly_def  using False  True 
-        by (metis ctrm_degree P.nat_pow_closed P.nat_pow_eone P_def R.one_closed UP_one_closed 
+        unfolding UP_to_IP_def X_poly_def  using False  True
+        by (metis ctrm_degree P.nat_pow_closed P.nat_pow_eone P_def R.one_closed UP_one_closed
             UP_r_one X_closed cfs_monom monom_one monom_rep_X_pow to_poly_inverse to_poly_mult_simp(2))
       thus ?thesis using True False
-        unfolding UP_to_IP_def X_poly_def 0 
+        unfolding UP_to_IP_def X_poly_def 0
         by (metis P_def R.one_closed cfs_monom)
     qed
   next
     case False
-    then have 0: "UP_to_IP R i (X_poly R [^]\<^bsub>UP R\<^esub> n) x = \<zero>" 
-      unfolding UP_to_IP_def 
+    then have 0: "UP_to_IP R i (X_poly R [^]\<^bsub>UP R\<^esub> n) x = \<zero>"
+      unfolding UP_to_IP_def
       by meson
     have "(pvar R i [^]\<^bsub>Pring R {i}\<^esub> n) = mset_to_IP R (nat_to_mset i n)"
       by (simp add: pvar_pow)
     hence "(pvar R i [^]\<^bsub>Pring R {i}\<^esub> n) x = \<zero>"
       by (metis False R.mset_to_IP_simp' count_eq_zero_iff nat_to_msetE' singleton_iff subsetI)
-    then show ?thesis using 0 
+    then show ?thesis using 0
       by presburger
   qed
 qed
 
 lemma one_var_indexed_poly_monom_simp:
   assumes "a \<in> carrier R"
   shows "(a \<odot>\<^bsub>Pring R {i}\<^esub> ((pvar R i) [^]\<^bsub>Pring R {i}\<^esub> n)) x = (if x = (nat_to_mset i n) then a else \<zero>)"
 proof-
-  have 0: "(a \<odot>\<^bsub>Pring R {i}\<^esub> ((pvar R i) [^]\<^bsub>Pring R {i}\<^esub> n)) x = 
+  have 0: "(a \<odot>\<^bsub>Pring R {i}\<^esub> ((pvar R i) [^]\<^bsub>Pring R {i}\<^esub> n)) x =
                 a \<otimes> (((pvar R i) [^]\<^bsub>Pring R {i}\<^esub> n) x)"
     using  Pring_smult_cfs Pring_var_closed assms cring_def is_cring monoid.nat_pow_closed ring.Pring_is_monoid singletonI
-    by (simp add: monoid.nat_pow_closed ring.Pring_is_monoid R.Pring_smult_cfs R.Pring_var_closed R.ring_axioms) 
+    by (simp add: monoid.nat_pow_closed ring.Pring_is_monoid R.Pring_smult_cfs R.Pring_var_closed R.ring_axioms)
   have 1: "(pvar R i) [^]\<^bsub>Pring R {i}\<^esub> n = mset_to_IP R (nat_to_mset i n)"
-   using insertI1 
-   by (simp add: pvar_pow)    
-  then have 1: "(a \<odot>\<^bsub>Pring R {i}\<^esub> ((pvar R i) [^]\<^bsub>Pring R {i}\<^esub> n)) x= 
+   using insertI1
+   by (simp add: pvar_pow)
+  then have 1: "(a \<odot>\<^bsub>Pring R {i}\<^esub> ((pvar R i) [^]\<^bsub>Pring R {i}\<^esub> n)) x=
               a \<otimes> (mset_to_IP R (nat_to_mset i n) x)"
-    using "0" by presburger   
-  show ?thesis 
-    using assms 1 unfolding mset_to_IP_def 
-    using r_null r_one by simp         
-qed    
+    using "0" by presburger
+  show ?thesis
+    using assms 1 unfolding mset_to_IP_def
+    using r_null r_one by simp
+qed
 
 
 lemma UP_to_IP_monom:
   assumes "a \<in> carrier R"
   shows "UP_to_IP R i (up_ring.monom (UP R) a n) = a \<odot>\<^bsub>Pring R {i}\<^esub> ((pvar R i)[^]\<^bsub>Pring R {i}\<^esub>n)"
 proof
   fix x
   show "UP_to_IP R i (up_ring.monom (UP R) a n) x = (a \<odot>\<^bsub>Pring R {i}\<^esub> ((pvar R i) [^]\<^bsub>Pring R {i}\<^esub> n)) x"
   proof(cases "set_mset x \<subseteq> {i}")
     case True
-    then show ?thesis 
+    then show ?thesis
     proof(cases "count x i = n")
       case T: True
       then have "x = nat_to_mset i n"
-        using True 
-        by (metis count_inI emptyE insert_iff multiset_eqI nat_to_msetE 
+        using True
+        by (metis count_inI emptyE insert_iff multiset_eqI nat_to_msetE
             nat_to_msetE' subsetD)
       then have LHS: " (a \<odot>\<^bsub>Pring R {i}\<^esub> ((pvar R i) [^]\<^bsub>Pring R {i}\<^esub> n)) x = a"
-        using assms 
-        by (simp add: one_var_indexed_poly_monom_simp)                                                        
-      then show ?thesis 
+        using assms
+        by (simp add: one_var_indexed_poly_monom_simp)
+      then show ?thesis
         unfolding UP_to_IP_def
         using T True  assms(1)
-        by (metis UP_ring.cfs_monom is_UP_ring)        
+        by (metis UP_ring.cfs_monom is_UP_ring)
     next
       case False
-      then show ?thesis using True 
+      then show ?thesis using True
         unfolding UP_to_IP_def
-        by (metis INTEG.R.nat_to_msetE P_def assms cfs_monom one_var_indexed_poly_monom_simp)                                            
+        by (metis INTEG.R.nat_to_msetE P_def assms cfs_monom one_var_indexed_poly_monom_simp)
     qed
   next
     case False
-    then show ?thesis 
-      unfolding UP_to_IP_def 
-      by (metis (no_types, opaque_lifting) one_var_indexed_poly_monom_simp assms 
-          count_eq_zero_iff  equalityD2 insert_subset nat_to_msetE' subsetI subset_eq)        
+    then show ?thesis
+      unfolding UP_to_IP_def
+      by (metis (no_types, opaque_lifting) one_var_indexed_poly_monom_simp assms
+          count_eq_zero_iff  equalityD2 insert_subset nat_to_msetE' subsetI subset_eq)
   qed
 qed
 
 lemma UP_to_IP_monom':
   assumes "a \<in> carrier R"
   shows "UP_to_IP R i (up_ring.monom (UP R) a n) = a \<odot>\<^bsub>Pring R {i}\<^esub> ((pvar R i)[^]\<^bsub>Pring R {i}\<^esub>n)"
   by (metis R.Pring_smult UP_to_IP_monom assms)
 
 lemma UP_to_IP_closed:
   assumes "p \<in> carrier P"
   shows "(UP_to_IP R i p) \<in> carrier (Pring R {i})"
   apply(rule poly_induct3[of ])
   using assms apply blast
   apply (metis P_def R.Pring_add_closed UP_to_IP_add)
 proof-
   fix a fix n::nat
   assume A0: "a \<in> carrier R"
   have "(pvar R i [^]\<^bsub>Pring R {i}\<^esub> n) \<in> carrier (Pring R {i})"
     using pvar_closed[of R ] monoid.nat_pow_closed[of "Pring R {i}"]
   proof -
     show ?thesis
       by (meson R.Pring_is_monoid R.Pring_var_closed monoid.nat_pow_closed singleton_iff)
   qed
   then show "a \<in> carrier R \<Longrightarrow>
            UP_to_IP R i (up_ring.monom P a n) \<in> carrier (Pring R {i})"
-    using A0 assms(1) UP_to_IP_monom[of a i n] cring.poly_scalar_mult_closed [of R a _ "{i}"] 
+    using A0 assms(1) UP_to_IP_monom[of a i n] cring.poly_scalar_mult_closed [of R a _ "{i}"]
     by (metis P_def R.Pring_smult_closed)
 qed
 
 lemma IP_to_UP_inv:
   assumes "p \<in> carrier P"
   shows "IP_to_UP i (UP_to_IP R i p) = p"
   apply(rule poly_induct3[of ])
   using assms apply linarith
 proof-
   show "\<And>p q. q \<in> carrier P \<Longrightarrow>
            p \<in> carrier P \<Longrightarrow>
            IP_to_UP i (UP_to_IP R i p) = p \<Longrightarrow>
            IP_to_UP i (UP_to_IP R i q) = q \<Longrightarrow>
            IP_to_UP i (UP_to_IP R i (p \<oplus>\<^bsub>P\<^esub> q)) = p \<oplus>\<^bsub>P\<^esub> q"
   proof-
-    fix p q assume A: 
+    fix p q assume A:
            "q \<in> carrier P"
            "p \<in> carrier P"
            "IP_to_UP i (UP_to_IP R i p) = p"
            "IP_to_UP i (UP_to_IP R i q) = q"
     show "IP_to_UP i (UP_to_IP R i (p \<oplus>\<^bsub>P\<^esub> q)) = p \<oplus>\<^bsub>P\<^esub> q"
       using A UP_to_IP_add[of p q i]
-                  UP_to_IP_closed 
-                  IP_to_UP_add 
-      unfolding P_def 
+                  UP_to_IP_closed
+                  IP_to_UP_add
+      unfolding P_def
       by metis
   qed
   show "\<And>a n. a \<in> carrier R \<Longrightarrow>
            IP_to_UP i (UP_to_IP R i (up_ring.monom P a n)) =
            up_ring.monom P a n"
   proof-
     fix a fix n::nat
     assume A0: "a \<in> carrier R"
     have A1: "pvar R i [^]\<^bsub>Pring R {i}\<^esub> n \<in> carrier (Pring R {i})"
-      using pvar_closed monoid.nat_pow_closed 
+      using pvar_closed monoid.nat_pow_closed
       by (metis R.Pring_is_monoid R_cring singletonI)
     have "UP_to_IP R i (up_ring.monom (UP R) a n) = a \<odot>\<^bsub>Pring R {i}\<^esub>  (pvar R i [^]\<^bsub>Pring R {i}\<^esub> n)"
-      by (meson A0 UP_to_IP_monom')               
+      by (meson A0 UP_to_IP_monom')
     then have A2: "IP_to_UP i (UP_to_IP R i (up_ring.monom (UP R) a n)) =
             IP_to_UP i (a \<odot>\<^bsub>Pring R {i}\<^esub> (pvar R i [^]\<^bsub>Pring R {i}\<^esub> n))"
-      by presburger 
+      by presburger
     have A3: "IP_to_UP i (pvar R i [^]\<^bsub>Pring R {i}\<^esub> n) =  (up_ring.monom P \<one> n)"
     proof(induction n)
       case 0
-      then show ?case 
-        by (metis Group.nat_pow_0 IP_to_UP_one P_def monom_one)     
+      then show ?case
+        by (metis Group.nat_pow_0 IP_to_UP_one P_def monom_one)
     next
       case (Suc n)
 
-      then show ?case 
-      using IP_to_UP_ring_hom[of i] 
+      then show ?case
+      using IP_to_UP_ring_hom[of i]
               ring_hom_mult[of "IP_to_UP i" "Pring R {i}" "UP R" "pvar R i" "pvar R i [^]\<^bsub>Pring R {i}\<^esub> n"]
-                ring_hom_ring.homh[of "Pring R {i}" "UP R"  "IP_to_UP i"] 
+                ring_hom_ring.homh[of "Pring R {i}" "UP R"  "IP_to_UP i"]
       by (metis IP_to_UP_monom P.l_one P.nat_pow_closed P_def R.one_closed UP_cring.ctrm_degree UP_cring.monom_rep_X_pow  UP_one_closed X_closed cfs_monom is_UP_cring monom_one pvar_pow singletonI to_poly_inverse to_poly_mult_simp(1))
-    qed      
+    qed
     then show "IP_to_UP i (UP_to_IP R i (up_ring.monom P a n)) =
            up_ring.monom P a n"
-      using A2 IP_to_UP_scalar_mult[of a "pvar R i [^]\<^bsub>Pring R {i}\<^esub> n" i] 
+      using A2 IP_to_UP_scalar_mult[of a "pvar R i [^]\<^bsub>Pring R {i}\<^esub> n" i]
             A0 A1 P_def monic_monom_smult by presburger
   qed
-qed  
+qed
 
 lemma UP_to_IP_mult:
   assumes "p \<in> carrier (UP R)"
   assumes "Q \<in> carrier (UP R)"
-  shows "UP_to_IP R i (p \<otimes>\<^bsub>UP R\<^esub> Q) = 
+  shows "UP_to_IP R i (p \<otimes>\<^bsub>UP R\<^esub> Q) =
          UP_to_IP R i p \<otimes>\<^bsub>Pring R {i}\<^esub> UP_to_IP R i Q"
 proof-
   have 0: "IP_to_UP i (UP_to_IP R i (p \<otimes>\<^bsub>UP R\<^esub> Q)) = (p \<otimes>\<^bsub>UP R\<^esub> Q)"
     by (meson UP_cring.IP_to_UP_inv UP_ring.UP_mult_closed assms(1) assms(2) is_UP_cring is_UP_ring)
   have 1: "IP_to_UP i (UP_to_IP R i p \<otimes>\<^bsub>Pring R {i}\<^esub> UP_to_IP R i Q) =
         IP_to_UP i (UP_to_IP R i p) \<otimes>\<^bsub>UP R\<^esub> IP_to_UP i ( UP_to_IP R i Q)"
-    using IP_to_UP_ring_hom[of  i] 
-          ring_hom_mult[of "IP_to_UP i"] 
+    using IP_to_UP_ring_hom[of  i]
+          ring_hom_mult[of "IP_to_UP i"]
           UP_to_IP_closed assms
     by (smt P_def ring_hom_ring.homh)
-  have 2: "IP_to_UP i (UP_to_IP R i (p \<otimes>\<^bsub>UP R\<^esub> Q)) = 
+  have 2: "IP_to_UP i (UP_to_IP R i (p \<otimes>\<^bsub>UP R\<^esub> Q)) =
         IP_to_UP i (UP_to_IP R i p \<otimes>\<^bsub>Pring R {i}\<^esub> UP_to_IP R i Q)"
-    using 0 1 assms 
-    by (metis UP_cring.IP_to_UP_inv is_UP_cring)     
-  then show ?thesis 
-    by (metis "0" P_def R.Pring_mult_closed R.ring_axioms assms(1) assms(2) ring.Pring_car UP_to_IP_closed UP_to_IP_inv)    
+    using 0 1 assms
+    by (metis UP_cring.IP_to_UP_inv is_UP_cring)
+  then show ?thesis
+    by (metis "0" P_def R.Pring_mult_closed R.ring_axioms assms(1) assms(2) ring.Pring_car UP_to_IP_closed UP_to_IP_inv)
 qed
 
 lemma UP_to_IP_ring_hom:
 shows "ring_hom_ring (UP R) (Pring R {i}) (UP_to_IP R i)"
   apply(rule ring_hom_ringI)
   using P_def UP_ring apply force
   apply (simp add: R.Pring_is_ring; fail)
   apply (metis P_def UP_to_IP_closed)
   apply (meson UP_to_IP_mult)
   apply (meson UP_to_IP_add)
   by (metis IP_to_UP_one R.Pring_car R.Pring_one_closed UP_to_IP_inv)
 
 end
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsubsection\<open>The isomorphism $R[I\cup J] \sim R[I][J]$, where $I$ and $J$ are disjoint variable sets\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 text\<open>
-  Given a ring $R$ and variable sets $I$ and $J$, we'd like to construct the canonical 
-  (iso)morphism $R[I\cup J] \to R[I][J]$. This can be done with the univeral property of the 
-  previous section. Let $\phi: R \to R[J]$ be the inclusion of constants, and $f: J \to R[I]$ be 
+  Given a ring $R$ and variable sets $I$ and $J$, we'd like to construct the canonical
+  (iso)morphism $R[I\cup J] \to R[I][J]$. This can be done with the univeral property of the
+  previous section. Let $\phi: R \to R[J]$ be the inclusion of constants, and $f: J \to R[I]$ be
   the map which sends the variable $i$ to the polynomial variable $i$ over the ring $R[I][J]$.
-  Then these are the two basic pieces of input required to give us a canonical homomoprhism 
-  $R[I \cup J] \to R[I][J]$ with the universal property. The first map $\phi$ will be 
-  "\texttt{dist\_varset\_morpshim}" below, and the second map will be 
-  "\texttt{dist\_varset\_var\_ass}". The desired induced isomorphism will be 
+  Then these are the two basic pieces of input required to give us a canonical homomoprhism
+  $R[I \cup J] \to R[I][J]$ with the universal property. The first map $\phi$ will be
+  "\texttt{dist\_varset\_morpshim}" below, and the second map will be
+  "\texttt{dist\_varset\_var\_ass}". The desired induced isomorphism will be
   called "\texttt{var\_factor}".   \<close>
 
-definition(in ring) dist_varset_morphism 
-  :: "'d set \<Rightarrow> 'd set \<Rightarrow> 
+definition(in ring) dist_varset_morphism
+  :: "'d set \<Rightarrow> 'd set \<Rightarrow>
               ('a, (('a, 'd) mvar_poly, 'd) mvar_poly) ring_hom" where
-  "dist_varset_morphism (I:: 'd set) (J:: 'd set) = 
+  "dist_varset_morphism (I:: 'd set) (J:: 'd set) =
         (ring.indexed_const (Pring R J) :: ('d multiset \<Rightarrow> 'a) \<Rightarrow> 'd multiset \<Rightarrow> ('d multiset \<Rightarrow> 'a))\<circ> (ring.indexed_const R ::'a \<Rightarrow> 'd multiset \<Rightarrow> 'a)"
 
 definition(in ring) dist_varset_var_ass
     :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'd \<Rightarrow> (('a, 'd) mvar_poly, 'd) mvar_poly"
     where
-"dist_varset_var_ass (I:: 'd set) (J:: 'd set)  = (\<lambda>i. if i \<in> J then ring.indexed_const (Pring R J) (pvar R i) else 
+"dist_varset_var_ass (I:: 'd set) (J:: 'd set)  = (\<lambda>i. if i \<in> J then ring.indexed_const (Pring R J) (pvar R i) else
                                                 pvar (Pring R J) i )"
 
 context cring
 begin
 
 lemma dist_varset_morphism_is_morphism:
   assumes "(I:: 'd set) \<subseteq> J0 \<union> J1"
   assumes "J1 \<subseteq> I"
   assumes "\<phi> = dist_varset_morphism I J0"
   shows "ring_hom_ring R (Pring (Pring R J0) J1) \<phi>"
 proof-
   have 0:"ring_hom_ring R (Pring R J0) indexed_const"
     by (simp add: indexed_const_ring_hom)
   have 1:"ring_hom_ring (Pring R J0) (Pring (Pring R J0) J1) (ring.indexed_const (Pring R J0))"
     by (simp add: cring.indexed_const_ring_hom is_cring local.ring_axioms ring.Pring_is_cring)
   show ?thesis using 0 1 assms ring_hom_trans[of "indexed_const" R "Pring R J0" "ring.indexed_const (Pring R J0)"
               "(Pring (Pring R J0) J1) "]
-    unfolding dist_varset_morphism_def 
+    unfolding dist_varset_morphism_def
   by (meson ring_hom_ring.axioms(1) ring_hom_ring.axioms(2) ring_hom_ring.homh ring_hom_ringI2)
 qed
 
-definition var_factor :: 
+definition var_factor ::
   "'d set \<Rightarrow> 'd set \<Rightarrow> 'd set \<Rightarrow>
          (('a, 'd) mvar_poly, (('a, 'd) mvar_poly, 'd) mvar_poly) ring_hom "where
-"var_factor (I:: 'd set) (J0:: 'd set) (J1:: 'd set) = indexed_poly_induced_morphism  I (Pring (Pring R J0) J1)                                                       
+"var_factor (I:: 'd set) (J0:: 'd set) (J1:: 'd set) = indexed_poly_induced_morphism  I (Pring (Pring R J0) J1)
                                                         (dist_varset_morphism I J0) (dist_varset_var_ass I J0)"
 
 lemma indexed_const_closed:
   assumes "x \<in> carrier R"
   shows "indexed_const x \<in> carrier (Pring R I)"
   using Pring_car assms indexed_pset.indexed_const by blast
 
 lemma var_factor_morphism:
   assumes "(I:: 'd set) \<subseteq> J0 \<union> J1"
   assumes "J1 \<subseteq> I"
   assumes "J1 \<inter> J0 = {}"
   assumes "g = dist_varset_var_ass I J0"
   assumes "\<phi> = dist_varset_morphism I J0"
   assumes "\<psi> = (var_factor I J0 J1)"
   shows "ring_hom_ring (Pring R I) (Pring (Pring R J0) J1) \<psi> "
         "\<And>i. i \<in> J0 \<inter> I \<Longrightarrow> \<psi> (pvar R i) = ring.indexed_const (Pring R J0) (pvar R i)"
         "\<And>i. i \<in> J1 \<Longrightarrow> \<psi> (pvar R i) = pvar (Pring R J0) i"
         "\<And>a. a \<in> carrier (Pring R (J0 \<inter> I)) \<Longrightarrow> \<psi> a = ring.indexed_const (Pring R J0) a"
 proof-
   have 0: "g \<in> I \<rightarrow> carrier (Pring (Pring R J0) J1)"
   proof
     fix x
     assume A0: "x \<in> I"
     then have A1: "x \<in> J0 \<or> x \<in> J1"
       by (meson UnE assms(1) subsetD)
     have A2: "x \<notin> J0 \<Longrightarrow> x \<in> J1"
       using A1 by blast
     show "g x \<in> carrier (Pring (Pring R J0) J1)"
       apply(cases "x \<in> J0")
-      using assms A0 A1 A2 pvar_closed[of R x J0] 
+      using assms A0 A1 A2 pvar_closed[of R x J0]
         pvar_closed[of "Pring R J0" x J1]
         cring.indexed_const_closed[of "Pring R J0" ]
-      unfolding dist_varset_var_ass_def 
+      unfolding dist_varset_var_ass_def
        apply (smt Pring_is_cring is_cring)
-      using assms A0 A1 A2 pvar_closed[of R x J0] 
+      using assms A0 A1 A2 pvar_closed[of R x J0]
         pvar_closed[of "Pring R J0" x J1]
         cring.indexed_const_closed[of "Pring R J0" ]
-      unfolding dist_varset_var_ass_def 
+      unfolding dist_varset_var_ass_def
       by (smt Pring_is_cring is_cring)
   qed
   have 1: " cring (Pring R J0)"
     by (simp add: Pring_is_cring is_cring)
   have 2: " cring (Pring (Pring R J0) J1)"
     by (simp add: "1" Pring_is_ring ring.Pring_is_cring)
-  show C0: "ring_hom_ring (Pring R I) (Pring (Pring R J0) J1) \<psi> " 
-    using 0 assms Pring_universal_prop[of "Pring (Pring R J0) J1" g I \<phi> \<psi>] 
+  show C0: "ring_hom_ring (Pring R I) (Pring (Pring R J0) J1) \<psi> "
+    using 0 assms Pring_universal_prop[of "Pring (Pring R J0) J1" g I \<phi> \<psi>]
         dist_varset_morphism_is_morphism[of I J0 J1]
     unfolding var_factor_def
     by (meson Pring_is_cring Pring_is_ring is_cring ring.Pring_is_cring)
   show C1: "\<And>i. i \<in> J0 \<inter> I \<Longrightarrow> \<psi> (pvar R i) = ring.indexed_const (Pring R J0) (pvar R i)"
-    using  0 1 2 assms Pring_universal_prop(2)[of "Pring (Pring R J0) J1" g I \<phi> \<psi>] 
-        dist_varset_morphism_is_morphism[of I J0 J1 \<phi>] dist_varset_var_ass_def 
+    using  0 1 2 assms Pring_universal_prop(2)[of "Pring (Pring R J0) J1" g I \<phi> \<psi>]
+        dist_varset_morphism_is_morphism[of I J0 J1 \<phi>] dist_varset_var_ass_def
         dist_varset_morphism_def unfolding var_factor_def  var_to_IP_def
     by (smt IntE dist_varset_var_ass_def inf_commute inf_le2 ring.Pring_is_cring subsetD var_to_IP_def)
   have 3: "\<And>i. i \<in> J1 \<Longrightarrow> g i = mset_to_IP (Pring R J0) {#i#} "
     using assms unfolding dist_varset_var_ass_def var_to_IP_def
     by (meson disjoint_iff_not_equal)
   show C2: "\<And>i. i \<in> J1 \<Longrightarrow> \<psi> (pvar R i) = pvar (Pring R J0) i"
-      using  0 1 2 3 assms Pring_universal_prop(2)[of "Pring (Pring R J0) J1" g I \<phi> \<psi>] 
-        dist_varset_morphism_is_morphism[of I J0 J1 \<phi>] 
+      using  0 1 2 3 assms Pring_universal_prop(2)[of "Pring (Pring R J0) J1" g I \<phi> \<psi>]
+        dist_varset_morphism_is_morphism[of I J0 J1 \<phi>]
          unfolding   var_factor_def  var_to_IP_def
          by (metis subsetD)
   have 4: "\<And>k. k \<in> carrier R \<Longrightarrow> \<psi> (indexed_const k) = ring.indexed_const (Pring R J0) (indexed_const k)"
-      using  0 1 2 3 assms Pring_universal_prop(3)[of "Pring (Pring R J0) J1" g I \<phi> \<psi>] 
+      using  0 1 2 3 assms Pring_universal_prop(3)[of "Pring (Pring R J0) J1" g I \<phi> \<psi>]
         dist_varset_morphism_is_morphism[of I J0 J1 \<phi>] comp_apply
       unfolding   var_factor_def  var_to_IP_def dist_varset_morphism_def
       by metis
   show C3:  "\<And>a. a \<in> carrier (Pring R (J0 \<inter> I)) \<Longrightarrow> \<psi> a = ring.indexed_const (Pring R J0) a"
   proof- fix a assume A: "a \<in> carrier (Pring R (J0 \<inter> I))"
     show " \<psi> a = ring.indexed_const (Pring R J0) a"
       apply(rule indexed_pset.induct[of a "J0 \<inter> I" "carrier R"])
       using A Pring_car apply blast
-      using 4  
+      using 4
       apply blast
-    proof- 
+    proof-
       show "\<And>P Q. P \<in> Pring_set R (J0 \<inter> I) \<Longrightarrow>
            \<psi> P = ring.indexed_const (Pring R J0) P \<Longrightarrow>
            Q \<in> Pring_set R (J0 \<inter> I) \<Longrightarrow> \<psi> Q = ring.indexed_const (Pring R J0) Q \<Longrightarrow> \<psi> (P \<Oplus> Q) = ring.indexed_const (Pring R J0) (P \<Oplus> Q)"
-      proof- fix P Q 
+      proof- fix P Q
         assume A0: "P \<in> Pring_set R (J0 \<inter> I)"
         assume A1: "\<psi> P = ring.indexed_const (Pring R J0) P"
         assume A2: "Q \<in> Pring_set R (J0 \<inter> I)"
         assume A3: "\<psi> Q = ring.indexed_const (Pring R J0) Q"
         have A0': "P \<in> Pring_set R I"
-          using A0  
+          using A0
           by (meson Int_lower2 Pring_carrier_subset subsetD)
         have A1': "Q \<in> Pring_set R I"
           by (meson A2 Int_lower2 Pring_carrier_subset subsetD)
         have B: "\<psi> (P \<Oplus> Q) = \<psi> P \<oplus>\<^bsub>Pring (Pring R J0) J1\<^esub> \<psi> Q"
-          using A0' A1' A2 A3 C0 assms ring_hom_add 
+          using A0' A1' A2 A3 C0 assms ring_hom_add
           by (metis (no_types, lifting) Pring_add local.ring_axioms ring.Pring_car ring_hom_ring.homh)
         have " ring.indexed_const (Pring R J0) (P \<Oplus> Q) =  ring.indexed_const (Pring R J0) P
                                  \<oplus>\<^bsub>Pring (Pring R J0) J1\<^esub>ring.indexed_const (Pring R J0) Q"
           by (simp add: Pring_add Pring_is_ring ring.Pring_add ring.indexed_padd_const)
         then show " \<psi> (P \<Oplus> Q) = ring.indexed_const (Pring R J0) (P \<Oplus> Q)"
-          using B 
+          using B
           by (simp add: A1 A3)
-      qed 
-      show "\<And>P i. P \<in> Pring_set R (J0 \<inter> I) \<Longrightarrow> \<psi> P = ring.indexed_const (Pring R J0) P \<Longrightarrow> 
+      qed
+      show "\<And>P i. P \<in> Pring_set R (J0 \<inter> I) \<Longrightarrow> \<psi> P = ring.indexed_const (Pring R J0) P \<Longrightarrow>
                     i \<in> J0 \<inter> I \<Longrightarrow> \<psi> (P \<Otimes> i) = ring.indexed_const (Pring R J0) (P \<Otimes> i)"
       proof- fix P i assume A0: "P \<in> Pring_set R (J0 \<inter> I)"
         assume A1: "\<psi> P = ring.indexed_const (Pring R J0) P"
         assume A2: "i \<in> J0 \<inter> I"
         have A0': "P \<in> carrier (Pring R I)"
-          using A0  Pring_carrier_subset 
+          using A0  Pring_carrier_subset
           by (metis (no_types, opaque_lifting) Pring_car in_mono inf_commute le_inf_iff subset_refl)
         have A0'': "P \<in> carrier (Pring R J0)"
-          using A0  Pring_carrier_subset 
+          using A0  Pring_carrier_subset
           by (metis (no_types, opaque_lifting) Pring_car in_mono inf_commute le_inf_iff subset_refl)
         have " \<psi> (P \<Otimes> i) = \<psi> P \<otimes>\<^bsub>Pring (Pring R J0) J1\<^esub>  ring.indexed_const (Pring R J0) (pvar R i)"
         proof-
           have "(P \<Otimes> i) = P \<otimes>\<^bsub>Pring R I\<^esub> pvar R i"
-            using A0' A2 unfolding var_to_IP_def 
-            by (metis A0 Pring_mult poly_index_mult)          
-          then show ?thesis 
-            using A0' C0 A2 C1[of i] ring_hom_ring.homh 
-                  ring_hom_mult[of \<psi> "Pring R I" "Pring (Pring R J0) J1" P "pvar R i"] 
+            using A0' A2 unfolding var_to_IP_def
+            by (metis A0 Pring_mult poly_index_mult)
+          then show ?thesis
+            using A0' C0 A2 C1[of i] ring_hom_ring.homh
+                  ring_hom_mult[of \<psi> "Pring R I" "Pring (Pring R J0) J1" P "pvar R i"]
           by (metis IntE Pring_var_closed)
         qed
-        then have" \<psi> (P \<Otimes> i) = ring.indexed_const (Pring R J0) P \<otimes>\<^bsub>Pring (Pring R J0) J1\<^esub> 
+        then have" \<psi> (P \<Otimes> i) = ring.indexed_const (Pring R J0) P \<otimes>\<^bsub>Pring (Pring R J0) J1\<^esub>
                            ring.indexed_const (Pring R J0) (pvar R i)"
           by (simp add: A1)
         then have " \<psi> (P \<Otimes> i) = ring.indexed_const (Pring R J0) (P \<otimes>\<^bsub>Pring R J0\<^esub> (pvar R i))"
-          using A0'' A2  cring.indexed_const_ring_hom[of "Pring R J0" J1] ring_hom_ring.homh 
-                ring_hom_mult[of "ring.indexed_const (Pring R J0)" "Pring R J0" _  P "pvar R i"] 
+          using A0'' A2  cring.indexed_const_ring_hom[of "Pring R J0" J1] ring_hom_ring.homh
+                ring_hom_mult[of "ring.indexed_const (Pring R J0)" "Pring R J0" _  P "pvar R i"]
           by (smt "1" IntE Pring_var_closed)
         then show "\<psi> (P \<Otimes> i) = ring.indexed_const (Pring R J0) (P \<Otimes> i)"
-          using poly_index_mult[of P J0 i] unfolding var_to_IP_def  
+          using poly_index_mult[of P J0 i] unfolding var_to_IP_def
           by (metis A0'' A2 IntE Pring_car Pring_mult)
       qed
     qed
   qed
 qed
 
 lemma var_factor_morphism':
   assumes "I = J0 \<union> J1"
   assumes "J1 \<subseteq> I"
   assumes "J1 \<inter> J0 = {}"
   assumes "\<psi> = (var_factor I J0 J1)"
   shows "ring_hom_ring (Pring R I) (Pring (Pring R J0) J1) \<psi> "
         "\<And>i. i \<in> J1 \<Longrightarrow> \<psi> (pvar R i) = pvar (Pring R J0) i"
         "\<And>a. a \<in> carrier (Pring R (J0 \<inter> I)) \<Longrightarrow> \<psi> a = ring.indexed_const (Pring R J0) a"
-  using assms var_factor_morphism 
+  using assms var_factor_morphism
   apply blast
    using assms var_factor_morphism(3)
   apply (metis subset_refl)
-     using assms var_factor_morphism(4) 
+     using assms var_factor_morphism(4)
   by (metis Un_subset_iff Un_upper1)
 
 text\<open>Constructing the inverse morphism for \texttt{var\_factor\_morphism} \<close>
 
 
 lemma pvar_ass_closed:
   assumes "J1 \<subseteq> I"
   shows "pvar R \<in> J1 \<rightarrow> carrier (Pring R I)"
   by (meson Pi_I Pring_var_closed assms subsetD)
 
 text\<open>The following function gives us the inverse morphism $R[I][J] \to R[I \cup J]$:\<close>
-definition var_factor_inv :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'd set \<Rightarrow> 
+definition var_factor_inv :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'd set \<Rightarrow>
   ((('a, 'd) mvar_poly, 'd) mvar_poly, ('a, 'd) mvar_poly) ring_hom" where
-"var_factor_inv (I:: 'd set) (J0:: 'd set) (J1:: 'd set) = indexed_poly_induced_morphism J1 (Pring R I)                                                      
+"var_factor_inv (I:: 'd set) (J0:: 'd set) (J1:: 'd set) = indexed_poly_induced_morphism J1 (Pring R I)
                                                         (id:: ('d multiset \<Rightarrow> 'a) \<Rightarrow> 'd multiset \<Rightarrow> 'a)
                                                        (pvar R)"
-  
+
 lemma var_factor_inv_morphism:
   assumes "I = J0 \<union> J1"
   assumes "J1 \<subseteq> I"
   assumes "J1 \<inter> J0 = {}"
   assumes "\<psi> = (var_factor_inv I J0 J1)"
   shows "ring_hom_ring (Pring (Pring R J0) J1) (Pring R I)  \<psi> "
         "\<And>i. i \<in> J1 \<Longrightarrow> \<psi> (pvar (Pring R J0) i) = pvar R i"
         "\<And>a. a \<in> carrier (Pring R J0) \<Longrightarrow> \<psi> (ring.indexed_const (Pring R J0) a) = a"
-proof-   
+proof-
   have 0:  "ring_hom_ring (Pring R J0) (Pring R I) id"
     apply(rule ring_hom_ringI)
-      apply (simp add: Pring_is_ring; fail) 
+      apply (simp add: Pring_is_ring; fail)
         apply (simp add: Pring_is_ring;fail )
        apply (metis Pring_car Pring_carrier_subset Un_upper1 assms(1) id_apply subsetD)
     apply (metis Pring_mult_eq id_apply)
      apply (metis Pring_add_eq id_apply)
   by (simp add: Pring_one_eq)
-  then show "ring_hom_ring (Pring (Pring R J0) J1) (Pring R I)  \<psi> " 
+  then show "ring_hom_ring (Pring (Pring R J0) J1) (Pring R I)  \<psi> "
     using cring.Pring_universal_prop(1)[of "Pring R J0" "Pring R I" "pvar R" J1 id \<psi>]
         pvar_ass_closed[of J0 I]
     by (metis Pring_is_cring assms(2) assms(4) is_cring pvar_ass_closed var_factor_inv_def)
   show "\<And>i. i \<in> J1 \<Longrightarrow> \<psi> (pvar (Pring R J0) i) = pvar R i"
-    using 0 assms pvar_ass_closed[of J0 I] 
+    using 0 assms pvar_ass_closed[of J0 I]
         cring.Pring_universal_prop(2)[of "Pring R J0" "Pring R I" "pvar R" J1 id \<psi>]
     by (metis Pring_is_cring is_cring pvar_ass_closed var_factor_inv_def var_to_IP_def)
   show "\<And>a. a \<in> carrier (Pring R J0) \<Longrightarrow> \<psi> (ring.indexed_const (Pring R J0) a) = a"
-    using 0 assms pvar_ass_closed[of J0 I] 
+    using 0 assms pvar_ass_closed[of J0 I]
         cring.Pring_universal_prop(3)[of "Pring R J0" "Pring R I" "pvar R" J1 id \<psi>]
     by (smt Pi_I Pring_is_cring Pring_var_closed id_def is_cring subsetD var_factor_inv_def)
 qed
 
 lemma var_factor_inv_inverse:
   assumes "I = J0 \<union> J1"
   assumes "J1 \<subseteq> I"
   assumes "J1 \<inter> J0 = {}"
   assumes "\<psi>1 = (var_factor_inv I J0 J1)"
   assumes "\<psi>0 = (var_factor I J0 J1)"
   assumes "P \<in> carrier (Pring R I)"
   shows "\<psi>1 (\<psi>0 P) = P"
   apply(rule indexed_pset.induct[of P I "carrier R"])
   using Pring_car assms(6) apply blast
     using var_factor_inv_morphism(3)[of I J0 J1 "\<psi>1"] var_factor_morphism'(3)[of I J0 J1 \<psi>0] assms
-    apply (metis indexed_const_closed inf_sup_absorb)    
+    apply (metis indexed_const_closed inf_sup_absorb)
 proof-
   have 0: "ring_hom_ring (Pring (Pring R J0) J1) (Pring R I) \<psi>1"
     by (simp add: assms(1) assms(3) assms(4) var_factor_inv_morphism(1))
   have 1: "ring_hom_ring (Pring R I) (Pring (Pring R J0) J1) \<psi>0"
     by (simp add: assms(1) assms(3) assms(5) var_factor_morphism'(1))
   have 2: "\<psi>1 \<circ> \<psi>0 \<in> ring_hom (Pring R I) (Pring R I)"
     using 0 1 ring_hom_trans[of \<psi>0 "Pring R I" "Pring (Pring R J0) J1" \<psi>1 "Pring R I"]
-          ring_hom_ring.homh[of "Pring R I" "Pring (Pring R J0) J1" \<psi>0] 
+          ring_hom_ring.homh[of "Pring R I" "Pring (Pring R J0) J1" \<psi>0]
           ring_hom_ring.homh[of "Pring (Pring R J0) J1" "Pring R I" \<psi>1]
     by blast
   show "\<And>P Q. P \<in> Pring_set R I \<Longrightarrow>
            \<psi>1 (\<psi>0 P) = P \<Longrightarrow> Q \<in> Pring_set R I \<Longrightarrow> \<psi>1 (\<psi>0 Q) = Q \<Longrightarrow> \<psi>1 (\<psi>0 (P \<Oplus> Q)) = P \<Oplus> Q"
   proof- fix P Q assume A0: "P \<in> Pring_set R I" "\<psi>1 (\<psi>0 P) = P"
     assume A1: "Q \<in> Pring_set R I" "\<psi>1 (\<psi>0 Q) = Q"
     show "\<psi>1 (\<psi>0 (P \<Oplus> Q)) = P \<Oplus> Q"
       using  A0 A1 2 ring_hom_add[of "\<psi>1 \<circ> \<psi>0" "Pring R I" "Pring R I" P Q] comp_apply[of \<psi>1 \<psi>0]
       by (simp add: "2" \<open>P \<in> Pring_set R I\<close> \<open>Q \<in> Pring_set R I\<close> Pring_add Pring_car)
   qed
   show "\<And>P i. P \<in> Pring_set R I \<Longrightarrow> \<psi>1 (\<psi>0 P) = P \<Longrightarrow> i \<in> I \<Longrightarrow> \<psi>1 (\<psi>0 (P \<Otimes> i)) = P \<Otimes> i"
   proof-
     fix P i assume A: "P \<in> Pring_set R I" "\<psi>1 (\<psi>0 P) = P" "i \<in> I"
     show "\<psi>1 (\<psi>0 (P \<Otimes> i)) = P \<Otimes> i"
     proof-
       have A0: "P \<Otimes> i = P \<otimes>\<^bsub>Pring R I\<^esub> pvar R i"
         by (metis A(1) A(3) Pring_mult local.ring_axioms ring.poly_index_mult var_to_IP_def)
       have A1: "\<psi>1 (\<psi>0 (pvar R i)) = pvar R i"
-       by (metis A(3) Int_iff Pring_var_closed UnE Un_subset_iff Un_upper1 assms(1) assms(2) 
-           assms(3) assms(4) assms(5) cring.var_factor_morphism(2) is_cring 
+       by (metis A(3) Int_iff Pring_var_closed UnE Un_subset_iff Un_upper1 assms(1) assms(2)
+           assms(3) assms(4) assms(5) cring.var_factor_morphism(2) is_cring
            var_factor_inv_morphism(2) var_factor_inv_morphism(3) var_factor_morphism'(2))
-     then show ?thesis 
+     then show ?thesis
        using 2 A0 A ring_hom_mult[of "\<psi>1 \<circ> \<psi>0" "Pring R I" _  P "pvar R i" ]
             Pring_car Pring_var_closed comp_apply[of \<psi>1 \<psi>0]
        by smt
    qed
  qed
 qed
 
 lemma var_factor_total_eval:
   assumes "I = J0 \<union> J1"
   assumes "J1 \<subseteq> I"
   assumes "J1 \<inter> J0 = {}"
   assumes "\<psi> = (var_factor I J0 J1)"
   assumes "closed_fun R g"
   assumes "P \<in> carrier (Pring R I)"
   shows "total_eval R g P = total_eval R g (total_eval (Pring R J0) (indexed_const \<circ> g) (\<psi> P))"
   apply(rule indexed_pset.induct[of P I "carrier R"])
   using Pring_car assms apply blast
-    apply (metis Pring_is_cring assms(1) assms(2) assms(3) assms(4) cring.total_eval_const 
+    apply (metis Pring_is_cring assms(1) assms(2) assms(3) assms(4) cring.total_eval_const
       indexed_const_closed is_cring var_factor_morphism'(3))
 proof-
   show "\<And>P Q. P \<in> Pring_set R I \<Longrightarrow>
            total_eval R g P = total_eval R g (total_eval (Pring R J0) (indexed_const \<circ> g) (\<psi> P)) \<Longrightarrow>
            Q \<in> Pring_set R I \<Longrightarrow>
            total_eval R g Q = total_eval R g (total_eval (Pring R J0) (indexed_const \<circ> g) (\<psi> Q)) \<Longrightarrow>
            total_eval R g (P \<Oplus> Q) = total_eval R g (total_eval (Pring R J0) (indexed_const \<circ> g) (\<psi> (P \<Oplus> Q)))"
-  proof- fix P Q 
+  proof- fix P Q
     assume A: " P \<in> Pring_set R I"
            "total_eval R g P = total_eval R g (total_eval (Pring R J0) (indexed_const \<circ> g) (\<psi> P))"
     assume B: " Q \<in> Pring_set R I"
            "total_eval R g Q = total_eval R g (total_eval (Pring R J0) (indexed_const \<circ> g) (\<psi> Q))"
     have 0: "\<psi> (P \<Oplus> Q) = \<psi> P \<oplus>\<^bsub>Pring (Pring R J0) J1\<^esub> \<psi> Q"
-      using A B assms var_factor_morphism'(1)[of I J0 J1 \<psi>] 
-            Pring_add[of I P Q] Pring_car[of I] 
+      using A B assms var_factor_morphism'(1)[of I J0 J1 \<psi>]
+            Pring_add[of I P Q] Pring_car[of I]
             ring_hom_ring.homh[of "Pring R I" "Pring (Pring R J0) J1" \<psi>]
             ring_hom_add[of \<psi> "Pring R I" "Pring (Pring R J0) J1" P Q]
-      by metis     
+      by metis
     have 1: "closed_fun (Pring R J0) (indexed_const \<circ> g)"
-      using assms comp_apply   
-      by (smt Pi_I closed_fun_simp indexed_const_closed)            
+      using assms comp_apply
+      by (smt Pi_I closed_fun_simp indexed_const_closed)
     have 2: "\<psi> P \<in> carrier (Pring (Pring R J0) J1)"
-      using assms A var_factor_morphism'(1)[of I J0 J1 \<psi>] 
-            ring_hom_ring.homh ring_hom_closed Pring_car  
-      by metis   
+      using assms A var_factor_morphism'(1)[of I J0 J1 \<psi>]
+            ring_hom_ring.homh ring_hom_closed Pring_car
+      by metis
     have 3: "\<psi> Q \<in> carrier (Pring (Pring R J0) J1)"
-      using assms B var_factor_morphism'(1)[of I J0 J1 \<psi>] 
-          ring_hom_ring.homh ring_hom_closed Pring_car 
-      by metis     
+      using assms B var_factor_morphism'(1)[of I J0 J1 \<psi>]
+          ring_hom_ring.homh ring_hom_closed Pring_car
+      by metis
     have 4: "(total_eval (Pring R J0) (indexed_const \<circ> g) (\<psi> (P \<Oplus> Q))) =
             (total_eval (Pring R J0) (indexed_const \<circ> g) (\<psi> P)) \<oplus>\<^bsub>Pring R J0\<^esub>
             (total_eval (Pring R J0) (indexed_const \<circ> g) (\<psi> Q))"
       using 0 1 2 3 A B assms cring.total_eval_add[of "Pring R J0" "\<psi> P" J1 "\<psi> Q" "indexed_const \<circ> g"]
       by (metis Pring_car Pring_is_cring is_cring)
     have 5: "(total_eval (Pring R J0) (indexed_const \<circ> g) (\<psi> P)) \<in> carrier (Pring R J0)"
-      using 3 assms cring.total_eval_closed "1" "2" Pring_is_cring is_cring 
+      using 3 assms cring.total_eval_closed "1" "2" Pring_is_cring is_cring
       by blast
     have 6: "(total_eval (Pring R J0) (indexed_const \<circ> g) (\<psi> Q)) \<in> carrier (Pring R J0)"
-      using 4 assms  "1" "3" Pring_is_cring cring.total_eval_closed is_cring 
+      using 4 assms  "1" "3" Pring_is_cring cring.total_eval_closed is_cring
       by blast
     show "total_eval R g (P \<Oplus> Q) = total_eval R g (total_eval (Pring R J0) (indexed_const \<circ> g) (\<psi> (P \<Oplus> Q)))"
-      using 5 6 4 assms 
+      using 5 6 4 assms
             total_eval_add[of  "(total_eval (Pring R J0) (indexed_const \<circ> g) (\<psi> p))" J0
                               "(total_eval (Pring R J0) (indexed_const \<circ> g) (\<psi> Q))" ]
-      by (smt A(1) A(2) B(1) B(2) Pring_add Pring_car in_mono indexed_pset_mono order_refl 
+      by (smt A(1) A(2) B(1) B(2) Pring_add Pring_car in_mono indexed_pset_mono order_refl
           subsetD subset_iff total_eval_add)
   qed
   show "\<And>P i. P \<in> Pring_set R I \<Longrightarrow>
            total_eval R g P = total_eval R g (total_eval (Pring R J0) (indexed_const \<circ> g) (\<psi> P)) \<Longrightarrow>
            i \<in> I \<Longrightarrow> total_eval R g (P \<Otimes> i) = total_eval R g (total_eval (Pring R J0) (indexed_const \<circ> g) (\<psi> (P \<Otimes> i)))"
-  proof- fix P i 
+  proof- fix P i
     assume A: "P \<in> Pring_set R I"
            "total_eval R g P = total_eval R g (total_eval (Pring R J0) (indexed_const \<circ> g) (\<psi> P))"
-           "i \<in> I" 
+           "i \<in> I"
     have 0: "(P \<Otimes> i) = P \<otimes>\<^bsub>Pring R I\<^esub> (pvar R i)"
-      using A poly_index_mult  
+      using A poly_index_mult
       by (metis Pring_mult var_to_IP_def)
     have 1: "\<psi> (P \<Otimes> i) = \<psi> P \<otimes>\<^bsub>Pring (Pring R J0) J1\<^esub> \<psi> (pvar R i)"
       using 0 A assms var_factor_morphism'(1)[of I J0 J1 \<psi>]
-            pvar_closed[of R i] ring_hom_mult ring_hom_ring.homh 
+            pvar_closed[of R i] ring_hom_mult ring_hom_ring.homh
       by (smt Pring_car Pring_var_closed)
     have 2: "\<psi> P \<in> carrier (Pring (Pring R J0) J1)"
-      using assms A var_factor_morphism'(1)[of I J0 J1 \<psi>] 
-            ring_hom_ring.homh ring_hom_closed Pring_car  
-      by metis   
+      using assms A var_factor_morphism'(1)[of I J0 J1 \<psi>]
+            ring_hom_ring.homh ring_hom_closed Pring_car
+      by metis
     have 3: "\<psi> (pvar R i) \<in> carrier (Pring (Pring R J0) J1)"
       using assms A var_factor_morphism'(1)[of I J0 J1 \<psi>]
-            ring_hom_ring.homh ring_hom_closed Pring_car pvar_closed[of R i I]  
+            ring_hom_ring.homh ring_hom_closed Pring_car pvar_closed[of R i I]
       by (metis is_cring)
     have 4: "closed_fun (Pring R J0) (indexed_const \<circ> g)"
       apply(rule cring.closed_funI)
       using Pring_is_cring is_cring apply blast
             using assms indexed_const_closed closed_fun_simp[of g] comp_apply[of indexed_const g]
           proof -
             fix x :: 'c
             show "(indexed_const \<circ> g) x \<in> carrier (Pring R J0)"
               by (metis (no_types) \<open>\<And>n. closed_fun R g \<Longrightarrow> g n \<in> carrier R\<close> \<open>\<And>x. (indexed_const \<circ> g) x = indexed_const (g x)\<close> \<open>closed_fun R g\<close> indexed_const_closed)
           qed
-    have 5: "(total_eval (Pring R J0) (indexed_const \<circ> g) (\<psi> (P \<Otimes> i))) = 
-              (total_eval (Pring R J0) (indexed_const \<circ> g) (\<psi> P)) \<otimes>\<^bsub>Pring R J0\<^esub> 
+    have 5: "(total_eval (Pring R J0) (indexed_const \<circ> g) (\<psi> (P \<Otimes> i))) =
+              (total_eval (Pring R J0) (indexed_const \<circ> g) (\<psi> P)) \<otimes>\<^bsub>Pring R J0\<^esub>
               (total_eval (Pring R J0) (indexed_const \<circ> g) (\<psi> (pvar R i)))"
-      using 2 3 4 cring.total_eval_ring_hom[of "Pring R J0" "indexed_const \<circ> g" J1]  
+      using 2 3 4 cring.total_eval_ring_hom[of "Pring R J0" "indexed_const \<circ> g" J1]
       by (metis "1" Pring_is_cring cring.total_eval_mult is_cring)
     have 6: "(total_eval (Pring R J0) (indexed_const \<circ> g) (\<psi> P)) \<in> carrier (Pring R J0)"
       using total_eval_closed "2" "4" Pring_is_cring cring.total_eval_closed is_cring
       by blast
     have 7: " (total_eval (Pring R J0) (indexed_const \<circ> g) (\<psi> (pvar R i))) \<in> carrier (Pring R J0)"
-      using "3" "4" Pring_is_cring cring.total_eval_closed is_cring 
+      using "3" "4" Pring_is_cring cring.total_eval_closed is_cring
       by blast
-    have 8: " total_eval R g (total_eval (Pring R J0) (indexed_const \<circ> g) (\<psi> (P \<Otimes> i))) = 
+    have 8: " total_eval R g (total_eval (Pring R J0) (indexed_const \<circ> g) (\<psi> (P \<Otimes> i))) =
               total_eval R g (total_eval (Pring R J0) (indexed_const \<circ> g) (\<psi> P)) \<otimes>\<^bsub>R\<^esub>
               total_eval R g (total_eval (Pring R J0) (indexed_const \<circ> g) (\<psi> (pvar R i)))"
-      using 6 7 
+      using 6 7
       by (metis "5" assms(5) cring.total_eval_mult is_cring)
-    have 9: " total_eval R g (total_eval (Pring R J0) (indexed_const \<circ> g) (\<psi> (P \<Otimes> i))) = 
+    have 9: " total_eval R g (total_eval (Pring R J0) (indexed_const \<circ> g) (\<psi> (P \<Otimes> i))) =
               total_eval R g P \<otimes>\<^bsub>R\<^esub>
               total_eval R g (total_eval (Pring R J0) (indexed_const \<circ> g) (\<psi> (pvar R i)))"
       using "8" A(2)
       by presburger
-    have 10: "total_eval R g (total_eval (Pring R J0) (indexed_const \<circ> g) (\<psi> (pvar R i))) = 
+    have 10: "total_eval R g (total_eval (Pring R J0) (indexed_const \<circ> g) (\<psi> (pvar R i))) =
               g i"
     proof(cases "i \<in> J0")
       case True
       then have "\<psi> (pvar R i) = ring.indexed_const (Pring R J0) (pvar R i)"
-        using assms  
+        using assms
         by (metis Pring_var_closed inf_sup_absorb var_factor_morphism'(3))
       then have "(total_eval (Pring R J0) (indexed_const \<circ> g) (\<psi> (pvar R i))) =  (pvar R i)"
         by (metis Pring_is_cring Pring_var_closed True cring.total_eval_const is_cring)
-      then show ?thesis 
-        using total_eval_var[of g i] assms var_to_IP_def  
-        by metis        
+      then show ?thesis
+        using total_eval_var[of g i] assms var_to_IP_def
+        by metis
     next
       case False
       then have "\<psi> (pvar R i) = (pvar  (Pring R J0)  i)"
         by (metis A(3) UnE assms(1) assms(2) assms(3) assms(4) cring.var_factor_morphism'(2) is_cring)
-      then have "total_eval (Pring R J0) (indexed_const \<circ> g) (\<psi> (pvar R i)) = 
+      then have "total_eval (Pring R J0) (indexed_const \<circ> g) (\<psi> (pvar R i)) =
                   indexed_const (g i)"
         using cring.total_eval_var[of "Pring R J0" "indexed_const \<circ> g" i] comp_apply
         by (metis "4" Pring_is_cring is_cring var_to_IP_def)
-      then show ?thesis 
-        using total_eval_const[of "g i" g] assms closed_fun_simp[of g i] 
-        by metis 
+      then show ?thesis
+        using total_eval_const[of "g i" g] assms closed_fun_simp[of g i]
+        by metis
     qed
     show "total_eval R g (P \<Otimes> i) = total_eval R g (total_eval (Pring R J0) (indexed_const \<circ> g) (\<psi> (P \<Otimes> i)))"
-      using 9 10 
+      using 9 10
       by (metis A(1) A(3) Pring_car assms(5) total_eval_indexed_pmult)
   qed
 qed
 
-lemma var_factor_closed: 
+lemma var_factor_closed:
   assumes "I = J0 \<union> J1"
   assumes "J1 \<subseteq> I"
   assumes "J1 \<inter> J0 = {}"
   assumes "P \<in> carrier (Pring R I)"
   shows "var_factor I J0 J1 P \<in> carrier (Pring (Pring R J0 ) J1)"
   using assms var_factor_morphism'[of I J0 J1] ring_hom_ring.homh
   by (metis ring_hom_closed)
 
-lemma var_factor_add: 
+lemma var_factor_add:
   assumes "I = J0 \<union> J1"
   assumes "J1 \<subseteq> I"
   assumes "J1 \<inter> J0 = {}"
   assumes "P \<in> carrier (Pring R I)"
   assumes "Q \<in> carrier (Pring R I)"
   shows "var_factor I J0 J1 (P \<oplus>\<^bsub>Pring R I\<^esub> Q) = var_factor I J0 J1 P \<oplus>\<^bsub>Pring (Pring R J0) J1\<^esub>
                                                  var_factor I J0 J1 Q"
   using assms var_factor_morphism'[of I J0 J1] ring_hom_ring.homh
   by (metis (no_types, lifting) ring_hom_add)
-  
-lemma var_factor_mult: 
+
+lemma var_factor_mult:
   assumes "I = J0 \<union> J1"
   assumes "J1 \<subseteq> I"
   assumes "J1 \<inter> J0 = {}"
   assumes "P \<in> carrier (Pring R I)"
   assumes "Q \<in> carrier (Pring R I)"
   shows "var_factor I J0 J1 (P \<otimes>\<^bsub>Pring R I\<^esub> Q) = var_factor I J0 J1 P \<otimes>\<^bsub>Pring (Pring R J0) J1\<^esub>
                                                  var_factor I J0 J1 Q"
   using assms var_factor_morphism'[of I J0 J1] ring_hom_ring.homh
   by (metis (no_types, lifting) ring_hom_mult)
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsubsection\<open>Viewing a Mulitvariable Polynomial as a Univariate Polynomial over a Multivariate Polynomial Base Ring\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
-definition multivar_poly_to_univ_poly :: 
+definition multivar_poly_to_univ_poly ::
  "'c set \<Rightarrow> 'c \<Rightarrow> ('a,'c) mvar_poly \<Rightarrow>
        (('a,'c) mvar_poly) u_poly  " where
 "multivar_poly_to_univ_poly I i P = ((IP_to_UP i) \<circ> (var_factor I (I - {i}) {i})) P"
 
-definition univ_poly_to_multivar_poly :: 
-  "'c set \<Rightarrow> 'c \<Rightarrow> (('a,'c) mvar_poly) u_poly \<Rightarrow> 
+definition univ_poly_to_multivar_poly ::
+  "'c set \<Rightarrow> 'c \<Rightarrow> (('a,'c) mvar_poly) u_poly \<Rightarrow>
         ('a,'c) mvar_poly"  where
 "univ_poly_to_multivar_poly I i P =((var_factor_inv I (I - {i}) {i}) \<circ> (UP_to_IP (Pring R (I - {i})) i)) P"
 
 lemma multivar_poly_to_univ_poly_is_hom:
   assumes "i \<in> I"
   shows "multivar_poly_to_univ_poly I i \<in> ring_hom (Pring R I) (UP (Pring R (I - {i})))"
   unfolding multivar_poly_to_univ_poly_def comp_apply
   apply(rule ring_hom_compose[of _ "Pring (Pring R (I - {i})) {i}" _ "var_factor I (I - {i}) {i}" "IP_to_UP i"])
        apply (simp add: Pring_is_ring; fail)
       apply (simp add: local.ring_axioms ring.Pring_is_ring; fail)
-     apply(rule UP_ring.UP_ring) unfolding UP_ring_def 
+     apply(rule UP_ring.UP_ring) unfolding UP_ring_def
      apply (simp add: Pring_is_ring; fail)
-    using assms var_factor_morphism'[of I "I - {i}" "{i}"] unfolding ring_hom_ring_def ring_hom_ring_axioms_def 
+    using assms var_factor_morphism'[of I "I - {i}" "{i}"] unfolding ring_hom_ring_def ring_hom_ring_axioms_def
     apply blast
-    using UP_cring.IP_to_UP_ring_hom[of "Pring R (I - {i})" i] 
-    unfolding ring_hom_ring_def ring_hom_ring_axioms_def UP_cring_def 
+    using UP_cring.IP_to_UP_ring_hom[of "Pring R (I - {i})" i]
+    unfolding ring_hom_ring_def ring_hom_ring_axioms_def UP_cring_def
     using Pring_is_cring is_cring apply blast
-    by blast 
+    by blast
 
 lemma multivar_poly_to_univ_poly_inverse:
   assumes "i \<in> I"
   assumes "\<psi>0 = multivar_poly_to_univ_poly I i"
   assumes "\<psi>1 = univ_poly_to_multivar_poly I i"
   assumes "P \<in> carrier (Pring R I)"
   shows "\<psi>1 (\<psi>0 P) = P"
 proof-
   have closed: "var_factor I (I - {i}) {i} P \<in> carrier (Pring (Pring R (I - {i})) {i})"
-    using assms var_factor_closed[of I "I - {i}" "{i}" P] 
+    using assms var_factor_closed[of I "I - {i}" "{i}" P]
     by blast
-  have cancel_1: "UP_to_IP (Pring R (I - {i})) i 
-                  (IP_to_UP i (var_factor I (I - {i}) {i} P)) = 
+  have cancel_1: "UP_to_IP (Pring R (I - {i})) i
+                  (IP_to_UP i (var_factor I (I - {i}) {i} P)) =
                         var_factor I (I - {i}) {i} P"
     using closed assms  ring.Pring_car
-          UP_cring.UP_to_IP_inv[of "Pring R (I - {i})" "var_factor I (I - {i}) {i} P" i] 
-          Pring_is_ring unfolding UP_cring_def 
-    using is_cring local.ring_axioms ring.Pring_is_cring 
-    by blast  
-  have "univ_poly_to_multivar_poly I i (multivar_poly_to_univ_poly I i P) = 
+          UP_cring.UP_to_IP_inv[of "Pring R (I - {i})" "var_factor I (I - {i}) {i} P" i]
+          Pring_is_ring unfolding UP_cring_def
+    using is_cring local.ring_axioms ring.Pring_is_cring
+    by blast
+  have "univ_poly_to_multivar_poly I i (multivar_poly_to_univ_poly I i P) =
         univ_poly_to_multivar_poly I i ((IP_to_UP i) (var_factor I (I - {i}) {i} P))"
     by (metis comp_apply multivar_poly_to_univ_poly_def)
-  then have "univ_poly_to_multivar_poly I i (multivar_poly_to_univ_poly I i P) = 
+  then have "univ_poly_to_multivar_poly I i (multivar_poly_to_univ_poly I i P) =
         ((var_factor_inv I (I - {i}) {i})  ((UP_to_IP (Pring R (I - {i})) i)
                  ((IP_to_UP i) (var_factor I (I - {i}) {i} P))))"
-    using comp_apply univ_poly_to_multivar_poly_def 
+    using comp_apply univ_poly_to_multivar_poly_def
     by metis
-  then have "univ_poly_to_multivar_poly I i (multivar_poly_to_univ_poly I i P) = 
+  then have "univ_poly_to_multivar_poly I i (multivar_poly_to_univ_poly I i P) =
         ((var_factor_inv I (I - {i}) {i})  (var_factor I (I - {i}) {i} P))"
-    using cancel_1 
+    using cancel_1
     by presburger
-  then show ?thesis using assms var_factor_inv_inverse[of I "I - {i}" "{i}" _ _ P] 
-    by (metis Diff_cancel Diff_disjoint Diff_eq_empty_iff Diff_partition Un_Diff_cancel 
+  then show ?thesis using assms var_factor_inv_inverse[of I "I - {i}" "{i}" _ _ P]
+    by (metis Diff_cancel Diff_disjoint Diff_eq_empty_iff Diff_partition Un_Diff_cancel
         Un_Diff_cancel2 Un_insert_right empty_Diff empty_subsetI insert_Diff_if insert_absorb )
 qed
 
 lemma multivar_poly_to_univ_poly_total_eval:
   assumes "i \<in> I"
   assumes "\<psi> = multivar_poly_to_univ_poly I i"
   assumes "P \<in> carrier (Pring R I)"
   assumes "closed_fun R g"
   shows "total_eval R g P = total_eval R g (to_function (Pring R (I - {i})) (\<psi> P) (indexed_const (g i)))"
 proof-
   have 0: "var_factor I (I - {i}) {i} P \<in> Pring_set (Pring R (I - {i})) {i}"
   proof-
     have 00: " ring (Pring R (I - {i}))"
-      using Pring_is_ring 
+      using Pring_is_ring
       by blast
-    then show ?thesis 
+    then show ?thesis
       using assms(1) assms(3) var_factor_closed[of I "I - {i}" "{i}" P] ring.Pring_car[of "Pring R (I - {i})" "{i}"]
       by blast
   qed
   have 1: "closed_fun (Pring R (I - {i})) (indexed_const \<circ> g)"
     using assms comp_apply  indexed_const_closed cring.closed_funI[of "Pring R (I - {i})"]
     by (metis Pring_is_cring closed_fun_simp is_cring)
   have 2: "cring (Pring R (I - {i}))"
     using Pring_is_cring is_cring by blast
-  have "(to_function (Pring R (I - {i})) (\<psi> P) (indexed_const (g i))) 
+  have "(to_function (Pring R (I - {i})) (\<psi> P) (indexed_const (g i)))
         =  total_eval (Pring R (I - {i})) (indexed_const \<circ> g) ((var_factor I (I - {i}) {i}) P) "
-    using assms 0 1 2 comp_apply UP_cring.IP_to_UP_poly_eval[of "Pring R (I - {i})" 
+    using assms 0 1 2 comp_apply UP_cring.IP_to_UP_poly_eval[of "Pring R (I - {i})"
                                              "(var_factor I (I - {i}) {i}) P" i "indexed_const \<circ> g"]
-    unfolding UP_cring_def multivar_poly_to_univ_poly_def 
-    by metis 
+    unfolding UP_cring_def multivar_poly_to_univ_poly_def
+    by metis
   then have 3: "total_eval R g ((to_function (Pring R (I - {i})) (\<psi> P) (indexed_const (g i))) )
         = total_eval R g ( total_eval (Pring R (I - {i})) (indexed_const \<circ> g) ((var_factor I (I - {i}) {i}) P)) "
     by presburger
   then have "total_eval R g P = total_eval R g (total_eval (Pring R (I - {i})) (indexed_const \<circ> g) ((var_factor I (I - {i}) {i}) P)) "
-    using assms var_factor_total_eval[of I "I - {i}" "{i}" "var_factor I (I - {i}) {i}" g P] 
+    using assms var_factor_total_eval[of I "I - {i}" "{i}" "var_factor I (I - {i}) {i}" g P]
     by blast
   then show ?thesis
-    using 3 
+    using 3
     by presburger
 qed
 
 text\<open>
   Induction for polynomial rings. Basically just \texttt{indexed\_pset.induct} with some
   boilerplate translations removed
 \<close>
 
 lemma(in ring) Pring_car_induct'':
   assumes "Q \<in> carrier (Pring R I)"
   assumes "\<And>c. c \<in> carrier R \<Longrightarrow> P (indexed_const c)"
   assumes "\<And>p q. p \<in> carrier (Pring R I) \<Longrightarrow> q \<in> carrier (Pring R I) \<Longrightarrow> P p \<Longrightarrow> P q \<Longrightarrow> P (p \<oplus>\<^bsub>Pring R I\<^esub> q)"
   assumes "\<And>p i. p \<in> carrier (Pring R I) \<Longrightarrow> i \<in> I \<Longrightarrow> P p \<Longrightarrow> P (p \<otimes>\<^bsub>Pring R I\<^esub> pvar R i)"
   shows "P Q"
   apply(rule indexed_pset.induct[of Q I "carrier R"])
   using Pring_car assms(1) apply blast
   using assms(2) apply blast
   apply (metis (full_types) Pring_add Pring_car assms(3))
 proof-
   fix Pa i assume A: "Pa \<in> Pring_set R I" "P Pa" "i \<in> I"
   then have 0: "Pa \<in> carrier (Pring R I)"
     using  assms A Pring_car
     by blast
   have "Pa \<Otimes> i = Pa \<otimes>\<^bsub>Pring R I\<^esub> pvar R i"
-    using 0 poly_index_mult assms A 
+    using 0 poly_index_mult assms A
     by (metis Pring_mult var_to_IP_def)
   then show "P (Pa \<Otimes> i)"
     by (simp add: "0" A(2) A(3) assms)
 qed
 
 subsubsection\<open>Application: A Polynomial Ring over a Domain is a Domain\<close>
 
 text \<open>
   In this section, we use the fact the UP \<open>R\<close> is a domain when \<open>R\<close> is a domain to show the analogous
-  result for indexed polynomial rings. We first prove it inductively for rings with a finite 
-  variable set, and then generalize to infinite variable sets using the fact that any two 
-  multivariable polynomials over an indexed polynomial ring must also lie in a finitely indexed 
+  result for indexed polynomial rings. We first prove it inductively for rings with a finite
+  variable set, and then generalize to infinite variable sets using the fact that any two
+  multivariable polynomials over an indexed polynomial ring must also lie in a finitely indexed
   polynomial subring.
 \<close>
 
 lemma indexed_const_mult:
   assumes "a \<in> carrier R"
   assumes "b \<in> carrier R"
   shows "indexed_const a \<otimes>\<^bsub> Pring R I\<^esub> indexed_const b  = indexed_const (a \<otimes>  b)"
-  by (metis Pring_mult assms(1) assms(2) indexed_const_P_mult_eq)  
+  by (metis Pring_mult assms(1) assms(2) indexed_const_P_mult_eq)
 
 lemma(in domain) Pring_fin_vars_is_domain:
   assumes "finite (I ::'c set)"
   shows "domain (Pring R I)"
 proof(rule finite_induct, rule assms)
   show "domain (Pring R ({}::'c set))"
   proof(rule domainI)
     show " cring (Pring R {})"
       by (simp add: Pring_is_cring is_cring)
     show "\<one>\<^bsub>Pring R {}\<^esub> \<noteq> \<zero>\<^bsub>Pring R {}\<^esub>"
     proof-
       have "\<one>\<^bsub>Pring R {}\<^esub> {#} \<noteq> \<zero>\<^bsub>Pring R {}\<^esub> {#}"
         by (metis Pring_one Pring_zero local.ring_axioms ring.indexed_const_def zero_not_one)
-      thus ?thesis 
+      thus ?thesis
         by metis
-    qed 
+    qed
     show "\<And>a b. a \<otimes>\<^bsub>Pring R ({}::'c set)\<^esub> b = \<zero>\<^bsub>Pring R {}\<^esub> \<Longrightarrow>
            a \<in> carrier (Pring R {}) \<Longrightarrow> b \<in> carrier (Pring R {}) \<Longrightarrow> a = \<zero>\<^bsub>Pring R {}\<^esub> \<or> b = \<zero>\<^bsub>Pring R {}\<^esub>"
-    proof- 
+    proof-
       fix a b assume A: "a \<otimes>\<^bsub>Pring R ({}::'c set)\<^esub> b = \<zero>\<^bsub>Pring R {}\<^esub>"
            "a \<in> carrier (Pring R {})" " b \<in> carrier (Pring R {})"
       have 0: "monomials_of R a \<subseteq> {{#}}"
         using A Pring_set_zero unfolding monomials_of_def Pring_car
         by blast
       have 1: "monomials_of R b \<subseteq> {{#}}"
         using A Pring_set_zero unfolding monomials_of_def Pring_car
         by blast
       obtain A where A_def: "A = a {#}"
-        by blast 
+        by blast
       obtain B where B_def: "B = b {#}"
-        by blast 
+        by blast
       have 2: "a = indexed_const A"
         unfolding A_def
         apply(rule ext)
         by (metis 0 complement_of_monomials_of empty_iff in_mono indexed_const_def insert_iff)
       have 3: "b = indexed_const B"
-        unfolding B_def 
+        unfolding B_def
         apply(rule ext)
         by (metis 1 complement_of_monomials_of empty_iff in_mono indexed_const_def insert_iff)
       have 4: "a \<otimes>\<^bsub>Pring R {}\<^esub> b = indexed_const (A \<otimes> B)"
-        using A unfolding 2 3 
+        using A unfolding 2 3
         by (metis "2" "3" A_def B_def Pring_carrier_coeff' indexed_const_mult)
       have 5: "A \<otimes> B = \<zero>"
         using A unfolding 4 by (metis Pring_zero indexed_const_def)
       have 6: "A = \<zero> \<or> B = \<zero>"
-        using 5 A_def B_def A 
+        using 5 A_def B_def A
         by (simp add: domain.integral_iff domain_axioms)
       show "a = \<zero>\<^bsub>Pring R {}\<^esub> \<or> b = \<zero>\<^bsub>Pring R {}\<^esub>"
-        unfolding 2 3 using 6 
+        unfolding 2 3 using 6
         by (metis Pring_zero)
     qed
   qed
   show "\<And>x F. finite (F:: 'c set) \<Longrightarrow> x \<notin> F \<Longrightarrow> domain (Pring R F) \<Longrightarrow> domain (Pring R (insert x F))"
   proof- fix S:: "'c set"  fix s assume A: "finite S" "s \<notin> S" "domain (Pring R S)"
     show "domain (Pring R (insert s S))"
     proof-
       have ring_hom: "multivar_poly_to_univ_poly (insert s S) s \<in> ring_hom (Pring R (insert s S)) (UP (Pring R S))"
-        using multivar_poly_to_univ_poly_is_hom 
-        by (metis A(2) Diff_insert_absorb insertI1) 
+        using multivar_poly_to_univ_poly_is_hom
+        by (metis A(2) Diff_insert_absorb insertI1)
       have domain: "domain (UP (Pring R S))"
         apply(rule UP_domain.UP_domain)
         unfolding UP_domain_def by(rule A)
       show "domain (Pring R (insert s S))"
         apply(rule ring_hom_ring.inj_on_domain[of _ "UP (Pring R S)" "multivar_poly_to_univ_poly (insert s S) s"])
           apply(rule ring_hom_ring.intro)
             apply (simp add: Pring_is_ring; fail)
            apply(rule UP_ring.UP_ring)
-        unfolding UP_ring_def 
+        unfolding UP_ring_def
         apply (simp add: Pring_is_ring; fail)
         unfolding ring_hom_ring_axioms_def apply(rule ring_hom)
       proof(rule inj_onI)
         fix x y assume A: "x \<in> carrier (Pring R (insert s S))"
            "y \<in> carrier (Pring R (insert s S))"
            "multivar_poly_to_univ_poly (insert s S) s x = multivar_poly_to_univ_poly (insert s S) s y"
         then have 0: "univ_poly_to_multivar_poly (insert s S) s (multivar_poly_to_univ_poly (insert s S) s x)
        = univ_poly_to_multivar_poly (insert s S) s (multivar_poly_to_univ_poly (insert s S) s y)"
-          by auto 
+          by auto
         have 1: "univ_poly_to_multivar_poly (insert s S) s (multivar_poly_to_univ_poly (insert s S) s x) = x"
-          using A by (meson cring.multivar_poly_to_univ_poly_inverse insertI1 is_cring) 
+          using A by (meson cring.multivar_poly_to_univ_poly_inverse insertI1 is_cring)
         have 2: "univ_poly_to_multivar_poly (insert s S) s (multivar_poly_to_univ_poly (insert s S) s y) = y"
-          using A  by (meson insertI1 multivar_poly_to_univ_poly_inverse) 
+          using A  by (meson insertI1 multivar_poly_to_univ_poly_inverse)
         show "x = y"
-          using 0 unfolding 1 2 by auto 
+          using 0 unfolding 1 2 by auto
       next
         show "domain (UP (Pring R S))"
           apply(rule UP_domain.UP_domain)
           unfolding UP_domain_def by(rule A)
       qed
     qed
   qed
 qed
 
 lemma locally_finite:
   assumes "a \<in> carrier (Pring R I)"
   shows "\<exists>J. J \<subseteq> I \<and> finite J \<and> a \<in> carrier (Pring R J)"
 proof(rule Pring_car_induct[of _ I], rule assms)
   have 0: "\<zero>\<^bsub>Pring R I\<^esub> = \<zero>\<^bsub>Pring R {}\<^esub>"
-    by (simp add: Pring_zero_eq) 
+    by (simp add: Pring_zero_eq)
   show "\<exists>J\<subseteq>I. finite J \<and> \<zero>\<^bsub>Pring R I\<^esub> \<in> carrier (Pring R J)"
-    unfolding 0 
-    by (meson Pring_zero_closed empty_subsetI finite.emptyI) 
+    unfolding 0
+    by (meson Pring_zero_closed empty_subsetI finite.emptyI)
 next
   fix m k assume A: "set_mset m \<subseteq> I \<and> k \<in> carrier R"
   obtain J where J_def: "J = set_mset m"
     by blast
   have 0: "k \<odot>\<^bsub>Pring R I\<^esub> mset_to_IP R m = k \<odot>\<^bsub>Pring R J\<^esub> mset_to_IP R m"
-    unfolding J_def by (simp add: Pring_smult) 
+    unfolding J_def by (simp add: Pring_smult)
   have 1: "k \<odot>\<^bsub>Pring R I\<^esub> mset_to_IP R m \<in> carrier (Pring R J)"
-    unfolding 0 J_def using A 
-    by (simp add: Pring_car Pring_smult mset_to_IP_closed poly_scalar_mult_closed) 
+    unfolding 0 J_def using A
+    by (simp add: Pring_car Pring_smult mset_to_IP_closed poly_scalar_mult_closed)
   show "\<exists>J\<subseteq>I. finite J \<and> k \<odot>\<^bsub>Pring R I\<^esub> mset_to_IP R m \<in> carrier (Pring R J)"
-    using J_def 1 A by blast 
+    using J_def 1 A by blast
 next
-  fix Q m k 
+  fix Q m k
   assume A: "Q \<in> carrier (Pring R I)\<and>(\<exists>J\<subseteq>I. finite J \<and> Q \<in> carrier (Pring R J)) \<and> set_mset m \<subseteq> I \<and> k \<in> carrier R"
   then obtain J where J_def: "J\<subseteq>I \<and> finite J \<and> Q \<in> carrier (Pring R J)"
-    by blast 
+    by blast
   obtain J' where J'_def: "J' = J \<union> set_mset m"
-    by blast 
+    by blast
   have 0: "finite J'"
-    unfolding J'_def using J_def by blast 
+    unfolding J'_def using J_def by blast
   have 1: "k \<odot>\<^bsub>Pring R I\<^esub> mset_to_IP R m = k \<odot>\<^bsub>Pring R J'\<^esub> mset_to_IP R m"
     unfolding J'_def using A by (simp add: Pring_smult)
   have 2: "Q \<in> carrier (Pring R J')"
-    using J_def unfolding J'_def 
-    by (metis Pring_car Pring_carrier_subset Un_upper1 subsetD) 
+    using J_def unfolding J'_def
+    by (metis Pring_car Pring_carrier_subset Un_upper1 subsetD)
   have 3: "Q \<Oplus> k \<odot>\<^bsub>Pring R I\<^esub> mset_to_IP R m \<in> carrier (Pring R J')"
-    using 0 1 2 J'_def A 
-    by (metis Pring_car Pring_smult_closed indexed_pset.indexed_padd mset_to_IP_closed sup.cobounded2) 
+    using 0 1 2 J'_def A
+    by (metis Pring_car Pring_smult_closed indexed_pset.indexed_padd mset_to_IP_closed sup.cobounded2)
   show "\<exists>J'\<subseteq>I. finite J' \<and> Q \<Oplus> k \<odot>\<^bsub>Pring R I\<^esub> mset_to_IP R m \<in> carrier (Pring R J')"
-    using 3 0 
-    by (metis A J'_def J_def Un_subset_iff) 
+    using 3 0
+    by (metis A J'_def J_def Un_subset_iff)
 qed
 
 lemma(in domain) Pring_is_domain:
   "domain (Pring R I)"
 proof(rule domainI, simp add: Pring_is_cring is_cring)
   have 0: "\<one>\<^bsub>Pring R I\<^esub> {#} = \<one>"
     by (simp add: Pring_one indexed_const_def)
   have 1: "\<zero>\<^bsub>Pring R I\<^esub> {#} = \<zero>"
-    by (simp add: Pring_zero indexed_const_def) 
+    by (simp add: Pring_zero indexed_const_def)
   have 2: "\<one> \<noteq> \<zero>"
     by simp
   show "\<one>\<^bsub>Pring R I\<^esub> \<noteq> \<zero>\<^bsub>Pring R I\<^esub>"
-    using 0 1 2 by auto 
+    using 0 1 2 by auto
 next
   fix a b assume A: "a \<otimes>\<^bsub>Pring R I\<^esub> b = \<zero>\<^bsub>Pring R I\<^esub>"
                     "a \<in> carrier (Pring R I)"
                     "b \<in> carrier (Pring R I)"
   obtain J0 where J0_def: "J0 \<subseteq> I \<and> finite J0 \<and> a \<in> carrier (Pring R J0)"
-    using A locally_finite by blast 
+    using A locally_finite by blast
   obtain J1 where J1_def: "J1 \<subseteq> I \<and> finite J1 \<and> b \<in> carrier (Pring R J1)"
-    using A locally_finite by blast 
+    using A locally_finite by blast
   obtain J where J_def: "J = J0 \<union> J1"
-    by blast 
+    by blast
   have J_finite: "finite J"
-    using J_def J0_def J1_def by blast 
+    using J_def J0_def J1_def by blast
   have 0: "a \<in> carrier (Pring R J)"
-    using J0_def unfolding J_def 
-    by (metis (no_types, lifting) Pring_car Pring_carrier_subset Un_upper1 subset_eq) 
+    using J0_def unfolding J_def
+    by (metis (no_types, lifting) Pring_car Pring_carrier_subset Un_upper1 subset_eq)
   have 1: "b \<in> carrier (Pring R J)"
-    using J1_def unfolding J_def 
-    by (metis Pring_car Pring_carrier_subset in_mono sup.cobounded2) 
+    using J1_def unfolding J_def
+    by (metis Pring_car Pring_carrier_subset in_mono sup.cobounded2)
   have 2: "a \<otimes>\<^bsub>Pring R J\<^esub> b = \<zero>\<^bsub>Pring R J\<^esub>"
     using A J_def 0 1 by (metis Pring_mult_eq Pring_zero_eq)
   have 3: "domain (Pring R J)"
-    using J_finite Pring_fin_vars_is_domain[of J] by blast 
+    using J_finite Pring_fin_vars_is_domain[of J] by blast
   have 4: "a = \<zero>\<^bsub>Pring R J\<^esub> \<or> b = \<zero>\<^bsub>Pring R J\<^esub>"
     using 3 2 0 1  by (simp add: domain.integral)
   thus "a = \<zero>\<^bsub>Pring R I\<^esub> \<or> b = \<zero>\<^bsub>Pring R I\<^esub>"
-    using Pring_zero_eq by blast 
+    using Pring_zero_eq by blast
 qed
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsubsection\<open>Relabelling of Variables for Indexed Polynomial Rings\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 definition relabel_vars :: "'d set \<Rightarrow> 'e set \<Rightarrow> ('d \<Rightarrow> 'e) \<Rightarrow>
          ('a, 'd) mvar_poly \<Rightarrow> ('a, 'e) mvar_poly" where
 "relabel_vars I J g = indexed_poly_induced_morphism I (Pring R J) indexed_const (\<lambda>i. pvar R (g i))"
 
-lemma relabel_vars_is_morphism: 
+lemma relabel_vars_is_morphism:
   assumes "g \<in> I \<rightarrow> J"
   shows "ring_hom_ring (Pring R I) (Pring R J) (relabel_vars I J g)"
         "\<And>i. i \<in> I \<Longrightarrow> relabel_vars I J g (pvar R i) = pvar R (g i)"
         "\<And>c. c \<in> carrier R \<Longrightarrow> relabel_vars I J g (indexed_const c) = indexed_const c"
   using Pring_universal_prop(1)[of "Pring R J" "\<lambda>i. pvar R (g i)" I indexed_const]
-    assms unfolding relabel_vars_def 
+    assms unfolding relabel_vars_def
     apply (meson Pi_iff Pring_is_cring Pring_var_closed indexed_const_ring_hom is_cring)
 proof-
   have 0: "cring (Pring R J)" " (\<lambda>i. mset_to_IP R {#g i#}) \<in> I \<rightarrow> carrier (Pring R J)"
     "ring_hom_ring R (Pring R J) indexed_const"
     using assms Pring_is_cring is_cring apply blast
     apply (smt Pi_iff Pring_var_closed assms var_to_IP_def)
     by (simp add: indexed_const_ring_hom)
-  show "\<And>i. i \<in> I \<Longrightarrow> indexed_poly_induced_morphism I (Pring R J) 
+  show "\<And>i. i \<in> I \<Longrightarrow> indexed_poly_induced_morphism I (Pring R J)
                         indexed_const (\<lambda>i. pvar R (g i)) (pvar R i) = pvar R (g i)"
     using 0 assms Pring_universal_prop(2)[of "Pring R J" "\<lambda>i. pvar R (g i)" I indexed_const]
-    unfolding relabel_vars_def var_to_IP_def 
+    unfolding relabel_vars_def var_to_IP_def
     by blast
-  show "\<And>c. c \<in> carrier R \<Longrightarrow> indexed_poly_induced_morphism I (Pring R J) 
+  show "\<And>c. c \<in> carrier R \<Longrightarrow> indexed_poly_induced_morphism I (Pring R J)
                           indexed_const (\<lambda>i. pvar R (g i)) (indexed_const c) = indexed_const c"
     using 0 assms Pring_universal_prop(3)[of "Pring R J" "\<lambda>i. pvar R (g i)" I indexed_const]
-    unfolding  var_to_IP_def 
+    unfolding  var_to_IP_def
     by blast
 qed
 
-lemma relabel_vars_add: 
+lemma relabel_vars_add:
   assumes "g \<in> I \<rightarrow> J"
   assumes "P \<in> carrier (Pring R I)"
   assumes "Q \<in> carrier (Pring R I)"
   shows "relabel_vars I J g (P \<oplus>\<^bsub>Pring R I\<^esub> Q) = relabel_vars I J g P \<oplus>\<^bsub>Pring R J\<^esub> relabel_vars I J g Q"
-  using assms relabel_vars_is_morphism(1)[of g I J] ring_hom_ring.homh ring_hom_add 
+  using assms relabel_vars_is_morphism(1)[of g I J] ring_hom_ring.homh ring_hom_add
   by (metis (no_types, lifting))
 
-lemma relabel_vars_mult: 
+lemma relabel_vars_mult:
   assumes "g \<in> I \<rightarrow> J"
   assumes "P \<in> carrier (Pring R I)"
   assumes "Q \<in> carrier (Pring R I)"
   shows "relabel_vars I J g (P \<otimes>\<^bsub>Pring R I\<^esub> Q) = relabel_vars I J g P \<otimes>\<^bsub>Pring R J\<^esub> relabel_vars I J g Q"
   using assms relabel_vars_is_morphism(1)[of g I J] ring_hom_ring.homh ring_hom_mult
   by (metis (no_types, lifting))
 
-lemma relabel_vars_closed: 
+lemma relabel_vars_closed:
   assumes "g \<in> I \<rightarrow> J"
   assumes "P \<in> carrier (Pring R I)"
   shows "relabel_vars I J g P \<in> carrier (Pring R J)"
-  using assms relabel_vars_is_morphism(1)[of g I J] ring_hom_ring.homh 
+  using assms relabel_vars_is_morphism(1)[of g I J] ring_hom_ring.homh
   by (metis ring_hom_closed)
 
-lemma relabel_vars_smult: 
+lemma relabel_vars_smult:
   assumes "g \<in> I \<rightarrow> J"
   assumes "P \<in> carrier (Pring R I)"
   assumes "a \<in> carrier R"
   shows "relabel_vars I J g (a \<odot>\<^bsub>Pring R I\<^esub>P) = a \<odot>\<^bsub>Pring R J\<^esub>relabel_vars I J g P"
 proof-
   have 0: "a \<odot>\<^bsub>Pring R I\<^esub>P = indexed_const a \<otimes>\<^bsub>Pring R I\<^esub> P"
     by (metis Pring_car Pring_mult Pring_smult assms(2) assms(3) poly_scalar_mult_eq)
   have 1: "a \<odot>\<^bsub>Pring R J\<^esub>relabel_vars I J g P = indexed_const a \<otimes>\<^bsub>Pring R J\<^esub> relabel_vars I J g P"
     by (metis Pring_car Pring_mult Pring_smult assms(1) assms(2) assms(3) poly_scalar_mult_eq relabel_vars_closed)
   show ?thesis using 0 1 assms relabel_vars_mult relabel_vars_is_morphism(3)[of g I J a]
     by (metis indexed_const_closed)
 qed
-  
-lemma relabel_vars_inverse: 
+
+lemma relabel_vars_inverse:
   assumes "g \<in> I \<rightarrow> J"
   assumes "h \<in> J \<rightarrow> I"
   assumes "\<And>i. i \<in> I \<Longrightarrow> h (g i) = i"
   assumes "P \<in> carrier (Pring R I)"
   shows  "relabel_vars J I h (relabel_vars I J g P) = P"
   apply(rule Pring_car_induct''[of _ I])
   using assms(4) apply blast
-  using assms 
+  using assms
   apply (metis relabel_vars_is_morphism(3))
 proof-
   show "\<And>p q. p \<in> carrier (Pring R I) \<Longrightarrow>
            q \<in> carrier (Pring R I) \<Longrightarrow>
            relabel_vars J I h (relabel_vars I J g p) = p \<Longrightarrow>
-           relabel_vars J I h (relabel_vars I J g q) = q \<Longrightarrow> 
+           relabel_vars J I h (relabel_vars I J g q) = q \<Longrightarrow>
            relabel_vars J I h (relabel_vars I J g (p \<oplus>\<^bsub>Pring R I\<^esub> q)) = p \<oplus>\<^bsub>Pring R I\<^esub> q"
   proof- fix p q assume A: " p \<in> carrier (Pring R I)" "q \<in> carrier (Pring R I)"
                 "relabel_vars J I h (relabel_vars I J g p) = p"
            "relabel_vars J I h (relabel_vars I J g q) = q"
     show "relabel_vars J I h (relabel_vars I J g (p \<oplus>\<^bsub>Pring R I\<^esub> q)) = p \<oplus>\<^bsub>Pring R I\<^esub> q"
     proof-
       have 0: "relabel_vars I J g (p \<oplus>\<^bsub>Pring R I\<^esub> q) = relabel_vars I J g p \<oplus>\<^bsub>Pring R J\<^esub> relabel_vars I J g q"
         using A(1) A(2) assms(1) relabel_vars_add by blast
       then show ?thesis
         using assms A relabel_vars_is_morphism(3)[of g I J] relabel_vars_is_morphism(3)[of h J I]
-              relabel_vars_closed[of g I J p] relabel_vars_closed[of g I J q] 
-              relabel_vars_add[of h J I "relabel_vars I J g p" "relabel_vars I J g q"] 
+              relabel_vars_closed[of g I J p] relabel_vars_closed[of g I J q]
+              relabel_vars_add[of h J I "relabel_vars I J g p" "relabel_vars I J g q"]
         by presburger
     qed
-  qed 
+  qed
   show "\<And>p i. p \<in> carrier (Pring R I) \<Longrightarrow>
            i \<in> I \<Longrightarrow>
-           relabel_vars J I h (relabel_vars I J g p) = p \<Longrightarrow> 
+           relabel_vars J I h (relabel_vars I J g p) = p \<Longrightarrow>
         relabel_vars J I h (relabel_vars I J g (p \<otimes>\<^bsub>Pring R I\<^esub> pvar R i)) = p \<otimes>\<^bsub>Pring R I\<^esub> pvar R i"
   proof- fix p i assume A: " p \<in> carrier (Pring R I)"
                          "relabel_vars J I h (relabel_vars I J g p) = p"
                          "i \<in> I"
     show " relabel_vars J I h (relabel_vars I J g (p \<otimes>\<^bsub>Pring R I\<^esub> pvar R i)) = p \<otimes>\<^bsub>Pring R I\<^esub> pvar R i"
-    proof- 
+    proof-
       have "relabel_vars J I h (relabel_vars I J g (pvar R i)) = pvar R i"
         using assms relabel_vars_is_morphism[of g I J] relabel_vars_is_morphism[of h J I]
         by (metis A(3) funcset_mem )
-      then show ?thesis 
+      then show ?thesis
         using assms  relabel_vars_is_morphism[of g I J] relabel_vars_is_morphism[of h J I]
-              relabel_vars_closed[of g I J p] relabel_vars_closed[of g I J "pvar R i"] 
-              relabel_vars_mult[of g I J "p" "pvar R i"] 
-              relabel_vars_mult[of h J I "relabel_vars I J g p" "relabel_vars I J g (pvar R i)"] 
+              relabel_vars_closed[of g I J p] relabel_vars_closed[of g I J "pvar R i"]
+              relabel_vars_mult[of g I J "p" "pvar R i"]
+              relabel_vars_mult[of h J I "relabel_vars I J g p" "relabel_vars I J g (pvar R i)"]
         by (metis A(1) A(2) A(3) Pring_var_closed)
     qed
   qed
 qed
 
 lemma relabel_vars_total_eval:
   assumes "g \<in> I \<rightarrow> J"
   assumes "P \<in> carrier (Pring R I)"
   assumes "closed_fun R f"
   shows "total_eval R (f \<circ> g) P = total_eval R f (relabel_vars I J g P)"
 proof(rule Pring_car_induct''[of P I])
   show "P \<in> carrier (Pring R I)"
     using assms(2) by blast
   show "\<And>c. c \<in> carrier R \<Longrightarrow> total_eval R (f \<circ> g) (indexed_const c) = total_eval R f (relabel_vars I J g (indexed_const c))"
     by (metis assms(1) relabel_vars_is_morphism(3) total_eval_const)
   show " \<And>p q. p \<in> carrier (Pring R I) \<Longrightarrow>
            q \<in> carrier (Pring R I) \<Longrightarrow>
            total_eval R (f \<circ> g) p = total_eval R f (relabel_vars I J g p) \<Longrightarrow>
            total_eval R (f \<circ> g) q = total_eval R f (relabel_vars I J g q) \<Longrightarrow>
            total_eval R (f \<circ> g) (p \<oplus>\<^bsub>Pring R I\<^esub> q) = total_eval R f (relabel_vars I J g (p \<oplus>\<^bsub>Pring R I\<^esub> q))"
 
   proof- fix p q assume A: "p \<in> carrier (Pring R I)" "q \<in> carrier (Pring R I)"
                            "total_eval R (f \<circ> g) p = total_eval R f (relabel_vars I J g p)"
                            "total_eval R (f \<circ> g) q = total_eval R f (relabel_vars I J g q)"
     have 0: "closed_fun R (f \<circ> g)"
-      apply(rule closed_funI) 
+      apply(rule closed_funI)
       using comp_apply[of f g] closed_fun_simp[of f] assms(3) by presburger
-    have 1: " (relabel_vars I J g (p \<oplus>\<^bsub>Pring R I\<^esub> q)) = 
+    have 1: " (relabel_vars I J g (p \<oplus>\<^bsub>Pring R I\<^esub> q)) =
                (relabel_vars I J g p) \<oplus>\<^bsub>Pring R J\<^esub> (relabel_vars I J g q)"
       using A(1) A(2) assms(1) relabel_vars_add by blast
-    have 2: "total_eval R f (relabel_vars I J g (p \<oplus>\<^bsub>Pring R I\<^esub> q)) = 
+    have 2: "total_eval R f (relabel_vars I J g (p \<oplus>\<^bsub>Pring R I\<^esub> q)) =
       (total_eval R f (relabel_vars I J g p)) \<oplus>\<^bsub>R\<^esub> (total_eval R f (relabel_vars I J g q))"
-      using total_eval_add[of _ J _ f] 
-      by (metis "1" A(1) A(2) A(3) A(4) assms(1) assms(3) relabel_vars_closed)      
+      using total_eval_add[of _ J _ f]
+      by (metis "1" A(1) A(2) A(3) A(4) assms(1) assms(3) relabel_vars_closed)
     show "total_eval R (f \<circ> g) (p \<oplus>\<^bsub>Pring R I\<^esub> q) = total_eval R f (relabel_vars I J g (p \<oplus>\<^bsub>Pring R I\<^esub> q))"
-      using A 0 1 2 
+      using A 0 1 2
       by (metis total_eval_add)
   qed
   show "\<And>p i. p \<in> carrier (Pring R I) \<Longrightarrow>
            i \<in> I \<Longrightarrow>
            total_eval R (f \<circ> g) p = total_eval R f (relabel_vars I J g p) \<Longrightarrow>
            total_eval R (f \<circ> g) (p \<otimes>\<^bsub>Pring R I\<^esub> pvar R i) = total_eval R f (relabel_vars I J g (p \<otimes>\<^bsub>Pring R I\<^esub> pvar R i))"
   proof- fix p i assume A: "p \<in> carrier (Pring R I)" " i \<in> I "
            "total_eval R (f \<circ> g) p = total_eval R f (relabel_vars I J g p)"
     have 0: "closed_fun R (f \<circ> g)"
-            apply(rule closed_funI) 
+            apply(rule closed_funI)
       using comp_apply[of f g] closed_fun_simp[of f] assms(3) by presburger
-    have 1: " (relabel_vars I J g (p \<otimes>\<^bsub>Pring R I\<^esub>  (pvar R i))) = 
+    have 1: " (relabel_vars I J g (p \<otimes>\<^bsub>Pring R I\<^esub>  (pvar R i))) =
                (relabel_vars I J g p) \<otimes>\<^bsub>Pring R J\<^esub> (relabel_vars I J g (pvar R i))"
-      by (meson A(1) A(2) assms(1) local.ring_axioms relabel_vars_mult ring.Pring_var_closed)      
-    have 2: "total_eval R f (relabel_vars I J g (p \<otimes>\<^bsub>Pring R I\<^esub>  (pvar R i))) = 
+      by (meson A(1) A(2) assms(1) local.ring_axioms relabel_vars_mult ring.Pring_var_closed)
+    have 2: "total_eval R f (relabel_vars I J g (p \<otimes>\<^bsub>Pring R I\<^esub>  (pvar R i))) =
       (total_eval R f (relabel_vars I J g p)) \<otimes>\<^bsub>R\<^esub> (total_eval R f (relabel_vars I J g (pvar R i)))"
       using total_eval_mult[of "relabel_vars I J g p" J "relabel_vars I J g (pvar R i)"]
       by (metis "1" A(1) A(2) A(3) assms(1) assms(3) is_cring pvar_closed relabel_vars_closed)
-    have 3: " total_eval R (f \<circ> g) (p \<otimes>\<^bsub>Pring R I\<^esub> pvar R i) = 
+    have 3: " total_eval R (f \<circ> g) (p \<otimes>\<^bsub>Pring R I\<^esub> pvar R i) =
                 total_eval R (f \<circ> g) p \<otimes>\<^bsub>R\<^esub> total_eval R (f \<circ> g)( pvar R i)"
       by (meson "0" A(1) A(2) Pring_var_closed total_eval_mult)
     have 4: "total_eval R (f \<circ> g)( pvar R i) = (total_eval R f (relabel_vars I J g (pvar R i)))"
-    proof- 
+    proof-
       have 40: "total_eval R (f \<circ> g)( pvar R i) = (f \<circ> g) i"
-        using total_eval_var[of "f \<circ>g" i] 
+        using total_eval_var[of "f \<circ>g" i]
         by (metis "0" var_to_IP_def)
       have 41: "relabel_vars I J g (pvar R i) = pvar R (g i)"
         by (simp add: A(2) assms(1) relabel_vars_is_morphism(2))
       have 42: "total_eval R f (relabel_vars I J g (pvar R i)) = f (g i)"
         using total_eval_var
         by (metis "41" assms(3) var_to_IP_def)
-      show ?thesis 
+      show ?thesis
         by (metis "40" "42" comp_apply)
     qed
-    show " total_eval R (f \<circ> g) (p \<otimes>\<^bsub>Pring R I\<^esub> pvar R i) = 
+    show " total_eval R (f \<circ> g) (p \<otimes>\<^bsub>Pring R I\<^esub> pvar R i) =
             total_eval R f (relabel_vars I J g (p \<otimes>\<^bsub>Pring R I\<^esub> pvar R i))"
       using A 0 1 2 3 4
       by presburger
   qed
 qed
 
 end
 
 
 end
 
diff --git a/thys/Padic_Field/Padic_Field_Polynomials.thy b/thys/Padic_Field/Padic_Field_Polynomials.thy
--- a/thys/Padic_Field/Padic_Field_Polynomials.thy
+++ b/thys/Padic_Field/Padic_Field_Polynomials.thy
@@ -1,1811 +1,1809 @@
 theory Padic_Field_Polynomials
   imports Padic_Fields
 
-begin 
+begin
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 section\<open>$p$-adic Univariate Polynomials and Hensel's Lemma\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 type_synonym padic_field_poly = "nat \<Rightarrow> padic_number"
 
 type_synonym padic_field_fun = "padic_number \<Rightarrow> padic_number"
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsection\<open>Gauss Norms of Polynomials\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 text \<open>
-  The Gauss norm of a polynomial is defined to be the minimum valuation of a coefficient of that 
-  polynomial. This induces a valuation on the ring of polynomials, and in particular it satisfies 
-  the ultrametric inequality. In addition, the Gauss norm of a polynomial $f(x)$ gives a lower 
-  bound for the value $\text{val } (f(a))$ in terms of $\text{val }(a)$, for a point 
-  $a \in \mathbb{Q}_p$. We introduce Gauss norms here as a useful tool for stating and proving 
-  Hensel's Lemma for the field $\mathbb{Q}_p$. We are abusing terminology slightly in calling 
-  this the Gauss norm, rather than the Gauss valuation, but this is just to conform with our 
-  decision to work exclusively with the $p$-adic valuation and not discuss the equivalent 
+  The Gauss norm of a polynomial is defined to be the minimum valuation of a coefficient of that
+  polynomial. This induces a valuation on the ring of polynomials, and in particular it satisfies
+  the ultrametric inequality. In addition, the Gauss norm of a polynomial $f(x)$ gives a lower
+  bound for the value $\text{val } (f(a))$ in terms of $\text{val }(a)$, for a point
+  $a \in \mathbb{Q}_p$. We introduce Gauss norms here as a useful tool for stating and proving
+  Hensel's Lemma for the field $\mathbb{Q}_p$. We are abusing terminology slightly in calling
+  this the Gauss norm, rather than the Gauss valuation, but this is just to conform with our
+  decision to work exclusively with the $p$-adic valuation and not discuss the equivalent
   real-valued $p$-adic norm. For a detailed treatment of Gauss norms one can see, for example
   \cite{engler2005valued}.
 \<close>
 context padic_fields
 begin
 
 no_notation Zp.to_fun (infixl\<open>\<bullet>\<close> 70)
 
 abbreviation(input) Q\<^sub>p_x where
 "Q\<^sub>p_x \<equiv> UP Q\<^sub>p"
 
 definition gauss_norm where
 "gauss_norm g = Min (val ` g ` {..degree g}) "
 
-lemma gauss_normE:  
+lemma gauss_normE:
   assumes "g \<in> carrier Q\<^sub>p_x"
-  shows "gauss_norm g \<le> val (g k)" 
+  shows "gauss_norm g \<le> val (g k)"
   apply(cases "k \<le> degree g")
-  unfolding gauss_norm_def 
-  using assms apply auto[1]  
+  unfolding gauss_norm_def
+  using assms apply auto[1]
 proof-
   assume "\<not> k \<le> degree g"
   then have "g k = \<zero>\<^bsub>Q\<^sub>p\<^esub> "
-    by (simp add: UPQ.deg_leE assms)    
+    by (simp add: UPQ.deg_leE assms)
   then show "Min (val ` g ` {..deg Q\<^sub>p g}) \<le> val (g k)"
-    by (simp add: local.val_zero)    
+    by (simp add: local.val_zero)
 qed
 
 lemma gauss_norm_geqI:
   assumes "g \<in> carrier (UP Q\<^sub>p)"
   assumes "\<And>n. val (g n) \<ge> \<alpha>"
   shows "gauss_norm g \<ge> \<alpha>"
-  unfolding gauss_norm_def using assms 
+  unfolding gauss_norm_def using assms
   by simp
 
 lemma gauss_norm_eqI:
   assumes "g \<in> carrier (UP Q\<^sub>p)"
   assumes "\<And>n. val (g n) \<ge> \<alpha>"
   assumes "val (g i) = \<alpha>"
   shows "gauss_norm g = \<alpha>"
-proof- 
+proof-
   have 0: "gauss_norm g \<le> \<alpha>"
     using assms gauss_normE gauss_norm_def by fastforce
   have 1: "gauss_norm g \<ge> \<alpha>"
-    using assms gauss_norm_geqI by auto 
-  show ?thesis using 0 1 by auto 
+    using assms gauss_norm_geqI by auto
+  show ?thesis using 0 1 by auto
 qed
 
 lemma nonzero_poly_nonzero_coeff:
   assumes "g \<in> carrier Q\<^sub>p_x"
   assumes "g \<noteq> \<zero>\<^bsub>Q\<^sub>p_x\<^esub>"
   shows "\<exists>k. k \<le>degree g \<and> g k \<noteq>\<zero>\<^bsub>Q\<^sub>p\<^esub>"
 proof(rule ccontr)
   assume "\<not> (\<exists>k\<le>degree g. g k \<noteq> \<zero>\<^bsub>Q\<^sub>p\<^esub>)"
   then have 0: "\<And>k. g k = \<zero>\<^bsub>Q\<^sub>p\<^esub>"
     by (meson UPQ.deg_leE assms(1) not_le_imp_less)
-  then show False 
+  then show False
     using assms  UPQ.cfs_zero by blast
-qed 
+qed
 
 lemma gauss_norm_prop:
   assumes "g \<in> carrier Q\<^sub>p_x"
   assumes "g \<noteq> \<zero>\<^bsub>Q\<^sub>p_x\<^esub>"
   shows "gauss_norm g \<noteq> \<infinity>"
-proof- 
+proof-
   obtain k where k_def: "k \<le>degree g \<and> g k \<noteq>\<zero>\<^bsub>Q\<^sub>p\<^esub>"
-    using assms nonzero_poly_nonzero_coeff 
+    using assms nonzero_poly_nonzero_coeff
     by blast
   then have 0: "gauss_norm g \<le> val (g k)"
     using assms(1) gauss_normE by blast
   have "g k \<in> carrier Q\<^sub>p"
-    using UPQ.cfs_closed assms(1) by blast    
+    using UPQ.cfs_closed assms(1) by blast
   hence "val (g k) < \<infinity>"
-    using k_def assms  
+    using k_def assms
     by (metis eint_ord_code(3) eint_ord_simps(4) val_ineq)
-  then show ?thesis 
-    using 0 not_le by fastforce   
+  then show ?thesis
+    using 0 not_le by fastforce
 qed
 
 lemma gauss_norm_coeff_norm:
   "\<exists>n \<le> degree g. (gauss_norm g) = val (g n)"
 proof-
   have "finite (val ` g ` {..deg Q\<^sub>p g})"
     by blast
   hence "\<exists>x \<in> (val ` g ` {..deg Q\<^sub>p g}). gauss_norm g = x"
   unfolding gauss_norm_def
-  by auto 
-  thus ?thesis unfolding gauss_norm_def 
-    by blast 
+  by auto
+  thus ?thesis unfolding gauss_norm_def
+    by blast
 qed
 
 lemma gauss_norm_smult_cfs:
   assumes "g \<in> carrier Q\<^sub>p_x"
   assumes "a \<in> carrier Q\<^sub>p"
   assumes "gauss_norm g = val (g k)"
   shows "gauss_norm (a \<odot>\<^bsub>Q\<^sub>p_x\<^esub> g) = val a + val (g k)"
 proof-
   obtain l where l_def: "gauss_norm (a \<odot>\<^bsub>Q\<^sub>p_x\<^esub> g) =  val ((a \<odot>\<^bsub>Q\<^sub>p_x\<^esub> g) l)"
-    using gauss_norm_coeff_norm 
+    using gauss_norm_coeff_norm
     by blast
   then have "gauss_norm (a \<odot>\<^bsub>Q\<^sub>p_x\<^esub> g) =  val (a \<otimes>\<^bsub>Q\<^sub>p\<^esub> (g l))"
-    using assms 
-    by simp   
+    using assms
+    by simp
   then have "gauss_norm (a \<odot>\<^bsub>Q\<^sub>p_x\<^esub> g) =  val a + val (g l)"
-    by (simp add: UPQ.cfs_closed assms(1) assms(2) val_mult)            
+    by (simp add: UPQ.cfs_closed assms(1) assms(2) val_mult)
   then have 0: "gauss_norm (a \<odot>\<^bsub>Q\<^sub>p_x\<^esub> g) \<le> val a +val (g k)"
     using assms  gauss_normE[of g l]
-    by (metis UPQ.UP_smult_closed UPQ.cfs_closed UPQ.cfs_smult gauss_normE val_mult)          
+    by (metis UPQ.UP_smult_closed UPQ.cfs_closed UPQ.cfs_smult gauss_normE val_mult)
   have "val a + val (g k) = val ((a \<odot>\<^bsub>Q\<^sub>p_x\<^esub> g) k)"
-    by (simp add: UPQ.cfs_closed assms(1) assms(2) val_mult)       
+    by (simp add: UPQ.cfs_closed assms(1) assms(2) val_mult)
   then have "gauss_norm (a \<odot>\<^bsub>Q\<^sub>p_x\<^esub> g) \<ge> val a + val (g k)"
-    by (metis \<open>gauss_norm (a \<odot>\<^bsub>UP Q\<^sub>p\<^esub> g) = val a + val (g l)\<close> add_left_mono assms(1) assms(3) gauss_normE)   
-  then show ?thesis 
-    using 0  by auto     
+    by (metis \<open>gauss_norm (a \<odot>\<^bsub>UP Q\<^sub>p\<^esub> g) = val a + val (g l)\<close> add_left_mono assms(1) assms(3) gauss_normE)
+  then show ?thesis
+    using 0  by auto
 qed
 
 lemma gauss_norm_smult:
   assumes "g \<in> carrier Q\<^sub>p_x"
   assumes "a \<in> carrier Q\<^sub>p"
   shows "gauss_norm (a \<odot>\<^bsub>Q\<^sub>p_x\<^esub> g) = val a + gauss_norm g"
-  using gauss_norm_smult_cfs[of g a] gauss_norm_coeff_norm[of g] assms 
+  using gauss_norm_smult_cfs[of g a] gauss_norm_coeff_norm[of g] assms
   by metis
 
 lemma gauss_norm_ultrametric:
   assumes "g \<in> carrier Q\<^sub>p_x"
   assumes "h \<in> carrier Q\<^sub>p_x"
   shows "gauss_norm (g \<oplus>\<^bsub>Q\<^sub>p_x\<^esub> h) \<ge> min (gauss_norm g) (gauss_norm h)"
 proof-
   obtain k where "gauss_norm (g \<oplus>\<^bsub>Q\<^sub>p_x\<^esub> h) = val ((g \<oplus>\<^bsub>Q\<^sub>p_x\<^esub> h) k)"
-    using gauss_norm_coeff_norm 
+    using gauss_norm_coeff_norm
     by blast
   then have 0: "gauss_norm (g \<oplus>\<^bsub>Q\<^sub>p_x\<^esub> h) = val (g k \<oplus>\<^bsub>Q\<^sub>p\<^esub> h k)"
-    by (simp add: assms(1) assms(2))      
+    by (simp add: assms(1) assms(2))
   have "min (val (g k)) (val (h k))\<ge> min (gauss_norm g) (gauss_norm h)"
-    using gauss_normE[of g k] gauss_normE[of h k]  assms(1) assms(2) min.mono 
-    by blast    
-  then show ?thesis 
-    using 0 val_ultrametric[of "g k" "h k"] assms(1) assms(2) dual_order.trans 
-    by (metis (no_types, lifting) UPQ.cfs_closed)         
+    using gauss_normE[of g k] gauss_normE[of h k]  assms(1) assms(2) min.mono
+    by blast
+  then show ?thesis
+    using 0 val_ultrametric[of "g k" "h k"] assms(1) assms(2) dual_order.trans
+    by (metis (no_types, lifting) UPQ.cfs_closed)
 qed
 
-lemma gauss_norm_a_inv: 
+lemma gauss_norm_a_inv:
   assumes "f \<in> carrier (UP Q\<^sub>p)"
   shows "gauss_norm (\<ominus>\<^bsub>UP Q\<^sub>p\<^esub>f) = gauss_norm f"
-proof- 
+proof-
   have 0: "\<And>n. ((\<ominus>\<^bsub>UP Q\<^sub>p\<^esub>f) n) = \<ominus> (f n)"
     using assms by simp
   have 1: "\<And>n. val ((\<ominus>\<^bsub>UP Q\<^sub>p\<^esub>f) n) = val (f n)"
     using 0 assms UPQ.UP_car_memE(1) val_minus by presburger
   obtain i where i_def: "gauss_norm f = val (f i)"
     using assms gauss_norm_coeff_norm by blast
   have 2: "\<And>k. val ((\<ominus>\<^bsub>UP Q\<^sub>p\<^esub>f) k) \<ge> val (f i)"
-    unfolding 1 
+    unfolding 1
     using i_def assms gauss_normE by fastforce
-  show ?thesis 
+  show ?thesis
     apply(rule gauss_norm_eqI[of _ _ i])
       apply (simp add: assms; fail)
     unfolding 1 using assms gauss_normE apply blast
-    unfolding i_def by blast 
+    unfolding i_def by blast
 qed
 
 lemma gauss_norm_ultrametric':
   assumes "f \<in> carrier (UP Q\<^sub>p)"
   assumes "g \<in> carrier (UP Q\<^sub>p)"
   shows "gauss_norm (f \<ominus>\<^bsub>UP Q\<^sub>p\<^esub> g) \<ge> min (gauss_norm f) (gauss_norm g)"
-  unfolding a_minus_def 
-  using assms gauss_norm_a_inv[of g] gauss_norm_ultrametric 
+  unfolding a_minus_def
+  using assms gauss_norm_a_inv[of g] gauss_norm_ultrametric
   by (metis UPQ.UP_a_inv_closed)
 
 lemma gauss_norm_finsum:
   assumes "f \<in> A \<rightarrow> carrier Q\<^sub>p_x"
   assumes "finite A"
   assumes "A \<noteq> {}"
-  shows " gauss_norm (\<Oplus>\<^bsub>Q\<^sub>p_x\<^esub>i\<in>A. f i) \<ge> Min (gauss_norm ` (f`A))"  
+  shows " gauss_norm (\<Oplus>\<^bsub>Q\<^sub>p_x\<^esub>i\<in>A. f i) \<ge> Min (gauss_norm ` (f`A))"
 proof-
   obtain k where k_def: "val ((\<Oplus>\<^bsub>Q\<^sub>p_x\<^esub>i\<in>A. f i) k) = gauss_norm (\<Oplus>\<^bsub>Q\<^sub>p_x\<^esub>i\<in>A. f i)"
     by (metis gauss_norm_coeff_norm)
   then have 0: "val (\<Oplus>\<^bsub>Q\<^sub>p\<^esub>i\<in>A. f i k) \<ge> Min (val ` (\<lambda> i. f i k) ` A)"
-    using finsum_val_ultrametric[of "\<lambda> i. f i k" A] assms 
-    by (simp add: \<open>\<lbrakk>(\<lambda>i. f i k) \<in> A \<rightarrow> carrier Q\<^sub>p; finite A; A \<noteq> {}\<rbrakk> \<Longrightarrow> Min (val ` (\<lambda>i. f i k) ` A) \<le> val (\<Oplus>i\<in>A. f i k)\<close> Pi_iff UPQ.cfs_closed)      
+    using finsum_val_ultrametric[of "\<lambda> i. f i k" A] assms
+    by (simp add: \<open>\<lbrakk>(\<lambda>i. f i k) \<in> A \<rightarrow> carrier Q\<^sub>p; finite A; A \<noteq> {}\<rbrakk> \<Longrightarrow> Min (val ` (\<lambda>i. f i k) ` A) \<le> val (\<Oplus>i\<in>A. f i k)\<close> Pi_iff UPQ.cfs_closed)
   have "(\<And>a. a \<in> A \<Longrightarrow> (val \<circ> (\<lambda>i. f i k)) a \<ge> gauss_norm (f a))"
     using gauss_normE assms
-    by (metis (no_types, lifting) Pi_split_insert_domain Set.set_insert comp_apply)  
-  then have "Min (val ` (\<lambda> i. f i k) ` A) \<ge> Min ((\<lambda> i. gauss_norm (f  i)) ` A)" 
-    using Min_mono'[of A] 
+    by (metis (no_types, lifting) Pi_split_insert_domain Set.set_insert comp_apply)
+  then have "Min (val ` (\<lambda> i. f i k) ` A) \<ge> Min ((\<lambda> i. gauss_norm (f  i)) ` A)"
+    using Min_mono'[of A]
     by (simp add: assms(2) image_comp)
   then have 1: "Min (val ` (\<lambda> i. f i k) ` A) \<ge> Min (gauss_norm ` f ` A)"
     by (metis image_image)
   have "f \<in> A \<rightarrow> carrier (UP Q\<^sub>p) \<longrightarrow> ((\<Oplus>\<^bsub>Q\<^sub>p_x\<^esub>i\<in>A. f i) \<in> carrier Q\<^sub>p_x \<and> ((\<Oplus>\<^bsub>Q\<^sub>p_x\<^esub>i\<in>A. f i) k) = (\<Oplus>\<^bsub>Q\<^sub>p\<^esub>i\<in>A. f i k)) "
     apply(rule finite.induct[of A])
       apply (simp add: assms(2); fail)
-     apply (metis (no_types, lifting) Pi_I Qp.add.finprod_one_eqI UPQ.P.finsum_closed UPQ.P.finsum_empty UPQ.cfs_zero empty_iff)        
+     apply (metis (no_types, lifting) Pi_I Qp.add.finprod_one_eqI UPQ.P.finsum_closed UPQ.P.finsum_empty UPQ.cfs_zero empty_iff)
   proof-
     fix a A assume A: "finite A" "f \<in> A \<rightarrow> carrier (UP Q\<^sub>p) \<longrightarrow> ( finsum (UP Q\<^sub>p) f A \<in> carrier (UP Q\<^sub>p) \<and> finsum (UP Q\<^sub>p) f A k = (\<Oplus>i\<in>A. f i k)) "
     show " f \<in> insert a A \<rightarrow> carrier (UP Q\<^sub>p) \<longrightarrow>  finsum (UP Q\<^sub>p) f (insert a A) \<in> carrier (UP Q\<^sub>p) \<and> finsum (UP Q\<^sub>p) f (insert a A) k = (\<Oplus>i\<in>insert a A. f i k)"
       apply(cases "a \<in> A")
-      using A 
+      using A
       apply (simp add: insert_absorb; fail)
     proof assume B: "a \<notin> A" " f \<in> insert a A \<rightarrow> carrier (UP Q\<^sub>p)"
       then have f_a: "f a \<in> carrier (UP Q\<^sub>p)"
-        by blast 
+        by blast
       have f_A: "f \<in> A \<rightarrow> carrier (UP Q\<^sub>p)"
-        using B by blast 
+        using B by blast
       have "finsum (UP Q\<^sub>p) f (insert a A) = f a \<oplus>\<^bsub>UP Q\<^sub>p\<^esub>finsum (UP Q\<^sub>p) f A"
-        using assms A B f_a f_A  finsum_insert by simp       
+        using assms A B f_a f_A  finsum_insert by simp
       then have 0: "finsum (UP Q\<^sub>p) f (insert a A) k = f a k \<oplus>\<^bsub>Q\<^sub>p\<^esub> (finsum (UP Q\<^sub>p) f A) k"
-        using f_a f_A A B 
+        using f_a f_A A B
         by simp
       have " ( \<lambda> a. f a k) \<in> A \<rightarrow> carrier Q\<^sub>p"
       proof fix a assume "a \<in> A"
         then have "f a \<in> carrier (UP Q\<^sub>p)"
-          using f_A by blast 
+          using f_A by blast
         then show "f a k \<in> carrier Q\<^sub>p"
-          using A cfs_closed by blast 
-      qed 
+          using A cfs_closed by blast
+      qed
       then have 0: "finsum (UP Q\<^sub>p) f (insert a A) k = (\<Oplus>i\<in>insert a A. f i k)"
-        using A B Qp.finsum_insert[of A a "\<lambda> a. f a k"] 
-        by (simp add: UPQ.cfs_closed)        
+        using A B Qp.finsum_insert[of A a "\<lambda> a. f a k"]
+        by (simp add: UPQ.cfs_closed)
       thus " finsum (UP Q\<^sub>p) f (insert a A) \<in> carrier (UP Q\<^sub>p) \<and> finsum (UP Q\<^sub>p) f (insert a A) k = (\<Oplus>i\<in>insert a A. f i k)"
         using B(2) UPQ.P.finsum_closed by blast
-    qed        
-  qed   
+    qed
+  qed
   then have "(\<Oplus>\<^bsub>Q\<^sub>p_x\<^esub>i\<in>A. f i) \<in> carrier Q\<^sub>p_x \<and> ((\<Oplus>\<^bsub>Q\<^sub>p_x\<^esub>i\<in>A. f i) k) = (\<Oplus>\<^bsub>Q\<^sub>p\<^esub>i\<in>A. f i k)"
-    using assms by blast 
+    using assms by blast
   hence 3: "gauss_norm (\<Oplus>\<^bsub>Q\<^sub>p_x\<^esub>i\<in>A. f i) \<ge> Min (val ` (\<lambda> i. f i k) ` A)"
-    using 0  k_def by auto 
-  thus ?thesis 
-    using 1 le_trans by auto 
+    using 0  k_def by auto
+  thus ?thesis
+    using 1 le_trans by auto
 qed
 
 lemma gauss_norm_monom:
   assumes "a \<in> carrier Q\<^sub>p"
   shows "gauss_norm (monom Q\<^sub>p_x a n) = val a"
 proof-
   have "val ((monom Q\<^sub>p_x a n) n) \<ge> gauss_norm (monom Q\<^sub>p_x a n)"
-    using assms gauss_normE[of "monom Q\<^sub>p_x a n" n] UPQ.monom_closed 
-    by blast   
-  then show ?thesis 
-    using gauss_norm_coeff_norm[of "monom Q\<^sub>p_x a n"] assms val_ineq UPQ.cfs_monom by fastforce     
+    using assms gauss_normE[of "monom Q\<^sub>p_x a n" n] UPQ.monom_closed
+    by blast
+  then show ?thesis
+    using gauss_norm_coeff_norm[of "monom Q\<^sub>p_x a n"] assms val_ineq UPQ.cfs_monom by fastforce
 qed
 
 lemma val_val_ring_prod:
   assumes "a \<in> \<O>\<^sub>p"
   assumes "b \<in> carrier Q\<^sub>p"
   shows "val (a \<otimes>\<^bsub>Q\<^sub>p\<^esub> b) \<ge> val b"
 proof-
   have 0: "val (a \<otimes>\<^bsub>Q\<^sub>p\<^esub> b) = val a + val b"
-    using assms val_ring_memE[of a] val_mult 
+    using assms val_ring_memE[of a] val_mult
     by blast
   have 1: " val a \<ge> 0"
-    using assms 
+    using assms
     by (simp add: val_ring_memE)
-  then show ?thesis 
-    using assms 0 
-    by simp   
+  then show ?thesis
+    using assms 0
+    by simp
 qed
 
 lemma val_val_ring_prod':
   assumes "a \<in> \<O>\<^sub>p"
   assumes "b \<in> carrier Q\<^sub>p"
   shows "val (b \<otimes>\<^bsub>Q\<^sub>p\<^esub> a) \<ge> val b"
   using val_val_ring_prod[of a b]
-  by (simp add: Qp.m_comm val_ring_memE assms(1) assms(2)) 
+  by (simp add: Qp.m_comm val_ring_memE assms(1) assms(2))
 
 lemma val_ring_nat_pow_closed:
   assumes "a \<in> \<O>\<^sub>p"
   shows "(a[^](n::nat)) \<in> \<O>\<^sub>p"
   apply(induction n)
   apply auto[1]
   using Qp.inv_one Z\<^sub>p_mem apply blast
   by (metis Qp.nat_pow_Suc Qp.nat_pow_closed val_ring_memE assms image_eqI inc_of_prod to_Zp_closed to_Zp_inc to_Zp_mult)
-  
+
 lemma val_ringI:
   assumes "a \<in> carrier Q\<^sub>p"
   assumes "val a \<ge>0"
   shows " a \<in> \<O>\<^sub>p"
   apply(rule val_ring_val_criterion)
-  using assms by auto 
+  using assms by auto
 
 notation UPQ.to_fun (infixl\<open>\<bullet>\<close> 70)
 
 lemma val_gauss_norm_eval:
   assumes "g \<in> carrier Q\<^sub>p_x"
   assumes "a \<in> \<O>\<^sub>p"
   shows "val (g \<bullet> a) \<ge> gauss_norm g"
 proof-
   have 0: "g\<bullet>a = (\<Oplus>\<^bsub>Q\<^sub>p\<^esub>i\<in>{..degree g}. (g i)\<otimes>\<^bsub>Q\<^sub>p\<^esub> (a[^]i))"
-    using val_ring_memE assms to_fun_formula[of g a] by auto 
-    
+    using val_ring_memE assms to_fun_formula[of g a] by auto
+
   have 1: "(\<lambda>i. g i \<otimes>\<^bsub>Q\<^sub>p\<^esub> (a[^]i)) \<in> {..degree g} \<rightarrow> carrier Q\<^sub>p"
-     using assms 
-    by (meson Pi_I val_ring_memE cfs_closed monom_term_car)    
+     using assms
+    by (meson Pi_I val_ring_memE cfs_closed monom_term_car)
   then have 2: "val (g\<bullet>a) \<ge> Min (val ` (\<lambda> i. ((g i)\<otimes>\<^bsub>Q\<^sub>p\<^esub> (a[^]i))) ` {..degree g})"
-    using 0 finsum_val_ultrametric[of "\<lambda> i. ((g i)\<otimes>\<^bsub>Q\<^sub>p\<^esub> (a[^]i))" "{..degree g}" ]  
+    using 0 finsum_val_ultrametric[of "\<lambda> i. ((g i)\<otimes>\<^bsub>Q\<^sub>p\<^esub> (a[^]i))" "{..degree g}" ]
     by (metis finite_atMost not_empty_eq_Iic_eq_empty)
   have 3: "\<And> i. val ((g i)\<otimes>\<^bsub>Q\<^sub>p\<^esub> (a[^]i)) = val (g i) + val (a[^]i)"
-    using assms val_mult 
-    by (simp add: val_ring_memE UPQ.cfs_closed)    
-  have 4: "\<And> i. val ((g i)\<otimes>\<^bsub>Q\<^sub>p\<^esub> (a[^]i)) \<ge> val (g i)"    
+    using assms val_mult
+    by (simp add: val_ring_memE UPQ.cfs_closed)
+  have 4: "\<And> i. val ((g i)\<otimes>\<^bsub>Q\<^sub>p\<^esub> (a[^]i)) \<ge> val (g i)"
   proof-
-    fix i 
+    fix i
     show "val ((g i)\<otimes>\<^bsub>Q\<^sub>p\<^esub> (a[^]i)) \<ge> val (g i)"
-      using val_val_ring_prod'[of "a[^]i" "g i" ] 
-        assms(1) assms(2) val_ring_nat_pow_closed cfs_closed 
-      by simp      
+      using val_val_ring_prod'[of "a[^]i" "g i" ]
+        assms(1) assms(2) val_ring_nat_pow_closed cfs_closed
+      by simp
   qed
   have "Min (val ` (\<lambda>i. g i \<otimes>\<^bsub>Q\<^sub>p\<^esub> (a[^]i)) ` {..degree g}) \<ge> Min ((\<lambda>i. val (g i)) ` {..degree g})"
-    using Min_mono'[of "{..degree g}" "\<lambda>i. val (g i)" "\<lambda>i. val (g i \<otimes>\<^bsub>Q\<^sub>p\<^esub> (a[^]i))" ] 4 2 
+    using Min_mono'[of "{..degree g}" "\<lambda>i. val (g i)" "\<lambda>i. val (g i \<otimes>\<^bsub>Q\<^sub>p\<^esub> (a[^]i))" ] 4 2
     by (metis finite_atMost image_image)
   then have "Min (val ` (\<lambda>i. g i \<otimes>\<^bsub>Q\<^sub>p\<^esub> (a[^]i)) ` {..degree g}) \<ge> Min (val ` g ` {..degree g})"
     by (metis  image_image)
   then have  "val (g\<bullet>a) \<ge> Min (val ` g ` {..degree g})"
-    using 2 
-    by (meson atMost_iff atMost_subset_iff in_mono)    
-  then show ?thesis 
+    using 2
+    by (meson atMost_iff atMost_subset_iff in_mono)
+  then show ?thesis
     by (simp add: \<open>val (g\<bullet>a) \<ge> Min (val ` g ` {..degree g})\<close> gauss_norm_def)
 qed
 
 lemma positive_gauss_norm_eval:
   assumes "g \<in> carrier Q\<^sub>p_x"
   assumes "gauss_norm g \<ge> 0"
   assumes "a \<in> \<O>\<^sub>p"
   shows "(g\<bullet>a) \<in> \<O>\<^sub>p"
   apply(rule val_ring_val_criterion[of "g\<bullet>a"])
-  using assms val_ring_memE 
+  using assms val_ring_memE
   using UPQ.to_fun_closed apply blast
-  using assms val_gauss_norm_eval[of g a] by auto 
-  
+  using assms val_gauss_norm_eval[of g a] by auto
+
 lemma positive_gauss_norm_valuation_ring_coeffs:
   assumes "g \<in> carrier Q\<^sub>p_x"
   assumes "gauss_norm g \<ge> 0"
   shows "g n \<in> \<O>\<^sub>p"
   apply(rule val_ringI)
   using cfs_closed assms(1) apply blast
-  using gauss_normE[of g n] assms by auto  
+  using gauss_normE[of g n] assms by auto
 
 lemma val_ring_cfs_imp_nonneg_gauss_norm:
   assumes "g \<in> carrier (UP Q\<^sub>p)"
   assumes "\<And>n. g n \<in> \<O>\<^sub>p"
   shows "gauss_norm g \<ge> 0"
   by(rule gauss_norm_geqI, rule assms, rule val_ring_memE, rule assms)
 
 lemma val_of_add_pow:
   assumes "a \<in> carrier Q\<^sub>p"
   shows "val ([(n::nat)]\<cdot>a) \<ge> val a"
 proof-
   have 0: "[(n::nat)]\<cdot>a = ([n]\<cdot>\<one>)\<otimes>a"
     using assms Qp.add_pow_ldistr Qp.cring_simprules(12) Qp.one_closed by presburger
   have 1: "val ([(n::nat)]\<cdot>a) = val ([n]\<cdot>\<one>) + val a"
     unfolding 0 by(rule val_mult, simp, rule assms)
-  show ?thesis unfolding 1 using assms 
+  show ?thesis unfolding 1 using assms
     by (simp add: val_of_nat_inc)
 qed
 
 lemma gauss_norm_pderiv:
   assumes "g \<in> carrier (UP Q\<^sub>p)"
   shows "gauss_norm g \<le> gauss_norm (pderiv g)"
   apply(rule gauss_norm_geqI)
   using UPQ.pderiv_closed assms apply blast
-  using gauss_normE pderiv_cfs val_of_add_pow 
+  using gauss_normE pderiv_cfs val_of_add_pow
   by (smt UPQ.cfs_closed assms dual_order.trans)
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsection\<open>Mapping Polynomials with Value Ring Coefficients to Polynomials over $\mathbb{Z}_p$\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 definition to_Zp_poly where
 "to_Zp_poly g = (\<lambda>n. to_Zp (g n))"
 
 lemma to_Zp_poly_closed:
   assumes "g \<in> carrier Q\<^sub>p_x"
   assumes "gauss_norm g \<ge> 0"
   shows "to_Zp_poly g \<in> carrier (UP Z\<^sub>p)"
 proof-
   have  "to_Zp_poly g \<in> up Z\<^sub>p"
     apply(rule mem_upI)
-   unfolding to_Zp_poly_def 
-   using cfs_closed[of g ] assms(1) to_Zp_closed[of ]  apply blast  
+   unfolding to_Zp_poly_def
+   using cfs_closed[of g ] assms(1) to_Zp_closed[of ]  apply blast
   proof-
     have "\<exists>n. bound \<zero>\<^bsub>Q\<^sub>p\<^esub> n g"
      using UPQ.deg_leE assms(1) by auto
     then obtain n where n_def: " bound \<zero>\<^bsub>Q\<^sub>p\<^esub> n g"
-      by blast 
+      by blast
     then have " bound \<zero>\<^bsub>Z\<^sub>p\<^esub> n (\<lambda>n. to_Zp (g n))"
-      unfolding bound_def 
+      unfolding bound_def
       by (simp add: to_Zp_zero)
     then show "\<exists>n. bound \<zero>\<^bsub>Z\<^sub>p\<^esub> n (\<lambda>n. to_Zp (g n))"
       by blast
   qed
   then show ?thesis using UP_def[of Z\<^sub>p]
     by simp
 qed
 
 definition poly_inc where
 "poly_inc g = (\<lambda>n::nat. \<iota> (g n))"
 
 lemma poly_inc_closed:
   assumes "g \<in> carrier (UP Z\<^sub>p)"
   shows "poly_inc g \<in> carrier Q\<^sub>p_x"
 proof-
   have "poly_inc g \<in> up Q\<^sub>p"
   proof(rule mem_upI)
     show "\<And>n. poly_inc g n \<in> carrier Q\<^sub>p"
     proof- fix n
       have "g n \<in> carrier Z\<^sub>p"
-        using assms UP_def 
+        using assms UP_def
         by (simp add: UP_def mem_upD)
       then show "poly_inc g n \<in> carrier Q\<^sub>p"
-        using assms poly_inc_def[of g] inc_def[of "g n" ] inc_closed 
-        by force                   
+        using assms poly_inc_def[of g] inc_def[of "g n" ] inc_closed
+        by force
     qed
     show "\<exists>n. bound \<zero>\<^bsub>Q\<^sub>p\<^esub> n (poly_inc g)"
     proof-
       obtain n where n_def: " bound \<zero>\<^bsub>Z\<^sub>p\<^esub> n g"
         using assms  bound_def[of "\<zero>\<^bsub>Z\<^sub>p\<^esub>" _ g]Zp.cring_axioms UP_cring.deg_leE[of Z\<^sub>p g]
-        unfolding UP_cring_def 
-        by metis 
+        unfolding UP_cring_def
+        by metis
       then have " bound \<zero>\<^bsub>Q\<^sub>p\<^esub> n (poly_inc g)"
-        unfolding poly_inc_def bound_def 
+        unfolding poly_inc_def bound_def
         by (metis Qp.nat_inc_zero Zp.nat_inc_zero inc_of_nat)
       then show ?thesis by blast
     qed
   qed
-  then show ?thesis 
+  then show ?thesis
     by (simp add: \<open>poly_inc g \<in> up Q\<^sub>p\<close> UP_def)
 qed
 
 lemma poly_inc_inverse_right:
   assumes "g \<in> carrier (UP Z\<^sub>p)"
   shows "to_Zp_poly (poly_inc g) = g"
 proof-
   have 0: "\<And>n. g n \<in> carrier Z\<^sub>p"
-    by (simp add: Zp.cfs_closed assms)    
-  show ?thesis 
+    by (simp add: Zp.cfs_closed assms)
+  show ?thesis
     unfolding to_Zp_poly_def poly_inc_def
   proof
     fix n
     show "to_Zp (\<iota> (g n)) = g n"
-      using 0 inc_to_Zp 
+      using 0 inc_to_Zp
       by auto
   qed
 qed
 
 lemma poly_inc_inverse_left:
   assumes "g \<in> carrier Q\<^sub>p_x"
   assumes "gauss_norm g \<ge>0"
   shows "poly_inc (to_Zp_poly g) = g"
 proof
   fix x
   show "poly_inc (to_Zp_poly g) x = g x"
-    using assms unfolding poly_inc_def to_Zp_poly_def 
-    by (simp add: positive_gauss_norm_valuation_ring_coeffs to_Zp_inc)    
+    using assms unfolding poly_inc_def to_Zp_poly_def
+    by (simp add: positive_gauss_norm_valuation_ring_coeffs to_Zp_inc)
 qed
 
-lemma poly_inc_plus: 
+lemma poly_inc_plus:
   assumes "f \<in> carrier (UP Z\<^sub>p)"
   assumes "g \<in> carrier (UP Z\<^sub>p)"
   shows "poly_inc (f \<oplus>\<^bsub>UP Z\<^sub>p\<^esub> g) = poly_inc f \<oplus>\<^bsub>UP Q\<^sub>p\<^esub> poly_inc g"
 proof
-  fix n 
+  fix n
   have 0: "poly_inc (f \<oplus>\<^bsub>UP Z\<^sub>p\<^esub> g) n = \<iota> (f n \<oplus>\<^bsub>Z\<^sub>p\<^esub> g n)"
     unfolding poly_inc_def using assms by auto
   have 1: "(poly_inc f \<oplus>\<^bsub>UP Q\<^sub>p\<^esub> poly_inc g) n = poly_inc f n \<oplus> poly_inc g n"
     by(rule cfs_add, rule poly_inc_closed, rule assms, rule poly_inc_closed, rule assms)
   show "poly_inc (f \<oplus>\<^bsub>UP Z\<^sub>p\<^esub> g) n = (poly_inc f \<oplus>\<^bsub>UP Q\<^sub>p\<^esub> poly_inc g) n"
-    unfolding 0 1 unfolding poly_inc_def 
+    unfolding 0 1 unfolding poly_inc_def
     apply(rule inc_of_sum)
     using assms apply (simp add: Zp.cfs_closed; fail)
         using assms by (simp add: Zp.cfs_closed)
 qed
 
 lemma poly_inc_monom:
   assumes "a \<in> carrier Z\<^sub>p"
   shows "poly_inc (monom (UP Z\<^sub>p) a m) = monom (UP Q\<^sub>p) (\<iota> a) m"
-proof fix n 
+proof fix n
   show "poly_inc (monom (UP Z\<^sub>p) a m) n = monom (UP Q\<^sub>p) (\<iota> a) m n"
     apply(cases "m = n")
-    using assms cfs_monom[of "\<iota> a"] Zp.cfs_monom[of a] unfolding poly_inc_def 
+    using assms cfs_monom[of "\<iota> a"] Zp.cfs_monom[of a] unfolding poly_inc_def
      apply (simp add: inc_closed; fail)
-    using assms cfs_monom[of "\<iota> a"] Zp.cfs_monom[of a] unfolding poly_inc_def 
+    using assms cfs_monom[of "\<iota> a"] Zp.cfs_monom[of a] unfolding poly_inc_def
     by (metis Qp.nat_mult_zero Zp_nat_inc_zero inc_closed inc_of_nat)
 qed
 
-lemma poly_inc_times: 
+lemma poly_inc_times:
   assumes "f \<in> carrier (UP Z\<^sub>p)"
   assumes "g \<in> carrier (UP Z\<^sub>p)"
   shows "poly_inc (f \<otimes>\<^bsub>UP Z\<^sub>p\<^esub> g) = poly_inc f \<otimes>\<^bsub>UP Q\<^sub>p\<^esub> poly_inc g"
   apply(rule UP_ring.poly_induct3[of Z\<^sub>p])
   apply (simp add: Zp.is_UP_ring; fail)
   using assms apply blast
-proof- 
-  fix p q 
+proof-
+  fix p q
   assume A: "q \<in> carrier (UP Z\<^sub>p)"  "p \<in> carrier (UP Z\<^sub>p)"
             "poly_inc (f \<otimes>\<^bsub>UP Z\<^sub>p\<^esub> p) = poly_inc f \<otimes>\<^bsub>UP Q\<^sub>p\<^esub> poly_inc p"
             "poly_inc (f \<otimes>\<^bsub>UP Z\<^sub>p\<^esub> q) = poly_inc f \<otimes>\<^bsub>UP Q\<^sub>p\<^esub> poly_inc q"
   have 0: "(f \<otimes>\<^bsub>UP Z\<^sub>p\<^esub> (p \<oplus>\<^bsub>UP Z\<^sub>p\<^esub> q)) = (f \<otimes>\<^bsub>UP Z\<^sub>p\<^esub> p) \<oplus>\<^bsub>UP Z\<^sub>p\<^esub> (f \<otimes>\<^bsub>UP Z\<^sub>p\<^esub> q)"
-    using assms(1) A 
+    using assms(1) A
     by (simp add: Zp.P.r_distr)
   have 1: "poly_inc (p \<oplus>\<^bsub>UP Z\<^sub>p\<^esub> q) = poly_inc p \<oplus>\<^bsub>UP Q\<^sub>p\<^esub> poly_inc q"
     by(rule poly_inc_plus, rule A, rule A)
   show "poly_inc (f \<otimes>\<^bsub>UP Z\<^sub>p\<^esub> (p \<oplus>\<^bsub>UP Z\<^sub>p\<^esub> q)) = poly_inc f \<otimes>\<^bsub>UP Q\<^sub>p\<^esub> poly_inc (p \<oplus>\<^bsub>UP Z\<^sub>p\<^esub> q)"
-    unfolding 0 1 using A poly_inc_closed poly_inc_plus 
+    unfolding 0 1 using A poly_inc_closed poly_inc_plus
     by (simp add: UPQ.P.r_distr assms(1))
 next
   fix a fix n::nat
   assume A: "a \<in> carrier Z\<^sub>p"
   show "poly_inc (f \<otimes>\<^bsub>UP Z\<^sub>p\<^esub> monom (UP Z\<^sub>p) a n) =
            poly_inc f \<otimes>\<^bsub>UP Q\<^sub>p\<^esub> poly_inc (monom (UP Z\<^sub>p) a n)"
   proof
-    fix m 
+    fix m
     show "poly_inc (f \<otimes>\<^bsub>UP Z\<^sub>p\<^esub> monom (UP Z\<^sub>p) a n) m =
          (poly_inc f \<otimes>\<^bsub>UP Q\<^sub>p\<^esub> poly_inc (monom (UP Z\<^sub>p) a n)) m"
     proof(cases "m < n")
       case True
       have T0: "(f \<otimes>\<^bsub>UP Z\<^sub>p\<^esub> monom (UP Z\<^sub>p) a n) m = \<zero>\<^bsub>Z\<^sub>p\<^esub>"
-        using True Zp.cfs_monom_mult[of f a m n] A assms 
+        using True Zp.cfs_monom_mult[of f a m n] A assms
         by blast
       have T1: "poly_inc (monom (UP Z\<^sub>p) a n) =  (monom (UP Q\<^sub>p) (\<iota> a) n)"
         by(rule poly_inc_monom , rule A)
       show ?thesis
-        unfolding T0 T1 using True 
+        unfolding T0 T1 using True
         by (metis A Q\<^sub>p_def T0 UPQ.cfs_monom_mult Zp_def assms(1) inc_closed padic_fields.to_Zp_zero padic_fields_axioms poly_inc_closed poly_inc_def to_Zp_inc zero_in_val_ring)
     next
       case False
       then have F0: "m \<ge> n"
-        using False by simp 
+        using False by simp
       have F1: "(f \<otimes>\<^bsub>UP Z\<^sub>p\<^esub> monom (UP Z\<^sub>p) a n) m = a \<otimes>\<^bsub>Z\<^sub>p\<^esub> f (m - n)"
-        using Zp.cfs_monom_mult_l' F0 A assms by simp 
+        using Zp.cfs_monom_mult_l' F0 A assms by simp
       have F2: "poly_inc (monom (UP Z\<^sub>p) a n)  = monom (UP Q\<^sub>p) (\<iota> a) n "
         by(rule poly_inc_monom, rule A)
-      have F3: "(poly_inc f \<otimes>\<^bsub>UP Q\<^sub>p\<^esub> poly_inc (monom (UP Z\<^sub>p) a n)) m 
+      have F3: "(poly_inc f \<otimes>\<^bsub>UP Q\<^sub>p\<^esub> poly_inc (monom (UP Z\<^sub>p) a n)) m
                 = (\<iota> a) \<otimes> (poly_inc f (m -n))"
-        using UPQ.cfs_monom_mult_l' F0 A assms poly_inc_closed 
+        using UPQ.cfs_monom_mult_l' F0 A assms poly_inc_closed
         by (simp add: F2 inc_closed)
-      show ?thesis 
-        unfolding F3 unfolding poly_inc_def F1 
+      show ?thesis
+        unfolding F3 unfolding poly_inc_def F1
         apply(rule inc_of_prod, rule A)
         using assms Zp.cfs_closed by blast
     qed
   qed
 qed
-      
+
 lemma poly_inc_one:
 "poly_inc (\<one>\<^bsub>UP Z\<^sub>p\<^esub>) = \<one>\<^bsub>UP Q\<^sub>p\<^esub>"
 apply(rule ext)
-  unfolding poly_inc_def 
-  using inc_of_one inc_of_zero  
+  unfolding poly_inc_def
+  using inc_of_one inc_of_zero
   by simp
 
 lemma poly_inc_zero:
 "poly_inc (\<zero>\<^bsub>UP Z\<^sub>p\<^esub>) = \<zero>\<^bsub>UP Q\<^sub>p\<^esub>"
 apply(rule ext)
-  unfolding poly_inc_def 
-  using inc_of_one inc_of_zero  
+  unfolding poly_inc_def
+  using inc_of_one inc_of_zero
   by simp
 
-lemma poly_inc_hom: 
+lemma poly_inc_hom:
 "poly_inc \<in> ring_hom (UP Z\<^sub>p) (UP Q\<^sub>p)"
   apply(rule ring_hom_memI)
      apply(rule poly_inc_closed, blast)
     apply(rule poly_inc_times, blast, blast)
    apply(rule poly_inc_plus, blast, blast)
   by(rule poly_inc_one)
 
 lemma poly_inc_as_poly_lift_hom:
   assumes "f \<in> carrier (UP Z\<^sub>p)"
   shows "poly_inc f = poly_lift_hom Z\<^sub>p Q\<^sub>p \<iota> f"
   apply(rule ext)
-  unfolding poly_inc_def 
+  unfolding poly_inc_def
   using Zp.poly_lift_hom_cf[of Q\<^sub>p \<iota> f] assms UPQ.R_cring local.inc_is_hom
   by blast
 
 lemma poly_inc_eval:
   assumes "g \<in> carrier (UP Z\<^sub>p)"
   assumes "a \<in> carrier Z\<^sub>p"
   shows "to_function Q\<^sub>p (poly_inc g) (\<iota> a) = \<iota> (to_function Z\<^sub>p g a)"
-proof- 
+proof-
   have 0: "poly_inc g = poly_lift_hom Z\<^sub>p Q\<^sub>p \<iota> g"
-    using assms poly_inc_as_poly_lift_hom[of g] by blast 
+    using assms poly_inc_as_poly_lift_hom[of g] by blast
   have 1: "to_function Q\<^sub>p (poly_lift_hom Z\<^sub>p Q\<^sub>p \<iota> g) (\<iota> a) = \<iota> (to_function Z\<^sub>p g a)"
     using Zp.poly_lift_hom_eval[of Q\<^sub>p \<iota> g a] assms inc_is_hom
-    unfolding to_fun_def Zp.to_fun_def 
+    unfolding to_fun_def Zp.to_fun_def
     using UPQ.R_cring by blast
-  show ?thesis unfolding 0 1 
-    by blast 
+  show ?thesis unfolding 0 1
+    by blast
 qed
 
 lemma val_ring_poly_eval:
   assumes "f \<in> carrier (UP Q\<^sub>p)"
   assumes "\<And> i. f i \<in> \<O>\<^sub>p"
   shows "\<And>x. x \<in> \<O>\<^sub>p \<Longrightarrow> f \<bullet> x \<in> \<O>\<^sub>p"
   apply(rule positive_gauss_norm_eval, rule assms)
   apply(rule val_ring_cfs_imp_nonneg_gauss_norm)
-  using assms by auto 
+  using assms by auto
 
 lemma Zp_res_of_pow:
   assumes "a \<in> carrier Z\<^sub>p"
   assumes "b \<in> carrier Z\<^sub>p"
   assumes "a n = b n"
   shows "(a[^]\<^bsub>Z\<^sub>p\<^esub>(k::nat)) n = (b[^]\<^bsub>Z\<^sub>p\<^esub>(k::nat)) n"
   apply(induction k)
-  using assms Group.nat_pow_0 to_Zp_one apply metis 
+  using assms Group.nat_pow_0 to_Zp_one apply metis
   using Zp.geometric_series_id[of a b] Zp_residue_mult_zero(1) assms(1) assms(2) assms(3)
-    pow_closed res_diff_zero_fact'' res_diff_zero_fact(1) by metis 
+    pow_closed res_diff_zero_fact'' res_diff_zero_fact(1) by metis
 
 lemma to_Zp_nat_pow:
   assumes "a \<in> \<O>\<^sub>p"
   shows "to_Zp (a[^](n::nat)) = (to_Zp a)[^]\<^bsub>Z\<^sub>p\<^esub>(n::nat)"
   apply(induction n)
   using assms Group.nat_pow_0 to_Zp_one apply metis
-  using assms to_Zp_mult[of a] Qp.m_comm Qp.nat_pow_Suc val_ring_memE pow_suc to_Zp_closed val_ring_nat_pow_closed 
-  by metis 
+  using assms to_Zp_mult[of a] Qp.m_comm Qp.nat_pow_Suc val_ring_memE pow_suc to_Zp_closed val_ring_nat_pow_closed
+  by metis
 
 lemma  to_Zp_res_of_pow:
   assumes "a \<in> \<O>\<^sub>p"
   assumes "b \<in> \<O>\<^sub>p"
   assumes "to_Zp a n = to_Zp b n"
   shows "to_Zp (a[^](k::nat)) n = to_Zp (b[^](k::nat)) n"
   using assms val_ring_memE Zp_res_of_pow to_Zp_closed to_Zp_nat_pow by presburger
 
 lemma poly_eval_cong:
   assumes "g \<in> carrier (UP Q\<^sub>p)"
   assumes "\<And>i. g i \<in> \<O>\<^sub>p"
   assumes "a \<in> \<O>\<^sub>p"
   assumes "b \<in> \<O>\<^sub>p"
   assumes "to_Zp a k = to_Zp b k"
   shows "to_Zp (g \<bullet> a) k = to_Zp (g \<bullet> b) k"
 proof-
   have "(\<forall>i. g i \<in> \<O>\<^sub>p) \<longrightarrow> to_Zp (g \<bullet> a) k = to_Zp (g \<bullet> b) k"
   proof(rule UPQ.poly_induct[of g])
     show " g \<in> carrier (UP Q\<^sub>p)"
       using assms by blast
     show "\<And>p. p \<in> carrier (UP Q\<^sub>p) \<Longrightarrow> deg Q\<^sub>p p = 0 \<Longrightarrow> (\<forall>i. p i \<in> \<O>\<^sub>p) \<longrightarrow> to_Zp (p \<bullet> a) k = to_Zp (p \<bullet> b) k"
     proof fix p assume A: "p \<in> carrier (UP Q\<^sub>p)" "deg Q\<^sub>p p = 0" "\<forall>i. p i \<in> \<O>\<^sub>p"
       obtain c where c_def: "c \<in> carrier Q\<^sub>p \<and> p = up_ring.monom (UP Q\<^sub>p) c 0"
-        using A 
+        using A
         by (metis UPQ.zcf_degree_zero UPQ.cfs_closed UPQ.trms_of_deg_leq_0 UPQ.trms_of_deg_leq_degree_f)
       have p_eq: "p = up_ring.monom (UP Q\<^sub>p) c 0"
-        using c_def by blast 
+        using c_def by blast
       have p_cfs: "p 0 = c"
         unfolding p_eq using c_def UP_ring.cfs_monom[of Q\<^sub>p c 0 0] UPQ.P_is_UP_ring by presburger
       have c_closed: "c \<in> \<O>\<^sub>p"
         using p_cfs A(3) by blast
       have 0: "(p \<bullet> a) = c"
         unfolding p_eq using c_def assms by (meson UPQ.to_fun_const val_ring_memE(2))
       have 1: "(p \<bullet> b) = c"
         unfolding p_eq using c_def assms UPQ.to_fun_const val_ring_memE(2) by presburger
       show " to_Zp (p \<bullet> a) k = to_Zp (p \<bullet> b) k"
-        unfolding 0 1 by blast 
+        unfolding 0 1 by blast
     qed
     show "\<And>p. (\<And>q. q \<in> carrier (UP Q\<^sub>p) \<Longrightarrow> deg Q\<^sub>p q < deg Q\<^sub>p p \<Longrightarrow> (\<forall>i. q i \<in> \<O>\<^sub>p) \<longrightarrow> to_Zp (q \<bullet> a) k = to_Zp (q \<bullet> b) k) \<Longrightarrow>
          p \<in> carrier (UP Q\<^sub>p) \<Longrightarrow> 0 < deg Q\<^sub>p p \<Longrightarrow> (\<forall>i. p i \<in> \<O>\<^sub>p) \<longrightarrow> to_Zp (p \<bullet> a) k = to_Zp (p \<bullet> b) k"
-    proof 
+    proof
       fix p assume A: "(\<And>q. q \<in> carrier (UP Q\<^sub>p) \<Longrightarrow> deg Q\<^sub>p q < deg Q\<^sub>p p \<Longrightarrow> (\<forall>i. q i \<in> \<O>\<^sub>p) \<longrightarrow> to_Zp (q \<bullet> a) k = to_Zp (q \<bullet> b) k)"
                       "p \<in> carrier (UP Q\<^sub>p)" "0 < deg Q\<^sub>p p " " \<forall>i. p i \<in> \<O>\<^sub>p"
       obtain q where q_def: "q \<in> carrier (UP Q\<^sub>p) \<and> deg Q\<^sub>p q < deg Q\<^sub>p p \<and> p = UPQ.ltrm p \<oplus>\<^bsub>UP Q\<^sub>p\<^esub>q"
         by (metis A(2) A(3) UPQ.ltrm_closed UPQ.ltrm_decomp UPQ.UP_a_comm)
       have 0: "\<And>i.  p i = q i \<oplus> UPQ.ltrm p i"
-        using q_def A 
+        using q_def A
         by (metis Qp.a_ac(2) UPQ.ltrm_closed UPQ.UP_car_memE(1) UPQ.cfs_add)
       have 1: "\<forall>i. q i \<in> \<O>\<^sub>p"
-      proof fix i 
+      proof fix i
         show "q i \<in> \<O>\<^sub>p"
           apply(cases "i < deg Q\<^sub>p p")
-          using 0[of i] A(4) A(2) q_def 
-          using UPQ.ltrm_closed UPQ.P.a_ac(2) UPQ.trunc_cfs UPQ.trunc_closed UPQ.trunc_simps(1) 
+          using 0[of i] A(4) A(2) q_def
+          using UPQ.ltrm_closed UPQ.P.a_ac(2) UPQ.trunc_cfs UPQ.trunc_closed UPQ.trunc_simps(1)
            apply (metis Qp.r_zero UPQ.ltrm_cfs UPQ.cfs_closed UPQ.deg_leE)
-          using q_def 
+          using q_def
           by (metis (no_types, opaque_lifting) A(2) A(4) UPQ.P.add.m_closed UPQ.coeff_of_sum_diff_degree0 UPQ.deg_leE UPQ.equal_deg_sum UPQ.equal_deg_sum' \<open>\<And>thesis. (\<And>q. q \<in> carrier (UP Q\<^sub>p) \<and> deg Q\<^sub>p q < deg Q\<^sub>p p \<and> p = up_ring.monom (UP Q\<^sub>p) (p (deg Q\<^sub>p p)) (deg Q\<^sub>p p) \<oplus>\<^bsub>UP Q\<^sub>p\<^esub> q \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close> lessI linorder_neqE_nat)
       qed
       have 2: "UPQ.lcf p \<in> \<O>\<^sub>p"
-        using A(4) by blast 
+        using A(4) by blast
       have 3: "UPQ.ltrm p \<bullet> a = UPQ.lcf p \<otimes> a[^] deg Q\<^sub>p p"
-        apply(rule UP_cring.to_fun_monom) unfolding UP_cring_def 
-        using Qp.cring apply blast
-        using A UPQ.lcf_closed apply blast
-        using assms val_ring_memE(2) by blast
+        apply(rule UP_cring.to_fun_monom) unfolding UP_cring_def
+          apply (simp add: UPQ.R_cring)
+         apply (simp add: A(2) UPQ.cfs_closed)
+        using assms(3) val_ring_memE(2) by blast
       have 4: "UPQ.ltrm p \<bullet> b = UPQ.lcf p \<otimes> b[^] deg Q\<^sub>p p"
-        apply(rule UP_cring.to_fun_monom) unfolding UP_cring_def 
-        using Qp.cring apply blast
-        using A UPQ.lcf_closed apply blast
+        apply(rule UP_cring.to_fun_monom) unfolding UP_cring_def
+          apply (simp add: UPQ.R_cring)
+         apply (simp add: A(2) UPQ.cfs_closed)
         using assms val_ring_memE(2) by blast
       have p_eq: "p = q \<oplus>\<^bsub>UP Q\<^sub>p\<^esub> UPQ.ltrm p"
         using q_def by (metis A(2) UPQ.ltrm_closed UPQ.UP_a_comm)
       have 5: "p \<bullet> a = q \<bullet> a \<oplus>  UPQ.lcf p \<otimes> a[^] deg Q\<^sub>p p"
-        using assms val_ring_memE(2) p_eq q_def UPQ.to_fun_plus[of q "UPQ.ltrm p" a] 
+        using assms val_ring_memE(2) p_eq q_def UPQ.to_fun_plus[of q "UPQ.ltrm p" a]
         by (metis "3" A(2) UPQ.ltrm_closed UPQ.to_fun_plus)
       have 6: "p \<bullet> b = q \<bullet> b \<oplus>  UPQ.lcf p \<otimes> b[^] deg Q\<^sub>p p"
-        using assms val_ring_memE(2) p_eq q_def UPQ.to_fun_plus[of q "UPQ.ltrm p" a] 
+        using assms val_ring_memE(2) p_eq q_def UPQ.to_fun_plus[of q "UPQ.ltrm p" a]
         by (metis "4" A(2) UPQ.ltrm_closed UPQ.to_fun_plus)
       have 7: "UPQ.lcf p \<otimes> b[^] deg Q\<^sub>p p \<in> \<O>\<^sub>p"
         apply(rule val_ring_times_closed)
         using "2" apply linarith
         by(rule val_ring_nat_pow_closed, rule assms)
       have 8: "UPQ.lcf p \<otimes> a[^] deg Q\<^sub>p p \<in> \<O>\<^sub>p"
         apply(rule val_ring_times_closed)
         using "2" apply linarith
         by(rule val_ring_nat_pow_closed, rule assms)
       have 9: "q \<bullet> a \<in> \<O>\<^sub>p"
         using q_def 1 assms(3) val_ring_poly_eval by blast
       have 10: "q \<bullet> b \<in> \<O>\<^sub>p"
-        using q_def 1 assms(4) val_ring_poly_eval by blast 
+        using q_def 1 assms(4) val_ring_poly_eval by blast
       have 11: "to_Zp (p \<bullet> a) = to_Zp (q \<bullet> a) \<oplus>\<^bsub>Z\<^sub>p\<^esub> to_Zp (UPQ.ltrm p \<bullet> a)"
         using 5 8 9 to_Zp_add 3 by presburger
       have 12: "to_Zp (p \<bullet> b) = to_Zp (q \<bullet> b) \<oplus>\<^bsub>Z\<^sub>p\<^esub> to_Zp (UPQ.ltrm p \<bullet> b)"
         using 6 10 7 to_Zp_add 4  by presburger
       have 13: "to_Zp (p \<bullet> a) k = to_Zp (q \<bullet> a) k \<oplus>\<^bsub>Zp_res_ring k\<^esub> to_Zp (UPQ.ltrm p \<bullet> a) k"
         unfolding 11 using residue_of_sum by blast
       have 14: "to_Zp (p \<bullet> b) k = to_Zp (q \<bullet> b) k \<oplus>\<^bsub>Zp_res_ring k\<^esub> to_Zp (UPQ.ltrm p \<bullet> b) k"
         unfolding 12 using residue_of_sum by blast
       have 15: "to_Zp (UPQ.ltrm p \<bullet> a) k = to_Zp (UPQ.ltrm p \<bullet> b) k"
       proof(cases "k = 0")
         case True
         have T0: "to_Zp (UPQ.ltrm p \<bullet> a) \<in> carrier Z\<^sub>p"
           unfolding 3 using 8  to_Zp_closed val_ring_memE(2) by blast
         have T1: "to_Zp (UPQ.ltrm p \<bullet> b) \<in> carrier Z\<^sub>p"
           unfolding 4 using 7 to_Zp_closed val_ring_memE(2) by blast
-        show ?thesis unfolding True using T0 T1 padic_integers.p_res_ring_0 
-          by (metis p_res_ring_0' residues_closed)       
+        show ?thesis unfolding True using T0 T1 padic_integers.p_res_ring_0
+          by (metis p_res_ring_0' residues_closed)
       next
         case False
         have k_pos: "k > 0"
-          using False by presburger 
+          using False by presburger
         have 150: "to_Zp (p (deg Q\<^sub>p p) \<otimes> a [^] deg Q\<^sub>p p) = to_Zp (p (deg Q\<^sub>p p)) \<otimes>\<^bsub>Z\<^sub>p\<^esub> to_Zp( a [^] deg Q\<^sub>p p)"
-         apply(rule to_Zp_mult) 
+         apply(rule to_Zp_mult)
           using "2" apply blast
          by(rule val_ring_nat_pow_closed, rule assms)
         have 151: "to_Zp (p (deg Q\<^sub>p p) \<otimes> b [^] deg Q\<^sub>p p) = to_Zp (p (deg Q\<^sub>p p)) \<otimes>\<^bsub>Z\<^sub>p\<^esub> to_Zp( b [^] deg Q\<^sub>p p)"
-         apply(rule to_Zp_mult) 
+         apply(rule to_Zp_mult)
           using "2" apply blast
          by(rule val_ring_nat_pow_closed, rule assms)
        have 152: "to_Zp (p (deg Q\<^sub>p p) \<otimes> a [^] deg Q\<^sub>p p) k = to_Zp (p (deg Q\<^sub>p p)) k \<otimes>\<^bsub>Zp_res_ring k\<^esub> to_Zp( a [^] deg Q\<^sub>p p) k"
          unfolding 150 using residue_of_prod by blast
        have 153: "to_Zp (p (deg Q\<^sub>p p) \<otimes> b [^] deg Q\<^sub>p p) k = to_Zp (p (deg Q\<^sub>p p)) k \<otimes>\<^bsub>Zp_res_ring k\<^esub> to_Zp( b [^] deg Q\<^sub>p p) k"
          unfolding 151 using residue_of_prod by blast
        have 154: "to_Zp( a [^] deg Q\<^sub>p p) k = to_Zp a k [^]\<^bsub>Zp_res_ring k\<^esub> deg Q\<^sub>p p"
-       proof- 
+       proof-
        have 01: "\<And>m::nat. to_Zp (a[^]m) k = to_Zp a k [^]\<^bsub>Zp_res_ring k\<^esub> m"
        proof-
          fix m::nat show "to_Zp (a [^] m) k = to_Zp a k [^]\<^bsub>Zp_res_ring k\<^esub> m"
        proof-
          have 00: "to_Zp (a[^]m) = to_Zp a [^]\<^bsub>Z\<^sub>p\<^esub> m"
          using assms to_Zp_nat_pow[of a "m"] by blast
        have 01: "to_Zp a \<in> carrier Z\<^sub>p"
-         using assms to_Zp_closed val_ring_memE(2) by blast 
+         using assms to_Zp_closed val_ring_memE(2) by blast
        have 02: "to_Zp a k \<in> carrier (Zp_res_ring k)"
          using 01 residues_closed by blast
        have 03: "cring (Zp_res_ring k)"
          using k_pos padic_integers.R_cring padic_integers_axioms by blast
        have 01: "(to_Zp a [^]\<^bsub>Z\<^sub>p\<^esub> m) k = (to_Zp a) k [^]\<^bsub>Zp_res_ring k\<^esub> m"
          apply(induction m)
          using 01 02 apply (metis Group.nat_pow_0 k_pos residue_of_one(1))
-         using residue_of_prod[of "to_Zp a [^]\<^bsub>Z\<^sub>p\<^esub> m" "to_Zp a" k] 01 02 03 
+         using residue_of_prod[of "to_Zp a [^]\<^bsub>Z\<^sub>p\<^esub> m" "to_Zp a" k] 01 02 03
        proof -
          fix ma :: nat
          assume "(to_Zp a [^]\<^bsub>Z\<^sub>p\<^esub> ma) k = to_Zp a k [^]\<^bsub>Zp_res_ring k\<^esub> ma"
          then show "(to_Zp a [^]\<^bsub>Z\<^sub>p\<^esub> Suc ma) k = to_Zp a k [^]\<^bsub>Zp_res_ring k\<^esub> Suc ma"
            by (metis (no_types) Group.nat_pow_Suc residue_of_prod)
        qed
-       show ?thesis unfolding 00 01 by blast 
+       show ?thesis unfolding 00 01 by blast
        qed
        qed
-       thus ?thesis by blast       
+       thus ?thesis by blast
        qed
        have 155: "to_Zp( b [^] deg Q\<^sub>p p) k = to_Zp b k [^]\<^bsub>Zp_res_ring k\<^esub> deg Q\<^sub>p p"
          using assms by (metis "154" to_Zp_res_of_pow)
        show ?thesis
-         unfolding 3 4 152 153 154 155 assms by blast 
+         unfolding 3 4 152 153 154 155 assms by blast
      qed
      show "to_Zp (p \<bullet> a) k = to_Zp (p \<bullet> b) k"
        unfolding 13 14 15 using A 1 q_def by presburger
    qed
   qed
-  thus ?thesis using assms by blast 
+  thus ?thesis using assms by blast
 qed
 
 lemma to_Zp_poly_eval:
   assumes "g \<in> carrier Q\<^sub>p_x"
   assumes "gauss_norm g \<ge> 0"
   assumes "a \<in> \<O>\<^sub>p"
   shows "to_Zp (to_function Q\<^sub>p g a) = to_function Z\<^sub>p (to_Zp_poly g) (to_Zp a)"
-proof- 
+proof-
   obtain h where h_def: "h = to_Zp_poly g"
-    by blast 
+    by blast
   obtain b where b_def: "b = to_Zp a"
-    by blast 
+    by blast
   have h_poly_inc: "poly_inc h = g"
-    unfolding h_def using assms 
+    unfolding h_def using assms
     by (simp add: poly_inc_inverse_left)
   have b_inc: "\<iota> b = a"
-    unfolding b_def using assms 
+    unfolding b_def using assms
     by (simp add: to_Zp_inc)
   have h_closed: "h \<in> carrier (UP Z\<^sub>p)"
-    unfolding h_def using assms 
+    unfolding h_def using assms
     by (simp add: to_Zp_poly_closed)
   have b_closed: "b \<in> carrier Z\<^sub>p"
-    unfolding b_def using assms 
+    unfolding b_def using assms
     by (simp add: to_Zp_closed val_ring_memE)
   have 0: "to_function Q\<^sub>p (poly_inc h) (\<iota> b) = \<iota> (to_function Z\<^sub>p h b)"
     apply(rule poly_inc_eval)
     using h_def assms apply (simp add: to_Zp_poly_closed; fail)
-    unfolding b_def using assms 
+    unfolding b_def using assms
     by (simp add: to_Zp_closed val_ring_memE)
   have 1: "to_Zp (to_function Q\<^sub>p (poly_inc h) (\<iota> b)) = to_function Z\<^sub>p h b"
-    unfolding 0 
+    unfolding 0
     using h_closed b_closed Zp.to_fun_closed Zp.to_fun_def inc_to_Zp by auto
-  show ?thesis 
-    using 1 unfolding h_poly_inc b_inc 
-    unfolding h_def b_def by blast 
+  show ?thesis
+    using 1 unfolding h_poly_inc b_inc
+    unfolding h_def b_def by blast
 qed
 
 lemma poly_eval_equal_val:
   assumes "g \<in> carrier (UP Q\<^sub>p)"
   assumes "\<And>x. g x \<in> \<O>\<^sub>p"
   assumes "a \<in> \<O>\<^sub>p"
   assumes "b \<in> \<O>\<^sub>p"
   assumes "val (g \<bullet> a) < eint n"
   assumes "to_Zp a n = to_Zp b n"
   shows "val (g \<bullet> b) = val (g \<bullet> a)"
 proof-
   have "(\<forall>x. g x \<in> \<O>\<^sub>p) \<longrightarrow> to_Zp (g \<bullet> b) n = to_Zp (g \<bullet> a) n"
   proof(rule poly_induct[of g])
     show "g \<in> carrier (UP Q\<^sub>p)"
       by (simp add: assms(1))
     show "\<And>p. p \<in> carrier (UP Q\<^sub>p) \<Longrightarrow> deg Q\<^sub>p p = 0 \<Longrightarrow> (\<forall>x. p x \<in> \<O>\<^sub>p) \<longrightarrow> to_Zp (p \<bullet> b) n = to_Zp (p \<bullet> a) n"
     proof fix p assume A: "p \<in> carrier (UP Q\<^sub>p)" " deg Q\<^sub>p p = 0 " "\<forall>x. p x \<in> \<O>\<^sub>p "
       show "to_Zp (p \<bullet> b) n = to_Zp (p \<bullet> a) n"
         using A  by (metis val_ring_memE UPQ.to_fun_ctrm UPQ.trms_of_deg_leq_0 UPQ.trms_of_deg_leq_degree_f assms(3) assms(4))
     qed
     show "\<And>p. (\<And>q. q \<in> carrier (UP Q\<^sub>p) \<Longrightarrow> deg Q\<^sub>p q < deg Q\<^sub>p p \<Longrightarrow> (\<forall>x. q x \<in> \<O>\<^sub>p) \<longrightarrow> to_Zp (q \<bullet> b) n = to_Zp (q \<bullet> a) n) \<Longrightarrow>
          p \<in> carrier (UP Q\<^sub>p) \<Longrightarrow> 0 < deg Q\<^sub>p p \<Longrightarrow> (\<forall>x. p x \<in> \<O>\<^sub>p) \<longrightarrow> to_Zp (p \<bullet> b) n = to_Zp (p \<bullet> a) n"
     proof fix p assume IH: "(\<And>q. q \<in> carrier (UP Q\<^sub>p) \<Longrightarrow> deg Q\<^sub>p q < deg Q\<^sub>p p \<Longrightarrow> (\<forall>x. q x \<in> \<O>\<^sub>p) \<longrightarrow> to_Zp (q \<bullet> b) n = to_Zp (q \<bullet> a) n)"
       assume A: "p \<in> carrier (UP Q\<^sub>p)" "0 < deg Q\<^sub>p p" "\<forall>x. p x \<in> \<O>\<^sub>p"
       show "to_Zp (p \<bullet> b) n = to_Zp (p \<bullet> a) n"
       proof-
         obtain q where q_def: "q \<in> carrier (UP Q\<^sub>p) \<and> deg Q\<^sub>p q < deg Q\<^sub>p p \<and>
                       p = q \<oplus>\<^bsub>UP Q\<^sub>p\<^esub> ltrm p"
           using A  by (meson UPQ.ltrm_decomp)
         have p_eq: "p = q \<oplus>\<^bsub>UP Q\<^sub>p\<^esub> ltrm p"
-          using q_def by blast 
+          using q_def by blast
         have "\<forall>x. q x \<in> \<O>\<^sub>p" proof fix x
           have px: "p x = (q \<oplus>\<^bsub>UP Q\<^sub>p\<^esub> ltrm p) x"
-            using p_eq by simp 
+            using p_eq by simp
           show "q x \<in> \<O>\<^sub>p"
           proof(cases "x \<le> deg Q\<^sub>p q")
             case True
-            then have "p x = q x"           
-              unfolding px using q_def A 
+            then have "p x = q x"
+              unfolding px using q_def A
               by (smt UPQ.ltrm_closed UPQ.P.add.right_cancel UPQ.coeff_of_sum_diff_degree0 UPQ.deg_ltrm UPQ.trunc_cfs UPQ.trunc_closed UPQ.trunc_simps(1) less_eq_Suc_le nat_neq_iff not_less_eq_eq)
-            then show ?thesis using A 
+            then show ?thesis using A
               by blast
           next
             case False
-            then show ?thesis 
+            then show ?thesis
               using q_def UPQ.deg_eqI eq_imp_le nat_le_linear zero_in_val_ring
               by (metis (no_types, lifting) UPQ.coeff_simp UPQ.deg_belowI)
           qed
         qed
         then have 0: " to_Zp (q \<bullet> b) n = to_Zp (q \<bullet> a) n"
           using IH q_def by blast
         have 1: "to_Zp (ltrm p \<bullet> b) n = to_Zp (ltrm p \<bullet> a) n"
         proof-
           have 10: "(ltrm p \<bullet> b) = (p (deg Q\<^sub>p p)) \<otimes> b[^] (deg Q\<^sub>p p)"
             using assms A  by (meson val_ring_memE UPQ.to_fun_monom)
           have 11: "(ltrm p \<bullet> a) = (p (deg Q\<^sub>p p)) \<otimes> a[^] (deg Q\<^sub>p p)"
             using assms A by (meson val_ring_memE UPQ.to_fun_monom)
           have 12: "to_Zp (b[^] (deg Q\<^sub>p p)) n = to_Zp (a[^] (deg Q\<^sub>p p)) n"
             using to_Zp_res_of_pow assms by metis
           have 13: "p (deg Q\<^sub>p p) \<in> \<O>\<^sub>p"
             using A(3) by blast
           have 14: "b[^] (deg Q\<^sub>p p) \<in> \<O>\<^sub>p"
             using assms(4) val_ring_nat_pow_closed by blast
           have 15: "a[^] (deg Q\<^sub>p p) \<in> \<O>\<^sub>p"
             using assms(3) val_ring_nat_pow_closed by blast
           have 16: "(ltrm p \<bullet> b) \<in> \<O>\<^sub>p"
             by (simp add: "10" "13" "14" val_ring_times_closed)
           have 17: "to_Zp (ltrm p \<bullet> b) n = to_Zp (p (deg Q\<^sub>p p)) n \<otimes>\<^bsub>Zp_res_ring n\<^esub> to_Zp (b[^] (deg Q\<^sub>p p)) n"
-            using 10 13 14 15 16 assms residue_of_prod to_Zp_mult by presburger 
+            using 10 13 14 15 16 assms residue_of_prod to_Zp_mult by presburger
           have 18: "(ltrm p \<bullet> a) \<in> \<O>\<^sub>p"
             by (simp add: "11" "15" A(3) val_ring_times_closed)
           have 19: "to_Zp (ltrm p \<bullet> a) n = to_Zp (p (deg Q\<^sub>p p)) n \<otimes>\<^bsub>Zp_res_ring n\<^esub> to_Zp (a[^] (deg Q\<^sub>p p)) n"
             using 10 13 14 15 16 17 18 assms residue_of_prod to_Zp_mult 11  by presburger
           show ?thesis using 12 17 19 by presburger
         qed
         have 2: "p (deg Q\<^sub>p p) \<in> \<O>\<^sub>p"
           using A(3) by blast
         have 3: "(ltrm p \<bullet> b) \<in> \<O>\<^sub>p"
-          using 2 assms  
-          by (metis A(1) Q\<^sub>p_def val_ring_memE val_ring_memE UPQ.ltrm_closed Zp_def \<iota>_def 
+          using 2 assms
+          by (metis A(1) Q\<^sub>p_def val_ring_memE val_ring_memE UPQ.ltrm_closed Zp_def \<iota>_def
               gauss_norm_monom padic_fields.positive_gauss_norm_eval padic_fields_axioms)
         have 4: "(ltrm p \<bullet> a) \<in> \<O>\<^sub>p"
-          using 2 assms 
+          using 2 assms
           by (metis A(1) Q\<^sub>p_def val_ring_memE val_ring_memE UPQ.ltrm_closed Zp_def \<iota>_def
               gauss_norm_monom padic_fields.positive_gauss_norm_eval padic_fields_axioms)
         have 5: "(q \<bullet> b) \<in> \<O>\<^sub>p"
-          using  \<open>\<forall>x. q x \<in> \<O>\<^sub>p\<close> assms(4) q_def     
+          using  \<open>\<forall>x. q x \<in> \<O>\<^sub>p\<close> assms(4) q_def
           by (metis gauss_norm_coeff_norm positive_gauss_norm_eval val_ring_memE(1))
         have 6: "(q \<bullet> a) \<in> \<O>\<^sub>p"
-          using  \<open>\<forall>x. q x \<in> \<O>\<^sub>p\<close> assms(3) q_def     
+          using  \<open>\<forall>x. q x \<in> \<O>\<^sub>p\<close> assms(3) q_def
           by (metis gauss_norm_coeff_norm positive_gauss_norm_eval val_ring_memE(1))
         have 7: "to_Zp (p \<bullet> b) = to_Zp (ltrm p \<bullet> b)  \<oplus>\<^bsub>Z\<^sub>p\<^esub> to_Zp (q \<bullet> b)"
           using 5 3 q_def by (metis (no_types, lifting) A(1) val_ring_memE UPQ.ltrm_closed UPQ.to_fun_plus add_comm assms(4) to_Zp_add)
         have 8: "to_Zp (p \<bullet> a) = to_Zp (ltrm p \<bullet> a)  \<oplus>\<^bsub>Z\<^sub>p\<^esub> to_Zp (q \<bullet> a)"
           using 4 6 q_def by (metis (no_types, lifting) A(1) val_ring_memE UPQ.ltrm_closed UPQ.to_fun_plus add_comm assms(3) to_Zp_add)
         have 9: "to_Zp (p \<bullet> b) \<in> carrier Z\<^sub>p"
           using A assms by (meson val_ring_memE UPQ.to_fun_closed to_Zp_closed)
         have 10: "to_Zp (p \<bullet> a) \<in> carrier Z\<^sub>p"
           using A assms val_ring_memE UPQ.to_fun_closed to_Zp_closed by presburger
         have 11: "to_Zp (p \<bullet> b) n = to_Zp (ltrm p \<bullet> b) n  \<oplus>\<^bsub>Zp_res_ring n\<^esub> to_Zp (q \<bullet> b) n"
           using 7 9 5 3 residue_of_sum by presburger
         have 12: "to_Zp (p \<bullet> a) n = to_Zp (ltrm p \<bullet> a) n \<oplus>\<^bsub>Zp_res_ring n\<^esub> to_Zp (q \<bullet> a) n"
           using 8 6 4 residue_of_sum by presburger
-        show ?thesis using 0 11 12 q_def assms 
+        show ?thesis using 0 11 12 q_def assms
           using "1" by presburger
       qed
     qed
   qed
   have "(\<forall>x. g x \<in> \<O>\<^sub>p) "
     using assms by blast
   hence 0: "to_Zp (g \<bullet> b) n = to_Zp (g \<bullet> a) n"
     using \<open>(\<forall>x. g x \<in> \<O>\<^sub>p) \<longrightarrow> to_Zp (g \<bullet> b) n = to_Zp (g \<bullet> a) n\<close> by blast
   have 1: "g \<bullet> a \<in> \<O>\<^sub>p"
-    using  assms(1) assms(2) assms(3) 
+    using  assms(1) assms(2) assms(3)
     by (metis gauss_norm_coeff_norm positive_gauss_norm_eval val_ring_memE(1))
   have 2: "g \<bullet> b \<in> \<O>\<^sub>p"
-    using  assms(1) assms(2) assms(4) 
+    using  assms(1) assms(2) assms(4)
     by (metis gauss_norm_coeff_norm positive_gauss_norm_eval val_ring_memE(1))
   have 3: "val (g \<bullet> b) < eint n"
   proof-
     have P0: "to_Zp (g \<bullet> a) \<in> carrier Z\<^sub>p"
       using 1 val_ring_memE to_Zp_closed by blast
     have P1: "to_Zp (g \<bullet> b) \<in> carrier Z\<^sub>p"
       using 2 val_ring_memE to_Zp_closed by blast
     have P2: "val_Zp (to_Zp (g \<bullet> a)) < n"
       using 1 assms to_Zp_val by presburger
     have P3: "to_Zp (g \<bullet> a) \<noteq> \<zero>\<^bsub>Z\<^sub>p\<^esub>"
       using P2 P0 unfolding val_Zp_def     by (metis P2 infinity_ilessE val_Zp_def)
     have P4: "(to_Zp (g \<bullet> a)) n \<noteq> 0"
-      using 1 P2 P3 above_ord_nonzero[of "to_Zp (g \<bullet> a)" n]  
+      using 1 P2 P3 above_ord_nonzero[of "to_Zp (g \<bullet> a)" n]
       by (metis P0 eint.inject less_eintE val_ord_Zp)
     then have "to_Zp (g \<bullet> b) n \<noteq> 0"
       using 0 by linarith
     then have "val_Zp (to_Zp (g \<bullet> b)) < n"
-      using P1 P0 
-      by (smt below_val_Zp_zero eint_ile eint_ord_simps(1) eint_ord_simps(2) nonzero_imp_ex_nonzero_res residue_of_zero(2) zero_below_val_Zp)  
-    then show ?thesis using 2 
+      using P1 P0
+      by (smt below_val_Zp_zero eint_ile eint_ord_simps(1) eint_ord_simps(2) nonzero_imp_ex_nonzero_res residue_of_zero(2) zero_below_val_Zp)
+    then show ?thesis using 2
       by (metis to_Zp_val)
   qed
   thus ?thesis using 0 1 2 assms val_ring_equal_res_imp_equal_val[of "g \<bullet> b" "g \<bullet> a" n] by blast
 qed
 
 lemma to_Zp_poly_monom:
   assumes "a \<in> \<O>\<^sub>p"
   shows "to_Zp_poly (monom (UP Q\<^sub>p) a n) = monom (UP Z\<^sub>p) (to_Zp a) n"
-  unfolding to_Zp_poly_def 
+  unfolding to_Zp_poly_def
   apply(rule ext)
-  using assms cfs_monom[of a n] Zp.cfs_monom[of "to_Zp a" n] 
+  using assms cfs_monom[of a n] Zp.cfs_monom[of "to_Zp a" n]
   by (simp add: to_Zp_closed to_Zp_zero val_ring_memE(2))
 
 lemma to_Zp_poly_add:
   assumes "f \<in> carrier (UP Q\<^sub>p)"
   assumes "gauss_norm f \<ge> 0"
   assumes "g \<in> carrier (UP Q\<^sub>p)"
   assumes "gauss_norm g \<ge> 0"
   shows "to_Zp_poly (f \<oplus>\<^bsub>UP Q\<^sub>p\<^esub> g) = to_Zp_poly f \<oplus>\<^bsub>UP Z\<^sub>p\<^esub> to_Zp_poly g"
-proof- 
+proof-
   obtain F where F_def: "F = to_Zp_poly f"
-    by blast 
+    by blast
   obtain G where G_def: "G = to_Zp_poly g"
-    by blast 
+    by blast
   have F_closed: "F \<in> carrier (UP Z\<^sub>p)"
-    unfolding F_def using assms 
+    unfolding F_def using assms
     by (simp add: to_Zp_poly_closed)
   have G_closed: "G \<in> carrier (UP Z\<^sub>p)"
-    unfolding G_def using assms 
+    unfolding G_def using assms
     by (simp add: to_Zp_poly_closed)
   have F_inc: "poly_inc F = f"
-    using assms unfolding F_def 
+    using assms unfolding F_def
     using poly_inc_inverse_left by blast
   have G_inc: "poly_inc G = g"
-    using assms unfolding G_def 
+    using assms unfolding G_def
     by (simp add: poly_inc_inverse_left)
   have 0: "poly_inc (F \<oplus>\<^bsub>UP Z\<^sub>p\<^esub> G) = poly_inc F \<oplus>\<^bsub>UP Q\<^sub>p\<^esub> poly_inc G"
-    using F_closed G_closed 
+    using F_closed G_closed
     by (simp add: poly_inc_plus)
   have 1: "to_Zp_poly (poly_inc (F \<oplus>\<^bsub>UP Z\<^sub>p\<^esub> G)) = F \<oplus>\<^bsub>UP Z\<^sub>p\<^esub> G"
-    using G_closed F_closed 
+    using G_closed F_closed
     by (simp add: poly_inc_inverse_right)
-  show ?thesis 
-    using  1 unfolding F_inc G_inc 0 unfolding F_def G_def 
-    by blast 
+  show ?thesis
+    using  1 unfolding F_inc G_inc 0 unfolding F_def G_def
+    by blast
 qed
 
 lemma to_Zp_poly_zero:
 "to_Zp_poly (\<zero>\<^bsub>UP Q\<^sub>p\<^esub>) = \<zero>\<^bsub>UP Z\<^sub>p\<^esub>"
-  unfolding to_Zp_poly_def 
+  unfolding to_Zp_poly_def
   apply(rule ext)
   by (simp add: to_Zp_zero)
 
 lemma to_Zp_poly_one:
 "to_Zp_poly (\<one>\<^bsub>UP Q\<^sub>p\<^esub>) = \<one>\<^bsub>UP Z\<^sub>p\<^esub>"
-  unfolding to_Zp_poly_def 
+  unfolding to_Zp_poly_def
   apply(rule ext)
-  by (metis Zp.UP_one_closed poly_inc_inverse_right poly_inc_one to_Zp_poly_def) 
+  by (metis Zp.UP_one_closed poly_inc_inverse_right poly_inc_one to_Zp_poly_def)
 
 lemma val_ring_add_pow:
   assumes "a \<in> carrier Q\<^sub>p"
   assumes "val a \<ge> 0"
   shows "val ([(n::nat)]\<cdot>a) \<ge> 0"
 proof-
   have 0: "[(n::nat)]\<cdot>a = ([n]\<cdot>\<one>)\<otimes>a"
     using assms Qp.add_pow_ldistr Qp.cring_simprules(12) Qp.one_closed by presburger
-  show ?thesis unfolding 0 using assms 
+  show ?thesis unfolding 0 using assms
     by (meson Qp.nat_inc_closed val_ring_memE val_of_nat_inc val_ringI val_ring_times_closed)
 qed
 
 lemma to_Zp_poly_pderiv:
   assumes "g \<in> carrier (UP Q\<^sub>p)"
   assumes "gauss_norm g \<ge> 0"
   shows "to_Zp_poly (pderiv g) = Zp.pderiv (to_Zp_poly g)"
-proof- 
+proof-
   have 0: "gauss_norm g \<ge> 0 \<longrightarrow> to_Zp_poly (pderiv g) = Zp.pderiv (to_Zp_poly g)"
   proof(rule poly_induct, rule assms, rule)
-    fix p 
+    fix p
     assume A: " p \<in> carrier (UP Q\<^sub>p)"
          "deg Q\<^sub>p p = 0"
          "0 \<le> gauss_norm p"
     obtain a where a_def: "a \<in> \<O>\<^sub>p \<and> p = monom (UP Q\<^sub>p) a 0"
-      using A 
+      using A
       by (metis UPQ.ltrm_deg_0 positive_gauss_norm_valuation_ring_coeffs)
     have p_eq: "p = monom (UP Q\<^sub>p) a 0"
-      using a_def by blast 
+      using a_def by blast
     have 0: "to_Zp_poly p = monom (UP Z\<^sub>p) (to_Zp a) 0"
       unfolding p_eq
       apply(rule to_Zp_poly_monom)
-      using a_def by blast 
+      using a_def by blast
     have 1: "UPQ.pderiv (monom (UP Q\<^sub>p) a 0) = \<zero>\<^bsub>UP Q\<^sub>p\<^esub>"
       using A(1) A(2) UPQ.pderiv_deg_0 p_eq by blast
     have 2: "Zp.pderiv (monom (UP Z\<^sub>p) (to_Zp a) 0) = \<zero>\<^bsub>UP Z\<^sub>p\<^esub>"
-      apply(rule Zp.pderiv_deg_0) 
+      apply(rule Zp.pderiv_deg_0)
        apply(rule Zp.monom_closed, rule to_Zp_closed)
-      using a_def 
+      using a_def
        apply (simp add: val_ring_memE(2); fail)
       apply(cases "to_Zp a = \<zero>\<^bsub>Z\<^sub>p\<^esub>")
       apply (simp; fail)
-      apply(rule Zp.deg_monom, blast) 
-      using a_def 
+      apply(rule Zp.deg_monom, blast)
+      using a_def
       by (simp add: to_Zp_closed val_ring_memE(2))
     show "to_Zp_poly (UPQ.pderiv p) = Zp.pderiv (to_Zp_poly p)"
-      unfolding 0 unfolding p_eq 
-      unfolding 1 2 to_Zp_poly_zero by blast 
-  next 
-    fix p 
+      unfolding 0 unfolding p_eq
+      unfolding 1 2 to_Zp_poly_zero by blast
+  next
+    fix p
     assume A: "\<And>q. q \<in> carrier (UP Q\<^sub>p) \<Longrightarrow>
               deg Q\<^sub>p q < deg Q\<^sub>p p \<Longrightarrow>
               0 \<le> gauss_norm q \<longrightarrow>
               to_Zp_poly (UPQ.pderiv q) = Zp.pderiv (to_Zp_poly q)"
               "p \<in> carrier (UP Q\<^sub>p)"
               " 0 < deg Q\<^sub>p p"
     show "0 \<le> gauss_norm p \<longrightarrow> to_Zp_poly (UPQ.pderiv p) = Zp.pderiv (to_Zp_poly p)"
-    proof 
+    proof
       assume B: "0 \<le> gauss_norm p"
       obtain q where q_def: "q = trunc p"
-        by blast 
+        by blast
       have p_eq: "p = q \<oplus>\<^bsub>UP Q\<^sub>p\<^esub> ltrm p"
         by (simp add: A(2) UPQ.trunc_simps(1) q_def)
       have q_gauss_norm:    "gauss_norm q \<ge> 0"
-        unfolding q_def 
+        unfolding q_def
         apply(rule gauss_norm_geqI)
         using A apply (simp add: UPQ.trunc_closed; fail)
-        using trunc_cfs[of p] A gauss_normE 
+        using trunc_cfs[of p] A gauss_normE
       proof -
         fix n :: nat
         have f1: "\<zero> = q (deg Q\<^sub>p p)"
           by (simp add: UPQ.deg_leE UPQ.trunc_closed UPQ.trunc_degree \<open>0 < deg Q\<^sub>p p\<close> \<open>p \<in> carrier (UP Q\<^sub>p)\<close> q_def)
         have "\<forall>n. 0 \<le> val (p n)"
           by (meson B \<open>p \<in> carrier (UP Q\<^sub>p)\<close> eint_ord_trans gauss_normE)
         then show "0 \<le> val (Cring_Poly.truncate Q\<^sub>p p n)"
           using f1 by (metis (no_types) Qp.nat_mult_zero UPQ.ltrm_closed UPQ.coeff_of_sum_diff_degree0 UPQ.deg_ltrm UPQ.trunc_closed \<open>\<And>n. \<lbrakk>p \<in> carrier (UP Q\<^sub>p); n < deg Q\<^sub>p p\<rbrakk> \<Longrightarrow> Cring_Poly.truncate Q\<^sub>p p n = p n\<close> \<open>p \<in> carrier (UP Q\<^sub>p)\<close> nat_neq_iff p_eq q_def val_of_nat_inc)
       qed
       have 0: "to_Zp_poly (UPQ.pderiv q) = Zp.pderiv (to_Zp_poly q)"
-        using A q_def q_gauss_norm 
+        using A q_def q_gauss_norm
         by (simp add: UPQ.trunc_closed UPQ.trunc_degree)
       have 1: "UPQ.pderiv (monom (UP Q\<^sub>p) (p (deg Q\<^sub>p p)) (deg Q\<^sub>p p)) =
                monom (UP Q\<^sub>p) ([deg Q\<^sub>p p] \<cdot> p (deg Q\<^sub>p p)) (deg Q\<^sub>p p - 1)"
         apply(rule pderiv_monom)
         using A by (simp add: UPQ.UP_car_memE(1))
       have 2: "Zp.pderiv (monom (UP Z\<^sub>p) (to_Zp (p (deg Q\<^sub>p p))) (deg Q\<^sub>p p)) =
     monom (UP Z\<^sub>p) ([deg Q\<^sub>p p] \<cdot>\<^bsub>Z\<^sub>p\<^esub> to_Zp ( p (deg Q\<^sub>p p))) (deg Q\<^sub>p p - 1)"
-        using A  Zp.pderiv_monom[of "to_Zp ( p (deg Q\<^sub>p p))" "deg Q\<^sub>p p"]  
+        using A  Zp.pderiv_monom[of "to_Zp ( p (deg Q\<^sub>p p))" "deg Q\<^sub>p p"]
         by (simp add: UPQ.lcf_closed to_Zp_closed)
       have 3: "to_Zp_poly (UPQ.pderiv (monom (UP Q\<^sub>p) (p (deg Q\<^sub>p p)) (deg Q\<^sub>p p))) = monom (UP Z\<^sub>p) (to_Zp ([deg Q\<^sub>p p] \<cdot> p (deg Q\<^sub>p p))) (deg Q\<^sub>p p - 1)"
         unfolding 1 apply(rule to_Zp_poly_monom)
         apply(rule val_ring_memI)
          apply (simp add: A(2) UPQ.UP_car_memE(1); fail)
         apply(rule val_ring_add_pow)
-        using A 
+        using A
         apply (simp add: UPQ.lcf_closed; fail)
-        using B A 
+        using B A
         by (simp add: positive_gauss_norm_valuation_ring_coeffs val_ring_memE(1))
       have 4: "to_Zp_poly (ltrm p) = monom (UP Z\<^sub>p) (to_Zp (p (deg Q\<^sub>p p))) (deg Q\<^sub>p p)"
-        apply(rule to_Zp_poly_monom) using A 
+        apply(rule to_Zp_poly_monom) using A
         by (simp add: B positive_gauss_norm_valuation_ring_coeffs)
       have 5: "to_Zp_poly (UPQ.pderiv (ltrm p)) = Zp.pderiv (to_Zp_poly (ltrm p))"
-        unfolding 3 4 2 
+        unfolding 3 4 2
         by (simp add: A(2) B positive_gauss_norm_valuation_ring_coeffs to_Zp_nat_add_pow)
       have 6: "pderiv p = pderiv q \<oplus>\<^bsub>UP Q\<^sub>p\<^esub> pderiv (ltrm p)"
-        using p_eq 
+        using p_eq
         by (metis A(2) UPQ.ltrm_closed UPQ.pderiv_add UPQ.trunc_closed p_eq q_def)
       have 7: "to_Zp_poly p = to_Zp_poly q \<oplus>\<^bsub>UP Z\<^sub>p\<^esub> to_Zp_poly (ltrm p)"
-        using p_eq 
+        using p_eq
         by (metis (no_types, lifting) A(2) B UPQ.ltrm_closed UPQ.cfs_closed UPQ.trunc_closed gauss_norm_monom positive_gauss_norm_valuation_ring_coeffs q_def q_gauss_norm to_Zp_poly_add val_ring_memE(1))
       have 8: "to_Zp_poly  (pderiv p) =
                 to_Zp_poly (UPQ.pderiv q) \<oplus>\<^bsub>UP Z\<^sub>p\<^esub>
                  to_Zp_poly (UPQ.pderiv (monom (UP Q\<^sub>p) (p (deg Q\<^sub>p p)) (deg Q\<^sub>p p)))"
         unfolding 6 apply(rule to_Zp_poly_add)
            apply (simp add: A(2) UPQ.pderiv_closed UPQ.trunc_closed q_def; fail)
           apply (metis A(2) UPQ.cfs_closed UPQ.pderiv_cfs UPQ.trunc_closed gauss_norm_coeff_norm positive_gauss_norm_valuation_ring_coeffs q_def q_gauss_norm val_ring_add_pow val_ring_memE(1))
          apply (simp add: A(2) UPQ.UP_car_memE(1) UPQ.pderiv_closed; fail)
         apply(rule eint_ord_trans[of _ "gauss_norm (monom (UP Q\<^sub>p) (p (deg Q\<^sub>p p)) (deg Q\<^sub>p p))"])
         apply (simp add: A(2) B UPQ.cfs_closed gauss_norm_monom positive_gauss_norm_valuation_ring_coeffs val_ring_memE(1); fail)
         apply(rule gauss_norm_pderiv)
         using A(2) UPQ.ltrm_closed by blast
       have 9: "Zp.pderiv  (to_Zp_poly p) =  Zp.pderiv (to_Zp_poly q) \<oplus>\<^bsub>UP Z\<^sub>p\<^esub>
          Zp.pderiv (to_Zp_poly (monom (UP Q\<^sub>p) (p (deg Q\<^sub>p p)) (deg Q\<^sub>p p)))"
           unfolding 7 apply(rule Zp.pderiv_add)
            apply(rule to_Zp_poly_closed)
             apply (simp add: A(2) UPQ.trunc_closed q_def; fail)
            apply (simp add: q_gauss_norm; fail)
            apply(rule to_Zp_poly_closed)
            apply (simp add: A(2) UPQ.UP_car_memE(1); fail)
           by (simp add: A(2) B UPQ.cfs_closed gauss_norm_monom positive_gauss_norm_valuation_ring_coeffs val_ring_memE(1))
       show "to_Zp_poly (UPQ.pderiv p) = Zp.pderiv (to_Zp_poly p)"
-          unfolding 9 8 5 0 by blast 
+          unfolding 9 8 5 0 by blast
     qed
   qed
-  thus ?thesis using assms by blast 
+  thus ?thesis using assms by blast
 qed
 
 lemma val_p_int_pow:
 "val (\<pp>[^]k) = eint (k)"
   by (simp add: ord_p_pow_int p_intpow_closed(2))
 
 definition int_gauss_norm where
 "int_gauss_norm g = (SOME n::int. eint n = gauss_norm g)"
 
-lemma int_gauss_norm_eq: 
+lemma int_gauss_norm_eq:
   assumes "g \<in> carrier (UP Q\<^sub>p)"
   assumes "g \<noteq> \<zero>\<^bsub>UP Q\<^sub>p\<^esub>"
   shows "eint (int_gauss_norm g) = gauss_norm g"
-proof- 
+proof-
   have 0: "gauss_norm g < \<infinity>"
     using assms by (simp add: gauss_norm_prop)
-  then show ?thesis unfolding int_gauss_norm_def 
-    using assms 
+  then show ?thesis unfolding int_gauss_norm_def
+    using assms
     by fastforce
 qed
 
 lemma int_gauss_norm_smult:
   assumes "g \<in> carrier (UP Q\<^sub>p)"
   assumes "g \<noteq> \<zero>\<^bsub>UP Q\<^sub>p\<^esub>"
   assumes "a \<in> nonzero Q\<^sub>p"
   shows "int_gauss_norm (a \<odot>\<^bsub>UP Q\<^sub>p\<^esub> g) = ord a + int_gauss_norm g"
-  using gauss_norm_smult[of g a] int_gauss_norm_eq val_ord assms 
+  using gauss_norm_smult[of g a] int_gauss_norm_eq val_ord assms
   by (metis (no_types, opaque_lifting) Qp.nonzero_closed UPQ.UP_smult_closed UPQ.cfs_zero
       eint.distinct(2) eint.inject gauss_norm_coeff_norm local.val_zero plus_eint_simps(1))
 
 definition normalize_poly where
 "normalize_poly g = (if g = \<zero>\<^bsub>UP Q\<^sub>p\<^esub> then g else (\<pp>[^](- int_gauss_norm g)) \<odot>\<^bsub>Q\<^sub>p_x\<^esub> g)"
 
-lemma normalize_poly_zero: 
+lemma normalize_poly_zero:
 "normalize_poly \<zero>\<^bsub>UP Q\<^sub>p\<^esub> = \<zero>\<^bsub>UP Q\<^sub>p\<^esub>"
-  unfolding normalize_poly_def by simp 
+  unfolding normalize_poly_def by simp
 
 lemma normalize_poly_nonzero_eq:
   assumes "g \<noteq> \<zero>\<^bsub>UP Q\<^sub>p\<^esub>"
   assumes "g \<in> carrier (UP Q\<^sub>p)"
   shows "normalize_poly g = (\<pp>[^](- int_gauss_norm g)) \<odot>\<^bsub>UP Q\<^sub>p\<^esub> g"
-  using assms unfolding normalize_poly_def by simp 
+  using assms unfolding normalize_poly_def by simp
 
 lemma int_gauss_norm_normalize_poly:
   assumes "g \<noteq> \<zero>\<^bsub>UP Q\<^sub>p\<^esub>"
   assumes "g \<in> carrier (UP Q\<^sub>p)"
   shows "int_gauss_norm (normalize_poly g) = 0"
-  using normalize_poly_nonzero_eq int_gauss_norm_smult assms 
+  using normalize_poly_nonzero_eq int_gauss_norm_smult assms
   by (simp add: ord_p_pow_int p_intpow_closed(2))
 
-lemma normalize_poly_closed: 
+lemma normalize_poly_closed:
   assumes "g \<in> carrier (UP Q\<^sub>p)"
   shows "normalize_poly g \<in> carrier (UP Q\<^sub>p)"
-  using assms unfolding normalize_poly_def 
+  using assms unfolding normalize_poly_def
   by (simp add: p_intpow_closed(1))
 
 lemma normalize_poly_nonzero:
   assumes "g \<noteq> \<zero>\<^bsub>UP Q\<^sub>p\<^esub>"
   assumes "g \<in> carrier (UP Q\<^sub>p)"
   shows "normalize_poly g \<noteq> \<zero>\<^bsub>UP Q\<^sub>p\<^esub>"
-  using assms normalize_poly_nonzero_eq 
+  using assms normalize_poly_nonzero_eq
   by (metis (no_types, lifting) UPQ.UP_smult_one UPQ.module_axioms UPQ.smult_r_null module.smult_assoc1 p_intpow_closed(1) p_intpow_inv')
 
 lemma gauss_norm_normalize_poly:
   assumes "g \<noteq> \<zero>\<^bsub>UP Q\<^sub>p\<^esub>"
   assumes "g \<in> carrier (UP Q\<^sub>p)"
   shows "gauss_norm (normalize_poly g) = 0"
-proof- 
+proof-
   have 0: "eint (int_gauss_norm (normalize_poly g)) = gauss_norm (normalize_poly g)"
-    by(rule int_gauss_norm_eq, rule normalize_poly_closed, rule assms, 
+    by(rule int_gauss_norm_eq, rule normalize_poly_closed, rule assms,
           rule normalize_poly_nonzero, rule assms, rule assms)
-  show ?thesis 
-    using 0 int_gauss_norm_normalize_poly assms 
+  show ?thesis
+    using 0 int_gauss_norm_normalize_poly assms
     by (simp add: zero_eint_def)
 qed
 
 lemma taylor_term_eval_eq:
   assumes "f \<in> carrier (UP Q\<^sub>p)"
   assumes "x \<in> carrier Q\<^sub>p"
   assumes "t \<in> carrier Q\<^sub>p"
   assumes "\<And>j. i \<noteq> j \<Longrightarrow> val (UPQ.taylor_term x f i \<bullet> t) < val (UPQ.taylor_term x f j \<bullet> t) "
   shows "val (f \<bullet> t) = val (UPQ.taylor_term x f i \<bullet> t)"
 proof-
   have 0: "f = finsum (UP Q\<^sub>p) (UPQ.taylor_term x f) {..deg Q\<^sub>p f}"
     by(rule UPQ.taylor_term_sum[of f "deg Q\<^sub>p f" x], rule assms, blast, rule assms)
-  show ?thesis 
+  show ?thesis
   proof(cases "i \<in> {..deg Q\<^sub>p f}")
     case True
     have T0: "finsum (UP Q\<^sub>p) (UPQ.taylor_term x f) {..deg Q\<^sub>p f} = UPQ.taylor_term x f i \<oplus>\<^bsub>UP Q\<^sub>p\<^esub> finsum (UP Q\<^sub>p) (UPQ.taylor_term x f) ({..deg Q\<^sub>p f} - {i})"
       apply(rule UPQ.P.finsum_remove[of "{..deg Q\<^sub>p f}" "UPQ.taylor_term x f" i])
       by(rule UPQ.taylor_term_closed, rule assms, rule assms, blast, rule True)
     have T1: "f = UPQ.taylor_term x f i \<oplus>\<^bsub>UP Q\<^sub>p\<^esub> finsum (UP Q\<^sub>p) (UPQ.taylor_term x f) ({..deg Q\<^sub>p f} - {i})"
       using 0 T0 by metis
     have T2: "finsum (UP Q\<^sub>p) (UPQ.taylor_term x f) ({..deg Q\<^sub>p f} - {i}) \<in> carrier (UP Q\<^sub>p)"
       apply(rule UPQ.P.finsum_closed)
       using UPQ.taylor_term_closed assms(1) assms(2) by blast
     have T3: "UPQ.taylor_term x f i \<in> carrier (UP Q\<^sub>p)"
       by(rule UPQ.taylor_term_closed, rule assms, rule assms )
-    obtain g where g_def: "g = f" 
-      by blast 
+    obtain g where g_def: "g = f"
+      by blast
     have T4: "g = UPQ.taylor_term x f i \<oplus>\<^bsub>UP Q\<^sub>p\<^esub> finsum (UP Q\<^sub>p) (UPQ.taylor_term x f) ({..deg Q\<^sub>p f} - {i})"
       unfolding g_def by(rule T1)
     have g_closed: "g \<in> carrier (UP Q\<^sub>p)"
       unfolding g_def by(rule assms)
     have T5: "g \<bullet> t = UPQ.taylor_term x f i \<bullet> t \<oplus> ( finsum (UP Q\<^sub>p) (UPQ.taylor_term x f) ({..deg Q\<^sub>p f} - {i})) \<bullet> t"
       unfolding T4 by(rule UPQ.to_fun_plus, rule T2, rule T3, rule assms)
-    have T6: "( finsum (UP Q\<^sub>p) (UPQ.taylor_term x f) ({..deg Q\<^sub>p f} - {i})) \<bullet> t = 
+    have T6: "( finsum (UP Q\<^sub>p) (UPQ.taylor_term x f) ({..deg Q\<^sub>p f} - {i})) \<bullet> t =
                 ( finsum Q\<^sub>p (\<lambda>i. UPQ.taylor_term x f i \<bullet> t) ({..deg Q\<^sub>p f} - {i}))"
       apply(rule UPQ.to_fun_finsum, blast)
       using assms UPQ.taylor_term_closed apply blast
-      using assms by blast 
+      using assms by blast
     have T7: "\<And>j. j \<in> {..deg Q\<^sub>p f} - {i} \<Longrightarrow> val (UPQ.taylor_term x f j \<bullet> t) > val (UPQ.taylor_term x f i \<bullet> t)"
       using assms  by (metis Diff_iff singletonI)
     have T8: "val (( finsum (UP Q\<^sub>p) (UPQ.taylor_term x f) ({..deg Q\<^sub>p f} - {i})) \<bullet> t) > val (UPQ.taylor_term x f i \<bullet> t)"
       unfolding T6
       apply(rule finsum_val_ultrametric'')
-      using UPQ.taylor_term_closed assms 
+      using UPQ.taylor_term_closed assms
       apply (metis (no_types, lifting) Pi_I UPQ.to_fun_closed)
         apply blast
       using assms T7 apply blast
       using assms(4)[of "Suc i"] using eint_ord_simps(4)
-        assms(4) eint_ord_code(6)  g_def gr_implies_not_zero less_one by smt 
+        assms(4) eint_ord_code(6)  g_def gr_implies_not_zero less_one by smt
     have T9: "val (g \<bullet> t) =  val (UPQ.taylor_term x f i \<bullet> t)"
-      unfolding T5 using T8 T2 T3 
+      unfolding T5 using T8 T2 T3
       by (metis (no_types, lifting) Qp.add.m_comm UPQ.to_fun_closed assms(3) val_ultrametric_noteq)
-    show ?thesis using T9 unfolding g_def by blast 
+    show ?thesis using T9 unfolding g_def by blast
   next
     case False
     have "i > deg Q\<^sub>p f"
       using False by simp
     hence "i > deg Q\<^sub>p (UPQ.taylor x f)"
       using assms UPQ.taylor_deg by presburger
     hence F0: "UPQ.taylor x f i = \<zero>"
       using assms UPQ.taylor_closed UPQ.deg_leE by blast
     have F1: "(UPQ.taylor_term x f i \<bullet> t) = \<zero>"
       using UPQ.to_fun_taylor_term[of f t x i]
       unfolding F0
       using assms Qp.cring_simprules(2) Qp.cring_simprules(4) Qp.integral_iff Qp.nat_pow_closed by presburger
-    show ?thesis 
-      using assms(4)[of "Suc i"] unfolding F1 
+    show ?thesis
+      using assms(4)[of "Suc i"] unfolding F1
       by (metis eint_ord_code(6) local.val_zero n_not_Suc_n)
   qed
 qed
 
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsection\<open>Hensel's Lemma for \<open>p\<close>-adic fields\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 theorem hensels_lemma:
   assumes "f \<in> carrier (UP Q\<^sub>p)"
   assumes "a \<in> \<O>\<^sub>p"
   assumes "gauss_norm f \<ge> 0"
   assumes "val (f\<bullet>a) > 2*val ((pderiv f)\<bullet>a)"
   shows "\<exists>!\<alpha> \<in> \<O>\<^sub>p. f\<bullet>\<alpha> = \<zero> \<and> val (a \<ominus> \<alpha>) > val ((pderiv f)\<bullet>a)"
-proof- 
+proof-
   have a_closed: "a \<in> carrier Q\<^sub>p"
-    using assms val_ring_memE by auto 
+    using assms val_ring_memE by auto
   have f_nonzero: "f \<noteq> \<zero>\<^bsub>UP Q\<^sub>p\<^esub>"
   proof(rule ccontr)
     assume N: "\<not> f \<noteq> \<zero>\<^bsub>UP Q\<^sub>p\<^esub>"
     then have 0: "pderiv f = \<zero>\<^bsub>UP Q\<^sub>p\<^esub>"
       using UPQ.deg_zero UPQ.pderiv_deg_0 by blast
     have 1: "f = \<zero>\<^bsub>UP Q\<^sub>p\<^esub>"
-      using N by auto 
+      using N by auto
     have 2: "eint 2 * val (UPQ.pderiv \<zero>\<^bsub>UP Q\<^sub>p\<^esub> \<bullet> a) = \<infinity>"
       by (simp add: UPQ.to_fun_zero local.a_closed local.val_zero)
     show False using assms a_closed
-      unfolding 2 1 
+      unfolding 2 1
       using eint_ord_simps(6) by blast
   qed
   obtain h where h_def: "h = to_Zp_poly f"
-    by blast 
+    by blast
   have h_closed: "h \<in> carrier (UP Z\<^sub>p)"
     unfolding h_def using assms
     by (simp add: to_Zp_poly_closed)
   have h_deriv: "Zp.pderiv h = to_Zp_poly (pderiv f)"
-    unfolding h_def 
-    using to_Zp_poly_pderiv[of f] assms by auto 
+    unfolding h_def
+    using to_Zp_poly_pderiv[of f] assms by auto
   have 0: "to_Zp (f\<bullet>a) = to_function Z\<^sub>p h (to_Zp a)"
-    unfolding h_def 
-    using assms a_closed 
+    unfolding h_def
+    using assms a_closed
     by (simp add: UPQ.to_fun_def to_Zp_poly_eval)
   have 1: "to_Zp ((pderiv f)\<bullet>a) = to_function Z\<^sub>p (Zp.pderiv h) (to_Zp a)"
-    unfolding h_deriv 
-    using assms a_closed  UPQ.pderiv_closed UPQ.to_fun_def eint_ord_trans gauss_norm_pderiv to_Zp_poly_eval 
+    unfolding h_deriv
+    using assms a_closed  UPQ.pderiv_closed UPQ.to_fun_def eint_ord_trans gauss_norm_pderiv to_Zp_poly_eval
     by presburger
   have 2: "val (f\<bullet>a) = val_Zp (to_function Z\<^sub>p h (to_Zp a))"
-  proof- 
+  proof-
     have 20: "f\<bullet>a \<in> \<O>\<^sub>p"
       using assms positive_gauss_norm_eval by blast
     have 21: "val (f\<bullet>a) = val_Zp (to_Zp (f\<bullet>a))"
       using 20 by (simp add: to_Zp_val)
-    show ?thesis unfolding 21 0 by blast 
+    show ?thesis unfolding 21 0 by blast
   qed
   have 3: "val ((pderiv f)\<bullet>a) = val_Zp ( to_function Z\<^sub>p (Zp.pderiv h) (to_Zp a))"
-  proof- 
+  proof-
     have 30: "(pderiv f)\<bullet>a \<in> \<O>\<^sub>p"
-      using positive_gauss_norm_eval assms gauss_norm_pderiv 
+      using positive_gauss_norm_eval assms gauss_norm_pderiv
       by (meson UPQ.pderiv_closed eint_ord_trans)
     have 31: "val ((pderiv f)\<bullet>a) = val_Zp (to_Zp ((pderiv f)\<bullet>a))"
       using 30 by (simp add: to_Zp_val)
-    show ?thesis unfolding 31 1 by blast 
+    show ?thesis unfolding 31 1 by blast
   qed
   have 4: "\<exists>!\<alpha>. \<alpha> \<in> carrier Z\<^sub>p \<and>
         Zp.to_fun (to_Zp_poly f) \<alpha> = \<zero>\<^bsub>Z\<^sub>p\<^esub> \<and>
         val_Zp (Zp.to_fun (Zp.pderiv (to_Zp_poly f)) (to_Zp a))
         < val_Zp (to_Zp a \<ominus>\<^bsub>Z\<^sub>p\<^esub> \<alpha>)"
     apply(rule hensels_lemma')
     using h_closed h_def apply blast
     using assms local.a_closed to_Zp_closed apply blast
-    using assms unfolding 2 3 h_def Zp.to_fun_def by blast  
+    using assms unfolding 2 3 h_def Zp.to_fun_def by blast
   obtain \<alpha> where \<alpha>_def: "\<alpha> \<in> carrier Z\<^sub>p \<and>
         Zp.to_fun (to_Zp_poly f) \<alpha> = \<zero>\<^bsub>Z\<^sub>p\<^esub> \<and>
         val_Zp (Zp.to_fun (Zp.pderiv (to_Zp_poly f)) (to_Zp a))
-        < val_Zp (to_Zp a \<ominus>\<^bsub>Z\<^sub>p\<^esub> \<alpha>) 
+        < val_Zp (to_Zp a \<ominus>\<^bsub>Z\<^sub>p\<^esub> \<alpha>)
         \<and> (\<forall>x.  x \<in> carrier Z\<^sub>p \<and>
         Zp.to_fun (to_Zp_poly f) x = \<zero>\<^bsub>Z\<^sub>p\<^esub> \<and>
         val_Zp (Zp.to_fun (Zp.pderiv (to_Zp_poly f)) (to_Zp a))
         < val_Zp (to_Zp a \<ominus>\<^bsub>Z\<^sub>p\<^esub> x) \<longrightarrow> x = \<alpha>)"
-    using 4 by blast 
+    using 4 by blast
   obtain \<beta> where \<beta>_def: "\<beta> = \<iota> \<alpha>"
-    by blast 
+    by blast
   have \<beta>_closed: "\<beta> \<in> \<O>\<^sub>p"
     using \<alpha>_def unfolding \<beta>_def by simp
   have 5: "(Zp.to_fun (to_Zp_poly f) \<alpha>) = to_Zp (f\<bullet>\<beta>)"
-    using \<beta>_closed to_Zp_poly_eval[of f \<beta>] assms 
-    unfolding \<beta>_def UPQ.to_fun_def 
+    using \<beta>_closed to_Zp_poly_eval[of f \<beta>] assms
+    unfolding \<beta>_def UPQ.to_fun_def
     by (simp add: Zp.to_fun_def \<alpha>_def inc_to_Zp)
   have 6: "to_Zp (f\<bullet>\<beta>) = \<zero>\<^bsub>Z\<^sub>p\<^esub>"
-    using 5 \<alpha>_def by auto 
+    using 5 \<alpha>_def by auto
   have \<beta>_closed: "\<beta> \<in> \<O>\<^sub>p"
     unfolding \<beta>_def using \<alpha>_def  by simp
   have 7: "(f\<bullet>\<beta>) = \<zero>"
-    using 6 assms unfolding \<beta>_def 
+    using 6 assms unfolding \<beta>_def
     by (metis \<beta>_closed \<beta>_def inc_of_zero positive_gauss_norm_eval to_Zp_inc)
   have 8: "\<alpha> = to_Zp \<beta>"
-    unfolding \<beta>_def using \<alpha>_def 
+    unfolding \<beta>_def using \<alpha>_def
     by (simp add: inc_to_Zp)
   have 9: "to_Zp a \<ominus>\<^bsub>Z\<^sub>p\<^esub> \<alpha> = to_Zp (a \<ominus> \<beta>)"
-    unfolding 8 using assms(2) \<beta>_closed 
+    unfolding 8 using assms(2) \<beta>_closed
     by (simp add: to_Zp_minus)
   have 10: "val (a \<ominus> \<beta>) = val_Zp (to_Zp a \<ominus>\<^bsub>Z\<^sub>p\<^esub> \<alpha>)"
-    unfolding 9 using \<beta>_closed assms(2) 
+    unfolding 9 using \<beta>_closed assms(2)
     to_Zp_val val_ring_minus_closed by presburger
   have 11: "val (a \<ominus> \<beta>) > val ((pderiv f)\<bullet>a)"
-    using \<alpha>_def unfolding 9 10 3 h_def 
+    using \<alpha>_def unfolding 9 10 3 h_def
     by (simp add: Zp.to_fun_def)
   have 12: "\<beta> \<in> \<O>\<^sub>p \<and> f \<bullet> \<beta> = \<zero> \<and> val (UPQ.pderiv f \<bullet> a) < val (a \<ominus> \<beta>)"
     using "11" "7" \<beta>_closed by linarith
-  have 13: "\<forall>x. x\<in> \<O>\<^sub>p \<and> f \<bullet> x = \<zero> \<and> val (UPQ.pderiv f \<bullet> a) < val (a \<ominus> x) 
+  have 13: "\<forall>x. x\<in> \<O>\<^sub>p \<and> f \<bullet> x = \<zero> \<and> val (UPQ.pderiv f \<bullet> a) < val (a \<ominus> x)
             \<longrightarrow> x = \<beta>"
   proof(rule, rule)
     fix x assume A: "x \<in> \<O>\<^sub>p \<and>  f \<bullet> x = \<zero> \<and> val (UPQ.pderiv f \<bullet> a) < val (a \<ominus> x)"
     obtain y where y_def: "y = to_Zp x"
-      by blast 
+      by blast
     have y_closed: "y \<in> carrier Z\<^sub>p"
-      unfolding y_def using A 
+      unfolding y_def using A
       by (simp add: to_Zp_closed val_ring_memE(2))
     have eval: "Zp.to_fun (to_Zp_poly f) y = \<zero>\<^bsub>Z\<^sub>p\<^esub>"
-      unfolding y_def using A assms 
+      unfolding y_def using A assms
       by (metis UPQ.to_fun_def Zp.to_fun_def to_Zp_poly_eval to_Zp_zero)
     have 0: "to_Zp a \<ominus>\<^bsub>Z\<^sub>p\<^esub> y = to_Zp (a \<ominus> x)"
-      unfolding y_def using A assms 
+      unfolding y_def using A assms
       by (simp add: to_Zp_minus)
     have q: " val_Zp (Zp.to_fun (Zp.pderiv (to_Zp_poly f)) (to_Zp a)) = val (UPQ.pderiv f \<bullet> a)"
       by (simp add: "3" Zp.to_fun_def h_def)
     have 1: "y \<in> carrier Z\<^sub>p \<and>
         Zp.to_fun (to_Zp_poly f) y = \<zero>\<^bsub>Z\<^sub>p\<^esub> \<and>
         val_Zp (Zp.to_fun (Zp.pderiv (to_Zp_poly f)) (to_Zp a))
         < val_Zp (to_Zp a \<ominus>\<^bsub>Z\<^sub>p\<^esub> y)"
-      unfolding 0 eval Zp.to_fun_def h_def 
+      unfolding 0 eval Zp.to_fun_def h_def
       apply(intro conjI y_closed)
       using eval Zp.to_fun_def apply (simp; fail)
-      using A unfolding 0 eval Zp.to_fun_def h_def 3 
+      using A unfolding 0 eval Zp.to_fun_def h_def 3
       using assms(2) to_Zp_val val_ring_minus_closed by presburger
     have 2: "y = \<alpha>"
       using 1 \<alpha>_def by blast
     show "x = \<beta>"
-      using y_def unfolding 2 8 using A \<beta>_closed 
+      using y_def unfolding 2 8 using A \<beta>_closed
       by (metis to_Zp_inc)
   qed
   show "\<exists>!\<alpha>. \<alpha> \<in> \<O>\<^sub>p \<and> f \<bullet> \<alpha> = \<zero> \<and> val (UPQ.pderiv f \<bullet> a) < val (a \<ominus> \<alpha>)"
     using 12 13 by metis
 qed
 
 lemma nth_root_poly_root_fixed:
   assumes "(n::nat) > 1"
   assumes "a \<in> \<O>\<^sub>p"
   assumes "val (\<one> \<ominus>\<^bsub>Q\<^sub>p\<^esub> a) > 2* val ([n]\<cdot>\<one>)"
   shows "(\<exists>! b \<in> \<O>\<^sub>p. (b[^]n) = a \<and>  val (b \<ominus> \<one>) > val ([n]\<cdot>\<one>))"
-proof- 
+proof-
   obtain f where f_def: "f = up_ring.monom (UP Q\<^sub>p) \<one> n \<ominus>\<^bsub>UP Q\<^sub>p\<^esub> up_ring.monom (UP Q\<^sub>p) a 0"
-    by blast 
+    by blast
   have f_closed: "f \<in> carrier (UP Q\<^sub>p)"
     unfolding f_def apply(rule UPQ.P.ring_simprules)
-     apply (simp; fail)   using assms 
+     apply (simp; fail)   using assms
     by (simp add: val_ring_memE(2))
   have 0: "UPQ.pderiv (up_ring.monom (UP Q\<^sub>p) a 0) = \<zero>\<^bsub>UP Q\<^sub>p\<^esub>"
-    using assms 
+    using assms
     by (simp add: val_ring_memE(2))
   have 1: "UPQ.pderiv (up_ring.monom (UP Q\<^sub>p) (\<one>) n) = (up_ring.monom (UP Q\<^sub>p) ([n]\<cdot>\<one>) (n-1)) "
     using UPQ.pderiv_monom by blast
   have 2: "up_ring.monom (UP Q\<^sub>p) \<one> n \<in> carrier (UP Q\<^sub>p)"
     by simp
   have 3: "up_ring.monom (UP Q\<^sub>p) a 0 \<in> carrier (UP Q\<^sub>p)"
-    using assms val_ring_memE by simp 
+    using assms val_ring_memE by simp
   have 4: "UPQ.pderiv f  = up_ring.monom (UP Q\<^sub>p) ([n] \<cdot> \<one>) (n - 1)  \<ominus>\<^bsub>UP Q\<^sub>p\<^esub> \<zero>\<^bsub>UP Q\<^sub>p\<^esub>"
     using 2 3 assms val_ring_memE UPQ.pderiv_minus[of "up_ring.monom (UP Q\<^sub>p) \<one> n" "up_ring.monom (UP Q\<^sub>p) a 0"]
     unfolding f_def 0 1 by blast
   have 5: "UPQ.pderiv f = (up_ring.monom (UP Q\<^sub>p) ([n]\<cdot>\<one>) (n-1))"
     unfolding 4 a_minus_def by simp
   have a_closed: "a \<in> carrier Q\<^sub>p"
-    using assms val_ring_memE by blast 
+    using assms val_ring_memE by blast
   have 6: "UPQ.pderiv f \<bullet> \<one> = [n]\<cdot>\<one> \<otimes> \<one>[^](n-1)"
-    unfolding 5 using a_closed 
+    unfolding 5 using a_closed
     by (simp add: UPQ.to_fun_monom)
   have 7: "val (\<one> \<ominus>\<^bsub>Q\<^sub>p\<^esub> a) > val \<one>"
-  proof- 
+  proof-
     have "eint 2 * val ([n] \<cdot> \<one>) \<ge> 0"
-      by (meson eint_ord_trans eint_pos_int_times_ge val_of_nat_inc zero_less_numeral) 
+      by (meson eint_ord_trans eint_pos_int_times_ge val_of_nat_inc zero_less_numeral)
     thus ?thesis
-      using assms unfolding val_one  
+      using assms unfolding val_one
       by (simp add: Q\<^sub>p_def)
   qed
   hence 8: "val a = val \<one>"
-    using a_closed 
+    using a_closed
     by (metis Qp.cring_simprules(6) ultrametric_equal_eq')
   have 9:"val (a [^] (n - 1)) = 0"
-    by (simp add: "8" local.a_closed val_zero_imp_val_pow_zero) 
+    by (simp add: "8" local.a_closed val_zero_imp_val_pow_zero)
   have 10: "val ([n]\<cdot>\<one> \<otimes> \<one>[^](n-1)) = val ([n]\<cdot>\<one>)"
     unfolding val_one 9 by simp
   have 11: "0 \<le> gauss_norm f"
-  proof- 
+  proof-
     have p0: "gauss_norm (up_ring.monom (UP Q\<^sub>p) \<one> n) \<ge> 0"
       using gauss_norm_monom by simp
     have p1: "gauss_norm (up_ring.monom (UP Q\<^sub>p) a 0) \<ge> 0"
       using gauss_norm_monom assms val_ring_memE by simp
     have p2: "min (gauss_norm (up_ring.monom (UP Q\<^sub>p) \<one> n)) (gauss_norm (up_ring.monom (UP Q\<^sub>p) a 0)) \<ge> 0"
-      using p0 p1 by simp 
+      using p0 p1 by simp
     have p3: "0 \<le> gauss_norm
       (up_ring.monom (UP Q\<^sub>p) \<one> n \<ominus>\<^bsub>UP Q\<^sub>p\<^esub> up_ring.monom (UP Q\<^sub>p) a 0)"
       using gauss_norm_ultrametric'[of "up_ring.monom (UP Q\<^sub>p) \<one> n" "up_ring.monom (UP Q\<^sub>p) a 0"]
             p2  "2" "3" eint_ord_trans  by blast
-    show ?thesis using p3 unfolding f_def by simp 
+    show ?thesis using p3 unfolding f_def by simp
   qed
   have 12: "\<And>\<alpha>. \<alpha> \<in> \<O>\<^sub>p \<Longrightarrow> f \<bullet> \<alpha> = \<alpha>[^]n \<ominus> a"
-    unfolding f_def using a_closed 
+    unfolding f_def using a_closed
     by (simp add: UPQ.to_fun_const UPQ.to_fun_diff UPQ.to_fun_monic_monom val_ring_memE(2))
   have 13: "\<exists>!\<alpha>. \<alpha> \<in> \<O>\<^sub>p \<and> f \<bullet> \<alpha> = \<zero> \<and> val (UPQ.pderiv f \<bullet> \<one>) < val (\<one> \<ominus> \<alpha>)"
-    apply(rule hensels_lemma, rule f_closed, rule one_in_val_ring, rule 11) 
-    unfolding 6 10  
-    using a_closed assms 12[of \<one>] assms(3) 
+    apply(rule hensels_lemma, rule f_closed, rule one_in_val_ring, rule 11)
+    unfolding 6 10
+    using a_closed assms 12[of \<one>] assms(3)
     by (simp add: one_in_val_ring)
   have 14: "\<And>\<alpha>. \<alpha> \<in> \<O>\<^sub>p \<Longrightarrow> \<alpha>[^]n = a \<longleftrightarrow> f \<bullet> \<alpha> = \<zero>"
     unfolding f_def using a_closed 12 f_def val_ring_memE(2) by auto
   have 15: "val (UPQ.pderiv f \<bullet> \<one>) = val ([n]\<cdot>\<one>)"
-    unfolding 6 10 by auto 
+    unfolding 6 10 by auto
   have 16: "\<And>\<alpha>. \<alpha> \<in> \<O>\<^sub>p \<Longrightarrow> val (\<one> \<ominus> \<alpha>) = val (\<alpha> \<ominus> \<one>)"
-  proof- 
+  proof-
     have 17: "\<And>\<alpha>. \<alpha> \<in> \<O>\<^sub>p \<Longrightarrow> (\<one> \<ominus> \<alpha>) = \<ominus> (\<alpha> \<ominus> \<one>)"
-      using val_ring_memE 
+      using val_ring_memE
       by (meson Qp.minus_a_inv Qp.one_closed)
     show "\<And>\<alpha>. \<alpha> \<in> \<O>\<^sub>p \<Longrightarrow> val (\<one> \<ominus> \<alpha>) = val (\<alpha> \<ominus> \<one>)"
-      unfolding 17 
+      unfolding 17
       using Qp.minus_closed Qp.one_closed val_minus val_ring_memE(2) by presburger
   qed
-  show ?thesis using 13 unfolding 15 using 14 16 Qp.one_closed val_ring_memE(2) by metis   
+  show ?thesis using 13 unfolding 15 using 14 16 Qp.one_closed val_ring_memE(2) by metis
 qed
 
-lemma mod_zeroE: 
+lemma mod_zeroE:
   assumes "(a::int) mod k  = 0"
   shows "\<exists>l. a = l*k"
-  using assms 
+  using assms
   using Groups.mult_ac(2) by blast
 
 lemma to_Zp_poly_closed':
   assumes "g \<in> carrier (UP Q\<^sub>p)"
   assumes "\<And>i. g i \<in> \<O>\<^sub>p"
   shows "to_Zp_poly g \<in> carrier (UP Z\<^sub>p)"
 proof(rule to_Zp_poly_closed)
   show "g \<in> carrier (UP Q\<^sub>p)"
-    using assms(1) by blast 
+    using assms(1) by blast
   show "0 \<le> gauss_norm g"
   proof-
     have "\<And>i. val (g i) \<ge> 0"
-      using assms val_ring_memE by blast 
-    thus ?thesis unfolding gauss_norm_def 
+      using assms val_ring_memE by blast
+    thus ?thesis unfolding gauss_norm_def
       by (metis  gauss_norm_coeff_norm gauss_norm_def)
   qed
 qed
 
 lemma to_Zp_poly_eval_to_Zp:
   assumes "g \<in> carrier (UP Q\<^sub>p)"
   assumes "\<And>i. g i \<in> \<O>\<^sub>p"
   assumes "a \<in> \<O>\<^sub>p"
   shows "to_function Z\<^sub>p (to_Zp_poly g) (to_Zp a) = to_Zp (g \<bullet> a)"
-proof- 
+proof-
   have "(\<forall>i. g i \<in> \<O>\<^sub>p) \<longrightarrow> to_function Z\<^sub>p (to_Zp_poly g) (to_Zp a) = to_Zp (g \<bullet> a)"
     apply(rule UPQ.poly_induct[of g]) using assms apply blast
-  proof 
+  proof
     fix p assume A: "p \<in> carrier (UP Q\<^sub>p)" "deg Q\<^sub>p p = 0" "\<forall>i. p i \<in> \<O>\<^sub>p"
     obtain c where c_def: "c \<in> carrier Q\<^sub>p \<and> p = up_ring.monom (UP Q\<^sub>p) c  0"
       using A  by (metis UPQ.ltrm_deg_0 val_ring_memE(2))
     have 0: "to_Zp_poly (up_ring.monom (UP Q\<^sub>p) c  0) = up_ring.monom (UP Z\<^sub>p) (to_Zp c) 0"
       unfolding to_Zp_poly_def proof fix n show " to_Zp (up_ring.monom (UP Q\<^sub>p) c 0 n) = up_ring.monom (UP Z\<^sub>p) (to_Zp c) 0 n"
         using UP_ring.cfs_monom[of Z\<^sub>p "to_Zp c" 0 n] UP_ring.cfs_monom[of Q\<^sub>p c 0 n] to_Zp_closed[of c ]
-        unfolding UP_ring_def 
+        unfolding UP_ring_def
         apply(cases "0 = n")
         using UPQ.cfs_monom Zp.cfs_monom c_def apply presburger
-         using UPQ.cfs_monom Zp.cfs_monom c_def 
+         using UPQ.cfs_monom Zp.cfs_monom c_def
          using to_Zp_zero by presburger
-    qed    
+    qed
     have p_eq: "p = up_ring.monom (UP Q\<^sub>p) c  0"
-      using c_def by blast 
+      using c_def by blast
     have 1: "(up_ring.monom (UP Q\<^sub>p) c 0 \<bullet> a) = c"
-      using UPQ.to_fun_to_poly[of c a]  c_def assms val_ring_memE 
-      unfolding to_polynomial_def  by blast 
+      using UPQ.to_fun_to_poly[of c a]  c_def assms val_ring_memE
+      unfolding to_polynomial_def  by blast
     show "to_function Z\<^sub>p (to_Zp_poly p) (to_Zp a) = to_Zp (p \<bullet> a)"
       using c_def assms(3) val_ring_memE(2)[of a]
       UP_cring.to_fun_to_poly[of Z\<^sub>p "to_Zp c" "to_Zp a"]
-      unfolding p_eq 0 1 Zp.to_fun_def to_polynomial_def  
+      unfolding p_eq 0 1 Zp.to_fun_def to_polynomial_def
       using Zp.UP_cring_axioms to_Zp_closed by blast
-  next  
+  next
     show "\<And>p. (\<And>q. q \<in> carrier (UP Q\<^sub>p) \<Longrightarrow>
               deg Q\<^sub>p q < deg Q\<^sub>p p \<Longrightarrow> (\<forall>i. q i \<in> \<O>\<^sub>p) \<longrightarrow> to_function Z\<^sub>p (to_Zp_poly q) (to_Zp a) = to_Zp (q \<bullet> a)) \<Longrightarrow>
          p \<in> carrier (UP Q\<^sub>p) \<Longrightarrow> 0 < deg Q\<^sub>p p \<Longrightarrow> (\<forall>i. p i \<in> \<O>\<^sub>p) \<longrightarrow> to_function Z\<^sub>p (to_Zp_poly p) (to_Zp a) = to_Zp (p \<bullet> a)"
-    proof  fix p 
+    proof  fix p
       assume A: "(\<And>q. q \<in> carrier (UP Q\<^sub>p) \<Longrightarrow>
               deg Q\<^sub>p q < deg Q\<^sub>p p \<Longrightarrow> (\<forall>i. q i \<in> \<O>\<^sub>p) \<longrightarrow> to_function Z\<^sub>p (to_Zp_poly q) (to_Zp a) = to_Zp (q \<bullet> a))"
             "p \<in> carrier (UP Q\<^sub>p)" "0 < deg Q\<^sub>p p" "\<forall>i. p i \<in> \<O>\<^sub>p"
       show "to_function Z\<^sub>p (to_Zp_poly p) (to_Zp a) = to_Zp (p \<bullet> a)"
-      proof- 
+      proof-
         obtain q where q_def: "q = truncate Q\<^sub>p  p"
-          by blast 
+          by blast
         have  q_closed: "q \<in> carrier (UP Q\<^sub>p)"
           unfolding q_def by(rule UPQ.trunc_closed, rule A)
         obtain c where c_def: "c = UPQ.lcf p"
-          by blast 
+          by blast
         obtain n where n_def: "n = deg Q\<^sub>p p"
-          by blast 
+          by blast
         have 0: "p = q \<oplus>\<^bsub>UP Q\<^sub>p\<^esub> up_ring.monom (UP Q\<^sub>p) c n"
-          unfolding c_def n_def q_def 
+          unfolding c_def n_def q_def
           using A(2) UPQ.trunc_simps(1) by blast
         have 1: "up_ring.monom (UP Q\<^sub>p) c n \<in> carrier  (UP Q\<^sub>p)"
           using A(2) UPQ.ltrm_closed c_def n_def by blast
         have 2: "p \<bullet> a  = q  \<bullet> a \<oplus> (c \<otimes> a[^]n)"
-          unfolding 0 using assms val_ring_memE  
+          unfolding 0 using assms val_ring_memE
           by (metis "1" A(4) UPQ.to_fun_monom UPQ.to_fun_plus c_def q_closed)
         have 3: "\<And>i. i < n \<Longrightarrow> q i = p i"
-          unfolding n_def q_def 
+          unfolding n_def q_def
           using A(2) UPQ.trunc_cfs by blast
         have 4: "deg Q\<^sub>p q < n"
           unfolding n_def q_def using A
           using UPQ.trunc_degree by presburger
         have 5: "\<And>i. i \<ge> n \<Longrightarrow> i >  deg Q\<^sub>p q"
-          using A[of ] less_le_trans[of "deg Q\<^sub>p q" "deg Q\<^sub>p p"] unfolding q_def n_def  
+          using A[of ] less_le_trans[of "deg Q\<^sub>p q" "deg Q\<^sub>p p"] unfolding q_def n_def
           using "4" n_def q_def by blast
         have 6: "\<And>i. i \<ge> n \<Longrightarrow> q i = \<zero>"
           using q_closed 5 UPQ.deg_leE by blast
         have 7: "(\<forall>i. q i \<in> \<O>\<^sub>p) \<longrightarrow> to_function Z\<^sub>p (to_Zp_poly q) (to_Zp a) = to_Zp (q \<bullet> a)"
-          apply(rule   A) unfolding q_def 
+          apply(rule   A) unfolding q_def
           using q_closed q_def apply blast
           using "4" n_def q_def by blast
         have 8: "(\<forall>i. q i \<in> \<O>\<^sub>p)"
-        proof fix i show "q i \<in> \<O>\<^sub>p" apply(cases "i < n") 
-            using 3 A(4) apply blast using 6[of i] 
+        proof fix i show "q i \<in> \<O>\<^sub>p" apply(cases "i < n")
+            using 3 A(4) apply blast using 6[of i]
             by (metis less_or_eq_imp_le linorder_neqE_nat zero_in_val_ring)
         qed
         have 9: "to_function Z\<^sub>p (to_Zp_poly q) (to_Zp a) = to_Zp (q \<bullet> a)"
-          using 7 8 by blast 
+          using 7 8 by blast
         have 10: "to_Zp_poly p = to_Zp_poly q \<oplus>\<^bsub>UP Z\<^sub>p\<^esub> to_Zp_poly (up_ring.monom (UP Q\<^sub>p) c n)"
-        proof fix x  
+        proof fix x
           have 100: "to_Zp_poly (up_ring.monom (UP Q\<^sub>p) c n) = (up_ring.monom (UP Z\<^sub>p) (to_Zp c) n)"
             using to_Zp_poly_monom[of c] A(4) c_def by blast
           have 101: "deg Z\<^sub>p (to_Zp_poly q) \<le> n-1"
               apply(rule  UP_cring.deg_leqI)
               unfolding UP_cring_def using Zp.R_cring apply auto[1]
               using to_Zp_poly_closed' 8 q_closed apply blast
-              unfolding to_Zp_poly_def using 4 6 
-              by (simp add: to_Zp_zero) 
+              unfolding to_Zp_poly_def using 4 6
+              by (simp add: to_Zp_zero)
           have 102: "(to_Zp_poly q) \<in> carrier (UP Z\<^sub>p)"
-            apply(rule to_Zp_poly_closed', rule q_closed) using 8 by blast 
+            apply(rule to_Zp_poly_closed', rule q_closed) using 8 by blast
           have 103: "deg Z\<^sub>p (to_Zp_poly q) < n"
             using 101 4 by linarith
-            have T0: "(to_Zp_poly q \<oplus>\<^bsub>UP Z\<^sub>p\<^esub> to_Zp_poly (up_ring.monom (UP Q\<^sub>p) c n)) x = 
+            have T0: "(to_Zp_poly q \<oplus>\<^bsub>UP Z\<^sub>p\<^esub> to_Zp_poly (up_ring.monom (UP Q\<^sub>p) c n)) x =
                     (to_Zp_poly q x) \<oplus>\<^bsub>Z\<^sub>p\<^esub> (to_Zp_poly (up_ring.monom (UP Q\<^sub>p) c n) x)"
               apply(rule  UP_ring.cfs_add)
-              unfolding UP_ring_def apply (simp add: Zp.is_ring)
-               apply(rule 102) unfolding 100 apply(rule UP_ring.monom_closed)
-              unfolding UP_ring_def apply (simp add: Zp.is_ring)
-              apply(rule  to_Zp_closed ) unfolding c_def 
-              using A(2) UPQ.UP_car_memE(1) by blast
+                apply (simp add: Zp.is_UP_ring)
+               apply (simp add: "102")
+              using "100" A(2) UPQ.lcf_closed c_def to_Zp_closed by auto
             have c_closed: "c \<in> \<O>\<^sub>p"
               unfolding c_def  using A(4) by blast
             have to_Zp_c_closed: "to_Zp c \<in> carrier Z\<^sub>p"
               using c_closed to_Zp_closed val_ring_memE(2) by blast
           show "to_Zp_poly p x = (to_Zp_poly q \<oplus>\<^bsub>UP Z\<^sub>p\<^esub> to_Zp_poly (up_ring.monom (UP Q\<^sub>p) c n)) x"
           proof(cases "x < n")
             case True
             have T1: "(to_Zp_poly (up_ring.monom (UP Q\<^sub>p) c n) x) = \<zero>\<^bsub>Z\<^sub>p\<^esub>"
-              using True UP_ring.cfs_monom[of Z\<^sub>p] unfolding UP_ring_def 
-              by (metis "100" A(2) UPQ.cfs_closed UPQ.deg_leE Zp.is_ring c_def n_def to_Zp_closed to_Zp_zero)
-            have T2: "to_Zp (p x) = to_Zp (q x)" using 3[of x] True by smt 
+              using True UP_ring.cfs_monom[of Z\<^sub>p] unfolding UP_ring_def
+              by (simp add: A(2) UPQ.ltrm_cfs c_def n_def to_Zp_poly_def to_Zp_zero)
+            have T2: "to_Zp (p x) = to_Zp (q x)" using 3[of x] True by smt
             have T3: "to_Zp (p x) \<in> carrier Z\<^sub>p"
               apply(rule to_Zp_closed) using A(2) UPQ.UP_car_memE(1) by blast
-            show ?thesis using T3 
-              unfolding T0  unfolding T1 unfolding  to_Zp_poly_def  T2  
+            show ?thesis using T3
+              unfolding T0  unfolding T1 unfolding  to_Zp_poly_def  T2
               using Zp.cring_simprules(8) add_comm by presburger
           next
             case False
             have F: "q x = \<zero> "
-              using False  
+              using False
               by (metis "6" less_or_eq_imp_le linorder_neqE_nat)
             have F': "(to_Zp_poly q) x = \<zero>\<^bsub>Z\<^sub>p\<^esub>"
              unfolding to_Zp_poly_def F using to_Zp_zero by blast
             show "to_Zp_poly p x = (to_Zp_poly q \<oplus>\<^bsub>UP Z\<^sub>p\<^esub> to_Zp_poly (up_ring.monom (UP Q\<^sub>p) c n)) x"
             proof(cases "x = n")
               case True
               have T1: "to_Zp (p x) \<in> carrier Z\<^sub>p"
                 apply(rule to_Zp_closed)
                 using A(2) UPQ.UP_car_memE(1) by blast
               have T2: "(to_Zp_poly (up_ring.monom (UP Q\<^sub>p) c n) x) = to_Zp c"
                 unfolding 100 using UP_ring.cfs_monom[of Z\<^sub>p "to_Zp c" n n] unfolding UP_ring_def True
-                using Zp.is_ring to_Zp_c_closed by presburger
+                using Zp.ring_axioms to_Zp_c_closed by presburger
               show ?thesis using to_Zp_c_closed unfolding T0 F' T2 unfolding to_Zp_poly_def True c_def n_def
                 using Zp.cring_simprules(8) by presburger
             next
               case FF: False
               have F0: "p x = \<zero>"
-                using FF False unfolding n_def 
+                using FF False unfolding n_def
                 using A(2) UPQ.UP_car_memE(2) linorder_neqE_nat by blast
               have F1: "q x = \<zero>"
                 using FF False F by linarith
               have F2: "(up_ring.monom (UP Q\<^sub>p) c n) x = \<zero>"
                 using FF False A(2) UPQ.cfs_closed UPQ.cfs_monom c_def by presburger
               show ?thesis unfolding T0 unfolding to_Zp_poly_def F0 F1 F2
                 using Zp.r_zero Zp.zero_closed to_Zp_zero by presburger
             qed
           qed
         qed
         have 11: "deg Z\<^sub>p (to_Zp_poly q) \<le> n-1"
           apply(rule  UP_cring.deg_leqI)
           unfolding UP_cring_def using Zp.R_cring apply auto[1]
           using to_Zp_poly_closed' 8 q_closed apply blast
-          unfolding to_Zp_poly_def using 4 6 
+          unfolding to_Zp_poly_def using 4 6
           by (smt diff_commute diff_diff_cancel less_one less_or_eq_imp_le linorder_neqE_nat to_Zp_zero zero_less_diff)
         have 12: "(to_Zp_poly q) \<in> carrier (UP Z\<^sub>p)"
-          apply(rule to_Zp_poly_closed', rule q_closed) using 8 by blast 
+          apply(rule to_Zp_poly_closed', rule q_closed) using 8 by blast
         have 13: "deg Z\<^sub>p (to_Zp_poly q) < n"
           using 11 4 by linarith
         have 14: "to_Zp_poly (up_ring.monom (UP Q\<^sub>p) c n) = (up_ring.monom (UP Z\<^sub>p) (to_Zp c) n)"
           using to_Zp_poly_monom[of c] A(4) c_def by blast
-        have 15: "Zp.to_fun (to_Zp_poly q \<oplus>\<^bsub>UP Z\<^sub>p\<^esub> to_Zp_poly (up_ring.monom (UP Q\<^sub>p) c n)) (to_Zp a)= 
+        have 15: "Zp.to_fun (to_Zp_poly q \<oplus>\<^bsub>UP Z\<^sub>p\<^esub> to_Zp_poly (up_ring.monom (UP Q\<^sub>p) c n)) (to_Zp a)=
             Zp.to_fun (to_Zp_poly q) (to_Zp a) \<oplus>\<^bsub>Z\<^sub>p\<^esub> Zp.to_fun (to_Zp_poly (up_ring.monom (UP Q\<^sub>p) c n)) (to_Zp a)"
           apply(rule Zp.to_fun_plus)
-          unfolding 14  apply(rule UP_ring.monom_closed) 
-          unfolding UP_ring_def using Zp.is_ring apply auto[1]
-          apply(rule to_Zp_closed) unfolding c_def 
-          using A(2) UPQ.cfs_closed apply blast
-          using 12 apply blast 
-          apply(rule to_Zp_closed) using assms val_ring_memE by blast 
+          unfolding 14  apply(rule UP_ring.monom_closed)
+          unfolding UP_ring_def
+             apply (simp add: Zp.ring_axioms)
+            apply (simp add: A(2) UPQ.cfs_closed c_def to_Zp_closed)
+           using "12" apply blast
+          apply(rule to_Zp_closed) using assms val_ring_memE by blast
         have 16: "to_Zp (q \<bullet> a \<oplus> c \<otimes> a [^] n) = to_Zp (q \<bullet> a)  \<oplus>\<^bsub>Z\<^sub>p\<^esub> to_Zp (c \<otimes> a [^] n)"
           apply(rule to_Zp_add)
-           apply(rule val_ring_poly_eval, rule q_closed) 
+           apply(rule val_ring_poly_eval, rule q_closed)
             using "8" apply blast
              apply(rule assms)
             apply(rule val_ring_times_closed)
             unfolding c_def using A(4) apply blast
           by(rule val_ring_nat_pow_closed, rule assms)
         have  17: " to_function Z\<^sub>p (up_ring.monom (UP Z\<^sub>p) (to_Zp c) n) (to_Zp a) =  to_Zp (c \<otimes> a [^] n)"
           proof-
             have 170: "to_Zp (c \<otimes> a [^] n) = to_Zp c \<otimes>\<^bsub>Z\<^sub>p\<^esub> to_Zp (a [^] n)"
               apply(rule  to_Zp_mult[of c "a[^]n"])
               unfolding c_def using A(4) apply blast
               by(rule val_ring_nat_pow_closed, rule assms)
             have 171: "to_Zp (a [^] n) = (to_Zp a [^]\<^bsub>Z\<^sub>p\<^esub>n)"
               by(rule to_Zp_nat_pow, rule assms)
             have 172: "to_Zp c \<in> carrier Z\<^sub>p "
-              apply(rule to_Zp_closed) unfolding c_def 
+              apply(rule to_Zp_closed) unfolding c_def
               using A(2) UPQ.UP_car_memE(1) by blast
             have 173: "to_Zp a \<in> carrier Z\<^sub>p "
-              apply(rule to_Zp_closed) using assms val_ring_memE by blast  
+              apply(rule to_Zp_closed) using assms val_ring_memE by blast
             show ?thesis
               using 172 173 Zp.to_fun_monom[of "to_Zp c" "to_Zp a" n] unfolding Zp.to_fun_def 170 171
-              by blast 
+              by blast
         qed
         show ?thesis
           using 15 unfolding Zp.to_fun_def 10 2 16 9 unfolding 14 17
-          by blast 
+          by blast
       qed
     qed
   qed
-  thus ?thesis using assms by blast 
+  thus ?thesis using assms by blast
 qed
 
 lemma inc_nat_pow:
   assumes "a \<in> carrier Z\<^sub>p"
   shows "\<iota> ([(n::nat)] \<cdot>\<^bsub>Z\<^sub>p\<^esub>a) = [n]\<cdot>(\<iota> a)"
   apply(induction n)
   apply (metis Q\<^sub>p_def Qp.int_inc_zero Qp.nat_mult_zero Zp.add.nat_pow_0 Zp_int_inc_zero' \<iota>_def frac_inc_of_int)
-  unfolding Qp.add.nat_pow_Suc Zp.add.nat_pow_Suc 
+  unfolding Qp.add.nat_pow_Suc Zp.add.nat_pow_Suc
   using Zp_nat_mult_closed assms inc_of_sum by presburger
-  
+
 lemma poly_inc_pderiv:
   assumes "g \<in> carrier (UP Z\<^sub>p)"
   shows "poly_inc (Zp.pderiv g) = UPQ.pderiv (poly_inc g)"
 proof fix x
   have 0: "UPQ.pderiv (poly_inc g) x = [Suc x] \<cdot> poly_inc g (Suc x)"
     apply(rule UPQ.pderiv_cfs[of "poly_inc g" x])
     by(rule poly_inc_closed, rule assms)
   have 1: "Zp.pderiv g x = [Suc x] \<cdot>\<^bsub>Z\<^sub>p\<^esub> g (Suc x)"
     by(rule Zp.pderiv_cfs[of g x], rule assms)
   show "poly_inc (Zp.pderiv g) x = UPQ.pderiv (poly_inc g) x"
     unfolding 0  unfolding poly_inc_def 1 apply(rule  inc_nat_pow)
     using Zp.UP_car_memE(1) assms by blast
 qed
 
 lemma Zp_hensels_lemma:
   assumes "f \<in> carrier Zp_x"
   assumes "a \<in> carrier Z\<^sub>p"
   assumes "Zp.to_fun (Zp.pderiv f) a \<noteq> \<zero>\<^bsub>Z\<^sub>p\<^esub>"
   assumes "Zp.to_fun f a \<noteq> \<zero>\<^bsub>Z\<^sub>p \<^esub>"
   assumes "val_Zp (Zp.to_fun f a) > eint 2 * val_Zp (Zp.to_fun (Zp.pderiv f) a)"
   obtains \<alpha> where
-       "Zp.to_fun f \<alpha> = \<zero>\<^bsub>Z\<^sub>p\<^esub>" and "\<alpha> \<in> carrier Z\<^sub>p" 
+       "Zp.to_fun f \<alpha> = \<zero>\<^bsub>Z\<^sub>p\<^esub>" and "\<alpha> \<in> carrier Z\<^sub>p"
        "val_Zp (a \<ominus>\<^bsub>Z\<^sub>p\<^esub> \<alpha>) > val_Zp (Zp.to_fun (Zp.pderiv f) a)"
        "val_Zp (a \<ominus>\<^bsub>Z\<^sub>p\<^esub> \<alpha>) = val_Zp (divide (Zp.to_fun f a) (Zp.to_fun (Zp.pderiv f) a))"
        "val_Zp (Zp.to_fun (Zp.pderiv f) \<alpha>) = val_Zp (Zp.to_fun (Zp.pderiv f) a)"
 proof-
   have "hensel p f a"
-    using assms 
+    using assms
     by (simp add: Zp_def hensel.intro hensel_axioms.intro padic_integers_axioms)
-  then show ?thesis 
-    using hensel.full_hensels_lemma[of p f a] that 
-    unfolding Zp_def    
-    by blast       
+  then show ?thesis
+    using hensel.full_hensels_lemma[of p f a] that
+    unfolding Zp_def
+    by blast
 qed
 
 end
 end
diff --git a/thys/Padic_Field/Padic_Field_Powers.thy b/thys/Padic_Field/Padic_Field_Powers.thy
--- a/thys/Padic_Field/Padic_Field_Powers.thy
+++ b/thys/Padic_Field/Padic_Field_Powers.thy
@@ -1,11440 +1,11439 @@
 theory Padic_Field_Powers
   imports Ring_Powers Padic_Field_Polynomials Generated_Boolean_Algebra
           Padic_Field_Topology
 
 
-begin 
+begin
 
 text\<open>This theory is intended to develop the necessary background on subsets of powers of a $p$-adic
 field to prove Macintyre's quantifier elimination theorem. In particular, we define semi-algebraic
 subsets of $\mathbb{Q}_p^n$, semi-algebraic functions $\mathbb{Q}_p^n \to \mathbb{Q}_p$, and semi-
-algebraic mappings $\mathbb{Q}_p^n \to \mathbb{Q}_p^m$  for arbitrary $n, m \in \mathbb{N}$. In 
-addition we prove that many common sets and functions are semi-algebraic. We are closely following 
+algebraic mappings $\mathbb{Q}_p^n \to \mathbb{Q}_p^m$  for arbitrary $n, m \in \mathbb{N}$. In
+addition we prove that many common sets and functions are semi-algebraic. We are closely following
 the paper \cite{denef1986} by Denef, where an algebraic proof of Mactinyre's theorem is developed.\<close>
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 section\<open>Cartesian Powers of $p$-adic Fields\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 lemma list_tl:
 "tl (t#x) = x"
-  using List.list.sel(3) by auto 
+  using List.list.sel(3) by auto
 
 lemma list_hd:
 "hd (t#x) = t"
-  unfolding List.list.sel(1) by auto 
+  unfolding List.list.sel(1) by auto
 
 
 sublocale padic_fields <  cring_coord_rings Q\<^sub>p "UP Q\<^sub>p"
-  unfolding  cring_coord_rings_axioms_def cring_coord_rings_def 
-  using Qp.zero_not_one UPQ.R_cring 
+  unfolding  cring_coord_rings_axioms_def cring_coord_rings_def
+  using Qp.zero_not_one UPQ.R_cring
   apply (simp add: UPQ.is_UP_cring)
-  by auto 
+  by auto
 
 sublocale padic_fields < Qp: domain_coord_rings Q\<^sub>p "UP Q\<^sub>p"
-  unfolding domain_coord_rings_def cring_coord_rings_axioms_def cring_coord_rings_def 
-  using Qp.domain_axioms Qp.zero_not_one UPQ.R_cring 
+  unfolding domain_coord_rings_def cring_coord_rings_axioms_def cring_coord_rings_def
+  using Qp.domain_axioms Qp.zero_not_one UPQ.R_cring
   apply (simp add: UPQ.UP_cring_axioms)
-  by auto 
+  by auto
 
 context padic_fields
 begin
 no_notation Zp.to_fun (infixl\<open>\<bullet>\<close> 70)
 no_notation ideal_prod (infixl "\<cdot>\<index>" 80)
 
 notation
-evimage (infixr "\<inverse>\<index>" 90) and 
+evimage (infixr "\<inverse>\<index>" 90) and
 euminus_set ("_ \<^sup>c\<index>" 70)
 
 
 type_synonym padic_tuple = "padic_number list"
 type_synonym padic_function = "padic_number \<Rightarrow> padic_number"
 type_synonym padic_nary_function = "padic_tuple \<Rightarrow> padic_number"
 type_synonym padic_function_tuple = "padic_nary_function list"
 type_synonym padic_nary_function_poly = "nat \<Rightarrow> padic_nary_function"
 
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsection\<open>Polynomials over $\mathbb{Q}_p$ and Polynomial Maps\<close>
 (**************************************************************************************************)
-(**************************************************************************************************) 
+(**************************************************************************************************)
 
 lemma last_closed':
   assumes "x@[t] \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "t \<in> carrier  Q\<^sub>p"
-  using assms last_closed[of n "x@[t]" Q\<^sub>p] 
-  by (metis (full_types) cartesian_power_car_memE gr0I last_snoc 
+  using assms last_closed[of n "x@[t]" Q\<^sub>p]
+  by (metis (full_types) cartesian_power_car_memE gr0I last_snoc
       length_append_singleton less_not_refl zero_less_Suc)
 
 lemma segment_in_car':
   assumes "x@[t] \<in> carrier (Q\<^sub>p\<^bsup>Suc n\<^esup>)"
   shows "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
 proof-
   have 0: "length x = n"
     by (metis Suc_inject assms cartesian_power_car_memE length_append_singleton)
   have  "set x \<subseteq> set (x@[t])"
     by (metis rotate1.simps(2) set_rotate1 set_subset_Cons)
   then have 1: "set x \<subseteq> carrier Q\<^sub>p"
     using assms cartesian_power_car_memE''[of "x@[t]" Q\<^sub>p "Suc n"]
     by blast
-  show ?thesis 
+  show ?thesis
     using 0 1 assms cartesian_power_car_memI[of x n Q\<^sub>p]
     by blast
 qed
 
 lemma Qp_zero:
 "Q\<^sub>p\<^bsup>0\<^esup> = nil_ring"
   unfolding cartesian_power_def
   by simp
 
 lemma Qp_zero_carrier:
 "carrier (Q\<^sub>p\<^bsup>0\<^esup>) = {[]}"
   by (simp add: Qp_zero)
 
 text\<open>Abbreviation for constant polynomials\<close>
 
-abbreviation(input) Qp_to_IP where 
+abbreviation(input) Qp_to_IP where
 "Qp_to_IP k \<equiv> Qp.indexed_const k"
 
 lemma Qp_to_IP_car:
   assumes "k \<in> carrier Q\<^sub>p"
   shows "Qp_to_IP k \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
-  using assms 
-  unfolding coord_ring_def  
-  using Qp.indexed_const_closed by blast 
+  using assms
+  unfolding coord_ring_def
+  using Qp.indexed_const_closed by blast
 
 lemma(in cring_coord_rings) smult_closed:
   assumes "a \<in> carrier R"
   assumes "q \<in> carrier (R[\<X>\<^bsub>n\<^esub>])"
   shows "a \<odot>\<^bsub>R[\<X>\<^bsub>n\<^esub>]\<^esub> q \<in> carrier (R[\<X>\<^bsub>n\<^esub>])"
-  using assms unfolding coord_ring_def 
-  using Pring_smult_closed 
+  using assms unfolding coord_ring_def
+  using Pring_smult_closed
   by (simp add: R.Pring_smult_closed)
-  
-  
+
+
 lemma Qp_poly_smult_cfs:
   assumes "a \<in> carrier Q\<^sub>p"
   assumes "P \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   shows "(a \<odot>\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> P) m = a \<otimes> (P m)"
-  using assms unfolding coord_ring_def 
+  using assms unfolding coord_ring_def
   using Qp.Pring_smult_cfs by blast
 
 lemma Qp_smult_r_distr:
   assumes "a \<in> carrier Q\<^sub>p"
   assumes "P \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   assumes "q \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   shows "a \<odot>\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> (P \<oplus>\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> q) = (a \<odot>\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> P)  \<oplus>\<^bsub> Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> (a \<odot>\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> q)"
-  using assms unfolding coord_ring_def 
+  using assms unfolding coord_ring_def
   using Qp.Pring_smult_r_distr by blast
 
 lemma Qp_smult_l_distr:
   assumes "a \<in> carrier Q\<^sub>p"
   assumes "b \<in> carrier Q\<^sub>p"
   assumes "P \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   shows "(a \<oplus> b) \<odot>\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> P = (a \<odot>\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> P)  \<oplus>\<^bsub> Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> (b \<odot>\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> P)"
-  using assms unfolding coord_ring_def 
-  using Qp.Pring_smult_l_distr by blast                
-   
-abbreviation(input) Qp_funs  where 
+  using assms unfolding coord_ring_def
+  using Qp.Pring_smult_l_distr by blast
+
+abbreviation(input) Qp_funs  where
 "Qp_funs n \<equiv> Fun\<^bsub>n\<^esub> Q\<^sub>p"
 
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsection\<open>Evaluation of Polynomials in $\mathbb{Q}_p$\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 
-abbreviation(input) Qp_ev where 
+abbreviation(input) Qp_ev where
 "Qp_ev P q \<equiv> (eval_at_point Q\<^sub>p q P)"
 
 lemma Qp_ev_one:
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
-  shows "Qp_ev \<one>\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> a = \<one>" unfolding coord_ring_def 
+  shows "Qp_ev \<one>\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> a = \<one>" unfolding coord_ring_def
   by (metis Qp.Pring_one eval_at_point_const Qp.one_closed assms)
 
 lemma Qp_ev_zero:
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
-  shows "Qp_ev \<zero>\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> a = \<zero>"unfolding coord_ring_def 
+  shows "Qp_ev \<zero>\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> a = \<zero>"unfolding coord_ring_def
   by (metis Qp.Pring_zero eval_at_point_const Qp.zero_closed assms)
 
 lemma Qp_eval_pvar_pow:
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "k < n"
   assumes "(m::nat) \<noteq> 0"
   shows "Qp_ev ((pvar Q\<^sub>p k)[^]\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> m) a = ((a!k)[^]m)"
   by (metis eval_at_point_nat_pow eval_pvar assms(1) assms(2) pvar_closed)
 
 text\<open>composition of polynomials over $\mathbb{Q}_p$\<close>
 
 definition Qp_poly_comp where
 "Qp_poly_comp m fs = poly_compose (length fs) m fs"
 
 text\<open>lemmas about polynomial maps\<close>
 
 lemma Qp_is_poly_tupleI:
   assumes "\<And>i. i < length fs\<Longrightarrow> fs!i \<in> carrier (Q\<^sub>p[\<X>\<^bsub>m\<^esub>])"
   shows "is_poly_tuple m fs"
-  unfolding is_poly_tuple_def using assms 
+  unfolding is_poly_tuple_def using assms
   using cartesian_power_car_memE'' cartesian_power_car_memI' by blast
 
 lemma Qp_is_poly_tuple_append:
   assumes "is_poly_tuple m fs"
   assumes "is_poly_tuple m gs"
   shows "is_poly_tuple m (fs@gs)"
 proof(rule Qp_is_poly_tupleI)
   show "\<And>i. i < length (fs @ gs) \<Longrightarrow> (fs @ gs) ! i \<in> carrier (Q\<^sub>p[\<X>\<^bsub>m\<^esub>])"
   proof- fix i assume A: "i < length (fs @ gs)"
   show "(fs @ gs) ! i \<in> carrier (Q\<^sub>p[\<X>\<^bsub>m\<^esub>])"
     apply(cases "i < length fs")
     using assms is_poly_tupleE[of m fs i] nth_append[of fs gs i]
      apply presburger
   proof-
     assume B: "\<not> i < length fs"
     then have C: "length fs \<le> i \<and> i < length (fs @ gs)"
-      using A  not_le 
+      using A  not_le
       by blast
     then have "i - length fs < length gs"
-      using length_append[of fs gs] 
+      using length_append[of fs gs]
       by linarith
-    then show ?thesis 
-      using A assms is_poly_tupleE[of m gs "i - length fs"] nth_append[of fs gs i] B 
+    then show ?thesis
+      using A assms is_poly_tupleE[of m gs "i - length fs"] nth_append[of fs gs i] B
       by presburger
   qed
   qed
 qed
 
 lemma Qp_poly_mapE:
   assumes "is_poly_tuple n fs"
   assumes "length fs = m"
   assumes "as \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "j < m"
   shows "(poly_map n fs as)!j \<in> carrier Q\<^sub>p"
   using assms  poly_map_closed cartesian_power_car_memE' by blast
-  
+
 lemma Qp_poly_mapE':
   assumes "as \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "length (poly_map n fs as) = length fs"
-  unfolding poly_map_def 
-  using Qp.cring_axioms poly_tuple_evalE' 
+  unfolding poly_map_def
+  using Qp.cring_axioms poly_tuple_evalE'
   by (metis assms restrict_def)
-    
+
 lemma Qp_poly_mapE'':
   assumes "is_poly_tuple n fs"
   assumes "length fs = m"
   assumes "n \<noteq> 0"
   assumes "as \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "j < m"
   shows "(poly_map n fs as)!j = (Qp_ev (fs!j) as)"
-  using assms 
+  using assms
   unfolding poly_map_def poly_tuple_eval_def
   by (metis (no_types, lifting) nth_map restrict_apply')
-  
+
 lemma poly_map_apply:
   assumes "as \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "poly_map n fs as = poly_tuple_eval fs as"
-  unfolding poly_map_def restrict_def 
+  unfolding poly_map_def restrict_def
   by (simp add: assms)
 
-lemma poly_map_pullbackI: 
+lemma poly_map_pullbackI:
   assumes "is_poly_tuple n fs"
   assumes "as \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "poly_map n fs as \<in> S"
   shows "as \<in> poly_map n fs \<inverse>\<^bsub>n\<^esub> S"
-  using assms poly_map_apply 
+  using assms poly_map_apply
   by blast
- 
-lemma poly_map_pullbackI': 
+
+lemma poly_map_pullbackI':
   assumes "is_poly_tuple n fs"
   assumes "as \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "poly_map n fs as \<in> S"
   shows "as \<in> ((poly_map n fs) -` S)"
   by (simp add: assms(3))
 
 text\<open>lemmas about polynomial composition\<close>
 lemma  poly_compose_ring_hom:
   assumes "is_poly_tuple m fs"
   assumes "length fs = n"
   shows  "(ring_hom_ring (Q\<^sub>p[\<X>\<^bsub>n\<^esub>]) (Q\<^sub>p[\<X>\<^bsub>m\<^esub>]) (Qp_poly_comp m fs))"
-  unfolding Qp_poly_comp_def  
+  unfolding Qp_poly_comp_def
   by (simp add: assms(1) assms(2)  poly_compose_ring_hom)
 
 lemma poly_compose_closed:
   assumes "is_poly_tuple m fs"
   assumes "length fs = n"
   assumes "f \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   shows "(Qp_poly_comp m fs f) \<in> carrier (Q\<^sub>p[\<X>\<^bsub>m\<^esub>])"
-  using Qp.cring_axioms assms 
-  unfolding Qp_poly_comp_def  
+  using Qp.cring_axioms assms
+  unfolding Qp_poly_comp_def
   using poly_compose_closed by blast
 
 
 lemma poly_compose_add:
   assumes "is_poly_tuple m fs"
   assumes "length fs = n"
   assumes "f \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   assumes "g \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   shows "Qp_poly_comp m fs (f \<oplus>\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> g) = (Qp_poly_comp m fs f) \<oplus>\<^bsub>Q\<^sub>p[\<X>\<^bsub>m\<^esub>]\<^esub>  (Qp_poly_comp m fs g)"
   using Qp.cring_axioms assms poly_compose_add
-  unfolding is_poly_tuple_def Qp_poly_comp_def  
+  unfolding is_poly_tuple_def Qp_poly_comp_def
   by blast
 
 lemma poly_compose_mult:
   assumes "is_poly_tuple m fs"
   assumes "length fs = n"
   assumes "f \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   assumes "g \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   shows "Qp_poly_comp m fs (f \<otimes>\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> g) = (Qp_poly_comp m fs f) \<otimes>\<^bsub>Q\<^sub>p[\<X>\<^bsub>m\<^esub>]\<^esub>  (Qp_poly_comp m fs g)"
   using Qp.cring_axioms assms poly_compose_mult
-  unfolding is_poly_tuple_def Qp_poly_comp_def  
+  unfolding is_poly_tuple_def Qp_poly_comp_def
   by blast
 
 lemma poly_compose_const:
   assumes "is_poly_tuple m fs"
   assumes "length fs = n"
   assumes "a \<in> carrier Q\<^sub>p"
   shows "Qp_poly_comp m fs (Qp_to_IP a) = Qp_to_IP a"
   using Qp.cring_axioms assms poly_compose_const
-  unfolding is_poly_tuple_def Qp_poly_comp_def  
+  unfolding is_poly_tuple_def Qp_poly_comp_def
   by metis
 
 lemma Qp_poly_comp_eval:
   assumes "is_poly_tuple m fs"
   assumes "length fs = n"
   assumes "f \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   assumes "as \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
   shows "Qp_ev (Qp_poly_comp m fs f) as = Qp_ev f (poly_map m fs as)"
 proof-
   have "(restrict (poly_tuple_eval fs) (carrier (Q\<^sub>p\<^bsup>m\<^esup>)) as) = poly_tuple_eval fs as"
-    unfolding restrict_def 
+    unfolding restrict_def
     by (simp add: assms)
-  thus ?thesis 
-  using assms Qp.cring_axioms poly_compose_eval 
-  unfolding Qp_poly_comp_def poly_map_def 
+  thus ?thesis
+  using assms Qp.cring_axioms poly_compose_eval
+  unfolding Qp_poly_comp_def poly_map_def
   by presburger
 qed
 
 
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsection\<open>Mapping Univariate Polynomials to Multivariable Polynomials in One Variable\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 
 abbreviation(input) to_Qp_x where
 "to_Qp_x \<equiv> (IP_to_UP (0::nat) :: (nat multiset \<Rightarrow> padic_number) \<Rightarrow> nat \<Rightarrow> padic_number)"
 
 abbreviation(input) from_Qp_x where
 "from_Qp_x \<equiv> UP_to_IP Q\<^sub>p (0::nat)"
 
 lemma from_Qp_x_closed:
   assumes "q \<in> carrier Q\<^sub>p_x"
   shows "from_Qp_x q \<in> carrier (Q\<^sub>p[\<X>\<^bsub>1\<^esub>])"
-  using assms UP_to_IP_closed unfolding coord_ring_def  
+  using assms UP_to_IP_closed unfolding coord_ring_def
   by (metis One_nat_def lessThan_0 lessThan_Suc)
 
 lemma to_Qp_x_closed:
   assumes "q \<in> carrier (Q\<^sub>p[\<X>\<^bsub>1\<^esub>])"
   shows "to_Qp_x q \<in> carrier Q\<^sub>p_x"
   using assms Qp.IP_to_UP_closed[of q "0::nat"] Qp.cring_axioms
-  unfolding coord_ring_def 
+  unfolding coord_ring_def
   by (metis One_nat_def lessThan_0 lessThan_Suc)
-  
+
 lemma to_Qp_x_from_Qp_x:
   assumes "q \<in> carrier (Q\<^sub>p[\<X>\<^bsub>1\<^esub>])"
   shows "from_Qp_x (to_Qp_x q) = q"
   using assms UP_to_IP_inv[of q "0::nat"] Qp.Pring_car
-  unfolding coord_ring_def 
+  unfolding coord_ring_def
     by (metis One_nat_def lessThan_0 lessThan_Suc)
 
 lemma from_Qp_x_to_Qp_x:
   assumes "q \<in> carrier Q\<^sub>p_x"
   shows "to_Qp_x (from_Qp_x q) = q"
   by (meson UPQ.IP_to_UP_inv assms)
 
 text\<open>ring hom properties of these maps\<close>
 
 lemma to_Qp_x_ring_hom:
 "to_Qp_x \<in> ring_hom (Q\<^sub>p[\<X>\<^bsub>1\<^esub>]) Q\<^sub>p_x"
   using IP_to_UP_ring_hom[of "0::nat"] ring_hom_ring.homh
-  unfolding coord_ring_def 
+  unfolding coord_ring_def
     by (metis One_nat_def lessThan_0 lessThan_Suc)
 
 lemma from_Qp_x_ring_hom:
 "from_Qp_x \<in> ring_hom Q\<^sub>p_x (Q\<^sub>p[\<X>\<^bsub>1\<^esub>])"
   using  UP_to_IP_ring_hom ring_hom_ring.homh
-  unfolding coord_ring_def 
+  unfolding coord_ring_def
     by (metis One_nat_def lessThan_0 lessThan_Suc)
 
 
 lemma from_Qp_x_add:
   assumes "a \<in> carrier Q\<^sub>p_x"
   assumes "b \<in> carrier Q\<^sub>p_x"
   shows "from_Qp_x (a \<oplus>\<^bsub>Q\<^sub>p_x\<^esub> b) = from_Qp_x a \<oplus>\<^bsub>Q\<^sub>p[\<X>\<^bsub>1\<^esub>]\<^esub> from_Qp_x b"
   by (metis (mono_tags, lifting) assms(1) assms(2) from_Qp_x_ring_hom ring_hom_add)
-  
+
 lemma from_Qp_x_mult:
   assumes "a \<in> carrier Q\<^sub>p_x"
   assumes "b \<in> carrier Q\<^sub>p_x"
   shows "from_Qp_x (a \<otimes>\<^bsub>Q\<^sub>p_x\<^esub> b) = from_Qp_x a \<otimes>\<^bsub>Q\<^sub>p[\<X>\<^bsub>1\<^esub>]\<^esub> from_Qp_x b"
   by (metis assms(1) assms(2) from_Qp_x_ring_hom ring_hom_mult)
 
 text\<open>equivalence of evaluation maps\<close>
 
 lemma Qp_poly_Qp_x_eval:
   assumes "P \<in> carrier (Q\<^sub>p[\<X>\<^bsub>1\<^esub>])"
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>)"
   shows "Qp_ev P a = (to_Qp_x P)\<bullet>(Qp.to_R a)"
 proof-
   have 0: "(IP_to_UP 0 P) \<bullet> (a ! 0) = ((IP_to_UP 0 P) \<bullet> (if 0 < length a then a ! 0 else \<zero>))"
-    using assms 
+    using assms
     by (metis (mono_tags, lifting) cartesian_power_car_memE gr0I zero_neq_one)
   have 1: "closed_fun Q\<^sub>p (\<lambda>n. if n < length a then a ! n else \<zero>)"
   proof
     fix n
     show "(if n < length a then a ! n else \<zero>) \<in> carrier Q\<^sub>p"
       apply(cases "n < length a")
-      using assms 
+      using assms
       apply (metis cartesian_power_car_memE cartesian_power_car_memE')
       by (meson Qp.cring_axioms cring.cring_simprules(2))
   qed
   have 2: " P \<in> Pring_set Q\<^sub>p {0::nat}"
-    using assms unfolding coord_ring_def 
-   by (metis Qp.Pring_car UPQ.UP_to_IP_closed assms(1) to_Qp_x_closed to_Qp_x_from_Qp_x)   
+    using assms unfolding coord_ring_def
+   by (metis Qp.Pring_car UPQ.UP_to_IP_closed assms(1) to_Qp_x_closed to_Qp_x_from_Qp_x)
   have 3: "total_eval Q\<^sub>p (\<lambda>i. if i < length a then a ! i else \<zero>) P = IP_to_UP 0 P \<bullet> (if 0 < length a then a ! 0 else \<zero>)"
-    using 1 2 assms IP_to_UP_poly_eval[of P "0::nat" "(\<lambda>i. if i < length a then a ! i else \<zero>)" ]          
-          UPQ.to_fun_def by presburger    
-  then show ?thesis 
-    using 0  
-    unfolding eval_at_point_def  
+    using 1 2 assms IP_to_UP_poly_eval[of P "0::nat" "(\<lambda>i. if i < length a then a ! i else \<zero>)" ]
+          UPQ.to_fun_def by presburger
+  then show ?thesis
+    using 0
+    unfolding eval_at_point_def
     by blast
 qed
 
 lemma Qp_x_Qp_poly_eval:
   assumes "P \<in> carrier Q\<^sub>p_x"
   assumes "a \<in> carrier Q\<^sub>p"
   shows "P \<bullet> a = Qp_ev (from_Qp_x P) (to_R1 a)"
 proof-
   have "Qp_ev (from_Qp_x P) (to_R1 a) = (to_Qp_x (from_Qp_x P))\<bullet>(Qp.to_R (Qp.to_R1 a))"
     using Qp_poly_Qp_x_eval assms(1) assms(2) from_Qp_x_closed Qp.to_R1_closed by blast
-  then show ?thesis using assms 
-    by (metis UPQ.IP_to_UP_inv Qp.to_R_to_R1)   
+  then show ?thesis using assms
+    by (metis UPQ.IP_to_UP_inv Qp.to_R_to_R1)
 qed
 
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsection\<open>$n^{th}$-Power Sets over $\mathbb{Q}_p$\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 definition P_set where
 "P_set (n::nat) = {a \<in> nonzero Q\<^sub>p.  (\<exists>y \<in> carrier Q\<^sub>p . (y[^] n) = a)}"
 
 lemma P_set_carrier:
 "P_set n \<subseteq> carrier Q\<^sub>p"
-  unfolding P_set_def nonzero_def 
-  by blast 
+  unfolding P_set_def nonzero_def
+  by blast
 
 lemma P_set_memI:
   assumes "a \<in> carrier Q\<^sub>p"
   assumes "a \<noteq> \<zero>"
   assumes "b \<in> carrier Q\<^sub>p"
   assumes "b[^](n::nat) = a"
   shows "a \<in> P_set n"
-  unfolding P_set_def 
-  using assms 
+  unfolding P_set_def
+  using assms
   by (metis (mono_tags, lifting) mem_Collect_eq not_nonzero_Qp)
 
 lemma P_set_nonzero:
 "P_set n \<subseteq> nonzero Q\<^sub>p"
-  unfolding P_set_def by blast 
+  unfolding P_set_def by blast
 
 
 lemma P_set_nonzero':
   assumes "a \<in> P_set n"
   shows "a \<in> nonzero Q\<^sub>p"
         "a \<in> carrier  Q\<^sub>p"
-  using assms P_set_nonzero P_set_carrier 
-   apply blast using assms P_set_carrier  by blast 
+  using assms P_set_nonzero P_set_carrier
+   apply blast using assms P_set_carrier  by blast
 
 lemma P_set_one:
   assumes "n \<noteq> 0"
-  shows "\<one> \<in> P_set (n::nat)" 
-proof-
-  have 0: "\<one>[^]n = \<one>" 
-    using Qp.nat_pow_one by blast    
+  shows "\<one> \<in> P_set (n::nat)"
+proof-
+  have 0: "\<one>[^]n = \<one>"
+    using Qp.nat_pow_one by blast
   have 1: "\<one> \<in> carrier Q\<^sub>p"
-    by blast        
-  then show ?thesis   
-    using one_nonzero unfolding P_set_def 
-    using 0 by blast       
+    by blast
+  then show ?thesis
+    using one_nonzero unfolding P_set_def
+    using 0 by blast
 qed
 
 lemma zeroth_P_set:
 "P_set 0 = {\<one>}"
 proof
   show "P_set 0 \<subseteq> {\<one>}"
-  unfolding P_set_def 
+  unfolding P_set_def
   proof
     fix x
     assume "x \<in> {a \<in> nonzero Q\<^sub>p. \<exists>y\<in>carrier Q\<^sub>p. (y[^](0::nat)) = a}"
     then have "\<exists>y\<in>carrier Q\<^sub>p. (y[^](0::nat)) = x"
-      by blast 
+      by blast
     then obtain a where a_def: "a \<in> carrier Q\<^sub>p \<and> (a[^](0::nat)) = x"
-      by blast 
+      by blast
     then show "x \<in> {\<one>}"
       using Qp.nat_pow_0 by blast
   qed
   show "{\<one>} \<subseteq> P_set 0"
-    using P_set_memI[of \<one> \<one> 0] Qp.nat_pow_one Qp.one_closed local.one_neq_zero 
+    using P_set_memI[of \<one> \<one> 0] Qp.nat_pow_one Qp.one_closed local.one_neq_zero
     by blast
 qed
 
 lemma P_set_mult_closed:
   assumes "n \<noteq> 0"
   assumes "a \<in> P_set n"
   assumes "b \<in> P_set n"
   shows "a \<otimes> b \<in> P_set n"
 proof-
   obtain a0 where a0_def: "a0 \<in> carrier Q\<^sub>p \<and> (a0 [^] n = a)"
     using assms(2)
     unfolding P_set_def
     by blast
   obtain b0 where b0_def: "b0 \<in> carrier Q\<^sub>p \<and> (b0 [^] n = b)"
     using assms(3)
     unfolding P_set_def
     by blast
   have "(a0 \<otimes> b0) [^] n = a0 [^] n \<otimes> b0 [^] n"
-    using a0_def b0_def Qp.nat_pow_distrib by blast    
+    using a0_def b0_def Qp.nat_pow_distrib by blast
   then have 0: "a \<otimes> b  = (a0 \<otimes> b0) [^] n"
-    using a0_def b0_def by blast    
+    using a0_def b0_def by blast
   have 1: "a0 \<otimes> b0 \<in> carrier Q\<^sub>p"
     by (meson Qp.cring_axioms a0_def b0_def cring.cring_simprules(5))
   have 2: "a \<otimes> b \<in> nonzero Q\<^sub>p"
-    using assms nonzero_is_submonoid  unfolding P_set_def 
-    by (metis (no_types, lifting) "0" "1" Qp.integral Qp_nat_pow_nonzero a0_def b0_def mem_Collect_eq not_nonzero_Qp)   
-  then show ?thesis 
-    using 0 1 assms 
-    unfolding P_set_def by blast 
-qed
-    
+    using assms nonzero_is_submonoid  unfolding P_set_def
+    by (metis (no_types, lifting) "0" "1" Qp.integral Qp_nat_pow_nonzero a0_def b0_def mem_Collect_eq not_nonzero_Qp)
+  then show ?thesis
+    using 0 1 assms
+    unfolding P_set_def by blast
+qed
+
 lemma P_set_inv_closed:
   assumes "a \<in> P_set n"
   shows "inv a \<in> P_set n"
 proof(cases "n = 0")
   case True
-  then show ?thesis 
+  then show ?thesis
     using assms zeroth_P_set
-    by (metis Qp.inv_one singletonD)    
+    by (metis Qp.inv_one singletonD)
 next
   case False
   then show ?thesis proof-
     obtain a0 where a0_def: "a0 \<in> carrier Q\<^sub>p \<and> a0[^]n = a"
-      using assms P_set_def[of n] by blast             
+      using assms P_set_def[of n] by blast
   have "a0 \<in> nonzero Q\<^sub>p"
     apply(rule ccontr)
   proof-
     assume "a0 \<notin> nonzero Q\<^sub>p "
     then have "a0 = \<zero>"
-      using a0_def 
-      by (meson not_nonzero_Qp)      
+      using a0_def
+      by (meson not_nonzero_Qp)
     then show False using a0_def assms
       by (metis (mono_tags, lifting) False P_set_def Qp.cring_axioms \<open>a0 \<notin> nonzero Q\<^sub>p\<close>
-          cring_def mem_Collect_eq neq0_conv ring.pow_zero) 
+          cring_def mem_Collect_eq neq0_conv ring.pow_zero)
   qed
   then have "(inv a0)[^]n = inv a"
-    using a0_def  \<open>a0 \<in> carrier Q\<^sub>p \<and> (a0[^]n) = a\<close> \<open>a0 \<in> nonzero Q\<^sub>p\<close> Units_nonzero 
-          monoid.nat_pow_of_inv[of Q\<^sub>p a n] Qp.nat_pow_of_inv Units_eq_nonzero by presburger    
-  then show ?thesis 
-    by (metis P_set_memI Qp.nat_pow_closed Qp.nonzero_memE(2) Qp.nonzero_pow_nonzero \<open>a0 \<in> nonzero Q\<^sub>p\<close> a0_def inv_in_frac(1) inv_in_frac(2))   
+    using a0_def  \<open>a0 \<in> carrier Q\<^sub>p \<and> (a0[^]n) = a\<close> \<open>a0 \<in> nonzero Q\<^sub>p\<close> Units_nonzero
+          monoid.nat_pow_of_inv[of Q\<^sub>p a n] Qp.nat_pow_of_inv Units_eq_nonzero by presburger
+  then show ?thesis
+    by (metis P_set_memI Qp.nat_pow_closed Qp.nonzero_memE(2) Qp.nonzero_pow_nonzero \<open>a0 \<in> nonzero Q\<^sub>p\<close> a0_def inv_in_frac(1) inv_in_frac(2))
   qed
 qed
 
 lemma P_set_val:
   assumes "a \<in> P_set (n::nat)"
   shows  "(ord a) mod n = 0"
 proof(cases "n = 0")
   case True
-  then show ?thesis 
-    using assms zeroth_P_set 
-    by (metis mod_by_0 of_nat_0 ord_one singletonD)     
+  then show ?thesis
+    using assms zeroth_P_set
+    by (metis mod_by_0 of_nat_0 ord_one singletonD)
 next
   case False
   then show ?thesis
   proof-
     obtain b where b_def: "b \<in> carrier Q\<^sub>p \<and> (b[^] n) = a"
-      using assms P_set_def  by blast 
+      using assms P_set_def  by blast
     have an: "a \<in> nonzero Q\<^sub>p"
       using P_set_def assms by blast
     have bn: "b \<in> nonzero Q\<^sub>p"
     proof(rule ccontr)
       assume "b \<notin> nonzero Q\<^sub>p"
       then have "b = \<zero>\<^bsub> Q\<^sub>p\<^esub>"
-        using b_def not_nonzero_Qp 
-        by metis        
+        using b_def not_nonzero_Qp
+        by metis
       then have "(b[^] n) = \<zero>"
         using False Qp.cring_axioms cring_def ring.pow_zero
         by blast
-      then show False 
-        using b_def an  Qp.not_nonzero_memI by blast        
-    qed
-    then have "ord a = n * (ord b)" 
-      using b_def an nonzero_nat_pow_ord 
-      by blast
-    then show ?thesis 
+      then show False
+        using b_def an  Qp.not_nonzero_memI by blast
+    qed
+    then have "ord a = n * (ord b)"
+      using b_def an nonzero_nat_pow_ord
+      by blast
+    then show ?thesis
       by (metis mod_mult_self1_is_0)
   qed
 qed
 
 
 lemma P_set_pow:
   assumes "n > 0"
   assumes "s \<in> P_set n"
   shows "s[^]k \<in> P_set (n*k)"
 proof-
   obtain y where y_def: "y \<in> carrier Q\<^sub>p \<and> y[^]n = s"
     using assms unfolding P_set_def by blast
   then have 0: "y \<in> nonzero Q\<^sub>p"
     using assms P_set_nonzero'(1) Qp_nonzero_nat_pow by blast
   have 1: "y[^](n*k) = s[^] k"
     using 0 y_def assms Qp.nat_pow_pow by blast
   hence 2: "s[^]k \<in> nonzero Q\<^sub>p"
     using 0 by (metis Qp_nat_pow_nonzero)
   thus ?thesis unfolding P_set_def using 1 y_def by blast
 qed
 
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsection\<open>Semialgebraic Sets\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 text\<open>
-  In this section we introduce the notion of a $p$-adic semialgebraic set. Intuitively, these are 
-  the subsets of $\mathbb{Q}_p^n$ which are definable by first order quantifier-free formulas in 
-  the standard first-order language of rings, with an additional relation symbol included for the 
-  relation $\text{ val}(x) \leq \text{ val}(y)$, interpreted according to the definiton of the 
-  $p$-adic valuation on $\mathbb{Q}_p$. In fact, by Macintyre's quantifier elimination theorem 
-  for the first-order theory of $\mathbb{Q}_p$ in this language, one can equivalently remove the 
-  ``quantifier-free" clause from the latter definition. The definition we give here is also 
-  equivalent, and due to Denef in \cite{denef1986}. The given definition here is desirable mainly 
+  In this section we introduce the notion of a $p$-adic semialgebraic set. Intuitively, these are
+  the subsets of $\mathbb{Q}_p^n$ which are definable by first order quantifier-free formulas in
+  the standard first-order language of rings, with an additional relation symbol included for the
+  relation $\text{ val}(x) \leq \text{ val}(y)$, interpreted according to the definiton of the
+  $p$-adic valuation on $\mathbb{Q}_p$. In fact, by Macintyre's quantifier elimination theorem
+  for the first-order theory of $\mathbb{Q}_p$ in this language, one can equivalently remove the
+  ``quantifier-free" clause from the latter definition. The definition we give here is also
+  equivalent, and due to Denef in \cite{denef1986}. The given definition here is desirable mainly
   for its utility in producing a proof of Macintyre's theorem, which is our overarching goal.
 \<close>
       (********************************************************************************************)
       (********************************************************************************************)
       subsubsection\<open>Defining Semialgebraic Sets\<close>
       (********************************************************************************************)
       (********************************************************************************************)
 
 definition basic_semialg_set where
 "basic_semialg_set (m::nat) (n::nat) P = {q \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). \<exists>y \<in> carrier Q\<^sub>p. Qp_ev P q = (y[^]n)}"
 
 lemma basic_semialg_set_zero_set:
   assumes "P \<in> carrier (Q\<^sub>p[\<X>\<^bsub>m\<^esub>])"
   assumes "q \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
   assumes "Qp_ev P q = \<zero>"
   assumes "n \<noteq>  0"
   shows "q \<in> basic_semialg_set (m::nat) (n::nat) P"
 proof-
   have "\<zero> = (\<zero>[^]n)"
-    using assms(4) Qp.nat_pow_zero by blast   
-  then show ?thesis 
-    unfolding basic_semialg_set_def 
-    using assms Qp.cring_axioms cring.cring_simprules(2) 
+    using assms(4) Qp.nat_pow_zero by blast
+  then show ?thesis
+    unfolding basic_semialg_set_def
+    using assms Qp.cring_axioms cring.cring_simprules(2)
     by blast
 qed
 
 lemma basic_semialg_set_def':
   assumes "n \<noteq> 0"
   assumes "P \<in> carrier (Q\<^sub>p[\<X>\<^bsub>m\<^esub>])"
   shows "basic_semialg_set (m::nat) (n::nat) P = {q \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). Qp_ev P q = \<zero> \<or> Qp_ev P q \<in> (P_set n)}"
 proof
   show "basic_semialg_set m n P \<subseteq> {q \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). Qp_ev P q = \<zero> \<or> Qp_ev P q \<in> P_set n}"
   proof
     fix x
     assume A: "x \<in> basic_semialg_set m n P"
      show "x \<in> {q \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). Qp_ev P q = \<zero> \<or> Qp_ev P q \<in> P_set n}"
       apply(cases "Qp_ev P x = \<zero>")
       using A basic_semialg_set_def apply blast
-      unfolding basic_semialg_set_def P_set_def 
+      unfolding basic_semialg_set_def P_set_def
     proof
       assume A0: "Qp_ev P x \<noteq> \<zero>"
       have A1: " \<exists>y\<in>carrier Q\<^sub>p. Qp_ev P x = (y[^]n)"
-        using A basic_semialg_set_def 
+        using A basic_semialg_set_def
         by blast
       have A2: "x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
-        using A basic_semialg_set_def 
+        using A basic_semialg_set_def
         by blast
       show " x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<and> (Qp_ev P x = \<zero> \<or> Qp_ev P x \<in> {a \<in> nonzero Q\<^sub>p. \<exists>y\<in>carrier Q\<^sub>p. (y[^]n) = a})"
-        by (metis (mono_tags, lifting) A1 A2 Qp.nonzero_memI assms(2) eval_at_point_closed mem_Collect_eq)        
+        by (metis (mono_tags, lifting) A1 A2 Qp.nonzero_memI assms(2) eval_at_point_closed mem_Collect_eq)
     qed
   qed
   show "{q \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). Qp_ev P q = \<zero> \<or> Qp_ev P q \<in> P_set n} \<subseteq> basic_semialg_set m n P"
   proof
     fix x
     assume A: " x \<in> {q \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). Qp_ev P q = \<zero> \<or> Qp_ev P q \<in> P_set n}"
     then have A':"x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
-      by blast 
+      by blast
     show "x \<in>  basic_semialg_set m n P"
       using A A'
       apply(cases "Qp_ev P x = \<zero>")
       using assms basic_semialg_set_zero_set[of P m x n]
        apply blast
     proof-
       assume B: "x \<in> {q \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). Qp_ev P q = \<zero> \<or> Qp_ev P q \<in> P_set n} "
       assume B': "x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
       assume B'': "Qp_ev P x \<noteq> \<zero> "
       show "x \<in> basic_semialg_set m n P"
-        unfolding basic_semialg_set_def  P_set_def 
+        unfolding basic_semialg_set_def  P_set_def
       proof
         have "\<exists>y\<in>carrier Q\<^sub>p. Qp_ev P x = (y[^]n) "
-          using A nonzero_def [of Q\<^sub>p] unfolding P_set_def  
+          using A nonzero_def [of Q\<^sub>p] unfolding P_set_def
         proof -
           assume "x \<in> {q \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). Qp_ev P q = \<zero> \<or> Qp_ev P q \<in> {a \<in> nonzero Q\<^sub>p. \<exists>y\<in>carrier Q\<^sub>p. (y[^]n) = a}}"
           then have "Qp_ev P x \<in> nonzero Q\<^sub>p \<and> (\<exists>r. r \<in> carrier Q\<^sub>p \<and> (r[^]n) = Qp_ev P x)"
             using B'' by blast
           then show ?thesis
             by blast
         qed
         then show "x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<and> (\<exists>y\<in>carrier Q\<^sub>p. Qp_ev P x = (y[^]n))"
-          using B' 
+          using B'
           by blast
       qed
     qed
   qed
 qed
 
 lemma basic_semialg_set_memI:
   assumes "q \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
-  assumes "y \<in> carrier Q\<^sub>p" 
+  assumes "y \<in> carrier Q\<^sub>p"
   assumes "Qp_ev P q = (y[^]n)"
   shows "q \<in> basic_semialg_set m n P"
-  using assms(1) assms(2) assms(3) basic_semialg_set_def 
+  using assms(1) assms(2) assms(3) basic_semialg_set_def
   by blast
 
 lemma basic_semialg_set_memE:
   assumes  "q \<in> basic_semialg_set m n P"
   shows "q \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
         "\<exists>y \<in> carrier Q\<^sub>p. Qp_ev P q = (y[^]n)"
   using assms basic_semialg_set_def apply blast
   using assms basic_semialg_set_def by blast
 
 definition is_basic_semialg :: "nat \<Rightarrow> ((nat \<Rightarrow> int) \<times> (nat \<Rightarrow> int)) set list set \<Rightarrow> bool" where
 "is_basic_semialg m S \<equiv> (\<exists> (n::nat) \<noteq> 0. (\<exists> P \<in> carrier (Q\<^sub>p[\<X>\<^bsub>m\<^esub>]). S = basic_semialg_set m n P))"
 
-abbreviation(input) basic_semialgs where 
+abbreviation(input) basic_semialgs where
 "basic_semialgs m \<equiv> {S. (is_basic_semialg m S)}"
 
 definition semialg_sets where
-"semialg_sets n = gen_boolean_algebra (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) (basic_semialgs n)" 
+"semialg_sets n = gen_boolean_algebra (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) (basic_semialgs n)"
 
 lemma carrier_is_semialg:
 "(carrier (Q\<^sub>p\<^bsup>n\<^esup>)) \<in> semialg_sets n "
-  unfolding semialg_sets_def 
+  unfolding semialg_sets_def
   using gen_boolean_algebra.universe by blast
 
 lemma empty_set_is_semialg:
 " {} \<in> semialg_sets n"
-  using carrier_is_semialg[of n] 
+  using carrier_is_semialg[of n]
   unfolding semialg_sets_def using gen_boolean_algebra.complement
   by (metis Diff_cancel)
 
 lemma semialg_intersect:
   assumes "A \<in> semialg_sets n"
   assumes "B \<in> semialg_sets n"
   shows "(A \<inter> B) \<in> semialg_sets n "
-  using assms(1) assms(2) gen_boolean_algebra_intersect semialg_sets_def 
+  using assms(1) assms(2) gen_boolean_algebra_intersect semialg_sets_def
   by blast
 
 lemma semialg_union:
   assumes "A \<in> semialg_sets n"
   assumes "B \<in> semialg_sets n"
   shows "(A \<union> B) \<in> semialg_sets n "
   using assms gen_boolean_algebra.union semialg_sets_def
   by blast
 
 lemma semialg_complement:
   assumes "A \<in> semialg_sets n"
   shows "(carrier (Q\<^sub>p\<^bsup>n\<^esup>) - A) \<in> semialg_sets n "
-  using assms gen_boolean_algebra.complement semialg_sets_def 
+  using assms gen_boolean_algebra.complement semialg_sets_def
   by blast
 
 lemma semialg_zero:
   assumes "A \<in> semialg_sets 0"
   shows "A = {[]} \<or> A = {}"
   using assms
-  unfolding semialg_sets_def cartesian_power_def 
+  unfolding semialg_sets_def cartesian_power_def
 proof-
   assume A0:  " A \<in> gen_boolean_algebra (carrier (RDirProd_list (R_list 0 Q\<^sub>p))) (basic_semialgs 0)"
   show " A = {[]} \<or> A = {}"
   proof-
     have "A \<noteq> {[]} \<longrightarrow> A = {}"
     proof
       assume A1: "A \<noteq> {[]}"
       show "A = {}"
       proof-
         have "(R_list 0 Q\<^sub>p) = []"
           by simp
         then have "(carrier (RDirProd_list (R_list 0 Q\<^sub>p))) = {[]}"
-          using RDirProd_list_nil 
+          using RDirProd_list_nil
           by simp
-        then show ?thesis 
-          using A0 A1 
+        then show ?thesis
+          using A0 A1
           by (metis gen_boolean_algebra_subset subset_singletonD)
       qed
     qed
-    then show ?thesis 
+    then show ?thesis
       by linarith
   qed
 qed
 
 lemma basic_semialg_is_semialg:
   assumes "is_basic_semialg n A"
   shows "A \<in> semialg_sets n"
-  by (metis (no_types, lifting) assms gen_boolean_algebra.simps inf_absorb1 
-      is_basic_semialg_def mem_Collect_eq basic_semialg_set_def 
+  by (metis (no_types, lifting) assms gen_boolean_algebra.simps inf_absorb1
+      is_basic_semialg_def mem_Collect_eq basic_semialg_set_def
       semialg_sets_def subsetI)
 
 lemma basic_semialg_is_semialg':
   assumes "f \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   assumes "m \<noteq>0"
   assumes "A = basic_semialg_set n m f"
   shows "A \<in> semialg_sets n"
-  using assms basic_semialg_is_semialg is_basic_semialg_def 
+  using assms basic_semialg_is_semialg is_basic_semialg_def
   by blast
 
 definition is_semialgebraic where
 "is_semialgebraic n S = (S \<in> semialg_sets n)"
 
 lemma is_semialgebraicE:
   assumes "is_semialgebraic n S"
   shows "S \<in> semialg_sets n"
   using assms is_semialgebraic_def by blast
 
 lemma is_semialgebraic_closed:
   assumes "is_semialgebraic n S"
   shows "S \<subseteq> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
-  using is_semialgebraicE[of n S] unfolding semialg_sets_def 
-  using assms gen_boolean_algebra_subset is_semialgebraicE semialg_sets_def 
+  using is_semialgebraicE[of n S] unfolding semialg_sets_def
+  using assms gen_boolean_algebra_subset is_semialgebraicE semialg_sets_def
   by blast
 
 lemma is_semialgebraicI:
   assumes "S \<in> semialg_sets n"
   shows "is_semialgebraic n S"
   by (simp add: assms is_semialgebraic_def)
 
 lemma basic_semialg_is_semialgebraic:
   assumes "is_basic_semialg n A"
   shows "is_semialgebraic n A"
   using assms basic_semialg_is_semialg is_semialgebraicI by blast
 
 lemma basic_semialg_is_semialgebraic':
   assumes "f \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   assumes "m \<noteq>0"
   assumes "A = basic_semialg_set n m f"
   shows "is_semialgebraic n A"
   using assms(1) assms(2) assms(3) basic_semialg_is_semialg' is_semialgebraicI by blast
 
 
       (********************************************************************************************)
       (********************************************************************************************)
       subsubsection\<open>Algebraic Sets over $p$-adic Fields\<close>
       (********************************************************************************************)
       (********************************************************************************************)
 
 lemma p_times_square_not_square:
   assumes "a \<in> nonzero Q\<^sub>p"
   shows "\<pp> \<otimes> (a [^] (2::nat)) \<notin> P_set (2::nat)"
 proof
   assume A: "\<pp> \<otimes> (a[^](2::nat)) \<in> P_set (2::nat)"
-  then have "\<pp> \<otimes> (a[^](2::nat)) \<in> nonzero Q\<^sub>p" 
-    unfolding P_set_def 
-    by blast    
+  then have "\<pp> \<otimes> (a[^](2::nat)) \<in> nonzero Q\<^sub>p"
+    unfolding P_set_def
+    by blast
   then obtain b where b_def: "b \<in> carrier Q\<^sub>p \<and> b[^](2::nat) = \<pp> \<otimes> (a[^](2::nat))"
-    using A P_set_def by blast 
+    using A P_set_def by blast
    have "b \<in> nonzero Q\<^sub>p"
-     apply(rule ccontr) using b_def assms 
-     by (metis A P_set_nonzero'(1) Qp.nat_pow_zero not_nonzero_Qp zero_neq_numeral)     
+     apply(rule ccontr) using b_def assms
+     by (metis A P_set_nonzero'(1) Qp.nat_pow_zero not_nonzero_Qp zero_neq_numeral)
    then have LHS: "ord (b[^](2::nat)) = 2* (ord b)"
      using nonzero_nat_pow_ord
      by presburger
    have "ord( \<pp> \<otimes> (a[^](2::nat))) = 1 + 2* ord a"
-     using assms nonzero_nat_pow_ord Qp_nat_pow_nonzero ord_mult ord_p p_nonzero 
-     by presburger      
-   then show False 
-     using b_def LHS 
+     using assms nonzero_nat_pow_ord Qp_nat_pow_nonzero ord_mult ord_p p_nonzero
+     by presburger
+   then show False
+     using b_def LHS
      by presburger
 qed
 
 lemma p_times_square_not_square':
   assumes "a \<in> carrier Q\<^sub>p"
   shows "\<pp> \<otimes> (a [^] (2::nat)) = \<zero> \<Longrightarrow> a = \<zero>"
-  by (metis Qp.integral Qp.nat_pow_closed Qp.nonzero_closed Qp.nonzero_memE(2) Qp.nonzero_pow_nonzero assms p_nonzero)  
+  by (metis Qp.integral Qp.nat_pow_closed Qp.nonzero_closed Qp.nonzero_memE(2) Qp.nonzero_pow_nonzero assms p_nonzero)
 
 lemma zero_set_semialg_set:
   assumes "q \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "Qp_ev q a = \<zero> \<longleftrightarrow>( \<exists>y \<in> carrier Q\<^sub>p. \<pp> \<otimes> ((Qp_ev q a) [^] (2::nat)) = y[^](2::nat)) "
 proof
   show "Qp_ev q a = \<zero> \<Longrightarrow> \<exists>y\<in>carrier Q\<^sub>p. \<pp> \<otimes> (Qp_ev q a[^] (2::nat)) = (y[^] (2::nat))"
   proof-
     assume "Qp_ev q a = \<zero>"
     then have "\<pp> \<otimes> (Qp_ev q a[^](2::nat)) = (\<zero>[^](2::nat))"
-      by (metis Qp.int_inc_closed Qp.nat_pow_zero Qp.r_null zero_neq_numeral)              
+      by (metis Qp.int_inc_closed Qp.nat_pow_zero Qp.r_null zero_neq_numeral)
     then have "\<zero> \<in> carrier Q\<^sub>p \<and> \<pp> \<otimes> (Qp_ev q a[^](2::nat)) = (\<zero>[^](2::nat))"
-      using Qp.cring_axioms cring.cring_simprules(2) 
+      using Qp.cring_axioms cring.cring_simprules(2)
       by blast
     then show "\<exists>y\<in>carrier Q\<^sub>p. \<pp> \<otimes> (Qp_ev q a[^] (2::nat)) = (y[^] (2::nat))"
-      by blast 
+      by blast
   qed
   show " \<exists>y\<in>carrier Q\<^sub>p. \<pp> \<otimes> (Qp_ev q a[^](2::nat)) = (y[^](2::nat)) \<Longrightarrow> Qp_ev q a = \<zero>"
   proof-
     assume A: " \<exists>y\<in>carrier Q\<^sub>p. \<pp> \<otimes> (Qp_ev q a[^](2::nat)) = (y[^](2::nat))"
     then obtain b where b_def: "b\<in>carrier Q\<^sub>p \<and> \<pp> \<otimes> (Qp_ev q a[^](2::nat)) = (b[^](2::nat))"
-      by blast 
+      by blast
     show "Qp_ev q a = \<zero>"
     proof(rule ccontr)
       assume " Qp_ev q a \<noteq> \<zero>"
-      then have " Qp_ev q a \<in> nonzero Q\<^sub>p" using assms eval_at_point_closed[of a n q]  nonzero_def 
+      then have " Qp_ev q a \<in> nonzero Q\<^sub>p" using assms eval_at_point_closed[of a n q]  nonzero_def
       proof -
         have "Qp_ev q a \<in> carrier Q\<^sub>p"
-          using \<open>\<lbrakk>a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>); q \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])\<rbrakk> \<Longrightarrow> 
-                  Qp_ev q a \<in> carrier Q\<^sub>p\<close> \<open>a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)\<close> \<open>q \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])\<close> 
+          using \<open>\<lbrakk>a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>); q \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])\<rbrakk> \<Longrightarrow>
+                  Qp_ev q a \<in> carrier Q\<^sub>p\<close> \<open>a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)\<close> \<open>q \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])\<close>
           by fastforce
         then have "Qp_ev q a \<in> {r \<in> carrier Q\<^sub>p. r \<noteq> \<zero>}"
           using \<open>Qp_ev q a \<noteq> \<zero>\<close> by force
         then show ?thesis
           by (metis nonzero_def )
       qed
       then have "\<pp> \<otimes> (Qp_ev q a[^](2::nat)) \<in> nonzero Q\<^sub>p"
-        by (metis Qp.nonzero_closed Qp.nonzero_mult_closed Qp_nat_pow_nonzero not_nonzero_Qp p_nonzero p_times_square_not_square')        
+        by (metis Qp.nonzero_closed Qp.nonzero_mult_closed Qp_nat_pow_nonzero not_nonzero_Qp p_nonzero p_times_square_not_square')
       then have "\<pp> \<otimes> (Qp_ev q a[^](2::nat)) \<in> P_set (2::nat)"
-        using b_def 
-        unfolding P_set_def 
-        by blast 
-      then show False 
-        using \<open>Qp_ev q a \<in> nonzero Q\<^sub>p\<close> p_times_square_not_square 
+        using b_def
+        unfolding P_set_def
+        by blast
+      then show False
+        using \<open>Qp_ev q a \<in> nonzero Q\<^sub>p\<close> p_times_square_not_square
         by blast
     qed
   qed
 qed
 
 lemma alg_as_semialg:
   assumes "P \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   assumes "q = \<pp> \<odot>\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> (P[^]\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> (2::nat))"
   shows "zero_set Q\<^sub>p n P = basic_semialg_set n (2::nat) q"
 proof
   have 00: "\<And>x. x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<Longrightarrow> Qp_ev q x = \<pp> \<otimes> (Qp_ev P x) [^] (2::nat)"
-    using assms eval_at_point_smult MP.nat_pow_closed Qp.int_inc_closed eval_at_point_nat_pow 
-    by presburger  
+    using assms eval_at_point_smult MP.nat_pow_closed Qp.int_inc_closed eval_at_point_nat_pow
+    by presburger
   show "V\<^bsub>Q\<^sub>p\<^esub> n P \<subseteq> basic_semialg_set n 2 q"
   proof
     fix x
     assume A: "x \<in> V\<^bsub>Q\<^sub>p\<^esub> n P "
     show "x \<in> basic_semialg_set n (2::nat) q "
     proof-
       have P: "Qp_ev P x = \<zero>"
-        using A zero_setE(2) 
+        using A zero_setE(2)
         by blast
       have "Qp_ev q x = \<zero>"
       proof-
         have "Qp_ev q x = \<pp> \<otimes> (Qp_ev (P[^]\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> (2::nat)) x)"
           using assms eval_at_point_smult[of x n "(P[^]\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> (2::nat))" \<pp>] basic_semialg_set_def
-          by (meson A MP.nat_pow_closed Qp.int_inc_closed zero_setE(1))                
-        then show ?thesis 
-          by (metis A P Qp.int_inc_closed Qp.integral_iff Qp.nat_pow_zero Qp.zero_closed assms(1) 
-              eval_at_point_nat_pow neq0_conv zero_less_numeral zero_setE(1))                        
+          by (meson A MP.nat_pow_closed Qp.int_inc_closed zero_setE(1))
+        then show ?thesis
+          by (metis A P Qp.int_inc_closed Qp.integral_iff Qp.nat_pow_zero Qp.zero_closed assms(1)
+              eval_at_point_nat_pow neq0_conv zero_less_numeral zero_setE(1))
       qed
       then have 0: "Qp_ev q x = \<zero> \<or> Qp_ev q x \<in> P_set (2::nat)"
         by blast
       have 1: "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
         using A zero_setE(1)
         by blast
-      then show ?thesis using 0 basic_semialg_set_def' 
-        by (metis (no_types, opaque_lifting) Qp.nat_pow_zero Qp.zero_closed 
-            \<open>eval_at_point Q\<^sub>p x q = \<zero>\<close> basic_semialg_set_memI zero_neq_numeral)                                            
+      then show ?thesis using 0 basic_semialg_set_def'
+        by (metis (no_types, opaque_lifting) Qp.nat_pow_zero Qp.zero_closed
+            \<open>eval_at_point Q\<^sub>p x q = \<zero>\<close> basic_semialg_set_memI zero_neq_numeral)
     qed
   qed
   show "basic_semialg_set n 2 q \<subseteq> V\<^bsub>Q\<^sub>p\<^esub> n P"
   proof
-    fix x 
+    fix x
     assume A: "x \<in> basic_semialg_set n 2 q"
     have 0: "\<not> Qp_ev q x \<in> P_set 2"
     proof
       assume "Qp_ev q x \<in> P_set 2"
       then have 0: "Qp_ev q x \<in> nonzero Q\<^sub>p \<and>  (\<exists>y \<in> carrier Q\<^sub>p . (y[^] (2::nat)) = Qp_ev q x)"
         using P_set_def by blast
       have "( \<exists>y \<in> carrier Q\<^sub>p. \<pp> \<otimes> ((Qp_ev P x) [^] (2::nat)) = y[^](2::nat))"
       proof-
         obtain y where y_def: "y \<in> carrier Q\<^sub>p \<and> (y[^] (2::nat)) = Qp_ev q x"
-          using 0 by blast 
+          using 0 by blast
         have "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
           using A basic_semialg_set_memE(1) by blast
         then have "Qp_ev q x = \<pp> \<otimes> ((Qp_ev P x) [^] (2::nat))"
-          using assms eval_at_point_scalar_mult 00 by blast         
+          using assms eval_at_point_scalar_mult 00 by blast
         then have "(y[^] (2::nat)) = \<pp> \<otimes> ((Qp_ev P x) [^] (2::nat))"
           using y_def by blast
-        then show ?thesis using y_def by blast 
+        then show ?thesis using y_def by blast
       qed
       then have "Qp_ev P x = \<zero>"
         by (metis (no_types, lifting) A assms(1) basic_semialg_set_def mem_Collect_eq zero_set_semialg_set)
       then have "Qp_ev q x = \<zero>"
-        using assms eval_at_point_smult 
-        by (metis "00" A Qp.int_inc_closed Qp.nat_pow_zero Qp.r_null basic_semialg_set_memE(1) zero_neq_numeral)                                    
-      then show False 
-        using 0 Qp.not_nonzero_memI by blast        
+        using assms eval_at_point_smult
+        by (metis "00" A Qp.int_inc_closed Qp.nat_pow_zero Qp.r_null basic_semialg_set_memE(1) zero_neq_numeral)
+      then show False
+        using 0 Qp.not_nonzero_memI by blast
     qed
     show " x \<in> V\<^bsub>Q\<^sub>p\<^esub> n P"
       apply(rule zero_setI)
       using A basic_semialg_set_memE(1) apply blast
-      using A 0 00[of x] 
+      using A 0 00[of x]
       by (metis assms(1) basic_semialg_set_memE(1) basic_semialg_set_memE(2) zero_set_semialg_set)
   qed
 qed
 
 lemma is_zero_set_imp_basic_semialg:
   assumes "P \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   assumes "S = zero_set Q\<^sub>p n P"
   shows "is_basic_semialg n S"
   unfolding is_basic_semialg_def
 proof-
   obtain q where q_def: "q = \<pp> \<odot>\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> (P[^]\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> (2::nat))"
     by blast
   have 0: "zero_set Q\<^sub>p n P = basic_semialg_set n (2::nat) q"
-    using alg_as_semialg[of P n q]  q_def assms(1) by linarith    
+    using alg_as_semialg[of P n q]  q_def assms(1) by linarith
   have "(P [^]\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> (2::nat)) \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
-    using assms(1) 
-    by blast    
+    using assms(1)
+    by blast
   then have "\<pp> \<odot>\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub>(P [^]\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> (2::nat)) \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
-    using assms q_def Qp.int_inc_closed local.smult_closed by blast 
+    using assms q_def Qp.int_inc_closed local.smult_closed by blast
   then have 1: "q \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
     by (metis q_def )
-  then show "\<exists>m. m \<noteq> 0 \<and> (\<exists>P\<in>carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>]). S = basic_semialg_set n m P)" 
+  then show "\<exists>m. m \<noteq> 0 \<and> (\<exists>P\<in>carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>]). S = basic_semialg_set n m P)"
     using 0 assms
-    by (metis zero_neq_numeral)   
+    by (metis zero_neq_numeral)
 qed
 
 lemma is_zero_set_imp_semialg:
   assumes "P \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   assumes "S = zero_set Q\<^sub>p n P"
   shows "is_semialgebraic n S"
   using assms(1) assms(2) basic_semialg_is_semialg is_semialgebraicI is_zero_set_imp_basic_semialg
   by blast
 
 text\<open>Algebraic sets are semialgebraic\<close>
 
 lemma is_algebraic_imp_is_semialg:
   assumes "is_algebraic Q\<^sub>p n S"
   shows "is_semialgebraic n S"
   proof(rule is_semialgebraicI)
     obtain ps where ps_def: "finite ps \<and> ps \<subseteq> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>]) \<and> S =  affine_alg_set Q\<^sub>p n ps"
-      using is_algebraicE 
+      using is_algebraicE
       by (metis  assms)
     have "ps \<subseteq> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>]) \<longrightarrow> affine_alg_set Q\<^sub>p n ps \<in> semialg_sets n"
       apply(rule finite.induct[of ps])
         apply (simp add: ps_def)
       using affine_alg_set_empty[of n]
        apply (simp add: carrier_is_semialg)
     proof
       fix A a
       assume IH: "A \<subseteq> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>]) \<longrightarrow> affine_alg_set Q\<^sub>p n A \<in> semialg_sets n"
       assume P: "insert a A \<subseteq> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
       have "A \<subseteq> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
         using P by blast
-      then 
+      then
       show "affine_alg_set Q\<^sub>p n (insert a A) \<in> semialg_sets n"
-        using IH P semialg_intersect[of "affine_alg_set Q\<^sub>p n A" n "affine_alg_set Q\<^sub>p n {a}" ] 
-               is_zero_set_imp_semialg affine_alg_set_insert[of n a A] 
+        using IH P semialg_intersect[of "affine_alg_set Q\<^sub>p n A" n "affine_alg_set Q\<^sub>p n {a}" ]
+               is_zero_set_imp_semialg affine_alg_set_insert[of n a A]
         by (metis Int_commute affine_alg_set_singleton insert_subset is_semialgebraicE)
     qed
-    then show "S \<in> semialg_sets n" 
+    then show "S \<in> semialg_sets n"
       using ps_def by blast
   qed
 
 
       (********************************************************************************************)
       (********************************************************************************************)
       subsubsection\<open>Basic Lemmas about the Semialgebraic Predicate\<close>
       (********************************************************************************************)
       (********************************************************************************************)
 
 
 text\<open>Finite and cofinite sets are semialgebraic\<close>
 
 lemma finite_is_semialg:
   assumes "F \<subseteq> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "finite F"
   shows "is_semialgebraic n F"
-  using Qp.finite_sets_are_algebraic is_algebraic_imp_is_semialg[of n F] 
-           assms(1) assms(2)  
+  using Qp.finite_sets_are_algebraic is_algebraic_imp_is_semialg[of n F]
+           assms(1) assms(2)
   by blast
 
 definition is_cofinite where
 "is_cofinite n F = finite (ring_pow_comp Q\<^sub>p n F)"
 
 lemma is_cofiniteE:
   assumes "F \<subseteq> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "is_cofinite n F"
   shows "finite (carrier (Q\<^sub>p\<^bsup>n\<^esup>) - F)"
-  using assms(2) is_cofinite_def 
+  using assms(2) is_cofinite_def
   by (simp add: ring_pow_comp_def)
-  
+
 lemma complement_is_semialg:
   assumes "is_semialgebraic n F"
   shows "is_semialgebraic n ((carrier (Q\<^sub>p\<^bsup>n\<^esup>)) - F)"
   using assms is_semialgebraic_def semialg_complement by blast
 
 lemma cofinite_is_semialgebraic:
   assumes "F \<subseteq> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "is_cofinite n F"
   shows "is_semialgebraic n F"
-  using assms ring_pow_comp_inv[of F Q\<^sub>p n] complement_is_semialg[of n "(carrier (Q\<^sub>p\<^bsup>n\<^esup>) - F)"] 
+  using assms ring_pow_comp_inv[of F Q\<^sub>p n] complement_is_semialg[of n "(carrier (Q\<^sub>p\<^bsup>n\<^esup>) - F)"]
         finite_is_semialg[of "(carrier (Q\<^sub>p\<^bsup>n\<^esup>) - F)"]  is_cofiniteE[of F]
   by (simp add: ring_pow_comp_def)
-  
+
 lemma diff_is_semialgebraic:
   assumes "is_semialgebraic n A"
   assumes "is_semialgebraic n B"
   shows "is_semialgebraic n (A - B)"
   apply(rule is_semialgebraicI)
-  using assms unfolding semialg_sets_def 
-  using gen_boolean_algebra_diff is_semialgebraicE semialg_sets_def 
+  using assms unfolding semialg_sets_def
+  using gen_boolean_algebra_diff is_semialgebraicE semialg_sets_def
   by blast
 
 lemma intersection_is_semialg:
   assumes "is_semialgebraic n A"
   assumes "is_semialgebraic n B"
   shows "is_semialgebraic n (A \<inter> B)"
-  using assms(1) assms(2) is_semialgebraicE is_semialgebraicI semialg_intersect 
+  using assms(1) assms(2) is_semialgebraicE is_semialgebraicI semialg_intersect
   by blast
 
 lemma union_is_semialgebraic:
   assumes "is_semialgebraic n A"
   assumes "is_semialgebraic n B"
   shows "is_semialgebraic n (A \<union> B)"
   using assms(1) assms(2) is_semialgebraicE is_semialgebraicI semialg_union by blast
 
 lemma carrier_is_semialgebraic:
 "is_semialgebraic n (carrier (Q\<^sub>p\<^bsup>n\<^esup>))"
-using carrier_is_semialg 
+using carrier_is_semialg
   by (simp add: carrier_is_semialg is_semialgebraic_def)
 
 lemma empty_is_semialgebraic:
 "is_semialgebraic n {}"
   by (simp add: empty_set_is_semialg is_semialgebraic_def)
 
 
       (********************************************************************************************)
       (********************************************************************************************)
       subsubsection\<open>One-Dimensional Semialgebraic Sets\<close>
       (********************************************************************************************)
       (********************************************************************************************)
 
 definition one_var_semialg where
 "one_var_semialg S =  ((to_R1 ` S) \<in> (semialg_sets 1))"
 
 definition univ_basic_semialg_set where
 "univ_basic_semialg_set (m::nat) P = {a \<in> carrier Q\<^sub>p. (\<exists>y \<in> carrier Q\<^sub>p. (P \<bullet> a = (y[^]m)))}"
 
 text\<open>Equivalence of univ\_basic\_semialg\_sets and semialgebraic subsets of $\mathbb{Q}^1$ \<close>
 
 lemma univ_basic_semialg_set_to_semialg_set:
   assumes "P \<in> carrier Q\<^sub>p_x"
   assumes "m \<noteq> 0"
   shows "to_R1 ` (univ_basic_semialg_set m P) = basic_semialg_set 1 m (from_Qp_x P)"
 proof
   show "(\<lambda>a. [a]) ` univ_basic_semialg_set m P \<subseteq> basic_semialg_set 1 m (from_Qp_x P)"
   proof fix x
     assume A: "x \<in> (\<lambda>a. [a]) ` univ_basic_semialg_set m P"
     then obtain b y where by_def:"b \<in> carrier Q\<^sub>p \<and> y \<in> carrier Q\<^sub>p \<and> (P \<bullet> b) = (y[^]m) \<and> x = [b]"
-      unfolding univ_basic_semialg_set_def 
+      unfolding univ_basic_semialg_set_def
       by blast
     then have "x \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>)"
       using A Qp.to_R1_closed[of b]
-      unfolding univ_basic_semialg_set_def 
+      unfolding univ_basic_semialg_set_def
       by blast
     then show "x \<in> basic_semialg_set 1 m (from_Qp_x P)"
       using by_def Qp_x_Qp_poly_eval assms
-      unfolding basic_semialg_set_def 
+      unfolding basic_semialg_set_def
       by blast
   qed
   show "basic_semialg_set 1 m (from_Qp_x P) \<subseteq> (\<lambda>a. [a]) ` univ_basic_semialg_set m P"
   proof
     fix x
     assume A: "x \<in> basic_semialg_set 1 m (from_Qp_x P)"
-    then obtain b where b_def:  "b \<in> carrier Q\<^sub>p \<and> x = [b]" 
-      unfolding basic_semialg_set_def 
+    then obtain b where b_def:  "b \<in> carrier Q\<^sub>p \<and> x = [b]"
+      unfolding basic_semialg_set_def
       by (metis (mono_tags, lifting) mem_Collect_eq Qp.to_R1_to_R Qp.to_R_pow_closed)
     obtain y where y_def: "y \<in> carrier Q\<^sub>p \<and> (Qp_ev (from_Qp_x P) [b]  = (y[^]m))"
       using A b_def
-      unfolding basic_semialg_set_def 
+      unfolding basic_semialg_set_def
       by blast
     have " P \<bullet> b = (y[^]m)"
-      using assms y_def b_def Qp_x_Qp_poly_eval by blast            
+      using assms y_def b_def Qp_x_Qp_poly_eval by blast
     then show " x \<in> (\<lambda>a. [a]) ` univ_basic_semialg_set m P"
-      using y_def b_def 
+      using y_def b_def
       unfolding basic_semialg_set_def univ_basic_semialg_set_def
       by blast
   qed
 qed
 
 definition is_univ_semialgebraic where
 "is_univ_semialgebraic S = (S \<subseteq> carrier Q\<^sub>p \<and> is_semialgebraic 1 (to_R1 ` S))"
 
 lemma is_univ_semialgebraicE:
   assumes "is_univ_semialgebraic S"
   shows "is_semialgebraic 1 (to_R1 ` S)"
   using assms is_univ_semialgebraic_def by blast
 
 lemma is_univ_semialgebraicI:
   assumes "is_semialgebraic 1 (to_R1 ` S)"
   shows "is_univ_semialgebraic S"
 proof-
   have "S \<subseteq> carrier Q\<^sub>p"
-  proof fix x assume "x \<in> S" 
-    then have "(to_R1 x) \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>)" 
-      using assms 
+  proof fix x assume "x \<in> S"
+    then have "(to_R1 x) \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>)"
+      using assms
       by (smt Collect_mono_iff gen_boolean_algebra_subset image_def is_semialgebraicE mem_Collect_eq semialg_sets_def Qp.to_R1_carrier)
-    then show "x \<in> carrier Q\<^sub>p" 
-      using assms 
+    then show "x \<in> carrier Q\<^sub>p"
+      using assms
       by (metis nth_Cons_0 Qp.to_R_pow_closed)
   qed
-  then show ?thesis 
-    using assms 
-    unfolding is_univ_semialgebraic_def  
-    by blast
-qed 
+  then show ?thesis
+    using assms
+    unfolding is_univ_semialgebraic_def
+    by blast
+qed
 
 lemma univ_basic_semialg_set_is_univ_semialgebraic:
   assumes "P \<in> carrier Q\<^sub>p_x"
   assumes "m \<noteq> 0"
   shows "is_univ_semialgebraic (univ_basic_semialg_set m P)"
-  using assms 
-  by (metis (mono_tags, lifting) basic_semialg_is_semialgebraic' 
-      from_Qp_x_closed is_univ_semialgebraic_def mem_Collect_eq subsetI 
+  using assms
+  by (metis (mono_tags, lifting) basic_semialg_is_semialgebraic'
+      from_Qp_x_closed is_univ_semialgebraic_def mem_Collect_eq subsetI
       univ_basic_semialg_set_def univ_basic_semialg_set_to_semialg_set)
 
 lemma intersection_is_univ_semialgebraic:
   assumes "is_univ_semialgebraic A"
   assumes "is_univ_semialgebraic B"
   shows "is_univ_semialgebraic (A \<inter> B)"
-  using assms intersection_is_semialg[of 1 "((\<lambda>a. [a]) ` A)" "((\<lambda>a. [a]) ` B)"]  
-  unfolding is_univ_semialgebraic_def 
+  using assms intersection_is_semialg[of 1 "((\<lambda>a. [a]) ` A)" "((\<lambda>a. [a]) ` B)"]
+  unfolding is_univ_semialgebraic_def
   by (metis le_infI1 Qp.to_R1_intersection)
 
 lemma union_is_univ_semialgebraic:
   assumes "is_univ_semialgebraic A"
   assumes "is_univ_semialgebraic B"
   shows "is_univ_semialgebraic (A \<union> B)"
-  using assms union_is_semialgebraic[of 1 "((\<lambda>a. [a]) ` A)" "((\<lambda>a. [a]) ` B)"]  
-  unfolding is_univ_semialgebraic_def 
+  using assms union_is_semialgebraic[of 1 "((\<lambda>a. [a]) ` A)" "((\<lambda>a. [a]) ` B)"]
+  unfolding is_univ_semialgebraic_def
   by (metis Un_subset_iff image_Un)
 
 lemma diff_is_univ_semialgebraic:
   assumes "is_univ_semialgebraic A"
   assumes "is_univ_semialgebraic B"
   shows "is_univ_semialgebraic (A - B)"
-  using assms diff_is_semialgebraic[of 1 "((\<lambda>a. [a]) ` A)" "((\<lambda>a. [a]) ` B)"]  
-  unfolding is_univ_semialgebraic_def 
+  using assms diff_is_semialgebraic[of 1 "((\<lambda>a. [a]) ` A)" "((\<lambda>a. [a]) ` B)"]
+  unfolding is_univ_semialgebraic_def
   by (smt Diff_subset subset_trans Qp.to_R1_diff)
 
 lemma finite_is_univ_semialgebraic:
   assumes "A \<subseteq> carrier Q\<^sub>p"
   assumes "finite A"
   shows "is_univ_semialgebraic A"
-  using assms finite_is_semialg[of "((\<lambda>a. [a]) ` A)" ]  to_R1_finite[of A] 
-  unfolding is_univ_semialgebraic_def 
-  by (metis Qp.to_R1_carrier Qp.to_R1_subset) 
+  using assms finite_is_semialg[of "((\<lambda>a. [a]) ` A)" ]  to_R1_finite[of A]
+  unfolding is_univ_semialgebraic_def
+  by (metis Qp.to_R1_carrier Qp.to_R1_subset)
 
 
       (********************************************************************************************)
       (********************************************************************************************)
       subsubsection\<open>Defining the $p$-adic Valuation Semialgebraically\<close>
       (********************************************************************************************)
       (********************************************************************************************)
 
 
 lemma Qp_square_root_criterion0:
   assumes "p \<noteq> 2"
   assumes "a \<in> carrier Q\<^sub>p"
   assumes "b \<in> carrier Q\<^sub>p"
   assumes "val a \<le> val b"
   assumes "a \<noteq> \<zero>"
   assumes "b \<noteq> \<zero>"
   assumes "val a \<ge> 0"
   shows "\<exists>y \<in> carrier Q\<^sub>p. a[^](2::nat) \<oplus>\<^bsub>Q\<^sub>p\<^esub> \<pp>\<otimes>b[^](2::nat) = (y [^] (2::nat))"
 proof-
   have 0: "(to_Zp a) \<in> carrier Z\<^sub>p"
-    using assms(2) to_Zp_closed 
+    using assms(2) to_Zp_closed
     by blast
   have 1: "(to_Zp b) \<in> carrier Z\<^sub>p"
-    using assms(3) to_Zp_closed 
+    using assms(3) to_Zp_closed
     by blast
   have 2: "a \<in> \<O>\<^sub>p"
     using val_ring_val_criterion assms(2) assms(5) assms(7) by blast
   have 3: "b \<in> \<O>\<^sub>p"
-    using assms val_ring_val_criterion[of b] dual_order.trans by blast 
+    using assms val_ring_val_criterion[of b] dual_order.trans by blast
   have 4: "val_Zp (to_Zp b) = val b"
-    using 3 Zp_def \<iota>_def padic_fields.to_Zp_val padic_fields_axioms by blast    
+    using 3 Zp_def \<iota>_def padic_fields.to_Zp_val padic_fields_axioms by blast
   have 5:  "val_Zp (to_Zp a) = val a"
     using Q\<^sub>p_def Zp_def assms(2) assms(7) padic_fields.Qp_val_ringI padic_fields.to_Zp_val padic_fields_axioms
-    by blast    
-  have "\<exists>y \<in> carrier Z\<^sub>p. (to_Zp a)[^]\<^bsub>Z\<^sub>p\<^esub>(2::nat) \<oplus>\<^bsub>Z\<^sub>p\<^esub> \<p> \<otimes>\<^bsub>Z\<^sub>p\<^esub>(to_Zp b)[^]\<^bsub>Z\<^sub>p\<^esub>(2::nat) = (y [^]\<^bsub>Z\<^sub>p\<^esub> (2::nat))" 
+    by blast
+  have "\<exists>y \<in> carrier Z\<^sub>p. (to_Zp a)[^]\<^bsub>Z\<^sub>p\<^esub>(2::nat) \<oplus>\<^bsub>Z\<^sub>p\<^esub> \<p> \<otimes>\<^bsub>Z\<^sub>p\<^esub>(to_Zp b)[^]\<^bsub>Z\<^sub>p\<^esub>(2::nat) = (y [^]\<^bsub>Z\<^sub>p\<^esub> (2::nat))"
     using 0 1 2 4 5 assms Zp_square_root_criterion[of "(to_Zp a)" "(to_Zp b)"]
-    by (metis "3" to_Zp_inc to_Zp_zero zero_in_val_ring)   
-  then obtain y where y_def: "y \<in> carrier Z\<^sub>p \<and> (to_Zp a)[^]\<^bsub>Z\<^sub>p\<^esub>(2::nat) \<oplus>\<^bsub>Z\<^sub>p\<^esub> \<p> \<otimes>\<^bsub>Z\<^sub>p\<^esub>(to_Zp b)[^]\<^bsub>Z\<^sub>p\<^esub>(2::nat) = (y [^]\<^bsub>Z\<^sub>p\<^esub> (2::nat))" 
-    by blast
-  have 6: "a[^](2::nat) \<oplus>\<^bsub>Q\<^sub>p\<^esub> \<pp> \<otimes>b[^](2::nat) = ((\<iota> y) [^] (2::nat))" 
+    by (metis "3" to_Zp_inc to_Zp_zero zero_in_val_ring)
+  then obtain y where y_def: "y \<in> carrier Z\<^sub>p \<and> (to_Zp a)[^]\<^bsub>Z\<^sub>p\<^esub>(2::nat) \<oplus>\<^bsub>Z\<^sub>p\<^esub> \<p> \<otimes>\<^bsub>Z\<^sub>p\<^esub>(to_Zp b)[^]\<^bsub>Z\<^sub>p\<^esub>(2::nat) = (y [^]\<^bsub>Z\<^sub>p\<^esub> (2::nat))"
+    by blast
+  have 6: "a[^](2::nat) \<oplus>\<^bsub>Q\<^sub>p\<^esub> \<pp> \<otimes>b[^](2::nat) = ((\<iota> y) [^] (2::nat))"
   proof-
     have 0: "\<iota> (y [^]\<^bsub>Z\<^sub>p\<^esub> (2::nat)) = ((\<iota> y) [^] (2::nat))"
       using Qp_nonzero_nat_pow nat_pow_closed inc_pow nat_inc_zero inc_is_hom \<iota>_def y_def ring_hom_nat_pow[of Z\<^sub>p Q\<^sub>p \<iota> y 2]
-            Q\<^sub>p_def Qp.ring_axioms Zp.ring_axioms 
+            Q\<^sub>p_def Qp.ring_axioms Zp.ring_axioms
       by blast
     have 1: "\<iota> (y [^]\<^bsub>Z\<^sub>p\<^esub> (2::nat)) = \<iota> ((to_Zp a)[^]\<^bsub>Z\<^sub>p\<^esub>(2::nat) \<oplus>\<^bsub>Z\<^sub>p\<^esub> \<p> \<otimes>\<^bsub>Z\<^sub>p\<^esub>(to_Zp b)[^]\<^bsub>Z\<^sub>p\<^esub>(2::nat))"
       using y_def by presburger
     have 2: "\<iota> (y [^]\<^bsub>Z\<^sub>p\<^esub> (2::nat)) = \<iota> ((to_Zp a)[^]\<^bsub>Z\<^sub>p\<^esub>(2::nat)) \<oplus>\<^bsub>Q\<^sub>p\<^esub> \<iota> ( \<p> \<otimes>\<^bsub>Z\<^sub>p\<^esub>(to_Zp b)[^]\<^bsub>Z\<^sub>p\<^esub>(2::nat))"
-      using "1" Zp.m_closed Zp_int_inc_closed assms(2) assms(3) inc_of_sum pow_closed to_Zp_closed by presburger       
+      using "1" Zp.m_closed Zp_int_inc_closed assms(2) assms(3) inc_of_sum pow_closed to_Zp_closed by presburger
     hence 3: "\<iota> (y [^]\<^bsub>Z\<^sub>p\<^esub> (2::nat)) = (\<iota> (to_Zp a))[^](2::nat) \<oplus> (\<iota>  \<p>) \<otimes> \<iota> ((to_Zp b)[^]\<^bsub>Z\<^sub>p\<^esub>(2::nat))"
-      using Qp_nonzero_nat_pow nat_pow_closed inc_pow nat_inc_zero inc_is_hom \<iota>_def y_def ring_hom_nat_pow[of Z\<^sub>p Q\<^sub>p \<iota> _ 2]           
-            Q\<^sub>p_def Qp.ring_axioms Zp.ring_axioms Zp_int_inc_closed assms(2) assms(3) inc_of_prod pow_closed to_Zp_closed 
-      by metis 
+      using Qp_nonzero_nat_pow nat_pow_closed inc_pow nat_inc_zero inc_is_hom \<iota>_def y_def ring_hom_nat_pow[of Z\<^sub>p Q\<^sub>p \<iota> _ 2]
+            Q\<^sub>p_def Qp.ring_axioms Zp.ring_axioms Zp_int_inc_closed assms(2) assms(3) inc_of_prod pow_closed to_Zp_closed
+      by metis
     then show ?thesis
-      using "0" "4" val_ring_ord_criterion assms(2) assms(3) assms(4) assms(5) 
+      using "0" "4" val_ring_ord_criterion assms(2) assms(3) assms(4) assms(5)
           assms(6) assms(7) inc_pow not_nonzero_Zp ord_of_nonzero(1) p_inc to_Zp_closed to_Zp_inc
       by (metis to_Zp_zero val_pos val_ringI zero_in_val_ring)
   qed
   have "(\<iota> y) \<in> carrier Q\<^sub>p"
-    using frac_closed local.inc_def y_def inc_closed by blast    
-  then show ?thesis 
-    using 6 
-    by blast 
+    using frac_closed local.inc_def y_def inc_closed by blast
+  then show ?thesis
+    using 6
+    by blast
 qed
 
 lemma eint_minus_ineq':
   assumes "(a::eint) \<ge> b"
   shows "a -b \<ge> 0"
   by (metis assms eint_minus_ineq eint_ord_simps(3) idiff_infinity idiff_self order_trans top.extremum_unique top_eint_def)
 
 lemma Qp_square_root_criterion:
   assumes "p \<noteq> 2"
   assumes "a \<in> carrier Q\<^sub>p"
   assumes "b \<in> carrier Q\<^sub>p"
   assumes "ord b \<ge> ord a"
   assumes "a \<noteq> \<zero>"
   assumes "b \<noteq> \<zero>"
   shows "\<exists>y \<in> carrier Q\<^sub>p. a[^](2::nat) \<oplus>\<^bsub>Q\<^sub>p\<^esub> \<pp>\<otimes>b[^](2::nat) = (y [^] (2::nat))"
 proof-
   have "\<exists>k::int. k \<le> min (ord a) (ord b) \<and> k mod 2 = 0"
   proof-
       let ?k = "if (min (ord a) (ord b)) mod 2 = 0 then min (ord a) (ord b)  else (min (ord a) (ord b)) - 1"
       have "?k \<le> min (ord a) (ord b) \<and> ?k mod 2 = 0"
         apply(cases "(min (ord a) (ord b)) mod 2 = 0 ")
        apply presburger
         by presburger
-      then show ?thesis 
+      then show ?thesis
         by meson
   qed
   then obtain k where k_def: "k \<le> min (ord a) (ord b) \<and> k mod 2 = 0"
-    by meson 
+    by meson
   obtain a0 where a0_def: "a0 = (\<pp>[^](-k)) \<otimes> a"
     by blast
   obtain b0 where b0_def: "b0 = (\<pp>[^](-k)) \<otimes> b"
     by blast
   have 0: "a0 \<in> nonzero Q\<^sub>p"
     using Qp.cring_axioms Qp.field_axioms Ring.integral a0_def assms(2) assms(5) cring_simprules(5)
-          not_nonzero_Qp p_intpow_closed(1) p_nonzero 
-    by (metis Qp_int_pow_nonzero cring.cring_simprules(5))         
+          not_nonzero_Qp p_intpow_closed(1) p_nonzero
+    by (metis Qp_int_pow_nonzero cring.cring_simprules(5))
   have 1: "val a0 =  val a - k"
     using a0_def assms(2) assms(5) val_mult p_nonzero  p_intpow_closed(1)
-    by (metis Qp.m_comm Qp_int_pow_nonzero p_intpow_inv'' val_fract val_p_int_pow)         
+    by (metis Qp.m_comm Qp_int_pow_nonzero p_intpow_inv'' val_fract val_p_int_pow)
   have 11: "val b0 =  val b - k"
-    using assms(3) assms(6) b0_def val_mult p_nonzero p_intpow_closed(1) 
-    by (metis Qp.m_lcomm Qp.one_closed Qp.r_one Qp_int_pow_nonzero p_intpow_inv'' val_fract val_p_int_pow)            
+    using assms(3) assms(6) b0_def val_mult p_nonzero p_intpow_closed(1)
+    by (metis Qp.m_lcomm Qp.one_closed Qp.r_one Qp_int_pow_nonzero p_intpow_inv'' val_fract val_p_int_pow)
   have A: "val a \<ge> k"
     using k_def val_ord assms by (smt eint_ord_simps(1) not_nonzero_Qp)
   have B: "val b \<ge> k"
-    using  k_def val_ord assms by (smt eint_ord_simps(1) not_nonzero_Qp)    
+    using  k_def val_ord assms by (smt eint_ord_simps(1) not_nonzero_Qp)
   then have 2: "val a0 \<ge> 0"
     using A 1 assms k_def eint_minus_ineq eint_ord_code(5) local.eint_minus_ineq' by presburger
   have 3: "val a0 \<le> val b0"
-    using 1 11 assms 
-    by (metis eint.distinct(2) eint_minus_ineq eint_ord_simps(1) val_def)   
+    using 1 11 assms
+    by (metis eint.distinct(2) eint_minus_ineq eint_ord_simps(1) val_def)
   have 4: "a0 \<noteq> \<zero>"
-    using a0_def "0" Qp.nonzero_memE(2) by blast    
+    using a0_def "0" Qp.nonzero_memE(2) by blast
   have 5: "b0 \<noteq> \<zero>"
-    using b0_def 
-    by (metis "4" Qp.integral_iff a0_def assms(2) assms(3) assms(6) p_intpow_closed(1))   
+    using b0_def
+    by (metis "4" Qp.integral_iff a0_def assms(2) assms(3) assms(6) p_intpow_closed(1))
   have "\<exists>y \<in> carrier Q\<^sub>p. a0[^](2::nat) \<oplus>\<^bsub>Q\<^sub>p\<^esub> \<pp>\<otimes>b0[^](2::nat) = (y [^] (2::nat))"
-    using Qp_square_root_criterion0[of a0 b0] assms 2 3 4 5 b0_def a0_def Qp.m_closed p_intpow_closed(1) 
-    by metis    
+    using Qp_square_root_criterion0[of a0 b0] assms 2 3 4 5 b0_def a0_def Qp.m_closed p_intpow_closed(1)
+    by metis
   then obtain y where y_def: " y \<in> carrier Q\<^sub>p \<and> a0[^](2::nat) \<oplus>\<^bsub>Q\<^sub>p\<^esub> \<pp>\<otimes>b0[^](2::nat) = (y [^] (2::nat))"
-    by blast 
+    by blast
   then have 6: " (\<pp>[^] (2 * k)) \<otimes>  (a0[^](2::nat) \<oplus>\<^bsub>Q\<^sub>p\<^esub> \<pp>\<otimes>b0[^](2::nat)) =  (\<pp>[^] (2 * k))  \<otimes> (y [^] (2::nat))"
     by presburger
   then have 8: "((\<pp>[^] (2 * k)) \<otimes>  (a0[^](2::nat))) \<oplus>\<^bsub>Q\<^sub>p\<^esub>((\<pp>[^] (2 * k)) \<otimes> (\<pp>\<otimes>b0[^](2::nat))) =  (\<pp>[^] (2 * k))  \<otimes> (y [^] (2::nat))"
-    using 6 Qp.r_distr[of "(a0[^](2::nat))" " (\<pp>\<otimes>b0[^](2::nat))" "(\<pp>[^] (2 * k))"] 
-    by (metis Qp.add.int_pow_closed Qp.m_closed Qp.nat_pow_closed Qp.one_closed a0_def assms(2) assms(3) b0_def p_inc p_intpow_closed(1) y_def)    
+    using 6 Qp.r_distr[of "(a0[^](2::nat))" " (\<pp>\<otimes>b0[^](2::nat))" "(\<pp>[^] (2 * k))"]
+    by (metis Qp.add.int_pow_closed Qp.m_closed Qp.nat_pow_closed Qp.one_closed a0_def assms(2) assms(3) b0_def p_inc p_intpow_closed(1) y_def)
   have 9: "(\<pp>[^](int 2*k)) = (\<pp>[^]k)[^](2::nat)"
     using Qp_int_nat_pow_pow[of \<pp> k 2]
     by (metis mult_of_nat_commute p_nonzero)
   then have "((\<pp>[^]k)[^](2::nat) \<otimes>  (a0[^](2::nat))) \<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp>[^]k)[^](2::nat) \<otimes> (\<pp>\<otimes>b0[^](2::nat)) =  (\<pp>[^]k)[^](2::nat) \<otimes> (y [^] (2::nat))"
     by (metis "8" int_eq_iff_numeral)
   then have "((\<pp>[^]k) \<otimes> a0)[^](2::nat) \<oplus>\<^bsub>Q\<^sub>p\<^esub>((\<pp>[^]k)[^](2::nat)) \<otimes> (\<pp>\<otimes>b0[^](2::nat)) = ((\<pp>[^]k)[^](2::nat))  \<otimes> (y [^] (2::nat))"
     by (metis Qp.cring_axioms a0_def assms(2) comm_monoid.nat_pow_distrib cring.cring_simprules(5) cring_def p_intpow_closed(1))
   then have 10:  "((\<pp>[^]k) \<otimes> a0)[^](2::nat) \<oplus>\<^bsub>Q\<^sub>p\<^esub>((\<pp>[^]k)[^](2::nat)) \<otimes> (\<pp>\<otimes>b0[^](2::nat)) =  ((\<pp>[^]k)  \<otimes> y) [^] (2::nat)"
-    using comm_monoid.nat_pow_distrib y_def 
+    using comm_monoid.nat_pow_distrib y_def
     by (metis Qp.comm_monoid_axioms p_intpow_closed(1))
   then have "((\<pp>[^]k) \<otimes> a0)[^](2::nat) \<oplus>\<^bsub>Q\<^sub>p\<^esub>((((\<pp>[^]k)[^](2::nat)) \<otimes> \<pp>)\<otimes>b0[^](2::nat)) =  ((\<pp>[^]k)  \<otimes> y) [^] (2::nat)"
     using 10 monoid.m_assoc[of Q\<^sub>p "((\<pp>[^]k)[^](2::nat))" \<pp> " b0[^](2::nat)"]
-    by (metis Qp.int_inc_closed Qp.m_assoc Qp.m_closed Qp.nat_pow_closed assms(3) b0_def p_intpow_closed(1))    
+    by (metis Qp.int_inc_closed Qp.m_assoc Qp.m_closed Qp.nat_pow_closed assms(3) b0_def p_intpow_closed(1))
   then have "((\<pp>[^]k) \<otimes> a0)[^](2::nat) \<oplus>\<^bsub>Q\<^sub>p\<^esub>((\<pp> \<otimes> ((\<pp>[^]k)[^](2::nat)) )\<otimes>b0[^](2::nat)) =  ((\<pp>[^]k)  \<otimes> y) [^] (2::nat)"
     by (metis Qp.group_commutes_pow Qp.int_inc_closed Qp.m_comm p_intpow_closed(1))
   then have "((\<pp>[^]k) \<otimes> a0)[^](2::nat) \<oplus>\<^bsub>Q\<^sub>p\<^esub>\<pp> \<otimes> (((\<pp>[^]k)[^](2::nat)) \<otimes>b0[^](2::nat)) =  ((\<pp>[^]k)  \<otimes> y) [^] (2::nat)"
     by (metis "10" Qp.int_inc_closed Qp.m_closed Qp.m_lcomm Qp.nat_pow_closed assms(3) b0_def p_intpow_closed(1))
   then have "((\<pp>[^]k) \<otimes> a0)[^](2::nat) \<oplus>\<^bsub>Q\<^sub>p\<^esub>\<pp> \<otimes> ((\<pp>[^]k) \<otimes>b0)[^](2::nat) =  ((\<pp>[^]k)  \<otimes> y) [^] (2::nat)"
     by (metis Qp.m_closed Qp.nat_pow_distrib assms(3) b0_def p_intpow_closed(1))
   then have "a[^](2::nat) \<oplus>\<^bsub>Q\<^sub>p\<^esub>\<pp> \<otimes> b[^](2::nat) =  ((\<pp>[^]k)  \<otimes> y) [^] (2::nat)"
     by (metis Qp.l_one Qp.m_assoc a0_def assms(2) assms(3) b0_def p_intpow_closed(1) p_intpow_inv)
-  then show ?thesis 
+  then show ?thesis
     by (meson Qp.cring_axioms cring.cring_simprules(5) p_intpow_closed(1) y_def)
 qed
 
 lemma Qp_val_ring_alt_def0:
   assumes "a \<in> nonzero Q\<^sub>p"
   assumes "ord a \<ge> 0"
   shows "\<exists>y \<in> carrier Q\<^sub>p. \<one> \<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp>[^](3::nat))\<otimes> (a[^](4::nat)) = (y[^](2::nat))"
 proof-
   have "\<exists>y \<in> carrier Z\<^sub>p. \<one>\<^bsub>Z\<^sub>p\<^esub> \<oplus>\<^bsub>Z\<^sub>p\<^esub> (\<p> [^]\<^bsub>Z\<^sub>p\<^esub> (3::nat))\<otimes>\<^bsub>Z\<^sub>p\<^esub> ((to_Zp a) [^]\<^bsub>Z\<^sub>p\<^esub> (4::nat)) = (y [^]\<^bsub>Z\<^sub>p\<^esub> (2::nat))"
-    using padic_integers.Zp_semialg_eq[of p "to_Zp a"] prime assms to_Zp_def 
+    using padic_integers.Zp_semialg_eq[of p "to_Zp a"] prime assms to_Zp_def
     by (metis (no_types, lifting) Qp.nonzero_closed Qp.not_nonzero_memI Zp_def val_ring_ord_criterion not_nonzero_Zp padic_integers_axioms to_Zp_closed to_Zp_inc to_Zp_zero zero_in_val_ring)
   then obtain y where y_def: "y \<in> carrier Z\<^sub>p \<and> \<one>\<^bsub>Z\<^sub>p\<^esub> \<oplus>\<^bsub>Z\<^sub>p\<^esub> (\<p> [^]\<^bsub>Z\<^sub>p\<^esub> (3::nat))\<otimes>\<^bsub>Z\<^sub>p\<^esub> ((to_Zp a) [^]\<^bsub>Z\<^sub>p\<^esub> (4::nat)) = (y [^]\<^bsub>Z\<^sub>p\<^esub> (2::nat))"
     by blast
   then have "\<one> \<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp>[^](3::nat))\<otimes> (a[^](4::nat)) = ((\<iota> y)[^](2::nat))"
-    using  Group.nat_pow_0 Group.nat_pow_Suc nonzero_def 
-        val_ring_ord_criterion  assms inc_of_nonzero inc_of_prod inc_of_sum inc_pow 
-        m_closed nat_inc_closed nat_pow_closed  not_nonzero_Zp 
+    using  Group.nat_pow_0 Group.nat_pow_Suc nonzero_def
+        val_ring_ord_criterion  assms inc_of_nonzero inc_of_prod inc_of_sum inc_pow
+        m_closed nat_inc_closed nat_pow_closed  not_nonzero_Zp
         numeral_2_eq_2 p_natpow_inc to_Zp_closed to_Zp_inc
     by (smt Qp.nonzero_closed Qp.nonzero_memE(2) Zp.monom_term_car p_pow_nonzero(1) pow_closed to_Zp_zero zero_in_val_ring)
   then have "(\<iota> y) \<in> carrier Q\<^sub>p \<and> \<one> \<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp>[^](3::nat))\<otimes> (a[^](4::nat)) = ((\<iota> y)[^](2::nat))"
-    using y_def inc_closed by blast        
-  then show ?thesis 
-    by blast 
+    using y_def inc_closed by blast
+  then show ?thesis
+    by blast
 qed
 
 text\<open>Defining the valuation semialgebraically for odd primes\<close>
 
 lemma P_set_ord_semialg_odd_p:
   assumes "p \<noteq> 2"
   assumes "a \<in> carrier Q\<^sub>p"
   assumes "b \<in> carrier Q\<^sub>p"
   shows  "val a \<le> val b \<longleftrightarrow> (\<exists>y \<in> carrier Q\<^sub>p. (a[^](2::nat)) \<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp> \<otimes> (b[^](2::nat))) = (y[^](2::nat)))"
 proof(cases "a = \<zero>")
   case True
   show "val a \<le> val b \<longleftrightarrow> (\<exists>y \<in> carrier Q\<^sub>p. (a[^](2::nat)) \<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp> \<otimes> (b[^](2::nat))) = (y[^](2::nat)))"
   proof
     show "val b \<ge> val a \<Longrightarrow> \<exists>y\<in>carrier Q\<^sub>p. (a[^](2::nat)) \<oplus>\<^bsub>Q\<^sub>p\<^esub> \<pp> \<otimes> (b[^](2::nat)) = (y[^](2::nat))"
     proof-
       assume A: "val b \<ge> val a"
       then have "val b \<ge> \<infinity>"
         by (metis True local.val_zero)
-      then have "b = \<zero>" 
-        using assms(3) local.val_zero val_ineq by presburger        
+      then have "b = \<zero>"
+        using assms(3) local.val_zero val_ineq by presburger
       then have "(a[^](2::nat)) \<oplus>\<^bsub>Q\<^sub>p\<^esub> \<pp> \<otimes> (b[^](2::nat)) = (\<zero>[^](2::nat))"
-        using True 
-        by (metis Qp.int_inc_zero Qp.int_nat_pow_rep Qp.nonzero_closed Qp.r_null Qp.r_zero assms(3) p_nonzero zero_power2)        
-      then show ?thesis 
+        using True
+        by (metis Qp.int_inc_zero Qp.int_nat_pow_rep Qp.nonzero_closed Qp.r_null Qp.r_zero assms(3) p_nonzero zero_power2)
+      then show ?thesis
         using \<open>b = \<zero>\<close> assms(3) by blast
     qed
     show "\<exists>y\<in>carrier Q\<^sub>p. (a[^](2::nat)) \<oplus>\<^bsub>Q\<^sub>p\<^esub> \<pp> \<otimes> (b[^](2::nat)) = (y[^](2::nat)) \<Longrightarrow> val b \<ge> val a"
     proof-
       assume "\<exists>y\<in>carrier Q\<^sub>p. (a[^](2::nat)) \<oplus>\<^bsub>Q\<^sub>p\<^esub> \<pp> \<otimes> (b[^](2::nat)) = (y[^](2::nat))"
       then obtain y where y_def: "y \<in> carrier Q\<^sub>p \<and>(a[^](2::nat)) \<oplus>\<^bsub>Q\<^sub>p\<^esub> \<pp> \<otimes> (b[^](2::nat)) = (y[^](2::nat))"
-        by blast 
+        by blast
       then have 0: "\<pp> \<otimes> (b[^](2::nat)) = (y[^](2::nat))"
-        by (metis (no_types, lifting) Qp.add.r_cancel_one' Qp.int_inc_closed Qp.nat_pow_closed 
-            Qp.not_nonzero_memI Qp_nonzero_nat_pow True assms(2) assms(3) local.monom_term_car not_nonzero_Qp zero_less_numeral)        
+        by (metis (no_types, lifting) Qp.add.r_cancel_one' Qp.int_inc_closed Qp.nat_pow_closed
+            Qp.not_nonzero_memI Qp_nonzero_nat_pow True assms(2) assms(3) local.monom_term_car not_nonzero_Qp zero_less_numeral)
       have "b = \<zero>"
         apply(rule ccontr)
-        using 0 assms y_def p_times_square_not_square[of b] 
-        unfolding P_set_def  
-        by (metis (no_types, opaque_lifting) P_set_memI Qp.nat_pow_closed True 
-            \<open>b \<in> nonzero Q\<^sub>p \<Longrightarrow> \<pp> \<otimes> b [^] 2 \<notin> P_set 2\<close> not_nonzero_Qp p_times_square_not_square')        
-      then show ?thesis 
-        using eint_ord_code(3) local.val_zero by presburger        
+        using 0 assms y_def p_times_square_not_square[of b]
+        unfolding P_set_def
+        by (metis (no_types, opaque_lifting) P_set_memI Qp.nat_pow_closed True
+            \<open>b \<in> nonzero Q\<^sub>p \<Longrightarrow> \<pp> \<otimes> b [^] 2 \<notin> P_set 2\<close> not_nonzero_Qp p_times_square_not_square')
+      then show ?thesis
+        using eint_ord_code(3) local.val_zero by presburger
     qed
   qed
 next
-  case False 
-  then show ?thesis 
+  case False
+  then show ?thesis
   proof(cases "b = \<zero>")
     case True
     then have "(a[^](2::nat)) \<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp> \<otimes> (b[^](2::nat))) =  (a[^](2::nat))"
-      by (metis Qp.add.l_cancel_one' Qp.int_inc_zero Qp.int_nat_pow_rep Qp.nat_pow_closed Qp.nonzero_closed Qp.r_null assms(2) assms(3) p_nonzero zero_power2)     
-    then have 0: "(\<exists>y \<in> carrier Q\<^sub>p. (a[^](2::nat)) \<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp> \<otimes> (b[^](2::nat))) = (y[^](2::nat)))" 
-      using assms(2) 
+      by (metis Qp.add.l_cancel_one' Qp.int_inc_zero Qp.int_nat_pow_rep Qp.nat_pow_closed Qp.nonzero_closed Qp.r_null assms(2) assms(3) p_nonzero zero_power2)
+    then have 0: "(\<exists>y \<in> carrier Q\<^sub>p. (a[^](2::nat)) \<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp> \<otimes> (b[^](2::nat))) = (y[^](2::nat)))"
+      using assms(2)
       by blast
     have 1: "val a \<le> val b"
-      using True assms local.val_zero  eint_ord_code(3) by presburger      
+      using True assms local.val_zero  eint_ord_code(3) by presburger
     show "val a \<le> val b \<longleftrightarrow> (\<exists>y \<in> carrier Q\<^sub>p. (a[^](2::nat)) \<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp> \<otimes> (b[^](2::nat))) = (y[^](2::nat)))"
-      using 0 1 
-      by blast
-  next 
-    case F: False 
+      using 0 1
+      by blast
+  next
+    case F: False
     show "val a \<le> val b \<longleftrightarrow> (\<exists>y \<in> carrier Q\<^sub>p. (a[^](2::nat)) \<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp> \<otimes> (b[^](2::nat))) = (y[^](2::nat)))"
     proof
       show "val b \<ge> val a \<Longrightarrow> \<exists>y\<in>carrier Q\<^sub>p. (a[^](2::nat)) \<oplus>\<^bsub>Q\<^sub>p\<^esub> \<pp> \<otimes> (b[^](2::nat)) = (y[^](2::nat))"
       proof-
         assume  "val b \<ge> val a "
         then have "ord b \<ge> ord a"
-          using F False 
-          by (metis eint_ord_simps(1) val_def)          
+          using F False
+          by (metis eint_ord_simps(1) val_def)
         then show "\<exists>y\<in>carrier Q\<^sub>p. (a[^](2::nat)) \<oplus>\<^bsub>Q\<^sub>p\<^esub> \<pp> \<otimes> (b[^](2::nat)) = (y[^](2::nat))"
-          using assms Qp_square_root_criterion[of a b] False F 
+          using assms Qp_square_root_criterion[of a b] False F
           by blast
       qed
       show "\<exists>y\<in>carrier Q\<^sub>p.(a[^](2::nat)) \<oplus>\<^bsub>Q\<^sub>p\<^esub> \<pp> \<otimes> (b[^](2::nat)) = (y[^](2::nat)) \<Longrightarrow> val b \<ge> val a"
       proof-
         assume "\<exists>y\<in>carrier Q\<^sub>p. (a[^](2::nat)) \<oplus>\<^bsub>Q\<^sub>p\<^esub> \<pp> \<otimes> (b[^](2::nat)) = (y[^](2::nat))"
         then obtain y where y_def: "y \<in> carrier Q\<^sub>p \<and>(a[^](2::nat)) \<oplus>\<^bsub>Q\<^sub>p\<^esub> \<pp> \<otimes> (b[^](2::nat)) = (y[^](2::nat))"
-          by blast 
+          by blast
         have 0: "ord (a[^](2::nat)) =  2* ord a"
-          by (metis (mono_tags, opaque_lifting) False Suc_1 assms(2) int_eq_iff_numeral nat_numeral 
+          by (metis (mono_tags, opaque_lifting) False Suc_1 assms(2) int_eq_iff_numeral nat_numeral
               nonzero_nat_pow_ord not_nonzero_Qp)
         have 1: "ord (\<pp> \<otimes> (b[^](2::nat))) = 1 + 2* ord b"
         proof-
           have 0: "ord (\<pp> \<otimes> (b[^](2::nat))) = ord \<pp> + ord  (b[^](2::nat))"
-            using F Qp_nat_pow_nonzero assms(3) not_nonzero_Qp ord_mult p_nonzero 
-            by metis           
+            using F Qp_nat_pow_nonzero assms(3) not_nonzero_Qp ord_mult p_nonzero
+            by metis
           have 1: "ord  (b[^](2::nat)) = 2* ord b"
-            using F assms 
-            by (metis (mono_tags, opaque_lifting)  Suc_1  int_eq_iff_numeral nat_numeral 
+            using F assms
+            by (metis (mono_tags, opaque_lifting)  Suc_1  int_eq_iff_numeral nat_numeral
               nonzero_nat_pow_ord not_nonzero_Qp)
-          then show ?thesis 
-            using "0" ord_p 
+          then show ?thesis
+            using "0" ord_p
             by linarith
         qed
         show "val b \<ge> val a"
         proof(rule ccontr)
           assume "\<not> val b \<ge> val a"
           then have "val b \<noteq> val a \<and> val a \<ge> val b"
-            by (metis linear)            
+            by (metis linear)
           then have "ord a > ord b"
-            using F False assms 
-            by (metis \<open>\<not> val a \<le> val b\<close> eint_ord_simps(1) le_less not_less_iff_gr_or_eq val_def)                                
+            using F False assms
+            by (metis \<open>\<not> val a \<le> val b\<close> eint_ord_simps(1) le_less not_less_iff_gr_or_eq val_def)
           then have "ord (a[^](2::nat)) > ord (\<pp> \<otimes> (b[^](2::nat)))"
-            using 0 1 
+            using 0 1
             by linarith
           then have "ord ((a[^](2::nat)) \<oplus>\<^bsub>Q\<^sub>p\<^esub> \<pp> \<otimes> (b[^](2::nat))) = ord (\<pp> \<otimes> (b[^](2::nat)))"
-            by (meson F False Qp.int_inc_closed Qp_nat_pow_nonzero assms(2) assms(3) 
+            by (meson F False Qp.int_inc_closed Qp_nat_pow_nonzero assms(2) assms(3)
                 local.monom_term_car not_nonzero_Qp ord_ultrametric_noteq p_times_square_not_square')
           then have A0: "ord (y[^](2::nat)) = 1 + 2* ord b"
             by (metis "1" \<open>y \<in> carrier Q\<^sub>p \<and> (a[^]2) \<oplus>\<^bsub>Q\<^sub>p\<^esub> \<pp> \<otimes> (b[^]2) = (y[^]2)\<close>)
           have A1: "(y[^](2::nat)) \<in> nonzero Q\<^sub>p"
-            using y_def 0 1  
+            using y_def 0 1
             by (smt F False Qp.nonzero_closed Qp_nat_pow_nonzero assms(2) assms(3) diff_ord_nonzero
-                local.monom_term_car not_nonzero_Qp p_nonzero p_times_square_not_square')            
+                local.monom_term_car not_nonzero_Qp p_nonzero p_times_square_not_square')
           have A2: "y \<in> nonzero Q\<^sub>p"
-            using A1 Qp_nonzero_nat_pow pos2 y_def by blast            
+            using A1 Qp_nonzero_nat_pow pos2 y_def by blast
           have A3: "ord (y[^](2::nat)) = 2* ord y"
-            using  A2  nonzero_nat_pow_ord 
+            using  A2  nonzero_nat_pow_ord
             by presburger
-          then show False using A0 
+          then show False using A0
             by presburger
         qed
       qed
     qed
   qed
 qed
 
 text\<open>Defining the valuation ring semialgebraically for all primes\<close>
 
 lemma Qp_val_ring_alt_def:
   assumes "a \<in> carrier Q\<^sub>p"
   shows "a \<in> \<O>\<^sub>p \<longleftrightarrow> (\<exists>y \<in> carrier Q\<^sub>p. \<one> \<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp>[^](3::nat))\<otimes> (a[^](4::nat)) = (y[^](2::nat)))"
 proof(cases "a = \<zero>")
   case True
   then have "\<one> \<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp>[^](3::nat))\<otimes> (a[^](4::nat)) = \<one>"
-    by (metis Qp.add.l_cancel_one' Qp.integral_iff Qp.nat_pow_closed Qp.not_nonzero_memI 
-        Qp.one_closed Qp_nonzero_nat_pow assms not_nonzero_Qp p_natpow_closed(1) zero_less_numeral)    
+    by (metis Qp.add.l_cancel_one' Qp.integral_iff Qp.nat_pow_closed Qp.not_nonzero_memI
+        Qp.one_closed Qp_nonzero_nat_pow assms not_nonzero_Qp p_natpow_closed(1) zero_less_numeral)
   then have "\<one> \<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp>[^](3::nat))\<otimes> (a[^](4::nat)) = (\<one>[^](2::nat))"
-    using Qp.nat_pow_one by blast        
-  then show ?thesis 
-    using True zero_in_val_ring by blast      
+    using Qp.nat_pow_one by blast
+  then show ?thesis
+    using True zero_in_val_ring by blast
 next
   case False
   show "a \<in> \<O>\<^sub>p \<longleftrightarrow> (\<exists>y \<in> carrier Q\<^sub>p. \<one> \<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp>[^](3::nat))\<otimes> (a[^](4::nat)) = (y[^](2::nat)))"
   proof
     show "a \<in> \<O>\<^sub>p \<Longrightarrow> (\<exists>y \<in> carrier Q\<^sub>p. \<one> \<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp>[^](3::nat))\<otimes> (a[^](4::nat)) = (y[^](2::nat)))"
-      using assms Qp_val_ring_alt_def0[of a] False   
+      using assms Qp_val_ring_alt_def0[of a] False
       by (meson not_nonzero_Qp ord_nonneg)
     show "(\<exists>y \<in> carrier Q\<^sub>p. \<one> \<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp>[^](3::nat))\<otimes> (a[^](4::nat)) = (y[^](2::nat))) \<Longrightarrow> a \<in> \<O>\<^sub>p"
     proof-
       assume "(\<exists>y \<in> carrier Q\<^sub>p. \<one> \<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp>[^](3::nat))\<otimes> (a[^](4::nat)) = (y[^](2::nat)))"
       then obtain y where y_def: "y \<in> carrier Q\<^sub>p \<and>\<one> \<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp>[^](3::nat))\<otimes> (a[^](4::nat)) = (y[^](2::nat))"
         by blast
       then have "(\<pp>[^](3::nat))\<otimes> (a[^](4::nat)) = (y[^](2::nat)) \<ominus>\<^bsub>Q\<^sub>p\<^esub> \<one>"
-        using Qp.ring_simprules 
-        by (smt Qp.nat_pow_closed assms p_natpow_closed(1))        
+        using Qp.ring_simprules
+        by (smt Qp.nat_pow_closed assms p_natpow_closed(1))
       then have "ord ((\<pp>[^](3::nat))\<otimes> (a[^](4::nat))) = ord ((y[^](2::nat)) \<ominus>\<^bsub>Q\<^sub>p\<^esub> \<one>)"
         by presburger
       then have "3 + ord (a[^](4::nat)) = ord ((y[^](2::nat)) \<ominus>\<^bsub>Q\<^sub>p\<^esub> \<one>)"
         by (metis False Qp_nat_pow_nonzero assms not_nonzero_Qp of_nat_numeral ord_mult ord_p_pow_nat p_nonzero)
       then have 0: "3 + 4* ord a = ord ((y[^](2::nat)) \<ominus>\<^bsub>Q\<^sub>p\<^esub> \<one>)"
         using assms False nonzero_nat_pow_ord[of a "(4::nat)"]
         by (metis nonzero_nat_pow_ord not_nonzero_Qp of_nat_numeral)
       have "ord a \<ge> 0"
       proof(rule ccontr)
         assume "\<not> 0 \<le> ord a"
         then have 00: "ord ((y[^](2::nat)) \<ominus>\<^bsub>Q\<^sub>p\<^esub> \<one>) < 0"
-          using 0 
+          using 0
           by linarith
         have yn: "y \<in> nonzero Q\<^sub>p"
           apply(rule ccontr)
-          using y_def 0 
-          by (metis "00" Qp.not_eq_diff_nonzero Qp.one_closed Qp.one_nonzero Qp.pow_zero 
-              \<open>\<pp> [^] 3 \<otimes> a [^] 4 = y [^] 2 \<ominus> \<one>\<close> diff_ord_nonzero less_numeral_extra(3) 
-              local.one_neq_zero not_nonzero_Qp ord_one zero_less_numeral)          
+          using y_def 0
+          by (metis "00" Qp.not_eq_diff_nonzero Qp.one_closed Qp.one_nonzero Qp.pow_zero
+              \<open>\<pp> [^] 3 \<otimes> a [^] 4 = y [^] 2 \<ominus> \<one>\<close> diff_ord_nonzero less_numeral_extra(3)
+              local.one_neq_zero not_nonzero_Qp ord_one zero_less_numeral)
         then have "ord ((y[^](2::nat)) \<ominus>\<^bsub>Q\<^sub>p\<^esub> \<one>) = ord (y[^](2::nat))"
           using y_def ord_ultrametric_noteq''[of  "(y[^](2::nat))" "\<one>" ]
           by (metis "00" False Qp.integral Qp.nat_pow_closed Qp.nonzero_closed Qp.nonzero_pow_nonzero
-              Qp.not_eq_diff_nonzero Qp.one_nonzero Qp.r_right_minus_eq \<open>\<pp> [^] 3 \<otimes> a [^] 4 = y [^] 2 \<ominus> \<one>\<close> 
-              assms ord_one ord_ultrametric_noteq p_nonzero)          
+              Qp.not_eq_diff_nonzero Qp.one_nonzero Qp.r_right_minus_eq \<open>\<pp> [^] 3 \<otimes> a [^] 4 = y [^] 2 \<ominus> \<one>\<close>
+              assms ord_one ord_ultrametric_noteq p_nonzero)
         then have "ord ((y[^](2::nat)) \<ominus>\<^bsub>Q\<^sub>p\<^esub> \<one>) = 2* ord y"
-          using y_def Qp_nat_pow_nonzero Qp_nonzero_nat_pow nonzero_nat_pow_ord[of y "(2::nat)"]  yn 
+          using y_def Qp_nat_pow_nonzero Qp_nonzero_nat_pow nonzero_nat_pow_ord[of y "(2::nat)"]  yn
           by linarith
         then have "3 + (4* ord a) = 2* ord y"
-          using "00" "0" 
-          by linarith      
-        then show False 
+          using "00" "0"
+          by linarith
+        then show False
           by presburger
       qed
       then show "a \<in> \<O>\<^sub>p"
         using False val_ring_ord_criterion assms by blast
     qed
   qed
 qed
 
 lemma Qp_val_alt_def:
   assumes "a \<in> carrier Q\<^sub>p"
   assumes "b \<in> carrier Q\<^sub>p"
   shows "val b \<le> val a \<longleftrightarrow> (\<exists>y \<in> carrier Q\<^sub>p. (b[^](4::nat)) \<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp>[^](3::nat))\<otimes> (a[^](4::nat)) = (y[^](2::nat)))"
 proof
   show "val a \<ge> val b \<Longrightarrow> \<exists>y\<in>carrier Q\<^sub>p. (b[^](4::nat)) \<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp>[^](3::nat))\<otimes> (a[^](4::nat)) = (y[^](2::nat))"
   proof-
     assume A: "val a \<ge> val b"
     show "\<exists>y\<in>carrier Q\<^sub>p. (b[^](4::nat)) \<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp>[^](3::nat))\<otimes> (a[^](4::nat)) = (y[^](2::nat))"
     proof(cases "b = \<zero>")
       case True
       then have "a = \<zero>"
-        using A assms(1) val_ineq 
+        using A assms(1) val_ineq
         by blast
       then have "(b[^](4::nat)) \<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp>[^](3::nat))\<otimes> (a[^](4::nat)) = (\<zero>[^](2::nat))"
-        by (metis Qp.nat_pow_zero Qp.r_null Qp.r_zero True assms(2) p_natpow_closed(1) zero_neq_numeral)        
-      then show ?thesis 
+        by (metis Qp.nat_pow_zero Qp.r_null Qp.r_zero True assms(2) p_natpow_closed(1) zero_neq_numeral)
+      then show ?thesis
         using True A assms(2)
-        by blast      
+        by blast
     next
       case False
       assume B: "b \<noteq> \<zero>"
       show "\<exists>y\<in>carrier Q\<^sub>p. (b[^](4::nat)) \<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp>[^](3::nat)) \<otimes> (a[^](4::nat)) = (y[^](2::nat))"
       proof(cases "a = \<zero>")
         case True
         then have "(b[^](4::nat)) \<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp>[^](3::nat))\<otimes> (a[^](4::nat)) = (b[^](4::nat))"
           using Qp.cring_axioms Qp.nat_pow_closed assms(2) cring_def p_natpow_closed(1) ring.pow_zero zero_less_numeral
           by (metis Qp.add.l_cancel_one' Qp.integral_iff assms(1))
         then have "(b[^](4::nat)) \<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp>[^](3::nat))\<otimes> (a[^](4::nat)) = ((b[^](2::nat))[^] (2::nat))"
           by (metis Qp_nat_pow_pow assms(2) mult_2_right numeral_Bit0)
         then have "(b[^](2::nat)) \<in> carrier Q\<^sub>p \<and> (b[^](4::nat)) \<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp>[^](3::nat))\<otimes> (a[^](4::nat)) = ((b[^](2::nat))[^] (2::nat))"
           using Qp.nat_pow_closed assms(2)
           by blast
-        then show ?thesis 
+        then show ?thesis
           by blast
       next
         case False
         have F0:  "b \<in> nonzero Q\<^sub>p"
-          using B assms(2) not_nonzero_Qp 
-          by metis          
+          using B assms(2) not_nonzero_Qp
+          by metis
         have F1: "a \<in> nonzero Q\<^sub>p"
-          using False assms(1) not_nonzero_Qp 
+          using False assms(1) not_nonzero_Qp
           by metis
         then have "(a \<div> b) \<in> nonzero Q\<^sub>p"
           using B
-          by (meson Localization.submonoid.m_closed Qp.nonzero_is_submonoid assms(2) inv_in_frac(3))                            
+          by (meson Localization.submonoid.m_closed Qp.nonzero_is_submonoid assms(2) inv_in_frac(3))
         then have "val a \<ge> val b"
           using F0 F1 A by blast
         then have "val (a \<div> b) \<ge> 0"
-          using F0 F1 val_fract assms(1) local.eint_minus_ineq' by presburger         
+          using F0 F1 val_fract assms(1) local.eint_minus_ineq' by presburger
         obtain y where y_def: "y \<in> carrier Q\<^sub>p \<and> \<one> \<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp>[^](3::nat))\<otimes> ((a \<div> b)[^](4::nat)) = (y[^](2::nat))"
-          using Qp_val_ring_alt_def0 
-          by (meson B False Qp.integral Qp.nonzero_closed \<open>(a \<div> b) \<in> nonzero Q\<^sub>p\<close> \<open>0 \<le> val (a \<div> b)\<close> 
-              assms(1) assms(2) inv_in_frac(1) inv_in_frac(2) ord_nonneg val_ringI)          
+          using Qp_val_ring_alt_def0
+          by (meson B False Qp.integral Qp.nonzero_closed \<open>(a \<div> b) \<in> nonzero Q\<^sub>p\<close> \<open>0 \<le> val (a \<div> b)\<close>
+              assms(1) assms(2) inv_in_frac(1) inv_in_frac(2) ord_nonneg val_ringI)
         then have "(b[^](4::nat)) \<otimes> (\<one> \<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp>[^](3::nat))\<otimes> ((a \<div> b)[^](4::nat))) =
                (b[^](4::nat)) \<otimes> (y[^](2::nat))"
           by presburger
         then have  F2: "(b[^](4::nat)) \<otimes> (\<one> \<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp>[^](3::nat))\<otimes> ((a \<div> b)[^](4::nat))) =
                ((b[^](2::nat)) [^] (2::nat)) \<otimes> (y[^](2::nat))"
-          by (metis Qp.nat_pow_pow assms(2) mult_2_right numeral_Bit0)          
+          by (metis Qp.nat_pow_pow assms(2) mult_2_right numeral_Bit0)
         have  F3: "((b[^](4::nat)) \<otimes> \<one>) \<oplus>\<^bsub>Q\<^sub>p\<^esub> ((b[^](4::nat)) \<otimes>((\<pp>[^](3::nat))\<otimes> ((a \<div> b)[^](4::nat)))) =
                ((b[^](2::nat))[^] (2::nat)) \<otimes> (y[^](2::nat))"
         proof-
           have 0: "(\<pp>[^](3::nat)) \<otimes> (a \<div> b[^](4::nat)) \<in> carrier Q\<^sub>p "
           proof-
             have "(a \<div> b[^](4::nat)) \<in> carrier Q\<^sub>p"
-              using F0 Qp.nat_pow_closed assms(1) fract_closed Qp_nat_pow_nonzero by presburger              
-            then show ?thesis 
+              using F0 Qp.nat_pow_closed assms(1) fract_closed Qp_nat_pow_nonzero by presburger
+            then show ?thesis
               by (meson Qp.cring_axioms cring.cring_simprules(5) p_natpow_closed(1))
           qed
           have 1: "(b[^](4::nat)) \<in> carrier Q\<^sub>p"
-            using Qp.nat_pow_closed assms(2) 
+            using Qp.nat_pow_closed assms(2)
             by blast
-          then show ?thesis 
-            using 0 F2  ring.ring_simprules(23)[of Q\<^sub>p "\<one>" "(\<pp>[^](3::nat))\<otimes> ((a \<div> b)[^](4::nat))" "(b[^](4::nat))"] 
-                  Qp.cring_axioms Qp.nonzero_mult_closed Qp.ring_axioms Qp_nat_pow_nonzero \<open>(a \<div> b) \<in> nonzero Q\<^sub>p\<close> p_nonzero 
-            by blast                      
+          then show ?thesis
+            using 0 F2  ring.ring_simprules(23)[of Q\<^sub>p "\<one>" "(\<pp>[^](3::nat))\<otimes> ((a \<div> b)[^](4::nat))" "(b[^](4::nat))"]
+                  Qp.cring_axioms Qp.nonzero_mult_closed Qp.ring_axioms Qp_nat_pow_nonzero \<open>(a \<div> b) \<in> nonzero Q\<^sub>p\<close> p_nonzero
+            by blast
         qed
         have F4: "(b[^](4::nat)) \<in> carrier Q\<^sub>p"
-          using Qp.nat_pow_closed assms(2) 
+          using Qp.nat_pow_closed assms(2)
           by blast
-        then have "((b[^](4::nat)) \<otimes> \<one>) = (b[^](4::nat))" 
-          using Qp.r_one by blast 
+        then have "((b[^](4::nat)) \<otimes> \<one>) = (b[^](4::nat))"
+          using Qp.r_one by blast
         then have F5:  "(b[^](4::nat))\<oplus>\<^bsub>Q\<^sub>p\<^esub> ((b[^](4::nat)) \<otimes>((\<pp>[^](3::nat))\<otimes> ((a \<div> b)[^](4::nat)))) =
                ((b[^](2::nat)) [^] (2::nat)) \<otimes> (y[^](2::nat))"
-          using F3 
-          by presburger          
+          using F3
+          by presburger
         have "((b[^](4::nat)) \<otimes>((\<pp>[^](3::nat))\<otimes> ((a \<div> b)[^](4::nat)))) = (\<pp>[^](3::nat))\<otimes>((b[^](4::nat)) \<otimes> ((a \<div> b)[^](4::nat)))"
         proof-
           have 0: "(b[^](4::nat)) \<in> carrier Q\<^sub>p"
-            using F4 by blast            
+            using F4 by blast
           have 1: "(\<pp>[^](3::nat)) \<in> carrier Q\<^sub>p"
-            by blast            
+            by blast
           have  2: "((a \<div> b)[^](4::nat)) \<in> carrier Q\<^sub>p"
-            using F0 Qp.nat_pow_closed assms(1) fract_closed 
+            using F0 Qp.nat_pow_closed assms(1) fract_closed
             by blast
-          show ?thesis using 0 1 2  monoid.m_assoc[of Q\<^sub>p] comm_monoid.m_comm[of Q\<^sub>p] 
-            using Qp.m_lcomm by presburger            
+          show ?thesis using 0 1 2  monoid.m_assoc[of Q\<^sub>p] comm_monoid.m_comm[of Q\<^sub>p]
+            using Qp.m_lcomm by presburger
         qed
         then have  "(b[^](4::nat))\<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp>[^](3::nat))\<otimes>((b[^](4::nat)) \<otimes> ((a \<div> b)[^](4::nat))) =
                ((b[^](2::nat)) [^] (2::nat)) \<otimes> (y[^](2::nat))"
           using F5  by presburger
         then have  "(b[^](4::nat))\<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp>[^](3::nat))\<otimes>((b \<otimes>(a \<div> b))[^](4::nat)) =
                ((b[^](2::nat)) [^] (2::nat)) \<otimes> (y[^](2::nat))"
-          using F0 Qp.nat_pow_distrib assms(1) assms(2) fract_closed by presburger          
+          using F0 Qp.nat_pow_distrib assms(1) assms(2) fract_closed by presburger
         then have  "(b[^](4::nat))\<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp>[^](3::nat))\<otimes>(a[^](4::nat)) =
                ((b[^](2::nat)) [^] (2::nat)) \<otimes> (y[^](2::nat))"
-          by (metis F0 assms(1) local.fract_cancel_right)          
+          by (metis F0 assms(1) local.fract_cancel_right)
         then have  "(b[^](4::nat))\<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp>[^](3::nat))\<otimes>(a[^](4::nat)) =
                (((b[^](2::nat))\<otimes> y)[^](2::nat))"
-          using Qp.nat_pow_closed Qp.nat_pow_distrib assms(2) y_def by blast          
+          using Qp.nat_pow_closed Qp.nat_pow_distrib assms(2) y_def by blast
         then have  "((b[^](2::nat))\<otimes> y) \<in> carrier Q\<^sub>p \<and> (b[^](4::nat))\<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp>[^](3::nat))\<otimes>(a[^](4::nat)) =
                (((b[^](2::nat))\<otimes> y)[^](2::nat))"
           by (meson Qp.cring_axioms Qp.nat_pow_closed assms(2) cring.cring_simprules(5) y_def)
-        then show ?thesis 
-          by blast       
+        then show ?thesis
+          by blast
       qed
     qed
   qed
   show "\<exists>y \<in> carrier Q\<^sub>p. (b[^](4::nat)) \<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp>[^](3::nat))\<otimes> (a[^](4::nat)) = (y[^](2::nat)) \<Longrightarrow> val a \<ge> val b"
   proof-
     assume A: "\<exists>y \<in> carrier Q\<^sub>p. (b[^](4::nat)) \<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp>[^](3::nat))\<otimes> (a[^](4::nat)) = (y[^](2::nat))"
     show "val a \<ge> val b"
     proof(cases "a = \<zero>")
       case True
-      then show ?thesis 
-        using eint_ord_code(3) local.val_zero by presburger        
+      then show ?thesis
+        using eint_ord_code(3) local.val_zero by presburger
     next
       case False
       have  "b \<noteq> \<zero>"
       proof(rule ccontr)
         assume  "\<not> b \<noteq> \<zero>"
         then have "\<exists>y \<in> carrier Q\<^sub>p.  (\<pp>[^](3::nat))\<otimes> (a[^](4::nat)) = (y[^](2::nat))"
-          using A 
-          by (metis (no_types, lifting) Qp.add.r_cancel_one' Qp.nat_pow_closed Qp.nonzero_memE(2) 
+          using A
+          by (metis (no_types, lifting) Qp.add.r_cancel_one' Qp.nat_pow_closed Qp.nonzero_memE(2)
               Qp_nonzero_nat_pow assms(1) assms(2) local.monom_term_car not_nonzero_Qp
-              p_natpow_closed(1) zero_less_numeral)          
+              p_natpow_closed(1) zero_less_numeral)
         then obtain y where y_def: "y \<in> carrier Q\<^sub>p \<and> (\<pp>[^](3::nat))\<otimes> (a[^](4::nat)) = (y[^](2::nat))"
-          by blast 
+          by blast
         have 0: "ord ((\<pp>[^](3::nat))\<otimes> (a[^](4::nat))) = 3 +  4* ord a"
         proof-
           have 00: "(\<pp>[^](3::nat)) \<in> nonzero Q\<^sub>p"
-            using Qp_nat_pow_nonzero p_nonzero by blast           
+            using Qp_nat_pow_nonzero p_nonzero by blast
           have 01: "(a[^](4::nat)) \<in> nonzero Q\<^sub>p"
-            using False Qp_nat_pow_nonzero assms(1) not_nonzero_Qp Qp.nonzero_memI by presburger            
-          then show ?thesis using ord_mult[of "\<pp>[^](3::nat)" "a[^](4::nat)"] 
+            using False Qp_nat_pow_nonzero assms(1) not_nonzero_Qp Qp.nonzero_memI by presburger
+          then show ?thesis using ord_mult[of "\<pp>[^](3::nat)" "a[^](4::nat)"]
             by (metis (no_types, lifting) "00" False assms(1) nonzero_nat_pow_ord
                 not_nonzero_Qp of_nat_numeral ord_p_pow_nat)
         qed
         have 1: "ord ((\<pp>[^](3::nat))\<otimes> (a[^](4::nat))) = 2* (ord y)"
         proof-
           have "y \<noteq> \<zero>"
           proof(rule ccontr)
             assume " \<not> y \<noteq> \<zero>"
             then have "(\<pp>[^](3::nat))\<otimes> (a[^](4::nat)) = \<zero>"
               using y_def  Qp.cring_axioms cring_def pos2 ring.pow_zero by blast
-            then show False 
-              by (metis False Qp.integral Qp.nat_pow_closed Qp.nonzero_pow_nonzero 
-                  Qp.not_nonzero_memI Qp_nat_pow_nonzero assms(1) p_natpow_closed(1) p_nonzero)             
+            then show False
+              by (metis False Qp.integral Qp.nat_pow_closed Qp.nonzero_pow_nonzero
+                  Qp.not_nonzero_memI Qp_nat_pow_nonzero assms(1) p_natpow_closed(1) p_nonzero)
           qed
-          then show ?thesis 
-            using y_def 
+          then show ?thesis
+            using y_def
             by (metis nonzero_nat_pow_ord not_nonzero_Qp of_nat_numeral)
         qed
-        then show False 
-          using 0 
+        then show False
+          using 0
           by presburger
       qed
       then have F0: "b \<in> nonzero Q\<^sub>p"
         using assms(2) not_nonzero_Qp  by metis
-        
+
       have F1: "a \<in> nonzero Q\<^sub>p"
-        using False assms(1) not_nonzero_Qp by metis 
+        using False assms(1) not_nonzero_Qp by metis
       obtain y where y_def: "y \<in> carrier Q\<^sub>p \<and> (b[^](4::nat)) \<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp>[^](3::nat))\<otimes> (a[^](4::nat)) = (y[^](2::nat))"
         using A by blast
-      show ?thesis 
+      show ?thesis
       proof(rule ccontr)
         assume " \<not> val a \<ge> val b "
         then have F2: "ord a < ord b"
-          using F0 F1 assms 
-          by (metis False \<open>b \<noteq> \<zero>\<close> eint_ord_simps(1) leI val_def)          
+          using F0 F1 assms
+          by (metis False \<open>b \<noteq> \<zero>\<close> eint_ord_simps(1) leI val_def)
         have 0: "ord ((\<pp>[^](3::nat))\<otimes> (a[^](4::nat))) = 3 + 4* ord a"
-          using F0 ord_mult F1 Qp_nat_pow_nonzero nonzero_nat_pow_ord ord_p_pow_nat p_natpow_closed(2) 
+          using F0 ord_mult F1 Qp_nat_pow_nonzero nonzero_nat_pow_ord ord_p_pow_nat p_natpow_closed(2)
           by presburger
         have 1: " ord (b[^](4::nat)) = 4* ord b"
-          using F0 nonzero_nat_pow_ord 
+          using F0 nonzero_nat_pow_ord
           by presburger
         have 2: "(4 * (ord b)) > 4 * (ord a)"
           using F2 by linarith
         have 3: "(4 * (ord b)) \<le> 3 + 4* ord a"
         proof(rule ccontr)
           assume "\<not> (4 * (ord b)) \<le> 3 + 4* ord a"
           then have "(4 * (ord b)) > 3 + 4* ord a"
             by linarith
           then have 30: "ord ((b[^](4::nat)) \<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp>[^](3::nat))\<otimes> (a[^](4::nat))) = 3 + 4* ord a"
-            using "0" "1" F0 F1  Qp_nat_pow_nonzero Qp.nat_pow_closed assms(1) monom_term_car not_nonzero_Qp ord_ultrametric_noteq 
-                p_natpow_closed(1) p_nonzero 
+            using "0" "1" F0 F1  Qp_nat_pow_nonzero Qp.nat_pow_closed assms(1) monom_term_car not_nonzero_Qp ord_ultrametric_noteq
+                p_natpow_closed(1) p_nonzero
             by (metis Qp.integral)
           have "y \<in> nonzero Q\<^sub>p"
           proof(rule ccontr)
             assume A: "y \<notin> nonzero Q\<^sub>p"
             then have "y = \<zero>"
               using y_def  Qp.nonzero_memI by blast
             then have "b [^] 4 \<oplus> \<pp> [^] 3 \<otimes> a [^] 4 = \<zero>"
-              by (smt "0" "1" A F0 False Qp.integral Qp.nat_pow_closed Qp.nonzero_closed 
+              by (smt "0" "1" A F0 False Qp.integral Qp.nat_pow_closed Qp.nonzero_closed
                 Qp.nonzero_mult_closed Qp.nonzero_pow_nonzero Qp.pow_zero assms(1) diff_ord_nonzero not_nonzero_Qp p_nonzero pos2 y_def)
-            then show False 
+            then show False
               by (smt "0" "1" A F0 F1 Qp.integral Qp.nat_pow_closed Qp.nonzero_mult_closed
                 Qp_nat_pow_nonzero assms(1) diff_ord_nonzero not_nonzero_Qp p_natpow_closed(1) p_nonzero y_def)
-          qed                        
+          qed
           then have 31: "ord ((b[^](4::nat)) \<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp>[^](3::nat))\<otimes> (a[^](4::nat))) = 2* ord y"
-            using nonzero_nat_pow_ord y_def 
+            using nonzero_nat_pow_ord y_def
             by presburger
           then show False using 30 by presburger
         qed
-        show False 
-          using 2 3 
-          by presburger 
+        show False
+          using 2 3
+          by presburger
       qed
     qed
   qed
 qed
 
 text\<open>The polynomial in two variables which semialgebraically defines the valuation relation\<close>
 
 definition Qp_val_poly where
 "Qp_val_poly = (pvar Q\<^sub>p 1)[^]\<^bsub>Q\<^sub>p[\<X>\<^bsub>2\<^esub>]\<^esub>(4::nat) \<oplus>\<^bsub>Q\<^sub>p[\<X>\<^bsub>2\<^esub>]\<^esub> (\<pp>[^](3::nat) \<odot>\<^bsub>Q\<^sub>p[\<X>\<^bsub>2\<^esub>]\<^esub> ((pvar Q\<^sub>p 0)[^]\<^bsub>Q\<^sub>p[\<X>\<^bsub>2\<^esub>]\<^esub>(4::nat)))"
 
 lemma Qp_val_poly_closed:
 "Qp_val_poly \<in> carrier (Q\<^sub>p[\<X>\<^bsub>2\<^esub>])"
 proof-
   have "(pvar Q\<^sub>p 1) \<in> carrier (Q\<^sub>p[\<X>\<^bsub>2\<^esub>])"
-    using local.pvar_closed one_less_numeral_iff semiring_norm(76) by blast          
+    using local.pvar_closed one_less_numeral_iff semiring_norm(76) by blast
   then have 0: "(pvar Q\<^sub>p 1)[^]\<^bsub>Q\<^sub>p[\<X>\<^bsub>2\<^esub>]\<^esub>(4::nat) \<in> carrier (Q\<^sub>p[\<X>\<^bsub>2\<^esub>])"
-    
-    using  ring.Pring_is_ring[of Q\<^sub>p "{0::nat..2-1}"] 
-          monoid.nat_pow_closed[of "coord_ring Q\<^sub>p 2"] Qp.cring_axioms cring.axioms(1) ring.Pring_is_monoid 
+
+    using  ring.Pring_is_ring[of Q\<^sub>p "{0::nat..2-1}"]
+          monoid.nat_pow_closed[of "coord_ring Q\<^sub>p 2"] Qp.cring_axioms cring.axioms(1) ring.Pring_is_monoid
     by blast
   have 1: "(pvar Q\<^sub>p 0)[^]\<^bsub>Q\<^sub>p[\<X>\<^bsub>2\<^esub>]\<^esub>(4::nat) \<in> carrier (Q\<^sub>p[\<X>\<^bsub>2\<^esub>])"
-    using local.pvar_closed pos2 by blast      
+    using local.pvar_closed pos2 by blast
   have 2: "\<pp>[^](3::nat) \<odot>\<^bsub>Q\<^sub>p[\<X>\<^bsub>2\<^esub>]\<^esub>(pvar Q\<^sub>p 0)[^]\<^bsub>Q\<^sub>p[\<X>\<^bsub>2\<^esub>]\<^esub>(4::nat) \<in> carrier (Q\<^sub>p[\<X>\<^bsub>2\<^esub>])"
-    using 1 local.smult_closed p_natpow_closed(1) by blast    
-  then show ?thesis 
-    unfolding Qp_val_poly_def 
-    using 0 by blast    
-qed 
+    using 1 local.smult_closed p_natpow_closed(1) by blast
+  then show ?thesis
+    unfolding Qp_val_poly_def
+    using 0 by blast
+qed
 
 lemma Qp_val_poly_eval:
   assumes "a \<in> carrier Q\<^sub>p"
   assumes "b \<in> carrier Q\<^sub>p"
   shows "Qp_ev Qp_val_poly [a, b] =  (b[^](4::nat)) \<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp>[^](3::nat))\<otimes> (a[^](4::nat))"
 proof-
   have 0: "[a,b] \<in> carrier (Q\<^sub>p\<^bsup>2\<^esup>)"
   proof(rule cartesian_power_car_memI)
     show "length [a, b] = 2"
       by simp
     have "set [a,b] = {a,b}"
       by auto
     then show "set [a, b] \<subseteq> carrier Q\<^sub>p"
-      using assms 
+      using assms
       by (simp add: \<open>a \<in> carrier Q\<^sub>p\<close> \<open>b \<in> carrier Q\<^sub>p\<close>)
-  qed 
+  qed
   obtain f where f_def: "f = ((pvar Q\<^sub>p 1)[^]\<^bsub>Q\<^sub>p[\<X>\<^bsub>2\<^esub>]\<^esub>(4::nat))"
-    by blast 
+    by blast
   obtain g where g_def: "g = (\<pp>[^](3::nat) \<odot>\<^bsub>Q\<^sub>p[\<X>\<^bsub>2\<^esub>]\<^esub> ((pvar Q\<^sub>p 0)[^]\<^bsub>Q\<^sub>p[\<X>\<^bsub>2\<^esub>]\<^esub>(4::nat)))"
-    by blast 
+    by blast
   have 1: "Qp_val_poly = f \<oplus>\<^bsub>Q\<^sub>p[\<X>\<^bsub>2\<^esub>]\<^esub> g"
-    unfolding Qp_val_poly_def 
-    using f_def g_def by blast 
+    unfolding Qp_val_poly_def
+    using f_def g_def by blast
   have 1: "Qp_ev (pvar Q\<^sub>p (0::nat)) [a,b] = a"
-    using eval_pvar 
+    using eval_pvar
     by (metis \<open>[a, b] \<in> carrier (Q\<^sub>p\<^bsup>2\<^esup>)\<close> nth_Cons_0 pos2)
   have 2: "Qp_ev (pvar Q\<^sub>p (1::nat)) [a,b] = b"
-    using eval_pvar 
-    by (metis (no_types, lifting) "0" One_nat_def add_diff_cancel_right' assms(2) 
+    using eval_pvar
+    by (metis (no_types, lifting) "0" One_nat_def add_diff_cancel_right' assms(2)
         cartesian_power_car_memE gr_zeroI less_numeral_extra(1) less_numeral_extra(4)
         list.size(4) nth_Cons_pos Qp.to_R1_closed Qp.to_R_to_R1 zero_less_diff)
   have 3: "Qp_ev ((pvar Q\<^sub>p 1)[^]\<^bsub>Q\<^sub>p[\<X>\<^bsub>2\<^esub>]\<^esub>(4::nat)) [a,b] = (b[^](4::nat))"
-    by (metis "0" "2" eval_at_point_nat_pow local.pvar_closed one_less_numeral_iff semiring_norm(76))    
+    by (metis "0" "2" eval_at_point_nat_pow local.pvar_closed one_less_numeral_iff semiring_norm(76))
   have 4: "Qp_ev ((pvar Q\<^sub>p 0)[^]\<^bsub>Q\<^sub>p[\<X>\<^bsub>2\<^esub>]\<^esub>(4::nat)) [a,b] = (a[^](4::nat))"
-    using "0" "1" eval_at_point_nat_pow local.pvar_closed pos2 by presburger    
+    using "0" "1" eval_at_point_nat_pow local.pvar_closed pos2 by presburger
   then have 5: "Qp_ev (poly_scalar_mult Q\<^sub>p (\<pp>[^](3::nat)) ((pvar Q\<^sub>p 0)[^]\<^bsub>Q\<^sub>p[\<X>\<^bsub>2\<^esub>]\<^esub>(4::nat))) [a,b] = (\<pp>[^](3::nat))\<otimes> (a[^](4::nat))"
-    using eval_at_point_smult[of "[a,b]" 2 "(pvar Q\<^sub>p 0)[^]\<^bsub>Q\<^sub>p[\<X>\<^bsub>2\<^esub>]\<^esub>(4::nat)" "\<pp>[^](3::nat)" ] 2 
-    by (metis "0" MP.nat_pow_closed eval_at_point_scalar_mult local.pvar_closed p_natpow_closed(1) zero_less_numeral)    
-  then show ?thesis 
+    using eval_at_point_smult[of "[a,b]" 2 "(pvar Q\<^sub>p 0)[^]\<^bsub>Q\<^sub>p[\<X>\<^bsub>2\<^esub>]\<^esub>(4::nat)" "\<pp>[^](3::nat)" ] 2
+    by (metis "0" MP.nat_pow_closed eval_at_point_scalar_mult local.pvar_closed p_natpow_closed(1) zero_less_numeral)
+  then show ?thesis
   proof-
     have 00: "[a, b] \<in> carrier (Q\<^sub>p\<^bsup>2\<^esup>)"
       by (simp add: "0")
     have 01: " pvar Q\<^sub>p 1 [^]\<^bsub>Q\<^sub>p[\<X>\<^bsub>2\<^esub>]\<^esub> (4::nat) \<in> carrier (Q\<^sub>p[\<X>\<^bsub>2\<^esub>])"
-      by (meson MP.nat_pow_closed local.pvar_closed one_less_numeral_iff semiring_norm(76))      
+      by (meson MP.nat_pow_closed local.pvar_closed one_less_numeral_iff semiring_norm(76))
     have 02: "\<pp>[^](3::nat) \<odot>\<^bsub>Q\<^sub>p[\<X>\<^bsub>2\<^esub>]\<^esub>  (pvar Q\<^sub>p 0 [^]\<^bsub>Q\<^sub>p[\<X>\<^bsub>2\<^esub>]\<^esub> (4::nat)) \<in> carrier (Q\<^sub>p[\<X>\<^bsub>2\<^esub>])"
-      by (meson MP.nat_pow_closed local.pvar_closed local.smult_closed p_natpow_closed(1) zero_less_numeral)      
+      by (meson MP.nat_pow_closed local.pvar_closed local.smult_closed p_natpow_closed(1) zero_less_numeral)
     then show ?thesis
-      unfolding Qp_val_poly_def 
-      using 00 01 02 
-      by (metis (no_types, lifting) "3" "4" MP.nat_pow_closed eval_at_point_add eval_at_point_smult 
-          local.pvar_closed p_natpow_closed(1) zero_less_numeral)      
+      unfolding Qp_val_poly_def
+      using 00 01 02
+      by (metis (no_types, lifting) "3" "4" MP.nat_pow_closed eval_at_point_add eval_at_point_smult
+          local.pvar_closed p_natpow_closed(1) zero_less_numeral)
   qed
 qed
 
 lemma Qp_2I:
   assumes "a \<in> carrier Q\<^sub>p"
   assumes "b \<in> carrier Q\<^sub>p"
   shows "[a,b] \<in> carrier (Q\<^sub>p\<^bsup>2\<^esup>)"
   apply(rule cartesian_power_car_memI)
-  using assms 
+  using assms
   apply (simp add: assms(1) assms(2))
-  using assms 
+  using assms
   by (simp add: assms(1) assms(2))
 
 lemma pair_id:
   assumes "length as = 2"
   shows "as = [as!0, as!1]"
-  using assms 
-  by (smt One_nat_def diff_Suc_1 length_Cons less_Suc0 less_SucE list.size(3) 
+  using assms
+  by (smt One_nat_def diff_Suc_1 length_Cons less_Suc0 less_SucE list.size(3)
       nth_Cons' nth_equalityI numeral_2_eq_2)
 
 lemma Qp_val_semialg:
   assumes "a \<in> carrier Q\<^sub>p"
   assumes "b \<in> carrier Q\<^sub>p"
   shows "val b \<le> val a \<longleftrightarrow> [a,b] \<in> basic_semialg_set 2 (2::nat) Qp_val_poly"
 proof
   show "val a \<ge> val b \<Longrightarrow> [a, b] \<in> basic_semialg_set 2 2 Qp_val_poly"
-    using Qp_val_alt_def[of a b] Qp_2I[of a b]  Qp_val_poly_eval[of a b] 
-    unfolding basic_semialg_set_def 
+    using Qp_val_alt_def[of a b] Qp_2I[of a b]  Qp_val_poly_eval[of a b]
+    unfolding basic_semialg_set_def
     by (metis (mono_tags, lifting) assms(1) assms(2) mem_Collect_eq)
   show "[a, b] \<in> basic_semialg_set 2 2 Qp_val_poly \<Longrightarrow> val a \<ge> val b"
-    using Qp_val_alt_def[of a b] Qp_2I[of a b]  Qp_val_poly_eval[of a b] 
-    unfolding basic_semialg_set_def 
-    using assms(1) assms(2) 
+    using Qp_val_alt_def[of a b] Qp_2I[of a b]  Qp_val_poly_eval[of a b]
+    unfolding basic_semialg_set_def
+    using assms(1) assms(2)
     by blast
 qed
 
 definition val_relation_set where
 "val_relation_set = {as \<in> carrier (Q\<^sub>p\<^bsup>2\<^esup>). val (as!1) \<le> val (as!0)}"
 
 lemma val_relation_setE:
   assumes "as \<in> val_relation_set"
   shows "as!0 \<in> carrier Q\<^sub>p \<and> as!1 \<in> carrier Q\<^sub>p \<and> as = [as!0,as!1] \<and> val (as!1) \<le> val (as!0)"
-  using assms  unfolding val_relation_set_def 
+  using assms  unfolding val_relation_set_def
   by (smt cartesian_power_car_memE cartesian_power_car_memE' mem_Collect_eq one_less_numeral_iff pair_id pos2 semiring_norm(76))
-  
+
 lemma val_relation_setI:
   assumes "as!0 \<in> carrier Q\<^sub>p"
   assumes "as!1 \<in> carrier Q\<^sub>p"
   assumes "length as = 2"
   assumes "val (as!1) \<le> val(as!0)"
   shows "as \<in> val_relation_set"
-  unfolding val_relation_set_def using assms Qp_2I[of "as!0" "as!1"] 
-  by (metis (no_types, lifting) mem_Collect_eq pair_id)  
+  unfolding val_relation_set_def using assms Qp_2I[of "as!0" "as!1"]
+  by (metis (no_types, lifting) mem_Collect_eq pair_id)
 
 lemma val_relation_semialg:
 "val_relation_set  = basic_semialg_set 2 (2::nat) Qp_val_poly"
 proof
   show "val_relation_set \<subseteq>  basic_semialg_set 2 (2::nat) Qp_val_poly"
   proof fix as
     assume A: "as \<in> val_relation_set"
     have 0: "length as = 2"
-      unfolding val_relation_set_def 
+      unfolding val_relation_set_def
       by (metis (no_types, lifting) A cartesian_power_car_memE mem_Collect_eq val_relation_set_def)
     have 1: "as = [as ! 0, as ! 1]"
       by (metis (no_types, lifting) A cartesian_power_car_memE mem_Collect_eq pair_id val_relation_set_def)
     show "as \<in>   basic_semialg_set 2 (2::nat) Qp_val_poly"
-      using A 1 val_relation_setE[of as]  Qp_val_semialg[of  "as!0" "as!1"] 
+      using A 1 val_relation_setE[of as]  Qp_val_semialg[of  "as!0" "as!1"]
       by presburger
   qed
   show "basic_semialg_set 2 (2::nat) Qp_val_poly \<subseteq> val_relation_set"
   proof
     fix as
     assume "as \<in> basic_semialg_set 2 (2::nat) Qp_val_poly"
     then show "as \<in> val_relation_set"
       using val_relation_setI[of as]
-      by (smt cartesian_power_car_memE cartesian_power_car_memE' mem_Collect_eq 
-          one_less_numeral_iff Qp_val_semialg basic_semialg_set_def 
+      by (smt cartesian_power_car_memE cartesian_power_car_memE' mem_Collect_eq
+          one_less_numeral_iff Qp_val_semialg basic_semialg_set_def
           val_relation_set_def padic_fields_axioms pair_id pos2 semiring_norm(76))
   qed
 qed
 
 lemma val_relation_is_semialgebraic:
 "is_semialgebraic 2 val_relation_set"
 proof -
 have "{rs \<in> carrier (Q\<^sub>p\<^bsup>2\<^esup>). val (rs ! 0) \<ge> val (rs ! 1)} = basic_semialg_set (Suc 1) (Suc 1) Qp_val_poly"
 using Suc_1 val_relation_semialg val_relation_set_def by presburger
   then show ?thesis
     by (metis (no_types) Qp_val_poly_closed Suc_1 basic_semialg_is_semialgebraic' val_relation_set_def zero_neq_numeral)
 qed
 
 lemma Qp_val_ring_is_semialg:
   obtains P where "P \<in> carrier Q\<^sub>p_x \<and> \<O>\<^sub>p = univ_basic_semialg_set 2 P"
 proof-
   obtain P where P_def: "P = (\<pp>[^](3::nat)) \<odot>\<^bsub>Q\<^sub>p_x \<^esub>(X_poly Q\<^sub>p) [^]\<^bsub>Q\<^sub>p_x\<^esub> (4::nat) \<oplus>\<^bsub>Q\<^sub>p_x\<^esub> \<one>\<^bsub>Q\<^sub>p_x\<^esub>"
     by blast
   have 0: "P \<in> carrier Q\<^sub>p_x"
   proof-
     have 0: "(X_poly Q\<^sub>p) \<in> carrier Q\<^sub>p_x"
       using UPQ.X_closed by blast
-    then show ?thesis 
-      using P_def UPQ.P.nat_pow_closed p_natpow_closed(1) by blast    
+    then show ?thesis
+      using P_def UPQ.P.nat_pow_closed p_natpow_closed(1) by blast
   qed
   have 1: "\<O>\<^sub>p = univ_basic_semialg_set 2 P"
   proof
     show "\<O>\<^sub>p \<subseteq> univ_basic_semialg_set 2 P"
     proof
       fix x
       assume A: "x \<in> \<O>\<^sub>p"
       show "x \<in> univ_basic_semialg_set 2 P"
       proof-
         have x_car: "x \<in> carrier Q\<^sub>p"
-          using A val_ring_memE by blast  
+          using A val_ring_memE by blast
         then have "(\<exists>y \<in> carrier Q\<^sub>p. \<one> \<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp>[^](3::nat))\<otimes> (x[^](4::nat)) = (y[^](2::nat)))"
-          using A Qp_val_ring_alt_def[of x] 
+          using A Qp_val_ring_alt_def[of x]
           by blast
         then obtain y where y_def: "y \<in> carrier Q\<^sub>p \<and> \<one> \<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp>[^](3::nat))\<otimes> (x[^](4::nat)) = (y[^](2::nat))"
-          by blast 
+          by blast
         have "y \<in> carrier Q\<^sub>p \<and> P \<bullet> x = (y[^](2::nat))"
         proof-
           have "P \<bullet> x = \<one> \<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<pp>[^](3::nat))\<otimes> (x[^](4::nat))"
           proof-
             have "((\<pp>[^](3::nat)) \<odot>\<^bsub>Q\<^sub>p_x\<^esub> (X_poly Q\<^sub>p) [^]\<^bsub>Q\<^sub>p_x\<^esub>  (4::nat)) \<in> carrier Q\<^sub>p_x"
-              using UPQ.monom_closed p_natpow_closed(1) by blast              
+              using UPQ.monom_closed p_natpow_closed(1) by blast
             then have "P \<bullet> x = (((\<pp>[^](3::nat)) \<odot>\<^bsub>Q\<^sub>p_x\<^esub> (X_poly Q\<^sub>p) [^]\<^bsub>Q\<^sub>p_x\<^esub>  (4::nat))\<bullet> x) \<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<one>\<^bsub>Q\<^sub>p_x\<^esub> \<bullet> x)"
-              using P_def x_car UPQ.to_fun_plus by blast                        
+              using P_def x_car UPQ.to_fun_plus by blast
             then have 0: "P \<bullet> x = (\<pp>[^](3::nat)) \<otimes>(( (X_poly Q\<^sub>p) [^]\<^bsub>Q\<^sub>p_x\<^esub>  (4::nat))\<bullet> x) \<oplus>\<^bsub>Q\<^sub>p\<^esub> (\<one>\<^bsub>Q\<^sub>p_x\<^esub> \<bullet> x)"
-              using UPQ.P.nat_pow_closed UPQ.X_closed UPQ.to_fun_smult p_natpow_closed(1) x_car by presburger              
+              using UPQ.P.nat_pow_closed UPQ.X_closed UPQ.to_fun_smult p_natpow_closed(1) x_car by presburger
             have "(( (X_poly Q\<^sub>p) [^]\<^bsub>Q\<^sub>p_x\<^esub>  (4::nat))\<bullet> x) = (x[^](4::nat))"
-              using UPQ.to_fun_X_pow x_car by blast              
+              using UPQ.to_fun_X_pow x_car by blast
             then have "P \<bullet> x = (\<pp>[^](3::nat)) \<otimes>(x[^](4::nat)) \<oplus>\<^bsub>Q\<^sub>p\<^esub> \<one>"
-              using "0" UPQ.to_fun_one x_car by presburger                                                        
-            then show ?thesis 
+              using "0" UPQ.to_fun_one x_car by presburger
+            then show ?thesis
               using y_def Qp.add.m_comm Qp.one_closed local.monom_term_car p_natpow_closed(1) x_car
-              by presburger              
+              by presburger
           qed
-          then show ?thesis 
-            using y_def 
+          then show ?thesis
+            using y_def
             by blast
         qed
-        then show ?thesis 
-          unfolding  univ_basic_semialg_set_def 
-          using x_car 
+        then show ?thesis
+          unfolding  univ_basic_semialg_set_def
+          using x_car
           by blast
       qed
     qed
     show "univ_basic_semialg_set 2 P \<subseteq> \<O>\<^sub>p"
     proof fix x
       assume A: "x \<in> univ_basic_semialg_set (2::nat) P"
       then obtain y where y_def: "y \<in> carrier Q\<^sub>p \<and> (P \<bullet> x) = (y[^](2::nat))"
         unfolding univ_basic_semialg_set_def
         by blast
       have x_car: "x \<in> carrier Q\<^sub>p"
-        using A 
+        using A
         by (metis (no_types, lifting) mem_Collect_eq univ_basic_semialg_set_def)
       have 0: "(P \<bullet> x) = (\<pp>[^](3::nat)) \<otimes> (x[^](4::nat)) \<oplus>\<^bsub>Q\<^sub>p\<^esub> \<one>"
-        using P_def x_car UPQ.UP_one_closed UPQ.monom_closed UPQ.monom_rep_X_pow UPQ.to_fun_monom 
-          UPQ.to_fun_one UPQ.to_fun_plus p_natpow_closed(1) by presburger                       
+        using P_def x_car UPQ.UP_one_closed UPQ.monom_closed UPQ.monom_rep_X_pow UPQ.to_fun_monom
+          UPQ.to_fun_one UPQ.to_fun_plus p_natpow_closed(1) by presburger
       have 1: "y \<in> carrier Q\<^sub>p \<and> (\<pp>[^](3::nat)) \<otimes> (x[^](4::nat)) \<oplus>\<^bsub>Q\<^sub>p\<^esub> \<one> = (y[^](2::nat))"
-        using "0" y_def 
+        using "0" y_def
         by blast
       then show "x \<in> \<O>\<^sub>p"
-        using x_car Qp_val_ring_alt_def[of x] y_def 
-        by (metis Qp.add.m_comm Qp.one_closed local.monom_term_car p_natpow_closed(1))        
-    qed
-  qed
-  show ?thesis 
-    using 0 1 that 
+        using x_car Qp_val_ring_alt_def[of x] y_def
+        by (metis Qp.add.m_comm Qp.one_closed local.monom_term_car p_natpow_closed(1))
+    qed
+  qed
+  show ?thesis
+    using 0 1 that
     by blast
 qed
 
 lemma Qp_val_ring_is_univ_semialgebraic:
 "is_univ_semialgebraic \<O>\<^sub>p"
 proof-
   obtain P where "P \<in> carrier Q\<^sub>p_x \<and> \<O>\<^sub>p = univ_basic_semialg_set 2 P"
     using Qp_val_ring_is_semialg by blast
-  then show ?thesis 
+  then show ?thesis
     by (metis univ_basic_semialg_set_is_univ_semialgebraic zero_neq_numeral)
 qed
 
 lemma Qp_val_ring_is_semialgebraic:
 "is_semialgebraic 1 (to_R1` \<O>\<^sub>p)"
   using Qp_val_ring_is_univ_semialgebraic is_univ_semialgebraic_def by blast
 
       (********************************************************************************************)
       (********************************************************************************************)
       subsubsection\<open>Inverse Images of Semialgebraic Sets by Polynomial Maps\<close>
       (********************************************************************************************)
       (********************************************************************************************)
 
 lemma basic_semialg_pullback:
   assumes "f \<in> carrier (Q\<^sub>p[\<X>\<^bsub>k\<^esub>])"
   assumes "is_poly_tuple n fs"
   assumes "length fs = k"
   assumes "S = basic_semialg_set k m f"
   assumes "m  \<noteq>0"
   shows "poly_map n fs  \<inverse>\<^bsub>n\<^esub> S = basic_semialg_set n m (Qp_poly_comp n fs f)"
 proof
   show "poly_map n fs  \<inverse>\<^bsub>n\<^esub> S \<subseteq> basic_semialg_set n m (Qp_poly_comp n fs f)"
   proof
     fix x
     assume A: "x \<in> poly_map n fs  \<inverse>\<^bsub>n\<^esub> S"
-    then have 0: "poly_map n fs x \<in> S"      
+    then have 0: "poly_map n fs x \<in> S"
     proof -
       have "\<exists>n f. {rs. rs \<in> S} \<subseteq> {rs \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). \<exists>r. r \<in> carrier Q\<^sub>p \<and> Qp_ev f rs = (r[^](n::nat))}"
         by (metis (no_types) Collect_mem_eq \<open>S = basic_semialg_set k m f\<close> basic_semialg_set_def eq_iff)
       then show ?thesis
         using A by blast
     qed
     have 1: "x \<in> carrier  (Q\<^sub>p\<^bsup>n\<^esup>)"
-      using A assms 
-      by (meson evimage_eq)      
+      using A assms
+      by (meson evimage_eq)
     have "\<exists>y \<in> (carrier Q\<^sub>p). Qp_ev f (poly_map n fs x) = (y[^]m)"
-      using A 0 assms basic_semialg_set_def 
+      using A 0 assms basic_semialg_set_def
       by blast
     then have "\<exists>y \<in> (carrier Q\<^sub>p). Qp_ev (Qp_poly_comp n fs f) x = (y[^]m)"
       using 1 assms Qp_poly_comp_eval
       by blast
     then show "x \<in> basic_semialg_set n m (Qp_poly_comp n fs f)"
-      using "1" basic_semialg_set_def 
+      using "1" basic_semialg_set_def
       by blast
   qed
   show "basic_semialg_set n m (Qp_poly_comp n fs f) \<subseteq> poly_map n fs  \<inverse>\<^bsub>n\<^esub> S"
   proof fix x
     assume A: "x \<in> basic_semialg_set n m (Qp_poly_comp n fs f)"
     have 0: "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
-      using A basic_semialg_set_def 
+      using A basic_semialg_set_def
       by blast
     have 1: "(poly_map n fs x) \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>)"
       using "0" poly_map_closed assms(2) assms(3)  by blast
     show "x \<in> poly_map n fs  \<inverse>\<^bsub>n\<^esub> S"
     proof-
       have "\<exists>y \<in> carrier Q\<^sub>p. Qp_ev (Qp_poly_comp n fs f) x = (y[^]m)"
-        using A basic_semialg_set_def 
+        using A basic_semialg_set_def
         by blast
       then have 2: "\<exists>y \<in> carrier Q\<^sub>p. Qp_ev f (poly_map n fs x) = (y[^]m)"
         using assms Qp_poly_comp_eval
         by (metis (no_types, lifting) A basic_semialg_set_def mem_Collect_eq)
       have 3: "poly_map n fs x \<in> S"
-        using assms 0 1 basic_semialg_set_def[of k m f] "2" 
+        using assms 0 1 basic_semialg_set_def[of k m f] "2"
         by blast
       show ?thesis
-        using "0" "3" by blast        
+        using "0" "3" by blast
     qed
   qed
 qed
 
 lemma basic_semialg_pullback':
   assumes "is_poly_tuple n fs"
   assumes "length fs = k"
   assumes "A \<in> basic_semialgs k"
   shows "poly_map n fs  \<inverse>\<^bsub>n\<^esub> A \<in> (basic_semialgs n)"
 proof-
   obtain f m where fm_def: "m \<noteq>0 \<and>f \<in> carrier (Q\<^sub>p[\<X>\<^bsub>k\<^esub>]) \<and> A = basic_semialg_set k m f"
-    using assms 
+    using assms
     by (metis is_basic_semialg_def mem_Collect_eq)
   then have "poly_map n fs  \<inverse>\<^bsub>n\<^esub> A  = basic_semialg_set n m (Qp_poly_comp n fs f)"
-    using assms basic_semialg_pullback[of f k n fs A m] 
-    by linarith    
-  then show ?thesis unfolding is_basic_semialg_def 
+    using assms basic_semialg_pullback[of f k n fs A m]
+    by linarith
+  then show ?thesis unfolding is_basic_semialg_def
     by (metis (mono_tags, lifting) assms(1) assms(2) fm_def mem_Collect_eq poly_compose_closed)
 qed
 
 lemma semialg_pullback:
   assumes "is_poly_tuple n fs"
   assumes "length fs = k"
   assumes "S \<in> semialg_sets k"
   shows "poly_map n fs  \<inverse>\<^bsub>n\<^esub> S \<in> semialg_sets n"
   unfolding semialg_sets_def
   apply(rule gen_boolean_algebra.induct[of S "(carrier (Q\<^sub>p\<^bsup>k\<^esup>))" "basic_semialgs k"])
   using assms semialg_sets_def apply blast
   apply (metis assms(1) assms(2) carrier_is_semialgebraic evimageI2 extensional_vimage_closed is_semialgebraicE poly_map_closed semialg_sets_def subsetI subset_antisym)
   apply (metis Int_absorb2 assms(1) assms(2) basic_semialg_is_semialg basic_semialg_is_semialgebraic basic_semialg_pullback' is_semialgebraic_closed mem_Collect_eq semialg_sets_def)
   apply (metis evimage_Un semialg_sets_def semialg_union)
   by (metis assms(1) assms(2) carrier_is_semialgebraic diff_is_semialgebraic evimage_Diff extensional_vimage_closed is_semialgebraicE is_semialgebraicI poly_map_closed poly_map_pullbackI semialg_sets_def subsetI subset_antisym)
 
 lemma pullback_is_semialg:
   assumes "is_poly_tuple n fs"
   assumes "length fs = k"
   assumes "S \<in> semialg_sets k"
   shows "is_semialgebraic n (poly_map n fs  \<inverse>\<^bsub>n\<^esub> S)"
-  using assms(1) assms(2) assms(3)  is_semialgebraicI padic_fields_axioms semialg_pullback 
+  using assms(1) assms(2) assms(3)  is_semialgebraicI padic_fields_axioms semialg_pullback
   by blast
 
 text\<open>Equality and inequality sets for a pair of polynomials\<close>
 
 definition val_ineq_set where
 "val_ineq_set n f g = {x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). val (Qp_ev f x) \<le> val (Qp_ev g x)}"
 
 lemma poly_map_length :
   assumes "length fs = m"
   assumes "as \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "length (poly_map n fs as) = m"
   using assms unfolding poly_map_def poly_tuple_eval_def
-  by (metis (no_types, lifting) length_map restrict_apply')  
+  by (metis (no_types, lifting) length_map restrict_apply')
 
 lemma val_ineq_set_pullback:
   assumes "f \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   assumes "g \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   shows "val_ineq_set n f g = poly_map n [g,f] \<inverse>\<^bsub>n\<^esub> val_relation_set "
 proof
   show "val_ineq_set n f g \<subseteq> poly_map n [g,f]  \<inverse>\<^bsub>n\<^esub> val_relation_set"
   proof
     fix x
     assume "x \<in> val_ineq_set n f g"
     then have 0: "x \<in>  carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<and> val (Qp_ev f x) \<le> val (Qp_ev g x)"
       by (metis (mono_tags, lifting) mem_Collect_eq val_ineq_set_def)
     have 1: "poly_map n [g,f] x = [Qp_ev g x, Qp_ev f x]"
-      unfolding  poly_map_def poly_tuple_eval_def  using 0 
+      unfolding  poly_map_def poly_tuple_eval_def  using 0
       by (metis (no_types, lifting) Cons_eq_map_conv list.simps(8) restrict_apply')
     have 2: "poly_map n [g,f] x \<in> val_relation_set"
       apply(rule val_relation_setI)
-      using 1 0 assms  apply (metis eval_at_point_closed nth_Cons_0)         
+      using 1 0 assms  apply (metis eval_at_point_closed nth_Cons_0)
       using 1 0 assms  apply (metis One_nat_def eval_at_point_closed diff_Suc_1 less_numeral_extra(1) nth_Cons_pos Qp.to_R_to_R1)
       using poly_map_length assms 0  apply (metis "1" Qp_2I cartesian_power_car_memE eval_at_point_closed)
-      by (metis "0" "1" One_nat_def nth_Cons_0 nth_Cons_Suc)    
+      by (metis "0" "1" One_nat_def nth_Cons_0 nth_Cons_Suc)
     have 3: "is_poly_tuple n [g, f]"
-      using assms 
+      using assms
       by (smt One_nat_def diff_Suc_1 Qp_is_poly_tupleI length_Suc_conv less_SucE less_one list.size(3) nth_Cons')
-    then show "x \<in> poly_map n [g,f]  \<inverse>\<^bsub>n\<^esub> val_relation_set" 
-      using 0 1 2 
-      by blast    
+    then show "x \<in> poly_map n [g,f]  \<inverse>\<^bsub>n\<^esub> val_relation_set"
+      using 0 1 2
+      by blast
   qed
   show "poly_map n [g,f] \<inverse>\<^bsub>n\<^esub> val_relation_set \<subseteq> val_ineq_set n f g"
   proof fix x
     have 0: "is_poly_tuple n [g, f]"
-      using Qp_is_poly_tupleI assms 
+      using Qp_is_poly_tupleI assms
       by (metis (no_types, lifting) diff_Suc_1 length_Cons less_Suc0 less_SucE list.size(3) nth_Cons')
     assume A: "x \<in> poly_map n [g,f] \<inverse>\<^bsub>n\<^esub> val_relation_set"
     then have 1: "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<and> poly_map n [g,f] x \<in> val_relation_set"
-      using 0 
-      by (meson evimageD extensional_vimage_closed subsetD)       
+      using 0
+      by (meson evimageD extensional_vimage_closed subsetD)
     have 2: "poly_map n [g,f] x = [Qp_ev g x, Qp_ev f x]"
-      by (metis "1" Qp_poly_mapE' length_0_conv poly_map_cons)    
+      by (metis "1" Qp_poly_mapE' length_0_conv poly_map_cons)
     show "x \<in> val_ineq_set n f g"
        using 0 1 2 unfolding val_ineq_set_def  val_relation_set_def
-       by (metis (no_types, lifting) "1" list.inject mem_Collect_eq nth_Cons_0 poly_map_apply val_relation_setE)       
+       by (metis (no_types, lifting) "1" list.inject mem_Collect_eq nth_Cons_0 poly_map_apply val_relation_setE)
    qed
 qed
 
 lemma val_ineq_set_is_semialg:
   assumes "f \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   assumes "g \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   shows "val_ineq_set n f g \<in> semialg_sets n"
 proof-
   have 0: "val_relation_set \<in> semialg_sets 2"
-    using val_relation_semialg basic_semialg_is_semialg' 
+    using val_relation_semialg basic_semialg_is_semialg'
     by (metis Qp_val_poly_closed zero_neq_numeral)
   show ?thesis using val_ineq_set_pullback semialg_pullback[of n "[g,f]" 2 "val_relation_set" ]
     by (metis (no_types, lifting) "0" assms(1) assms(2) diff_Suc_1 Qp_is_poly_tupleI
         length_Cons less_Suc0 less_SucE list.size(3) nth_Cons_0 nth_Cons_pos numeral_2_eq_2
         zero_neq_numeral)
 qed
 
 lemma val_ineq_set_is_semialgebraic:
   assumes "f \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   assumes "g \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   shows "is_semialgebraic n (val_ineq_set n f g)"
   using assms(1) assms(2) is_semialgebraicI val_ineq_set_is_semialg by blast
 
 lemma val_ineq_setI:
   assumes "f \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   assumes "g \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   assumes "x \<in> (val_ineq_set n f g)"
   shows "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
         "val (Qp_ev f x) \<le> val (Qp_ev g x)"
   using assms unfolding val_ineq_set_def  apply blast
-  using assms unfolding val_ineq_set_def by blast 
-  
+  using assms unfolding val_ineq_set_def by blast
+
 lemma val_ineq_setE:
   assumes "f \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   assumes "g \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes      "val (Qp_ev f x) \<le> val (Qp_ev g x)"
   shows "x \<in> (val_ineq_set n f g)"
-  using assms unfolding val_ineq_set_def 
+  using assms unfolding val_ineq_set_def
   by blast
 
 lemma val_ineq_set_is_semialgebraic':
   assumes "f \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   assumes "g \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   shows "is_semialgebraic n {x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). val (Qp_ev f x) \<le> val (Qp_ev g x)}"
-  using assms val_ineq_set_is_semialgebraic unfolding val_ineq_set_def by blast 
+  using assms val_ineq_set_is_semialgebraic unfolding val_ineq_set_def by blast
 
 lemma val_eq_set_is_semialgebraic:
   assumes "f \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   assumes "g \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   shows "is_semialgebraic n {x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). val (Qp_ev f x) = val (Qp_ev g x)}"
 proof-
-  have 0: "is_semialgebraic n {x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). val (Qp_ev f x) \<le> val (Qp_ev g x)}" 
-    using assms val_ineq_set_is_semialgebraic unfolding val_ineq_set_def 
-    by blast 
-  have 1: "is_semialgebraic n {x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). val (Qp_ev g x) \<le> val (Qp_ev f x)}" 
-    using assms val_ineq_set_is_semialgebraic unfolding val_ineq_set_def 
-    by blast 
+  have 0: "is_semialgebraic n {x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). val (Qp_ev f x) \<le> val (Qp_ev g x)}"
+    using assms val_ineq_set_is_semialgebraic unfolding val_ineq_set_def
+    by blast
+  have 1: "is_semialgebraic n {x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). val (Qp_ev g x) \<le> val (Qp_ev f x)}"
+    using assms val_ineq_set_is_semialgebraic unfolding val_ineq_set_def
+    by blast
   have 2: "{x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). val (Qp_ev f x) = val (Qp_ev g x)} = {x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). val (Qp_ev f x) \<le> val (Qp_ev g x)} \<inter>
                                                    {x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). val (Qp_ev g x) \<le>  val (Qp_ev f x)}"
-    apply(rule equalityI, rule  subsetI , rule IntI) unfolding mem_Collect_eq 
+    apply(rule equalityI, rule  subsetI , rule IntI) unfolding mem_Collect_eq
     using le_less apply blast apply (metis order_refl)
-    apply(rule  subsetI, erule IntE) unfolding mem_Collect_eq 
+    apply(rule  subsetI, erule IntE) unfolding mem_Collect_eq
     by (meson less_le_trans not_less_iff_gr_or_eq)
   show ?thesis unfolding 2 apply(rule intersection_is_semialg)
-    using 0 apply blast using 1 by blast  
+    using 0 apply blast using 1 by blast
 qed
 
 lemma equalityI'':
   assumes "\<And>x. A x \<Longrightarrow> B x"
   assumes "\<And>x. B x \<Longrightarrow> A x"
   shows "{x. A x} = {x. B x}"
-  using assms by blast 
+  using assms by blast
 
 lemma val_strict_ineq_set_is_semialgebraic:
   assumes "f \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   assumes "g \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   shows "is_semialgebraic n {x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). val (Qp_ev f x) < val (Qp_ev g x)}"
 proof-
   have 0: "{x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). val (Qp_ev f x) < val (Qp_ev g x)} =
              {x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). val (Qp_ev f x) \<le> val (Qp_ev g x)} - {x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). val (Qp_ev f x) = val (Qp_ev g x)}"
-    apply(rule equalityI', rule DiffI) unfolding le_less mem_Collect_eq  apply blast 
+    apply(rule equalityI', rule DiffI) unfolding le_less mem_Collect_eq  apply blast
     unfolding mem_Collect_eq using neq_iff apply blast
-    apply(erule DiffE) unfolding mem_Collect_eq by blast 
-  show ?thesis unfolding 0 
-    apply(rule diff_is_semialgebraic) 
-    using assms val_ineq_set_is_semialgebraic[of f n g] unfolding val_ineq_set_def apply blast 
-    using assms val_eq_set_is_semialgebraic[of f n g] unfolding val_ineq_set_def by blast 
+    apply(erule DiffE) unfolding mem_Collect_eq by blast
+  show ?thesis unfolding 0
+    apply(rule diff_is_semialgebraic)
+    using assms val_ineq_set_is_semialgebraic[of f n g] unfolding val_ineq_set_def apply blast
+    using assms val_eq_set_is_semialgebraic[of f n g] unfolding val_ineq_set_def by blast
 qed
 
 lemma constant_poly_val_exists:
   shows "\<exists>g \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>]). (\<forall> x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). val (Qp_ev g x) = c)"
 proof-
   obtain a where a_def: "a \<in> carrier Q\<^sub>p \<and> val a = c"
     by (meson Qp.minus_closed Qp.nonzero_closed dist_nonempty' p_nonzero)
   obtain g where g_def: "g = coord_const a"
-    by blast 
-  show ?thesis using a_def g_def Qp_to_IP_car 
+    by blast
+  show ?thesis using a_def g_def Qp_to_IP_car
     by (metis (no_types, opaque_lifting) Qp_to_IP_car a_def eval_at_point_const g_def le_less subset_iff)
 qed
 
 lemma val_ineq_set_is_semialgebraic'':
   assumes "f \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   shows "is_semialgebraic n {x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). val (Qp_ev f x) \<le> c}"
 proof-
   obtain g where g_def: "g \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>]) \<and> (\<forall> x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). val (Qp_ev g x) = c)"
     using constant_poly_val_exists by blast
   have 0: "is_semialgebraic n {x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). val (Qp_ev f x) \<le> val (Qp_ev g x)}"
     apply(rule val_ineq_set_is_semialgebraic')
-    using assms apply blast using  g_def by blast 
+    using assms apply blast using  g_def by blast
   have 1: "{x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). val (Qp_ev f x) \<le> val (Qp_ev g x)} = {x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). val (Qp_ev f x) \<le> c}"
     apply(rule equalityI'') using g_def apply fastforce using g_def by fastforce
-  show ?thesis using 0 unfolding 1 by blast 
+  show ?thesis using 0 unfolding 1 by blast
 qed
 
 lemma val_ineq_set_is_semialgebraic''':
   assumes "f \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   shows "is_semialgebraic n {x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). c \<le> val (Qp_ev f x)}"
 proof-
   obtain g where g_def: "g \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>]) \<and> (\<forall> x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). val (Qp_ev g x) = c)"
     using constant_poly_val_exists by blast
   have 0: "is_semialgebraic n {x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). val (Qp_ev g x) \<le> val (Qp_ev f x)}"
     apply(rule val_ineq_set_is_semialgebraic')
      using  g_def apply blast using assms by blast
   have 1: "{x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). val (Qp_ev g x) \<le> val (Qp_ev f x)} = {x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>).  c \<le> val (Qp_ev f x)}"
     apply(rule equalityI'') using g_def apply fastforce using g_def by fastforce
-  show ?thesis using 0 unfolding 1 by blast 
+  show ?thesis using 0 unfolding 1 by blast
 qed
 
 lemma val_eq_set_is_semialgebraic':
   assumes "f \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   shows "is_semialgebraic n {x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). val (Qp_ev f x) = c}"
 proof-
   obtain g where g_def: "g \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>]) \<and> (\<forall> x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). val (Qp_ev g x) = c)"
     using constant_poly_val_exists by blast
   have 0: "is_semialgebraic n {x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). val (Qp_ev f x) = val (Qp_ev g x)}"
     apply(rule val_eq_set_is_semialgebraic)
-    using assms apply blast using  g_def by blast 
+    using assms apply blast using  g_def by blast
   have 1: "{x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). val (Qp_ev f x) = val (Qp_ev g x)} = {x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). val (Qp_ev f x) = c}"
     apply(rule equalityI'') using g_def apply fastforce using g_def by metis
-  show ?thesis using 0 unfolding 1 by blast 
+  show ?thesis using 0 unfolding 1 by blast
 qed
 
 lemma val_strict_ineq_set_is_semialgebraic':
   assumes "f \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   shows "is_semialgebraic n {x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). val (Qp_ev f x) < c}"
 proof-
   obtain g where g_def: "g \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>]) \<and> (\<forall> x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). val (Qp_ev g x) = c)"
     using constant_poly_val_exists by blast
   have 0: "is_semialgebraic n {x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). val (Qp_ev f x) < val (Qp_ev g x)}"
     apply(rule val_strict_ineq_set_is_semialgebraic)
-    using assms apply blast using  g_def by blast 
+    using assms apply blast using  g_def by blast
   have 1: "{x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). val (Qp_ev f x) < val (Qp_ev g x)} = {x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). val (Qp_ev f x) < c}"
-    apply(rule equalityI'') using g_def apply fastforce using g_def 
+    apply(rule equalityI'') using g_def apply fastforce using g_def
     by fastforce
-  show ?thesis using 0 g_def unfolding 1 
+  show ?thesis using 0 g_def unfolding 1
     by blast
 qed
 
 lemma val_strict_ineq_set_is_semialgebraic'':
   assumes "f \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   shows "is_semialgebraic n {x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). c < val (Qp_ev f x)}"
 proof-
   obtain g where g_def: "g \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>]) \<and> (\<forall> x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). val (Qp_ev g x) = c)"
     using constant_poly_val_exists by blast
   have 0: "is_semialgebraic n {x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>).  val (Qp_ev g x) < val (Qp_ev f x)}"
     apply(rule val_strict_ineq_set_is_semialgebraic)
-    using   g_def apply blast using assms  by blast 
+    using   g_def apply blast using assms  by blast
   have 1: "{x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). val (Qp_ev g x) < val (Qp_ev f x)} = {x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). c < val (Qp_ev f x)}"
     apply(rule equalityI'') using assms g_def apply fastforce using assms g_def by fastforce
-  show ?thesis using 0 g_def unfolding 1 
+  show ?thesis using 0 g_def unfolding 1
     by blast
 qed
 
 lemma(in cring) R1_memE:
   assumes "x \<in> carrier (R\<^bsup>1\<^esup>)"
   shows "x = [(hd x)]"
-  using assms cartesian_power_car_memE 
+  using assms cartesian_power_car_memE
   by (metis diff_is_0_eq' hd_conv_nth le_eq_less_or_eq length_0_conv length_tl list.exhaust list.sel(3) normalize.cases nth_Cons_0 zero_neq_one)
 
 lemma(in cring) R1_memE':
- assumes "x \<in> carrier (R\<^bsup>1\<^esup>)" 
- shows "hd x \<in> carrier R" 
+ assumes "x \<in> carrier (R\<^bsup>1\<^esup>)"
+ shows "hd x \<in> carrier R"
   using R1_memE assms cartesian_power_car_memE[of x R 1] cartesian_power_car_memE'[of x R 1 0]
   by (metis hd_conv_nth less_numeral_extra(1) list.size(3) zero_neq_one)
 
 lemma univ_val_ineq_set_is_univ_semialgebraic:
 "is_univ_semialgebraic {x \<in> carrier Q\<^sub>p. val x \<le> c}"
 proof-
   have 0: "is_semialgebraic 1 {x \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>). val (Qp_ev (pvar Q\<^sub>p 0) x) \<le> c}"
     apply(rule val_ineq_set_is_semialgebraic'')
-    using pvar_closed by blast 
+    using pvar_closed by blast
   have 1: "{x \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>). val (Qp_ev (pvar Q\<^sub>p 0) x) \<le> c} = to_R1 ` {x \<in> carrier Q\<^sub>p. val x \<le> c}"
   proof(rule equalityI')
     show " \<And>x. x \<in> {x \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>). val (eval_at_point Q\<^sub>p x (pvar Q\<^sub>p 0)) \<le> c} \<Longrightarrow> x \<in> (\<lambda>a. [a]) ` {x \<in> carrier Q\<^sub>p. val x \<le> c}"
     proof- fix x assume A: "x \<in> {x \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>). val (eval_at_point Q\<^sub>p x (pvar Q\<^sub>p 0)) \<le> c}"
       then have 0: "x = [(hd x)] \<and> hd x \<in> carrier Q\<^sub>p"
         using Qp.R1_memE Qp.R1_memE' by blast
       have 1: "eval_at_point Q\<^sub>p x (pvar Q\<^sub>p 0) = hd x"
-        using A 0 
+        using A 0
         by (metis (no_types, lifting) One_nat_def eval_pvar lessI nth_Cons_0 Qp.to_R1_closed)
       then show "x \<in> (\<lambda>a. [a]) ` {x \<in> carrier Q\<^sub>p. val x \<le> c}"
-        using A 0 unfolding mem_Collect_eq   
+        using A 0 unfolding mem_Collect_eq
         by (metis (no_types, lifting) image_iff mem_Collect_eq)
     qed
     show "\<And>x. x \<in> (\<lambda>a. [a]) ` {x \<in> carrier Q\<^sub>p. val x \<le> c} \<Longrightarrow> x \<in> {x \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>). val (eval_at_point Q\<^sub>p x (pvar Q\<^sub>p 0)) \<le> c}"
     proof fix x assume A: "x \<in> (\<lambda>a. [a]) ` {x \<in> carrier Q\<^sub>p. val x \<le> c} "
       then obtain a where a_def: "x = [a] \<and> a \<in> carrier Q\<^sub>p \<and> val a \<le> c"
         by blast
       then have 0: "x \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>)"
         using cartesian_power_car_memI Qp.to_R1_closed by presburger
       then have 1: "(eval_at_point Q\<^sub>p x (pvar Q\<^sub>p 0)) = a"
         using a_def  by (metis eval_pvar less_one Qp.to_R_to_R1)
       show "x \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>) \<and> val (eval_at_point Q\<^sub>p x (pvar Q\<^sub>p 0)) \<le> c"
-        unfolding 1 using a_def 0 by blast 
-    qed
-  qed
-  show ?thesis using 0 unfolding 1 
+        unfolding 1 using a_def 0 by blast
+    qed
+  qed
+  show ?thesis using 0 unfolding 1
     using is_univ_semialgebraicI by blast
 qed
 
 lemma univ_val_strict_ineq_set_is_univ_semialgebraic:
 "is_univ_semialgebraic {x \<in> carrier Q\<^sub>p. val x < c}"
 proof-
   have 0: "is_semialgebraic 1 {x \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>). val (Qp_ev (pvar Q\<^sub>p 0) x) <c}"
     apply(rule val_strict_ineq_set_is_semialgebraic')
-    using pvar_closed by blast 
+    using pvar_closed by blast
   have 1: "{x \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>). val (Qp_ev (pvar Q\<^sub>p 0) x) < c} = to_R1 ` {x \<in> carrier Q\<^sub>p. val x < c}"
   proof(rule equalityI')
     show " \<And>x. x \<in> {x \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>). val (eval_at_point Q\<^sub>p x (pvar Q\<^sub>p 0)) < c} \<Longrightarrow> x \<in> (\<lambda>a. [a]) ` {x \<in> carrier Q\<^sub>p. val x < c}"
     proof- fix x assume A: "x \<in> {x \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>). val (eval_at_point Q\<^sub>p x (pvar Q\<^sub>p 0)) < c}"
       then have 0: "x = [(hd x)] \<and> hd x \<in> carrier Q\<^sub>p"
         using Qp.R1_memE Qp.R1_memE' by blast
       have 1: "eval_at_point Q\<^sub>p x (pvar Q\<^sub>p 0) = hd x"
-        using A 0 
+        using A 0
         by (metis (no_types, lifting) One_nat_def eval_pvar lessI nth_Cons_0 Qp.to_R1_closed)
       then show "x \<in> (\<lambda>a. [a]) ` {x \<in> carrier Q\<^sub>p. val x < c}"
-        using A 0 unfolding mem_Collect_eq   
+        using A 0 unfolding mem_Collect_eq
         by (metis (no_types, lifting) image_iff mem_Collect_eq)
     qed
     show "\<And>x. x \<in> (\<lambda>a. [a]) ` {x \<in> carrier Q\<^sub>p. val x < c} \<Longrightarrow> x \<in> {x \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>). val (eval_at_point Q\<^sub>p x (pvar Q\<^sub>p 0)) < c}"
     proof fix x assume A: "x \<in> (\<lambda>a. [a]) ` {x \<in> carrier Q\<^sub>p. val x < c} "
       then obtain a where a_def: "x = [a] \<and> a \<in> carrier Q\<^sub>p \<and> val a < c"
         by blast
       then have 0: "x \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>)"
         using cartesian_power_car_memI Qp.to_R1_closed by presburger
       then have 1: "(eval_at_point Q\<^sub>p x (pvar Q\<^sub>p 0)) = a"
         using a_def  by (metis eval_pvar less_one Qp.to_R_to_R1)
       show "x \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>) \<and> val (eval_at_point Q\<^sub>p x (pvar Q\<^sub>p 0)) < c"
-        unfolding 1 using a_def 0 by blast 
-    qed
-  qed
-  show ?thesis using 0 unfolding 1 
+        unfolding 1 using a_def 0 by blast
+    qed
+  qed
+  show ?thesis using 0 unfolding 1
     using is_univ_semialgebraicI by blast
 qed
 
 lemma univ_val_eq_set_is_univ_semialgebraic:
 "is_univ_semialgebraic {x \<in> carrier Q\<^sub>p. val x = c}"
 proof-
   have 0: "is_semialgebraic 1 {x \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>). val (Qp_ev (pvar Q\<^sub>p 0) x) = c}"
     apply(rule val_eq_set_is_semialgebraic')
-    using pvar_closed by blast 
+    using pvar_closed by blast
   have 1: "{x \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>). val (Qp_ev (pvar Q\<^sub>p 0) x) = c} = to_R1 ` {x \<in> carrier Q\<^sub>p. val x = c}"
   proof(rule equalityI')
     show " \<And>x. x \<in> {x \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>). val (eval_at_point Q\<^sub>p x (pvar Q\<^sub>p 0)) = c} \<Longrightarrow> x \<in> (\<lambda>a. [a]) ` {x \<in> carrier Q\<^sub>p. val x = c}"
     proof- fix x assume A: "x \<in> {x \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>). val (eval_at_point Q\<^sub>p x (pvar Q\<^sub>p 0)) = c}"
       then have 0: "x = [(hd x)] \<and> hd x \<in> carrier Q\<^sub>p"
         using Qp.R1_memE Qp.R1_memE' by blast
       have 1: "eval_at_point Q\<^sub>p x (pvar Q\<^sub>p 0) = hd x"
-        using A 0 
+        using A 0
         by (metis (no_types, lifting) One_nat_def eval_pvar lessI nth_Cons_0 Qp.to_R1_closed)
       show "x \<in> (\<lambda>a. [a]) ` {x \<in> carrier Q\<^sub>p. val x = c}"
-        using A 0 unfolding mem_Collect_eq 1 by blast 
+        using A 0 unfolding mem_Collect_eq 1 by blast
     qed
     show "\<And>x. x \<in> (\<lambda>a. [a]) ` {x \<in> carrier Q\<^sub>p. val x = c} \<Longrightarrow> x \<in> {x \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>). val (eval_at_point Q\<^sub>p x (pvar Q\<^sub>p 0)) = c}"
     proof fix x assume A: "x \<in> (\<lambda>a. [a]) ` {x \<in> carrier Q\<^sub>p. val x = c} "
       then obtain a where a_def: "x = [a] \<and> a \<in> carrier Q\<^sub>p \<and> val a = c"
         by blast
       then have 0: "x \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>)"
         using cartesian_power_car_memI Qp.to_R1_closed by presburger
       then have 1: "(eval_at_point Q\<^sub>p x (pvar Q\<^sub>p 0)) = a"
         using a_def  by (metis eval_pvar less_one Qp.to_R_to_R1)
       show "x \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>) \<and> val (eval_at_point Q\<^sub>p x (pvar Q\<^sub>p 0)) = c"
-        unfolding 1 using a_def 0 by blast 
-    qed
-  qed
-  show ?thesis using 0 unfolding 1 
+        unfolding 1 using a_def 0 by blast
+    qed
+  qed
+  show ?thesis using 0 unfolding 1
     using is_univ_semialgebraicI by blast
 qed
 
 
       (********************************************************************************************)
       (********************************************************************************************)
       subsubsection\<open>One Dimensional $p$-adic Balls are Semialgebraic\<close>
       (********************************************************************************************)
       (********************************************************************************************)
 
 lemma coord_ring_one_def:
 "Pring Q\<^sub>p {(0::nat)} = (Q\<^sub>p[\<X>\<^bsub>1\<^esub>])"
 proof-
   have "{(0::nat)} = {..<1}"
-    by auto 
-  thus ?thesis 
-  unfolding coord_ring_def 
+    by auto
+  thus ?thesis
+  unfolding coord_ring_def
   by auto
 qed
 
 lemma times_p_pow_val:
   assumes "a \<in> carrier Q\<^sub>p"
   assumes "b = \<pp>[^]n \<otimes> a"
   shows "val b = val a + n"
   using  val_mult[of "\<pp>[^]n" a] assms unfolding assms(2) val_p_int_pow
   by (metis add.commute p_intpow_closed(1))
 
 lemma times_p_pow_neg_val:
   assumes "a \<in> carrier Q\<^sub>p"
   assumes "b = \<pp>[^]-n \<otimes> a"
   shows "val b = val a - n"
   by (metis Qp.m_comm Qp_int_pow_nonzero assms(1) assms(2) p_intpow_closed(1) p_intpow_inv'' p_nonzero val_fract val_p_int_pow)
 
 lemma eint_minus_int_pos:
   assumes "a - eint n \<ge> 0"
   shows "a \<ge> n"
   using assms apply(induction a)
   apply (metis diff_ge_0_iff_ge eint_ord_simps(1) idiff_eint_eint zero_eint_def)
   by simp
 
 text\<open>\<open>p\<close>-adic balls as pullbacks of polynomial maps\<close>
 
 lemma balls_as_pullbacks:
   assumes "c \<in> carrier Q\<^sub>p"
   shows "\<exists>P \<in> carrier (Q\<^sub>p[\<X>\<^bsub>1\<^esub>]). to_R1` B\<^bsub>n\<^esub>[c] = poly_map 1 [P]  \<inverse>\<^bsub>1\<^esub> (to_R1 ` \<O>\<^sub>p)"
 proof-
   obtain P0 where P0_def: "P0 = (to_poly (\<pp>[^](-n))) \<otimes>\<^bsub>Q\<^sub>p_x\<^esub>((X_poly Q\<^sub>p) \<ominus>\<^bsub>Q\<^sub>p_x\<^esub> to_poly c)"
     by blast
   have 0: "P0 \<in> carrier Q\<^sub>p_x"
   proof-
     have P0: "(X_poly Q\<^sub>p) \<ominus>\<^bsub>Q\<^sub>p_x\<^esub> to_poly c \<in> carrier Q\<^sub>p_x"
-      using UPQ.X_closed UPQ.to_poly_closed assms by blast      
+      using UPQ.X_closed UPQ.to_poly_closed assms by blast
     have P1: "(to_poly (\<pp>[^](-n))) \<in> carrier Q\<^sub>p_x"
-      using UPQ.to_poly_closed p_intpow_closed(1) by blast      
-    then show ?thesis 
-      using P0_def P0 P1 
-      by blast     
+      using UPQ.to_poly_closed p_intpow_closed(1) by blast
+    then show ?thesis
+      using P0_def P0 P1
+      by blast
   qed
   have 1: "\<And>x. x \<in> carrier Q\<^sub>p \<Longrightarrow> P0 \<bullet> x = (\<pp>[^](-n)) \<otimes>  (x \<ominus>\<^bsub>Q\<^sub>p\<^esub> c)"
   proof- fix x assume A: "x \<in> carrier Q\<^sub>p"
     have P0: "(to_poly (\<pp>[^](-n))) \<bullet> x = (\<pp>[^](-n))"
-      using A UPQ.to_fun_to_poly p_intpow_closed(1) by blast      
+      using A UPQ.to_fun_to_poly p_intpow_closed(1) by blast
     have P1: "((X_poly Q\<^sub>p) \<ominus>\<^bsub>Q\<^sub>p_x\<^esub> to_poly c) \<bullet> x =  (x \<ominus>\<^bsub>Q\<^sub>p\<^esub> c)"
-      by (metis A UPQ.to_fun_X_minus X_poly_minus_def assms)      
+      by (metis A UPQ.to_fun_X_minus X_poly_minus_def assms)
     have P2: "to_poly (\<pp>[^](-n)) \<in> carrier Q\<^sub>p_x"
-      using UPQ.to_poly_closed p_intpow_closed(1) by blast      
+      using UPQ.to_poly_closed p_intpow_closed(1) by blast
     have P3: "((X_poly Q\<^sub>p) \<ominus>\<^bsub>Q\<^sub>p_x\<^esub> to_poly c) \<in> carrier Q\<^sub>p_x"
-      using UPQ.X_closed UPQ.to_poly_closed assms by blast     
+      using UPQ.X_closed UPQ.to_poly_closed assms by blast
     have "to_poly (\<pp>[^]- n) \<otimes>\<^bsub>Q\<^sub>p_x\<^esub> ((X_poly Q\<^sub>p) \<ominus>\<^bsub>Q\<^sub>p_x\<^esub> to_poly c) \<bullet> x = to_poly (\<pp>[^]- n) \<bullet> x \<otimes> (((X_poly Q\<^sub>p) \<ominus>\<^bsub>Q\<^sub>p_x\<^esub> to_poly c) \<bullet> x)"
       using A P0_def P0 P1 P2 P3 to_fun_mult[of "to_poly (\<pp>[^](-n))" "(X_poly Q\<^sub>p) \<ominus>\<^bsub>Q\<^sub>p_x\<^esub> to_poly c" x] UPQ.to_fun_mult
-      by blast     
+      by blast
     then have "to_poly (\<pp>[^]- n) \<otimes>\<^bsub>Q\<^sub>p_x\<^esub> ((X_poly Q\<^sub>p) \<ominus>\<^bsub>Q\<^sub>p_x\<^esub> to_poly c) \<bullet> x = (\<pp>[^](-n)) \<otimes>  (x \<ominus>\<^bsub>Q\<^sub>p\<^esub> c) "
       by (metis P0 P1)
     then show "P0 \<bullet> x = (\<pp>[^](-n)) \<otimes>  (x \<ominus>\<^bsub>Q\<^sub>p\<^esub> c)"
-      using P0_def by metis 
+      using P0_def by metis
   qed
   have 2: " (\<lambda>a. [a]) ` B\<^bsub>n\<^esub>[c] = poly_map 1  [from_Qp_x P0]  \<inverse>\<^bsub>1\<^esub>  ((\<lambda>a. [a]) ` \<O>\<^sub>p)"
   proof
     show "(\<lambda>a. [a]) ` B\<^bsub>n\<^esub>[c] \<subseteq> poly_map 1  [from_Qp_x P0]  \<inverse>\<^bsub>1\<^esub>  ((\<lambda>a. [a]) ` \<O>\<^sub>p)"
     proof
       fix x
       assume A: "x \<in> (\<lambda>a. [a]) ` B\<^bsub>n\<^esub>[c]"
       then obtain a where a_def: "x = [a] \<and> a \<in> B\<^bsub>n\<^esub>[c]"
         by blast
       have P0: "P0 \<bullet> a \<in> \<O>\<^sub>p"
       proof-
         have "B\<^bsub>n\<^esub>[c] \<subseteq> carrier Q\<^sub>p"
-          using c_ball_in_Qp by blast         
+          using c_ball_in_Qp by blast
         hence a_closed: "a \<in> carrier Q\<^sub>p"
-          using a_def by blast 
+          using a_def by blast
         have P0: "P0 \<bullet> a = (\<pp>[^](-n)) \<otimes>  (a \<ominus> c)"
-          using 1 a_def c_ballE(1) 
+          using 1 a_def c_ballE(1)
           by blast
         then have P1: "val (P0 \<bullet> a) = val (\<pp>[^](-n)) + val (a \<ominus> c)"
-          using val_mult[of "\<pp>[^]-n" "a \<ominus> c"] a_closed assms Qp.minus_closed p_intpow_closed(1) 
-          by presburger              
+          using val_mult[of "\<pp>[^]-n" "a \<ominus> c"] a_closed assms Qp.minus_closed p_intpow_closed(1)
+          by presburger
          then have P2: "val (P0 \<bullet> a) = val (a \<ominus>\<^bsub>Q\<^sub>p\<^esub> c) - n"
            by (metis P0 Qp.m_comm Qp.minus_closed Qp_int_pow_nonzero assms local.a_closed
-               p_intpow_closed(1) p_intpow_inv'' p_nonzero val_fract val_p_int_pow)                         
-         have P3: "val (a \<ominus>\<^bsub>Q\<^sub>p\<^esub> c) \<ge> n" 
-           using a_def c_ballE(2) 
+               p_intpow_closed(1) p_intpow_inv'' p_nonzero val_fract val_p_int_pow)
+         have P3: "val (a \<ominus>\<^bsub>Q\<^sub>p\<^esub> c) \<ge> n"
+           using a_def c_ballE(2)
            by blast
-         then have "val (P0 \<bullet> a) \<ge> -n + n" 
-           using P2 by (metis add.commute diff_conv_add_uminus diff_self local.eint_minus_ineq' zero_eint_def)           
-         then have P4: "val (P0 \<bullet> a) \<ge> 0" 
-           by (metis add.commute add.right_inverse zero_eint_def)           
-         have P5: "P0 \<bullet> a \<in> carrier Q\<^sub>p" 
-           using "0" UPQ.to_fun_closed local.a_closed by blast           
-         then show ?thesis using P4   
-           using val_ring_val_criterion 
-           by blast            
+         then have "val (P0 \<bullet> a) \<ge> -n + n"
+           using P2 by (metis add.commute diff_conv_add_uminus diff_self local.eint_minus_ineq' zero_eint_def)
+         then have P4: "val (P0 \<bullet> a) \<ge> 0"
+           by (metis add.commute add.right_inverse zero_eint_def)
+         have P5: "P0 \<bullet> a \<in> carrier Q\<^sub>p"
+           using "0" UPQ.to_fun_closed local.a_closed by blast
+         then show ?thesis using P4
+           using val_ring_val_criterion
+           by blast
       qed
       have "poly_map 1 [from_Qp_x P0] x = [Qp_ev (from_Qp_x P0) [a]] "
-        using a_def poly_map_def[of 1 "[from_Qp_x P0]"] poly_tuple_eval_def[of ] 
-        by (metis Qp_poly_mapE' c_ballE(1) length_0_conv poly_map_cons Qp.to_R1_closed)       
+        using a_def poly_map_def[of 1 "[from_Qp_x P0]"] poly_tuple_eval_def[of ]
+        by (metis Qp_poly_mapE' c_ballE(1) length_0_conv poly_map_cons Qp.to_R1_closed)
       then have "poly_map 1 [from_Qp_x P0] x = [P0 \<bullet> a] "
         using Qp_x_Qp_poly_eval[of P0 a]
         by (metis "0" a_def c_ballE(1))
       then have P1: "poly_map 1 [from_Qp_x P0] x \<in> ((\<lambda>a. [a]) ` \<O>\<^sub>p)"
-        using P0 
-        by blast 
+        using P0
+        by blast
       have P2: "x \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>)"
-        using A c_ballE(1) Qp.to_R1_closed 
+        using A c_ballE(1) Qp.to_R1_closed
         by blast
       have P3: "is_poly_tuple 1 [from_Qp_x P0]"
         apply(rule Qp_is_poly_tupleI)
-        by (metis "0" Qp_is_poly_tupleI from_Qp_x_closed gr_implies_not0 is_poly_tupleE is_poly_tuple_Cons list.size(3) zero_neq_one)        
+        by (metis "0" Qp_is_poly_tupleI from_Qp_x_closed gr_implies_not0 is_poly_tupleE is_poly_tuple_Cons list.size(3) zero_neq_one)
       show "x \<in> poly_map 1 [UP_to_IP Q\<^sub>p 0 P0] \<inverse>\<^bsub>1\<^esub> (\<lambda>a. [a]) ` \<O>\<^sub>p"
-        using P3 P2 P1 unfolding evimage_def poly_map_def  
-        by blast  
+        using P3 P2 P1 unfolding evimage_def poly_map_def
+        by blast
     qed
     have 20: "is_poly_tuple 1 [from_Qp_x P0]"
-      using 0 UP_to_IP_closed[of P0 "0::nat"] 
-      unfolding is_poly_tuple_def 
+      using 0 UP_to_IP_closed[of P0 "0::nat"]
+      unfolding is_poly_tuple_def
       by (metis (no_types, lifting) empty_set from_Qp_x_closed list.simps(15) singletonD subset_code(1))
     show "poly_map 1 [UP_to_IP Q\<^sub>p 0 P0] \<inverse>\<^bsub>1\<^esub> (\<lambda>a. [a]) ` \<O>\<^sub>p \<subseteq> (\<lambda>a. [a]) ` B\<^bsub>n\<^esub>[c]"
     proof fix x assume A: "x \<in> poly_map 1 [UP_to_IP Q\<^sub>p 0 P0] \<inverse>\<^bsub>1\<^esub> ((\<lambda>a. [a]) ` \<O>\<^sub>p)"
       have A0: "(\<lambda>a. [a]) ` \<O>\<^sub>p \<subseteq> carrier (Q\<^sub>p\<^bsup>1\<^esup>)"
-        using Qp_val_ring_is_univ_semialgebraic is_univ_semialgebraic_def Qp.to_R1_car_subset 
+        using Qp_val_ring_is_univ_semialgebraic is_univ_semialgebraic_def Qp.to_R1_car_subset
               Qp_val_ring_is_semialgebraic is_semialgebraic_closed by presburger
       have "poly_map 1 [from_Qp_x P0] x \<in>  ((\<lambda>a. [a]) ` \<O>\<^sub>p)"
         using A0 A 20  by blast
       then obtain a where a_def:  "a \<in> \<O>\<^sub>p \<and> (poly_map 1 [from_Qp_x P0] x) = [a]"
         by blast
       have x_closed: "x \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>)"
-        using A 
-        by (meson evimage_eq) 
+        using A
+        by (meson evimage_eq)
       then obtain y where y_def: "x = [y] \<and> y \<in> carrier Q\<^sub>p"
-        using A 
+        using A
         by (metis Qp.to_R1_to_R Qp.to_R_pow_closed)
       have "(poly_map 1 [from_Qp_x P0] x) = [(Qp_ev (from_Qp_x P0) [y])]"
-        unfolding poly_map_def poly_tuple_eval_def using x_closed 
-        by (smt "20" One_nat_def length_Suc_conv list.size(3) nth_Cons_0 nth_map 
+        unfolding poly_map_def poly_tuple_eval_def using x_closed
+        by (smt "20" One_nat_def length_Suc_conv list.size(3) nth_Cons_0 nth_map
             poly_tuple_eval_closed poly_tuple_eval_def restrict_apply' Qp.to_R1_to_R y_def zero_less_Suc)
       then have "(poly_map 1 [from_Qp_x P0] x) = [P0 \<bullet> y]"
-        by (metis "0" Qp_x_Qp_poly_eval y_def)        
+        by (metis "0" Qp_x_Qp_poly_eval y_def)
       then have "[a] = [P0 \<bullet> y]"
-        using a_def 
+        using a_def
         by presburger
-      then have A1: "a =  (\<pp>[^](-n)) \<otimes>  (y \<ominus>\<^bsub>Q\<^sub>p\<^esub> c)" 
-        using 1[of y] y_def  
+      then have A1: "a =  (\<pp>[^](-n)) \<otimes>  (y \<ominus>\<^bsub>Q\<^sub>p\<^esub> c)"
+        using 1[of y] y_def
         by blast
       have "y \<in> B\<^bsub>n\<^esub>[c]"
       proof-
         have B0: "val a =  val (y \<ominus>\<^bsub>Q\<^sub>p\<^esub> c) - n"
-          using A1 y_def Qp.minus_closed assms times_p_pow_neg_val by blast               
+          using A1 y_def Qp.minus_closed assms times_p_pow_neg_val by blast
         have B1: "val a \<ge> 0"
-          using a_def val_ring_memE by blast   
+          using a_def val_ring_memE by blast
         then have  "val (y \<ominus>\<^bsub>Q\<^sub>p\<^esub> c) - n \<ge> 0"
-          using B0 
+          using B0
           by metis
         then have  "val (y \<ominus>\<^bsub>Q\<^sub>p\<^esub> c) \<ge> n"
-          using eint_minus_int_pos by blast          
+          using eint_minus_int_pos by blast
         then show "y \<in> B\<^bsub>n\<^esub>[c]"
           using c_ballI y_def by blast
       qed
       then show "x \<in> (\<lambda>a. [a]) ` B\<^bsub>n\<^esub>[c]"
-        using y_def by blast 
-    qed
-  qed
-  then show ?thesis 
+        using y_def by blast
+    qed
+  qed
+  then show ?thesis
     by (meson "0" from_Qp_x_closed)
 qed
 
 lemma ball_is_semialgebraic:
   assumes "c \<in> carrier Q\<^sub>p"
   shows "is_semialgebraic 1 (to_R1` B\<^bsub>n\<^esub>[c])"
 proof-
   obtain P where P_def: "P \<in> carrier (Q\<^sub>p[\<X>\<^bsub>1\<^esub>]) \<and> to_R1` B\<^bsub>n\<^esub>[c] = poly_map 1 [P]  \<inverse>\<^bsub>1\<^esub> (to_R1 ` \<O>\<^sub>p) "
     using assms balls_as_pullbacks[of c n] by meson
   have "is_poly_tuple 1 [P]"
-    using P_def unfolding is_poly_tuple_def 
+    using P_def unfolding is_poly_tuple_def
     by (metis (no_types, opaque_lifting) list.inject list.set_cases neq_Nil_conv subset_code(1))
   then show ?thesis
-    using assms P_def pullback_is_semialg[of 1 "[P]" 1 "((\<lambda>a. [a]) ` \<O>\<^sub>p) "]  
-    by (metis (mono_tags, lifting) One_nat_def 
+    using assms P_def pullback_is_semialg[of 1 "[P]" 1 "((\<lambda>a. [a]) ` \<O>\<^sub>p) "]
+    by (metis (mono_tags, lifting) One_nat_def
         Qp_val_ring_is_semialgebraic is_semialgebraic_def length_Suc_conv
         list.distinct(1) list.size(3))
 qed
-    
+
 lemma ball_is_univ_semialgebraic:
   assumes "c \<in> carrier Q\<^sub>p"
-  shows "is_univ_semialgebraic (B\<^bsub>n\<^esub>[c])"  
-  using assms ball_is_semialgebraic c_ball_in_Qp is_univ_semialgebraic_def 
-  by presburger  
+  shows "is_univ_semialgebraic (B\<^bsub>n\<^esub>[c])"
+  using assms ball_is_semialgebraic c_ball_in_Qp is_univ_semialgebraic_def
+  by presburger
 
 abbreviation Qp_to_R1_set  where
 "Qp_to_R1_set S \<equiv> to_R1 ` S"
 
 
       (********************************************************************************************)
       (********************************************************************************************)
       subsubsection\<open>Finite Unions and Intersections of Semialgebraic Sets\<close>
       (********************************************************************************************)
       (********************************************************************************************)
 
 
-definition are_semialgebraic where 
+definition are_semialgebraic where
 "are_semialgebraic n Xs = (\<forall> x. x \<in> Xs \<longrightarrow> is_semialgebraic n x)"
 
 lemma are_semialgebraicI:
   assumes "\<And>x. x \<in> Xs \<Longrightarrow> is_semialgebraic n x "
   shows "are_semialgebraic n Xs"
   using are_semialgebraic_def assms by blast
 
 lemma are_semialgebraicE:
   assumes "are_semialgebraic n Xs"
   assumes "x \<in> Xs"
   shows "is_semialgebraic n x"
   using are_semialgebraic_def assms(1) assms(2) by blast
 
-definition are_univ_semialgebraic where 
+definition are_univ_semialgebraic where
 "are_univ_semialgebraic Xs = (\<forall> x. x \<in> Xs \<longrightarrow> is_univ_semialgebraic x)"
 
 lemma are_univ_semialgebraicI:
   assumes "\<And>x. x \<in> Xs \<Longrightarrow> is_univ_semialgebraic x "
   shows "are_univ_semialgebraic Xs"
   using are_univ_semialgebraic_def assms by blast
 
 lemma are_univ_semialgebraicE:
   assumes "are_univ_semialgebraic Xs"
   assumes "x \<in> Xs"
   shows "is_univ_semialgebraic x"
   using are_univ_semialgebraic_def assms(1) assms(2) by blast
 
 lemma are_univ_semialgebraic_semialgebraic:
   assumes "are_univ_semialgebraic Xs"
   shows "are_semialgebraic 1 (Qp_to_R1_set ` Xs)"
   apply(rule  are_semialgebraicI)
-  using  are_univ_semialgebraicE assms image_iff is_univ_semialgebraicE 
+  using  are_univ_semialgebraicE assms image_iff is_univ_semialgebraicE
   by (metis (no_types, lifting))
-  
+
 lemma to_R1_set_union:
 "to_R1 ` (\<Union> Xs) = \<Union> (Qp_to_R1_set ` Xs)"
   using image_iff by blast
 
 lemma to_R1_inter:
   assumes "Xs \<noteq> {}"
   shows "to_R1 ` (\<Inter> Xs) = \<Inter> (Qp_to_R1_set ` Xs)"
 proof
   show "to_R1 ` (\<Inter> Xs)  \<subseteq> \<Inter> (Qp_to_R1_set ` Xs)"
     by blast
   show "\<Inter> (Qp_to_R1_set ` Xs) \<subseteq> to_R1 ` (\<Inter> Xs)"
   proof fix x
     assume  A: "x \<in> \<Inter> (Qp_to_R1_set ` Xs)"
     then have 0: "\<And>S. S \<in> Xs \<Longrightarrow> x \<in> (Qp_to_R1_set S)"
       by blast
     obtain S where "S \<in> Xs \<and> x \<in> (Qp_to_R1_set S)"
-      using assms 0 
+      using assms 0
       by blast
     then obtain b where b_def: "b \<in> S \<and> x = [b]"
-      by blast  
+      by blast
     have "b \<in> (\<Inter> Xs)"
       using "0" b_def by blast
-    then show "x \<in> to_R1 ` (\<Inter> Xs)" 
+    then show "x \<in> to_R1 ` (\<Inter> Xs)"
       using b_def by blast
   qed
-qed    
+qed
 
 lemma finite_union_is_semialgebraic:
   assumes "finite Xs"
   shows "Xs \<subseteq> semialg_sets n \<longrightarrow> is_semialgebraic n (\<Union> Xs)"
   apply(rule finite.induct[of Xs])
     apply (simp add: assms)
   apply (simp add: empty_is_semialgebraic)
   by (metis Sup_insert insert_subset is_semialgebraicI union_is_semialgebraic)
-    
+
 lemma finite_union_is_semialgebraic':
   assumes "finite Xs"
   assumes "Xs \<subseteq> semialg_sets n "
   shows "is_semialgebraic n (\<Union> Xs)"
   using assms(1) assms(2) finite_union_is_semialgebraic by blast
 
 lemma(in padic_fields) finite_union_is_semialgebraic'':
   assumes "finite S"
   assumes  "\<And>x. x \<in> S \<Longrightarrow> is_semialgebraic m (F x)"
   shows "is_semialgebraic m (\<Union> x \<in> S. F x)"
   using assms  finite_union_is_semialgebraic[of "F`S" m] unfolding is_semialgebraic_def
   by blast
 
 lemma finite_union_is_univ_semialgebraic':
   assumes "finite Xs"
   assumes "are_univ_semialgebraic Xs"
   shows "is_univ_semialgebraic (\<Union> Xs)"
 proof-
   have "is_semialgebraic 1 (Qp_to_R1_set (\<Union> Xs))"
     using assms finite_union_is_semialgebraic'[of "((`) (\<lambda>a. [a]) ` Xs)"] to_R1_set_union[of Xs]
-    by (metis (no_types, lifting) are_semialgebraicE are_univ_semialgebraic_semialgebraic 
+    by (metis (no_types, lifting) are_semialgebraicE are_univ_semialgebraic_semialgebraic
         finite_imageI is_semialgebraicE subsetI)
-  then show ?thesis 
+  then show ?thesis
     using is_univ_semialgebraicI by blast
 qed
 
 lemma finite_intersection_is_semialgebraic:
   assumes "finite Xs"
   shows "Xs \<subseteq> semialg_sets n \<and> Xs \<noteq>{} \<longrightarrow> is_semialgebraic n (\<Inter> Xs)"
   apply(rule finite.induct[of Xs])
   apply (simp add: assms)
   apply auto[1]
 proof fix A::"((nat \<Rightarrow> int) \<times> (nat \<Rightarrow> int)) set list set set" fix a
   assume 0: "finite A"
   assume 1: "A \<subseteq> semialg_sets n \<and> A \<noteq> {} \<longrightarrow> is_semialgebraic n (\<Inter> A) "
   assume 2: "insert a A \<subseteq> semialg_sets n \<and> insert a A \<noteq> {}"
   show "is_semialgebraic n (\<Inter> (insert a A))"
   proof(cases "A = {}")
     case True
     then have "insert a A = {a}"
       by simp
-    then show ?thesis 
+    then show ?thesis
       by (metis "2" cInf_singleton insert_subset is_semialgebraicI)
   next
     case False
     then have "A \<subseteq> semialg_sets n \<and> A \<noteq> {}"
       using "2" by blast
     then have "is_semialgebraic n (\<Inter> A) "
       using "1" by linarith
-    then show ?thesis 
-      using 0 1 2 intersection_is_semialg 
-      by (metis Inf_insert insert_subset is_semialgebraicI)      
+    then show ?thesis
+      using 0 1 2 intersection_is_semialg
+      by (metis Inf_insert insert_subset is_semialgebraicI)
   qed
 qed
 
 lemma finite_intersection_is_semialgebraic':
   assumes "finite Xs"
   assumes "Xs \<subseteq> semialg_sets n \<and> Xs \<noteq>{}"
   shows  " is_semialgebraic n (\<Inter> Xs)"
   by (simp add: assms(1) assms(2) finite_intersection_is_semialgebraic)
 
 lemma finite_intersection_is_semialgebraic'':
   assumes "finite Xs"
   assumes "are_semialgebraic n Xs \<and> Xs \<noteq>{}"
   shows  " is_semialgebraic n (\<Inter> Xs)"
-  by (meson are_semialgebraicE assms(1) assms(2) 
+  by (meson are_semialgebraicE assms(1) assms(2)
       finite_intersection_is_semialgebraic' is_semialgebraicE subsetI)
 
 lemma finite_intersection_is_univ_semialgebraic:
   assumes "finite Xs"
   assumes "are_univ_semialgebraic Xs"
   assumes "Xs \<noteq> {}"
   shows  "is_univ_semialgebraic (\<Inter> Xs)"
 proof-
   have "are_semialgebraic 1 (Qp_to_R1_set ` Xs)"
     using are_univ_semialgebraic_semialgebraic assms(2) by blast
   then have  "is_semialgebraic 1 (\<Inter> (Qp_to_R1_set ` Xs))"
     using assms finite_intersection_is_semialgebraic''[of "Qp_to_R1_set ` Xs" 1]
     by blast
   then  have "is_semialgebraic 1 (Qp_to_R1_set (\<Inter> Xs))"
     using assms  to_R1_inter[of Xs]
-    by presburger    
-  then show ?thesis 
+    by presburger
+  then show ?thesis
     using is_univ_semialgebraicI by blast
 qed
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsection\<open>Cartesian Products of Semialgebraic Sets\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 lemma Qp_times_basic_semialg_right:
   assumes "a \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   shows "cartesian_product (basic_semialg_set n k a) (carrier (Q\<^sub>p\<^bsup>m\<^esup>)) = basic_semialg_set (n+ m) k a"
 proof
   show "cartesian_product (basic_semialg_set n k a) (carrier (Q\<^sub>p\<^bsup>m\<^esup>)) \<subseteq> basic_semialg_set (n + m) k a"
-  proof fix x 
+  proof fix x
     assume "x \<in> cartesian_product (basic_semialg_set n k a) (carrier (Q\<^sub>p\<^bsup>m\<^esup>))"
     then obtain as bs where as_bs_def: "as \<in> (basic_semialg_set n k a) \<and> bs \<in> (carrier (Q\<^sub>p\<^bsup>m\<^esup>)) \<and> x = as@bs"
       using cartesian_product_memE[of x "basic_semialg_set n k a" "carrier (Q\<^sub>p\<^bsup>m\<^esup>)" Q\<^sub>p n]
-            basic_semialg_set_def    
+            basic_semialg_set_def
       by (metis (no_types, lifting) append_take_drop_id basic_semialg_set_memE(1) subsetI)
     have 0: "x \<in> carrier (Q\<^sub>p\<^bsup>n+m\<^esup>)"
-      using as_bs_def basic_semialg_set_memE(1) cartesian_product_closed' 
+      using as_bs_def basic_semialg_set_memE(1) cartesian_product_closed'
       by blast
     have 1: "(Qp_ev a x = Qp_ev a as)"
-      using as_bs_def  poly_eval_cartesian_prod[of as n bs m a] assms basic_semialg_set_memE(1) by blast      
+      using as_bs_def  poly_eval_cartesian_prod[of as n bs m a] assms basic_semialg_set_memE(1) by blast
     obtain y where y_def: "y \<in> carrier Q\<^sub>p \<and> (Qp_ev a as = (y[^]k))"
-      using as_bs_def using basic_semialg_set_memE(2)[of as n k a] by blast 
+      using as_bs_def using basic_semialg_set_memE(2)[of as n k a] by blast
     show "x \<in> basic_semialg_set (n + m) k a"
       apply(rule basic_semialg_set_memI[of _ _ y])
         apply (simp add: "0")
        apply (simp add: y_def)
-      using "1" y_def by blast     
+      using "1" y_def by blast
   qed
   show "basic_semialg_set (n + m) k a \<subseteq> cartesian_product (basic_semialg_set n k a) (carrier (Q\<^sub>p\<^bsup>m\<^esup>))"
-  proof fix x 
+  proof fix x
     assume A: "x \<in> basic_semialg_set (n + m) k a"
     have A0: "x \<in> carrier (Q\<^sub>p\<^bsup>n+m\<^esup>)"
       using A basic_semialg_set_memE(1) by blast
     have A1: "set x \<subseteq> carrier Q\<^sub>p"
-      using A0  
+      using A0
       by (metis (no_types, lifting) cartesian_power_car_memE cartesian_power_car_memE' in_set_conv_nth subsetI)
     have A2: "length x = n + m"
-      using A0 cartesian_power_car_memE 
+      using A0 cartesian_power_car_memE
       by blast
     obtain as where as_def: "as = take n x"
       by blast
     obtain bs where bs_def: "bs = drop n x"
-      by blast 
-    have 0:  "x = as@bs" 
-      using A as_def bs_def 
+      by blast
+    have 0:  "x = as@bs"
+      using A as_def bs_def
         by (metis append_take_drop_id)
       have 1: "as \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
         apply(rule cartesian_power_car_memI)
-        using as_def A2 
+        using as_def A2
         apply (simp add: A2 min.absorb2)
         by (metis (no_types, lifting) A1 as_def dual_order.trans set_take_subset)
       have 2:  "bs \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
         apply(rule cartesian_power_car_memI)
-        using bs_def A2  
+        using bs_def A2
         apply (simp add: A2)
         by (metis A1 bs_def order.trans set_drop_subset)
       obtain y where y_def: "y \<in> carrier Q\<^sub>p \<and>  Qp_ev a x = (y[^]k)"
-        using basic_semialg_set_memE A by meson        
+        using basic_semialg_set_memE A by meson
       have 3: "as \<in> basic_semialg_set n k a"
         apply(rule basic_semialg_set_memI[of _ _ y])
           apply (simp add: "1")
         using \<open>y \<in> carrier Q\<^sub>p \<and> Qp_ev a x = (y[^]k)\<close> apply blast
-        using y_def A 1 0 2 assms(1) poly_eval_cartesian_prod by blast        
+        using y_def A 1 0 2 assms(1) poly_eval_cartesian_prod by blast
       show " x \<in> cartesian_product (basic_semialg_set n k a) (carrier (Q\<^sub>p\<^bsup>m\<^esup>))"
-        using 3 2 "0" 
+        using 3 2 "0"
         by (metis (mono_tags, lifting) as_def basic_semialg_set_memE(1) bs_def cartesian_product_memI subsetI)
     qed
 qed
 
 lemma Qp_times_basic_semialg_right_is_semialgebraic:
   assumes "k > 0"
   assumes "a \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   shows "is_semialgebraic (n + m) (cartesian_product (basic_semialg_set n k a) (carrier (Q\<^sub>p\<^bsup>m\<^esup>)))"
 proof-
   have 0: "cartesian_product (basic_semialg_set n k a) (carrier (Q\<^sub>p\<^bsup>m\<^esup>)) = basic_semialg_set (n+ m) k a"
-    using Qp_times_basic_semialg_right assms 
+    using Qp_times_basic_semialg_right assms
     by presburger
-  have 1: " a \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n+m\<^esub>])" 
-    using assms poly_ring_car_mono'(2) by blast    
+  have 1: " a \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n+m\<^esub>])"
+    using assms poly_ring_car_mono'(2) by blast
   have 2: "is_semialgebraic (n + m) (basic_semialg_set (n + m) k a)"
-    using  assms basic_semialg_is_semialgebraic'[of a "n+m" k "basic_semialg_set (n + m) k a"] 
+    using  assms basic_semialg_is_semialgebraic'[of a "n+m" k "basic_semialg_set (n + m) k a"]
           "1" by blast
   show ?thesis
-    using 0 2  
-    by metis 
-qed      
+    using 0 2
+    by metis
+qed
 
 lemma Qp_times_basic_semialg_right_is_semialgebraic':
-  assumes "A \<in> basic_semialgs n" 
+  assumes "A \<in> basic_semialgs n"
   shows "is_semialgebraic (n + m) (cartesian_product A (carrier (Q\<^sub>p\<^bsup>m\<^esup>)))"
 proof-
   obtain k P where "k \<noteq> 0 \<and> P\<in>carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])\<and> A = basic_semialg_set n k P"
-    using assms  is_basic_semialg_def 
+    using assms  is_basic_semialg_def
     by (metis mem_Collect_eq)
-  then show ?thesis using 
-      Qp_times_basic_semialg_right_is_semialgebraic[of  k P] 
+  then show ?thesis using
+      Qp_times_basic_semialg_right_is_semialgebraic[of  k P]
     using assms(1) by blast
 qed
 
 lemma cartesian_product_memE':
   assumes "x \<in> cartesian_product A B"
   obtains a b where "a \<in> A \<and> b \<in> B \<and> x = a@b"
-  using assms unfolding cartesian_product_def by blast 
+  using assms unfolding cartesian_product_def by blast
 
 lemma Qp_times_basic_semialg_left:
   assumes "a \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   shows "cartesian_product (carrier (Q\<^sub>p\<^bsup>m\<^esup>)) (basic_semialg_set n k a)  = basic_semialg_set (n+m) k (shift_vars n m a)"
 proof
   show "cartesian_product (carrier (Q\<^sub>p\<^bsup>m\<^esup>)) (basic_semialg_set n k a) \<subseteq> basic_semialg_set (n + m) k (shift_vars n m a)"
   proof fix x
     assume A: "x \<in> cartesian_product (carrier (Q\<^sub>p\<^bsup>m\<^esup>)) (basic_semialg_set n k a)"
     then obtain as bs where as_bs_def: "as \<in> (carrier (Q\<^sub>p\<^bsup>m\<^esup>)) \<and> bs \<in> (basic_semialg_set n k a) \<and> x = as@bs "
      using cartesian_product_memE' by blast
     have 0: "Qp_ev (shift_vars n m a) x = Qp_ev a bs"
-      using A as_bs_def assms shift_vars_eval[of a n as m bs ] 
-      by (metis (no_types, lifting) basic_semialg_set_memE(1)) 
+      using A as_bs_def assms shift_vars_eval[of a n as m bs ]
+      by (metis (no_types, lifting) basic_semialg_set_memE(1))
     obtain y where y_def: "y \<in> carrier Q\<^sub>p \<and> Qp_ev a bs = (y[^]k)"
-      using as_bs_def basic_semialg_set_memE(2) 
+      using as_bs_def basic_semialg_set_memE(2)
       by blast
     have 1: "x \<in> carrier (Q\<^sub>p\<^bsup>n+m\<^esup>)"
-      using A as_bs_def 
+      using A as_bs_def
       by (metis (no_types, lifting) add.commute basic_semialg_set_memE(1) cartesian_product_closed')
     show "x \<in> basic_semialg_set (n + m) k (shift_vars n m a)"
       apply(rule basic_semialg_set_memI[of _ _ y])
         apply (simp add: "1")
           using y_def apply blast
-           using "0" y_def by blast     
+           using "0" y_def by blast
   qed
   show "basic_semialg_set (n + m) k (shift_vars n m a) \<subseteq> cartesian_product (carrier (Q\<^sub>p\<^bsup>m\<^esup>)) (basic_semialg_set n k a) "
-  proof fix x 
+  proof fix x
     assume A: "x \<in> basic_semialg_set (n + m) k (shift_vars n m a)"
     then obtain y where y_def: "y \<in> carrier Q\<^sub>p \<and> Qp_ev (shift_vars n m a) x = (y[^]k)"
-      using assms basic_semialg_set_memE[of x "n + m" k "shift_vars n m a"] 
-            shift_vars_closed[of a m] Qp.cring_axioms       
+      using assms basic_semialg_set_memE[of x "n + m" k "shift_vars n m a"]
+            shift_vars_closed[of a m] Qp.cring_axioms
       by blast
     have "x \<in> carrier (Q\<^sub>p\<^bsup>m+n\<^esup>)"
-      using A basic_semialg_set_memE(1) 
+      using A basic_semialg_set_memE(1)
       by (metis add.commute)
     then have "x \<in> cartesian_product (carrier (Q\<^sub>p\<^bsup>m\<^esup>)) (carrier (Q\<^sub>p\<^bsup>n\<^esup>))"
       using cartesian_product_carrier by blast
     then obtain as bs where as_bs_def: "x = as@bs \<and> as \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<and> bs \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
-      by (meson cartesian_product_memE')      
+      by (meson cartesian_product_memE')
     have "bs \<in> (basic_semialg_set n k a)"
     apply(rule basic_semialg_set_memI[of _ _ y])
       using as_bs_def apply blast
        apply (simp add: y_def)
-        using y_def shift_vars_eval[of a n as m bs ] as_bs_def assms(1) 
-        by metis 
+        using y_def shift_vars_eval[of a n as m bs ] as_bs_def assms(1)
+        by metis
     then show "x \<in> cartesian_product (carrier (Q\<^sub>p\<^bsup>m\<^esup>)) (basic_semialg_set n k a)"
-      using as_bs_def unfolding cartesian_product_def 
-      by blast        
+      using as_bs_def unfolding cartesian_product_def
+      by blast
   qed
 qed
 
 lemma Qp_times_basic_semialg_left_is_semialgebraic:
   assumes "k > 0"
   assumes "a \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   shows "is_semialgebraic (n + m) (cartesian_product (carrier (Q\<^sub>p\<^bsup>m\<^esup>)) (basic_semialg_set n k a))"
   using basic_semialg_is_semialgebraic'[of a "n+m" k] Qp_times_basic_semialg_left
   by (metis assms(1) assms(2) basic_semialg_is_semialgebraic is_basic_semialg_def neq0_conv shift_vars_closed)
 
 lemma Qp_times_basic_semialg_left_is_semialgebraic':
-  assumes "A \<in> basic_semialgs n" 
+  assumes "A \<in> basic_semialgs n"
   shows "is_semialgebraic (n + m) (cartesian_product (carrier (Q\<^sub>p\<^bsup>m\<^esup>)) A)"
 proof-
   obtain k P where "k \<noteq> 0 \<and> P\<in>carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])\<and> A = basic_semialg_set n k P"
-    using assms  is_basic_semialg_def 
+    using assms  is_basic_semialg_def
     by (metis mem_Collect_eq)
-  then show ?thesis using 
-      Qp_times_basic_semialg_left_is_semialgebraic[of k P] 
+  then show ?thesis using
+      Qp_times_basic_semialg_left_is_semialgebraic[of k P]
     using assms(1)  by blast
 qed
 
 lemma product_of_basic_semialgs_is_semialg:
   assumes "k > 0"
   assumes "l > 0"
   assumes "a \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   assumes "b \<in> carrier (Q\<^sub>p[\<X>\<^bsub>m\<^esub>])"
   shows "is_semialgebraic (n + m) (cartesian_product (basic_semialg_set n k a) (basic_semialg_set m l b))"
 proof-
   have 0:  "cartesian_product (basic_semialg_set n k a) (carrier (Q\<^sub>p\<^bsup>m\<^esup>)) =  basic_semialg_set (n+ m) k a"
     using Qp_times_basic_semialg_right assms  by presburger
   have 1:  "cartesian_product (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) (basic_semialg_set m l b)  = basic_semialg_set (m + n) l (shift_vars m n b)"
     using Qp_times_basic_semialg_left assms by blast
-  have 2: "(cartesian_product (basic_semialg_set n k a) (basic_semialg_set m l b)) = 
-                cartesian_product (basic_semialg_set n k a) (carrier (Q\<^sub>p\<^bsup>m\<^esup>)) \<inter> 
+  have 2: "(cartesian_product (basic_semialg_set n k a) (basic_semialg_set m l b)) =
+                cartesian_product (basic_semialg_set n k a) (carrier (Q\<^sub>p\<^bsup>m\<^esup>)) \<inter>
                  cartesian_product (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) (basic_semialg_set m l b)"
   proof-
     have 0: "basic_semialg_set n k a \<subseteq> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
       using basic_semialg_set_memE(1) by blast
     have 1: "carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<subseteq> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
       by simp
     have 2: "carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<subseteq> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
       by simp
     have 3: "basic_semialg_set m l b \<subseteq> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
       using basic_semialg_set_memE(1) by blast
-    show ?thesis 
-      using 0 1 2 3 cartesian_product_intersection[of "(basic_semialg_set n k a)" Q\<^sub>p n 
+    show ?thesis
+      using 0 1 2 3 cartesian_product_intersection[of "(basic_semialg_set n k a)" Q\<^sub>p n
                                               "(carrier (Q\<^sub>p\<^bsup>m\<^esup>))" m
-                                              "(carrier (Q\<^sub>p\<^bsup>n\<^esup>))" "(basic_semialg_set m l b)"] 
+                                              "(carrier (Q\<^sub>p\<^bsup>n\<^esup>))" "(basic_semialg_set m l b)"]
       by (smt Collect_cong inf.absorb_iff1 inf.absorb_iff2)
   qed
   then show ?thesis
-    using Qp_times_basic_semialg_left_is_semialgebraic 
-          Qp_times_basic_semialg_right_is_semialgebraic  assms   
+    using Qp_times_basic_semialg_left_is_semialgebraic
+          Qp_times_basic_semialg_right_is_semialgebraic  assms
     by (metis (no_types, lifting) add.commute intersection_is_semialg)
 qed
 
 lemma product_of_basic_semialgs_is_semialg':
   assumes "A \<in> (basic_semialgs n)"
   assumes "B \<in> (basic_semialgs m)"
   shows "is_semialgebraic (n + m) (cartesian_product A B)"
 proof-
   obtain k a where ka_def: "k > 0 \<and> a \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>]) \<and> A = (basic_semialg_set n k a)"
-    using assms 
+    using assms
     by (metis is_basic_semialg_def mem_Collect_eq neq0_conv)
   obtain l b where lb_def: "l > 0 \<and> b \<in> carrier (Q\<^sub>p[\<X>\<^bsub>m\<^esub>]) \<and> B = (basic_semialg_set m l b)"
-    by (metis assms(2) gr_zeroI is_basic_semialg_def mem_Collect_eq)   
+    by (metis assms(2) gr_zeroI is_basic_semialg_def mem_Collect_eq)
   show ?thesis using ka_def lb_def assms product_of_basic_semialgs_is_semialg
   by blast
 qed
 
 lemma car_times_semialg_is_semialg:
   assumes "is_semialgebraic m B"
   shows "is_semialgebraic (n + m) (cartesian_product (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) B)"
   apply(rule gen_boolean_algebra.induct[of B "carrier (Q\<^sub>p\<^bsup>m\<^esup>)""basic_semialgs m"])
   using assms is_semialgebraic_def semialg_sets_def apply blast
-  apply (simp add: carrier_is_semialgebraic cartesian_product_carrier)  
+  apply (simp add: carrier_is_semialgebraic cartesian_product_carrier)
 proof-
   show "\<And>A. A \<in> basic_semialgs m \<Longrightarrow> is_semialgebraic (n + m) (cartesian_product (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) (A \<inter> carrier (Q\<^sub>p\<^bsup>m\<^esup>)))"
   proof-
-    fix A assume A: "A \<in> basic_semialgs m " 
+    fix A assume A: "A \<in> basic_semialgs m "
     then have " (A \<inter> carrier (Q\<^sub>p\<^bsup>m\<^esup>)) = A"
       by (metis basic_semialg_set_memE(1) inf_absorb1 is_basic_semialg_def mem_Collect_eq subsetI)
     then show "is_semialgebraic (n + m) (cartesian_product (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) (A \<inter> carrier (Q\<^sub>p\<^bsup>m\<^esup>)))"
       using add.commute[of n m] assms A
-            Qp_times_basic_semialg_left_is_semialgebraic' 
-      by (simp add: \<open>n + m = m + n\<close>)      
+            Qp_times_basic_semialg_left_is_semialgebraic'
+      by (simp add: \<open>n + m = m + n\<close>)
   qed
   show "\<And>A C. A \<in> gen_boolean_algebra (carrier (Q\<^sub>p\<^bsup>m\<^esup>)) (basic_semialgs m) \<Longrightarrow>
            is_semialgebraic (n + m) (cartesian_product (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) A) \<Longrightarrow>
            C \<in> gen_boolean_algebra (carrier (Q\<^sub>p\<^bsup>m\<^esup>)) (basic_semialgs m) \<Longrightarrow>
            is_semialgebraic (n + m) (cartesian_product (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) C) \<Longrightarrow>
            is_semialgebraic (n + m) (cartesian_product (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) (A \<union> C))"
   proof- fix A C assume A: " A \<in> gen_boolean_algebra (carrier (Q\<^sub>p\<^bsup>m\<^esup>)) (basic_semialgs m)"
            "is_semialgebraic (n + m) (cartesian_product (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) A)"
            "C \<in> gen_boolean_algebra (carrier (Q\<^sub>p\<^bsup>m\<^esup>)) (basic_semialgs m)"
-          " is_semialgebraic (n + m) (cartesian_product (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) C)" 
+          " is_semialgebraic (n + m) (cartesian_product (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) C)"
     then have B: "is_semialgebraic (n + m) (cartesian_product (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) A \<union> cartesian_product (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) C)"
       using union_is_semialgebraic by blast
     show "is_semialgebraic (n + m) (cartesian_product (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) (A \<union> C))"
     proof-
       have 0: "A \<subseteq> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
-        using A(1) gen_boolean_algebra_subset 
+        using A(1) gen_boolean_algebra_subset
         by blast
-      have  1: " C \<subseteq> carrier (Q\<^sub>p\<^bsup>m\<^esup>)" 
+      have  1: " C \<subseteq> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
         using A(3) gen_boolean_algebra_subset
         by blast
       then show ?thesis using 0 A B
         using cartesian_product_binary_union_right[of A Q\<^sub>p m C "(carrier (Q\<^sub>p\<^bsup>n\<^esup>))"]
-        unfolding is_semialgebraic_def semialg_sets_def 
+        unfolding is_semialgebraic_def semialg_sets_def
         by presburger
     qed
   qed
   show "\<And>A. A \<in> gen_boolean_algebra (carrier (Q\<^sub>p\<^bsup>m\<^esup>)) (basic_semialgs m) \<Longrightarrow>
          is_semialgebraic (n + m) (cartesian_product (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) A) \<Longrightarrow>
          is_semialgebraic (n + m) (cartesian_product (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) (carrier (Q\<^sub>p\<^bsup>m\<^esup>) - A))"
   proof- fix A assume A: "A \<in> gen_boolean_algebra (carrier (Q\<^sub>p\<^bsup>m\<^esup>)) (basic_semialgs m)"
                       "is_semialgebraic (n + m) (cartesian_product (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) A)"
     then have "A \<subseteq> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
-      using gen_boolean_algebra_subset 
+      using gen_boolean_algebra_subset
       by blast
     then show "is_semialgebraic (n + m) (cartesian_product (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) (carrier (Q\<^sub>p\<^bsup>m\<^esup>) - A))"
       using A cartesian_product_car_complement_right[of A Q\<^sub>p m n]
-      unfolding is_semialgebraic_def semialg_sets_def 
+      unfolding is_semialgebraic_def semialg_sets_def
       by (metis (mono_tags, lifting) semialg_complement semialg_sets_def)
   qed
 qed
 
 lemma basic_semialg_times_semialg_is_semialg:
-  assumes "A \<in> basic_semialgs n" 
+  assumes "A \<in> basic_semialgs n"
   assumes "is_semialgebraic m B"
   shows " is_semialgebraic (n + m) (cartesian_product A B)"
   apply(rule gen_boolean_algebra.induct[of B "carrier (Q\<^sub>p\<^bsup>m\<^esup>)""basic_semialgs m"])
   using assms(2) is_semialgebraic_def semialg_sets_def apply blast
   using Qp_times_basic_semialg_right_is_semialgebraic' assms(1) apply blast
   apply (metis assms(1) basic_semialg_is_semialgebraic inf.absorb1 is_semialgebraic_closed mem_Collect_eq product_of_basic_semialgs_is_semialg')
     apply (metis (no_types, lifting) cartesian_product_binary_union_right is_semialgebraicI is_semialgebraic_closed semialg_sets_def union_is_semialgebraic)
 proof-
   show "\<And>Aa. Aa \<in> gen_boolean_algebra (carrier (Q\<^sub>p\<^bsup>m\<^esup>)) (basic_semialgs m) \<Longrightarrow>
           is_semialgebraic (n + m) (cartesian_product A Aa) \<Longrightarrow> is_semialgebraic (n + m) (cartesian_product A (carrier (Q\<^sub>p\<^bsup>m\<^esup>) - Aa))"
    proof- fix B assume A:  "B \<in> gen_boolean_algebra (carrier (Q\<^sub>p\<^bsup>m\<^esup>)) (basic_semialgs m)"
                          "is_semialgebraic (n + m) (cartesian_product A B)"
     show "is_semialgebraic (n + m) (cartesian_product A (carrier (Q\<^sub>p\<^bsup>m\<^esup>) - B))"
-    using A assms cartesian_product_complement_right[of B Q\<^sub>p m A n] add.commute[of n m] 
+    using A assms cartesian_product_complement_right[of B Q\<^sub>p m A n] add.commute[of n m]
   proof -
    have f1: "\<forall>n B. \<not> is_semialgebraic n B \<or> B \<subseteq> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
      by (meson is_semialgebraic_closed)
    have "is_basic_semialg n A"
      using \<open>A \<in> {S. is_basic_semialg n S}\<close> by blast
    then have f2: "is_semialgebraic n A"
      using padic_fields.basic_semialg_is_semialgebraic padic_fields_axioms by blast
    have "B \<in> semialg_sets m"
      using \<open>B \<in> gen_boolean_algebra (carrier (Q\<^sub>p\<^bsup>m\<^esup>)) {S. is_basic_semialg m S}\<close> semialg_sets_def by blast
    then have "is_semialgebraic m B"
      by (meson padic_fields.is_semialgebraicI padic_fields_axioms)
    then show ?thesis
      using f2 f1 by (metis (no_types) Qp_times_basic_semialg_right_is_semialgebraic' \<open>A \<in> {S. is_basic_semialg n S}\<close> \<open>\<lbrakk>B \<subseteq> carrier (Q\<^sub>p\<^bsup>m\<^esup>); A \<subseteq> carrier (Q\<^sub>p\<^bsup>n\<^esup>)\<rbrakk> \<Longrightarrow> cartesian_product A (carrier (Q\<^sub>p\<^bsup>m\<^esup>)) - cartesian_product A B = cartesian_product A (carrier (Q\<^sub>p\<^bsup>m\<^esup>) - B)\<close> \<open>is_semialgebraic (n + m) (cartesian_product A B)\<close> diff_is_semialgebraic)
   qed
-  qed  
+  qed
 qed
 
 text\<open>Semialgebraic sets are closed under cartesian products\<close>
 
 lemma cartesian_product_is_semialgebraic:
   assumes "is_semialgebraic n A"
   assumes "is_semialgebraic m B"
   shows "is_semialgebraic (n + m) (cartesian_product A B)"
   apply(rule gen_boolean_algebra.induct[of A "carrier (Q\<^sub>p\<^bsup>n\<^esup>)""basic_semialgs n"])
   using assms is_semialgebraicE semialg_sets_def apply blast
-  using assms car_times_semialg_is_semialg  apply blast 
-  using assms basic_semialg_times_semialg_is_semialg  
+  using assms car_times_semialg_is_semialg  apply blast
+  using assms basic_semialg_times_semialg_is_semialg
   apply (simp add: Int_absorb2 basic_semialg_is_semialgebraic is_semialgebraic_closed)
 proof-
   show "\<And>A C. A \<in> gen_boolean_algebra (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) (basic_semialgs n) \<Longrightarrow>
            is_semialgebraic (n + m) (cartesian_product A B) \<Longrightarrow>
            C \<in> gen_boolean_algebra (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) (basic_semialgs n) \<Longrightarrow>
            is_semialgebraic (n + m) (cartesian_product C B) \<Longrightarrow> is_semialgebraic (n + m) (cartesian_product (A \<union> C) B)"
   proof- fix A C assume A: "A \<in> gen_boolean_algebra (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) (basic_semialgs n)"
                            "is_semialgebraic (n + m) (cartesian_product A B)"
                            "C \<in> gen_boolean_algebra (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) (basic_semialgs n)"
                            "is_semialgebraic (n + m) (cartesian_product C B)"
     show "is_semialgebraic (n + m) (cartesian_product (A \<union> C) B)"
-      using A cartesian_product_binary_union_left[of A Q\<^sub>p n C B]  
+      using A cartesian_product_binary_union_left[of A Q\<^sub>p n C B]
       by (metis (no_types, lifting) gen_boolean_algebra_subset union_is_semialgebraic)
   qed
   show "\<And>A. A \<in> gen_boolean_algebra (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) (basic_semialgs n) \<Longrightarrow>
          is_semialgebraic (n + m) (cartesian_product A B) \<Longrightarrow> is_semialgebraic (n + m) (cartesian_product (carrier (Q\<^sub>p\<^bsup>n\<^esup>) - A) B)"
-  proof- fix A assume A: "A \<in> gen_boolean_algebra (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) (basic_semialgs n)" 
+  proof- fix A assume A: "A \<in> gen_boolean_algebra (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) (basic_semialgs n)"
                         "is_semialgebraic (n + m) (cartesian_product A B)"
     show "is_semialgebraic (n + m) (cartesian_product (carrier (Q\<^sub>p\<^bsup>n\<^esup>) - A) B)"
       using assms  A cartesian_product_complement_left[of A Q\<^sub>p n B m]
-      unfolding is_semialgebraic_def semialg_sets_def 
-      by (metis car_times_semialg_is_semialg diff_is_semialgebraic is_semialgebraicE is_semialgebraicI 
-          is_semialgebraic_closed semialg_sets_def)     
+      unfolding is_semialgebraic_def semialg_sets_def
+      by (metis car_times_semialg_is_semialg diff_is_semialgebraic is_semialgebraicE is_semialgebraicI
+          is_semialgebraic_closed semialg_sets_def)
   qed
 qed
 
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsection\<open>$N^{th}$ Power Residues\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 
 definition nth_root_poly where
 "nth_root_poly (n::nat) a = ((X_poly Q\<^sub>p) [^]\<^bsub>Q\<^sub>p_x\<^esub>  n) \<ominus>\<^bsub>Q\<^sub>p_x\<^esub> (to_poly a)"
 
 lemma nth_root_poly_closed:
   assumes "a \<in> carrier Q\<^sub>p"
   shows "nth_root_poly n a \<in> carrier Q\<^sub>p_x"
-  using assms unfolding nth_root_poly_def 
+  using assms unfolding nth_root_poly_def
   by (meson UPQ.P.minus_closed UPQ.P.nat_pow_closed UPQ.X_closed UPQ.to_poly_closed)
 
 lemma nth_root_poly_eval:
   assumes "a \<in> carrier Q\<^sub>p"
   assumes "b \<in> carrier Q\<^sub>p"
   shows "(nth_root_poly n a) \<bullet> b = (b[^]n) \<ominus>\<^bsub>Q\<^sub>p\<^esub> a"
-  using assms unfolding nth_root_poly_def 
+  using assms unfolding nth_root_poly_def
   using UPQ.P.nat_pow_closed UPQ.X_closed UPQ.to_fun_X_pow UPQ.to_fun_diff UPQ.to_fun_to_poly UPQ.to_poly_closed by presburger
 
 text\<open>Hensel's lemma gives us this criterion for the existence of \<open>n\<close>-th roots\<close>
 
 lemma nth_root_poly_root:
   assumes "(n::nat) > 1"
   assumes "a \<in> \<O>\<^sub>p"
   assumes "a \<noteq> \<one>"
   assumes "val (\<one> \<ominus>\<^bsub>Q\<^sub>p\<^esub> a) > 2* val ([n]\<cdot>\<one>)"
   shows "(\<exists> b \<in> \<O>\<^sub>p. ((b[^]n) = a))"
 proof-
   obtain \<alpha> where alpha_def: "\<alpha> \<in> carrier Z\<^sub>p \<and> \<iota> \<alpha> = a"
     using assms(2) by blast
   have 0: "\<alpha> \<in> carrier Z\<^sub>p"
     by (simp add: alpha_def)
   have 1: "\<alpha> \<noteq> \<one>\<^bsub>Z\<^sub>p\<^esub>"
-    using assms alpha_def inc_of_one 
+    using assms alpha_def inc_of_one
     by blast
   obtain N where N_def: "N = [n]\<cdot>\<^bsub>Z\<^sub>p\<^esub> \<one>\<^bsub>Z\<^sub>p\<^esub>"
     by blast
   have N_closed: "N \<in> carrier Z\<^sub>p"
-    using N_def Zp_nat_mult_closed 
-    by blast   
+    using N_def Zp_nat_mult_closed
+    by blast
   have 2: "\<iota> ([n]\<cdot>\<^bsub>Z\<^sub>p\<^esub> \<one>\<^bsub>Z\<^sub>p\<^esub>) = ([n]\<cdot> \<one>)"
   proof(induction n)
-    case 0 
+    case 0
     have 00: "[(0::nat)] \<cdot>\<^bsub>Z\<^sub>p\<^esub> \<one>\<^bsub>Z\<^sub>p\<^esub> = \<zero>\<^bsub>Z\<^sub>p\<^esub>"
       using Zp_nat_inc_zero by blast
     have 01: "[(0::nat)] \<cdot>\<^bsub>Q\<^sub>p\<^esub> \<one> = \<zero>"
-      using Qp.nat_inc_zero by blast                
+      using Qp.nat_inc_zero by blast
     then show ?case
       using 00 inc_of_nat by blast
   next
-    case (Suc n)    
+    case (Suc n)
     then show ?case
-      using inc_of_nat by blast    
+      using inc_of_nat by blast
   qed
   have 3: "val_Zp (\<one>\<^bsub>Z\<^sub>p\<^esub> \<ominus>\<^bsub>Z\<^sub>p\<^esub> \<alpha>) = val (\<one> \<ominus>\<^bsub>Q\<^sub>p\<^esub> a)"
-    using alpha_def Zp.one_closed ring_hom_one[of \<iota> Z\<^sub>p Q\<^sub>p]  inc_is_hom Zp.ring_hom_minus[of Q\<^sub>p \<iota> "\<one>\<^bsub>Z\<^sub>p\<^esub>" \<alpha> ] 
-    Qp.ring_axioms 
-    unfolding \<iota>_def 
+    using alpha_def Zp.one_closed ring_hom_one[of \<iota> Z\<^sub>p Q\<^sub>p]  inc_is_hom Zp.ring_hom_minus[of Q\<^sub>p \<iota> "\<one>\<^bsub>Z\<^sub>p\<^esub>" \<alpha> ]
+    Qp.ring_axioms
+    unfolding \<iota>_def
     by (metis Q\<^sub>p_def Zp.minus_closed Zp_def padic_fields.val_of_inc padic_fields_axioms)
   have 4: "([n]\<cdot>\<^bsub>Z\<^sub>p\<^esub> \<one>\<^bsub>Z\<^sub>p\<^esub>) \<in> nonzero Z\<^sub>p"
-  proof- 
+  proof-
     have 40: "int n \<ge> 0"
       using of_nat_0_le_iff by blast
     have "nat (int n) = n"
       using nat_int by blast
     hence "[n]\<cdot>\<^bsub>Z\<^sub>p\<^esub> \<one>\<^bsub>Z\<^sub>p\<^esub> = [int n]\<cdot>\<^bsub>Z\<^sub>p\<^esub> \<one>\<^bsub>Z\<^sub>p\<^esub>"
-      using 40  unfolding add_pow_def int_pow_def nat_pow_def 
+      using 40  unfolding add_pow_def int_pow_def nat_pow_def
       proof -
         have "(if int n < 0 then inv\<^bsub>add_monoid Z\<^sub>p\<^esub> rec_nat \<one>\<^bsub>add_monoid Z\<^sub>p\<^esub> (\<lambda>n f. f \<otimes>\<^bsub>add_monoid Z\<^sub>p\<^esub> \<one>\<^bsub>Z\<^sub>p\<^esub>) 0 else rec_nat \<one>\<^bsub>add_monoid Z\<^sub>p\<^esub> (\<lambda>n f. f \<otimes>\<^bsub>add_monoid Z\<^sub>p\<^esub> \<one>\<^bsub>Z\<^sub>p\<^esub>) n) = rec_nat \<one>\<^bsub>add_monoid Z\<^sub>p\<^esub> (\<lambda>n f. f \<otimes>\<^bsub>add_monoid Z\<^sub>p\<^esub> \<one>\<^bsub>Z\<^sub>p\<^esub>) n"
           by (meson of_nat_less_0_iff)
         then show "rec_nat \<one>\<^bsub>add_monoid Z\<^sub>p\<^esub> (\<lambda>n f. f \<otimes>\<^bsub>add_monoid Z\<^sub>p\<^esub> \<one>\<^bsub>Z\<^sub>p\<^esub>) n = (let f = rec_nat \<one>\<^bsub>add_monoid Z\<^sub>p\<^esub> (\<lambda>n f. f \<otimes>\<^bsub>add_monoid Z\<^sub>p\<^esub> \<one>\<^bsub>Z\<^sub>p\<^esub>) in if int n < 0 then inv\<^bsub>add_monoid Z\<^sub>p\<^esub> f (nat (- int n)) else f (nat (int n)))"
           using \<open>nat (int n) = n\<close> by presburger
       qed
-      thus ?thesis 
+      thus ?thesis
         using Zp_char_0[of n] Zp.not_nonzero_memE Zp_char_0' assms(1) gr_implies_not_zero by blast
     qed
   then have 5: "val_Zp ([n]\<cdot>\<^bsub>Z\<^sub>p\<^esub> \<one>\<^bsub>Z\<^sub>p\<^esub>) = val ([n]\<cdot>\<^bsub>Q\<^sub>p\<^esub> (\<one>))"
-    using 2 ord_of_inc 
+    using 2 ord_of_inc
     by (metis N_closed N_def val_of_inc)
   then have 6: "(val_Zp (\<one>\<^bsub>Z\<^sub>p\<^esub> \<ominus>\<^bsub>Z\<^sub>p\<^esub> \<alpha>)) > 2*(val_Zp ([n]\<cdot>\<^bsub>Z\<^sub>p\<^esub> \<one>\<^bsub>Z\<^sub>p\<^esub>))"
-    using assms 3  by presburger   
+    using assms 3  by presburger
   have "\<exists> b \<in> carrier Z\<^sub>p. (b[^]\<^bsub>Z\<^sub>p\<^esub>n= \<alpha>)"
-    using Zp_nth_root_lemma[of \<alpha> n] assms "0" "1" "6" by blast   
+    using Zp_nth_root_lemma[of \<alpha> n] assms "0" "1" "6" by blast
   then obtain b where b_def: "b \<in> carrier Z\<^sub>p \<and>    (b[^]\<^bsub>Z\<^sub>p\<^esub>n= \<alpha>)"
-    by blast 
+    by blast
   then have "\<iota> (b [^]\<^bsub>Z\<^sub>p\<^esub>n) = a"
-    using alpha_def by blast    
+    using alpha_def by blast
   then have "(\<iota> b) [^] n = a"
-    by (metis Qp.nat_inc_zero Q\<^sub>p_def Qp.nat_pow_zero Zp.nat_pow_0 Zp.nat_pow_zero 
+    by (metis Qp.nat_inc_zero Q\<^sub>p_def Qp.nat_pow_zero Zp.nat_pow_0 Zp.nat_pow_zero
         Zp_nat_inc_zero \<iota>_def alpha_def assms(3) b_def frac_inc_of_nat inc_of_one inc_pow not_nonzero_Qp)
-  then show ?thesis                       
+  then show ?thesis
     using b_def by blast
 qed
 
 text\<open>All points sufficiently close to 1 have nth roots\<close>
 
 lemma eint_nat_times_2:
 "2*(n::nat) = 2*eint n"
-  using times_eint_simps(1) 
+  using times_eint_simps(1)
   by (metis mult.commute mult_2_right of_nat_add)
 
 lemma P_set_of_one:
 "P_set 1 = nonzero Q\<^sub>p"
   apply(rule equalityI) apply(rule subsetI)
-  unfolding P_set_def nonzero_def mem_Collect_eq  apply blast 
-  apply(rule subsetI)  unfolding P_set_def nonzero_def mem_Collect_eq  
+  unfolding P_set_def nonzero_def mem_Collect_eq  apply blast
+  apply(rule subsetI)  unfolding P_set_def nonzero_def mem_Collect_eq
   using Qp.nat_pow_eone by blast
 
-lemma nth_power_fact: 
+lemma nth_power_fact:
   assumes "(n::nat) \<ge> 1"
   shows "\<exists> (m::nat) > 0. \<forall> u \<in> carrier Q\<^sub>p. ac m u = 1 \<and> val u = 0 \<longrightarrow> u \<in> P_set n"
 proof(cases "n = 1")
-  case True 
+  case True
   have "\<forall> u \<in> carrier Q\<^sub>p. ac 1 u = 1 \<and> val u = 0 \<longrightarrow> u \<in> P_set n"
-    unfolding True P_set_of_one 
+    unfolding True P_set_of_one
     by (metis iless_Suc_eq padic_fields.val_ring_memE padic_fields.zero_in_val_ring padic_fields_axioms val_nonzero zero_eint_def)
   then show ?thesis  by blast
 next
   case F: False
   obtain m where m_def: "m = 1 + nat ( 2*(ord ([n]\<cdot>\<^bsub>Q\<^sub>p\<^esub> (\<one>))))"
-    by blast 
+    by blast
   then have m_pos: "m > 0"
     by linarith
   have "\<forall> u \<in> carrier Q\<^sub>p. ac m u = 1 \<and> val u = 0 \<longrightarrow> u \<in> P_set n"
   proof
     fix u
     assume A: "u \<in> carrier Q\<^sub>p"
     show " ac m u = 1 \<and> val u = 0 \<longrightarrow> u \<in> P_set n"
     proof
       assume B: "ac m u = 1 \<and> val u = 0"
       then have 0: "val u = val \<one>"
-        by (smt A ac_def not_nonzero_Qp val_one val_ord zero_eint_def)        
+        by (smt A ac_def not_nonzero_Qp val_one val_ord zero_eint_def)
       have 1: "ac m u = ac m \<one>"
-        by (metis B Qp.one_nonzero ac_p ac_p_int_pow_factor angular_component_factors_x angular_component_p inc_of_one m_pos p_nonzero)        
+        by (metis B Qp.one_nonzero ac_p ac_p_int_pow_factor angular_component_factors_x angular_component_p inc_of_one m_pos p_nonzero)
       have 2: "u \<in> nonzero Q\<^sub>p"
       proof-
         have "ac m \<zero> = 0"
-          by (meson ac_def)         
+          by (meson ac_def)
         then have "u \<noteq> \<zero>"
           by (metis B zero_neq_one)
-        then show ?thesis 
-          using A not_nonzero_Qp Qp.nonzero_memI by presburger          
+        then show ?thesis
+          using A not_nonzero_Qp Qp.nonzero_memI by presburger
       qed
       then have 3: "val (\<one> \<ominus>\<^bsub>Q\<^sub>p\<^esub> u) \<ge> m" using m_pos 0 1 ac_ord_prop[of "\<one>" u "0::int" m]
-        by (metis B Qp.one_nonzero add.right_neutral eint.inject val_ord zero_eint_def)                   
+        by (metis B Qp.one_nonzero add.right_neutral eint.inject val_ord zero_eint_def)
       show "u \<in> P_set n"
       proof(cases "u = \<one>")
         case True
-        then show ?thesis 
+        then show ?thesis
           by (metis P_set_one insert_iff zeroth_P_set)
       next
         case False
         have F0: "u \<in> \<O>\<^sub>p"
           apply(rule val_ring_memI, rule A)
-          unfolding 0 val_one by auto 
+          unfolding 0 val_one by auto
         have F1: "val (\<one> \<ominus>\<^bsub>Q\<^sub>p\<^esub> u) \<ge> m"
-          using False 3 by blast         
+          using False 3 by blast
         have "ord (\<one> \<ominus> u) \<ge> m"
           by (metis A F1 False Qp.not_eq_diff_nonzero Qp.one_closed eint_ord_simps(1) val_ord)
         hence F2: "ord (\<one> \<ominus>\<^bsub>Q\<^sub>p\<^esub> u) > 2*(ord ([n]\<cdot> \<one>))"
           using m_def F1 A False Qp.not_eq_diff_nonzero Qp.one_closed eint_ord_simps(1)
             int_nat_eq of_nat_1 of_nat_add val_ord[of "\<one> \<ominus> u"] eint_nat_times_2
-          by linarith     
+          by linarith
         have "val (\<one> \<ominus>\<^bsub>Q\<^sub>p\<^esub> u) > 2*(val ([n]\<cdot> \<one>))"
-        proof-    
+        proof-
           have 0: "val (\<one> \<ominus>\<^bsub>Q\<^sub>p\<^esub> u) > 2*(ord ([n]\<cdot> \<one>))"
             using F2 val_ord[of "\<one> \<ominus> u"] A False Qp.not_eq_diff_nonzero Qp.one_closed eint_ord_simps(2) by presburger
           have "n > 0"
             using assms  by linarith
           hence "eint (ord ([n] \<cdot> \<one>)) = val ([n] \<cdot> \<one>)"
-            using val_ord_nat_inc[of n]  
+            using val_ord_nat_inc[of n]
             by blast
           hence "2*ord ([n]\<cdot> \<one>) = 2*val ([n]\<cdot> \<one>)"
             by (metis inc_of_nat times_eint_simps(1))
-          thus ?thesis 
-            using 0 val_ord[of "\<one> \<ominus> u"] assms  
+          thus ?thesis
+            using 0 val_ord[of "\<one> \<ominus> u"] assms
             by presburger
         qed
         then have "(\<exists> b \<in> \<O>\<^sub>p. ((b[^]n) = u))"
-          using m_def False nth_root_poly_root[of n u] F0 assms  F by linarith          
+          using m_def False nth_root_poly_root[of n u] F0 assms  F by linarith
         then have "(\<exists> b \<in> carrier Q\<^sub>p. ((b[^]n) = u))"
           using val_ring_memE by blast
         then show "u \<in> P_set n"
-          using P_set_def[of n] 2 
+          using P_set_def[of n] 2
           by blast
       qed
     qed
   qed
-  then show ?thesis using m_pos by blast 
+  then show ?thesis using m_pos by blast
 qed
 
 definition pow_res where
 "pow_res (n::nat) x = {a. a \<in> carrier Q\<^sub>p \<and>  (\<exists>y \<in> nonzero Q\<^sub>p. (a = x \<otimes> (y[^]n)))}"
 
 lemma nonzero_pow_res:
   assumes "x \<in> nonzero Q\<^sub>p"
   shows "pow_res (n::nat) x \<subseteq> nonzero Q\<^sub>p"
 proof
-  fix a 
+  fix a
   assume "a \<in> pow_res n x"
   then obtain y where y_def: "y \<in> nonzero Q\<^sub>p \<and> (a = x \<otimes> (y[^]n))"
-    using pow_res_def by blast 
+    using pow_res_def by blast
   then show "a \<in> nonzero Q\<^sub>p"
-    using assms Qp.Units_m_closed Qp_nat_pow_nonzero Units_eq_nonzero by blast    
-qed
-
-lemma pow_res_of_zero: 
+    using assms Qp.Units_m_closed Qp_nat_pow_nonzero Units_eq_nonzero by blast
+qed
+
+lemma pow_res_of_zero:
   shows "pow_res n \<zero> = {\<zero>}"
   unfolding pow_res_def  apply(rule equalityI)
   apply(rule subsetI)
-  unfolding mem_Collect_eq 
+  unfolding mem_Collect_eq
   apply (metis Qp.integral_iff Qp.nat_pow_closed Qp.nonzero_closed Qp.zero_closed insertCI)
   apply(rule subsetI) unfolding mem_Collect_eq
-  by (metis Qp.nat_pow_one Qp.one_nonzero Qp.r_one Qp.zero_closed equals0D insertE)   
+  by (metis Qp.nat_pow_one Qp.one_nonzero Qp.r_one Qp.zero_closed equals0D insertE)
 
 lemma equal_pow_resI:
   assumes "x \<in> carrier Q\<^sub>p"
   assumes "y \<in> pow_res n x"
   shows "pow_res n x = pow_res n y"
 proof
   have y_closed: "y \<in> carrier Q\<^sub>p"
-    using assms unfolding pow_res_def by blast 
+    using assms unfolding pow_res_def by blast
   obtain c where c_def: "c \<in> nonzero Q\<^sub>p \<and> y = x \<otimes> (c[^]n)"
-    using assms pow_res_def by blast 
+    using assms pow_res_def by blast
   have "((inv c)[^]n) = inv (c[^]n)"
-    using c_def Qp.field_axioms Qp.nat_pow_of_inv Units_eq_nonzero by blast     
+    using c_def Qp.field_axioms Qp.nat_pow_of_inv Units_eq_nonzero by blast
   then have "y \<otimes> ((inv c)[^]n) = x"
-    using y_closed c_def assms Qp.inv_cancelL(2) Qp.nonzero_closed Qp_nat_pow_nonzero Units_eq_nonzero 
-    by presburger   
+    using y_closed c_def assms Qp.inv_cancelL(2) Qp.nonzero_closed Qp_nat_pow_nonzero Units_eq_nonzero
+    by presburger
   then have P0: "(inv c) \<in> nonzero Q\<^sub>p \<and> x =y \<otimes> ((inv c)[^]n) "
-    using c_def  nonzero_inverse_Qp by blast  
+    using c_def  nonzero_inverse_Qp by blast
   show "pow_res n x \<subseteq> pow_res n y"
   proof
     fix a
     assume A: "a \<in> pow_res n x"
     then have "a \<in> carrier Q\<^sub>p"
       by (metis (no_types, lifting) mem_Collect_eq pow_res_def)
     obtain b  where b_def: "b \<in> nonzero Q\<^sub>p \<and> a = x \<otimes> (b[^]n)"
-      using A pow_res_def by blast 
+      using A pow_res_def by blast
     then have 0: "b \<in> nonzero Q\<^sub>p \<and> a = y \<otimes> ((inv c)[^]n) \<otimes> (b[^]n)"
-      using \<open>y \<otimes> inv c [^] n = x\<close> by blast      
+      using \<open>y \<otimes> inv c [^] n = x\<close> by blast
     have "b \<in> nonzero Q\<^sub>p \<and> a = y \<otimes> (((inv c)  \<otimes> b)[^]n)"
     proof-
       have "(inv c)[^]n \<otimes> (b[^]n) = ((inv c)  \<otimes> b)[^]n"
-        using c_def b_def assms P0 Qp.nat_pow_distrib Qp.nonzero_closed by presburger        
+        using c_def b_def assms P0 Qp.nat_pow_distrib Qp.nonzero_closed by presburger
       then have " y \<otimes> (((inv c)[^]n) \<otimes> (b[^]n)) = y \<otimes> (((inv c)  \<otimes> b)[^]n)"
-        by presburger        
-      then show ?thesis 
-        using y_closed 0 P0 Qp.m_assoc Qp.nat_pow_closed Qp.nonzero_closed assms(1) by presburger      
+        by presburger
+      then show ?thesis
+        using y_closed 0 P0 Qp.m_assoc Qp.nat_pow_closed Qp.nonzero_closed assms(1) by presburger
     qed
     then have "((inv c)  \<otimes> b) \<in> nonzero Q\<^sub>p \<and> a = y \<otimes> (((inv c)  \<otimes> b)[^]n)"
-      by (metis P0 Qp.integral_iff Qp.nonzero_closed Qp.nonzero_mult_closed not_nonzero_Qp)              
-    then show "a \<in> pow_res n y" using pow_res_def \<open>a \<in> carrier Q\<^sub>p\<close> by blast      
+      by (metis P0 Qp.integral_iff Qp.nonzero_closed Qp.nonzero_mult_closed not_nonzero_Qp)
+    then show "a \<in> pow_res n y" using pow_res_def \<open>a \<in> carrier Q\<^sub>p\<close> by blast
   qed
   show "pow_res n y \<subseteq> pow_res n x"
   proof
     fix a
-    assume A: "a \<in> pow_res n y" 
+    assume A: "a \<in> pow_res n y"
     then have 0: "a \<in> carrier Q\<^sub>p"
       by (metis (no_types, lifting) mem_Collect_eq pow_res_def)
     obtain b  where b_def: "b \<in> nonzero Q\<^sub>p \<and> a = y \<otimes> (b[^]n)"
-      using A pow_res_def by blast 
+      using A pow_res_def by blast
     then have "a = (x \<otimes> (c[^]n)) \<otimes> (b[^]n)"
-      using c_def by blast      
+      using c_def by blast
     then have "a = x \<otimes> ((c[^]n) \<otimes> (b[^]n))"
-      by (meson Qp.m_assoc Qp.nat_pow_closed Qp.nonzero_closed assms(1) b_def c_def)         
+      by (meson Qp.m_assoc Qp.nat_pow_closed Qp.nonzero_closed assms(1) b_def c_def)
     then have "a = x \<otimes> ((c \<otimes> b)[^]n)"
-      using Qp.nat_pow_distrib Qp.nonzero_closed b_def c_def by presburger    
+      using Qp.nat_pow_distrib Qp.nonzero_closed b_def c_def by presburger
     then have "(c \<otimes> b) \<in> nonzero Q\<^sub>p \<and>   a = x \<otimes> ((c \<otimes> b)[^]n)"
-      by (metis Qp.integral_iff Qp.nonzero_closed Qp.nonzero_mult_closed b_def c_def not_nonzero_Qp)        
-    then show "a \<in> pow_res n x" 
-      using pow_res_def 0  by blast 
+      by (metis Qp.integral_iff Qp.nonzero_closed Qp.nonzero_mult_closed b_def c_def not_nonzero_Qp)
+    then show "a \<in> pow_res n x"
+      using pow_res_def 0  by blast
   qed
 qed
 
 lemma zeroth_pow_res:
   assumes "x \<in> carrier Q\<^sub>p"
   shows "pow_res 0 x = {x}"
-  apply(rule equalityI) 
+  apply(rule equalityI)
   apply(rule subsetI)
   unfolding pow_res_def mem_Collect_eq
   using assms apply (metis Qp.nat_pow_0 Qp.r_one singletonI)
   apply(rule subsetI)
   unfolding pow_res_def mem_Collect_eq
   using assms by (metis Qp.nat_pow_0 Qp.one_nonzero Qp.r_one equals0D insertE)
 
 lemma Zp_car_zero_res: assumes "x \<in> carrier Z\<^sub>p"
   shows "x 0 = 0"
-  using assms unfolding Zp_def  
-  using Zp_def Zp_defs(3) padic_set_zero_res prime by blast 
+  using assms unfolding Zp_def
+  using Zp_def Zp_defs(3) padic_set_zero_res prime by blast
 
 lemma zeroth_ac:
   assumes "x \<in> carrier Q\<^sub>p"
   shows "ac 0 x = 0"
   apply(cases "x = \<zero> ")
-  unfolding ac_def apply presburger 
+  unfolding ac_def apply presburger
   using assms angular_component_closed[of x] Zp_car_zero_res unfolding nonzero_def mem_Collect_eq
   by presburger
 
 lemma nonzero_ac_imp_nonzero:
   assumes "x \<in> carrier Q\<^sub>p"
   assumes "ac m x \<noteq> 0"
   shows "x \<in> nonzero Q\<^sub>p"
-  using assms unfolding ac_def nonzero_def mem_Collect_eq 
+  using assms unfolding ac_def nonzero_def mem_Collect_eq
   by presburger
 
 lemma nonzero_ac_val_ord:
   assumes "x \<in> carrier Q\<^sub>p"
   assumes "ac m x \<noteq> 0"
   shows "val x = ord x"
   using nonzero_ac_imp_nonzero assms val_ord by blast
 
 lemma pow_res_equal_ord:
   assumes "n > 0"
   shows  "\<exists>m > 0. \<forall>x y. x \<in> nonzero Q\<^sub>p \<and> y \<in> nonzero Q\<^sub>p \<and> ac m x = ac m y \<and> ord x = ord y \<longrightarrow> pow_res n x = pow_res n y"
-proof- 
+proof-
   obtain m where m_def_0: "m > 0 \<and> ( \<forall> u \<in> carrier Q\<^sub>p. ac m u = 1 \<and> val u = 0 \<longrightarrow> u \<in> P_set n)"
-    using assms nth_power_fact[of n] 
-    by (metis less_imp_le_nat less_one linorder_neqE_nat nat_le_linear zero_less_iff_neq_zero)    
-  then have  m_def: "m > 0 \<and> ( \<forall> u \<in> carrier Q\<^sub>p. ac m u = 1 \<and> ord u = 0 \<longrightarrow> u \<in> P_set n)"   
+    using assms nth_power_fact[of n]
+    by (metis less_imp_le_nat less_one linorder_neqE_nat nat_le_linear zero_less_iff_neq_zero)
+  then have  m_def: "m > 0 \<and> ( \<forall> u \<in> carrier Q\<^sub>p. ac m u = 1 \<and> ord u = 0 \<longrightarrow> u \<in> P_set n)"
     by (smt nonzero_ac_val_ord zero_eint_def)
   have  "\<forall>x y. x \<in> nonzero Q\<^sub>p \<and> y \<in> nonzero Q\<^sub>p \<and> ac m x = ac m y \<and> ord x = ord y \<longrightarrow> pow_res n x = pow_res n y"
   proof
-    fix x 
+    fix x
     show "\<forall>y. x \<in> nonzero Q\<^sub>p \<and> y \<in> nonzero Q\<^sub>p \<and> ac m x = ac m y \<and> ord x = ord y \<longrightarrow> pow_res n x = pow_res n y"
     proof fix y
       show "x \<in> nonzero Q\<^sub>p \<and> y \<in> nonzero Q\<^sub>p \<and> ac m x = ac m y \<and> ord x = ord y \<longrightarrow> pow_res n x = pow_res n y"
       proof
         assume A: "x \<in> nonzero Q\<^sub>p \<and> y \<in> nonzero Q\<^sub>p \<and> ac m x = ac m y \<and> ord x = ord y "
         then have 0: "ac m (x \<div> y) = 1"
-          using ac_inv[of y m] ac_mult 
+          using ac_inv[of y m] ac_mult
           by (metis ac_inv'''(1) ac_mult'  m_def nonzero_inverse_Qp)
         have 1: "ord (x \<div> y) = 0"
           using A  ord_fract by presburger
         have 2: "(x \<div> y) \<in> nonzero Q\<^sub>p"
-          using A 
-          by (metis Qp.nonzero_closed Qp.nonzero_mult_closed local.fract_cancel_right nonzero_inverse_Qp not_nonzero_Qp zero_fract)          
+          using A
+          by (metis Qp.nonzero_closed Qp.nonzero_mult_closed local.fract_cancel_right nonzero_inverse_Qp not_nonzero_Qp zero_fract)
         have 3: "(x \<div> y) \<in> P_set n"
-          using m_def 0 1 2 nonzero_def 
-          by (smt Qp.nonzero_closed)          
+          using m_def 0 1 2 nonzero_def
+          by (smt Qp.nonzero_closed)
         then obtain b where b_def: "b \<in> carrier Q\<^sub>p \<and> (b[^]n) = (x \<div> y)"
-          using P_set_def 
-          by blast 
+          using P_set_def
+          by blast
         then have "(x \<div> y) \<otimes> y = (b[^]n) \<otimes> y"
-          by presburger          
+          by presburger
         then have "x = (b[^]n) \<otimes> y"
-          using A b_def 
+          using A b_def
           by (metis Qp.nonzero_closed local.fract_cancel_left)
         then have "x = y \<otimes>(b[^]n)"
           using A b_def
           by (metis Qp.nonzero_closed local.fract_cancel_right)
         then have "x \<in> pow_res n y"
-          unfolding pow_res_def using A b_def 
-          by (metis (mono_tags, lifting) "2" Qp.nat_pow_0 Qp.nonzero_closed Qp_nonzero_nat_pow mem_Collect_eq not_gr_zero)        
+          unfolding pow_res_def using A b_def
+          by (metis (mono_tags, lifting) "2" Qp.nat_pow_0 Qp.nonzero_closed Qp_nonzero_nat_pow mem_Collect_eq not_gr_zero)
         then show "pow_res n x = pow_res n y"
-          using  A equal_pow_resI[of x y n] unfolding nonzero_def 
-          by (metis (mono_tags, lifting) A Qp.nonzero_closed equal_pow_resI)           
+          using  A equal_pow_resI[of x y n] unfolding nonzero_def
+          by (metis (mono_tags, lifting) A Qp.nonzero_closed equal_pow_resI)
       qed
     qed
   qed
-  then show ?thesis using m_def by blast 
+  then show ?thesis using m_def by blast
 qed
 
 lemma pow_res_equal:
   assumes "n > 0"
   shows  "\<exists>m> 0. \<forall>x y. x \<in> nonzero Q\<^sub>p \<and> y \<in> nonzero Q\<^sub>p \<and> ac m x = ac m y \<and> ord x = (ord y mod n) \<longrightarrow> pow_res n x = pow_res n y"
 proof-
   obtain m where m_def: "m > 0 \<and> (\<forall>x y. x \<in> nonzero Q\<^sub>p \<and> y \<in> nonzero Q\<^sub>p \<and> ac m x = ac m y \<and> ord x = ord y \<longrightarrow> pow_res n x = pow_res n y)"
-    using assms pow_res_equal_ord 
+    using assms pow_res_equal_ord
     by meson
   have "\<forall>x y. x \<in> nonzero Q\<^sub>p \<and> y \<in> nonzero Q\<^sub>p \<and> ac m x = ac m y \<and> ord x = ord y mod int n \<longrightarrow> pow_res n x = pow_res n y"
   proof fix x
     show  "\<forall>y. x \<in> nonzero Q\<^sub>p \<and> y \<in> nonzero Q\<^sub>p \<and> ac m x = ac m y \<and> ord x = ord y mod int n \<longrightarrow> pow_res n x = pow_res n y"
     proof fix y
       show "x \<in> nonzero Q\<^sub>p \<and> y \<in> nonzero Q\<^sub>p \<and> ac m x = ac m y \<and> ord x = ord y mod int n \<longrightarrow> pow_res n x = pow_res n y"
       proof
         assume A: "x \<in> nonzero Q\<^sub>p \<and> y \<in> nonzero Q\<^sub>p \<and> ac m x = ac m y \<and> ord x = ord y mod int n"
         show "pow_res n x = pow_res n y"
         proof-
           have A0: "x \<in> nonzero Q\<^sub>p \<and> y \<in> nonzero Q\<^sub>p"
-            using A by blast 
+            using A by blast
           have A1: "ac m x = ac m y"
-            using A by blast 
+            using A by blast
           have A2: "ord x = ord y mod int n"
-            using A by blast 
+            using A by blast
           obtain k where k_def: "k = ord x"
-            by blast 
+            by blast
           obtain l where l_def: "ord y = ord x + (l:: int)*(int n)"
-            using assms A2 
+            using assms A2
             by (metis A k_def mod_eqE mod_mod_trivial mult_of_nat_commute)
           have m_def': "\<And>x y. x \<in> nonzero Q\<^sub>p \<and> y \<in> nonzero Q\<^sub>p \<and> ac m x = ac m y \<and> ord x = ord y \<Longrightarrow>  pow_res n x = pow_res n y"
-            using m_def 
+            using m_def
             by blast
           have 0: "ord (y \<otimes> (\<pp>[^](- l*n))) = ord x"
           proof-
             have 0: "ord (y \<otimes> (\<pp>[^](- l*n))) = ord y + (ord  (\<pp>[^](- l*n)))"
-              using ord_mult p_nonzero  A0 Qp_int_pow_nonzero 
+              using ord_mult p_nonzero  A0 Qp_int_pow_nonzero
               by blast
             have 1: "ord (\<pp>[^](- l*n)) = - l*n"
               using ord_p_pow_int[of "-l*n"]
-              by blast 
+              by blast
             then have "ord (y \<otimes> (\<pp>[^](- l*n))) = ord y - l*n"
-              using 0 
+              using 0
               by linarith
-            then show ?thesis 
+            then show ?thesis
               using k_def l_def by linarith
-          qed          
+          qed
           have 1: "ac m (y \<otimes> (\<pp>[^](- l*n))) = ac m y"
-            using assms ac_p_int_pow_factor_right[of ] m_def A Qp.nonzero_closed by presburger   
+            using assms ac_p_int_pow_factor_right[of ] m_def A Qp.nonzero_closed by presburger
           have 2: "y \<otimes> (\<pp>[^](- l*n)) \<in> nonzero Q\<^sub>p"
-            using A0 Qp_int_pow_nonzero[of \<pp> "- l*n"] Qp.cring_axioms nonzero_def cring.cring_simprules(5) 
-              fract_cancel_left not_nonzero_Qp p_intpow_inv'' p_nonzero zero_fract Qp.integral_iff 
-              Qp.nonzero_closed Qp.nonzero_memE(2) Qp.nonzero_memI Qp.nonzero_mult_closed 
-              minus_mult_commute mult_minus_right p_intpow_closed(1) p_intpow_closed(2) 
-            by presburger              
+            using A0 Qp_int_pow_nonzero[of \<pp> "- l*n"] Qp.cring_axioms nonzero_def cring.cring_simprules(5)
+              fract_cancel_left not_nonzero_Qp p_intpow_inv'' p_nonzero zero_fract Qp.integral_iff
+              Qp.nonzero_closed Qp.nonzero_memE(2) Qp.nonzero_memI Qp.nonzero_mult_closed
+              minus_mult_commute mult_minus_right p_intpow_closed(1) p_intpow_closed(2)
+            by presburger
           then have 3: "pow_res n (y \<otimes> (\<pp>[^](- l*n))) = pow_res n x"
-            using 2 A0 m_def'[of "y \<otimes> (\<pp>[^](- l*n))" x] "0" "1" A1 
+            using 2 A0 m_def'[of "y \<otimes> (\<pp>[^](- l*n))" x] "0" "1" A1
             by linarith
           have 4: "(y \<otimes> (\<pp>[^](- l*n))) = (y \<otimes> ((\<pp>[^]- l)[^]n))"
-            using Qp_int_nat_pow_pow[of \<pp> "-l" n] p_nonzero  
-            by presburger 
+            using Qp_int_nat_pow_pow[of \<pp> "-l" n] p_nonzero
+            by presburger
           have "y \<otimes> (\<pp>[^](- l*n))\<in> pow_res n y "
-            using "2" "4" Qp_int_pow_nonzero nonzero_def p_nonzero 
+            using "2" "4" Qp_int_pow_nonzero nonzero_def p_nonzero
             unfolding pow_res_def nonzero_def
           proof -
             assume a1: "\<And>x n. x \<in> {a \<in> carrier Q\<^sub>p. a \<noteq> \<zero>} \<Longrightarrow> x [^] (n::int) \<in> {a \<in> carrier Q\<^sub>p. a \<noteq> \<zero>}"
             assume a2: "\<pp> \<in> {a \<in> carrier Q\<^sub>p. a \<noteq> \<zero>}"
             assume a3: "y \<otimes> \<pp> [^] (- l * int n) \<in> {a \<in> carrier Q\<^sub>p. a \<noteq> \<zero>}"
             have f4: "\<pp> [^] (- 1 * l) \<in> {r \<in> carrier Q\<^sub>p. r \<noteq> \<zero>}"
               using a2 a1 by presburger
             have f5: "- l = - 1 * l"
               by linarith
             then have f6: "y \<otimes> \<pp> [^] (- 1 * l * int n) = y \<otimes> (\<pp> [^] (- 1 * l)) [^] n"
               using \<open>y \<otimes> \<pp> [^] (- l * int n) = y \<otimes> (\<pp> [^] - l) [^] n\<close> by presburger
             then have "y \<otimes> (\<pp> [^] (- 1 * l)) [^] n \<in> {r \<in> carrier Q\<^sub>p. r \<noteq> \<zero>}"
               using f5 a3 by presburger
             then have "y \<otimes> (\<pp> [^] (- 1 * l)) [^] n \<in> {r \<in> carrier Q\<^sub>p. \<exists>ra. ra \<in> {r \<in> carrier Q\<^sub>p. r \<noteq> \<zero>} \<and> r = y \<otimes> ra [^] n}"
               using f4 by blast
             then have "y \<otimes> \<pp> [^] (- l * int n) \<in> {r \<in> carrier Q\<^sub>p. \<exists>ra. ra \<in> {r \<in> carrier Q\<^sub>p. r \<noteq> \<zero>} \<and> r = y \<otimes> ra [^] n}"
               using f6 f5 by presburger
             then show "y \<otimes> \<pp> [^] (- l * int n) \<in> {r \<in> carrier Q\<^sub>p. \<exists>ra\<in>{r \<in> carrier Q\<^sub>p. r \<noteq> \<zero>}. r = y \<otimes> ra [^] n}"
               by meson
-          qed 
+          qed
           then have "pow_res n (y \<otimes> (\<pp>[^](- l*n))) = pow_res n y"
-            using equal_pow_resI[of "(y \<otimes> (\<pp>[^](- l*n)))" y n] "2" A0 assms  
-                  Qp.nonzero_mult_closed p_intpow_closed(2) 
-            by (metis (mono_tags, opaque_lifting) "3" A Qp.nonzero_closed equal_pow_resI)                            
-          then show ?thesis using 3 by blast 
+            using equal_pow_resI[of "(y \<otimes> (\<pp>[^](- l*n)))" y n] "2" A0 assms
+                  Qp.nonzero_mult_closed p_intpow_closed(2)
+            by (metis (mono_tags, opaque_lifting) "3" A Qp.nonzero_closed equal_pow_resI)
+          then show ?thesis using 3 by blast
         qed
       qed
     qed
   qed
-  then show ?thesis 
+  then show ?thesis
     using m_def
     by blast
 qed
-    
+
 definition pow_res_classes where
 "pow_res_classes n = {S. \<exists>x \<in> nonzero Q\<^sub>p. S = pow_res n x }"
 
 lemma pow_res_semialg_def:
   assumes "x \<in> nonzero Q\<^sub>p"
   assumes "n \<ge> 1"
   shows "\<exists>P \<in> carrier Q\<^sub>p_x. pow_res n x = (univ_basic_semialg_set n P) - {\<zero>}"
 proof-
   have 0: "pow_res n x = {a \<in> carrier Q\<^sub>p. \<exists>y\<in>nonzero Q\<^sub>p. (inv x) \<otimes> a = (y[^]n)}"
   proof
     show "pow_res n x \<subseteq> {a \<in> carrier Q\<^sub>p. \<exists>y\<in>nonzero Q\<^sub>p. inv x \<otimes> a = (y[^]n)}"
     proof
       fix a
       assume A: "a \<in> pow_res n x"
-      then have "a \<in> carrier Q\<^sub>p \<and>  (\<exists>y\<in>nonzero Q\<^sub>p. a = x \<otimes> (y[^]n))" 
-        unfolding pow_res_def 
-        by blast 
+      then have "a \<in> carrier Q\<^sub>p \<and>  (\<exists>y\<in>nonzero Q\<^sub>p. a = x \<otimes> (y[^]n))"
+        unfolding pow_res_def
+        by blast
       then obtain y where y_def: "y \<in> nonzero Q\<^sub>p \<and>a = x \<otimes> (y[^]n)"
-        by blast 
+        by blast
       then have "y \<in> nonzero Q\<^sub>p \<and> inv x \<otimes> a = (y[^]n)"
       proof -
         show ?thesis
-          by (metis (no_types, opaque_lifting) Qp.m_assoc Qp.m_comm Qp.nat_pow_closed Qp.nonzero_closed 
-              \<open>a \<in> carrier Q\<^sub>p \<and> (\<exists>y\<in>nonzero Q\<^sub>p. a = x \<otimes> y [^] n)\<close> assms(1) local.fract_cancel_right nonzero_inverse_Qp y_def)         
-      qed       
+          by (metis (no_types, opaque_lifting) Qp.m_assoc Qp.m_comm Qp.nat_pow_closed Qp.nonzero_closed
+              \<open>a \<in> carrier Q\<^sub>p \<and> (\<exists>y\<in>nonzero Q\<^sub>p. a = x \<otimes> y [^] n)\<close> assms(1) local.fract_cancel_right nonzero_inverse_Qp y_def)
+      qed
       then show "a \<in> {a \<in> carrier Q\<^sub>p. \<exists>y\<in>nonzero Q\<^sub>p. inv x \<otimes> a = (y[^]n)}"
-        using assms \<open>a \<in> carrier Q\<^sub>p \<and> (\<exists>y\<in>nonzero Q\<^sub>p. a = x \<otimes> (y[^]n))\<close> 
+        using assms \<open>a \<in> carrier Q\<^sub>p \<and> (\<exists>y\<in>nonzero Q\<^sub>p. a = x \<otimes> (y[^]n))\<close>
         by blast
     qed
 
     show "{a \<in> carrier Q\<^sub>p. \<exists>y\<in>nonzero Q\<^sub>p. inv x \<otimes> a = (y[^]n)} \<subseteq> pow_res n x"
     proof
-      fix a 
+      fix a
       assume A: "a \<in> {a \<in> carrier Q\<^sub>p. \<exists>y\<in>nonzero Q\<^sub>p. inv x \<otimes> a = (y[^]n)}"
       show "a \<in> pow_res n x"
       proof-
         have "a \<in> carrier Q\<^sub>p \<and> (\<exists>y\<in>nonzero Q\<^sub>p. inv x \<otimes> a = (y[^]n))"
-          using A by blast 
+          using A by blast
         then obtain y where y_def: "y\<in>nonzero Q\<^sub>p \<and> inv x \<otimes> a = (y[^]n)"
-          by blast 
+          by blast
         then have "y\<in>nonzero Q\<^sub>p \<and>  a = x \<otimes>(y[^]n)"
           by (metis Qp.l_one Qp.m_assoc Qp.nonzero_closed Qp.not_nonzero_memI
-              \<open>a \<in> carrier Q\<^sub>p \<and> (\<exists>y\<in>nonzero Q\<^sub>p. inv x \<otimes> a = y [^] n)\<close> assms(1) field_inv(2) inv_in_frac(1))          
-        then show ?thesis 
+              \<open>a \<in> carrier Q\<^sub>p \<and> (\<exists>y\<in>nonzero Q\<^sub>p. inv x \<otimes> a = y [^] n)\<close> assms(1) field_inv(2) inv_in_frac(1))
+        then show ?thesis
           by (metis (mono_tags, lifting) \<open>a \<in> carrier Q\<^sub>p \<and> (\<exists>y\<in>nonzero Q\<^sub>p. inv x \<otimes> a = (y[^]n))\<close> mem_Collect_eq pow_res_def)
       qed
     qed
   qed
   obtain P where P_def: "P = up_ring.monom Q\<^sub>p_x (inv x) 1"
-    by blast    
-  have P_closed: "P \<in> carrier Q\<^sub>p_x" 
-    using P_def Qp.nonzero_closed Qp.nonzero_memE(2) UPQ.is_UP_monomE(1) UPQ.is_UP_monomI assms(1) inv_in_frac(1) by presburger   
+    by blast
+  have P_closed: "P \<in> carrier Q\<^sub>p_x"
+    using P_def Qp.nonzero_closed Qp.nonzero_memE(2) UPQ.is_UP_monomE(1) UPQ.is_UP_monomI assms(1) inv_in_frac(1) by presburger
   have P_eval: "\<And>a. a \<in> carrier Q\<^sub>p \<Longrightarrow> (P \<bullet> a) = (inv x) \<otimes> a"
-    using P_def to_fun_monom[of ] 
-    by (metis Qp.nat_pow_eone Qp.nonzero_closed Qp.not_nonzero_memI assms(1) inv_in_frac(1))   
+    using P_def to_fun_monom[of ]
+    by (metis Qp.nat_pow_eone Qp.nonzero_closed Qp.not_nonzero_memI assms(1) inv_in_frac(1))
   have 0: "pow_res n x = {a \<in> carrier Q\<^sub>p. \<exists>y\<in>nonzero Q\<^sub>p. (P \<bullet> a) = (y[^]n)}"
   proof
     show "pow_res n x \<subseteq> {a \<in> carrier Q\<^sub>p. \<exists>y\<in>nonzero Q\<^sub>p. P \<bullet> a = (y[^]n)}"
     proof fix a
       assume "a \<in> pow_res n x"
       then have "a \<in> carrier Q\<^sub>p \<and> (\<exists>y\<in>nonzero Q\<^sub>p. inv x \<otimes> a = (y[^]n))"
-        using 0 
+        using 0
         by blast
       then show "a \<in> {a \<in> carrier Q\<^sub>p. \<exists>y\<in>nonzero Q\<^sub>p. P \<bullet> a = (y[^]n)}"
-        using P_eval 
+        using P_eval
         by (metis (mono_tags, lifting) mem_Collect_eq)
      qed
     show "{a \<in> carrier Q\<^sub>p. \<exists>y\<in>nonzero Q\<^sub>p. P \<bullet> a = (y[^]n)} \<subseteq> pow_res n x"
     proof fix a
-      assume "a \<in> {a \<in> carrier Q\<^sub>p. \<exists>y\<in>nonzero Q\<^sub>p. P \<bullet> a = (y[^]n)}" 
+      assume "a \<in> {a \<in> carrier Q\<^sub>p. \<exists>y\<in>nonzero Q\<^sub>p. P \<bullet> a = (y[^]n)}"
       then obtain y where y_def: "y\<in>nonzero Q\<^sub>p \<and> P \<bullet> a = (y[^]n)"
-        by blast 
+        by blast
       then have "y\<in>nonzero Q\<^sub>p \<and> inv x \<otimes> a = (y[^]n)"
-        using P_eval \<open>a \<in> {a \<in> carrier Q\<^sub>p. \<exists>y\<in>nonzero Q\<^sub>p. P \<bullet> a = (y[^]n)}\<close> 
+        using P_eval \<open>a \<in> {a \<in> carrier Q\<^sub>p. \<exists>y\<in>nonzero Q\<^sub>p. P \<bullet> a = (y[^]n)}\<close>
         by blast
       then have "a \<in> carrier Q\<^sub>p \<and> (\<exists>y\<in>nonzero Q\<^sub>p. inv x \<otimes> a = (y[^]n))"
          using \<open>a \<in> {a \<in> carrier Q\<^sub>p. \<exists>y\<in>nonzero Q\<^sub>p. P \<bullet> a = (y[^]n)}\<close> by blast
       then show "a \<in> pow_res n x"
-      using 0  
+      using 0
       by blast
     qed
   qed
   have 1: "univ_basic_semialg_set n P - {\<zero>} = {a \<in> carrier Q\<^sub>p. \<exists>y\<in>nonzero Q\<^sub>p. (P \<bullet> a) = (y[^]n)}"
   proof
     show "univ_basic_semialg_set n P - {\<zero>} \<subseteq> {a \<in> carrier Q\<^sub>p. \<exists>y\<in>nonzero Q\<^sub>p. P \<bullet> a = (y[^]n)}"
     proof
-      fix a 
+      fix a
       assume A: "a \<in> univ_basic_semialg_set n P - {\<zero>}"
       then have A0: "a \<in> carrier Q\<^sub>p \<and> (\<exists>y\<in>carrier Q\<^sub>p. P \<bullet> a = (y[^]n))"
-        unfolding univ_basic_semialg_set_def by blast         
+        unfolding univ_basic_semialg_set_def by blast
       then have A0': "a \<in> nonzero Q\<^sub>p \<and> (\<exists>y\<in>carrier Q\<^sub>p. P \<bullet> a = (y[^]n))"
-        using A 
+        using A
         by (metis DiffD2 not_nonzero_Qp singletonI)
       then obtain y where y_def: "y\<in>carrier Q\<^sub>p \<and> P \<bullet> a = (y[^]n)"
         by blast
       have A1: "(P \<bullet> a) \<noteq> \<zero>"
-        using P_eval A0' Qp.integral_iff Qp.nonzero_closed Qp.nonzero_memE(2) assms(1) inv_in_frac(1) inv_in_frac(2) by presburger                      
-      have A2: "y \<in> nonzero Q\<^sub>p" 
+        using P_eval A0' Qp.integral_iff Qp.nonzero_closed Qp.nonzero_memE(2) assms(1) inv_in_frac(1) inv_in_frac(2) by presburger
+      have A2: "y \<in> nonzero Q\<^sub>p"
       proof-
         have A20: "(y[^]n) \<noteq>\<zero>"
-          using A1 y_def 
+          using A1 y_def
           by blast
         have "y \<noteq> \<zero>"
-          apply(rule ccontr) using A20 assms 
-          by (metis Qp.nat_pow_eone Qp.semiring_axioms Qp.zero_closed le_zero_eq semiring.nat_pow_zero)                        
-        then show ?thesis 
-          using y_def A1 not_nonzero_Qp Qp.not_nonzero_memE by blast          
+          apply(rule ccontr) using A20 assms
+          by (metis Qp.nat_pow_eone Qp.semiring_axioms Qp.zero_closed le_zero_eq semiring.nat_pow_zero)
+        then show ?thesis
+          using y_def A1 not_nonzero_Qp Qp.not_nonzero_memE by blast
       qed
-      then have "y \<in> nonzero  Q\<^sub>p \<and> P \<bullet> a = (y[^]n)" using y_def 
-        by blast      
-      then show "a \<in> {a \<in> carrier Q\<^sub>p. \<exists>y\<in>nonzero Q\<^sub>p. P \<bullet> a = (y[^]n)}"  
-        using A0 nonzero_def 
+      then have "y \<in> nonzero  Q\<^sub>p \<and> P \<bullet> a = (y[^]n)" using y_def
+        by blast
+      then show "a \<in> {a \<in> carrier Q\<^sub>p. \<exists>y\<in>nonzero Q\<^sub>p. P \<bullet> a = (y[^]n)}"
+        using A0 nonzero_def
         by blast
     qed
     show "{a \<in> carrier Q\<^sub>p. \<exists>y\<in>nonzero Q\<^sub>p. P \<bullet> a = (y[^]n)} \<subseteq> univ_basic_semialg_set n P - {\<zero>}"
     proof
-      fix a 
+      fix a
       assume A: "a \<in> {a \<in> carrier Q\<^sub>p. \<exists>y\<in>nonzero Q\<^sub>p. P \<bullet> a = (y[^]n)}"
       then obtain y where y_def: "y\<in>nonzero Q\<^sub>p \<and> P \<bullet> a = (y[^]n)"
-        by blast 
+        by blast
       then have "y \<noteq>\<zero> \<and> y\<in> carrier Q\<^sub>p \<and> P \<bullet> a = (y[^]n)"
-        by (metis (mono_tags, opaque_lifting) Qp.nonzero_closed Qp.not_nonzero_memI)        
+        by (metis (mono_tags, opaque_lifting) Qp.nonzero_closed Qp.not_nonzero_memI)
       then have "a \<noteq>\<zero>"
-        using  P_eval 
-        by (metis Qp.m_comm Qp.nonzero_closed Qp.nonzero_memE(2) Qp.nonzero_pow_nonzero Qp.zero_closed assms(1) inv_in_frac(1) zero_fract)        
+        using  P_eval
+        by (metis Qp.m_comm Qp.nonzero_closed Qp.nonzero_memE(2) Qp.nonzero_pow_nonzero Qp.zero_closed assms(1) inv_in_frac(1) zero_fract)
       then show "a \<in> univ_basic_semialg_set n P - {\<zero>}"
-        unfolding univ_basic_semialg_set_def 
-        using A \<open>y \<noteq> \<zero> \<and> y \<in> carrier Q\<^sub>p \<and> P \<bullet> a = (y[^]n)\<close> 
+        unfolding univ_basic_semialg_set_def
+        using A \<open>y \<noteq> \<zero> \<and> y \<in> carrier Q\<^sub>p \<and> P \<bullet> a = (y[^]n)\<close>
         by blast
     qed
   qed
-  show ?thesis using 0 1 
+  show ?thesis using 0 1
     by (metis (no_types, lifting) P_closed)
 qed
 
 lemma pow_res_is_univ_semialgebraic:
   assumes "x \<in> carrier Q\<^sub>p"
   shows "is_univ_semialgebraic (pow_res n x)"
 proof(cases "n = 0")
   case True
   have T0: "pow_res n x = {x}"
-    unfolding True using assms 
+    unfolding True using assms
     by (simp add: assms zeroth_pow_res)
   have "[x] \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>)"
     using assms Qp.to_R1_closed by blast
   hence "is_semialgebraic 1 {[x]}"
     using is_algebraic_imp_is_semialg singleton_is_algebraic by blast
-  thus ?thesis unfolding T0 using assms  
-    by (simp add: \<open>x \<in> carrier Q\<^sub>p\<close> finite_is_univ_semialgebraic)   
+  thus ?thesis unfolding T0 using assms
+    by (simp add: \<open>x \<in> carrier Q\<^sub>p\<close> finite_is_univ_semialgebraic)
 next
   case False
   show ?thesis
   proof(cases "x = \<zero>")
     case True
-    then show ?thesis using finite_is_univ_semialgebraic False pow_res_of_zero 
+    then show ?thesis using finite_is_univ_semialgebraic False pow_res_of_zero
       by (metis Qp.zero_closed empty_subsetI finite.emptyI finite.insertI insert_subset)
   next
     case F: False
-    then show ?thesis 
-      using False pow_res_semialg_def[of x n] diff_is_univ_semialgebraic[of _ "{\<zero>}"]  finite_is_univ_semialgebraic[of "{\<zero>}"] 
+    then show ?thesis
+      using False pow_res_semialg_def[of x n] diff_is_univ_semialgebraic[of _ "{\<zero>}"]  finite_is_univ_semialgebraic[of "{\<zero>}"]
       by (metis Qp.zero_closed assms empty_subsetI finite.emptyI finite.insertI insert_subset less_one less_or_eq_imp_le linorder_neqE_nat not_nonzero_Qp univ_basic_semialg_set_is_univ_semialgebraic)
   qed
 qed
 
 lemma pow_res_is_semialg:
   assumes "x \<in> carrier Q\<^sub>p"
   shows "is_semialgebraic 1 (to_R1 ` (pow_res n x))"
-  using assms pow_res_is_univ_semialgebraic is_univ_semialgebraicE 
+  using assms pow_res_is_univ_semialgebraic is_univ_semialgebraicE
   by blast
 
 lemma pow_res_refl:
   assumes "x \<in> carrier Q\<^sub>p"
   shows "x \<in> pow_res n x"
 proof-
   have "x = x \<otimes> (\<one> [^]n)"
     using assms Qp.nat_pow_one Qp.r_one by presburger
-  thus ?thesis 
+  thus ?thesis
     using assms unfolding pow_res_def mem_Collect_eq
     using Qp.one_nonzero by blast
 qed
 
 lemma equal_pow_resE:
   assumes "a \<in> carrier Q\<^sub>p"
   assumes "b \<in> carrier Q\<^sub>p"
   assumes "n > 0"
   assumes "pow_res n a = pow_res n b"
   shows "\<exists> s \<in> P_set n. a = b \<otimes> s"
 proof-
   have "a \<in> pow_res n b"
     using assms pow_res_refl by blast
   then obtain y where y_def: " y \<in> nonzero Q\<^sub>p \<and> a = b \<otimes> y[^]n"
     unfolding pow_res_def by blast
-  thus ?thesis  unfolding P_set_def 
+  thus ?thesis  unfolding P_set_def
     using Qp.nonzero_closed Qp_nat_pow_nonzero by blast
 qed
 
 lemma pow_res_one:
   assumes "x \<in> nonzero Q\<^sub>p"
   shows "pow_res 1 x = nonzero Q\<^sub>p"
 proof show "pow_res 1 x \<subseteq> nonzero Q\<^sub>p"
       using  assms nonzero_pow_res[of x 1] by blast
     show "nonzero Q\<^sub>p \<subseteq> pow_res 1 x"
     proof fix y assume A: "y \<in> nonzero Q\<^sub>p"
       then have 0: "\<one> \<in> nonzero Q\<^sub>p \<and> y = x \<otimes> ((inv x)\<otimes> y)[^](1::nat)"
-        using assms  Qp.m_comm Qp.nat_pow_eone Qp.nonzero_closed Qp.nonzero_mult_closed 
+        using assms  Qp.m_comm Qp.nat_pow_eone Qp.nonzero_closed Qp.nonzero_mult_closed
           Qp.one_nonzero local.fract_cancel_right nonzero_inverse_Qp by presburger
       have 1: "(inv x)\<otimes> y \<in> nonzero Q\<^sub>p"
         using A assms by (metis Qp.Units_m_closed Units_eq_nonzero nonzero_inverse_Qp)
       then show "y \<in> pow_res 1 x"
-        unfolding pow_res_def using 0 1 A Qp.nonzero_closed by blast  
-    qed
-qed
-
-
-lemma pow_res_zero: 
+        unfolding pow_res_def using 0 1 A Qp.nonzero_closed by blast
+    qed
+qed
+
+
+lemma pow_res_zero:
   assumes "n > 0"
-  shows "pow_res n \<zero> = {\<zero>}" 
+  shows "pow_res n \<zero> = {\<zero>}"
 proof
   show "pow_res n \<zero> \<subseteq> {\<zero>}"
-  unfolding pow_res_def 
+  unfolding pow_res_def
   using Qp.l_null Qp.nat_pow_closed Qp.nonzero_closed by blast
   show "{\<zero>} \<subseteq> pow_res n \<zero>"
-    using assms unfolding pow_res_def 
+    using assms unfolding pow_res_def
     using Qp.l_null Qp.one_closed Qp.one_nonzero empty_subsetI insert_subset by blast
 qed
 
 
 lemma equal_pow_resI':
   assumes "a \<in> carrier Q\<^sub>p"
   assumes "b \<in> carrier Q\<^sub>p"
   assumes "c \<in> P_set n"
   assumes "a = b \<otimes> c"
   assumes "n > 0"
-  shows "pow_res n a = pow_res n b" 
+  shows "pow_res n a = pow_res n b"
 proof-
   obtain y where y_def: "c = y[^]n \<and> y \<in> carrier Q\<^sub>p"
-    using assms unfolding P_set_def by blast 
+    using assms unfolding P_set_def by blast
   have c_nonzero: "c \<in> nonzero Q\<^sub>p"
     using P_set_nonzero'(1) assms(3) by blast
-  have y_nonzero: "y \<in> nonzero Q\<^sub>p" 
+  have y_nonzero: "y \<in> nonzero Q\<^sub>p"
     using y_def c_nonzero  Qp_nonzero_nat_pow assms(5) by blast
   have 0: "a \<in> pow_res n b"
-    using assms y_nonzero y_def unfolding pow_res_def  
-    by blast
-  show ?thesis  
+    using assms y_nonzero y_def unfolding pow_res_def
+    by blast
+  show ?thesis
     apply(cases "b = \<zero>")
     using pow_res_zero Qp.l_null Qp.nonzero_closed assms(4) c_nonzero apply presburger
     by (metis "0" assms(1) assms(2) assms(5) not_nonzero_Qp equal_pow_resI)
 qed
 
 lemma equal_pow_resI'':
   assumes "n > 0"
   assumes "a \<in> nonzero Q\<^sub>p"
   assumes "b \<in> nonzero Q\<^sub>p"
   assumes "a \<otimes> inv b \<in> P_set n"
   shows "pow_res n a = pow_res n b"
-  using assms equal_pow_resI'[of a b "a \<otimes> inv b" n] Qp.nonzero_closed local.fract_cancel_right 
+  using assms equal_pow_resI'[of a b "a \<otimes> inv b" n] Qp.nonzero_closed local.fract_cancel_right
   by blast
-  
+
 lemma equal_pow_resI''':
   assumes "n > 0"
   assumes "a \<in> nonzero Q\<^sub>p"
   assumes "b \<in> nonzero Q\<^sub>p"
   assumes "c \<in> nonzero Q\<^sub>p"
   assumes "pow_res n (c \<otimes> a) = pow_res n (c \<otimes> b)"
   shows "pow_res n a = pow_res n b"
 proof-
   have 0: "c \<otimes>a \<in> nonzero Q\<^sub>p"
     by (meson Localization.submonoid.m_closed Qp.nonzero_is_submonoid assms(2) assms(4))
   have 1: "c \<otimes>b \<in> nonzero Q\<^sub>p"
     by (meson Localization.submonoid.m_closed Qp.nonzero_is_submonoid assms(3) assms(4))
   have "c\<otimes>a \<in> pow_res n (c\<otimes>b)"
   proof(cases "n = 1")
     case True
-    then show ?thesis     
+    then show ?thesis
       using assms 0 1 pow_res_one[of "c\<otimes>b"] by blast
   next
     case False
     then have "n \<ge> 2"
       using assms(1) by linarith
-    then show ?thesis 
+    then show ?thesis
       using assms 0 1 pow_res_refl[of "c\<otimes>a" n] unfolding nonzero_def
       by blast
   qed
   then obtain y where y_def: "y \<in> nonzero Q\<^sub>p \<and> (c \<otimes> a) = (c \<otimes> b)\<otimes>y[^]n"
     using assms unfolding pow_res_def by blast
   then have "a = b\<otimes>y[^]n"
-    using assms 
+    using assms
     by (metis Qp.m_assoc Qp.m_lcancel Qp.nonzero_closed Qp.nonzero_mult_closed Qp.not_nonzero_memI Qp_nat_pow_nonzero)
-  then show ?thesis 
+  then show ?thesis
     by (metis P_set_memI Qp.nonzero_closed Qp.nonzero_mult_closed Qp.not_nonzero_memI Qp_nat_pow_nonzero assms(1) assms(3) equal_pow_resI' y_def)
 qed
 
 lemma equal_pow_resI'''':
   assumes "n > 0"
   assumes "a \<in> carrier Q\<^sub>p"
   assumes "b \<in> carrier Q\<^sub>p"
   assumes "a = b \<otimes> u"
   assumes "u \<in> P_set n"
   shows "pow_res n a = pow_res n b"
 proof(cases "a = \<zero>")
   case True
   then have "b = \<zero>"
-    using assms unfolding P_set_def 
+    using assms unfolding P_set_def
     by (metis (no_types, lifting) Qp.integral Qp.nonzero_closed Qp.not_nonzero_memI mem_Collect_eq)
-  then show ?thesis using pow_res_zero 
+  then show ?thesis using pow_res_zero
     using True by blast
 next
   case False
   then have 0: "a \<in> nonzero Q\<^sub>p"
-    using Qp.not_nonzero_memE assms(2) by blast  
+    using Qp.not_nonzero_memE assms(2) by blast
   have 1: "b \<in> nonzero Q\<^sub>p"
-    using 0 assms unfolding P_set_def 
+    using 0 assms unfolding P_set_def
     by (metis (no_types, lifting) Qp.integral_iff Qp.nonzero_closed mem_Collect_eq not_nonzero_Qp)
   have 2: "a \<otimes> (inv b)\<in> P_set n"
-    using assms 0 1 
+    using assms 0 1
     by (metis P_set_nonzero'(2) Qp.inv_cancelR(1) Qp.m_comm Qp.nonzero_memE(2) Units_eq_nonzero inv_in_frac(1))
   then show ?thesis using equal_pow_resI''
     by (meson "0" "1" assms(1) equal_pow_resI)
 qed
 
 
 lemma Zp_Units_ord_zero:
   assumes "a \<in> Units Z\<^sub>p"
   shows "ord_Zp a = 0"
 proof-
   have "inv\<^bsub>Z\<^sub>p\<^esub> a \<in> nonzero Z\<^sub>p"
     apply(rule Zp.nonzero_memI, rule Zp.Units_inv_closed, rule assms)
     using assms Zp.Units_inverse in_Units_imp_not_zero by blast
   then have "ord_Zp (a \<otimes>\<^bsub>Z\<^sub>p\<^esub> inv \<^bsub>Z\<^sub>p\<^esub> a) = ord_Zp a + ord_Zp (inv \<^bsub>Z\<^sub>p\<^esub> a)"
     using assms ord_Zp_mult Zp.Units_nonzero zero_not_one
-    by (metis Zp.zero_not_one)    
-  then show ?thesis 
+    by (metis Zp.zero_not_one)
+  then show ?thesis
     by (smt Zp.Units_closed Zp.Units_r_inv Zp.integral_iff Zp.nonzero_closed \<open>inv\<^bsub>Z\<^sub>p\<^esub> a \<in> nonzero Z\<^sub>p\<close> assms ord_Zp_one ord_pos)
 qed
 
 lemma pow_res_nth_pow:
   assumes "a \<in> nonzero Q\<^sub>p"
   assumes "n > 0"
   shows "pow_res n (a[^]n) = pow_res n \<one>"
 proof
   show "pow_res n (a [^] n) \<subseteq> pow_res n \<one>"
   proof fix x assume A: "x \<in> pow_res n (a [^] n)"
     then show "x \<in> pow_res n \<one>"
-      by (metis P_set_memI Qp.l_one Qp.nat_pow_closed Qp.nonzero_closed Qp.nonzero_memE(2) 
+      by (metis P_set_memI Qp.l_one Qp.nat_pow_closed Qp.nonzero_closed Qp.nonzero_memE(2)
           Qp.nonzero_pow_nonzero Qp.one_closed assms(1) assms(2) equal_pow_resI')
   qed
   show "pow_res n \<one> \<subseteq> pow_res n (a [^] n)"
   proof fix x assume A: "x \<in> pow_res n \<one>"
     then obtain y where y_def: "y \<in> nonzero Q\<^sub>p \<and> x = \<one>\<otimes>y[^]n"
       unfolding pow_res_def by blast
     then have 0: "x = y[^]n"
       using Qp.l_one Qp.nonzero_closed by blast
     have "y[^]n = a[^]n \<otimes> (inv a \<otimes> y)[^]n"
     proof-
       have "a[^]n \<otimes> (inv a \<otimes> y)[^]n = a[^]n \<otimes> (inv a)[^]n \<otimes> y[^]n"
         using Qp.Units_inv_closed Qp.m_assoc Qp.nat_pow_closed Qp.nat_pow_distrib Qp.nonzero_closed Units_eq_nonzero assms(1) y_def by presburger
-      then show ?thesis 
+      then show ?thesis
         by (metis Qp.Units_inv_inv Qp.inv_cancelR(1) Qp.nat_pow_distrib Qp.nonzero_closed Qp.nonzero_mult_closed Units_eq_nonzero assms(1) nonzero_inverse_Qp y_def)
     qed
     then show "x \<in> pow_res n (a [^] n)"
-      using y_def  A assms unfolding pow_res_def mem_Collect_eq  
+      using y_def  A assms unfolding pow_res_def mem_Collect_eq
       by (metis "0" Qp.integral Qp.m_closed Qp.nonzero_closed Qp.not_nonzero_memI inv_in_frac(1) inv_in_frac(2) not_nonzero_Qp)
   qed
 qed
 
 lemma pow_res_of_p_pow:
   assumes "n > 0"
   shows "pow_res n (\<pp>[^]((l::int)*n)) = pow_res n \<one>"
 proof-
   have 0: "\<pp>[^]((l::int)*n) = (\<pp>[^]l)[^]n"
     using Qp_p_int_nat_pow_pow by blast
   have "\<pp>[^]((l::int)*n) \<in> P_set n"
-    using  P_set_memI[of _ "\<pp>[^]l"]  
+    using  P_set_memI[of _ "\<pp>[^]l"]
     by (metis "0" Qp.not_nonzero_memI Qp_int_pow_nonzero p_intpow_closed(1) p_nonzero)
-  thus ?thesis 
-    using "0" assms p_intpow_closed(2) pow_res_nth_pow by presburger  
-qed
-
-lemma pow_res_nonzero: 
+  thus ?thesis
+    using "0" assms p_intpow_closed(2) pow_res_nth_pow by presburger
+qed
+
+lemma pow_res_nonzero:
   assumes "n > 0"
   assumes "a \<in> nonzero Q\<^sub>p"
   assumes "b \<in> carrier Q\<^sub>p"
   assumes "pow_res n a = pow_res n b"
   shows "b \<in> nonzero Q\<^sub>p"
-  using assms nonzero_pow_res[of a n] pow_res_zero[of n] 
+  using assms nonzero_pow_res[of a n] pow_res_zero[of n]
   by (metis insert_subset not_nonzero_Qp)
 
 lemma pow_res_mult:
   assumes "n > 0"
   assumes "a \<in> carrier Q\<^sub>p"
   assumes "b \<in> carrier Q\<^sub>p"
   assumes "c \<in> carrier Q\<^sub>p"
   assumes "d \<in> carrier Q\<^sub>p"
   assumes "pow_res n a = pow_res n c"
   assumes "pow_res n b = pow_res n d"
   shows "pow_res n (a \<otimes> b) = pow_res n (c \<otimes> d)"
 proof(cases "a \<in> nonzero Q\<^sub>p")
   case True
   then have "c \<in> nonzero Q\<^sub>p"
     using assms pow_res_nonzero by blast
   then obtain \<alpha> where alpha_def: "\<alpha> \<in> nonzero Q\<^sub>p \<and> a = c \<otimes> \<alpha>[^]n"
-    using  assms True pow_res_refl[of a n] unfolding assms unfolding pow_res_def   
+    using  assms True pow_res_refl[of a n] unfolding assms unfolding pow_res_def
      by blast
-  show ?thesis 
+  show ?thesis
   proof(cases "b \<in> nonzero Q\<^sub>p")
     case T: True
     then have "d \<in> nonzero Q\<^sub>p"
       using assms pow_res_nonzero by blast
     then obtain \<beta> where beta_def: "\<beta> \<in> nonzero Q\<^sub>p \<and> b = d \<otimes> \<beta>[^]n"
-      using  T pow_res_refl[of b n] unfolding assms unfolding pow_res_def   
+      using  T pow_res_refl[of b n] unfolding assms unfolding pow_res_def
       using assms by blast
     then have "a \<otimes> b = (c \<otimes> d) \<otimes> (\<alpha>[^]n \<otimes> \<beta>[^] n)"
       using Qp.m_assoc Qp.m_lcomm Qp.nonzero_closed Qp.nonzero_mult_closed Qp_nat_pow_nonzero alpha_def assms(3) assms(4) assms(5) by presburger
     then have 0: "a \<otimes> b = (c \<otimes> d) \<otimes> ((\<alpha> \<otimes> \<beta>)[^] n)"
         by (metis Qp.nat_pow_distrib Qp.nonzero_closed alpha_def beta_def)
-      show ?thesis 
+      show ?thesis
         apply(intro equal_pow_resI'[of _ _  "(\<alpha> \<otimes> \<beta>)[^] n"] Qp.ring_simprules assms
-                    P_set_memI[of _ "\<alpha> \<otimes> \<beta>"] Qp.nat_pow_closed nonzero_memE 0 Qp_nat_pow_nonzero 
+                    P_set_memI[of _ "\<alpha> \<otimes> \<beta>"] Qp.nat_pow_closed nonzero_memE 0 Qp_nat_pow_nonzero
                     )
-        using alpha_def beta_def apply auto 
-        apply(intro nonzero_memI Qp.nonzero_mult_closed)  
-        using alpha_def beta_def nonzero_memE  apply auto 
-        by (meson Qp.integral_iff)        
+        using alpha_def beta_def apply auto
+        apply(intro nonzero_memI Qp.nonzero_mult_closed)
+        using alpha_def beta_def nonzero_memE  apply auto
+        by (meson Qp.integral_iff)
   next
     case False
     then have "b = \<zero>"
       by (meson assms not_nonzero_Qp)
     then have "d = \<zero>"
       using assms by (metis False not_nonzero_Qp pow_res_nonzero)
-    then show ?thesis 
-      using Qp.r_null \<open>b = \<zero>\<close> assms  by presburger   
+    then show ?thesis
+      using Qp.r_null \<open>b = \<zero>\<close> assms  by presburger
   qed
 next
-  case False    
+  case False
   then have "a = \<zero>"
       by (meson assms  not_nonzero_Qp)
     then have "c = \<zero>"
       using assms by (metis False not_nonzero_Qp pow_res_nonzero)
-    then show ?thesis 
-      using Qp.r_null \<open>a = \<zero>\<close> assms Qp.l_null by presburger       
+    then show ?thesis
+      using Qp.r_null \<open>a = \<zero>\<close> assms Qp.l_null by presburger
   qed
 
 lemma pow_res_p_pow_factor:
   assumes "n > 0"
   assumes "a \<in> carrier Q\<^sub>p"
   shows "pow_res n a = pow_res n (\<pp>[^]((l::int)*n) \<otimes> a)"
 proof(cases "a  = \<zero>")
   case True
-  then show ?thesis 
-    using Qp.r_null p_intpow_closed(1) by presburger 
+  then show ?thesis
+    using Qp.r_null p_intpow_closed(1) by presburger
 next
   case False
-  then show ?thesis using assms pow_res_of_p_pow 
+  then show ?thesis using assms pow_res_of_p_pow
     using Qp.m_comm Qp.one_closed Qp.r_one p_intpow_closed(1) pow_res_mult by presburger
 qed
 
 lemma pow_res_classes_finite:
   assumes "n \<ge> 1"
   shows "finite (pow_res_classes n)"
 proof(cases "n = 1")
   case True
   have "pow_res_classes n = {(nonzero Q\<^sub>p)}"
-    using True pow_res_one unfolding pow_res_classes_def 
+    using True pow_res_one unfolding pow_res_classes_def
     using Qp.one_nonzero by blast
-  then show ?thesis by auto 
+  then show ?thesis by auto
 next
   case False
   then have n_bound: "n \<ge> 2"
     using assms by linarith
   obtain m where m_def: "m > 0 \<and> (\<forall>x y. x \<in> nonzero Q\<^sub>p \<and> y \<in> nonzero Q\<^sub>p \<and> ac m x = ac m y \<and> ord x = ord y \<longrightarrow> pow_res n x = pow_res n y)"
     using assms  False pow_res_equal_ord n_bound
-    by (metis gr_zeroI le_numeral_extra(2))   
+    by (metis gr_zeroI le_numeral_extra(2))
   obtain f where f_def: "f  = (\<lambda> \<eta> \<nu>. (SOME y. y \<in> (pow_res_classes n) \<and> (\<exists> x \<in> y. ac m x = \<eta> \<and> ord x = \<nu>)))"
-    by blast 
+    by blast
   have 0: "\<And>x. x \<in> nonzero Q\<^sub>p \<Longrightarrow> pow_res n x = f (ac m x) (ord x)"
   proof- fix x assume A: "x \<in> nonzero Q\<^sub>p"
     obtain \<eta> where eta_def: "\<eta> = ac m x"
-      by blast 
+      by blast
     obtain \<nu> where nu_def: "\<nu> = ord x"
-      by blast 
+      by blast
     have "\<exists>y \<in>pow_res n x. ac m y = ac m x \<and> ord y = ord x"
-      using pow_res_refl A assms neq0_conv Qp.nonzero_closed by blast      
+      using pow_res_refl A assms neq0_conv Qp.nonzero_closed by blast
     hence "pow_res n x \<in> (pow_res_classes n) \<and> (\<exists> y \<in> (pow_res n x). ac m y = \<eta> \<and> ord y = \<nu>)"
-      unfolding nu_def eta_def using assms unfolding pow_res_classes_def 
+      unfolding nu_def eta_def using assms unfolding pow_res_classes_def
       using A by blast
     then have 0: "(\<exists> y. y \<in> (pow_res_classes n) \<and> (\<exists> x \<in> y. ac m x = \<eta> \<and> ord x = \<nu>))"
-      by blast 
+      by blast
     have  "f \<eta> \<nu> = (SOME y. y \<in> (pow_res_classes n) \<and> (\<exists> x \<in> y. ac m x = \<eta> \<and> ord x = \<nu>))"
-      using f_def by blast 
+      using f_def by blast
     then have 1: "f \<eta> \<nu> \<in> (pow_res_classes n) \<and> ((\<exists> y \<in> (f \<eta> \<nu>).  ac m y = \<eta> \<and> ord y = \<nu>))"
       using  0  someI_ex[of "\<lambda> y. y \<in> (pow_res_classes n) \<and> (\<exists> x \<in> y. ac m x = \<eta> \<and> ord x = \<nu>)"]
-      unfolding f_def by blast 
+      unfolding f_def by blast
     then obtain y where y_def: "y \<in> (f \<eta> \<nu>) \<and> ac m y = ac m x \<and> ord y =  ord x"
-      unfolding nu_def eta_def by blast 
+      unfolding nu_def eta_def by blast
     obtain a where a_def: "a \<in> nonzero Q\<^sub>p \<and> (f \<eta> \<nu>) = pow_res n a"
       using 1 unfolding pow_res_classes_def by blast
     then have 2: "y \<in> pow_res n a"
-      using y_def by blast 
+      using y_def by blast
     have 3: "y \<in> nonzero Q\<^sub>p"
       using y_def nonzero_pow_res[of a n] a_def by blast
     then have 4: "pow_res n y = pow_res n a"
-      using 3 y_def a_def equal_pow_resI[of y a n]  n_bound Qp.nonzero_closed 
-      by (metis equal_pow_resI)     
+      using 3 y_def a_def equal_pow_resI[of y a n]  n_bound Qp.nonzero_closed
+      by (metis equal_pow_resI)
     have 5: "pow_res n y = f \<eta> \<nu>"
-      using 4 a_def by blast 
+      using 4 a_def by blast
     then show "pow_res n x = f (ac m x) (ord x)"
-      unfolding eta_def nu_def 
+      unfolding eta_def nu_def
       using "3" A m_def y_def by blast
   qed
   obtain N where N_def: "N > 0 \<and> (\<forall> u \<in> carrier Q\<^sub>p. ac N u = 1 \<and> val u = 0 \<longrightarrow> u \<in> P_set n)"
-    using n_bound nth_power_fact assms by blast   
+    using n_bound nth_power_fact assms by blast
   have 1: "\<And>x. x \<in> nonzero Q\<^sub>p \<Longrightarrow> (\<exists>y \<in> nonzero Q\<^sub>p. ord y \<ge> 0 \<and> ord y < n \<and> pow_res n x = pow_res n y)"
   proof- fix x assume x_def: "x \<in> nonzero Q\<^sub>p"
     then obtain k where k_def: "k = ord x mod n"
-      by blast 
+      by blast
     then obtain l where l_def: "ord x = (int n)*l + k"
-      using cancel_div_mod_rules(2)[of n "ord x"0] unfolding k_def 
+      using cancel_div_mod_rules(2)[of n "ord x"0] unfolding k_def
       by (metis group_add_class.add.right_cancel)
     have "x = (\<pp>[^](ord x)) \<otimes> \<iota> (angular_component x)"
       using x_def angular_component_factors_x by blast
     then have "x = (\<pp>[^](n*l + k)) \<otimes> \<iota> (angular_component x)"
-      unfolding l_def by blast 
+      unfolding l_def by blast
     hence "x = \<pp>[^](int n*l) \<otimes> (\<pp>[^] k) \<otimes> \<iota> (angular_component x)"
       by (metis p_intpow_add)
     hence 0: "x = (\<pp>[^]l)[^]n \<otimes> (\<pp>[^] k) \<otimes> \<iota> (angular_component x)"
       using p_pow_factor[of n l k] \<open>x = \<pp> [^] (int n * l + k) \<otimes> \<iota> (angular_component x)\<close> by presburger
     have 1: "\<iota> (angular_component x) \<in> carrier Q\<^sub>p"
-      using x_def angular_component_closed inc_closed by blast 
+      using x_def angular_component_closed inc_closed by blast
     hence 2: "x = (\<pp>[^]l)[^]n \<otimes> ((\<pp>[^] k) \<otimes> \<iota> (angular_component x))"
       using 0 by (metis Qp.m_assoc Qp.nat_pow_closed p_intpow_closed(1))
     obtain a where a_def: "a = (\<pp>[^] k) \<otimes> \<iota> (angular_component x)"
-      by blast 
+      by blast
     have 30: "angular_component x \<in> Units Z\<^sub>p"
-      using angular_component_unit x_def by blast 
+      using angular_component_unit x_def by blast
     then have 3: "\<iota> (angular_component x) \<in> Units Q\<^sub>p"
       by (metis Units_eq_nonzero Zp.Units_closed in_Units_imp_not_zero inc_of_nonzero not_nonzero_Qp)
     have 4: "\<iota> (angular_component x) \<in> nonzero Q\<^sub>p"
-      using 3 Units_nonzero_Qp by blast  
+      using 3 Units_nonzero_Qp by blast
     have a_nonzero: "a \<in> nonzero Q\<^sub>p"
-      unfolding a_def  4 
+      unfolding a_def  4
       by (meson "3" Qp.UnitsI(1) Qp.Units_m_closed Units_nonzero_Qp p_intpow_closed(1) p_intpow_inv)
     have 5: "x = a \<otimes>(\<pp>[^]l)[^]n"
-      using 2 a_nonzero unfolding a_def 
+      using 2 a_nonzero unfolding a_def
       using Qp.m_comm Qp.nat_pow_closed Qp.nonzero_closed p_intpow_closed(1) by presburger
     hence "x \<in> pow_res n a"
-      unfolding pow_res_def 
+      unfolding pow_res_def
       using Qp.nonzero_closed Qp_int_pow_nonzero p_nonzero x_def by blast
     hence 6:"pow_res n a = pow_res n x"
-      using x_def a_def equal_pow_resI[of x  a n]  a_nonzero n_bound Qp.nonzero_closed equal_pow_resI 
-      by blast      
+      using x_def a_def equal_pow_resI[of x  a n]  a_nonzero n_bound Qp.nonzero_closed equal_pow_resI
+      by blast
     have 7: "ord (\<iota> (angular_component x)) = 0"
     proof-
       have "ord_Zp (angular_component x) = 0" using 30 Zp_Units_ord_zero by blast
       then have "val_Zp (angular_component x) = 0"
         using "30" unit_imp_val_Zp0 by blast
       then have "val (\<iota> (angular_component x)) = 0"
         by (metis angular_component_closed val_of_inc x_def)
-      then show ?thesis using angular_component_closed x_def  
+      then show ?thesis using angular_component_closed x_def
         by (metis "30" Zp.Units_closed \<open>ord_Zp (angular_component x) = 0\<close> in_Units_imp_not_zero not_nonzero_Qp ord_of_inc)
     qed
     have 8: "ord a = k"
       unfolding a_def using 3 4 7 ord_mult[of "\<pp> [^] k" "\<iota> (angular_component x)"] ord_p_pow_int[of k]
-                p_pow_nonzero 
+                p_pow_nonzero
       using Qp_int_pow_nonzero p_nonzero by presburger
     have 9: "k < n"
-      unfolding k_def 
-      using assms by auto     
+      unfolding k_def
+      using assms by auto
     show " \<exists>y\<in>nonzero Q\<^sub>p. 0 \<le> ord y \<and> ord y < int n \<and> pow_res n x = pow_res n y"
       by (metis "6" "8" a_nonzero assms k_def less_imp_of_nat_less of_nat_eq_0_iff pos_mod_conj rel_simps(45) zero_less_iff_neq_zero)
   qed
   have 2: "\<And>x. x \<in> (pow_res_classes n) \<Longrightarrow> \<exists> \<eta> \<nu>. \<eta> \<in> Units (Zp_res_ring m) \<and> \<nu> \<in> {0..<int n} \<and> x = f \<eta> \<nu>"
   proof- fix a assume A: "a \<in> (pow_res_classes n)"
     then obtain x where x_def: "x \<in> nonzero Q\<^sub>p \<and> a = pow_res n x"
-      unfolding pow_res_classes_def by blast 
+      unfolding pow_res_classes_def by blast
     then obtain x' where x'_def: "x' \<in> nonzero Q\<^sub>p \<and> ord x' \<ge> 0 \<and> ord x' < n \<and> pow_res n x' = a"
-      using 1[of x] unfolding x_def by blast 
+      using 1[of x] unfolding x_def by blast
     hence 20: "f (ac m x') (ord x') = a"
-      using 0 by blast 
+      using 0 by blast
     have 21: "ac m x' \<in> Units (Zp_res_ring m)"
       using x'_def ac_units m_def by presburger
     then have 22: "ac m x' \<in> Units (Zp_res_ring m) \<and> (ord x') \<in> ({0..<n}::int set ) \<and> a = f (ac m x') (ord x')"
       using x'_def 20 atLeastLessThan_iff by blast
-    then show "\<exists> \<eta> \<nu>. \<eta> \<in> Units (Zp_res_ring m) \<and> \<nu> \<in> {0..<int n} \<and> a = f \<eta> \<nu>" by blast 
+    then show "\<exists> \<eta> \<nu>. \<eta> \<in> Units (Zp_res_ring m) \<and> \<nu> \<in> {0..<int n} \<and> a = f \<eta> \<nu>" by blast
   qed
   obtain F where F_def: "F = (\<lambda>ps. f (fst ps) (snd ps))"
-    by blast 
+    by blast
   have 3: "\<And>x. x \<in> (pow_res_classes n) \<Longrightarrow> \<exists> ps \<in> Units (Zp_res_ring m) \<times> {0..<int n}.  x = f (fst ps) (snd ps)"
   proof- fix x assume A: "x \<in> pow_res_classes n"
     obtain \<eta> \<nu> where eta_nu_def: " \<eta> \<in> Units (Zp_res_ring m) \<and> \<nu> \<in> {0..<int n} \<and> x = f \<eta> \<nu>"
-      using 2 A by blast 
+      using 2 A by blast
     then have "F (\<eta>, \<nu>) = x"
       unfolding F_def by (metis fst_conv snd_conv)
     then show " \<exists> ps \<in> Units (Zp_res_ring m) \<times> {0..<int n}.  x = f (fst ps) (snd ps)"
       using eta_nu_def local.F_def by blast
   qed
   have 4: "pow_res_classes n \<subseteq> F ` (Units (Zp_res_ring m) \<times> {0..<int n})"
-    unfolding F_def using 3 
-    by blast 
+    unfolding F_def using 3
+    by blast
   have "finite (Units (Zp_res_ring m))"
-    using m_def residues.finite_Units unfolding residues_def 
+    using m_def residues.finite_Units unfolding residues_def
     by (metis Qp.one_nonzero ac_in_res_ring ac_one' p_res_ring_one p_residue_ring_car_memE(1))
   hence "finite (Units (Zp_res_ring m) \<times> {0..<int n})"
     by blast
   then show "finite (pow_res_classes n)"
     using 4 by (meson finite_surj)
 qed
 
 lemma pow_res_classes_univ_semialg:
   assumes "S \<in> pow_res_classes n"
   shows "is_univ_semialgebraic S"
 proof-
   obtain x where x_def: "x\<in>nonzero Q\<^sub>p \<and> S = pow_res n x"
-  using assms unfolding pow_res_classes_def by blast 
-  then show ?thesis using pow_res_is_univ_semialgebraic  
-    using Qp.nonzero_closed by blast 
+  using assms unfolding pow_res_classes_def by blast
+  then show ?thesis using pow_res_is_univ_semialgebraic
+    using Qp.nonzero_closed by blast
 qed
 
 lemma pow_res_classes_semialg:
   assumes "S \<in> pow_res_classes n"
   shows "is_semialgebraic 1 (to_R1` S)"
   using pow_res_classes_univ_semialg
     assms(1)  is_univ_semialgebraicE by blast
 
 definition nth_pow_wits where
 "nth_pow_wits n = (\<lambda> S. (SOME x. x \<in> (S \<inter> \<O>\<^sub>p)))` (pow_res_classes n)"
 
 lemma nth_pow_wits_finite:
   assumes "n > 0"
   shows "finite (nth_pow_wits n)"
 proof-
   have "n \<ge> 1"
     by (simp add: assms leI)
-  thus ?thesis 
-    unfolding nth_pow_wits_def using assms pow_res_classes_finite[of n] by blast 
-qed 
+  thus ?thesis
+    unfolding nth_pow_wits_def using assms pow_res_classes_finite[of n] by blast
+qed
 
 lemma nth_pow_wits_covers:
   assumes "n > 0"
-  assumes "x \<in> nonzero Q\<^sub>p" 
+  assumes "x \<in> nonzero Q\<^sub>p"
   shows "\<exists>y \<in> (nth_pow_wits n). y \<in> nonzero Q\<^sub>p \<and> y \<in> \<O>\<^sub>p \<and> x \<in> pow_res n y"
 proof-
   have PP: "(pow_res n x) \<in> pow_res_classes n"
-    unfolding pow_res_classes_def using assms by blast 
+    unfolding pow_res_classes_def using assms by blast
   obtain k where k_def: "val x = eint k"
     using assms val_ord by blast
-  obtain N::int where N_def: "N = (if  k < 0 then -k else k)" by blast 
+  obtain N::int where N_def: "N = (if  k < 0 then -k else k)" by blast
   then have N_nonneg: "N \<ge> 0"
-    unfolding N_def 
-    by presburger 
+    unfolding N_def
+    by presburger
   have 0: "int n \<ge> 1"
-    using assms by linarith 
+    using assms by linarith
   have "N*(int n) + k \<ge> 0"
   proof(cases "k<0")
-    case True then have "N = -k" unfolding N_def 
+    case True then have "N = -k" unfolding N_def
       by presburger
     then have "N*n + k = k*(1- int n)"
       using distrib_left[of k 1 "-int n"] mult_cancel_left2 mult_minus_left
       by (metis add.inverse_inverse diff_minus_eq_add minus_mult_minus neg_equal_iff_equal uminus_add_conv_diff)
     then show ?thesis using 0 True  zero_less_mult_iff[of k "1 - int n"]
    proof -
      have "0 \<le> N * (int n - 1)"
-       by (meson "0" N_nonneg diff_ge_0_iff_ge zero_le_mult_iff)      
+       by (meson "0" N_nonneg diff_ge_0_iff_ge zero_le_mult_iff)
      then show ?thesis
        by (metis (no_types) \<open>N = - k\<close> add.commute distrib_left minus_add_cancel mult_minus1_right uminus_add_conv_diff)
-   qed     
+   qed
   next
     case False
-    then have "N = k" unfolding N_def 
+    then have "N = k" unfolding N_def
       by presburger
-    then show ?thesis using 0 False 
+    then show ?thesis using 0 False
       by (metis N_nonneg add_increasing2 mult_nonneg_nonneg of_nat_0_le_iff)
   qed
   then have 1: "ord (\<pp>[^](N*n)\<otimes>x) \<ge> 0"
-    using ord_mult k_def val_ord assms 
+    using ord_mult k_def val_ord assms
     by (metis Qp_int_pow_nonzero eint.inject ord_p_pow_int p_nonzero)
   have 2: "\<pp>[^](N*n)\<otimes>x \<in> pow_res n x"
   proof-
     have "\<pp>[^](N*n) = (\<pp>[^]N)[^]n"
       using Qp_p_int_nat_pow_pow by blast
     then have "\<pp>[^]N \<in> nonzero Q\<^sub>p  \<and> \<pp>[^](N*n)\<otimes>x = x \<otimes> (\<pp>[^]N)[^]n"
       by (metis Qp.m_comm Qp.nonzero_closed Qp_int_pow_nonzero assms(2) p_nonzero)
-    then show ?thesis unfolding pow_res_def 
+    then show ?thesis unfolding pow_res_def
       by (metis (mono_tags, lifting) Qp.m_closed Qp.nonzero_closed assms(2) mem_Collect_eq p_intpow_closed(1))
   qed
   have 3: "\<pp>[^](N*n)\<otimes>x \<in> \<O>\<^sub>p"
-    using 1 assms 
+    using 1 assms
     by (metis Q\<^sub>p_def Qp.nonzero_mult_closed Qp_int_pow_nonzero Z\<^sub>p_def val_ring_ord_criterion \<iota>_def p_nonzero padic_fields.zero_in_val_ring padic_fields_axioms)
   have 4: "x \<in> pow_res n (\<pp>[^](N*n)\<otimes>x)"
     using 2 equal_pow_resI[of x "\<pp>[^](N*n)\<otimes>x" n] pow_res_refl[of "\<pp>[^](N*n)\<otimes>x" n] assms
-         Qp.nonzero_mult_closed p_intpow_closed(2) pow_res_refl Qp.nonzero_closed by metis         
+         Qp.nonzero_mult_closed p_intpow_closed(2) pow_res_refl Qp.nonzero_closed by metis
   have 5: "\<pp>[^](N*n)\<otimes>x \<in> (pow_res n x \<inter> \<O>\<^sub>p)"
-    using 2 3 by blast 
-  have 6: "(SOME z. z \<in> (pow_res n x) \<inter> \<O>\<^sub>p) \<in> pow_res n x \<inter> \<O>\<^sub>p" using 5 
+    using 2 3 by blast
+  have 6: "(SOME z. z \<in> (pow_res n x) \<inter> \<O>\<^sub>p) \<in> pow_res n x \<inter> \<O>\<^sub>p" using 5
     by (meson someI)
   obtain y where y_def: "y = (SOME z. z \<in> (pow_res n x) \<inter> \<O>\<^sub>p)"
-    by blast 
+    by blast
   then have A: "y \<in> pow_res n x"
     using "6" by blast
   then have "pow_res n x = pow_res n y"
     using equal_pow_resI[of x y n] assms y_def Qp.nonzero_closed nonzero_pow_res by blast
   then have 7: "x \<in> pow_res n y"
     using pow_res_refl[of x n] assms unfolding nonzero_def by blast
   have 8: "y \<in> nonzero Q\<^sub>p  "
-    using y_def PP 6 A nonzero_pow_res[of x n] assms  
+    using y_def PP 6 A nonzero_pow_res[of x n] assms
     by blast
   have 9: "y \<in> \<O>\<^sub>p"
     using y_def "6" by blast
   have "y\<in>(\<lambda>S. SOME x. x \<in> S \<inter> \<O>\<^sub>p) ` pow_res_classes n \<and> y \<in> nonzero Q\<^sub>p \<and> y \<in> \<O>\<^sub>p \<and> x \<in> pow_res n y"
-        using y_def PP 6 7 8 9 A nonzero_pow_res[of x n] assms   
-        by blast     
-  then show ?thesis unfolding nth_pow_wits_def by blast 
-qed
-  
+        using y_def PP 6 7 8 9 A nonzero_pow_res[of x n] assms
+        by blast
+  then show ?thesis unfolding nth_pow_wits_def by blast
+qed
+
 lemma nth_pow_wits_closed:
   assumes "n > 0"
   assumes "x \<in> nth_pow_wits n"
   shows "x \<in> carrier Q\<^sub>p" "x \<in> \<O>\<^sub>p" "x \<in> nonzero Q\<^sub>p" "\<exists> y \<in> pow_res_classes n. y = pow_res n x"
 proof-
   obtain c where c_def: "c \<in> pow_res_classes n \<and> x =   (SOME x. x \<in> (c \<inter> \<O>\<^sub>p))"
     by (metis (no_types, lifting) assms(2) image_iff nth_pow_wits_def)
   then obtain y where y_def: "y \<in> nonzero Q\<^sub>p \<and> c = pow_res n y"
     unfolding pow_res_classes_def by blast
   then obtain a where a_def: "a \<in> (nth_pow_wits n) \<and> a \<in> nonzero Q\<^sub>p \<and> a \<in> \<O>\<^sub>p \<and> y \<in> pow_res n a"
     using nth_pow_wits_covers[of n y] assms(1) by blast
   have 00: "pow_res n a = c"
-    using equal_pow_resI[of a y n] y_def assms a_def unfolding nonzero_def by blast 
+    using equal_pow_resI[of a y n] y_def assms a_def unfolding nonzero_def by blast
   then have P :"a \<in> c \<inter> \<O>\<^sub>p"
-    using pow_res_refl[of a n] assms a_def unfolding 00 nonzero_def by blast     
-  then show 0: "x \<in> \<O>\<^sub>p" using c_def  
+    using pow_res_refl[of a n] assms a_def unfolding 00 nonzero_def by blast
+  then show 0: "x \<in> \<O>\<^sub>p" using c_def
     by (metis Collect_mem_eq Int_Collect tfl_some)
   then show "x \<in> carrier Q\<^sub>p"
     using val_ring_memE by blast
   have 1: "c \<subseteq> nonzero Q\<^sub>p"
-    using c_def nonzero_pow_res[of y n] unfolding pow_res_classes_def 
+    using c_def nonzero_pow_res[of y n] unfolding pow_res_classes_def
     using assms(1) y_def by blast
   have "(SOME x. x \<in> (c \<inter> \<O>\<^sub>p)) \<in> (c \<inter> \<O>\<^sub>p)"
-    using P tfl_some 
+    using P tfl_some
     by (smt Int_def someI_ex)
   then have 2: "x \<in> c"
-    using c_def by blast 
+    using c_def by blast
   thus "x \<in> nonzero Q\<^sub>p"
-    using 1  by blast 
+    using 1  by blast
   show "\<exists> y \<in> pow_res_classes n. y = pow_res n x"
     using 00 2 c_def P a_def equal_pow_resI[of a x n] 0 val_ring_memE assms(1) by blast
 qed
 
 lemma finite_extensional_funcset:
   assumes "finite A"
   assumes "finite (B::'b set)"
   shows "finite (A \<rightarrow>\<^sub>E B)"
-  using finite_PiE[of A "\<lambda>_. B"] assms by blast 
+  using finite_PiE[of A "\<lambda>_. B"] assms by blast
 
 lemma nth_pow_wits_exists:
   assumes  "m > 0"
   assumes "c \<in> pow_res_classes m"
   shows "\<exists>x. x \<in> c \<inter> \<O>\<^sub>p"
 proof-
   obtain x where x_def: "x \<in> nonzero Q\<^sub>p \<and> pow_res m x = c"
-    using assms unfolding pow_res_classes_def by blast 
+    using assms unfolding pow_res_classes_def by blast
   obtain y where y_def: "y \<in> (nth_pow_wits m) \<and> y \<in> nonzero Q\<^sub>p \<and> y \<in> \<O>\<^sub>p \<and> x \<in> pow_res m y"
-    using nth_pow_wits_covers assms x_def 
+    using nth_pow_wits_covers assms x_def
     by blast
   have 0: "pow_res m x = pow_res m y"
     using x_def y_def equal_pow_resI Qp.nonzero_closed assms(1) by blast
   then have 1: "y \<in> pow_res m  x"
-    using pow_res_refl[of y m ] y_def assms unfolding nonzero_def  by blast      
-  thus ?thesis using x_def y_def assms 
-    by blast
-qed    
+    using pow_res_refl[of y m ] y_def assms unfolding nonzero_def  by blast
+  thus ?thesis using x_def y_def assms
+    by blast
+qed
 
 lemma pow_res_classes_mem_eq:
   assumes "m > 0"
   assumes "a \<in> pow_res_classes m"
   assumes "x \<in> a"
   shows  "a = pow_res m x"
 proof-
   obtain y where y_def: "y \<in> nonzero Q\<^sub>p \<and> a = pow_res m y"
-    using assms unfolding pow_res_classes_def by blast 
-  then show ?thesis using assms equal_pow_resI[of y x m] 
+    using assms unfolding pow_res_classes_def by blast
+  then show ?thesis using assms equal_pow_resI[of y x m]
     by (meson Qp.nonzero_closed nonzero_pow_res equal_pow_resI subsetD)
-qed  
+qed
 
 lemma nth_pow_wits_neq_pow_res:
   assumes  "m > 0"
   assumes "x \<in> nth_pow_wits m"
   assumes "y \<in> nth_pow_wits m"
   assumes "x \<noteq> y"
   shows "pow_res m x \<noteq> pow_res m y"
 proof-
   obtain a where a_def: "a \<in> pow_res_classes m \<and> x = (\<lambda> S. (SOME x. x \<in> (S \<inter> \<O>\<^sub>p))) a"
-    using assms unfolding nth_pow_wits_def by blast 
+    using assms unfolding nth_pow_wits_def by blast
   obtain b where b_def: "b \<in> pow_res_classes m \<and> y = (\<lambda> S. (SOME x. x \<in> (S \<inter> \<O>\<^sub>p))) b"
-    using assms unfolding nth_pow_wits_def by blast 
+    using assms unfolding nth_pow_wits_def by blast
   have a_neq_b: "a \<noteq> b"
-    using assms a_def b_def by blast   
+    using assms a_def b_def by blast
   have 0: "x \<in> a \<inter> \<O>\<^sub>p"
-    using a_def nth_pow_wits_exists[of m a] assms  
+    using a_def nth_pow_wits_exists[of m a] assms
     by (meson someI_ex)
   have 1: "y \<in> b \<inter> \<O>\<^sub>p"
-    using b_def nth_pow_wits_exists[of m b] assms  
-    by (meson someI_ex)    
+    using b_def nth_pow_wits_exists[of m b] assms
+    by (meson someI_ex)
   have 2: "pow_res m x = a"
     using a_def  pow_res_classes_mem_eq[of m a x] assms 0
     by blast
   have 3: "pow_res m y = b"
     using b_def  pow_res_classes_mem_eq[of m b y] assms 1
     by blast
-  show ?thesis 
+  show ?thesis
     by (simp add: "2" "3" a_neq_b)
-qed  
+qed
 
 lemma nth_pow_wits_disjoint_pow_res:
   assumes  "m > 0"
   assumes "x \<in> nth_pow_wits m"
   assumes "y \<in> nth_pow_wits m"
   assumes "x \<noteq> y"
   shows "pow_res m x \<inter> pow_res m y =  {}"
   using assms nth_pow_wits_neq_pow_res disjoint_iff_not_equal
   by (metis (no_types, opaque_lifting) nth_pow_wits_closed(4) pow_res_classes_mem_eq)
 
 lemma nth_power_fact':
   assumes "0 < (n::nat)"
   shows "\<exists>m>0. \<forall>u\<in>carrier Q\<^sub>p. ac m u = 1 \<and> val u = 0 \<longrightarrow> u \<in> P_set n"
-  using nth_power_fact[of n] assms 
+  using nth_power_fact[of n] assms
   by (metis less_one less_or_eq_imp_le linorder_neqE_nat neq0_conv)
 
 lemma equal_pow_res_criterion:
   assumes "N > 0"
   assumes "n > 0"
   assumes "\<forall> u \<in> carrier Q\<^sub>p. ac N u = 1 \<and> val u = 0 \<longrightarrow> u \<in> P_set n"
-  assumes "a \<in> carrier Q\<^sub>p" 
+  assumes "a \<in> carrier Q\<^sub>p"
   assumes "b \<in> carrier Q\<^sub>p"
   assumes "c \<in> carrier Q\<^sub>p"
   assumes "a = b \<otimes> (\<one> \<oplus> c)"
   assumes "val c \<ge> N"
   shows "pow_res n a = pow_res n b"
 proof(cases "b = \<zero>")
   case True
   then have "a = \<zero>"
     using assms Qp.add.m_closed Qp.l_null Qp.one_closed by presburger
-  then show ?thesis using True 
+  then show ?thesis using True
     by blast
 next
   case False
   then have F0: "a \<div> b = \<one> \<oplus> c"
-    by (metis Qp.Units_one_closed Qp.add.m_closed Qp.inv_cancelR(2) Qp.one_closed Qp.unit_factor assms(4) assms(5) assms(6) assms(7) field_inv(2) inv_in_frac(1))   
+    by (metis Qp.Units_one_closed Qp.add.m_closed Qp.inv_cancelR(2) Qp.one_closed Qp.unit_factor assms(4) assms(5) assms(6) assms(7) field_inv(2) inv_in_frac(1))
   have "0 < eint N"
     using assms by (metis eint_ord_simps(2) of_nat_0_less_iff zero_eint_def)
   hence F1: "val \<one> < val c"
-    using assms less_le_trans[of 0 N "val c"] unfolding val_one 
+    using assms less_le_trans[of 0 N "val c"] unfolding val_one
     by blast
   hence F2: " val \<one> = val (\<one> \<oplus> c)"
-    using assms val_one one_nonzero Qp.add.m_comm Qp.one_closed val_ultrametric_noteq by metis  
+    using assms val_one one_nonzero Qp.add.m_comm Qp.one_closed val_ultrametric_noteq by metis
   have "val \<one> + eint (int N) \<le> val (\<one> \<ominus> (\<one> \<oplus> c))"
   proof-
     have "val (\<one> \<ominus> (\<one> \<oplus> c)) = val c"
       using Qp.add.inv_closed Qp.minus_eq Qp.minus_sum Qp.one_closed Qp.r_neg2 assms(6) val_minus by presburger
-    thus ?thesis 
+    thus ?thesis
     unfolding val_one using assms F1 by (metis add.left_neutral)
   qed
   hence F3: "ac N \<one> = ac N (\<one> \<oplus> c)"
-    using F2  F1 assms ac_val[of \<one> "\<one> \<oplus> c" N] 
+    using F2  F1 assms ac_val[of \<one> "\<one> \<oplus> c" N]
     by (metis Qp.add.m_closed Qp.one_closed val_nonzero)
   have F4: "\<one> \<oplus> c \<in> P_set n"
-    using assms F1 F2 F3 val_one ac_one 
+    using assms F1 F2 F3 val_one ac_one
     by (metis Qp.add.m_closed Qp.one_closed Qp.one_nonzero ac_inv'' ac_inv'''(1) ac_one')
-  then show ?thesis 
+  then show ?thesis
     using assms(2) assms(4) assms(5) assms(7) equal_pow_resI' by blast
 qed
 
 
 
 lemma pow_res_nat_pow:
   assumes "n > 0"
   assumes "a \<in> carrier Q\<^sub>p"
   assumes "b \<in> carrier Q\<^sub>p"
   assumes "pow_res n a = pow_res n b"
   shows "pow_res n (a[^](k::nat)) = pow_res n (b[^]k)"
   apply(induction k)
   using assms apply (metis Group.nat_pow_0)
   using assms pow_res_mult by (smt Qp.nat_pow_Suc2 Qp.nat_pow_closed)
 
 lemma pow_res_mult':
   assumes "n > 0"
   assumes "a \<in> carrier Q\<^sub>p"
   assumes "b \<in> carrier Q\<^sub>p"
   assumes "c \<in> carrier Q\<^sub>p"
   assumes "d \<in> carrier Q\<^sub>p"
   assumes "e \<in> carrier Q\<^sub>p"
   assumes "f \<in> carrier Q\<^sub>p"
   assumes "pow_res n a = pow_res n d"
   assumes "pow_res n b = pow_res n e"
   assumes "pow_res n c = pow_res n f"
   shows "pow_res n (a \<otimes> b \<otimes> c) = pow_res n (d \<otimes> e \<otimes> f)"
 proof-
   have "pow_res n (a \<otimes> b) = pow_res n (d \<otimes> e)"
     using pow_res_mult assms  by meson
-  then show ?thesis using pow_res_mult assms 
+  then show ?thesis using pow_res_mult assms
     by (meson Qp.m_closed)
 qed
 
 lemma pow_res_disjoint:
   assumes "n > 0"
   assumes "a \<in> nonzero Q\<^sub>p"
   assumes "a \<notin> pow_res n \<one>"
   shows "\<not> (\<exists>y \<in> nonzero Q\<^sub>p. a = y[^]n)"
-  using assms unfolding pow_res_def 
+  using assms unfolding pow_res_def
   using Qp.l_one Qp.nonzero_closed by blast
 
 lemma pow_res_disjoint':
   assumes "n > 0"
   assumes "a \<in> nonzero Q\<^sub>p"
   assumes "pow_res n a \<noteq> pow_res n \<one>"
   shows "\<not> (\<exists>y \<in> nonzero Q\<^sub>p. a = y[^]n)"
-using assms pow_res_disjoint pow_res_refl 
+using assms pow_res_disjoint pow_res_refl
   by (metis pow_res_nth_pow)
 
 lemma pow_res_one_imp_nth_pow:
   assumes "n > 0"
   assumes "a \<in> pow_res n \<one>"
   shows "\<exists>y \<in> nonzero Q\<^sub>p. a = y[^]n"
-  using assms unfolding pow_res_def 
+  using assms unfolding pow_res_def
   using Qp.l_one Qp.nat_pow_closed Qp.nonzero_closed by blast
 
 lemma pow_res_eq:
   assumes "n > 0"
   assumes "a \<in> carrier Q\<^sub>p"
   assumes "b \<in> pow_res n a"
   shows "pow_res n b = pow_res n a"
 proof(cases "a = \<zero>")
   case True
   then show ?thesis using assms by (metis pow_res_zero singletonD)
 next
   case False
   then have a_nonzero: "a \<in> nonzero Q\<^sub>p" using Qp.not_nonzero_memE assms(2) by blast
-  show ?thesis 
+  show ?thesis
   proof(cases "n = 1")
     case True
-    then show ?thesis using a_nonzero assms 
-      using pow_res_one Q\<^sub>p_def Zp_def padic_fields_axioms by blast    
+    then show ?thesis using a_nonzero assms
+      using pow_res_one Q\<^sub>p_def Zp_def padic_fields_axioms by blast
   next
     case False
     then have "n \<ge> 2"
       using assms(1) by linarith
-    then show ?thesis using False a_nonzero assms Qp.nonzero_closed nonzero_pow_res equal_pow_resI 
+    then show ?thesis using False a_nonzero assms Qp.nonzero_closed nonzero_pow_res equal_pow_resI
       by blast
   qed
 qed
 
 lemma pow_res_classes_n_eq_one:
 "pow_res_classes 1 = {nonzero Q\<^sub>p}"
   unfolding pow_res_classes_def using pow_res_one Qp.one_nonzero by blast
 
 lemma nth_pow_wits_closed':
   assumes "n > 0"
   assumes "x \<in> nth_pow_wits n"
   shows "x \<in> \<O>\<^sub>p \<and> x \<in> nonzero Q\<^sub>p" using nth_pow_wits_closed
-  assms  by blast 
+  assms  by blast
 
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsection\<open>Semialgebraic Sets Defined by Congruences\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
       (********************************************************************************************)
       (********************************************************************************************)
       subsubsection\<open>$p$-adic ord Congruence Sets\<close>
       (********************************************************************************************)
       (********************************************************************************************)
 
 lemma carrier_is_univ_semialgebraic:
 "is_univ_semialgebraic (carrier Q\<^sub>p)"
   apply(rule is_univ_semialgebraicI)
-  using Qp.to_R1_carrier carrier_is_semialgebraic 
+  using Qp.to_R1_carrier carrier_is_semialgebraic
   by presburger
 
 lemma nonzero_is_univ_semialgebraic:
 "is_univ_semialgebraic (nonzero Q\<^sub>p)"
 proof-
   have "nonzero Q\<^sub>p = carrier Q\<^sub>p - {\<zero>}"
-    unfolding nonzero_def by blast 
+    unfolding nonzero_def by blast
   then show ?thesis using  diff_is_univ_semialgebraic[of "carrier Q\<^sub>p" "{\<zero>}"]
-    by (metis Diff_empty Diff_insert0 carrier_is_univ_semialgebraic empty_subsetI 
+    by (metis Diff_empty Diff_insert0 carrier_is_univ_semialgebraic empty_subsetI
         finite.emptyI finite.insertI finite_is_univ_semialgebraic insert_subset)
 qed
 
 definition ord_congruence_set where
 "ord_congruence_set n a = {x \<in> nonzero Q\<^sub>p. ord x mod n = a}"
 
 lemma ord_congruence_set_nonzero:
 "ord_congruence_set n a \<subseteq> nonzero Q\<^sub>p"
   by (metis (no_types, lifting) mem_Collect_eq ord_congruence_set_def subsetI)
 
 lemma ord_congruence_set_closed:
 "ord_congruence_set n a \<subseteq> carrier Q\<^sub>p"
-  using nonzero_def ord_congruence_set_nonzero 
-  unfolding nonzero_def 
+  using nonzero_def ord_congruence_set_nonzero
+  unfolding nonzero_def
   by (meson Qp.nonzero_closed ord_congruence_set_nonzero subset_iff)
-  
+
 lemma ord_congruence_set_memE:
   assumes "x \<in> ord_congruence_set n a"
   shows "x \<in> nonzero Q\<^sub>p"
         "ord x mod n = a"
   using assms ord_congruence_set_nonzero apply blast
   by (metis (mono_tags, lifting) assms mem_Collect_eq ord_congruence_set_def)
 
 lemma ord_congruence_set_memI:
   assumes "x \<in> nonzero Q\<^sub>p"
   assumes "ord x mod n = a"
   shows "x \<in> ord_congruence_set n a"
-  using assms 
+  using assms
   by (metis (mono_tags, lifting) mem_Collect_eq ord_congruence_set_def)
 
 text\<open>
   We want to prove that ord\_congruence\_set is a finite union of semialgebraic sets,
   hence is also semialgebraic.
 \<close>
 
 lemma pow_res_ord_cong:
   assumes "x \<in> carrier Q\<^sub>p"
   assumes "x \<in> ord_congruence_set n a"
   shows "pow_res n x \<subseteq> ord_congruence_set n a"
 proof fix y
   assume A: "y  \<in> pow_res n x"
   show "y \<in> ord_congruence_set (int n) a"
   proof-
     obtain a where a_def: "a \<in> nonzero Q\<^sub>p \<and> y = x \<otimes> (a[^]n)"
-      using A pow_res_def[of n x] by blast    
+      using A pow_res_def[of n x] by blast
     have 0: "x \<in> nonzero Q\<^sub>p"
-      using assms(2) ord_congruence_set_memE(1) 
+      using assms(2) ord_congruence_set_memE(1)
       by blast
     have 1: "y \<in> nonzero Q\<^sub>p"
-      using A  
-      by (metis "0" Qp.integral Qp.nonzero_closed Qp.nonzero_mult_closed Qp_nat_pow_nonzero a_def not_nonzero_Qp)     
+      using A
+      by (metis "0" Qp.integral Qp.nonzero_closed Qp.nonzero_mult_closed Qp_nat_pow_nonzero a_def not_nonzero_Qp)
     have 2: "ord y = ord x + n* ord  a"
-      using a_def 0 1 Qp_nat_pow_nonzero nonzero_nat_pow_ord ord_mult 
+      using a_def 0 1 Qp_nat_pow_nonzero nonzero_nat_pow_ord ord_mult
       by presburger
     show ?thesis
       apply(rule ord_congruence_set_memI)
-      using assms ord_congruence_set_memE 2 1 
+      using assms ord_congruence_set_memE 2 1
        apply blast
-      using "2" assms(2) ord_congruence_set_memE(2) 
+      using "2" assms(2) ord_congruence_set_memE(2)
       by presburger
   qed
-qed  
+qed
 
 lemma pow_res_classes_are_univ_semialgebraic:
   shows "are_univ_semialgebraic (pow_res_classes n)"
   apply(rule are_univ_semialgebraicI)
   using pow_res_classes_univ_semialg by blast
 
 lemma ord_congruence_set_univ_semialg:
   assumes "n \<ge> 0"
   shows "is_univ_semialgebraic (ord_congruence_set n a)"
 proof(cases "n = 0")
   case True
   have T0: "ord_congruence_set n a = {x \<in> nonzero Q\<^sub>p. ord x = a}"
     unfolding ord_congruence_set_def True by presburger
   have T1: "{x \<in> nonzero Q\<^sub>p. ord x = a} = {x \<in> nonzero Q\<^sub>p. val x = a}"
     apply(rule equalityI'')
     using val_ord apply blast
-    using val_ord 
+    using val_ord
     by (metis eint.inject)
   have T2: "{x \<in> nonzero Q\<^sub>p. val x = a} = {x \<in> carrier Q\<^sub>p. val x = a}"
     apply(rule equalityI'')
     using Qp.nonzero_closed apply blast
     by (metis iless_Suc_eq val_nonzero val_val_ring_prod zero_in_val_ring)
-  show ?thesis unfolding T0 T1 T2 using univ_val_eq_set_is_univ_semialgebraic by blast  
+  show ?thesis unfolding T0 T1 T2 using univ_val_eq_set_is_univ_semialgebraic by blast
 next
   case False
   obtain F where F_def: "F = {S \<in> (pow_res_classes (nat n)). S \<subseteq>(ord_congruence_set n a) }"
     by blast
   have 0: "F \<subseteq> pow_res_classes (nat n)"
     using F_def by blast
   have 1: "finite F"
     using 0 False nat_mono[of 1 n] nat_numeral[] pow_res_classes_finite[of "nat n"] rev_finite_subset
-    by (smt assms nat_one_as_int)        
+    by (smt assms nat_one_as_int)
   have 2: "are_univ_semialgebraic F"
     apply(rule are_univ_semialgebraicI) using 0 pow_res_classes_are_univ_semialgebraic
     by (metis (mono_tags) are_univ_semialgebraicE are_univ_semialgebraic_def assms  nat_mono nat_numeral subset_iff)
   have 3: "\<Union> F = (ord_congruence_set n a)"
   proof
     show "\<Union> F \<subseteq> ord_congruence_set n a"
-      using F_def 
+      using F_def
       by blast
     show "ord_congruence_set n a \<subseteq> \<Union> F"
     proof fix x
       assume A: "x \<in> ord_congruence_set n a"
       have x_nonzero: "x \<in> nonzero Q\<^sub>p"
         using A ord_congruence_set_memE(1) by blast
       have 0: "pow_res (nat n) x \<in> F"
-        using A pow_res_classes_def F_def 
+        using A pow_res_classes_def F_def
         by (smt nonzero_def assms mem_Collect_eq nat_0_le ord_congruence_set_memE(1) pow_res_ord_cong)
       have 1: "x \<in> pow_res (nat n) x" using False x_nonzero assms pow_res_refl[of x "nat n"]
         using Qp.nonzero_closed by blast
       show "x \<in> \<Union> F"
-        using 0 1 
-        by blast        
-    qed
-  qed
-  then show ?thesis 
+        using 0 1
+        by blast
+    qed
+  qed
+  then show ?thesis
     using "1" "2" finite_union_is_univ_semialgebraic'
     by fastforce
 qed
 
 lemma ord_congruence_set_is_semialg:
   assumes "n \<ge> 0"
   shows "is_semialgebraic 1 (Qp_to_R1_set (ord_congruence_set n a))"
-  using assms is_univ_semialgebraicE ord_congruence_set_univ_semialg 
+  using assms is_univ_semialgebraicE ord_congruence_set_univ_semialg
   by blast
 
       (********************************************************************************************)
       (********************************************************************************************)
       subsubsection\<open>Congruence Sets for the order of the Evaluation of a Polynomial\<close>
       (********************************************************************************************)
       (********************************************************************************************)
 
 
 lemma poly_map_singleton:
   assumes "f \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "poly_map n [f] x = [(Qp_ev f x)]"
-  unfolding poly_map_def poly_tuple_eval_def 
-  using assms 
+  unfolding poly_map_def poly_tuple_eval_def
+  using assms
   by (metis (no_types, lifting) Cons_eq_map_conv list.simps(8) restrict_apply')
-  
+
 definition poly_cong_set where
 "poly_cong_set n f m a = {x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). (Qp_ev f x) \<noteq> \<zero>  \<and> (ord (Qp_ev f x) mod m = a)}"
 
 lemma poly_cong_set_as_pullback:
   assumes "f \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
-  shows "poly_cong_set n f m a = poly_map n [f]  \<inverse>\<^bsub>n\<^esub>(Qp_to_R1_set (ord_congruence_set m a))" 
+  shows "poly_cong_set n f m a = poly_map n [f]  \<inverse>\<^bsub>n\<^esub>(Qp_to_R1_set (ord_congruence_set m a))"
 proof
   show "poly_cong_set n f m a \<subseteq> poly_map n [f]  \<inverse>\<^bsub>n\<^esub> ((\<lambda>a. [a]) ` ord_congruence_set m a)"
-  proof fix x 
+  proof fix x
     assume A: "x \<in> poly_cong_set n f m a"
     then have 0: "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
       by (metis (no_types, lifting) mem_Collect_eq poly_cong_set_def)
     have 1: "(Qp_ev f x) \<noteq> \<zero> "
       by (metis (mono_tags, lifting) A mem_Collect_eq poly_cong_set_def)
     have 2: "(ord (Qp_ev f x) mod m = a)"
       by (metis (mono_tags, lifting) A mem_Collect_eq poly_cong_set_def)
     have 3: "(Qp_ev f x) \<in> (ord_congruence_set m a)"
-      using "0" "1" "2" eval_at_point_closed assms not_nonzero_Qp ord_congruence_set_memI 
+      using "0" "1" "2" eval_at_point_closed assms not_nonzero_Qp ord_congruence_set_memI
       by metis
     show "x \<in> poly_map n [f]  \<inverse>\<^bsub>n\<^esub> ((\<lambda>a. [a]) ` ord_congruence_set m a)"
     proof-
       have 00: "poly_map n [f] x = [(Qp_ev f x)]"
         using "0" assms poly_map_singleton by blast
       have 01: "[eval_at_point Q\<^sub>p x f] \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>)"
         using "0" assms eval_at_point_closed Qp.to_R1_closed by blast
       hence 02: "poly_map n [f] x \<in> (\<lambda>a. [a]) ` ord_congruence_set m a"
-        using 3 "00" by blast      
+        using 3 "00" by blast
       then show "x \<in> poly_map n [f]  \<inverse>\<^bsub>n\<^esub> ((\<lambda>a. [a]) ` ord_congruence_set m a)"
-        using 0 unfolding evimage_def 
-        by blast        
+        using 0 unfolding evimage_def
+        by blast
     qed
   qed
   show "poly_map n [f] \<inverse>\<^bsub>n\<^esub> (\<lambda>a. [a]) ` ord_congruence_set m a
               \<subseteq> poly_cong_set n f m a"
   proof fix x
     assume A: "x \<in> poly_map n [f] \<inverse>\<^bsub>n\<^esub> ((\<lambda>a. [a]) ` (ord_congruence_set m a))"
     have 0: "((\<lambda>a. [a]) ` ord_congruence_set m a) \<subseteq> carrier (Q\<^sub>p\<^bsup>1\<^esup>)"
       using ord_congruence_set_closed Qp.to_R1_carrier by blast
     have "is_poly_tuple n [f]"
-      using assms unfolding is_poly_tuple_def 
+      using assms unfolding is_poly_tuple_def
       by (simp add: assms)
-    then have 1:"poly_map n [f] \<inverse>\<^bsub>n\<^esub>((\<lambda>a. [a]) ` ord_congruence_set m a) \<subseteq> carrier (Q\<^sub>p\<^bsup>n\<^esup>)" 
-      using 0 A assms One_nat_def 
+    then have 1:"poly_map n [f] \<inverse>\<^bsub>n\<^esub>((\<lambda>a. [a]) ` ord_congruence_set m a) \<subseteq> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
+      using 0 A assms One_nat_def
       by (metis extensional_vimage_closed)
    then have 2: "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
-     using A unfolding evimage_def by blast     
-    then have 3: "poly_map n [f] x \<in> ((\<lambda>a. [a]) ` ord_congruence_set m a)" 
-      using A assms 0 One_nat_def 
+     using A unfolding evimage_def by blast
+    then have 3: "poly_map n [f] x \<in> ((\<lambda>a. [a]) ` ord_congruence_set m a)"
+      using A assms 0 One_nat_def
       by blast
     have "poly_map n [f] x = [(Qp_ev f x)]"
       using "2" assms poly_map_singleton by blast
     then have "Qp_ev f x \<in> ord_congruence_set m a"
-      using 3 
+      using 3
       by (metis (mono_tags, lifting) image_iff list.inject)
     then show "x \<in> poly_cong_set n f m a"
-      unfolding poly_cong_set_def 
-      by (metis (mono_tags, lifting) "2" Qp.nonzero_memE(2) 
+      unfolding poly_cong_set_def
+      by (metis (mono_tags, lifting) "2" Qp.nonzero_memE(2)
           mem_Collect_eq ord_congruence_set_memE(1) ord_congruence_set_memE(2))
   qed
 qed
 
 lemma singleton_poly_tuple:
 "is_poly_tuple n [f] \<longleftrightarrow> f \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   unfolding is_poly_tuple_def
   by (metis (no_types, lifting) list.distinct(1) list.set_cases list.set_intros(1) set_ConsD subset_code(1))
 
 lemma poly_cong_set_is_semialgebraic:
   assumes "m \<ge> 0"
   assumes "f \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   shows "is_semialgebraic n (poly_cong_set n f m a)"
 proof-
   have 0: "(\<lambda>a. [a]) ` ord_congruence_set m a \<in> semialg_sets 1"
-  using assms 
-        ord_congruence_set_is_semialg[of m a] 
-  unfolding is_semialgebraic_def 
+  using assms
+        ord_congruence_set_is_semialg[of m a]
+  unfolding is_semialgebraic_def
   by blast
   have 1: "length [f] = 1"
   by simp
   hence " poly_map n [f] \<inverse>\<^bsub>n\<^esub> (\<lambda>a. [a]) ` ord_congruence_set m a \<in> semialg_sets n"
     using 0 singleton_poly_tuple[of n f] zero_neq_one assms
-    pullback_is_semialg[of n "[f]" 1 "(\<lambda>a. [a]) ` ord_congruence_set m a"]  
-  unfolding is_semialgebraic_def 
+    pullback_is_semialg[of n "[f]" 1 "(\<lambda>a. [a]) ` ord_congruence_set m a"]
+  unfolding is_semialgebraic_def
   by blast
-  thus ?thesis using assms poly_cong_set_as_pullback[of f n m a] 
-  unfolding is_semialgebraic_def 
+  thus ?thesis using assms poly_cong_set_as_pullback[of f n m a]
+  unfolding is_semialgebraic_def
   by presburger
-qed  
+qed
 
       (********************************************************************************************)
       (********************************************************************************************)
       subsubsection\<open>Congruence Sets for Angular Components\<close>
       (********************************************************************************************)
       (********************************************************************************************)
 
 
 text\<open>If a set is a union of \<open>n\<close>-th power residues, then it is semialgebraic.\<close>
 
 lemma pow_res_union_imp_semialg:
   assumes "n \<ge> 1"
   assumes "S \<subseteq> nonzero Q\<^sub>p"
   assumes "\<And>x. x \<in> S \<Longrightarrow> pow_res n x \<subseteq> S"
   shows "is_univ_semialgebraic S"
 proof-
   obtain F where F_def: "F = {T. T \<in> pow_res_classes n \<and> T \<subseteq> S}"
     by blast
   have 0: "F \<subseteq> pow_res_classes n"
     using F_def by blast
   have 1: "finite F"
-    using 0 pow_res_classes_finite[of n] assms(1) finite_subset 
-    by auto   
+    using 0 pow_res_classes_finite[of n] assms(1) finite_subset
+    by auto
   have 2: "are_univ_semialgebraic F"
-    using 0 
-    by (meson are_univ_semialgebraicE are_univ_semialgebraicI assms(1) 
+    using 0
+    by (meson are_univ_semialgebraicE are_univ_semialgebraicI assms(1)
         pow_res_classes_are_univ_semialgebraic padic_fields_axioms subsetD)
   have 3: "S = \<Union> F"
   proof
     show "S \<subseteq> \<Union> F"
     proof fix x
       assume A: "x \<in> S"
       then have "pow_res n x \<subseteq> S"
-        using assms(3) by blast   
+        using assms(3) by blast
       then have "pow_res n x \<in> F"
-        using A assms(2) F_def pow_res_classes_def 
+        using A assms(2) F_def pow_res_classes_def
         by (smt mem_Collect_eq subsetD)
       then have "pow_res n x \<subseteq> \<Union> F"
-        by blast        
+        by blast
       then show "x \<in> \<Union> F"
-        using A assms(1) assms(2) pow_res_refl[of x n] unfolding nonzero_def by blast         
-    qed      
+        using A assms(1) assms(2) pow_res_refl[of x n] unfolding nonzero_def by blast
+    qed
     show "\<Union> F \<subseteq> S"
-      using F_def 
-      by blast
-  qed
-  show ?thesis 
-    using 1 2 3 finite_union_is_univ_semialgebraic' 
-    by blast    
+      using F_def
+      by blast
+  qed
+  show ?thesis
+    using 1 2 3 finite_union_is_univ_semialgebraic'
+    by blast
 qed
 
 definition ac_cong_set1 where
 "ac_cong_set1 n y = {x \<in> carrier Q\<^sub>p. x \<noteq> \<zero> \<and> ac n x = ac n y}"
 
 lemma ac_cong_set1_is_univ_semialg:
   assumes "n > 0"
   assumes "b \<in> nonzero Q\<^sub>p"
   assumes "b \<in> \<O>\<^sub>p"
   shows "is_univ_semialgebraic (ac_cong_set1 n b)"
 proof(cases "n = 1 \<and> p = 2")
   case True
   have "(ac_cong_set1 n b) = nonzero Q\<^sub>p"
   proof
     have 0: "Units (Zp_res_ring n) = {1}"
     proof show "Units (Zp_res_ring n) \<subseteq> {1}"
       proof fix x assume A: "x \<in> Units (Zp_res_ring n)"
         have 0: "carrier (Zp_res_ring n) = {0..(int 2) - 1}"
-          using True 
-          by (metis assms(1) int_ops(3) p_residues power_one_right residues.res_carrier_eq)          
+          using True
+          by (metis assms(1) int_ops(3) p_residues power_one_right residues.res_carrier_eq)
         have 1: "carrier (Zp_res_ring n) = {0..(1::int)}"
         proof- have "int 2 - 1 = (1::int)"
                     by linarith
-                  then show ?thesis 
-                    using 0  
+                  then show ?thesis
+                    using 0
                     by presburger
         qed
         have 15: "{0..(1::int)} = {0, (1::int)}"
           using atLeastAtMostPlus1_int_conv [of 0 "0::int"]
           by (smt atLeastAtMost_singleton insert_commute)
         have 2: "carrier (Zp_res_ring n) = {0,(1::int)}"
-          using "1" "15" 
+          using "1" "15"
           by blast
         have 3: "0 \<notin> Units (Zp_res_ring n)"
-          using True zero_not_in_residue_units by blast  
-        have "x \<in> carrier (Zp_res_ring n)" 
-          using A unfolding Units_def by blast 
+          using True zero_not_in_residue_units by blast
+        have "x \<in> carrier (Zp_res_ring n)"
+          using A unfolding Units_def by blast
         then have "x = 1" using A  2 3
-          by (metis "1" atLeastAtMost_iff atLeastatMost_empty 
+          by (metis "1" atLeastAtMost_iff atLeastatMost_empty
               atLeastatMost_empty_iff2 linorder_neqE_linordered_idom mod_by_1 mod_pos_pos_trivial )
         then show "x \<in> {1}"
           by simp
       qed
       show "{1} \<subseteq> Units (Zp_res_ring n)"
-        by (meson assms(1) empty_subsetI insert_subset residue_1_unit(1))        
-    qed 
+        by (meson assms(1) empty_subsetI insert_subset residue_1_unit(1))
+    qed
     show "ac_cong_set1 n b \<subseteq> nonzero Q\<^sub>p"
       by (metis (mono_tags, lifting) ac_cong_set1_def mem_Collect_eq not_nonzero_Qp subsetI)
     show "nonzero Q\<^sub>p \<subseteq> ac_cong_set1 n b"
-    proof fix x 
+    proof fix x
       assume A: "x \<in> nonzero Q\<^sub>p"
       then have P0: "ac n x = 1"
         using 0 ac_units assms(1) by blast
       have P1: "ac n b = 1"
         using assms 0 ac_units assms(1) by blast
       then have "ac n x = ac n b"
-        using P0 by metis 
+        using P0 by metis
       then show " x \<in> ac_cong_set1 n b"
-        unfolding ac_cong_set1_def using A 
+        unfolding ac_cong_set1_def using A
       proof -
         have "x \<in> {r \<in> carrier Q\<^sub>p. r \<noteq> \<zero>}"
           by (metis (no_types) \<open>x \<in> nonzero Q\<^sub>p\<close> nonzero_def )
         then show "x \<in> {r \<in> carrier Q\<^sub>p. r \<noteq> \<zero> \<and> ac n r = ac n b}"
           using \<open>ac n x = ac n b\<close> by force
       qed
     qed
   qed
   then show "is_univ_semialgebraic (ac_cong_set1 n b)"
     by (simp add: nonzero_is_univ_semialgebraic)
 next
   case F: False
   have F0: "2 \<le> card (Units (Zp_res_ring n))"
   proof(cases "n = 1")
     case True
     then have "field (Zp_res_ring n)"
-       using p_res_ring_1_field by blast      
+       using p_res_ring_1_field by blast
     then have F00: "Units (Zp_res_ring n) = carrier (Zp_res_ring n) - {\<zero>\<^bsub>Zp_res_ring n\<^esub>}"
       using field.field_Units by blast
     have F01: "\<zero>\<^bsub>Zp_res_ring n\<^esub> \<in> carrier (Zp_res_ring n)"
       using assms(1) cring.cring_simprules(2) padic_integers.R_cring padic_integers_axioms by blast
     have F02: "card (carrier (Zp_res_ring n)) = p \<and> finite (carrier (Zp_res_ring n))"
-      by (smt F01 True nat_eq_iff2 p_res_ring_zero p_residue_ring_car_memE(1) power_one_right residue_ring_card)      
+      by (smt F01 True nat_eq_iff2 p_res_ring_zero p_residue_ring_car_memE(1) power_one_right residue_ring_card)
     have F03: "\<zero>\<^bsub>residue_ring (p ^ n)\<^esub> \<in> carrier (residue_ring (p ^ n)) "
       using F01 by blast
     have F04: "int (card (carrier (residue_ring (p ^ n)))) \<ge> int (card {\<zero>\<^bsub>residue_ring (p ^ n)\<^esub>}) "
-      by (smt F02 F03 nat_int of_nat_0_le_iff of_nat_1 of_nat_power p_res_ring_0 p_res_ring_zero 
+      by (smt F02 F03 nat_int of_nat_0_le_iff of_nat_1 of_nat_power p_res_ring_0 p_res_ring_zero
           p_residue_ring_car_memE(1) power_increasing power_one_right residue_ring_card)
     have "card (carrier (residue_ring (p ^ n))) - 1 = p - 1"
-      using F02 prime 
+      using F02 prime
       by (metis Totient.of_nat_eq_1_iff True less_imp_le_nat less_one nat_int nat_less_eq_zless
           of_nat_1 of_nat_diff of_nat_zero_less_power_iff p_residues pos_int_cases
           power_0 power_one_right residue_ring_card residues.m_gt_one zero_le_one)
     hence F05: "card (carrier (residue_ring (p ^ n)) - {\<zero>\<^bsub>residue_ring (p ^ n)\<^esub>}) = p - 1"
       using F02 F03 F04 card_Diff_singleton_if[of "(carrier (Zp_res_ring n))" "\<zero>\<^bsub>residue_ring (p^n)\<^esub>"]
-           True int_ops(6)[of "card (carrier (residue_ring (p ^ n)))" "card {\<zero>\<^bsub>residue_ring (p ^ n)\<^esub>}"] 
-           p_res_ring_zero p_residue_ring_car_memE(1) 
+           True int_ops(6)[of "card (carrier (residue_ring (p ^ n)))" "card {\<zero>\<^bsub>residue_ring (p ^ n)\<^esub>}"]
+           p_res_ring_zero p_residue_ring_car_memE(1)
       by (metis)
   hence F06: "card (Units (Zp_res_ring n)) = p -1"
       using True F02 F01 F00
-      by (metis p_res_ring_zero)        
+      by (metis p_res_ring_zero)
     have F04: "p - 1 \<ge>2 "
-      using F prime 
-      by (meson True linorder_cases not_less prime_ge_2_int zle_diff1_eq)     
-    then show ?thesis 
+      using F prime
+      by (meson True linorder_cases not_less prime_ge_2_int zle_diff1_eq)
+    then show ?thesis
       using F03 F06
-      by linarith      
+      by linarith
   next
     case False
-    then show ?thesis 
-      by (metis assms(1) less_imp_le_nat mod2_gr_0 mod_less nat_le_linear nat_neq_iff residue_units_card_geq_2)      
-  qed
-  show ?thesis 
+    then show ?thesis
+      by (metis assms(1) less_imp_le_nat mod2_gr_0 mod_less nat_le_linear nat_neq_iff residue_units_card_geq_2)
+  qed
+  show ?thesis
   apply(rule pow_res_union_imp_semialg[of "card (Units (Zp_res_ring n))"])
-  using F0 assms apply linarith  
+  using F0 assms apply linarith
   apply (metis (mono_tags, lifting) ac_cong_set1_def mem_Collect_eq not_nonzero_Qp subsetI)
 proof-
-  fix x 
-  assume AA: "x \<in> ac_cong_set1 n b" 
+  fix x
+  assume AA: "x \<in> ac_cong_set1 n b"
   show "pow_res (card (Units (Zp_res_ring n))) x \<subseteq> ac_cong_set1 n b"
   proof
     fix y
     assume A: "y \<in> pow_res (card (Units (Zp_res_ring n))) x"
     show "y \<in> ac_cong_set1 n b"
     proof-
       obtain k where k_def: "k = card (Units (Zp_res_ring n))"
         by blast
       have "k \<ge>2"
-        using assms k_def  F F0 by blast          
+        using assms k_def  F F0 by blast
       then obtain a where a_def: "a \<in> nonzero Q\<^sub>p \<and> y = x \<otimes> (a[^]k)"
-        using  k_def A pow_res_def[of k x] 
-        by blast 
+        using  k_def A pow_res_def[of k x]
+        by blast
       have 0: "x \<in> nonzero Q\<^sub>p"
         using AA ac_cong_set1_def
         by (metis (mono_tags, lifting) mem_Collect_eq not_nonzero_Qp)
       have 1: "y \<in> nonzero Q\<^sub>p"
-        by (metis "0" Qp.Units_m_closed Qp_nat_pow_nonzero Units_eq_nonzero \<open>\<And>thesis. (\<And>a. a \<in> nonzero Q\<^sub>p \<and> y = x \<otimes> a [^] k \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>)        
+        by (metis "0" Qp.Units_m_closed Qp_nat_pow_nonzero Units_eq_nonzero \<open>\<And>thesis. (\<And>a. a \<in> nonzero Q\<^sub>p \<and> y = x \<otimes> a [^] k \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>)
       have "ac n y = ac n x \<otimes>\<^bsub>Zp_res_ring n\<^esub> ac n (a[^]k)"
-        using a_def 0 1 Qp_nat_pow_nonzero ac_mult' 
+        using a_def 0 1 Qp_nat_pow_nonzero ac_mult'
         by blast
       then have 2: "ac n y = ac n x \<otimes>\<^bsub>Zp_res_ring n\<^esub> (ac n a)[^]\<^bsub>Zp_res_ring n\<^esub> k"
       proof-
         have "ac n (a[^]k) = ac n a [^]\<^bsub>Zp_res_ring n\<^esub> k"
-          using a_def assms(1) ac_nat_pow'[of a n k]         
+          using a_def assms(1) ac_nat_pow'[of a n k]
           by linarith
-        then show ?thesis 
-          using \<open>ac n y = ac n x \<otimes>\<^bsub>Zp_res_ring n\<^esub> ac n (a[^]k)\<close> 
+        then show ?thesis
+          using \<open>ac n y = ac n x \<otimes>\<^bsub>Zp_res_ring n\<^esub> ac n (a[^]k)\<close>
           by presburger
-      qed  
+      qed
       then have "ac n y = ac n x"
       proof-
         have "(ac n a) \<in> Units (Zp_res_ring n)"
           by (metis (mono_tags, opaque_lifting) a_def ac_units assms(1) )
         then have "(ac n a)^k mod (p^n) = 1"
           using k_def a_def ac_nat_pow ac_nat_pow' assms(1) residue_units_nilpotent
           using neq0_conv by presburger
         then have 00: "(ac n a)[^]\<^bsub>Zp_res_ring n\<^esub> k = 1"
-          by (metis a_def ac_nat_pow ac_nat_pow' mod_by_1 power_0 
+          by (metis a_def ac_nat_pow ac_nat_pow' mod_by_1 power_0
               zero_neq_one)
         have "ac n x \<otimes>\<^bsub>residue_ring (p ^ n)\<^esub> ac n a [^]\<^bsub>residue_ring (p ^ n)\<^esub> k = ac n x \<otimes>\<^bsub>Zp_res_ring n\<^esub> \<one>\<^bsub>Zp_res_ring n\<^esub>"
           using 00  assms(1) p_res_ring_one by presburger
         hence "ac n x \<otimes>\<^bsub>residue_ring (p ^ n)\<^esub> ac n a [^]\<^bsub>residue_ring (p ^ n)\<^esub> k = ac n x"
-          by (metis "0" Qp.nonzero_closed Qp.one_nonzero Qp.r_one ac_mult' ac_one' assms(1)) 
-        then show ?thesis 
-          using 2 "0" 00 
-          by linarith      
+          by (metis "0" Qp.nonzero_closed Qp.one_nonzero Qp.r_one ac_mult' ac_one' assms(1))
+        then show ?thesis
+          using 2 "0" 00
+          by linarith
       qed
-      then show ?thesis 
-        using  "1" AA nonzero_def 
-            ac_cong_set1_def[of n b] mem_Collect_eq 
-        by smt            
+      then show ?thesis
+        using  "1" AA nonzero_def
+            ac_cong_set1_def[of n b] mem_Collect_eq
+        by smt
     qed
   qed
 qed
 qed
 
 definition ac_cong_set where
 "ac_cong_set n k = {x \<in> carrier Q\<^sub>p. x \<noteq> \<zero> \<and> ac n x = k}"
 
 lemma ac_cong_set_is_univ_semialg:
   assumes "n >0 "
   assumes "k \<in> Units (Zp_res_ring n)"
   shows "is_univ_semialgebraic (ac_cong_set n k)"
 proof-
   have "k \<in> carrier (Zp_res_ring n)"
-    using assms(2) Units_def[of "Zp_res_ring n"] 
-    by blast  
+    using assms(2) Units_def[of "Zp_res_ring n"]
+    by blast
   then have k_n: "([k]\<cdot>\<^bsub>Z\<^sub>p\<^esub>\<one>\<^bsub>Z\<^sub>p\<^esub>) n = k"
     using assms
-    by (metis Zp_int_inc_res mod_pos_pos_trivial p_residue_ring_car_memE(1) p_residue_ring_car_memE(2))            
+    by (metis Zp_int_inc_res mod_pos_pos_trivial p_residue_ring_car_memE(1) p_residue_ring_car_memE(2))
   obtain b where b_def: "b = \<iota> ([k]\<cdot>\<^bsub>Z\<^sub>p\<^esub>\<one>\<^bsub>Z\<^sub>p\<^esub>)"
     by blast
   have 0: "k mod p \<noteq> 0"
-    using assms residue_UnitsE[of n k] 
-    by (metis le_eq_less_or_eq le_refl less_one nat_le_linear p_residues power_0 
+    using assms residue_UnitsE[of n k]
+    by (metis le_eq_less_or_eq le_refl less_one nat_le_linear p_residues power_0
         power_one_right residues.mod_in_res_units residues_def zero_less_one
-        zero_neq_one zero_not_in_residue_units zero_power)     
+        zero_neq_one zero_not_in_residue_units zero_power)
   then have "val_Zp ([k]\<cdot>\<^bsub>Z\<^sub>p\<^esub>\<one>\<^bsub>Z\<^sub>p\<^esub>) = 0"
     using val_Zp_p_int_unit by blast
   then have 1: "val b = 0"
     by (metis Zp_int_inc_closed b_def val_of_inc)
   have 2: "b \<in> \<O>\<^sub>p"
-    using b_def Zp_int_mult_closed 
-    by blast    
-  have "ord_Zp ([k] \<cdot>\<^bsub>Z\<^sub>p\<^esub> \<one>\<^bsub>Z\<^sub>p\<^esub>) = 0" 
+    using b_def Zp_int_mult_closed
+    by blast
+  have "ord_Zp ([k] \<cdot>\<^bsub>Z\<^sub>p\<^esub> \<one>\<^bsub>Z\<^sub>p\<^esub>) = 0"
     using 0 ord_Zp_p_int_unit by blast
   have "ac_Zp ([k]\<cdot>\<^bsub>Z\<^sub>p\<^esub>\<one>\<^bsub>Z\<^sub>p\<^esub>) = ([k]\<cdot>\<^bsub>Z\<^sub>p\<^esub>\<one>\<^bsub>Z\<^sub>p\<^esub>)"
-    using "0" Zp_int_inc_closed ac_Zp_of_Unit ord_Zp_p_int_unit \<open>val_Zp ([k] \<cdot>\<^bsub>Z\<^sub>p\<^esub> \<one>\<^bsub>Z\<^sub>p\<^esub>) = 0\<close> 
-    by blast    
+    using "0" Zp_int_inc_closed ac_Zp_of_Unit ord_Zp_p_int_unit \<open>val_Zp ([k] \<cdot>\<^bsub>Z\<^sub>p\<^esub> \<one>\<^bsub>Z\<^sub>p\<^esub>) = 0\<close>
+    by blast
   then have "(angular_component b) = ([k]\<cdot>\<^bsub>Z\<^sub>p\<^esub>\<one>\<^bsub>Z\<^sub>p\<^esub>)"
     using b_def 1 2 angular_component_ord_zero[of b]
-    by (metis Qp.int_inc_zero Qp.one_closed val_ring_memE Zp.int_inc_zero Zp.one_closed 
-        Zp.one_nonzero Zp_int_inc_closed angular_component_of_inclusion inc_closed inc_of_int 
-        inc_of_one inc_to_Zp local.val_zero not_nonzero_Qp val_ineq val_one zero_in_val_ring)    
+    by (metis Qp.int_inc_zero Qp.one_closed val_ring_memE Zp.int_inc_zero Zp.one_closed
+        Zp.one_nonzero Zp_int_inc_closed angular_component_of_inclusion inc_closed inc_of_int
+        inc_of_one inc_to_Zp local.val_zero not_nonzero_Qp val_ineq val_one zero_in_val_ring)
   then have "ac n b = k"
-    using ac_def[of n b] k_n 
-    by (metis Qp_char_0_int Zp_defs(1) ac_def b_def inc_of_int inc_of_one)    
+    using ac_def[of n b] k_n
+    by (metis Qp_char_0_int Zp_defs(1) ac_def b_def inc_of_int inc_of_one)
   then have 3: "(ac_cong_set n k) = (ac_cong_set1 n b)"
-    unfolding ac_cong_set_def ac_cong_set1_def 
+    unfolding ac_cong_set_def ac_cong_set1_def
     by meson
   have 4: "b \<in> nonzero Q\<^sub>p"
-    using 1 2 val_nonzero 
-    by (metis Qp.one_closed val_ring_memE Zp_def \<iota>_def local.one_neq_zero 
+    using 1 2 val_nonzero
+    by (metis Qp.one_closed val_ring_memE Zp_def \<iota>_def local.one_neq_zero
         not_nonzero_Qp padic_fields.val_ring_memE padic_fields_axioms val_ineq val_one)
-  then show ?thesis 
+  then show ?thesis
     using 1 2 3 assms ac_cong_set1_is_univ_semialg[of n b] val_nonzero[of b 1]
-    by presburger    
+    by presburger
 qed
 
 definition val_ring_constant_ac_set where
 "val_ring_constant_ac_set n k = {a \<in> \<O>\<^sub>p. val a = 0 \<and> ac n a = k}"
 
 lemma val_nonzero':
   assumes "a \<in> carrier Q\<^sub>p"
   assumes "val a = eint k"
   shows "a \<in> nonzero Q\<^sub>p"
   using val_nonzero[of a "k + 1"]
   by (metis Suc_ile_eq assms(1) assms(2) eint_ord_code(3) val_nonzero)
 
 lemma val_ord':
   assumes "a \<in> carrier Q\<^sub>p"
   assumes "a \<noteq>\<zero>"
   shows "val a = ord a"
   by (meson assms(1) assms(2) not_nonzero_Qp val_ord)
 
 lemma val_ring_constant_ac_set_is_univ_semialgebraic:
   assumes "n > 0"
   assumes "k \<noteq> 0"
-  shows "is_univ_semialgebraic (val_ring_constant_ac_set n k)" 
+  shows "is_univ_semialgebraic (val_ring_constant_ac_set n k)"
 proof(cases "val_ring_constant_ac_set n k = {}")
   case True
-  then show ?thesis 
+  then show ?thesis
     by (metis equals0D order_refl pow_res_union_imp_semialg subsetI)
 next
   case False
   then obtain b where b_def: "b \<in> val_ring_constant_ac_set n k"
     by blast
   have 0: "val_ring_constant_ac_set n k = q_ball n k 0 \<zero>"
   proof
     show "val_ring_constant_ac_set n k \<subseteq> q_ball n k 0 \<zero>"
-    proof fix x assume A: "x \<in> val_ring_constant_ac_set n k" then 
+    proof fix x assume A: "x \<in> val_ring_constant_ac_set n k" then
       show "x \<in> q_ball n k 0 \<zero>"
-      proof-         
+      proof-
         have 0: "x \<in> \<O>\<^sub>p \<and> val x = 0 \<and> ac n x = k"
-          using A 
-          unfolding val_ring_constant_ac_set_def 
+          using A
+          unfolding val_ring_constant_ac_set_def
           by blast
         then have x_car: "x \<in> carrier Q\<^sub>p"
-          using val_ring_memE 
+          using val_ring_memE
           by blast
         then have 00: "x = x \<ominus> \<zero>"
-          using Qp.ring_simprules by metis 
+          using Qp.ring_simprules by metis
         then have 1: "ac n (x \<ominus>\<^bsub>Q\<^sub>p\<^esub> \<zero>) = k"
-          using 0 
+          using 0
           by presburger
-        have 2: "val (x \<ominus>\<^bsub>Q\<^sub>p\<^esub> \<zero>) = 0" 
-          using 0 00 
-          by metis  
-        have 3: "x \<in> nonzero Q\<^sub>p" 
+        have 2: "val (x \<ominus>\<^bsub>Q\<^sub>p\<^esub> \<zero>) = 0"
+          using 0 00
+          by metis
+        have 3: "x \<in> nonzero Q\<^sub>p"
         proof(rule ccontr)
           assume " x \<notin> nonzero Q\<^sub>p "
           then have "x = \<zero>"
-            using Qp.nonzero_memI x_car by blast            
-          then show False 
-            using 0 val_zero 
-            by (metis ac_def assms(2))            
+            using Qp.nonzero_memI x_car by blast
+          then show False
+            using 0 val_zero
+            by (metis ac_def assms(2))
         qed
         have 4: "ord (x \<ominus>\<^bsub>Q\<^sub>p\<^esub> \<zero>) = 0"
         proof(rule ccontr)
           assume "ord (x \<ominus> \<zero>) \<noteq> 0"
           then have "val (x \<ominus> \<zero>) \<noteq> 0"
-            by (metis "00" "3" Qp.one_closed equal_val_imp_equal_ord(1) ord_one val_one)  
-          then show False 
-            using "2" 
+            by (metis "00" "3" Qp.one_closed equal_val_imp_equal_ord(1) ord_one val_one)
+          then show False
+            using "2"
             by blast
         qed
-        show ?thesis 
-          using 0 1 4 
-          unfolding  q_ball_def 
+        show ?thesis
+          using 0 1 4
+          unfolding  q_ball_def
           using x_car by blast
       qed
     qed
     show "q_ball n k 0 \<zero> \<subseteq> val_ring_constant_ac_set n k"
     proof fix x
       assume A: "x \<in> q_ball n k 0 \<zero>"
       then have 0: "ac n (x \<ominus>\<^bsub>Q\<^sub>p\<^esub> \<zero>) = k"
         using q_ballE'(1) by blast
       have 1: "ord (x \<ominus>\<^bsub>Q\<^sub>p\<^esub> \<zero>) = 0"
-        using q_ball_def A 
+        using q_ball_def A
         by blast
       have 2: "x \<in> carrier Q\<^sub>p"
-        using A q_ball_def by blast 
+        using A q_ball_def by blast
       have 3: "ord x = 0"
-        using 2 1 ring.ring_simprules[of Q\<^sub>p]  
-        by (metis Qp.ring_axioms)        
+        using 2 1 ring.ring_simprules[of Q\<^sub>p]
+        by (metis Qp.ring_axioms)
       have 4: "ac n x = k"
-        using 0 2 1  cring.axioms(1)[of Q\<^sub>p] ring.ring_simprules[of Q\<^sub>p] 
-         by (metis Qp.ring_axioms)         
-       have 5: "x \<in> \<O>\<^sub>p" 
-         using Qp_val_ringI[of x] 2 3 val_ord val_nonzero' 
+        using 0 2 1  cring.axioms(1)[of Q\<^sub>p] ring.ring_simprules[of Q\<^sub>p]
+         by (metis Qp.ring_axioms)
+       have 5: "x \<in> \<O>\<^sub>p"
+         using Qp_val_ringI[of x] 2 3 val_ord val_nonzero'
          by (metis Qp.integral_iff val_ring_memE Zp.nonzero_closed angular_component_closed
              angular_component_ord_zero image_eqI local.numer_denom_facts(1) local.numer_denom_facts(2)
-             local.numer_denom_facts(4) not_nonzero_Qp)                  
+             local.numer_denom_facts(4) not_nonzero_Qp)
        have 6: "x \<noteq> \<zero>"
-         using 4 assms ac_def[of n x] 
+         using 4 assms ac_def[of n x]
          by meson
        have 7: "val x = 0"
-         using 6 3 2 assms val_ord' zero_eint_def by presburger         
-       show " x \<in> val_ring_constant_ac_set n k" 
+         using 6 3 2 assms val_ord' zero_eint_def by presburger
+       show " x \<in> val_ring_constant_ac_set n k"
          unfolding val_ring_constant_ac_set_def
-         using 7 6 5 4  
-         by blast     
+         using 7 6 5 4
+         by blast
     qed
   qed
   obtain b where b_def: "b \<in> q_ball n k (0::int) \<zero>"
-    using "0" b_def by blast 
+    using "0" b_def by blast
   have 1: "b \<in> carrier Q\<^sub>p \<and> ac n b = k"
-    using b_def unfolding q_ball_def 
+    using b_def unfolding q_ball_def
     by (metis (mono_tags, lifting) "0" b_def mem_Collect_eq val_ring_constant_ac_set_def)
   then have 2: "b \<in> nonzero Q\<^sub>p"
-    using 1 assms 
-    by (metis ac_def not_nonzero_Qp)    
+    using 1 assms
+    by (metis ac_def not_nonzero_Qp)
   have "q_ball n k 0 \<zero> = B\<^bsub>0 + int n\<^esub>[b]"
-    using  1 b_def nonzero_def [of Q\<^sub>p] assms 0 2 c_ball_q_ball[of b n k "\<zero>" b 0] 
+    using  1 b_def nonzero_def [of Q\<^sub>p] assms 0 2 c_ball_q_ball[of b n k "\<zero>" b 0]
     by (meson Qp.cring_axioms cring.cring_simprules(2))
-  then have "is_univ_semialgebraic (q_ball n k (0::int) \<zero>) "   
-    using 1 ball_is_univ_semialgebraic[of b "0 + int n"] 
-    by  metis 
+  then have "is_univ_semialgebraic (q_ball n k (0::int) \<zero>) "
+    using 1 ball_is_univ_semialgebraic[of b "0 + int n"]
+    by  metis
   then show ?thesis
-    using 0 by presburger    
+    using 0 by presburger
 qed
 
 definition val_ring_constant_ac_sets where
 "val_ring_constant_ac_sets n = val_ring_constant_ac_set n ` (Units (Zp_res_ring n))"
 
 lemma val_ring_constant_ac_sets_are_univ_semialgebraic:
   assumes "n > 0"
   shows "are_univ_semialgebraic (val_ring_constant_ac_sets n)"
 proof(rule are_univ_semialgebraicI)
-  have 0: "\<not> coprime 0 p"  
-    using coprime_0_right_iff[of p] coprime_commute[of p 0] coprime_int_iff[of "nat p" 0] 
+  have 0: "\<not> coprime 0 p"
+    using coprime_0_right_iff[of p] coprime_commute[of p 0] coprime_int_iff[of "nat p" 0]
         nat_dvd_1_iff_1 prime_gt_1_nat zdvd1_eq
     by (metis not_prime_unit prime)
   have "(0::int) \<notin>(Units (Zp_res_ring n))"
     apply(rule ccontr)
-    using 0 assms residues.cring[of "p ^ n"] unfolding residues_def 
-      by (smt less_one not_gr_zero power_le_imp_le_exp power_less_imp_less_exp residue_UnitsE)      
-  fix x     
+    using 0 assms residues.cring[of "p ^ n"] unfolding residues_def
+      by (smt less_one not_gr_zero power_le_imp_le_exp power_less_imp_less_exp residue_UnitsE)
+  fix x
   assume A: "x \<in> val_ring_constant_ac_sets n"
   then obtain k where k_def: "x = val_ring_constant_ac_set n k \<and> k \<in> Units (Zp_res_ring n)"
     by (metis image_iff val_ring_constant_ac_sets_def)
   then show "is_univ_semialgebraic x"
-    using assms 
+    using assms
     by (metis \<open>0 \<notin> Units (Zp_res_ring n)\<close> val_ring_constant_ac_set_is_univ_semialgebraic)
-qed            
+qed
 
 definition ac_cong_set3 where
 "ac_cong_set3 n  = {as. \<exists> a b. a \<in> nonzero Q\<^sub>p \<and> b \<in> \<O>\<^sub>p \<and> val b = 0 \<and> (ac n a = ac n b) \<and> as = [a, b] }"
 
 definition ac_cong_set2 where
 "ac_cong_set2 n k = {as. \<exists> a b. a \<in> nonzero Q\<^sub>p \<and> b \<in> \<O>\<^sub>p \<and> val b = 0 \<and> (ac n a = k) \<and> (ac n b) = k \<and> as = [a, b] }"
 
 lemma ac_cong_set2_cartesian_product:
   assumes "k \<in> Units (Zp_res_ring n)"
   assumes "n > 0"
   shows "ac_cong_set2 n k = cartesian_product (to_R1` (ac_cong_set n k)) (to_R1` (val_ring_constant_ac_set n k))"
 proof
   show "ac_cong_set2 n k \<subseteq> cartesian_product ((\<lambda>a. [a]) ` ac_cong_set n k) ((\<lambda>a. [a]) ` val_ring_constant_ac_set n k)"
   proof fix x
     assume A: "x \<in> ac_cong_set2 n k"
     show "x \<in> (cartesian_product ((\<lambda>a. [a]) ` ac_cong_set n k) ((\<lambda>a. [a]) ` val_ring_constant_ac_set n k))"
       unfolding ac_cong_set_def val_ring_constant_ac_set_def ac_cong_set2_def
       apply(rule cartesian_product_memI[of _ Q\<^sub>p 1 _ 1])
        apply (metis (mono_tags, lifting) mem_Collect_eq subsetI Qp.to_R1_car_subset)
         apply (metis (no_types, lifting) val_ring_memE mem_Collect_eq subsetI Qp.to_R1_car_subset)
      proof-
-       obtain a b where ab_def: "x = [a,b] \<and> a \<in> nonzero Q\<^sub>p \<and> b \<in> \<O>\<^sub>p \<and> val b = 0 \<and> (ac n a = k) \<and> (ac n b) = k" 
+       obtain a b where ab_def: "x = [a,b] \<and> a \<in> nonzero Q\<^sub>p \<and> b \<in> \<O>\<^sub>p \<and> val b = 0 \<and> (ac n a = k) \<and> (ac n b) = k"
          using A
          unfolding ac_cong_set_def val_ring_constant_ac_set_def ac_cong_set2_def
          by blast
        have 0: "take 1 x = [a]"
          by (simp add: ab_def)
        have 1: "drop 1 x = [b]"
          by (simp add: ab_def)
        have 2: "a \<in> {x \<in> carrier Q\<^sub>p. x \<noteq> \<zero> \<and> ac n x = k}"
-         using ab_def nonzero_def 
-         by (smt mem_Collect_eq)    
-       have 3: "b \<in>  {a \<in> \<O>\<^sub>p. val a = 0 \<and> ac n a = k}" 
-         using ab_def 
+         using ab_def nonzero_def
+         by (smt mem_Collect_eq)
+       have 3: "b \<in>  {a \<in> \<O>\<^sub>p. val a = 0 \<and> ac n a = k}"
+         using ab_def
          by blast
        show "take 1 x \<in> (\<lambda>a. [a]) ` {x \<in> carrier Q\<^sub>p. x \<noteq> \<zero> \<and> ac n x = k}"
-         using  0 2 by blast  
-       show "drop 1 x \<in> (\<lambda>a. [a]) ` {a \<in> \<O>\<^sub>p. val a = 0 \<and> ac n a = k}" 
-         using 1 3 by blast 
+         using  0 2 by blast
+       show "drop 1 x \<in> (\<lambda>a. [a]) ` {a \<in> \<O>\<^sub>p. val a = 0 \<and> ac n a = k}"
+         using 1 3 by blast
      qed
   qed
   show "cartesian_product ((\<lambda>a. [a]) ` ac_cong_set n k) ((\<lambda>a. [a]) ` val_ring_constant_ac_set n k) \<subseteq> ac_cong_set2 n k"
   proof fix x
     have 0: "(\<lambda>a. [a]) ` ac_cong_set n k \<subseteq> carrier (Q\<^sub>p\<^bsup>1\<^esup>)"
-      using assms 
+      using assms
       by (metis (no_types, lifting) ac_cong_set_def mem_Collect_eq subsetI Qp.to_R1_car_subset)
     have 1: "((\<lambda>a. [a]) ` val_ring_constant_ac_set n k) \<subseteq> carrier (Q\<^sub>p\<^bsup>1\<^esup>)"
       by (smt val_ring_memE mem_Collect_eq subsetI Qp.to_R1_carrier Qp.to_R1_subset val_ring_constant_ac_set_def)
     assume A: "x \<in> cartesian_product ((\<lambda>a. [a]) ` ac_cong_set n k) ((\<lambda>a. [a]) ` val_ring_constant_ac_set n k)"
     then have "length x = 2"
-      using 0 1 A cartesian_product_closed[of "((\<lambda>a. [a]) ` ac_cong_set n k)" Q\<^sub>p 1 "((\<lambda>a. [a]) ` val_ring_constant_ac_set n k)" 1] 
+      using 0 1 A cartesian_product_closed[of "((\<lambda>a. [a]) ` ac_cong_set n k)" Q\<^sub>p 1 "((\<lambda>a. [a]) ` val_ring_constant_ac_set n k)" 1]
       by (metis (no_types, lifting) cartesian_power_car_memE one_add_one subset_iff)
     then obtain a b where ab_def: "take 1 x = [a] \<and> drop 1 x = [b]"
       by (metis One_nat_def add_diff_cancel_left' drop0 drop_Cons_numeral numerals(1) pair_id plus_1_eq_Suc take0 take_Cons_numeral)
     have 2: " a \<in> (ac_cong_set n k) \<and> b \<in> (val_ring_constant_ac_set n k)"
     proof-
       have P0: "take 1 x \<in> (\<lambda>a. [a]) ` ac_cong_set n k"
-        using 0 A cartesian_product_memE[of x "((\<lambda>a. [a]) ` ac_cong_set n k) " " ((\<lambda>a. [a]) ` val_ring_constant_ac_set n k)" Q\<^sub>p 1] 
+        using 0 A cartesian_product_memE[of x "((\<lambda>a. [a]) ` ac_cong_set n k) " " ((\<lambda>a. [a]) ` val_ring_constant_ac_set n k)" Q\<^sub>p 1]
         by blast
       have P1: "drop 1 x \<in> (\<lambda>a. [a]) ` val_ring_constant_ac_set n k"
-        using 0 A cartesian_product_memE[of x "((\<lambda>a. [a]) ` ac_cong_set n k) " " ((\<lambda>a. [a]) ` val_ring_constant_ac_set n k)" Q\<^sub>p 1] 
-        by blast 
+        using 0 A cartesian_product_memE[of x "((\<lambda>a. [a]) ` ac_cong_set n k) " " ((\<lambda>a. [a]) ` val_ring_constant_ac_set n k)" Q\<^sub>p 1]
+        by blast
       have P2: "[a] \<in> (\<lambda>a. [a]) ` ac_cong_set n k"
-        using P0 ab_def 
-        by metis 
+        using P0 ab_def
+        by metis
       have P3: "[b] \<in> (\<lambda>a. [a]) ` val_ring_constant_ac_set n k"
         using P1 ab_def by metis
-      show ?thesis 
-        using P2 P3 
-        by blast    
+      show ?thesis
+        using P2 P3
+        by blast
     qed
     have 3: "a \<in> nonzero Q\<^sub>p"
-      using 2 assms nonzero_def [of Q\<^sub>p]  ac_cong_set_def[of n k]   
+      using 2 assms nonzero_def [of Q\<^sub>p]  ac_cong_set_def[of n k]
       by blast
     have 4: "x = [a,b]"
       by (metis (no_types, lifting) \<open>length x = 2\<close> ab_def less_numeral_extra(1) nth_Cons_0 nth_take nth_via_drop pair_id)
     then have "\<exists>a b. a \<in> nonzero Q\<^sub>p \<and> b \<in> \<O>\<^sub>p \<and> val b = 0 \<and> ac n a = k \<and> ac n b = k \<and> x = [a, b]"
-      using 2 3 ab_def  unfolding val_ring_constant_ac_set_def ac_cong_set_def  
+      using 2 3 ab_def  unfolding val_ring_constant_ac_set_def ac_cong_set_def
       by blast
     then show "x \<in> ac_cong_set2 n k"
       unfolding ac_cong_set2_def val_ring_constant_ac_set_def ac_cong_set_def
       by blast
   qed
 qed
 
 lemma ac_cong_set2_is_semialg:
   assumes "k \<in> Units (Zp_res_ring n)"
   assumes "n > 0"
-  shows "is_semialgebraic 2 (ac_cong_set2 n k)" 
-  using  ac_cong_set_is_univ_semialg ac_cong_set2_cartesian_product[of k n] 
+  shows "is_semialgebraic 2 (ac_cong_set2 n k)"
+  using  ac_cong_set_is_univ_semialg ac_cong_set2_cartesian_product[of k n]
       cartesian_product_is_semialgebraic[of 1 "((\<lambda>a. [a]) ` ac_cong_set n k)" 1 " ((\<lambda>a. [a]) ` val_ring_constant_ac_set n k)"]
-  by (metis assms(1) assms(2) is_univ_semialgebraicE less_one less_or_eq_imp_le nat_neq_iff 
+  by (metis assms(1) assms(2) is_univ_semialgebraicE less_one less_or_eq_imp_le nat_neq_iff
       one_add_one val_ring_constant_ac_set_is_univ_semialgebraic zero_not_in_residue_units)
 
 lemma ac_cong_set3_as_union:
   assumes "n > 0"
   shows "ac_cong_set3 n = \<Union> (ac_cong_set2 n ` (Units (Zp_res_ring n)) )"
-proof 
+proof
   show "ac_cong_set3 n \<subseteq> \<Union> (ac_cong_set2 n ` Units (Zp_res_ring n))"
   proof fix x assume A: "x \<in> ac_cong_set3 n"
     then have 0: "x \<in> (ac_cong_set2 n (ac n (x!0)))"
       unfolding ac_cong_set2_def ac_cong_set3_def
       by (smt mem_Collect_eq nth_Cons_0)
     have 1: "(ac n (x!0)) \<in> Units (Zp_res_ring n)"
-      using A unfolding ac_cong_set3_def 
+      using A unfolding ac_cong_set3_def
       by (smt ac_units assms mem_Collect_eq nth_Cons_0)
     then show "x \<in> \<Union> (ac_cong_set2 n ` Units (Zp_res_ring n))"
-      using 0 
+      using 0
       by blast
   qed
   show "\<Union> (ac_cong_set2 n ` Units (Zp_res_ring n)) \<subseteq> ac_cong_set3 n"
   proof fix x assume A: "x \<in> \<Union> (ac_cong_set2 n ` Units (Zp_res_ring n))"
     obtain k where k_def: "x \<in> (ac_cong_set2 n k) \<and> k \<in> (Units (Zp_res_ring n))"
       using A by blast
     have 0: "k mod p \<noteq> 0"
-      using k_def One_nat_def  Suc_le_eq assms less_numeral_extra(1) 
-          power_one_right residues.m_gt_one residues.mod_in_res_units 
+      using k_def One_nat_def  Suc_le_eq assms less_numeral_extra(1)
+          power_one_right residues.m_gt_one residues.mod_in_res_units
       by (metis p_residues residue_UnitsE zero_not_in_residue_units)
     obtain b where b_def: "b = ([k]\<cdot>\<^bsub>Z\<^sub>p\<^esub>\<one>\<^bsub>Z\<^sub>p\<^esub>)"
       by blast
     have "k \<noteq>0"
-      using 0 mod_0 
+      using 0 mod_0
       by blast
     then have 1: "b \<in> nonzero Z\<^sub>p"
-      using 0 b_def int_unit 
-      by (metis Zp.Units_nonzero Zp.zero_not_one)      
-    have 10: "ord_Zp b = 0" using 0 1 
+      using 0 b_def int_unit
+      by (metis Zp.Units_nonzero Zp.zero_not_one)
+    have 10: "ord_Zp b = 0" using 0 1
       using b_def ord_Zp_p_int_unit by blast
     have 2: "\<iota> b \<in> nonzero Q\<^sub>p" using k_def
       using "1" inc_of_nonzero by blast
     have 3: "angular_component (\<iota> b) = ac_Zp b"
-      using "1" angular_component_of_inclusion 
+      using "1" angular_component_of_inclusion
       by blast
     have 4: "ac_Zp b = b"
-      using 1 10 
-      by (metis "3" Zp.r_one ac_Zp_factors' angular_component_closed inc_of_nonzero int_pow_0 mult_comm ord_Zp_def)      
+      using 1 10
+      by (metis "3" Zp.r_one ac_Zp_factors' angular_component_closed inc_of_nonzero int_pow_0 mult_comm ord_Zp_def)
     have 5: "ac_Zp b n = k"
     proof-
       have "k \<in> carrier (Zp_res_ring n)"
-        using k_def unfolding Units_def by blast 
-      then show ?thesis 
+        using k_def unfolding Units_def by blast
+      then show ?thesis
         using b_def k_def 4 Zp_int_inc_res  mod_pos_pos_trivial
         by (metis p_residue_ring_car_memE(1) p_residue_ring_car_memE(2))
     qed
     then have "ac n (\<iota> b) = k"
-      using 10 1 2 3 4 unfolding ac_def  
-      using Qp.not_nonzero_memI by metis       
-    then show "x \<in> ac_cong_set3 n" 
-      unfolding ac_cong_set3_def 
-      using k_def unfolding ac_cong_set2_def 
+      using 10 1 2 3 4 unfolding ac_def
+      using Qp.not_nonzero_memI by metis
+    then show "x \<in> ac_cong_set3 n"
+      unfolding ac_cong_set3_def
+      using k_def unfolding ac_cong_set2_def
       by (smt mem_Collect_eq)
   qed
 qed
-        
+
 lemma ac_cong_set3_is_semialgebraic:
   assumes "n > 0"
   shows "is_semialgebraic 2 (ac_cong_set3 n)"
 proof-
   have 0: "finite (ac_cong_set2 n ` (Units (Zp_res_ring n)) )"
-    using assms residues.finite_Units[of "p^n"] unfolding residues_def 
-    using p_residues residues.finite_Units by blast    
+    using assms residues.finite_Units[of "p^n"] unfolding residues_def
+    using p_residues residues.finite_Units by blast
   have 1: "are_semialgebraic 2 (ac_cong_set2 n ` (Units (Zp_res_ring n)) )"
-    apply(rule are_semialgebraicI) 
+    apply(rule are_semialgebraicI)
     using ac_cong_set2_is_semialg assms by blast
-  show ?thesis 
+  show ?thesis
     using 0 1 ac_cong_set3_as_union
     by (metis (no_types, lifting) are_semialgebraicE assms finite_union_is_semialgebraic' is_semialgebraicE subsetI)
 qed
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsection\<open>Permutations of indices of semialgebraic sets\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 
 lemma fun_inv_permute:
   assumes "\<sigma> permutes {..<n}"
   shows "fun_inv \<sigma> permutes {..<n}"
         "\<sigma> \<circ> (fun_inv \<sigma>) = id"
         "(fun_inv \<sigma>) \<circ> \<sigma> = id"
-  using assms unfolding fun_inv_def 
-  using permutes_inv apply blast 
+  using assms unfolding fun_inv_def
+  using permutes_inv apply blast
   using assms permutes_inv_o(1) apply blast
   using assms permutes_inv_o(2) by blast
 
 lemma poly_tuple_pullback_eq_poly_map_vimage:
   assumes "is_poly_tuple n fs"
   assumes "length fs = m"
   assumes "S \<subseteq> carrir (Q\<^sub>p\<^bsup>m\<^esup>)"
   shows "poly_map n fs  \<inverse>\<^bsub>n\<^esub> S = poly_tuple_pullback n S fs"
-  unfolding poly_map_def poly_tuple_pullback_def evimage_def restrict_def 
-  using assms 
+  unfolding poly_map_def poly_tuple_pullback_def evimage_def restrict_def
+  using assms
   by (smt vimage_inter_cong)
 
 lemma permutation_is_semialgebraic:
   assumes "is_semialgebraic n S"
   assumes "\<sigma> permutes {..<n}"
   shows "is_semialgebraic n (permute_list \<sigma> ` S)"
 proof-
   have "S \<subseteq> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
-    using assms gen_boolean_algebra_subset is_semialgebraic_def semialg_sets_def 
+    using assms gen_boolean_algebra_subset is_semialgebraic_def semialg_sets_def
     by blast
   then have "(permute_list \<sigma> ` S) = poly_tuple_pullback n S (permute_list (fun_inv \<sigma>) (pvar_list Q\<^sub>p n))"
     using Qp.cring_axioms assms pullback_by_permutation_of_poly_list'[of  \<sigma> n S] unfolding poly_map_def
-    by blast 
+    by blast
   then have 0: "(permute_list \<sigma> ` S) = poly_tuple_pullback n S (permute_list (fun_inv \<sigma>) (pvar_list Q\<^sub>p n))"
     using poly_tuple_pullback_def
     by blast
   have 1: "(fun_inv \<sigma>) permutes {..<n}"
-    using assms  unfolding fun_inv_def 
+    using assms  unfolding fun_inv_def
     using permutes_inv by blast
   then show ?thesis using 1 pullback_is_semialg[of n "(permute_list (fun_inv \<sigma>) (pvar_list Q\<^sub>p n))"]
                          permutation_of_poly_list_is_poly_list[of  n "(pvar_list Q\<^sub>p n)" "fun_inv \<sigma>"]
                          pvar_list_is_poly_tuple[of n] assms poly_tuple_pullback_eq_poly_map_vimage
   by (metis "0" \<open>S \<subseteq> carrier (Q\<^sub>p\<^bsup>n\<^esup>)\<close> is_semialgebraic_def length_permute_list pvar_list_length)
 qed
 
 lemma permute_list_closed:
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "\<sigma> permutes {..<n}"
   shows "permute_list \<sigma> a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   apply(rule cartesian_power_car_memI)
   using assms cartesian_power_car_memE length_permute_list apply blast
   using assms cartesian_power_car_memE'' permute_list_set by blast
-  
+
 lemma permute_list_closed':
   assumes "\<sigma> permutes {..<n}"
   assumes "permute_list \<sigma> a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   apply(rule cartesian_power_car_memI)
   apply (metis assms(2) cartesian_power_car_memE length_permute_list)
-  using assms cartesian_power_car_memE'[of "permute_list \<sigma> a" Q\<^sub>p n] 
+  using assms cartesian_power_car_memE'[of "permute_list \<sigma> a" Q\<^sub>p n]
   by (metis cartesian_power_car_memE in_set_conv_nth length_permute_list set_permute_list subsetI)
 
 lemma permute_list_compose_inv:
   assumes "\<sigma> permutes {..<n}"
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "permute_list \<sigma> (permute_list (fun_inv \<sigma>) a) = a"
         "permute_list (fun_inv \<sigma>) (permute_list \<sigma> a) = a"
   using assms apply (metis cartesian_power_car_memE fun_inv_permute(3) permute_list_compose permute_list_id)
   using assms by (metis cartesian_power_car_memE fun_inv_permute(2) fun_inv_permute(1) permute_list_compose permute_list_id)
 
 lemma permutation_is_semialgebraic_imp_is_semialgebraic:
   assumes "is_semialgebraic n (permute_list \<sigma> ` S)"
   assumes "\<sigma> permutes {..<n}"
   shows "is_semialgebraic n S"
 proof-
   have "permute_list (fun_inv \<sigma>) ` (permute_list \<sigma> ` S) = S"
   proof-
     have 0: "(permute_list \<sigma> ` S) \<subseteq> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
-      using assms unfolding is_semialgebraic_def semialg_sets_def  
+      using assms unfolding is_semialgebraic_def semialg_sets_def
       using gen_boolean_algebra_subset by blast
     have 1: "S \<subseteq> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
     proof fix x assume "x \<in> S" then show "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
-        using 0 assms 
+        using 0 assms
         by (meson image_subset_iff permute_list_closed')
     qed
     show ?thesis
     proof show "permute_list (fun_inv \<sigma>) ` permute_list \<sigma> ` S \<subseteq> S"
         using 0 assms permute_list_compose_inv[of \<sigma>]  "1" image_iff image_subset_iff subsetD
         by smt
         show "S \<subseteq> permute_list (fun_inv \<sigma>) ` permute_list \<sigma> ` S"
-        using 0 assms permute_list_compose_inv[of \<sigma>] 
+        using 0 assms permute_list_compose_inv[of \<sigma>]
         by (smt "1" image_iff subset_eq)
     qed
   qed
   then show ?thesis using permutation_is_semialgebraic
     by (metis assms(1) assms(2)  fun_inv_permute(1))
-qed      
+qed
 
 lemma split_cartesian_product_is_semialgebraic:
   assumes "i \<le> n"
   assumes "is_semialgebraic n A"
   assumes "is_semialgebraic m B"
   shows "is_semialgebraic (n + m) (split_cartesian_product n m i A B)"
   using assms cartesian_product_is_semialgebraic scp_permutes[of i n m]
-        permutation_is_semialgebraic[of "n + m" "cartesian_product A B" "(scp_permutation n m i)"] 
-  unfolding split_cartesian_product_def 
+        permutation_is_semialgebraic[of "n + m" "cartesian_product A B" "(scp_permutation n m i)"]
+  unfolding split_cartesian_product_def
   by blast
 
 definition reverse_val_relation_set where
 "reverse_val_relation_set = {as \<in> carrier (Q\<^sub>p\<^bsup>2\<^esup>). val (as ! 0) \<le> val (as ! 1)}"
 
 lemma  Qp_2_car_memE:
   assumes  "x \<in> carrier (Q\<^sub>p\<^bsup>2\<^esup>)"
   shows "x = [x!0, x!1]"
 proof-
   have "length x = 2"
     using assms cartesian_power_car_memE by blast
-  then show ?thesis 
+  then show ?thesis
     using pair_id by blast
 qed
 
 definition flip where
 "flip =  (\<lambda>i::nat. (if i = 0 then 1 else (if i = 1 then 0 else i)))"
 
 lemma flip_permutes:
 "flip permutes {0,1}"
- unfolding permutes_def flip_def 
+ unfolding permutes_def flip_def
  by (smt mem_simps(1))
 
-lemma flip_eval: 
-"flip 0 = 1" 
+lemma flip_eval:
+"flip 0 = 1"
 "flip 1 = 0"
-  unfolding flip_def 
-  by auto 
+  unfolding flip_def
+  by auto
 
 lemma flip_x:
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>2\<^esup>)"
   shows "permute_list flip x = [x!1, x!0]"
 proof-
   have 0: "x = [x!0, x!1]"
-    using assms Qp_2_car_memE by blast 
-  have 1: "length (permute_list flip x) = length [x!1, x!0]" 
-    using 0 unfolding permute_list_def  
+    using assms Qp_2_car_memE by blast
+  have 1: "length (permute_list flip x) = length [x!1, x!0]"
+    using 0 unfolding permute_list_def
     by (metis length_Cons length_map map_nth)
   have 2: "\<And>i. i < 2 \<Longrightarrow> permute_list flip x ! i = [x!1, x!0] ! i"
   proof- fix i::nat assume A: "i < 2"
     show "permute_list flip x ! i = [x!1, x!0] ! i"
-      using 0 unfolding permute_list_def 
+      using 0 unfolding permute_list_def
       by (smt flip_eval(1) flip_eval(2) length_Cons length_greater_0_conv list.simps(8) map_upt_Suc numeral_nat(7) upt_rec)
   qed
   have "\<And>i. i < length x \<Longrightarrow> permute_list flip x ! i = [x!1, x!0] ! i"
   proof-
   have 0:  "length x = 2"
       using assms cartesian_power_car_memE by blast
-      show  "\<And>i. i < length x \<Longrightarrow> permute_list flip x ! i = [x!1, x!0] ! i" using 2 unfolding 0 
-        by blast 
-  qed
-  thus ?thesis using 1 
+      show  "\<And>i. i < length x \<Longrightarrow> permute_list flip x ! i = [x!1, x!0] ! i" using 2 unfolding 0
+        by blast
+  qed
+  thus ?thesis using 1
     by (metis length_permute_list nth_equalityI)
 qed
 
-lemma permute_with_flip_closed: 
+lemma permute_with_flip_closed:
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>2::nat\<^esup>)"
   shows  "permute_list  flip x \<in> carrier (Q\<^sub>p\<^bsup>2::nat\<^esup>)"
-  apply(rule permute_list_closed) 
+  apply(rule permute_list_closed)
   using assms apply blast
-proof- 
+proof-
   have "{0::nat, 1} = {..<2::nat}"
-    by auto 
-  thus "flip permutes {..<2}" 
+    by auto
+  thus "flip permutes {..<2}"
   using flip_permutes
-  by auto  
+  by auto
 qed
 
 lemma reverse_val_relation_set_semialg:
 "is_semialgebraic 2 reverse_val_relation_set"
 proof-
   have 1: "reverse_val_relation_set = permute_list flip `  val_relation_set"
     apply(rule equalityI')
   proof-
     show " \<And>x. x \<in> reverse_val_relation_set \<Longrightarrow> x \<in> permute_list flip ` val_relation_set"
       proof- fix x assume A: "x \<in> reverse_val_relation_set"
     have 0: "permute_list flip x = [x ! 1, x ! 0]"
-      using flip_x[of x] A unfolding reverse_val_relation_set_def 
+      using flip_x[of x] A unfolding reverse_val_relation_set_def
       by blast
     have 1: "permute_list flip x \<in> carrier (Q\<^sub>p\<^bsup>2\<^esup>)"
-      apply(rule permute_with_flip_closed) using A unfolding reverse_val_relation_set_def by blast 
+      apply(rule permute_with_flip_closed) using A unfolding reverse_val_relation_set_def by blast
     have 2: "permute_list flip x \<in> val_relation_set"
-      using 1 A unfolding 0 reverse_val_relation_set_def val_relation_set_def mem_Collect_eq 
+      using 1 A unfolding 0 reverse_val_relation_set_def val_relation_set_def mem_Collect_eq
       by (metis Qp_2_car_memE list_hd list_tl)
     show "x \<in> permute_list flip ` val_relation_set"
-      using flip_x[of x] A unfolding reverse_val_relation_set_def val_relation_set_def mem_Collect_eq 
+      using flip_x[of x] A unfolding reverse_val_relation_set_def val_relation_set_def mem_Collect_eq
       by (metis (no_types, lifting) "1" "2" Qp_2_car_memE flip_x image_eqI list_tl nth_Cons_0 val_relation_set_def)
     qed
     show "\<And>x. x \<in> permute_list flip ` val_relation_set \<Longrightarrow> x \<in> reverse_val_relation_set"
     proof- fix x assume a: " x \<in> permute_list flip ` val_relation_set"
       then obtain y where y_def: "y \<in> val_relation_set \<and>x = permute_list flip y"
-        by blast 
+        by blast
       have y_closed: "y \<in> carrier (Q\<^sub>p\<^bsup>2\<^esup>)"
         using y_def basic_semialg_set_memE(1) val_relation_semialg by blast
       have y_length: " length y = 2"
-        using y_def basic_semialg_set_memE  val_relation_semialg 
+        using y_def basic_semialg_set_memE  val_relation_semialg
         by (metis cartesian_power_car_memE)
       obtain a b where ab_def: "y = [a,b]"
         using y_length pair_id by blast
       have 0: "a = y!0"
-        using ab_def 
+        using ab_def
         by (metis nth_Cons_0)
       have 1: "b = y!1"
-        using ab_def 
+        using ab_def
         by (metis cancel_comm_monoid_add_class.diff_cancel eq_numeral_extra(2) nth_Cons')
       have a_closed: "a \<in> carrier Q\<^sub>p"
-        using 0 y_closed unfolding 0 
-        by (meson cartesian_power_car_memE' rel_simps(75) zero_order(5)) 
+        using 0 y_closed unfolding 0
+        by (meson cartesian_power_car_memE' rel_simps(75) zero_order(5))
       have b_closed: "b \<in> carrier Q\<^sub>p"
       proof-
-        have "1 < (2::nat)" by linarith 
-        thus ?thesis 
-        using y_closed unfolding 1 
-        by (meson cartesian_power_car_memE') 
+        have "1 < (2::nat)" by linarith
+        thus ?thesis
+        using y_closed unfolding 1
+        by (meson cartesian_power_car_memE')
       qed
-      have 2: "x = [b, a]" using flip_x[of y] y_def y_closed unfolding  ab_def unfolding 0 1  
+      have 2: "x = [b, a]" using flip_x[of y] y_def y_closed unfolding  ab_def unfolding 0 1
         using \<open>y \<in> carrier (Q\<^sub>p\<^bsup>2\<^esup>) \<Longrightarrow> permute_list flip y = [y ! 1, y ! 0]\<close> y_closed y_def by presburger
       have x_closed: "x \<in> carrier (Q\<^sub>p\<^bsup>2\<^esup>)"
         using  y_def unfolding val_relation_set_def using permute_with_flip_closed[of y]
         by blast
       show " x \<in> reverse_val_relation_set"
-        using x_closed  y_def 
-        unfolding val_relation_set_def reverse_val_relation_set_def  mem_Collect_eq 2 0 1 
+        using x_closed  y_def
+        unfolding val_relation_set_def reverse_val_relation_set_def  mem_Collect_eq 2 0 1
         by (metis Qp_2_car_memE list_hd list_tl)
     qed
   qed
-  show ?thesis unfolding 1 
+  show ?thesis unfolding 1
     apply(rule permutation_is_semialgebraic)
     using val_relation_is_semialgebraic apply blast
-    using flip_permutes 
-    by (metis Suc_1 insert_commute lessThan_0 lessThan_Suc numeral_nat(7))     
+    using flip_permutes
+    by (metis Suc_1 insert_commute lessThan_0 lessThan_Suc numeral_nat(7))
 qed
 
 definition strict_val_relation_set where
 "strict_val_relation_set = {as \<in> carrier (Q\<^sub>p\<^bsup>2\<^esup>). val (as ! 0) < val (as ! 1)}"
 
 definition val_diag where
 "val_diag = {as \<in> carrier (Q\<^sub>p\<^bsup>2\<^esup>). val (as ! 0) = val (as ! 1)}"
 
 lemma val_diag_semialg:
 "is_semialgebraic 2 val_diag"
 proof-
   have "val_diag = val_relation_set \<inter>reverse_val_relation_set"
     apply(rule equalityI')
     apply(rule IntI)
-    unfolding val_diag_def val_relation_set_def reverse_val_relation_set_def mem_Collect_eq 
-  apply simp 
-  apply simp 
-      apply(erule IntE) unfolding mem_Collect_eq 
+    unfolding val_diag_def val_relation_set_def reverse_val_relation_set_def mem_Collect_eq
+  apply simp
+  apply simp
+      apply(erule IntE) unfolding mem_Collect_eq
     using basic_trans_rules(24) by blast
-  then show ?thesis using intersection_is_semialg  
+  then show ?thesis using intersection_is_semialg
     by (simp add: reverse_val_relation_set_semialg val_relation_is_semialgebraic)
 qed
 
-lemma strict_val_relation_set_is_semialg:  
+lemma strict_val_relation_set_is_semialg:
 "is_semialgebraic 2 strict_val_relation_set"
 proof-
   have 0: "strict_val_relation_set = reverse_val_relation_set - val_diag"
     apply(rule equalityI')
     apply(rule DiffI)
-    unfolding strict_val_relation_set_def val_diag_def val_relation_set_def reverse_val_relation_set_def mem_Collect_eq 
+    unfolding strict_val_relation_set_def val_diag_def val_relation_set_def reverse_val_relation_set_def mem_Collect_eq
     using order_le_less apply blast
   proof
     show "\<And>x. x \<in> carrier (Q\<^sub>p\<^bsup>2\<^esup>) \<and> val (x ! 0) < val (x ! 1) \<Longrightarrow> x \<in> carrier (Q\<^sub>p\<^bsup>2\<^esup>) \<and> val (x ! 0) = val (x ! 1) \<Longrightarrow> False"
-      using order_less_le by blast 
+      using order_less_le by blast
     show " \<And>x. x \<in> {as \<in> carrier (Q\<^sub>p\<^bsup>2\<^esup>). val (as ! 0) \<le> val (as ! 1)} - {as \<in> carrier (Q\<^sub>p\<^bsup>2\<^esup>). val (as ! 0) = val (as ! 1)} \<Longrightarrow>
          x \<in> carrier (Q\<^sub>p\<^bsup>2\<^esup>) \<and> val (x ! 0) < val (x ! 1)"
-      apply(erule DiffE) unfolding mem_Collect_eq using order_le_less by blast 
-  qed
-  show ?thesis unfolding 0 
-    apply(rule diff_is_semialgebraic ) 
+      apply(erule DiffE) unfolding mem_Collect_eq using order_le_less by blast
+  qed
+  show ?thesis unfolding 0
+    apply(rule diff_is_semialgebraic )
     using reverse_val_relation_set_semialg apply blast
         using val_diag_semialg by blast
 qed
 
 lemma singleton_length:
   "length [a] = 1"
-  by auto 
-
-lemma take_closed': 
+  by auto
+
+lemma take_closed':
   assumes "m > 0"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>m+l\<^esup>)"
   shows "take m x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
   apply(rule take_closed[of m "m+l"])
-  apply simp using assms by blast  
+  apply simp using assms by blast
 
 lemma triple_val_ineq_set_semialg:
   shows "is_semialgebraic 3 {as \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>). val (as!0) \<le> val (as!1) \<and> val (as!1) \<le> val (as!2)}"
 proof-
   have 0: "is_semialgebraic 3 {as \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>). val (as!0) \<le> val (as!1)}"
   proof-
     have 0: "{as \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>). val (as!0) \<le> val (as!1)} = cartesian_product (reverse_val_relation_set) (carrier (Q\<^sub>p\<^bsup>1\<^esup>))"
     proof(rule equalityI')
       show " \<And>x. x \<in> {as \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>). val (as ! 0) \<le> val (as ! 1)} \<Longrightarrow> x \<in> cartesian_product reverse_val_relation_set (carrier (Q\<^sub>p\<^bsup>1\<^esup>))"
       proof- fix x assume A: " x \<in> {as \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>). val (as ! 0) \<le> val (as ! 1)}"
-        then have 0: "length x = 3" unfolding mem_Collect_eq 
+        then have 0: "length x = 3" unfolding mem_Collect_eq
           using cartesian_power_car_memE by blast
         obtain a where a_def: "a = [x!0, x!1]"
-          by blast 
+          by blast
         have a_length: "length a = 2"
         proof-
           have "a = x!0 #[x!1]"
-            unfolding a_def 
+            unfolding a_def
             by blast
-          thus ?thesis using  length_Cons[of "x!0" "[x!1]"] unfolding singleton_length[of "x!1"] 
+          thus ?thesis using  length_Cons[of "x!0" "[x!1]"] unfolding singleton_length[of "x!1"]
             by presburger
         qed
         obtain b where b_def: "b = [x!2]"
-          by blast 
+          by blast
         have b_length: "length b = 1"
-          unfolding b_def singleton_length by auto 
+          unfolding b_def singleton_length by auto
         have a_closed: "a \<in> reverse_val_relation_set"
         proof-
           have 0: "a = take 2 x"
-            apply(rule nth_equalityI)  
-            unfolding a_length  0 length_take[of 2 x] 
+            apply(rule nth_equalityI)
+            unfolding a_length  0 length_take[of 2 x]
              apply linarith
           proof- fix i::nat assume  a: "i < 2" show "a ! i = take 2 x ! i "
               apply(cases "i = 0")
                apply (metis a_def nth_Cons_0 nth_take zero_less_numeral)
               by (smt "0" \<open>length (take 2 x) = min (length x) 2\<close> a_def linorder_neqE_nat min.commute min.strict_order_iff nth_take numeral_eq_iff one_less_numeral_iff pair_id pos2 rel_simps(22) rel_simps(48) rel_simps(9) semiring_norm(81))
           qed
           have 1: "a \<in> carrier (Q\<^sub>p\<^bsup>2\<^esup>)"
             apply(rule cartesian_power_car_memI')
-             apply (simp add: a_length) 
-            unfolding 0 using A unfolding mem_Collect_eq 
-            using cartesian_power_car_memE' by fastforce 
+             apply (simp add: a_length)
+            unfolding 0 using A unfolding mem_Collect_eq
+            using cartesian_power_car_memE' by fastforce
           show ?thesis using 1 A unfolding a_def reverse_val_relation_set_def A mem_Collect_eq
              by (metis Qp_2_car_memE list_tl nth_Cons_0)
         qed
         have b_closed: "b \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>)"
-          apply(rule cartesian_power_car_memI) 
-          unfolding b_length apply blast 
+          apply(rule cartesian_power_car_memI)
+          unfolding b_length apply blast
           apply(rule subsetI)
           unfolding b_def using A unfolding mem_Collect_eq using cartesian_power_car_memE'[of x Q\<^sub>p "3::nat"  "2::nat"]
-          by (metis in_set_conv_nth le_numeral_extra(4) less_one numeral_le_iff numeral_less_iff padic_fields.singleton_length padic_fields_axioms semiring_norm(173) semiring_norm(79) Qp.to_R_to_R1)
-        have 2: "x = a@b" 
+          by simp
+        have 2: "x = a@b"
           apply(rule nth_equalityI)
-          using 0 unfolding a_length b_length  length_append[of a b] apply presburger 
+          using 0 unfolding a_length b_length  length_append[of a b] apply presburger
         proof- fix i assume A: "i < length x"
           then have A1: "i < 3"
-            unfolding 0 by blast 
+            unfolding 0 by blast
           show "x ! i = (a @ b) ! i"
             apply(cases "i = 0")
              apply (metis a_def append.simps(2) nth_Cons_0)
             apply(cases "(i:: nat) = 1")
-             apply (simp add: a_def) 
+             apply (simp add: a_def)
 
           proof- assume a: "i \<noteq>0" "i \<noteq> 1"
             then have "i = 2"
-              using A1 by presburger 
-            thus ?thesis 
-              by (metis a_length b_def nth_append_length) 
+              using A1 by presburger
+            thus ?thesis
+              by (metis a_length b_def nth_append_length)
           qed
         qed
         have 3: "a = take 2 x"
-          apply(rule nth_equalityI)  
-          unfolding a_length  0 length_take[of 2 x] 
+          apply(rule nth_equalityI)
+          unfolding a_length  0 length_take[of 2 x]
            apply linarith
           proof- fix i::nat assume  a: "i < 2" show "a ! i = take 2 x ! i "
               apply(cases "i = 0")
                apply (metis a_def nth_Cons_0 nth_take zero_less_numeral)
               by (smt "0" \<open>length (take 2 x) = min (length x) 2\<close> a_def linorder_neqE_nat min.commute min.strict_order_iff nth_take numeral_eq_iff one_less_numeral_iff pair_id pos2 rel_simps(22) rel_simps(48) rel_simps(9) semiring_norm(81))
           qed
         show " x \<in> cartesian_product reverse_val_relation_set (carrier (Q\<^sub>p\<^bsup>1\<^esup>))"
-          apply(rule cartesian_product_memI[of _ Q\<^sub>p 2 _ 1]) 
+          apply(rule cartesian_product_memI[of _ Q\<^sub>p 2 _ 1])
              apply (simp add: is_semialgebraic_closed reverse_val_relation_set_semialg)
-            apply blast 
-          using 3  a_closed apply blast 
-        proof- 
+            apply blast
+          using 3  a_closed apply blast
+        proof-
           have "drop 2 x = b"
-            unfolding 2 unfolding 3 using 0 
+            unfolding 2 unfolding 3 using 0
             by simp
           then show "drop 2 x \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>)"
-            using b_closed by blast 
+            using b_closed by blast
         qed
       qed
       show "\<And>x. x \<in> cartesian_product reverse_val_relation_set (carrier (Q\<^sub>p\<^bsup>1\<^esup>)) \<Longrightarrow> x \<in> {as \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>). val (as ! 0) \<le> val (as ! 1)}"
       proof fix x assume A: "x \<in> cartesian_product reverse_val_relation_set (carrier (Q\<^sub>p\<^bsup>1\<^esup>))"
         then obtain a b where ab_def: "a \<in> reverse_val_relation_set" "b \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>)" "x = a@b"
-          using cartesian_product_memE'[of x reverse_val_relation_set "carrier (Q\<^sub>p\<^bsup>1\<^esup>)"] 
-          by metis 
+          using cartesian_product_memE'[of x reverse_val_relation_set "carrier (Q\<^sub>p\<^bsup>1\<^esup>)"]
+          by metis
         have a_length: "length a = 2"
-          using ab_def unfolding reverse_val_relation_set_def 
+          using ab_def unfolding reverse_val_relation_set_def
           using cartesian_power_car_memE by blast
-        have "(0::nat)< 2" by presburger  
+        have "(0::nat)< 2" by presburger
         hence  0: "x!0 = a!0"
-          unfolding ab_def using a_length 
+          unfolding ab_def using a_length
           by (metis append.simps(2) nth_Cons_0 pair_id)
-        have "(1::nat)< 2" by presburger  
+        have "(1::nat)< 2" by presburger
         hence 1: "x!1 = a!1"
-          unfolding ab_def using  a_length 
+          unfolding ab_def using  a_length
           by (metis append.simps(2) less_2_cases nth_Cons_0 nth_Cons_Suc pair_id)
         obtain b' where b'_def: "b = [b']"
-          using ab_def cartesian_power_car_memE 
+          using ab_def cartesian_power_car_memE
           by (metis (no_types, opaque_lifting) append_Cons append_Nil append_eq_append_conv min_list.cases singleton_length)
         have b'_closed: "b' \<in> carrier Q\<^sub>p"
-          using b'_def ab_def cartesian_power_car_memE 
+          using b'_def ab_def cartesian_power_car_memE
           by (metis Qp.R1_memE' list_hd)
         have x_closed: "x \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>)"
-          using  ab_def cartesian_power_append[of a Q\<^sub>p 2 b'] b'_def b'_closed 
-          unfolding b'_def ab_def(3) reverse_val_relation_set_def mem_Collect_eq   
-          by (metis Suc_1 eval_nat_numeral(3) plus_1_eq_Suc semiring_norm(164))
+          using  ab_def cartesian_power_append[of a Q\<^sub>p 2 b'] b'_def b'_closed
+          unfolding b'_def ab_def(3) reverse_val_relation_set_def mem_Collect_eq
+          by simp
         show "x \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>) \<and> val (x ! 0) \<le> val (x ! 1)"
-          using x_closed ab_def unfolding reverse_val_relation_set_def mem_Collect_eq  0 1 by blast 
+          using x_closed ab_def unfolding reverse_val_relation_set_def mem_Collect_eq  0 1 by blast
       qed
     qed
-    show ?thesis unfolding 0 
+    show ?thesis unfolding 0
       using cartesian_product_is_semialgebraic[of 2 reverse_val_relation_set 1 "carrier (Q\<^sub>p\<^bsup>1\<^esup>)"]
       by (simp add: carrier_is_semialgebraic reverse_val_relation_set_semialg)
   qed
   have 1: "is_semialgebraic 3 {as \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>). val (as!1) \<le> val (as!2)}"
   proof-
     have 0: "{as \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>). val (as!1) \<le> val (as!2)} = cartesian_product (carrier (Q\<^sub>p\<^bsup>1\<^esup>)) (reverse_val_relation_set)"
     proof(rule equalityI')
       show "\<And>x. x \<in> {as \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>). val (as ! 1) \<le> val (as ! 2)} \<Longrightarrow> x \<in> cartesian_product (carrier (Q\<^sub>p\<^bsup>1\<^esup>)) reverse_val_relation_set"
       proof-
        fix x assume A: " x \<in> {as \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>). val (as ! 1) \<le> val (as ! 2)}"
-        then have 0: "length x = 3" unfolding mem_Collect_eq 
+        then have 0: "length x = 3" unfolding mem_Collect_eq
           using cartesian_power_car_memE by blast
         obtain a where a_def: "a = [x!1, x!2]"
-          by blast 
+          by blast
         have a_length: "length a = 2"
         proof-
           have "a = x!1 #[x!2]"
-            unfolding a_def 
+            unfolding a_def
             by blast
-          thus ?thesis using  length_Cons[of "x!1" "[x!2]"] unfolding singleton_length[of "x!2"] 
+          thus ?thesis using  length_Cons[of "x!1" "[x!2]"] unfolding singleton_length[of "x!2"]
             by presburger
         qed
         obtain b where b_def: "b = [x!0]"
-          by blast 
+          by blast
         have b_length: "length b = 1"
-          unfolding b_def singleton_length by auto 
+          unfolding b_def singleton_length by auto
         have a_closed: "a \<in> reverse_val_relation_set"
         proof-
           have 0: "a = drop 1 x"
-            apply(rule nth_equalityI)  
-            unfolding a_length  0 length_drop[of 1 x] 
+            apply(rule nth_equalityI)
+            unfolding a_length  0 length_drop[of 1 x]
              apply linarith
           proof- fix i::nat assume  a: "i < 2" show " a ! i = drop 1 x ! i"
               apply(cases "i = 0")
-              unfolding a_def using nth_drop[of 1 x i] 
+              unfolding a_def using nth_drop[of 1 x i]
               apply (metis (no_types, opaque_lifting) "0" a_def arith_extra_simps(6) diff_is_0_eq' eq_imp_le eq_numeral_extra(1) flip_def flip_eval(1) less_numeral_extra(1) less_one less_or_eq_imp_le nat_add_left_cancel_le nat_le_linear nat_less_le nth_Cons_0 nth_drop numeral_neq_zero trans_less_add2 zero_less_diff)
               apply(cases "i = 1")
-              using nth_drop[of 1 x i] unfolding 0 
+              using nth_drop[of 1 x i] unfolding 0
               apply (metis "0" a_def a_length list.simps(1) nat_1_add_1 nth_drop one_le_numeral pair_id semiring_norm(3))
-              using a by presburger 
+              using a by presburger
           qed
            have 1: "a \<in> carrier (Q\<^sub>p\<^bsup>2\<^esup>)"
-             using a_def  A drop_closed[of 1 3 x Q\<^sub>p] unfolding 0 mem_Collect_eq 
+             using a_def  A drop_closed[of 1 3 x Q\<^sub>p] unfolding 0 mem_Collect_eq
               by (metis One_nat_def Suc_1 diff_Suc_1 numeral_3_eq_3 rel_simps(49) semiring_norm(77))
            show ?thesis using 1 A unfolding a_def reverse_val_relation_set_def A mem_Collect_eq
              by (metis Qp_2_car_memE list_tl nth_Cons_0)
         qed
         have b_closed: "b \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>)"
-          apply(rule cartesian_power_car_memI) 
-          unfolding b_length apply blast 
+          apply(rule cartesian_power_car_memI)
+          unfolding b_length apply blast
           apply(rule subsetI)
           unfolding b_def using A unfolding mem_Collect_eq using cartesian_power_car_memE'[of x Q\<^sub>p "3::nat"  "0::nat"]
           by (metis b_def b_length in_set_conv_nth less_one Qp.to_R_to_R1 zero_less_numeral)
         have 2: "x = b@a"
           apply(rule nth_equalityI)
-          using 0 unfolding a_length b_length  length_append[of b a] apply presburger 
+          using 0 unfolding a_length b_length  length_append[of b a] apply presburger
         proof- fix i assume A: "i < length x"
           then have A1: "i < 3"
-            unfolding 0 by blast 
+            unfolding 0 by blast
           show "x ! i = (b @ a) ! i"
             apply(cases "i = 0")
-             apply (metis append.simps(2) b_def nth_Cons_0)         
+             apply (metis append.simps(2) b_def nth_Cons_0)
             apply(cases "(i:: nat) = (1::nat)")
-            using  append.simps a_def nth_Cons  
+            using  append.simps a_def nth_Cons
              apply (metis b_length nth_append_length)
             apply(cases "(i:: nat) = (2::nat)")
-            using A unfolding 0 
+            using A unfolding 0
             apply (metis a_def a_length arith_special(3) b_length list.inject nth_append_length_plus pair_id)
           proof-  assume A0: "i \<noteq>0" "i \<noteq> 1" "i \<noteq>2"
-            then have "i \<ge> 3" by presburger 
+            then have "i \<ge> 3" by presburger
             then show "x ! i = (b @ a) ! i"
-              using A unfolding 0 by presburger               
+              using A unfolding 0 by presburger
           qed
         qed
         have 3: "a = drop 1 x"
-            apply(rule nth_equalityI)  
-            unfolding a_length  0 length_drop[of 1 x] 
+            apply(rule nth_equalityI)
+            unfolding a_length  0 length_drop[of 1 x]
              apply linarith
         proof- fix i::nat assume  a: "i < 2" show " a ! i = drop 1 x ! i"
               apply(cases "i = 0")
-              unfolding a_def using nth_drop[of 1 x i] 
+              unfolding a_def using nth_drop[of 1 x i]
               apply (metis (no_types, opaque_lifting) "0" a_def arith_extra_simps(6) diff_is_0_eq' eq_imp_le eq_numeral_extra(1) flip_def flip_eval(1) less_numeral_extra(1) less_one less_or_eq_imp_le nat_add_left_cancel_le nat_le_linear nat_less_le nth_Cons_0 nth_drop numeral_neq_zero trans_less_add2 zero_less_diff)
               apply(cases "i = 1")
-              using nth_drop[of 1 x i] unfolding 0 
+              using nth_drop[of 1 x i] unfolding 0
               apply (metis "0" a_def a_length list.simps(1) nat_1_add_1 nth_drop one_le_numeral pair_id semiring_norm(3))
-              using a by presburger 
+              using a by presburger
         qed
         show "x \<in> cartesian_product (carrier (Q\<^sub>p\<^bsup>1\<^esup>)) reverse_val_relation_set"
-          apply(rule cartesian_product_memI[of _ Q\<^sub>p 1 _ 2]) 
+          apply(rule cartesian_product_memI[of _ Q\<^sub>p 1 _ 2])
              apply (simp add: is_semialgebraic_closed reverse_val_relation_set_semialg)
-          using reverse_val_relation_set_def apply blast 
-          using take_closed[of 1 3 x] A unfolding mem_Collect_eq apply auto[1]          
-          using a_closed unfolding 3 by blast 
+          using reverse_val_relation_set_def apply blast
+          using take_closed[of 1 3 x] A unfolding mem_Collect_eq apply auto[1]
+          using a_closed unfolding 3 by blast
       qed
       show "\<And>x. x \<in> cartesian_product (carrier (Q\<^sub>p\<^bsup>1\<^esup>)) reverse_val_relation_set \<Longrightarrow> x \<in> {as \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>). val (as ! 1) \<le> val (as ! 2)}"
       proof fix x assume A: "x \<in>  cartesian_product (carrier (Q\<^sub>p\<^bsup>1\<^esup>)) reverse_val_relation_set "
         then obtain a b where ab_def: "a \<in> reverse_val_relation_set" "b \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>)" "x = b@a"
-          using cartesian_product_memE'[of x  "carrier (Q\<^sub>p\<^bsup>1\<^esup>)" reverse_val_relation_set] 
-          by metis 
+          using cartesian_product_memE'[of x  "carrier (Q\<^sub>p\<^bsup>1\<^esup>)" reverse_val_relation_set]
+          by metis
         have a_length: "length a = 2"
-          using ab_def unfolding reverse_val_relation_set_def 
+          using ab_def unfolding reverse_val_relation_set_def
           using cartesian_power_car_memE by blast
         obtain b' where b'_def: "b = [b']"
-          using ab_def cartesian_power_car_memE 
+          using ab_def cartesian_power_car_memE
           by (metis (no_types, opaque_lifting) append_Cons append_Nil append_eq_append_conv min_list.cases singleton_length)
         have b'_closed: "b' \<in> carrier Q\<^sub>p"
-          using b'_def ab_def cartesian_power_car_memE 
+          using b'_def ab_def cartesian_power_car_memE
           by (metis Qp.R1_memE' list_hd)
         have b_length: "length b = 1"
           by (simp add: b'_def)
         have x_id: "x = b'#a"
-          unfolding ab_def b'_def by auto 
-        have "(1::nat)< 2" by presburger  
+          unfolding ab_def b'_def by auto
+        have "(1::nat)< 2" by presburger
         hence  0: "x!1 = a!0"
-          unfolding ab_def b'_def using a_length 
+          unfolding ab_def b'_def using a_length
           by (metis b'_def b_length nth_append_length pair_id)
         have 00: "2 = Suc 1"
-          by auto 
+          by auto
         have 1: "x!2 = a!1"
           using a_length nth_Cons[of b' a "2::nat"]
-          unfolding x_id 00 
+          unfolding x_id 00
           by (meson nth_Cons_Suc)
         have x_closed: "x \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>)"
-          unfolding x_id b'_def using b'_closed cartesian_power_cons[of a Q\<^sub>p 2 b'] ab_def 
-          unfolding reverse_val_relation_set_def mem_Collect_eq 
-          using eval_nat_numeral(3) semiring_norm(175) by presburger
+          unfolding x_id b'_def using b'_closed cartesian_power_cons[of a Q\<^sub>p 2 b'] ab_def
+          unfolding reverse_val_relation_set_def mem_Collect_eq
+          by simp
 
         show "x \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>) \<and> val (x ! 1) \<le> val (x ! 2)"
-          using x_closed ab_def unfolding reverse_val_relation_set_def mem_Collect_eq  0 1 by blast 
+          using x_closed ab_def unfolding reverse_val_relation_set_def mem_Collect_eq  0 1 by blast
       qed
     qed
-    show ?thesis unfolding 0 
+    show ?thesis unfolding 0
       using cartesian_product_is_semialgebraic[of 2 reverse_val_relation_set 1 "carrier (Q\<^sub>p\<^bsup>1\<^esup>)"]
       by (metis add_num_simps(2) car_times_semialg_is_semialg one_plus_numeral reverse_val_relation_set_semialg)
   qed
-  have 2: "{as \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>). val (as!0) \<le> val (as!1) \<and> val (as!1) \<le> val (as!2)}= 
+  have 2: "{as \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>). val (as!0) \<le> val (as!1) \<and> val (as!1) \<le> val (as!2)}=
           {as \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>). val (as!0) \<le> val (as!1)} \<inter> {as \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>). val (as!1) \<le> val (as!2)}"
-    by blast 
-  show ?thesis using intersection_is_semialg 0 1 unfolding 2 by blast 
+    by blast
+  show ?thesis using intersection_is_semialg 0 1 unfolding 2 by blast
 qed
 
 lemma triple_val_ineq_set_semialg':
   shows "is_semialgebraic 3 {as \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>). val (as!0) \<le> val (as!1) \<and> val (as!1) < val (as!2)}"
 proof-
   have 0: "is_semialgebraic 3 {as \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>). val (as!0) \<le> val (as!1)}"
   proof-
     have 0: "{as \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>). val (as!0) \<le> val (as!1)} = cartesian_product (reverse_val_relation_set) (carrier (Q\<^sub>p\<^bsup>1\<^esup>))"
     proof(rule equalityI')
       show " \<And>x. x \<in> {as \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>). val (as ! 0) \<le> val (as ! 1)} \<Longrightarrow> x \<in> cartesian_product reverse_val_relation_set (carrier (Q\<^sub>p\<^bsup>1\<^esup>))"
       proof- fix x assume A: " x \<in> {as \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>). val (as ! 0) \<le> val (as ! 1)}"
-        then have 0: "length x = 3" unfolding mem_Collect_eq 
+        then have 0: "length x = 3" unfolding mem_Collect_eq
           using cartesian_power_car_memE by blast
         obtain a where a_def: "a = [x!0, x!1]"
-          by blast 
+          by blast
         have a_length: "length a = 2"
         proof-
           have "a = x!0 #[x!1]"
-            unfolding a_def 
+            unfolding a_def
             by blast
-          thus ?thesis using  length_Cons[of "x!0" "[x!1]"] unfolding singleton_length[of "x!1"] 
+          thus ?thesis using  length_Cons[of "x!0" "[x!1]"] unfolding singleton_length[of "x!1"]
             by presburger
         qed
         obtain b where b_def: "b = [x!2]"
-          by blast 
+          by blast
         have b_length: "length b = 1"
-          unfolding b_def singleton_length by auto 
+          unfolding b_def singleton_length by auto
         have a_closed: "a \<in> reverse_val_relation_set"
         proof-
           have 0: "a = take 2 x"
-            apply(rule nth_equalityI)  
-            unfolding a_length  0 length_take[of 2 x] 
+            apply(rule nth_equalityI)
+            unfolding a_length  0 length_take[of 2 x]
              apply linarith
           proof- fix i::nat assume  a: "i < 2" show "a ! i = take 2 x ! i "
               apply(cases "i = 0")
                apply (metis a_def nth_Cons_0 nth_take zero_less_numeral)
               by (smt "0" \<open>length (take 2 x) = min (length x) 2\<close> a_def linorder_neqE_nat min.commute min.strict_order_iff nth_take numeral_eq_iff one_less_numeral_iff pair_id pos2 rel_simps(22) rel_simps(48) rel_simps(9) semiring_norm(81))
           qed
            have 1: "a \<in> carrier (Q\<^sub>p\<^bsup>2\<^esup>)"
              using a_def 0  A unfolding mem_Collect_eq
-             by (metis le_numeral_extra(4) take_closed numeral_le_iff semiring_norm(173) semiring_norm(72))
+             by (meson Qp_2I cartesian_power_car_memE' rel_simps(49) rel_simps(51) semiring_norm(77))
            show ?thesis using 1 A unfolding a_def reverse_val_relation_set_def A mem_Collect_eq
              by (metis Qp_2_car_memE list_tl nth_Cons_0)
         qed
         have b_closed: "b \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>)"
-          apply(rule cartesian_power_car_memI) 
-          unfolding b_length apply blast 
+          apply(rule cartesian_power_car_memI)
+          unfolding b_length apply blast
           apply(rule subsetI)
           unfolding b_def using A unfolding mem_Collect_eq using cartesian_power_car_memE'[of x Q\<^sub>p "3::nat"  "2::nat"]
-          by (metis in_set_conv_nth le_numeral_extra(4) less_one numeral_le_iff numeral_less_iff padic_fields.singleton_length padic_fields_axioms semiring_norm(173) semiring_norm(79) Qp.to_R_to_R1)
-        have 2: "x = a@b" 
+          by simp
+        have 2: "x = a@b"
           apply(rule nth_equalityI)
-          using 0 unfolding a_length b_length  length_append[of a b] apply presburger 
+          using 0 unfolding a_length b_length  length_append[of a b] apply presburger
         proof- fix i assume A: "i < length x"
           then have A1: "i < 3"
-            unfolding 0 by blast 
+            unfolding 0 by blast
           show "x ! i = (a @ b) ! i"
             apply(cases "i = 0")
              apply (metis a_def append.simps(2) nth_Cons_0)
             apply(cases "(i:: nat) = 1")
-             apply (simp add: a_def) 
+             apply (simp add: a_def)
           proof- assume a: "i \<noteq>0" "i \<noteq> 1"
             then have "i = 2"
-              using A1 by presburger 
-            thus ?thesis 
-              by (metis a_length b_def nth_append_length) 
+              using A1 by presburger
+            thus ?thesis
+              by (metis a_length b_def nth_append_length)
           qed
         qed
         have 3: "a = take 2 x"
-          apply(rule nth_equalityI)  
-          unfolding a_length  0 length_take[of 2 x] 
+          apply(rule nth_equalityI)
+          unfolding a_length  0 length_take[of 2 x]
            apply linarith
           proof- fix i::nat assume  a: "i < 2" show "a ! i = take 2 x ! i "
               apply(cases "i = 0")
                apply (metis a_def nth_Cons_0 nth_take zero_less_numeral)
               by (smt "0" \<open>length (take 2 x) = min (length x) 2\<close> a_def linorder_neqE_nat min.commute min.strict_order_iff nth_take numeral_eq_iff one_less_numeral_iff pair_id pos2 rel_simps(22) rel_simps(48) rel_simps(9) semiring_norm(81))
           qed
         show " x \<in> cartesian_product reverse_val_relation_set (carrier (Q\<^sub>p\<^bsup>1\<^esup>))"
-          apply(rule cartesian_product_memI[of _ Q\<^sub>p 2 _ 1]) 
+          apply(rule cartesian_product_memI[of _ Q\<^sub>p 2 _ 1])
              apply (simp add: is_semialgebraic_closed reverse_val_relation_set_semialg)
-            apply blast 
-          using 3  a_closed apply blast 
-        proof- 
+            apply blast
+          using 3  a_closed apply blast
+        proof-
           have "drop 2 x = b"
-            unfolding 2 unfolding 3 using 0 
+            unfolding 2 unfolding 3 using 0
             by simp
           then show "drop 2 x \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>)"
-            using b_closed by blast 
+            using b_closed by blast
         qed
       qed
       show "\<And>x. x \<in> cartesian_product reverse_val_relation_set (carrier (Q\<^sub>p\<^bsup>1\<^esup>)) \<Longrightarrow> x \<in> {as \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>). val (as ! 0) \<le> val (as ! 1)}"
       proof fix x assume A: "x \<in> cartesian_product reverse_val_relation_set (carrier (Q\<^sub>p\<^bsup>1\<^esup>))"
         then obtain a b where ab_def: "a \<in> reverse_val_relation_set" "b \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>)" "x = a@b"
-          using cartesian_product_memE'[of x reverse_val_relation_set "carrier (Q\<^sub>p\<^bsup>1\<^esup>)"] 
-          by metis 
+          using cartesian_product_memE'[of x reverse_val_relation_set "carrier (Q\<^sub>p\<^bsup>1\<^esup>)"]
+          by metis
         have a_length: "length a = 2"
-          using ab_def unfolding reverse_val_relation_set_def 
+          using ab_def unfolding reverse_val_relation_set_def
           using cartesian_power_car_memE by blast
-        have "(0::nat)< 2" by presburger  
+        have "(0::nat)< 2" by presburger
         hence  0: "x!0 = a!0"
-          unfolding ab_def using a_length 
+          unfolding ab_def using a_length
           by (metis append.simps(2) nth_Cons_0 pair_id)
-        have "(1::nat)< 2" by presburger  
+        have "(1::nat)< 2" by presburger
         hence 1: "x!1 = a!1"
-          unfolding ab_def using  a_length 
+          unfolding ab_def using  a_length
           by (metis append.simps(2) less_2_cases nth_Cons_0 nth_Cons_Suc pair_id)
         obtain b' where b'_def: "b = [b']"
-          using ab_def cartesian_power_car_memE 
+          using ab_def cartesian_power_car_memE
           by (metis (no_types, opaque_lifting) append_Cons append_Nil append_eq_append_conv min_list.cases singleton_length)
         have b'_closed: "b' \<in> carrier Q\<^sub>p"
-          using b'_def ab_def cartesian_power_car_memE 
+          using b'_def ab_def cartesian_power_car_memE
           by (metis Qp.R1_memE' list_hd)
         have x_closed: "x \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>)"
-          using  ab_def cartesian_power_append[of a Q\<^sub>p 2 b'] b'_def b'_closed 
-          unfolding b'_def ab_def(3) reverse_val_relation_set_def mem_Collect_eq   
-          by (metis Suc_1 eval_nat_numeral(3) plus_1_eq_Suc semiring_norm(164))
+          using  ab_def cartesian_power_append[of a Q\<^sub>p 2 b'] b'_def b'_closed
+          unfolding b'_def ab_def(3) reverse_val_relation_set_def mem_Collect_eq
+          by simp
         show "x \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>) \<and> val (x ! 0) \<le> val (x ! 1)"
-          using x_closed ab_def unfolding reverse_val_relation_set_def mem_Collect_eq  0 1 by blast 
+          using x_closed ab_def unfolding reverse_val_relation_set_def mem_Collect_eq  0 1 by blast
       qed
     qed
-    show ?thesis unfolding 0 
+    show ?thesis unfolding 0
       using cartesian_product_is_semialgebraic[of 2 reverse_val_relation_set 1 "carrier (Q\<^sub>p\<^bsup>1\<^esup>)"]
       by (simp add: carrier_is_semialgebraic reverse_val_relation_set_semialg)
   qed
   have 1: "is_semialgebraic 3 {as \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>). val (as!1) < val (as!2)}"
   proof-
     have 0: "{as \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>). val (as!1) < val (as!2)} = cartesian_product (carrier (Q\<^sub>p\<^bsup>1\<^esup>)) (strict_val_relation_set)"
     proof(rule equalityI')
       show "\<And>x. x \<in> {as \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>). val (as ! 1) < val (as ! 2)} \<Longrightarrow> x \<in> cartesian_product (carrier (Q\<^sub>p\<^bsup>1\<^esup>)) strict_val_relation_set"
       proof- fix x assume A: " x \<in> {as \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>). val (as ! 1) < val (as ! 2)}"
-        then have 0: "length x = 3" unfolding mem_Collect_eq 
+        then have 0: "length x = 3" unfolding mem_Collect_eq
           using cartesian_power_car_memE by blast
         obtain a where a_def: "a = [x!1, x!2]"
-          by blast 
+          by blast
         have a_length: "length a = 2"
         proof-
           have "a = x!1 #[x!2]"
-            unfolding a_def 
+            unfolding a_def
             by blast
-          thus ?thesis using  length_Cons[of "x!1" "[x!2]"] unfolding singleton_length[of "x!2"] 
+          thus ?thesis using  length_Cons[of "x!1" "[x!2]"] unfolding singleton_length[of "x!2"]
             by presburger
         qed
         obtain b where b_def: "b = [x!0]"
-          by blast 
+          by blast
         have b_length: "length b = 1"
-          unfolding b_def singleton_length by auto 
+          unfolding b_def singleton_length by auto
         have a_closed: "a \<in> strict_val_relation_set"
         proof-
           have 0: "a = drop 1 x"
-            apply(rule nth_equalityI)  
-            unfolding a_length  0 length_drop[of 1 x] 
+            apply(rule nth_equalityI)
+            unfolding a_length  0 length_drop[of 1 x]
              apply linarith
           proof- fix i::nat assume  a: "i < 2" show " a ! i = drop 1 x ! i"
               apply(cases "i = 0")
-              unfolding a_def using nth_drop[of 1 x i] 
+              unfolding a_def using nth_drop[of 1 x i]
               apply (metis (no_types, opaque_lifting) "0" a_def arith_extra_simps(6) diff_is_0_eq' eq_imp_le eq_numeral_extra(1) flip_def flip_eval(1) less_numeral_extra(1) less_one less_or_eq_imp_le nat_add_left_cancel_le nat_le_linear nat_less_le nth_Cons_0 nth_drop numeral_neq_zero trans_less_add2 zero_less_diff)
               apply(cases "i = 1")
-              using nth_drop[of 1 x i] unfolding 0 
+              using nth_drop[of 1 x i] unfolding 0
               apply (metis "0" a_def a_length list.simps(1) nat_1_add_1 nth_drop one_le_numeral pair_id semiring_norm(3))
-              using a by presburger 
+              using a by presburger
           qed
            have 1: "a \<in> carrier (Q\<^sub>p\<^bsup>2\<^esup>)"
-             using a_def  A drop_closed[of 1 3 x Q\<^sub>p] unfolding 0 mem_Collect_eq 
+             using a_def  A drop_closed[of 1 3 x Q\<^sub>p] unfolding 0 mem_Collect_eq
               by (metis One_nat_def Suc_1 diff_Suc_1 numeral_3_eq_3 rel_simps(49) semiring_norm(77))
            show ?thesis using 1 A unfolding a_def strict_val_relation_set_def A mem_Collect_eq
              by (metis Qp_2_car_memE list_tl nth_Cons_0)
         qed
         have b_closed: "b \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>)"
-          apply(rule cartesian_power_car_memI) 
-          unfolding b_length apply blast 
+          apply(rule cartesian_power_car_memI)
+          unfolding b_length apply blast
           apply(rule subsetI)
           unfolding b_def using A unfolding mem_Collect_eq using cartesian_power_car_memE'[of x Q\<^sub>p "3::nat"  "0::nat"]
           by (metis b_def b_length in_set_conv_nth less_one Qp.to_R_to_R1 zero_less_numeral)
         have 2: "x = b@a"
           apply(rule nth_equalityI)
-          using 0 unfolding a_length b_length  length_append[of b a] apply presburger 
+          using 0 unfolding a_length b_length  length_append[of b a] apply presburger
         proof- fix i assume A: "i < length x"
           then have A1: "i < 3"
-            unfolding 0 by blast 
+            unfolding 0 by blast
           show "x ! i = (b @ a) ! i"
             apply(cases "i = 0")
-             apply (metis append.simps(2) b_def nth_Cons_0)         
+             apply (metis append.simps(2) b_def nth_Cons_0)
             apply(cases "(i:: nat) = (1::nat)")
-            using  append.simps a_def nth_Cons  
+            using  append.simps a_def nth_Cons
              apply (metis b_length nth_append_length)
             apply(cases "(i:: nat) = (2::nat)")
-            using A unfolding 0 
+            using A unfolding 0
             apply (metis a_def a_length arith_special(3) b_length list.inject nth_append_length_plus pair_id)
           proof-  assume A0: "i \<noteq>0" "i \<noteq> 1" "i \<noteq>2"
-            then have "i \<ge> 3" by presburger 
+            then have "i \<ge> 3" by presburger
             then show "x ! i = (b @ a) ! i"
-              using A unfolding 0 by presburger               
+              using A unfolding 0 by presburger
           qed
         qed
         have 3: "a = drop 1 x"
-            apply(rule nth_equalityI)  
-            unfolding a_length  0 length_drop[of 1 x] 
+            apply(rule nth_equalityI)
+            unfolding a_length  0 length_drop[of 1 x]
              apply linarith
         proof- fix i::nat assume  a: "i < 2" show " a ! i = drop 1 x ! i"
               apply(cases "i = 0")
-              unfolding a_def using nth_drop[of 1 x i] 
+              unfolding a_def using nth_drop[of 1 x i]
               apply (metis (no_types, opaque_lifting) "0" a_def arith_extra_simps(6) diff_is_0_eq' eq_imp_le eq_numeral_extra(1) flip_def flip_eval(1) less_numeral_extra(1) less_one less_or_eq_imp_le nat_add_left_cancel_le nat_le_linear nat_less_le nth_Cons_0 nth_drop numeral_neq_zero trans_less_add2 zero_less_diff)
               apply(cases "i = 1")
-              using nth_drop[of 1 x i] unfolding 0 
+              using nth_drop[of 1 x i] unfolding 0
               apply (metis "0" a_def a_length list.simps(1) nat_1_add_1 nth_drop one_le_numeral pair_id semiring_norm(3))
-              using a by presburger 
+              using a by presburger
         qed
         show "x \<in> cartesian_product (carrier (Q\<^sub>p\<^bsup>1\<^esup>)) strict_val_relation_set"
-          apply(rule cartesian_product_memI[of _ Q\<^sub>p 1 _ 2]) 
+          apply(rule cartesian_product_memI[of _ Q\<^sub>p 1 _ 2])
              apply (simp add: is_semialgebraic_closed strict_val_relation_set_is_semialg)
-          using strict_val_relation_set_def apply blast 
-          using take_closed[of 1 3 x Q\<^sub>p] A unfolding mem_Collect_eq apply auto[1]            
-          using a_closed unfolding 3 by blast 
+          using strict_val_relation_set_def apply blast
+          using take_closed[of 1 3 x Q\<^sub>p] A unfolding mem_Collect_eq apply auto[1]
+          using a_closed unfolding 3 by blast
       qed
       show "\<And>x. x \<in> cartesian_product (carrier (Q\<^sub>p\<^bsup>1\<^esup>)) strict_val_relation_set \<Longrightarrow> x \<in> {as \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>). val (as ! 1) < val (as ! 2)}"
       proof fix x assume A: "x \<in>  cartesian_product (carrier (Q\<^sub>p\<^bsup>1\<^esup>)) strict_val_relation_set "
         then obtain a b where ab_def: "a \<in> strict_val_relation_set" "b \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>)" "x = b@a"
-          using cartesian_product_memE'[of x  "carrier (Q\<^sub>p\<^bsup>1\<^esup>)" strict_val_relation_set] 
-          by metis 
+          using cartesian_product_memE'[of x  "carrier (Q\<^sub>p\<^bsup>1\<^esup>)" strict_val_relation_set]
+          by metis
         have a_length: "length a = 2"
-          using ab_def unfolding strict_val_relation_set_def 
+          using ab_def unfolding strict_val_relation_set_def
           using cartesian_power_car_memE by blast
         obtain b' where b'_def: "b = [b']"
-          using ab_def cartesian_power_car_memE 
+          using ab_def cartesian_power_car_memE
           by (metis (no_types, opaque_lifting) append_Cons append_Nil append_eq_append_conv min_list.cases singleton_length)
         have b'_closed: "b' \<in> carrier Q\<^sub>p"
-          using b'_def ab_def cartesian_power_car_memE 
+          using b'_def ab_def
           by (metis Qp.R1_memE' list_hd)
         have b_length: "length b = 1"
           by (simp add: b'_def)
         have x_id: "x = b'#a"
-          unfolding ab_def b'_def by auto 
-        have "(1::nat)< 2" by presburger  
+          unfolding ab_def b'_def by auto
+        have "(1::nat)< 2" by presburger
         hence  0: "x!1 = a!0"
-          unfolding ab_def b'_def using a_length 
+          unfolding ab_def b'_def using a_length
           by (metis b'_def b_length nth_append_length pair_id)
         have 00: "2 = Suc 1"
-          by auto 
+          by auto
         have 1: "x!2 = a!1"
           using a_length nth_Cons[of b' a "2::nat"]
-          unfolding x_id 00 
+          unfolding x_id 00
           by (meson nth_Cons_Suc)
         have x_closed: "x \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>)"
-          unfolding x_id b'_def using b'_closed cartesian_power_cons[of a Q\<^sub>p 2 b'] ab_def 
-          unfolding strict_val_relation_set_def mem_Collect_eq 
-          using eval_nat_numeral(3) semiring_norm(175) by presburger
-
+          unfolding x_id b'_def using b'_closed cartesian_power_cons[of a Q\<^sub>p 2 b'] ab_def
+          unfolding strict_val_relation_set_def mem_Collect_eq
+          by simp
         show "x \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>) \<and> val (x ! 1) < val (x ! 2)"
-          using x_closed ab_def unfolding strict_val_relation_set_def mem_Collect_eq  0 1 by blast 
+          using x_closed ab_def unfolding strict_val_relation_set_def mem_Collect_eq  0 1 by blast
       qed
     qed
-    show ?thesis unfolding 0 
+    show ?thesis unfolding 0
       using cartesian_product_is_semialgebraic[of 2 reverse_val_relation_set 1 "carrier (Q\<^sub>p\<^bsup>1\<^esup>)"]
       by (metis add_num_simps(2) car_times_semialg_is_semialg one_plus_numeral strict_val_relation_set_is_semialg)
   qed
-  have 2: "{as \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>). val (as!0) \<le> val (as!1) \<and> val (as!1) < val (as!2)}= 
+  have 2: "{as \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>). val (as!0) \<le> val (as!1) \<and> val (as!1) < val (as!2)}=
           {as \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>). val (as!0) \<le> val (as!1)} \<inter> {as \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>). val (as!1) < val (as!2)}"
-    by blast 
-  show ?thesis using intersection_is_semialg 0 1 unfolding 2 by blast 
+    by blast
+  show ?thesis using intersection_is_semialg 0 1 unfolding 2 by blast
 qed
 
 lemma triple_val_ineq_set_semialg'':
   shows "is_semialgebraic 3 {as \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>). val (as!1) < val (as!2)}"
 proof-
     have 0: "{as \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>). val (as!1) < val (as!2)} = cartesian_product (carrier (Q\<^sub>p\<^bsup>1\<^esup>)) (strict_val_relation_set)"
     proof(rule equalityI')
       show "\<And>x. x \<in> {as \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>). val (as ! 1) < val (as ! 2)} \<Longrightarrow> x \<in> cartesian_product (carrier (Q\<^sub>p\<^bsup>1\<^esup>)) strict_val_relation_set"
       proof- fix x assume A: " x \<in> {as \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>). val (as ! 1) < val (as ! 2)}"
-        then have 0: "length x = 3" unfolding mem_Collect_eq 
+        then have 0: "length x = 3" unfolding mem_Collect_eq
           using cartesian_power_car_memE by blast
         obtain a where a_def: "a = [x!1, x!2]"
-          by blast 
+          by blast
         have a_length: "length a = 2"
         proof-
           have "a = x!1 #[x!2]"
-            unfolding a_def 
+            unfolding a_def
             by blast
-          thus ?thesis using  length_Cons[of "x!1" "[x!2]"] unfolding singleton_length[of "x!2"] 
+          thus ?thesis using  length_Cons[of "x!1" "[x!2]"] unfolding singleton_length[of "x!2"]
             by presburger
         qed
         obtain b where b_def: "b = [x!0]"
-          by blast 
+          by blast
         have b_length: "length b = 1"
-          unfolding b_def singleton_length by auto 
+          unfolding b_def singleton_length by auto
         have a_closed: "a \<in> strict_val_relation_set"
         proof-
           have 0: "a = drop 1 x"
-            apply(rule nth_equalityI)  
-            unfolding a_length  0 length_drop[of 1 x] 
+            apply(rule nth_equalityI)
+            unfolding a_length  0 length_drop[of 1 x]
              apply linarith
           proof- fix i::nat assume  a: "i < 2" show " a ! i = drop 1 x ! i"
               apply(cases "i = 0")
-              unfolding a_def using nth_drop[of 1 x i] 
+              unfolding a_def using nth_drop[of 1 x i]
               apply (metis (no_types, opaque_lifting) "0" a_def arith_extra_simps(6) diff_is_0_eq' eq_imp_le eq_numeral_extra(1) flip_def flip_eval(1) less_numeral_extra(1) less_one less_or_eq_imp_le nat_add_left_cancel_le nat_le_linear nat_less_le nth_Cons_0 nth_drop numeral_neq_zero trans_less_add2 zero_less_diff)
               apply(cases "i = 1")
-              using nth_drop[of 1 x i] unfolding 0 
+              using nth_drop[of 1 x i] unfolding 0
               apply (metis "0" a_def a_length list.simps(1) nat_1_add_1 nth_drop one_le_numeral pair_id semiring_norm(3))
-              using a by presburger 
+              using a by presburger
           qed
            have 1: "a \<in> carrier (Q\<^sub>p\<^bsup>2\<^esup>)"
-             using a_def  A drop_closed[of 1 3 x Q\<^sub>p] unfolding 0 mem_Collect_eq 
+             using a_def  A drop_closed[of 1 3 x Q\<^sub>p] unfolding 0 mem_Collect_eq
               by (metis One_nat_def Suc_1 diff_Suc_1 numeral_3_eq_3 rel_simps(49) semiring_norm(77))
            show ?thesis using 1 A unfolding a_def strict_val_relation_set_def A mem_Collect_eq
              by (metis Qp_2_car_memE list_tl nth_Cons_0)
         qed
         have b_closed: "b \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>)"
-          apply(rule cartesian_power_car_memI) 
-          unfolding b_length apply blast 
+          apply(rule cartesian_power_car_memI)
+          unfolding b_length apply blast
           apply(rule subsetI)
           unfolding b_def using A unfolding mem_Collect_eq using cartesian_power_car_memE'[of x Q\<^sub>p "3::nat"  "0::nat"]
           by (metis b_def b_length in_set_conv_nth less_one Qp.to_R_to_R1 zero_less_numeral)
         have 2: "x = b@a"
           apply(rule nth_equalityI)
-          using 0 unfolding a_length b_length  length_append[of b a] apply presburger 
+          using 0 unfolding a_length b_length  length_append[of b a] apply presburger
         proof- fix i assume A: "i < length x"
           then have A1: "i < 3"
-            unfolding 0 by blast 
+            unfolding 0 by blast
           show "x ! i = (b @ a) ! i"
             apply(cases "i = 0")
-             apply (metis append.simps(2) b_def nth_Cons_0)         
+             apply (metis append.simps(2) b_def nth_Cons_0)
             apply(cases "(i:: nat) = (1::nat)")
-            using  append.simps a_def nth_Cons  
+            using  append.simps a_def nth_Cons
              apply (metis b_length nth_append_length)
             apply(cases "(i:: nat) = (2::nat)")
-            using A unfolding 0 
+            using A unfolding 0
             apply (metis a_def a_length arith_special(3) b_length list.inject nth_append_length_plus pair_id)
           proof-  assume A0: "i \<noteq>0" "i \<noteq> 1" "i \<noteq>2"
-            then have "i \<ge> 3" by presburger 
+            then have "i \<ge> 3" by presburger
             then show "x ! i = (b @ a) ! i"
-              using A unfolding 0 by presburger               
+              using A unfolding 0 by presburger
           qed
         qed
         have 3: "a = drop 1 x"
-            apply(rule nth_equalityI)  
-            unfolding a_length  0 length_drop[of 1 x] 
+            apply(rule nth_equalityI)
+            unfolding a_length  0 length_drop[of 1 x]
              apply linarith
         proof- fix i::nat assume  a: "i < 2" show " a ! i = drop 1 x ! i"
               apply(cases "i = 0")
-              unfolding a_def using nth_drop[of 1 x i] 
+              unfolding a_def using nth_drop[of 1 x i]
               apply (metis (no_types, opaque_lifting) "0" a_def arith_extra_simps(6) diff_is_0_eq' eq_imp_le eq_numeral_extra(1) flip_def flip_eval(1) less_numeral_extra(1) less_one less_or_eq_imp_le nat_add_left_cancel_le nat_le_linear nat_less_le nth_Cons_0 nth_drop numeral_neq_zero trans_less_add2 zero_less_diff)
               apply(cases "i = 1")
-              using nth_drop[of 1 x i] unfolding 0 
+              using nth_drop[of 1 x i] unfolding 0
               apply (metis "0" a_def a_length list.simps(1) nat_1_add_1 nth_drop one_le_numeral pair_id semiring_norm(3))
-              using a by presburger 
+              using a by presburger
         qed
         show "x \<in> cartesian_product (carrier (Q\<^sub>p\<^bsup>1\<^esup>)) strict_val_relation_set"
-          apply(rule cartesian_product_memI[of _ Q\<^sub>p 1 _ 2]) 
+          apply(rule cartesian_product_memI[of _ Q\<^sub>p 1 _ 2])
              apply (simp add: is_semialgebraic_closed strict_val_relation_set_is_semialg)
-          using strict_val_relation_set_def apply blast 
-          using take_closed[of 1 3 x] A unfolding mem_Collect_eq 
-          using one_le_numeral apply blast           
-          using a_closed unfolding 3 by blast 
+          using strict_val_relation_set_def apply blast
+          using take_closed[of 1 3 x] A unfolding mem_Collect_eq
+          using one_le_numeral apply blast
+          using a_closed unfolding 3 by blast
       qed
       show "\<And>x. x \<in> cartesian_product (carrier (Q\<^sub>p\<^bsup>1\<^esup>)) strict_val_relation_set \<Longrightarrow> x \<in> {as \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>). val (as ! 1) < val (as ! 2)}"
       proof fix x assume A: "x \<in>  cartesian_product (carrier (Q\<^sub>p\<^bsup>1\<^esup>)) strict_val_relation_set "
         then obtain a b where ab_def: "a \<in> strict_val_relation_set" "b \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>)" "x = b@a"
-          using cartesian_product_memE'[of x  "carrier (Q\<^sub>p\<^bsup>1\<^esup>)" strict_val_relation_set] 
-          by metis 
+          using cartesian_product_memE'[of x  "carrier (Q\<^sub>p\<^bsup>1\<^esup>)" strict_val_relation_set]
+          by metis
         have a_length: "length a = 2"
-          using ab_def unfolding strict_val_relation_set_def 
+          using ab_def unfolding strict_val_relation_set_def
           using cartesian_power_car_memE by blast
         obtain b' where b'_def: "b = [b']"
-          using ab_def cartesian_power_car_memE 
+          using ab_def cartesian_power_car_memE
           by (metis (no_types, opaque_lifting) append_Cons append_Nil append_eq_append_conv min_list.cases singleton_length)
         have b'_closed: "b' \<in> carrier Q\<^sub>p"
-          using b'_def ab_def cartesian_power_car_memE 
+          using b'_def ab_def cartesian_power_car_memE
           by (metis Qp.R1_memE' list_hd)
         have b_length: "length b = 1"
           by (simp add: b'_def)
         have x_id: "x = b'#a"
-          unfolding ab_def b'_def by auto 
-        have "(1::nat)< 2" by presburger  
+          unfolding ab_def b'_def by auto
+        have "(1::nat)< 2" by presburger
         hence  0: "x!1 = a!0"
-          unfolding ab_def b'_def using a_length 
+          unfolding ab_def b'_def using a_length
           by (metis b'_def b_length nth_append_length pair_id)
         have 00: "2 = Suc 1"
-          by auto 
+          by auto
         have 1: "x!2 = a!1"
           using a_length nth_Cons[of b' a "2::nat"]
-          unfolding x_id 00 
+          unfolding x_id 00
           by (meson nth_Cons_Suc)
         have x_closed: "x \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>)"
-          unfolding x_id b'_def using b'_closed cartesian_power_cons[of a Q\<^sub>p 2 b'] ab_def 
-          unfolding strict_val_relation_set_def mem_Collect_eq 
-          using eval_nat_numeral(3) semiring_norm(175) by presburger
+          unfolding x_id b'_def using b'_closed cartesian_power_cons[of a Q\<^sub>p 2 b'] ab_def
+          unfolding strict_val_relation_set_def mem_Collect_eq
+          by simp
 
         show "x \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>) \<and> val (x ! 1) < val (x ! 2)"
-          using x_closed ab_def unfolding strict_val_relation_set_def mem_Collect_eq  0 1 by blast 
+          using x_closed ab_def unfolding strict_val_relation_set_def mem_Collect_eq  0 1 by blast
       qed
     qed
-    show ?thesis unfolding 0 
+    show ?thesis unfolding 0
       using cartesian_product_is_semialgebraic[of 2 reverse_val_relation_set 1 "carrier (Q\<^sub>p\<^bsup>1\<^esup>)"]
       by (metis add_num_simps(2) car_times_semialg_is_semialg one_plus_numeral strict_val_relation_set_is_semialg)
 qed
 
 lemma triple_val_ineq_set_semialg''':
   shows "is_semialgebraic 3 {as \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>). val (as!1) \<le> val (as!2)}"
 proof-
     have 0: "{as \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>). val (as!1) \<le> val (as!2)} = cartesian_product (carrier (Q\<^sub>p\<^bsup>1\<^esup>)) (reverse_val_relation_set)"
     proof(rule equalityI')
       show "\<And>x. x \<in> {as \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>). val (as ! 1) \<le> val (as ! 2)} \<Longrightarrow> x \<in> cartesian_product (carrier (Q\<^sub>p\<^bsup>1\<^esup>)) reverse_val_relation_set"
       proof- fix x assume A: " x \<in> {as \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>). val (as ! 1) \<le> val (as ! 2)}"
-        then have 0: "length x = 3" unfolding mem_Collect_eq 
+        then have 0: "length x = 3" unfolding mem_Collect_eq
           using cartesian_power_car_memE by blast
         obtain a where a_def: "a = [x!1, x!2]"
-          by blast 
+          by blast
         have a_length: "length a = 2"
         proof-
           have "a = x!1 #[x!2]"
-            unfolding a_def 
+            unfolding a_def
             by blast
-          thus ?thesis using  length_Cons[of "x!1" "[x!2]"] unfolding singleton_length[of "x!2"] 
+          thus ?thesis using  length_Cons[of "x!1" "[x!2]"] unfolding singleton_length[of "x!2"]
             by presburger
         qed
         obtain b where b_def: "b = [x!0]"
-          by blast 
+          by blast
         have b_length: "length b = 1"
-          unfolding b_def singleton_length by auto 
+          unfolding b_def singleton_length by auto
         have a_closed: "a \<in> reverse_val_relation_set"
         proof-
           have 0: "a = drop 1 x"
-            apply(rule nth_equalityI)  
-            unfolding a_length  0 length_drop[of 1 x] 
+            apply(rule nth_equalityI)
+            unfolding a_length  0 length_drop[of 1 x]
              apply linarith
           proof- fix i::nat assume  a: "i < 2" show " a ! i = drop 1 x ! i"
               apply(cases "i = 0")
-              unfolding a_def using nth_drop[of 1 x i] 
+              unfolding a_def using nth_drop[of 1 x i]
               apply (metis (no_types, opaque_lifting) "0" a_def arith_extra_simps(6) diff_is_0_eq' eq_imp_le eq_numeral_extra(1) flip_def flip_eval(1) less_numeral_extra(1) less_one less_or_eq_imp_le nat_add_left_cancel_le nat_le_linear nat_less_le nth_Cons_0 nth_drop numeral_neq_zero trans_less_add2 zero_less_diff)
               apply(cases "i = 1")
-              using nth_drop[of 1 x i] unfolding 0 
+              using nth_drop[of 1 x i] unfolding 0
               apply (metis "0" a_def a_length list.simps(1) nat_1_add_1 nth_drop one_le_numeral pair_id semiring_norm(3))
-              using a by presburger 
+              using a by presburger
           qed
            have 1: "a \<in> carrier (Q\<^sub>p\<^bsup>2\<^esup>)"
-             using a_def  A drop_closed[of 1 3 x Q\<^sub>p] unfolding 0 mem_Collect_eq 
+             using a_def  A drop_closed[of 1 3 x Q\<^sub>p] unfolding 0 mem_Collect_eq
               by (metis One_nat_def Suc_1 diff_Suc_1 numeral_3_eq_3 rel_simps(49) semiring_norm(77))
            show ?thesis using 1 A unfolding a_def reverse_val_relation_set_def A mem_Collect_eq
              by (metis Qp_2_car_memE list_tl nth_Cons_0)
         qed
         have b_closed: "b \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>)"
-          apply(rule cartesian_power_car_memI) 
-          unfolding b_length apply blast 
+          apply(rule cartesian_power_car_memI)
+          unfolding b_length apply blast
           apply(rule subsetI)
           unfolding b_def using A unfolding mem_Collect_eq using cartesian_power_car_memE'[of x Q\<^sub>p "3::nat"  "0::nat"]
           by (metis b_def b_length in_set_conv_nth less_one Qp.to_R_to_R1 zero_less_numeral)
         have 2: "x = b@a"
           apply(rule nth_equalityI)
-          using 0 unfolding a_length b_length  length_append[of b a] apply presburger 
+          using 0 unfolding a_length b_length  length_append[of b a] apply presburger
         proof- fix i assume A: "i < length x"
           then have A1: "i < 3"
-            unfolding 0 by blast 
+            unfolding 0 by blast
           show "x ! i = (b @ a) ! i"
             apply(cases "i = 0")
-             apply (metis append.simps(2) b_def nth_Cons_0)         
+             apply (metis append.simps(2) b_def nth_Cons_0)
             apply(cases "(i:: nat) = (1::nat)")
-            using  append.simps a_def nth_Cons  
+            using  append.simps a_def nth_Cons
             apply (metis b_length nth_append_length)
             apply(cases "(i:: nat) = (2::nat)")
-            using A unfolding 0 
+            using A unfolding 0
             apply (metis a_def a_length arith_special(3) b_length list.inject nth_append_length_plus pair_id)
           proof-  assume A0: "i \<noteq>0" "i \<noteq> 1" "i \<noteq>2"
-            then have "i \<ge> 3" by presburger 
+            then have "i \<ge> 3" by presburger
             then show "x ! i = (b @ a) ! i"
-              using A unfolding 0 by presburger               
+              using A unfolding 0 by presburger
           qed
         qed
         have 3: "a = drop 1 x"
-            apply(rule nth_equalityI)  
-            unfolding a_length  0 length_drop[of 1 x] 
+            apply(rule nth_equalityI)
+            unfolding a_length  0 length_drop[of 1 x]
              apply linarith
         proof- fix i::nat assume  a: "i < 2" show " a ! i = drop 1 x ! i"
               apply(cases "i = 0")
-              unfolding a_def using nth_drop[of 1 x i] 
+              unfolding a_def using nth_drop[of 1 x i]
               apply (metis (no_types, opaque_lifting) "0" a_def arith_extra_simps(6) diff_is_0_eq' eq_imp_le eq_numeral_extra(1) flip_def flip_eval(1) less_numeral_extra(1) less_one less_or_eq_imp_le nat_add_left_cancel_le nat_le_linear nat_less_le nth_Cons_0 nth_drop numeral_neq_zero trans_less_add2 zero_less_diff)
               apply(cases "i = 1")
-              using nth_drop[of 1 x i] unfolding 0 
+              using nth_drop[of 1 x i] unfolding 0
               apply (metis "0" a_def a_length list.simps(1) nat_1_add_1 nth_drop one_le_numeral pair_id semiring_norm(3))
-              using a by presburger 
+              using a by presburger
         qed
         show "x \<in> cartesian_product (carrier (Q\<^sub>p\<^bsup>1\<^esup>)) reverse_val_relation_set"
-          apply(rule cartesian_product_memI[of _ Q\<^sub>p 1 _ 2]) 
+          apply(rule cartesian_product_memI[of _ Q\<^sub>p 1 _ 2])
              apply (simp add: is_semialgebraic_closed reverse_val_relation_set_semialg)
-          using reverse_val_relation_set_def apply blast 
-          using take_closed[of 1 3 x] A unfolding mem_Collect_eq  apply auto[1] 
-          using a_closed unfolding 3 by blast 
+          using reverse_val_relation_set_def apply blast
+          using take_closed[of 1 3 x] A unfolding mem_Collect_eq  apply auto[1]
+          using a_closed unfolding 3 by blast
       qed
       show "\<And>x. x \<in> cartesian_product (carrier (Q\<^sub>p\<^bsup>1\<^esup>)) reverse_val_relation_set \<Longrightarrow> x \<in> {as \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>). val (as ! 1) \<le> val (as ! 2)}"
       proof fix x assume A: "x \<in>  cartesian_product (carrier (Q\<^sub>p\<^bsup>1\<^esup>)) reverse_val_relation_set "
         then obtain a b where ab_def: "a \<in> reverse_val_relation_set" "b \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>)" "x = b@a"
-          using cartesian_product_memE'[of x  "carrier (Q\<^sub>p\<^bsup>1\<^esup>)" reverse_val_relation_set] 
-          by metis 
+          using cartesian_product_memE'[of x  "carrier (Q\<^sub>p\<^bsup>1\<^esup>)" reverse_val_relation_set]
+          by metis
         have a_length: "length a = 2"
-          using ab_def unfolding reverse_val_relation_set_def 
+          using ab_def unfolding reverse_val_relation_set_def
           using cartesian_power_car_memE by blast
         obtain b' where b'_def: "b = [b']"
-          using ab_def cartesian_power_car_memE 
+          using ab_def cartesian_power_car_memE
           by (metis (no_types, opaque_lifting) append_Cons append_Nil append_eq_append_conv min_list.cases singleton_length)
         have b'_closed: "b' \<in> carrier Q\<^sub>p"
-          using b'_def ab_def cartesian_power_car_memE 
+          using b'_def ab_def cartesian_power_car_memE
           by (metis Qp.R1_memE' list_hd)
         have b_length: "length b = 1"
           by (simp add: b'_def)
         have x_id: "x = b'#a"
-          unfolding ab_def b'_def by auto 
-        have "(1::nat)< 2" by presburger  
+          unfolding ab_def b'_def by auto
+        have "(1::nat)< 2" by presburger
         hence  0: "x!1 = a!0"
-          unfolding ab_def b'_def using a_length 
+          unfolding ab_def b'_def using a_length
           by (metis b'_def b_length nth_append_length pair_id)
         have 00: "2 = Suc 1"
-          by auto 
+          by auto
         have 1: "x!2 = a!1"
           using a_length nth_Cons[of b' a "2::nat"]
-          unfolding x_id 00 
+          unfolding x_id 00
           by (meson nth_Cons_Suc)
         have x_closed: "x \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>)"
-          unfolding x_id b'_def using b'_closed cartesian_power_cons[of a Q\<^sub>p 2 b'] ab_def 
-          unfolding reverse_val_relation_set_def mem_Collect_eq 
-          using eval_nat_numeral(3) semiring_norm(175) by presburger
+          unfolding x_id b'_def using b'_closed cartesian_power_cons[of a Q\<^sub>p 2 b'] ab_def
+          unfolding reverse_val_relation_set_def mem_Collect_eq
+          by simp
 
         show "x \<in> carrier (Q\<^sub>p\<^bsup>3\<^esup>) \<and> val (x ! 1) \<le> val (x ! 2)"
-          using x_closed ab_def unfolding reverse_val_relation_set_def mem_Collect_eq  0 1 by blast 
+          using x_closed ab_def unfolding reverse_val_relation_set_def mem_Collect_eq  0 1 by blast
       qed
     qed
-    show ?thesis unfolding 0 
+    show ?thesis unfolding 0
       using cartesian_product_is_semialgebraic[of 2 reverse_val_relation_set 1 "carrier (Q\<^sub>p\<^bsup>1\<^esup>)"]
       by (metis add_num_simps(2) car_times_semialg_is_semialg one_plus_numeral reverse_val_relation_set_semialg)
 qed
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsection\<open>Semialgebraic Functions\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 text\<open>
-  The most natural way to define a semialgebraic function $f: \mathbb{Q}_p^n \to \mathbb{Q}_p$ is a 
-  function whose graph is a semialgebraic subset of $\mathbb{Q}_p^{n+1}$. However, the definition 
-  given here is slightly different, and devised by Denef in \cite{denef1986} in order to prove 
-  Macintyre's theorem. As Denef notes, we can use Macintyre's theorem to deduce that the given 
+  The most natural way to define a semialgebraic function $f: \mathbb{Q}_p^n \to \mathbb{Q}_p$ is a
+  function whose graph is a semialgebraic subset of $\mathbb{Q}_p^{n+1}$. However, the definition
+  given here is slightly different, and devised by Denef in \cite{denef1986} in order to prove
+  Macintyre's theorem. As Denef notes, we can use Macintyre's theorem to deduce that the given
   definition perfectly aligns with the intuitive one.
 \<close>
 
       (********************************************************************************************)
       (********************************************************************************************)
       subsubsection\<open>Defining Semialgebraic Functions\<close>
       (********************************************************************************************)
       (********************************************************************************************)
 
 
 text\<open>Apply a function f to the tuple consisting of the first n indices, leaving the remaining indices
 unchanged\<close>
 
 definition partial_image where
 "partial_image m f xs = (f (take m xs))#(drop m xs)"
 
 definition partial_pullback where
 "partial_pullback m f l S = (partial_image m f)  \<inverse>\<^bsub>m+l\<^esub> S "
 
-lemma partial_pullback_memE: 
+lemma partial_pullback_memE:
   assumes "as \<in> partial_pullback m f l S"
   shows "as \<in> carrier (Q\<^sub>p\<^bsup>m + l\<^esup>)" "partial_image m f as \<in> S"
   using assms apply (metis evimage_eq partial_pullback_def)
-  using assms unfolding partial_pullback_def   
+  using assms unfolding partial_pullback_def
   by blast
 
-lemma partial_pullback_closed: 
-"partial_pullback m f l S \<subseteq> carrier (Q\<^sub>p\<^bsup>m + l\<^esup>)" 
+lemma partial_pullback_closed:
+"partial_pullback m f l S \<subseteq> carrier (Q\<^sub>p\<^bsup>m + l\<^esup>)"
   using partial_pullback_memE(1) by blast
 
-lemma partial_pullback_memI: 
+lemma partial_pullback_memI:
   assumes "as \<in> carrier (Q\<^sub>p\<^bsup>m + k\<^esup>)"
-  assumes "(f (take m as))#(drop m as) \<in> S" 
+  assumes "(f (take m as))#(drop m as) \<in> S"
   shows "as \<in> partial_pullback m f k S"
-  using assms unfolding partial_pullback_def partial_image_def evimage_def  
-  by blast 
+  using assms unfolding partial_pullback_def partial_image_def evimage_def
+  by blast
 
 lemma partial_image_eq:
   assumes "as \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "bs \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>)"
   assumes "x = as @ bs"
   shows "partial_image n f x = (f as)#bs"
 proof-
   have 0: "(take n x) = as"
     by (metis append_eq_conv_conj assms(1) assms(3) cartesian_power_car_memE)
   have 1: "drop n x = bs"
     by (metis "0" append_take_drop_id assms(3) same_append_eq)
-  show ?thesis using 0 1 unfolding partial_image_def 
-    by blast 
-qed
-  
+  show ?thesis using 0 1 unfolding partial_image_def
+    by blast
+qed
+
 lemma partial_pullback_memE':
   assumes "as \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "bs \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>)"
   assumes "x = as @ bs"
   assumes "x \<in> partial_pullback n f k S"
-  shows "(f as)#bs \<in> S" 
-  using partial_pullback_memE[of x n f k S] partial_image_def[of n f x] 
+  shows "(f as)#bs \<in> S"
+  using partial_pullback_memE[of x n f k S] partial_image_def[of n f x]
   by (metis assms(1) assms(2) assms(3) assms(4) partial_image_eq)
 
 text\<open>Partial pullbacks have the same algebraic properties as pullbacks\<close>
 
 lemma partial_pullback_intersect:
 "partial_pullback m f l (S1 \<inter> S2) = (partial_pullback m f l S1) \<inter> (partial_pullback m f l S2)"
-  unfolding partial_pullback_def 
+  unfolding partial_pullback_def
   by simp
- 
+
 lemma partial_pullback_union:
 "partial_pullback m f l (S1 \<union> S2) = (partial_pullback m f l S1) \<union> (partial_pullback m f l S2)"
-  unfolding partial_pullback_def 
+  unfolding partial_pullback_def
   by simp
 
 lemma cartesian_power_drop:
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>n+l\<^esup>)"
   shows "drop n x \<in> carrier (Q\<^sub>p\<^bsup>l\<^esup>)"
   apply(rule cartesian_power_car_memI)
-  using assms cartesian_power_car_memE 
+  using assms cartesian_power_car_memE
    apply (metis add_diff_cancel_left' length_drop)
   using assms cartesian_power_car_memE''
   by (metis order.trans set_drop_subset)
 
 lemma partial_pullback_complement:
   assumes "f \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<rightarrow> carrier Q\<^sub>p"
   shows "partial_pullback m f l (carrier (Q\<^sub>p\<^bsup>Suc l\<^esup>) - S) = carrier (Q\<^sub>p\<^bsup>m + l\<^esup>) - (partial_pullback m f l S) "
   apply(rule equalityI)
-  using partial_pullback_def[of m f l "(carrier (Q\<^sub>p\<^bsup>Suc l\<^esup>) - S)"] 
-        partial_pullback_def[of m f l S] 
-   apply (smt Diff_iff evimage_Diff partial_pullback_memE(1) subsetI)   
+  using partial_pullback_def[of m f l "(carrier (Q\<^sub>p\<^bsup>Suc l\<^esup>) - S)"]
+        partial_pullback_def[of m f l S]
+   apply (smt Diff_iff evimage_Diff partial_pullback_memE(1) subsetI)
 proof fix x assume A: " x \<in> carrier (Q\<^sub>p\<^bsup>m + l\<^esup>) - partial_pullback m f l S"
   show " x \<in> partial_pullback m f l (carrier (Q\<^sub>p\<^bsup>Suc l\<^esup>) - S) "
     apply(rule partial_pullback_memI)
-    using A 
+    using A
      apply blast
   proof
     have 00: "Suc l = l + 1"
-      by auto 
+      by auto
     have 0: "drop m x \<in> carrier (Q\<^sub>p\<^bsup>l\<^esup>)"
       by (meson A DiffD1 cartesian_power_drop)
     have 1: "take m x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
-      using A by (meson DiffD1 le_add1 take_closed)  
+      using A by (meson DiffD1 le_add1 take_closed)
     have "f (take m x) # drop m x \<in> carrier (Q\<^sub>p\<^bsup>l+1\<^esup>) "
-      using assms 0 1 00 cartesian_power_cons[of "drop m x" Q\<^sub>p l "f (take m x)"] 
-      by blast 
+      using assms 0 1 00 cartesian_power_cons[of "drop m x" Q\<^sub>p l "f (take m x)"]
+      by blast
     thus "f (take m x) # drop m x \<in> carrier (Q\<^sub>p\<^bsup>Suc l\<^esup>) "
-      using 00 by metis                   
+      using 00 by metis
     show "f (take m x) # drop m x \<notin> S"
-      using A unfolding partial_pullback_def partial_image_def 
-      by blast
-  qed
-qed     
+      using A unfolding partial_pullback_def partial_image_def
+      by blast
+  qed
+qed
 
 lemma partial_pullback_carrier:
   assumes "f \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<rightarrow> carrier Q\<^sub>p"
   shows "partial_pullback m f l (carrier (Q\<^sub>p\<^bsup>Suc l\<^esup>)) = carrier (Q\<^sub>p\<^bsup>m + l\<^esup>)"
   apply(rule equalityI)
   using partial_pullback_memE(1) apply blast
 proof fix x assume A: "x \<in> carrier (Q\<^sub>p\<^bsup>m + l\<^esup>)"
   show "x \<in> partial_pullback m f l (carrier (Q\<^sub>p\<^bsup>Suc l\<^esup>))"
     apply(rule partial_pullback_memI)
   using A cartesian_power_drop[of x m l] assms
-   apply blast  
+   apply blast
 proof-
   have "f (take m x) \<in> carrier Q\<^sub>p"
-    using A  assms take_closed[of m "m+l" x Q\<^sub>p]  
-    by (meson Pi_mem le_add1)    
-  then show "f (take m x) # drop m x \<in> carrier (Q\<^sub>p\<^bsup>Suc l\<^esup>)" 
+    using A  assms take_closed[of m "m+l" x Q\<^sub>p]
+    by (meson Pi_mem le_add1)
+  then show "f (take m x) # drop m x \<in> carrier (Q\<^sub>p\<^bsup>Suc l\<^esup>)"
     using cartesian_power_drop[of x m l]
     by (metis A add.commute cartesian_power_cons plus_1_eq_Suc)
 qed
 qed
 
 text\<open>Definition 1.4 from Denef\<close>
 
 definition is_semialg_function where
-"is_semialg_function m f = ((f \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<rightarrow> carrier Q\<^sub>p)  \<and> 
+"is_semialg_function m f = ((f \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<rightarrow> carrier Q\<^sub>p)  \<and>
                           (\<forall>l \<ge> 0. \<forall>S \<in> semialg_sets (1 + l). is_semialgebraic (m + l) (partial_pullback m f l S)))"
 
 lemma is_semialg_function_closed:
   assumes "is_semialg_function m f"
   shows "f \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<rightarrow> carrier Q\<^sub>p"
-  using is_semialg_function_def assms by blast 
+  using is_semialg_function_def assms by blast
 
 lemma is_semialg_functionE:
   assumes "is_semialg_function m f"
   assumes "is_semialgebraic (1 + k) S"
   shows " is_semialgebraic (m + k) (partial_pullback m f k S)"
-  using is_semialg_function_def assms 
+  using is_semialg_function_def assms
   by (meson is_semialgebraicE le0)
 
 lemma is_semialg_functionI:
   assumes "f \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<rightarrow> carrier Q\<^sub>p"
   assumes "\<And>k  S. S \<in> semialg_sets (1 + k) \<Longrightarrow> is_semialgebraic (m + k) (partial_pullback m f k S)"
   shows "is_semialg_function m f"
-  using assms unfolding is_semialg_function_def 
+  using assms unfolding is_semialg_function_def
   by blast
 
 text\<open>Semialgebraicity for functions can be verified on basic semialgebraic sets \<close>
 
 lemma is_semialg_functionI':
   assumes "f \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<rightarrow> carrier Q\<^sub>p"
   assumes "\<And>k  S. S \<in> basic_semialgs (1 + k) \<Longrightarrow> is_semialgebraic (m + k) (partial_pullback m f k S)"
   shows "is_semialg_function m f"
   apply(rule is_semialg_functionI)
   using assms(1) apply blast
 proof-
   show "\<And>k S. S \<in> semialg_sets (1 + k) \<Longrightarrow> is_semialgebraic (m + k) (partial_pullback m f k S)"
   proof- fix k S assume A: "S \<in> semialg_sets (1 + k)"
     show "is_semialgebraic (m + k) (partial_pullback m f k S)"
       apply(rule gen_boolean_algebra.induct[of S "carrier (Q\<^sub>p\<^bsup>1+k\<^esup>)" "basic_semialgs (1 + k)"])
-        using A unfolding semialg_sets_def 
-          apply blast          
+        using A unfolding semialg_sets_def
+          apply blast
         using partial_pullback_carrier assms carrier_is_semialgebraic plus_1_eq_Suc apply presburger
-        apply (metis assms(1) assms(2) carrier_is_semialgebraic intersection_is_semialg partial_pullback_carrier partial_pullback_intersect plus_1_eq_Suc)        
+        apply (metis assms(1) assms(2) carrier_is_semialgebraic intersection_is_semialg partial_pullback_carrier partial_pullback_intersect plus_1_eq_Suc)
         using partial_pullback_union union_is_semialgebraic apply presburger
         using assms(1) complement_is_semialg partial_pullback_complement plus_1_eq_Suc by presburger
   qed
-qed      
+qed
 
 text\<open>Graphs of semialgebraic functions are semialgebraic\<close>
-abbreviation graph where 
+abbreviation graph where
 "graph \<equiv> fun_graph Q\<^sub>p"
 
 lemma graph_memE:
   assumes "f \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<rightarrow> carrier Q\<^sub>p"
   assumes "x \<in> graph m f"
   shows "f (take m x) = x!m"
         "x = (take m x)@[f (take m x)]"
         "take m x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
 proof-
   obtain a where a_def:  "a\<in>carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<and> x = a @ [f a]"
-    using assms 
-    unfolding fun_graph_def 
-    by blast 
+    using assms
+    unfolding fun_graph_def
+    by blast
   then have 0: "a = take m x"
     by (metis append_eq_conv_conj cartesian_power_car_memE)
   then show "f (take m x) = x!m"
     by (metis a_def cartesian_power_car_memE nth_append_length)
   show "x = (take m x)@[f (take m x)]"
-    using "0" a_def 
+    using "0" a_def
     by blast
   show "take m x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
     using "0" a_def by blast
 qed
 
 lemma graph_memI:
   assumes "f \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<rightarrow> carrier Q\<^sub>p"
   assumes "f (take m x) = x!m"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>m+1\<^esup>)"
   shows "x \<in> graph m f"
 proof-
   have 0: "take m x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
     apply(rule take_closed[of _ "m + 1"])
-     apply simp 
+     apply simp
        using assms(3) by blast
   have "x = (take m x)@[x!m]"
-  by (metis \<open>take m x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)\<close> add.commute 
+  by (metis \<open>take m x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)\<close> add.commute
       assms(3) cartesian_power_car_memE length_append_singleton lessI
       nth_equalityI nth_take plus_1_eq_Suc take_Suc_conv_app_nth)
   then have "x = (take m x)@[f (take m x)]"
-    using assms(2) 
+    using assms(2)
     by presburger
-  then show ?thesis 
-    using assms 0 
-    unfolding fun_graph_def 
+  then show ?thesis
+    using assms 0
+    unfolding fun_graph_def
     by blast
 qed
 
 lemma graph_mem_closed:
   assumes "f \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<rightarrow> carrier Q\<^sub>p"
   assumes "x \<in> graph m f"
   shows "x \<in> carrier (Q\<^sub>p\<^bsup>m+1\<^esup>)"
 proof(rule cartesian_power_car_memI')
   show "length x = m + 1"
-    using assms graph_memE[of f m x] 
+    using assms graph_memE[of f m x]
     by (smt Groups.add_ac(2) cartesian_power_car_memE fun_graph_def length_append_singleton mem_Collect_eq plus_1_eq_Suc)
   show "\<And>i. i < m + 1 \<Longrightarrow> x ! i \<in> carrier Q\<^sub>p"
   proof- fix i assume A: "i < m + 1"
     then show "x ! i \<in> carrier Q\<^sub>p"
     proof(cases "i = m")
       case True
-      then show ?thesis using graph_memE[of f m x] 
+      then show ?thesis using graph_memE[of f m x]
         by (metis PiE assms(1) assms(2))
     next
       case False
-      then show ?thesis using graph_memE[of f m x] 
+      then show ?thesis using graph_memE[of f m x]
         by (metis \<open>i < m + 1\<close> add.commute assms(1) assms(2) cartesian_power_car_memE' less_SucE nth_take plus_1_eq_Suc)
     qed
   qed
 qed
 
 lemma graph_closed:
   assumes "f \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<rightarrow> carrier Q\<^sub>p"
   shows "graph m f \<subseteq> carrier (Q\<^sub>p\<^bsup>m+1\<^esup>)"
-  using assms graph_mem_closed 
+  using assms graph_mem_closed
   by blast
 
 text\<open>The \<open>m\<close>-dimensional diagonal set is semialgebraic\<close>
 
-notation diagonal ("\<Delta> ") 
+notation diagonal ("\<Delta> ")
 
 lemma diag_is_algebraic:
   shows "is_algebraic Q\<^sub>p (n + n) (\<Delta> n)"
-  using Qp.cring_axioms diagonal_is_algebraic  
+  using Qp.cring_axioms diagonal_is_algebraic
   by blast
 
 lemma diag_is_semialgebraic:
   shows "is_semialgebraic (n + n) (\<Delta> n)"
-  using diag_is_algebraic is_algebraic_imp_is_semialg  
+  using diag_is_algebraic is_algebraic_imp_is_semialg
   by blast
 
 text\<open>Transposition permutations\<close>
 
 definition transpose where
 "transpose i j = (Fun.swap i j id)"
 
 lemma transpose_permutes:
   assumes "i< n"
   assumes "j < n"
   shows "transpose i j permutes {..<n}"
-  unfolding permutes_def transpose_def 
+  unfolding permutes_def transpose_def
 proof
   show "\<forall>x. x \<notin> {..<n} \<longrightarrow> Fun.swap i j id x = x"
     using assms by (auto simp: Transposition.transpose_def)
   show "\<forall>y. \<exists>!x. Fun.swap i j id x = y"
   proof  fix y show "\<exists>!x. Fun.swap i j id x = y"
       using swap_id_eq[of i j y]
   by (metis eq_id_iff swap_apply(1) swap_apply(2) swap_id_eq swap_self)
 qed
 qed
 
 lemma transpose_alt_def:
 "transpose a b x = (if x = a then b else if x = b then a else x)"
   using swap_id_eq
   by (simp add: transpose_def)
 
-definition last_to_first where 
+definition last_to_first where
 "last_to_first n = (\<lambda>i. if i = (n-1) then 0 else if i < n-1 then i + 1 else i)"
 
-definition first_to_last where 
+definition first_to_last where
 "first_to_last n = fun_inv (last_to_first n)"
 
 lemma last_to_first_permutes:
   assumes "(n::nat) > 0"
   shows "last_to_first n permutes {..<n}"
-  unfolding permutes_def 
+  unfolding permutes_def
 proof
   show "\<forall>x. x \<notin> {..<n} \<longrightarrow> last_to_first n x = x"
   proof fix x show " x \<notin> {..<n} \<longrightarrow> last_to_first n x = x"
     proof assume A: "x \<notin> {..<n}" then have "\<not> x < n"
         by blast then have "x \<ge> n" by linarith
       then show  "last_to_first n x = x"
-        unfolding last_to_first_def using assms 
+        unfolding last_to_first_def using assms
         by auto
     qed
   qed
   show "\<forall>y. \<exists>!x. last_to_first n x = y"
   proof fix y
     show "\<exists>!x. last_to_first n x = y"
     proof(cases "y = 0")
       case True
       then have 0: "last_to_first n (n-1) = y"
-        using last_to_first_def 
+        using last_to_first_def
         by (simp add: last_to_first_def)
       have 1: "\<And>x. last_to_first n x = y \<Longrightarrow> x = n-1"
-        unfolding last_to_first_def using True 
+        unfolding last_to_first_def using True
         by (metis add_gr_0 less_numeral_extra(1) not_gr_zero)
-      show ?thesis 
-        using 0 1 
+      show ?thesis
+        using 0 1
         by blast
     next
       case False
       then show ?thesis
       proof(cases "y < n")
         case True
         then have 0: "last_to_first n (y-1) = y"
-          using False True 
-          unfolding last_to_first_def  
+          using False True
+          unfolding last_to_first_def
           using add.commute by auto
         have 1: "\<And>x. last_to_first n x = y \<Longrightarrow> x =(y-1)"
-          unfolding last_to_first_def 
-          using True False 
+          unfolding last_to_first_def
+          using True False
           by auto
-        show ?thesis using 0 1 by blast 
+        show ?thesis using 0 1 by blast
       next
         case F: False
         then have 0: "y \<ge> n"
           using not_less by blast
         then have 1: "last_to_first n y = y"
           by (simp add: \<open>\<forall>x. x \<notin> {..<n} \<longrightarrow> last_to_first n x = x\<close>)
         have 2: "\<And>x. last_to_first n x = y \<Longrightarrow> x =y"
-          using 0 unfolding last_to_first_def 
+          using 0 unfolding last_to_first_def
           using False by presburger
-        then show ?thesis using 1 2 by blast 
+        then show ?thesis using 1 2 by blast
       qed
     qed
   qed
 qed
 
 definition graph_swap where
 "graph_swap n f = permute_list ((first_to_last (n+1))) ` (graph n f)"
 
 lemma last_to_first_eq:
   assumes "length as = n"
   shows "permute_list (last_to_first (n+1)) (a#as)  = (as@[a])"
 proof-
   have 0: "\<And>i. i < (n+1) \<Longrightarrow> permute_list (last_to_first (n + 1)) (a # as) ! i = (as@[a]) ! i"
   proof-
     fix i assume A: "i < n+1"
     show "permute_list (last_to_first (n + 1)) (a # as) ! i = (as @ [a]) ! i"
     proof(cases "i = n")
       case True
       have 0: "(as @ [a]) ! i = a"
         by (metis True assms nth_append_length)
       have 1: "length (a#as) = n + 1"
         by (simp add: assms)
       have 2: "i < length (a # as)"
         using "1" A by linarith
       have 3: "last_to_first (n + 1) permutes {..<length (a # as)}"
         by (metis "1" add_gr_0 last_to_first_permutes less_numeral_extra(1))
       have 4: "permute_list (last_to_first (n + 1)) (a # as) ! i = (a # as) ! last_to_first (n + 1) i"
         using  2 3  permute_list_nth[of "last_to_first (n + 1)" "a#as" i]
         by blast
       have 5: "permute_list (last_to_first (n + 1)) (a # as) ! i = (a # as) ! 0"
-        using 4 unfolding last_to_first_def 
-        by (simp add: True)      
+        using 4 unfolding last_to_first_def
+        by (simp add: True)
       have 6: "permute_list (last_to_first (n + 1)) (a # as) ! i = a"
-        using 5 
+        using 5
         by simp
-      then show ?thesis using 0   by auto     
+      then show ?thesis using 0   by auto
     next
       case False
-      then show ?thesis 
-        by (smt A add.commute add.right_neutral add_diff_cancel_right' add_gr_0 
-            add_less_cancel_left append.simps(1) append.simps(2) assms last_to_first_def 
-            last_to_first_permutes less_SucE less_numeral_extra(1) list.size(3) list.size(4) 
-            nth_append permute_list_nth plus_1_eq_Suc)     
+      then show ?thesis
+        by (smt A add.commute add.right_neutral add_diff_cancel_right' add_gr_0
+            add_less_cancel_left append.simps(1) append.simps(2) assms last_to_first_def
+            last_to_first_permutes less_SucE less_numeral_extra(1) list.size(3) list.size(4)
+            nth_append permute_list_nth plus_1_eq_Suc)
     qed
   qed
   have 1: "length (a#as) = n + 1"
     by (simp add: assms)
   have 2: "length (permute_list (last_to_first (n+1)) (a#as)) = n + 1"
     by (metis "1" length_permute_list)
   have 3: "length (as@[a]) = n + 1"
     by (simp add: assms)
-  then show ?thesis using 0 2  
+  then show ?thesis using 0 2
     by (metis nth_equalityI)
-qed 
+qed
 
 lemma first_to_last_eq:
   assumes "as \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "a \<in> carrier Q\<^sub>p"
   shows "permute_list (first_to_last (n+1)) (as@[a])  = (a#as)"
 proof-
   have "length as = n"
     using assms(1) cartesian_power_car_memE by blast
-  then show ?thesis 
-  using last_to_first_eq last_to_first_permutes[of n] 
-        permute_list_compose_inv(2)[of  "(last_to_first (n + 1))" n "a # as"] 
-  unfolding first_to_last_def 
-  by (metis add_gr_0 assms(1) assms(2) cartesian_power_append last_to_first_permutes 
+  then show ?thesis
+  using last_to_first_eq last_to_first_permutes[of n]
+        permute_list_compose_inv(2)[of  "(last_to_first (n + 1))" n "a # as"]
+  unfolding first_to_last_def
+  by (metis add_gr_0 assms(1) assms(2) cartesian_power_append last_to_first_permutes
       less_one permute_list_closed' permute_list_compose_inv(2))
 qed
 
 lemma graph_swapI:
   assumes "as \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "f \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<rightarrow> carrier Q\<^sub>p"
   shows "(f as)#as \<in> graph_swap n f"
 proof-
   have 0: "as@[f as] \<in> graph n f"
-    using assms using graph_memI[of f n] fun_graph_def 
+    using assms using graph_memI[of f n] fun_graph_def
     by blast
   have 1: "f as \<in> carrier Q\<^sub>p"
-    using assms 
-    by blast
-  then show ?thesis 
+    using assms
+    by blast
+  then show ?thesis
   using assms 0 first_to_last_eq[of as "n" "f as"]
-  unfolding graph_swap_def 
+  unfolding graph_swap_def
   by (metis image_eqI)
 qed
 
 lemma graph_swapE:
   assumes "x \<in> graph_swap n f"
   assumes "f \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<rightarrow> carrier Q\<^sub>p"
   shows "hd x = f (tl x)"
 proof-
   obtain y where y_def: "y \<in> graph n f \<and> x = permute_list (first_to_last (n+1)) y"
-    using assms graph_swap_def 
+    using assms graph_swap_def
     by (smt image_def mem_Collect_eq)
   then have "take n y \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
-    using assms(2) graph_memE(3) 
-    by blast
-  then show "hd x = f (tl x)" 
-    by (metis (no_types, lifting) add.commute assms(2) cartesian_power_car_memE' 
-        first_to_last_eq graph_memE(1) graph_memE(2) graph_mem_closed lessI list.sel(1) 
+    using assms(2) graph_memE(3)
+    by blast
+  then show "hd x = f (tl x)"
+    by (metis (no_types, lifting) add.commute assms(2) cartesian_power_car_memE'
+        first_to_last_eq graph_memE(1) graph_memE(2) graph_mem_closed lessI list.sel(1)
         list.sel(3) plus_1_eq_Suc y_def)
 qed
 
 text\<open>Semialgebraic functions have semialgebraic graphs\<close>
 
 lemma graph_as_partial_pullback:
   assumes "f \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<rightarrow> carrier Q\<^sub>p"
   shows "partial_pullback n f 1 (\<Delta> 1) = graph n f"
 proof
   show "partial_pullback n f 1 (\<Delta> 1) \<subseteq> graph n f"
   proof fix x assume A: "x \<in> partial_pullback n f 1 (\<Delta> 1)"
     then have 0: "f (take n x) # drop n x \<in> \<Delta>  1"
       by (metis local.partial_image_def partial_pullback_memE(2))
     then have 1: "length (f (take n x) # drop n x)  = 2"
       using diagonal_def
       by (metis (no_types, lifting) cartesian_power_car_memE mem_Collect_eq one_add_one)
     then obtain b where b_def: "[b] = drop n x"
       by (metis list.inject pair_id)
     then have "[f (take n x), b] \<in> \<Delta>  1"
-      using "0" 
+      using "0"
       by presburger
     then have "b = f (take n x)"
-      using 0 
+      using 0
       by (smt One_nat_def Qp.cring_axioms diagonal_def drop0 drop_Suc_Cons list.inject mem_Collect_eq take_Suc_Cons)
     then have "x = (take n x)@[f (take n x)]"
       by (metis append_take_drop_id b_def)
-    then show "x \<in> graph n f" using graph_memI[of f n x] 
-      by (metis (no_types, lifting) A \<open>b = f (take n x)\<close> 
+    then show "x \<in> graph n f" using graph_memI[of f n x]
+      by (metis (no_types, lifting) A \<open>b = f (take n x)\<close>
           assms b_def nth_via_drop partial_pullback_memE(1))
   qed
   show "graph n f \<subseteq> partial_pullback n f 1 (\<Delta>  1)"
   proof fix x
     assume A: "x \<in> graph n f "
     then have 0: "x \<in> carrier (Q\<^sub>p\<^bsup>n+1\<^esup>)"
       using assms graph_mem_closed by blast
     have "x = (take n x) @ [f (take n x)]"
-      using A graph_memE(2)[of f n x] assms 
+      using A graph_memE(2)[of f n x] assms
       by blast
     then have "partial_image n f x =  [f (take n x), f (take n x)]"
       by (metis append_take_drop_id local.partial_image_def same_append_eq)
     then have "partial_image n f x \<in> \<Delta> 1"
       using assms 0 diagonal_def[of 1] Qp.cring_axioms diagonalI[of  "partial_image n f x"]
-       by (metis (no_types, lifting) A append_Cons append_eq_conv_conj 
-           cartesian_power_car_memE cartesian_power_car_memE' graph_memE(1) 
+       by (metis (no_types, lifting) A append_Cons append_eq_conv_conj
+           cartesian_power_car_memE cartesian_power_car_memE' graph_memE(1)
            less_add_one self_append_conv2 Qp.to_R1_closed)
     then show "x \<in> partial_pullback n f 1 (\<Delta>  1)"
-      unfolding partial_pullback_def using 0 
-      by blast
-  qed
-qed      
+      unfolding partial_pullback_def using 0
+      by blast
+  qed
+qed
 
 lemma semialg_graph:
   assumes "is_semialg_function n f"
   shows "is_semialgebraic (n + 1) (graph n f)"
-  using assms graph_as_partial_pullback[of f n] unfolding is_semialg_function_def 
+  using assms graph_as_partial_pullback[of f n] unfolding is_semialg_function_def
   by (metis diag_is_semialgebraic is_semialgebraicE less_imp_le_nat less_numeral_extra(1))
 
 text\<open>Functions induced by polynomials are semialgebraic\<close>
 
 definition var_list_segment where
 "var_list_segment i j = map (\<lambda>i. pvar Q\<^sub>p i) [i..< j]"
 
 lemma var_list_segment_length:
   assumes "i \<le> j"
   shows "length (var_list_segment i j) = j - i"
-  using assms var_list_segment_def 
+  using assms var_list_segment_def
   by fastforce
 
 lemma var_list_segment_entry:
   assumes "k < j - i"
   assumes "i \<le> j"
   shows "var_list_segment i j ! k = pvar Q\<^sub>p (i + k)"
-  using assms var_list_segment_length 
-  unfolding var_list_segment_def  
+  using assms var_list_segment_length
+  unfolding var_list_segment_def
   using nth_map_upt by blast
 
 lemma var_list_segment_is_poly_tuple:
   assumes "i \<le>j"
   assumes "j \<le> n"
   shows "is_poly_tuple n (var_list_segment i j)"
-  apply(rule Qp_is_poly_tupleI) 
+  apply(rule Qp_is_poly_tupleI)
   using assms var_list_segment_entry var_list_segment_length  Qp.cring_axioms pvar_closed[of  _ n]
-  by (metis (no_types, opaque_lifting)  add.commute add_lessD1 diff_add_inverse le_Suc_ex 
+  by (metis (no_types, opaque_lifting)  add.commute add_lessD1 diff_add_inverse le_Suc_ex
       less_diff_conv)
- 
+
 lemma map_by_var_list_segment:
   assumes "as \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "j \<le> n"
   assumes "i \<le> j"
   shows "poly_map n (var_list_segment i j) as = list_segment i j as"
   apply(rule nth_equalityI )
   unfolding poly_map_def var_list_segment_def  list_segment_def restrict_def poly_tuple_eval_def
   apply (metis (full_types) assms(1) length_map)
-  using assms  eval_pvar[of  _ n as] Qp.cring_axioms length_map add.commute  
-        length_upt less_diff_conv less_imp_add_positive nth_map nth_upt 
-        trans_less_add2 
+  using assms  eval_pvar[of  _ n as] Qp.cring_axioms length_map add.commute
+        length_upt less_diff_conv less_imp_add_positive nth_map nth_upt
+        trans_less_add2
   by (smt le_add_diff_inverse2)
 
 lemma map_by_var_list_segment_to_length:
   assumes "as \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "i \<le> n"
   shows "poly_map n (var_list_segment i n) as = drop i as"
   apply(rule nth_equalityI )
-  apply (metis Qp_poly_mapE' assms(1) assms(2) cartesian_power_car_memE length_drop var_list_segment_length)  
+  apply (metis Qp_poly_mapE' assms(1) assms(2) cartesian_power_car_memE length_drop var_list_segment_length)
   using assms map_by_var_list_segment[of as n n i] list_segment_drop[of i as]  cartesian_power_car_memE[of as Q\<^sub>p n]
         map_nth[of ] nth_drop nth_map[of _ "[i..<n]" "(pvar Q\<^sub>p)" ] nth_map[of _ "map (pvar Q\<^sub>p) [i..<n]" "eval_at_point Q\<^sub>p as"]
-  unfolding poly_map_def poly_tuple_eval_def var_list_segment_def restrict_def list_segment_def    
+  unfolding poly_map_def poly_tuple_eval_def var_list_segment_def restrict_def list_segment_def
   by (smt add.commute add_eq_self_zero drop_map drop_upt le_Suc_ex le_refl)
 
 lemma map_tail_by_var_list_segment:
   assumes "as \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
-  assumes "a \<in> carrier Q\<^sub>p"  
+  assumes "a \<in> carrier Q\<^sub>p"
   assumes "i < n"
   shows "poly_map (n+1) (var_list_segment 1 (n+1)) (a#as) = as"
 proof-
   have 0: "(a#as) \<in> carrier (Q\<^sub>p\<^bsup>n+1\<^esup>)"
-    using assms 
+    using assms
     by (meson cartesian_power_cons)
   have 1: "length as = n"
-    using assms cartesian_power_car_memE 
+    using assms cartesian_power_car_memE
     by blast
   have 2: "drop 1 (a # as) = as"
-    using 0 1 using list_segment_drop[of 1 "a#as"] 
+    using 0 1 using list_segment_drop[of 1 "a#as"]
     by (metis One_nat_def drop0 drop_Suc_Cons )
-  have "1 \<le>n + 1" by auto 
+  have "1 \<le>n + 1" by auto
   then show ?thesis
     using 0 2  map_by_var_list_segment_to_length[of "a#as" "n+1" 1]
-  by presburger   
+  by presburger
 qed
 
 lemma Qp_poly_tuple_Cons:
   assumes "is_poly_tuple n fs"
   assumes "f \<in> carrier (Q\<^sub>p[\<X>\<^bsub>k\<^esub>])"
   assumes "k \<le>n"
   shows "is_poly_tuple n (f#fs)"
-  using is_poly_tuple_Cons[of n fs f] poly_ring_car_mono[of  k n] assms 
-  by blast 
-  
+  using is_poly_tuple_Cons[of n fs f] poly_ring_car_mono[of  k n] assms
+  by blast
+
 lemma poly_map_Cons:
   assumes "is_poly_tuple n fs"
   assumes "f \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "poly_map n (f#fs) a = (Qp_ev f a)#poly_map n fs a"
-  using assms poly_map_cons by blast 
+  using assms poly_map_cons by blast
 
 lemma poly_map_append':
   assumes "is_poly_tuple n fs"
   assumes "is_poly_tuple n gs"
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "poly_map n (fs@gs) a = poly_map n fs a @ poly_map n gs a"
-  using assms(3) poly_map_append by blast  
+  using assms(3) poly_map_append by blast
 
 lemma partial_pullback_by_poly:
   assumes "f \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   assumes "S \<subseteq> carrier (Q\<^sub>p\<^bsup>1+k\<^esup>)"
   shows "partial_pullback n (Qp_ev f) k S = poly_tuple_pullback (n+k) S (f# (var_list_segment n (n+k)))"
 proof
   show "partial_pullback n (Qp_ev f) k S \<subseteq> poly_tuple_pullback (n+k) S (f # var_list_segment n (n + k))"
   proof fix x assume A: " x \<in> partial_pullback n (Qp_ev f) k S"
     then obtain as bs where as_bs_def: "as \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<and> bs \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>) \<and> x = as @ bs"
-      using  partial_pullback_memE(1)[of x n "(Qp_ev f)" k S] cartesian_power_decomp 
-      by metis       
-    then have 0: "(Qp_ev f as#bs) \<in> S" 
+      using  partial_pullback_memE(1)[of x n "(Qp_ev f)" k S] cartesian_power_decomp
+      by metis
+    then have 0: "(Qp_ev f as#bs) \<in> S"
       using A partial_pullback_memE'
       by blast
     have 1: "Qp_ev f as = Qp_ev f (as@bs)"
-      using assms as_bs_def poly_eval_cartesian_prod[of as n bs k f] 
+      using assms as_bs_def poly_eval_cartesian_prod[of as n bs k f]
         Qp.cring_axioms [of ]
-      by metis 
-    then have 2: "((Qp_ev f x) #bs) \<in> S" 
-      using "0" as_bs_def 
+      by metis
+    then have 2: "((Qp_ev f x) #bs) \<in> S"
+      using "0" as_bs_def
       by presburger
     have 3: "bs = list_segment n (n+k) x"
-      using as_bs_def list_segment_drop[of n x]  
+      using as_bs_def list_segment_drop[of n x]
       by (metis (no_types, lifting) add_cancel_right_right add_diff_cancel_left'
-          append_eq_append_conv append_take_drop_id cartesian_power_car_memE 
-          length_0_conv length_append length_map length_upt linorder_neqE_nat 
-          list_segment_def not_add_less1)      
+          append_eq_append_conv append_take_drop_id cartesian_power_car_memE
+          length_0_conv length_append length_map length_upt linorder_neqE_nat
+          list_segment_def not_add_less1)
     have 4: "is_poly_tuple (n+k) (f # var_list_segment n (n + k))"
       using Qp_poly_tuple_Cons
-        var_list_segment_is_poly_tuple 
+        var_list_segment_is_poly_tuple
       by (metis add.commute assms(1) dual_order.refl le_add2)
     have 5: "f \<in> carrier (Q\<^sub>p [\<X>\<^bsub>n + k\<^esub>])"
       using poly_ring_car_mono[of n "n + k"] assms le_add1 by blast
     have 6: "is_poly_tuple (n + k) (var_list_segment n (n + k))"
       by (simp add: var_list_segment_is_poly_tuple)
     have 7: "x \<in> carrier (Q\<^sub>p\<^bsup>n + k\<^esup>)"
       using as_bs_def cartesian_power_concat(1) by blast
     hence 8: "poly_map (n+k) (f # var_list_segment n (n + k)) x = (Qp_ev f x)#poly_map (n+k) (var_list_segment n (n + k)) x"
-      using 5 6 7 A poly_map_Cons[of "n + k" "var_list_segment n (n + k)" f x] 4 
-      unfolding partial_pullback_def evimage_def 
-      by blast           
+      using 5 6 7 A poly_map_Cons[of "n + k" "var_list_segment n (n + k)" f x] 4
+      unfolding partial_pullback_def evimage_def
+      by blast
     hence 6: "poly_map (n+k) (f # var_list_segment n (n + k)) x = (Qp_ev f x)#bs"
-      using 3 "7" le_add1 le_refl map_by_var_list_segment by presburger    
+      using 3 "7" le_add1 le_refl map_by_var_list_segment by presburger
     show " x \<in> poly_tuple_pullback (n+k) S (f # var_list_segment n (n + k))"
-      unfolding poly_tuple_pullback_def using 6 
+      unfolding poly_tuple_pullback_def using 6
       by (metis "2" "7" IntI poly_map_apply vimage_eq)
   qed
   show "poly_tuple_pullback (n + k) S (f # var_list_segment n (n + k)) \<subseteq> partial_pullback n (Qp_ev f) k S"
   proof fix x
     assume A: "x \<in> poly_tuple_pullback (n + k) S (f # var_list_segment n (n + k))"
     have 0: "is_poly_tuple (n+k) (f # var_list_segment n (n + k))"
-      using Qp_poly_tuple_Cons assms(1) le_add1 var_list_segment_is_poly_tuple 
+      using Qp_poly_tuple_Cons assms(1) le_add1 var_list_segment_is_poly_tuple
       by blast
     have 1: "x \<in> carrier (Q\<^sub>p\<^bsup>n+k\<^esup>)"
-      using A unfolding poly_tuple_pullback_def 
-      by blast 
+      using A unfolding poly_tuple_pullback_def
+      by blast
     have 2: "poly_map (n+k) (f # var_list_segment n (n + k)) x \<in> S"
-      using 1 assms A unfolding poly_map_def poly_tuple_pullback_def restrict_def 
+      using 1 assms A unfolding poly_map_def poly_tuple_pullback_def restrict_def
       by (metis (no_types, opaque_lifting) Int_commute add.commute evimage_def evimage_eq)
     have 3: "poly_map (n+k) (f # var_list_segment n (n + k)) x = (Qp_ev f x)#(drop n x)"
-      using poly_map_Cons[of "n + k" "var_list_segment n (n + k)" f x] 1 assms(1) map_by_var_list_segment_to_length 
-            le_add1 poly_map_cons by presburger      
+      using poly_map_Cons[of "n + k" "var_list_segment n (n + k)" f x] 1 assms(1) map_by_var_list_segment_to_length
+            le_add1 poly_map_cons by presburger
     have 4: "poly_map (n+k) (f # var_list_segment n (n + k)) x = (Qp_ev f (take n x))#(drop n x)"
-      using assms 1 3 eval_at_points_higher_pow[of  f n "n + k" "x"] le_add1 
-      by (metis  nat_le_iff_add)      
+      using assms 1 3 eval_at_points_higher_pow[of  f n "n + k" "x"] le_add1
+      by (metis  nat_le_iff_add)
     show "x \<in> partial_pullback n (Qp_ev f) k S"
       apply(rule partial_pullback_memI)
-      using 1 apply blast 
-      using 2 3 4 by metis       
+      using 1 apply blast
+      using 2 3 4 by metis
   qed
 qed
 
 lemma poly_is_semialg:
   assumes "f \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   shows "is_semialg_function n (Qp_ev f)"
 proof(rule is_semialg_functionI)
   show "Qp_ev f \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<rightarrow> carrier Q\<^sub>p"
-    using assms 
+    using assms
     by (meson Pi_I eval_at_point_closed)
   show "\<And>k S. S \<in> semialg_sets (1 + k) \<Longrightarrow> is_semialgebraic (n + k) (partial_pullback n (Qp_ev f) k S)"
   proof- fix k::nat fix S
     assume A: "S \<in> semialg_sets (1 + k)"
     have 0: "is_poly_tuple (n + k) (f # var_list_segment n (n + k))"
-      by (metis add.commute assms  le_add2 order_refl Qp_poly_tuple_Cons 
+      by (metis add.commute assms  le_add2 order_refl Qp_poly_tuple_Cons
           var_list_segment_is_poly_tuple)
     have 1: "length (f # var_list_segment n (n + k)) = k + 1"
       by (metis add.commute add_diff_cancel_left' le_add1 length_Cons
           plus_1_eq_Suc var_list_segment_length)
     have 2: "partial_pullback n (Qp_ev f) k S = poly_tuple_pullback (n + k) S (f # var_list_segment n (n + k))"
-      using A assms partial_pullback_by_poly[of f n S k] 
-      unfolding semialg_sets_def 
-      using gen_boolean_algebra_subset 
+      using A assms partial_pullback_by_poly[of f n S k]
+      unfolding semialg_sets_def
+      using gen_boolean_algebra_subset
       by blast
     then show "is_semialgebraic (n + k) (partial_pullback n (Qp_ev f) k S)"
       using  add.commute[of 1 k] 0 1 assms(1)
-            pullback_is_semialg[of "n+k" "(f # var_list_segment n (n + k))" "k+1" S] 
-      by (metis A is_semialgebraicI is_semialgebraic_closed poly_tuple_pullback_eq_poly_map_vimage)      
+            pullback_is_semialg[of "n+k" "(f # var_list_segment n (n + k))" "k+1" S]
+      by (metis A is_semialgebraicI is_semialgebraic_closed poly_tuple_pullback_eq_poly_map_vimage)
   qed
 qed
 
 text\<open>Families of polynomials defined by semialgebraic coefficient functions\<close>
 
 lemma semialg_function_on_carrier:
   assumes "is_semialg_function n f"
   assumes "restrict f (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) = restrict g (carrier (Q\<^sub>p\<^bsup>n\<^esup>))"
   shows "is_semialg_function n g"
 proof(rule is_semialg_functionI)
   have 0: "f \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<rightarrow> carrier Q\<^sub>p"
-    using assms(1) is_semialg_function_closed 
+    using assms(1) is_semialg_function_closed
     by blast
   show "g \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<rightarrow> carrier Q\<^sub>p"
   proof fix x assume A: "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)" then show " g x \<in> carrier Q\<^sub>p"
-      using assms(2) 0 
+      using assms(2) 0
       by (metis (no_types, lifting) PiE restrict_Pi_cancel)
-  qed    
+  qed
   show "\<And>k S. S \<in> semialg_sets (1 + k) \<Longrightarrow> is_semialgebraic (n + k) (partial_pullback n g k S)"
-  proof- fix k S 
+  proof- fix k S
     assume A: "S \<in> semialg_sets (1 + k)"
     have 1: "is_semialgebraic (n + k) (partial_pullback n f k S)"
-      using A assms(1) is_semialg_functionE is_semialgebraicI 
+      using A assms(1) is_semialg_functionE is_semialgebraicI
       by blast
     have 2: "(partial_pullback n f k S) = (partial_pullback n g k S)"
       unfolding partial_pullback_def partial_image_def evimage_def
     proof
       show "(\<lambda>xs. f (take n xs) # drop n xs) -` S \<inter> carrier (Q\<^sub>p\<^bsup>n+k\<^esup>) \<subseteq> (\<lambda>xs. g (take n xs) # drop n xs) -` S \<inter> carrier (Q\<^sub>p\<^bsup>n+k\<^esup>)"
       proof fix x assume "x \<in> (\<lambda>xs. f (take n xs) # drop n xs) -` S \<inter> carrier (Q\<^sub>p\<^bsup>n+k\<^esup>) "
         have "(take n x) \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
-          using assms 
-          by (meson \<open>x \<in> (\<lambda>xs. f (take n xs) # drop n xs) -` S \<inter> carrier (Q\<^sub>p\<^bsup>n+k\<^esup>)\<close> 
+          using assms
+          by (meson \<open>x \<in> (\<lambda>xs. f (take n xs) # drop n xs) -` S \<inter> carrier (Q\<^sub>p\<^bsup>n+k\<^esup>)\<close>
               inf_le2 le_add1 subset_iff take_closed)
         then have "f (take n x) = g (take n x)"
           using assms unfolding  restrict_def
-          by meson 
+          by meson
         then show " x \<in> (\<lambda>xs. g (take n xs) # drop n xs) -` S \<inter> carrier (Q\<^sub>p\<^bsup>n+k\<^esup>)"
-          using assms \<open>x \<in> (\<lambda>xs. f (take n xs) # drop n xs) -` S \<inter> carrier (Q\<^sub>p\<^bsup>n+k\<^esup>)\<close> 
+          using assms \<open>x \<in> (\<lambda>xs. f (take n xs) # drop n xs) -` S \<inter> carrier (Q\<^sub>p\<^bsup>n+k\<^esup>)\<close>
           by blast
       qed
       show "(\<lambda>xs. g (take n xs) # drop n xs) -` S \<inter> carrier (Q\<^sub>p\<^bsup>n+k\<^esup>) \<subseteq> (\<lambda>xs. f (take n xs) # drop n xs) -` S \<inter> carrier (Q\<^sub>p\<^bsup>n+k\<^esup>)"
       proof fix x assume A: "x \<in> (\<lambda>xs. g (take n xs) # drop n xs) -` S \<inter> carrier (Q\<^sub>p\<^bsup>n+k\<^esup>)"
         have "(take n x) \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
-          using assms 
+          using assms
           by (meson A  inf_le2 le_add1 subset_iff take_closed)
         then have "f (take n x) = g (take n x)"
           using assms unfolding  restrict_def
-          by meson 
+          by meson
         then show "x \<in> (\<lambda>xs. f (take n xs) # drop n xs) -` S \<inter> carrier (Q\<^sub>p\<^bsup>n+k\<^esup>)"
           using A by blast
       qed
     qed
-    then show "is_semialgebraic (n + k) (partial_pullback n g k S)" 
-      using 1 by auto 
+    then show "is_semialgebraic (n + k) (partial_pullback n g k S)"
+      using 1 by auto
   qed
 qed
 
 lemma semialg_function_on_carrier':
   assumes "is_semialg_function n f"
   assumes "\<And>a. a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<Longrightarrow> f a = g a"
   shows "is_semialg_function n g"
-  using assms semialg_function_on_carrier unfolding restrict_def 
+  using assms semialg_function_on_carrier unfolding restrict_def
   by (meson restrict_ext semialg_function_on_carrier)
 
 lemma constant_function_is_semialg:
   assumes "n > 0"
   assumes "x \<in> carrier Q\<^sub>p"
   assumes "\<And> a. a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<Longrightarrow> f a = x"
   shows "is_semialg_function n f"
 proof(rule semialg_function_on_carrier[of _ "Qp_ev (Qp_to_IP x)"])
   show "is_semialg_function n (Qp_ev (Qp_to_IP x))"
-    using assms poly_is_semialg[of "(Qp_to_IP x)"] Qp_to_IP_car 
+    using assms poly_is_semialg[of "(Qp_to_IP x)"] Qp_to_IP_car
     by blast
   have 0: "\<And> a. a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<Longrightarrow> f a = Qp_ev (Qp_to_IP x) a"
-    using eval_at_point_const assms 
+    using eval_at_point_const assms
     by blast
   then show "restrict (Qp_ev (Qp_to_IP x)) (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) = restrict f (carrier (Q\<^sub>p\<^bsup>n\<^esup>))"
     by (metis (no_types, lifting) restrict_ext)
 qed
 
 lemma cartesian_product_singleton_factor_projection_is_semialg:
   assumes "A \<subseteq> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
   assumes "b \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "is_semialgebraic (m+n) (cartesian_product A {b})"
   shows "is_semialgebraic m A"
 proof-
   obtain f where f_def: "f = map (pvar Q\<^sub>p) [0..<m]"
-    by blast  
+    by blast
   have 0: "is_poly_tuple m f"
-    using assms var_list_segment_is_poly_tuple[of 0 m m] 
-    unfolding var_list_segment_def f_def by blast  
+    using assms var_list_segment_is_poly_tuple[of 0 m m]
+    unfolding var_list_segment_def f_def by blast
   have 4: "length f = m"
-    unfolding f_def using length_map[of "pvar Q\<^sub>p" "[0..<m]"] by auto 
+    unfolding f_def using length_map[of "pvar Q\<^sub>p" "[0..<m]"] by auto
   obtain g where g_def: "(g::(nat multiset \<Rightarrow> ((nat \<Rightarrow> int) \<times> (nat \<Rightarrow> int)) set) list) = map (\<lambda>i::nat. Qp.indexed_const (b ! i)) [(0::nat)..<n]"
-    by blast 
+    by blast
   have 1: "is_poly_tuple m g"
   proof-
     have 0: "set [0::nat..< n] = {..<n}"
       using atLeast_upt by blast
     then have "\<And>i. i \<in> set [0::nat..< n] \<Longrightarrow> b!i \<in> carrier Q\<^sub>p"
       using assms(2) cartesian_power_car_memE'[of b Q\<^sub>p n]  by blast
     hence 1: "\<And>i. i \<in> set [0::nat..< n] \<Longrightarrow> Qp.indexed_const (b ! i) \<in> carrier (Q\<^sub>p[\<X>\<^bsub>m\<^esub>])"
       using assms Qp_to_IP_car by blast
-    show ?thesis 
-      unfolding is_poly_tuple_def g_def 
+    show ?thesis
+      unfolding is_poly_tuple_def g_def
       apply(rule subsetI)
-       using set_map[of "\<lambda>i. Qp.indexed_const (b ! i)" "[0..<n]"]  1 unfolding 0 
-       by (smt image_iff)    
+       using set_map[of "\<lambda>i. Qp.indexed_const (b ! i)" "[0..<n]"]  1 unfolding 0
+       by (smt image_iff)
   qed
   have 2: "is_poly_tuple m (f@g)"
     using 0 1 Qp_is_poly_tuple_append assms(3) by blast
   have 3: "\<And>x. x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<Longrightarrow> poly_tuple_eval (f@g) x = x@b"
   proof- fix x assume A: "x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
     have 30: "poly_tuple_eval f x = x"
     proof-
       have 300: "length (poly_tuple_eval f x) = length x"
-        unfolding poly_tuple_eval_def using cartesian_power_car_memE 
+        unfolding poly_tuple_eval_def using cartesian_power_car_memE
         by (metis "4" A length_map)
-      have "\<And>i. i < length x \<Longrightarrow> poly_tuple_eval f x ! i = x ! i"  
-        unfolding f_def poly_tuple_eval_def using nth_map 
+      have "\<And>i. i < length x \<Longrightarrow> poly_tuple_eval f x ! i = x ! i"
+        unfolding f_def poly_tuple_eval_def using nth_map
         by (metis "4" A add_cancel_right_left cartesian_power_car_memE eval_pvar f_def length_map nth_upt)
-      thus ?thesis using 300 
+      thus ?thesis using 300
         by (metis nth_equalityI)
     qed
     have 31: "poly_tuple_eval g x = b"
     proof-
       have 310: "length (poly_tuple_eval g x) = length b"
-        unfolding poly_tuple_eval_def g_def  using cartesian_power_car_memE 
+        unfolding poly_tuple_eval_def g_def  using cartesian_power_car_memE
         by (metis assms(2) length_map map_nth)
-      have 311: "length b = n" using assms cartesian_power_car_memE by blast 
+      have 311: "length b = n" using assms cartesian_power_car_memE by blast
       hence "\<And>i. i < n \<Longrightarrow> poly_tuple_eval g x ! i = b ! i" proof- fix i assume "i < n"
         thus "poly_tuple_eval g x ! i = b ! i"
-          unfolding g_def poly_tuple_eval_def  using eval_at_point_const[of "b!i" x m] 310 nth_map 
+          unfolding g_def poly_tuple_eval_def  using eval_at_point_const[of "b!i" x m] 310 nth_map
           by (metis "311" A assms(2) cartesian_power_car_memE' length_map map_nth)
       qed
-      thus ?thesis using 311 310 nth_equalityI 
+      thus ?thesis using 311 310 nth_equalityI
         by (metis list_eq_iff_nth_eq)
     qed
     have 32: "poly_tuple_eval (f @ g) x = poly_map  m (f@g) x"
-      unfolding poly_map_def restrict_def using A 
+      unfolding poly_map_def restrict_def using A
       by (simp add: A)
     have 33: "poly_tuple_eval f x = poly_map  m f x"
-      unfolding poly_map_def restrict_def using A 
+      unfolding poly_map_def restrict_def using A
       by (simp add: A)
     have 34: "poly_tuple_eval g x = poly_map  m g x"
-      unfolding poly_map_def restrict_def using A 
+      unfolding poly_map_def restrict_def using A
       by (simp add: A)
-    show "poly_tuple_eval (f @ g) x = x @ b" 
-      using assms 1 2 30 31  poly_map_append[of x m f g] A unfolding 32 33 34  
+    show "poly_tuple_eval (f @ g) x = x @ b"
+      using assms 1 2 30 31  poly_map_append[of x m f g] A unfolding 32 33 34
       by (simp add: A \<open>b \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)\<close>)
   qed
   have 4: "A = (poly_tuple_eval (f@g)  \<inverse>\<^bsub>m\<^esub> (cartesian_product A {b}))"
   proof
     show "A \<subseteq> poly_tuple_eval (f @ g) \<inverse>\<^bsub>m\<^esub> cartesian_product A {b}"
-    proof(rule subsetI) fix x assume A: "x \<in> A" 
+    proof(rule subsetI) fix x assume A: "x \<in> A"
       then have 0: "poly_tuple_eval (f@g) x = x@b"
         using 3 assms by blast
       then show " x \<in> poly_tuple_eval (f @ g) \<inverse>\<^bsub>m\<^esub> cartesian_product A {b}"
-        using A cartesian_product_memE 
+        using A cartesian_product_memE
         by (smt Un_upper1 assms(1) assms(2) cartesian_product_memI' evimageI2 in_mono insert_is_Un mk_disjoint_insert singletonI)
     qed
     show "poly_tuple_eval (f @ g) \<inverse>\<^bsub>m\<^esub> cartesian_product A {b} \<subseteq> A"
     proof(rule subsetI) fix x assume A: "x \<in> (poly_tuple_eval (f @ g) \<inverse>\<^bsub>m\<^esub> cartesian_product A {b})"
       then have "poly_tuple_eval (f @ g) x \<in> cartesian_product A {b}"
-        by blast 
+        by blast
       then have "x@b \<in> cartesian_product A {b}"
         using A 3 by (metis evimage_eq)
       then show "x \<in> A"
-        using A 
-        by (metis append_same_eq cartesian_product_memE' singletonD)        
+        using A
+        by (metis append_same_eq cartesian_product_memE' singletonD)
     qed
   qed
   have 5: "A = poly_map m (f@g)  \<inverse>\<^bsub>m\<^esub> (cartesian_product A {b})"
   proof
     show "A \<subseteq> poly_map m (f @ g) \<inverse>\<^bsub>m\<^esub> cartesian_product A {b}"
-      unfolding poly_map_def evimage_def restrict_def using 4    
-      by (smt IntI assms(1) evimageD in_mono subsetI vimageI)    
+      unfolding poly_map_def evimage_def restrict_def using 4
+      by (smt IntI assms(1) evimageD in_mono subsetI vimageI)
     show "poly_map m (f @ g) \<inverse>\<^bsub>m\<^esub> cartesian_product A {b} \<subseteq> A"
-      unfolding poly_map_def evimage_def restrict_def using 4    
+      unfolding poly_map_def evimage_def restrict_def using 4
       by (smt Int_iff evimageI2 subsetI vimage_eq)
   qed
   have 6: "length (f @ g) = m + n"
     unfolding f_def g_def by (metis index_list_length length_append length_map map_nth)
-  show ?thesis using 2 5 6 assms pullback_is_semialg[of m "f@g" "m+n" "cartesian_product A {b}"] 
+  show ?thesis using 2 5 6 assms pullback_is_semialg[of m "f@g" "m+n" "cartesian_product A {b}"]
     by (metis is_semialgebraicE zero_eq_add_iff_both_eq_0)
 qed
 
 lemma cartesian_product_factor_projection_is_semialg:
   assumes "A \<subseteq> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
   assumes "B \<subseteq> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "B \<noteq> {}"
   assumes "is_semialgebraic (m+n) (cartesian_product A B)"
   shows "is_semialgebraic m A"
 proof-
   obtain b where b_def: "b \<in> B"
     using assms by blast
   have "is_semialgebraic n {b}"
     using assms b_def is_algebraic_imp_is_semialg singleton_is_algebraic by blast
   hence 0: "is_semialgebraic (m+n) (cartesian_product (carrier (Q\<^sub>p\<^bsup>m\<^esup>)) {b})"
     using car_times_semialg_is_semialg assms(4)  by blast
-  have "(cartesian_product (carrier (Q\<^sub>p\<^bsup>m\<^esup>)) {b}) \<inter> (cartesian_product A B) 
+  have "(cartesian_product (carrier (Q\<^sub>p\<^bsup>m\<^esup>)) {b}) \<inter> (cartesian_product A B)
           = (cartesian_product A {b})"
-    using assms b_def cartesian_product_intersection[of "carrier (Q\<^sub>p\<^bsup>m\<^esup>)" Q\<^sub>p m "{b}" n A B] 
+    using assms b_def cartesian_product_intersection[of "carrier (Q\<^sub>p\<^bsup>m\<^esup>)" Q\<^sub>p m "{b}" n A B]
     by (metis (no_types, lifting) Int_absorb1 Int_empty_left Int_insert_left_if1 \<open>is_semialgebraic n {b}\<close> is_semialgebraic_closed set_eq_subset)
   hence "is_semialgebraic (m+n) (cartesian_product A {b})"
     using assms 0  intersection_is_semialg by metis
   thus ?thesis using assms cartesian_product_singleton_factor_projection_is_semialg
     by (meson \<open>is_semialgebraic n {b}\<close> insert_subset is_semialgebraic_closed)
 qed
 
 lemma partial_pullback_cartesian_product:
   assumes "\<xi> \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<rightarrow> carrier Q\<^sub>p"
   assumes "S \<subseteq> carrier (Q\<^sub>p\<^bsup>1\<^esup>)"
   shows "cartesian_product (partial_pullback m \<xi> 0 S) (carrier (Q\<^sub>p\<^bsup>1\<^esup>)) = partial_pullback m \<xi> 1 (cartesian_product S (carrier (Q\<^sub>p\<^bsup>1\<^esup>))) "
 proof
   show "cartesian_product (partial_pullback m \<xi> 0 S) (carrier (Q\<^sub>p\<^bsup>1\<^esup>)) \<subseteq> partial_pullback m \<xi> 1 (cartesian_product S (carrier (Q\<^sub>p\<^bsup>1\<^esup>)))"
   proof fix x assume A: "x \<in> cartesian_product (partial_pullback m \<xi> 0 S) (carrier (Q\<^sub>p\<^bsup>1\<^esup>))"
     then obtain y t where yt_def: "x = y@[t] \<and> y \<in> partial_pullback m \<xi> 0 S \<and> t \<in> carrier Q\<^sub>p"
       by (metis cartesian_product_memE' Qp.to_R1_to_R Qp.to_R_pow_closed)
     then have "[\<xi> y] \<in> S"
-      using partial_pullback_memE unfolding partial_image_def 
+      using partial_pullback_memE unfolding partial_image_def
       by (metis (no_types, lifting) add.right_neutral append.right_neutral cartesian_power_drop le_zero_eq take_closed partial_pullback_memE' take_eq_Nil)
     then have 0: "[\<xi> y]@[t] \<in> cartesian_product S (carrier (Q\<^sub>p\<^bsup>1\<^esup>))"
-      using cartesian_product_memI' yt_def 
+      using cartesian_product_memI' yt_def
       by (metis assms(2) carrier_is_semialgebraic is_semialgebraic_closed Qp.to_R1_closed)
     have 1: " x \<in> carrier (Q\<^sub>p\<^bsup>m + 1\<^esup>)"
-      using A yt_def 
+      using A yt_def
       by (metis add.right_neutral cartesian_power_append partial_pullback_memE(1))
     show "x \<in> partial_pullback m \<xi> 1 (cartesian_product S (carrier (Q\<^sub>p\<^bsup>1\<^esup>)))"
       apply(rule partial_pullback_memI)
       using "1" apply blast
-      using yt_def 0 
+      using yt_def 0
       by (smt Cons_eq_appendI add.right_neutral local.partial_image_def partial_image_eq partial_pullback_memE(1) self_append_conv2 Qp.to_R1_closed)
   qed
   show "partial_pullback m \<xi> 1 (cartesian_product S (carrier (Q\<^sub>p\<^bsup>1\<^esup>))) \<subseteq> cartesian_product (partial_pullback m \<xi> 0 S) (carrier (Q\<^sub>p\<^bsup>1\<^esup>))"
   proof(rule subsetI) fix x assume A: "x \<in> partial_pullback m \<xi> 1 (cartesian_product S (carrier (Q\<^sub>p\<^bsup>1\<^esup>)))"
-    then have 0: "x \<in> carrier (Q\<^sub>p\<^bsup>m + 1\<^esup>)" 
+    then have 0: "x \<in> carrier (Q\<^sub>p\<^bsup>m + 1\<^esup>)"
       using assms partial_pullback_memE[of x m \<xi> 1 "cartesian_product S (carrier (Q\<^sub>p\<^bsup>1\<^esup>))"]
-      by blast 
+      by blast
     have 1: "\<xi> (take m x) # drop m x \<in> cartesian_product S (carrier (Q\<^sub>p\<^bsup>1\<^esup>))"
       using A assms partial_pullback_memE[of x m \<xi> 1 "cartesian_product S (carrier (Q\<^sub>p\<^bsup>1\<^esup>))"]
-      unfolding partial_image_def 
-      by blast 
+      unfolding partial_image_def
+      by blast
     have 2: "\<xi> (take m (take m x)) # drop m (take m x)  = [\<xi> (take m x)]"
-      using 0 1 
+      using 0 1
       by (metis add.commute add.right_neutral append.right_neutral append_take_drop_id take0 take_drop)
     show "x \<in> cartesian_product (partial_pullback m \<xi> 0 S) (carrier (Q\<^sub>p\<^bsup>1\<^esup>))"
       apply(rule cartesian_product_memI[of _  Q\<^sub>p m _ 1])
          apply (metis add_cancel_right_right partial_pullback_closed)
-        apply blast 
-        apply(rule partial_pullback_memI[of _ m 0 \<xi> S]) using 0 
+        apply blast
+        apply(rule partial_pullback_memI[of _ m 0 \<xi> S]) using 0
        apply (metis Nat.add_0_right le_iff_add take_closed)
       using 2 apply (metis (no_types, lifting) "1" add.commute add.right_neutral assms(2) cartesian_product_memE(1) list.inject plus_1_eq_Suc take_Suc_Cons take_drop)
     using 0  cartesian_power_drop by blast
-  qed      
+  qed
 qed
 
 lemma cartesian_product_swap:
   assumes "A \<subseteq> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "B \<subseteq> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
   assumes "is_semialgebraic (m+n) (cartesian_product A B)"
   shows "is_semialgebraic (m+n) (cartesian_product B A)"
 proof-
   obtain f where f_def: "f = (\<lambda>i. (if i < m then n + i else (if i < m+n then i - m else i)))"
-    by blast 
+    by blast
   have 0: "\<And>i. i \<in> {..<m} \<longrightarrow> f i \<in> {n..<m+n}"
     unfolding f_def by simp
   have 1: "\<And>i. i \<in> {m..<m+n} \<longrightarrow> f i \<in> {..<n}"
     unfolding f_def by (simp add: less_diff_conv2)
   have 2: "\<And>i. i \<notin> {..<m + n} \<longrightarrow> f i \<notin> {..<m + n}"
     unfolding f_def by simp
   have f_permutes: "f permutes {..<m+n}"
     unfolding permutes_def
   proof
     show "\<forall>x. x \<notin> {..<m + n} \<longrightarrow> f x = x"
       unfolding f_def by simp
     show "\<forall>y. \<exists>!x. f x = y"
     proof fix y
       show "\<exists>!x. f x = y"
       proof(cases "y < n")
         case True
         have T0: "f (y+m) = y"
-          unfolding f_def using True  
+          unfolding f_def using True
           by simp
         have "\<And>i. f i = y \<Longrightarrow> i \<in> {m..<m+n}"
           using 0 1 2 True f_def nat_neq_iff by fastforce
         hence "\<And>i. f i = y \<Longrightarrow> i = y+m"
           using T0 unfolding f_def by auto
-        thus ?thesis using T0 by blast 
+        thus ?thesis using T0 by blast
       next
         case False
         show ?thesis
         proof(cases  "y \<in> {n..<m+n}")
           case True
           have T0: "f (y-n) = y"
             using True unfolding f_def by auto
           have "\<And>i. f i = y \<Longrightarrow> i \<in> {..<m}"
-            using 0 1 2 True f_def 
-            by (metis False atLeastLessThan_iff diff_add_inverse2 diff_diff_cancel diff_le_self 
-                lessThan_iff less_imp_diff_less linordered_semidom_class.add_diff_inverse nat_neq_iff not_add_less1)            
+            using 0 1 2 True f_def
+            by (metis False atLeastLessThan_iff diff_add_inverse2 diff_diff_cancel diff_le_self
+                lessThan_iff less_imp_diff_less linordered_semidom_class.add_diff_inverse nat_neq_iff not_add_less1)
           hence "\<And>i. f i = y \<Longrightarrow> i = y- n"
             using f_def by force
-          then show ?thesis using T0 by blast 
+          then show ?thesis using T0 by blast
         next
           case F: False
-          then show ?thesis using 0 1 2 unfolding f_def 
+          then show ?thesis using 0 1 2 unfolding f_def
             using False add_diff_inverse_nat lessThan_iff by auto
         qed
       qed
     qed
   qed
   have "permute_list f ` (cartesian_product A B) = (cartesian_product B A)"
   proof
     show "permute_list f ` cartesian_product A B \<subseteq> cartesian_product B A"
     proof fix x assume A: " x \<in> permute_list f ` cartesian_product A B"
       then obtain a b where ab_def: "a \<in> A \<and>b \<in> B \<and> x = permute_list f (a@b)"
         by (metis (mono_tags, lifting) cartesian_product_memE' image_iff)
       have 0: "x = permute_list f (a@b)"
-        using ab_def by blast 
+        using ab_def by blast
       have 1: "length a = n"
         using ab_def assms cartesian_power_car_memE[of a Q\<^sub>p n] by blast
       have 2: "length b = m"
         using ab_def assms cartesian_power_car_memE[of b Q\<^sub>p m] by blast
       have 3: "length x = m + n"
         using 1 2 0 f_permutes by simp
       have 4: "\<And>i. i < m \<Longrightarrow> x ! i = (a@b) ! (f i)"
-        unfolding 0 using permute_list_nth  
+        unfolding 0 using permute_list_nth
         by (metis "0" "3" f_permutes length_permute_list trans_less_add1)
       hence 5: "\<And>i. i < m \<Longrightarrow> x ! i = b!i"
-        unfolding f_def using 1 2 
+        unfolding f_def using 1 2
         by (metis "4" f_def nth_append_length_plus)
       have 6: "\<And>i. i \<in> {m..<m+n} \<Longrightarrow> x ! i = (a@b) ! (i - m)"
-        unfolding 0 using f_def permute_list_nth  f_permutes 
-        by (metis (no_types, lifting) "0" "3" atLeastLessThan_iff length_permute_list not_add_less2 
+        unfolding 0 using f_def permute_list_nth  f_permutes
+        by (metis (no_types, lifting) "0" "3" atLeastLessThan_iff length_permute_list not_add_less2
             ordered_cancel_comm_monoid_diff_class.diff_add)
       have 7: "x = b@a"
       proof(rule nth_equalityI)
         show "length x = length (b @ a)"
           using 1 2 3 by simp
         show "\<And>i. i < length x \<Longrightarrow> x ! i = (b @ a) ! i"
-          unfolding 3 using 1 2 4 5 
+          unfolding 3 using 1 2 4 5
           by (smt "0" add.commute add_diff_inverse_nat f_def f_permutes length_append nat_add_left_cancel_less nth_append permute_list_nth)
       qed
-      show "x \<in> cartesian_product B A" unfolding 7 using ab_def unfolding cartesian_product_def by blast    
+      show "x \<in> cartesian_product B A" unfolding 7 using ab_def unfolding cartesian_product_def by blast
     qed
     show "cartesian_product B A \<subseteq> permute_list f ` cartesian_product A B"
     proof fix y assume A: "y \<in> cartesian_product B A"
       then obtain b a where ab_def: "b \<in> B \<and> a \<in> A \<and> y = b@a"
         using cartesian_product_memE' by blast
       obtain x where 0: "x = permute_list f (a@b)"
-        by blast 
+        by blast
       have 1: "length a = n"
         using ab_def assms cartesian_power_car_memE[of a Q\<^sub>p n] by blast
       have 2: "length b = m"
         using ab_def assms cartesian_power_car_memE[of b Q\<^sub>p m] by blast
       have 3: "length x = m + n"
         using 1 2 0 f_permutes by simp
       have 4: "\<And>i. i < m \<Longrightarrow> x ! i = (a@b) ! (f i)"
-        unfolding 0 using permute_list_nth  
+        unfolding 0 using permute_list_nth
         by (metis "0" "3" f_permutes length_permute_list trans_less_add1)
       hence 5: "\<And>i. i < m \<Longrightarrow> x ! i = b!i"
-        unfolding f_def using 1 2 
+        unfolding f_def using 1 2
         by (metis "4" f_def nth_append_length_plus)
       have 6: "\<And>i. i \<in> {m..<m+n} \<Longrightarrow> x ! i = (a@b) ! (i - m)"
-        unfolding 0 using f_def permute_list_nth  f_permutes 
-        by (metis (no_types, lifting) "0" "3" atLeastLessThan_iff length_permute_list not_add_less2 
+        unfolding 0 using f_def permute_list_nth  f_permutes
+        by (metis (no_types, lifting) "0" "3" atLeastLessThan_iff length_permute_list not_add_less2
             ordered_cancel_comm_monoid_diff_class.diff_add)
       have 7: "x = b@a"
       proof(rule nth_equalityI)
         show "length x = length (b @ a)"
           using 1 2 3 by simp
         show "\<And>i. i < length x \<Longrightarrow> x ! i = (b @ a) ! i"
-          unfolding 3 using 1 2 4 5 
+          unfolding 3 using 1 2 4 5
           by (smt "0" add.commute add_diff_inverse_nat f_def f_permutes length_append nat_add_left_cancel_less nth_append permute_list_nth)
       qed
       show "y \<in> permute_list f ` cartesian_product A B"
-        using ab_def 7 cartesian_product_memI'[of _ Q\<^sub>p] unfolding 0  
-        by (metis assms(1) assms(2) image_eqI)       
-    qed
-  qed
-  thus ?thesis using assms f_permutes permutation_is_semialgebraic 
+        using ab_def 7 cartesian_product_memI'[of _ Q\<^sub>p] unfolding 0
+        by (metis assms(1) assms(2) image_eqI)
+    qed
+  qed
+  thus ?thesis using assms f_permutes permutation_is_semialgebraic
     by metis
 qed
 
 lemma Qp_zero_subset_is_semialg:
   assumes "S \<subseteq> carrier (Q\<^sub>p\<^bsup>0\<^esup>)"
   shows "is_semialgebraic 0 S"
 proof(cases "S = {}")
   case True
-  then show ?thesis 
-    by (simp add: empty_is_semialgebraic)  
+  then show ?thesis
+    by (simp add: empty_is_semialgebraic)
 next
   case False
   then have "S = carrier (Q\<^sub>p\<^bsup>0\<^esup>)"
-    using assms unfolding Qp_zero_carrier  by blast 
-  then show ?thesis 
-    by (simp add: carrier_is_semialgebraic)  
+    using assms unfolding Qp_zero_carrier  by blast
+  then show ?thesis
+    by (simp add: carrier_is_semialgebraic)
 qed
 
 lemma cartesian_product_empty_list:
 "cartesian_product A {[]} = A"
 "cartesian_product {[]} A = A"
 proof
   show "cartesian_product A {[]} \<subseteq> A"
     apply(rule subsetI)
-    unfolding cartesian_product_def 
+    unfolding cartesian_product_def
     by (smt append_Nil2 empty_iff insert_iff mem_Collect_eq)
   show "A \<subseteq> cartesian_product A {[]}"
     apply(rule subsetI)
-    unfolding cartesian_product_def 
+    unfolding cartesian_product_def
     by (smt append_Nil2 empty_iff insert_iff mem_Collect_eq)
   show "cartesian_product {[]} A = A"
-  proof 
+  proof
     show "cartesian_product {[]} A \<subseteq> A"
     apply(rule subsetI)
-    unfolding cartesian_product_def 
+    unfolding cartesian_product_def
     by (smt append_self_conv2 bex_empty insert_compr mem_Collect_eq)
   show "A \<subseteq> cartesian_product {[]} A"
     apply(rule subsetI)
-    unfolding cartesian_product_def 
+    unfolding cartesian_product_def
     by blast
   qed
 qed
 
 lemma cartesian_product_singleton_factor_projection_is_semialg':
   assumes "A \<subseteq> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
   assumes "b \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "is_semialgebraic (m+n) (cartesian_product A {b})"
   shows "is_semialgebraic m A"
 proof(cases "n > 0")
   case True
-  show ?thesis 
+  show ?thesis
   proof(cases "m > 0")
     case T: True
-    then show ?thesis 
+    then show ?thesis
       using assms True cartesian_product_singleton_factor_projection_is_semialg by blast
   next
     case False
-    then show ?thesis using Qp_zero_subset_is_semialg assms by blast  
+    then show ?thesis using Qp_zero_subset_is_semialg assms by blast
   qed
 next
   case False
   then have F0: "b = []"
     using assms Qp_zero_carrier by blast
   have "cartesian_product A {b} = A"
-    unfolding F0 
+    unfolding F0
     by (simp add: cartesian_product_empty_list(1))
-  then show ?thesis using assms False 
+  then show ?thesis using assms False
     by (metis add.right_neutral gr0I)
 qed
 
 (**************************************************************************************************)
 (**************************************************************************************************)
-subsection \<open>More on graphs of functions\<close>   
+subsection \<open>More on graphs of functions\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 
-text\<open>This section lays the groundwork for showing that semialgebraic functions are closed under 
+text\<open>This section lays the groundwork for showing that semialgebraic functions are closed under
      various algebraic operations\<close>
 
 text\<open>The take and drop functions on lists are polynomial maps\<close>
 
 lemma function_restriction:
   assumes "g \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<rightarrow> S"
   assumes "n \<le> k"
   shows "(g \<circ> (take n)) \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>) \<rightarrow> S"
-proof fix x 
+proof fix x
   assume "x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>)"
   then have "take n x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
     using assms(2) take_closed
     by blast
   then show "(g \<circ> take n) x \<in> S"
-    using assms comp_apply 
+    using assms comp_apply
     by (metis Pi_iff comp_def)
 qed
 
 lemma partial_pullback_restriction:
   assumes "g \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<rightarrow> carrier Q\<^sub>p"
   assumes "n < k"
-  shows "partial_pullback k (g \<circ> take n) m S = 
+  shows "partial_pullback k (g \<circ> take n) m S =
          split_cartesian_product  (n + m) (k - n) n (partial_pullback n g m S) (carrier (Q\<^sub>p\<^bsup>k - n\<^esup>))"
 proof(rule equalityI)
   show "partial_pullback k (g \<circ> take n) m S \<subseteq> split_cartesian_product (n + m) (k - n) n (partial_pullback n g m S) (carrier (Q\<^sub>p\<^bsup>k - n\<^esup>))"
   proof fix x assume A: "x \<in> partial_pullback k (g \<circ> take n) m S"
     obtain as bs where asbs_def: "x = as@bs \<and> as \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>) \<and> bs \<in> carrier  (Q\<^sub>p\<^bsup>m\<^esup>)"
       using partial_pullback_memE[of x k "g \<circ> take n" m S] A cartesian_power_decomp[of x Q\<^sub>p k m]
       by metis
     have 0: "((g \<circ> (take n)) as)#bs \<in> S"
-      using asbs_def partial_pullback_memE'[of as k bs m x] A 
+      using asbs_def partial_pullback_memE'[of as k bs m x] A
       by blast
     have 1: "(g (take n as))#bs \<in> S"
-      using 0 
+      using 0
       by (metis comp_apply)
     have 2: "take n as @ bs \<in> carrier (Q\<^sub>p\<^bsup>n+m\<^esup>)"
       by (meson asbs_def assms(2) cartesian_power_concat(1) less_imp_le_nat take_closed)
     have 3: "(take n as)@bs \<in> (partial_pullback n g m S)"
       using 1 2 partial_pullback_memI[of "(take n as)@bs" n m g S]
-      by (metis (mono_tags, opaque_lifting) asbs_def assms(2) local.partial_image_def nat_less_le 
+      by (metis (mono_tags, opaque_lifting) asbs_def assms(2) local.partial_image_def nat_less_le
           partial_image_eq subsetD  subset_refl take_closed)
     have 4: "drop n as \<in> (carrier (Q\<^sub>p\<^bsup>k - n\<^esup>))"
       using asbs_def assms(2) drop_closed
       by blast
     show " x \<in> split_cartesian_product (n + m) (k - n) n (partial_pullback n g m S) (carrier (Q\<^sub>p\<^bsup>k - n\<^esup>))"
-      using split_cartesian_product_memI[of "take n as" bs 
-                                        "partial_pullback n g m S" "drop n as" 
-                                         "carrier (Q\<^sub>p\<^bsup>k - n\<^esup>)" Q\<^sub>p "n + m" "k - n" n ]  4 
+      using split_cartesian_product_memI[of "take n as" bs
+                                        "partial_pullback n g m S" "drop n as"
+                                         "carrier (Q\<^sub>p\<^bsup>k - n\<^esup>)" Q\<^sub>p "n + m" "k - n" n ]  4
       by (metis (no_types, lifting) "3" append.assoc append_take_drop_id
-          asbs_def assms(2) cartesian_power_car_memE less_imp_le_nat partial_pullback_memE(1) 
+          asbs_def assms(2) cartesian_power_car_memE less_imp_le_nat partial_pullback_memE(1)
           subsetI take_closed)
-  qed  
+  qed
   show "split_cartesian_product (n + m) (k - n) n (partial_pullback n g m S) (carrier (Q\<^sub>p\<^bsup>k - n\<^esup>)) \<subseteq> partial_pullback k (g \<circ> take n) m S"
   proof fix x assume A: "x \<in> split_cartesian_product (n + m) (k - n) n (partial_pullback n g m S) (carrier (Q\<^sub>p\<^bsup>k - n\<^esup>))"
     show "x \<in> partial_pullback k (g \<circ> take n) m S"
     proof(rule partial_pullback_memI)
       have 0: "(partial_pullback n g m S) \<subseteq> carrier (Q\<^sub>p\<^bsup>n+m\<^esup>)"
         using partial_pullback_closed by blast
       then  have "split_cartesian_product (n + m) (k - n) n (partial_pullback n g m S) (carrier (Q\<^sub>p\<^bsup>k - n\<^esup>)) \<subseteq> carrier (Q\<^sub>p\<^bsup>n + m + (k - n)\<^esup>)"
         using assms A  split_cartesian_product_closed[of "partial_pullback n g m S" Q\<^sub>p "n + m"
-                                                "carrier (Q\<^sub>p\<^bsup>k - n\<^esup>)" "k - n" n] 
+                                                "carrier (Q\<^sub>p\<^bsup>k - n\<^esup>)" "k - n" n]
         using le_add1 by blast
       then show P: "x \<in> carrier (Q\<^sub>p\<^bsup>k+m\<^esup>)"
         by (smt A Nat.add_diff_assoc2 add.commute add_diff_cancel_left' assms(2) le_add1 less_imp_le_nat subsetD)
       have "take n x @ drop (n + (k - n)) x \<in> partial_pullback n g m S"
         using 0 A split_cartesian_product_memE[of x "n + m" "k - n" n "partial_pullback n g m S" "carrier (Q\<^sub>p\<^bsup>k - n\<^esup>)" Q\<^sub>p]
              le_add1 by blast
       have 1: "g (take n x) # drop k x \<in> S"
-        using  partial_pullback_memE   
+        using  partial_pullback_memE
         by (metis (no_types, lifting) \<open>take n x @ drop (n + (k - n)) x \<in> partial_pullback n g m S\<close>
-              \<open>x \<in> carrier (Q\<^sub>p\<^bsup>k+m\<^esup>)\<close> add.assoc assms(2) cartesian_power_drop le_add1 
+              \<open>x \<in> carrier (Q\<^sub>p\<^bsup>k+m\<^esup>)\<close> add.assoc assms(2) cartesian_power_drop le_add1
               le_add_diff_inverse less_imp_le_nat partial_pullback_memE' take_closed)
       have 2: "g (take n x) = (g \<circ> take n) (take k x)"
-        using assms P comp_apply[of g "take n" "take k x"] 
+        using assms P comp_apply[of g "take n" "take k x"]
         by (metis add.commute append_same_eq append_take_drop_id less_imp_add_positive take_add take_drop)
       then show "(g \<circ> take n) (take k x) # drop k x \<in> S"
         using "1" by presburger
     qed
   qed
 qed
 
 lemma comp_take_is_semialg:
   assumes "is_semialg_function n g"
   assumes "n < k"
   assumes "0 < n"
   shows "is_semialg_function k (g \<circ> (take n))"
 proof(rule is_semialg_functionI)
   show "g \<circ> take n \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>) \<rightarrow> carrier Q\<^sub>p"
-    using assms function_restriction[of g n "carrier Q\<^sub>p" k]  dual_order.strict_implies_order 
-        is_semialg_function_closed 
+    using assms function_restriction[of g n "carrier Q\<^sub>p" k]  dual_order.strict_implies_order
+        is_semialg_function_closed
     by blast
   show "\<And>ka S. S \<in> semialg_sets (1 + ka) \<Longrightarrow> is_semialgebraic (k + ka) (partial_pullback k (g \<circ> take n) ka S)"
   proof- fix l S assume A: "S \<in> semialg_sets (1 + l)"
     have 0: "is_semialgebraic (n + l) (partial_pullback n g l S) "
-      using assms A is_semialg_functionE is_semialgebraicI 
-      by blast      
+      using assms A is_semialg_functionE is_semialgebraicI
+      by blast
     have "is_semialgebraic (n + l + (k - n)) (split_cartesian_product (n + l) (k - n) n (partial_pullback n g l S) (carrier (Q\<^sub>p\<^bsup>k - n\<^esup>)))"
-      using A 0 split_cartesian_product_is_semialgebraic[of _ _ 
+      using A 0 split_cartesian_product_is_semialgebraic[of _ _
                                                   "partial_pullback n g l S" _ "carrier (Q\<^sub>p\<^bsup>k - n\<^esup>)"]
-            add_gr_0 assms(2) assms(3) carrier_is_semialgebraic le_add1 zero_less_diff 
+            add_gr_0 assms(2) assms(3) carrier_is_semialgebraic le_add1 zero_less_diff
       by presburger
     then  show "is_semialgebraic (k + l) (partial_pullback k (g \<circ> take n) l S)"
-      using partial_pullback_restriction[of g n k l S] 
+      using partial_pullback_restriction[of g n k l S]
   by (metis (no_types, lifting) add.assoc add.commute assms(1) assms(2) is_semialg_function_closed le_add_diff_inverse less_imp_le_nat)
   qed
 qed
 
 text\<open>Restriction of a graph to a semialgebraic domain\<close>
 
 lemma graph_formula:
   assumes "g \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<rightarrow> carrier Q\<^sub>p"
   shows "graph n g = {as \<in> carrier (Q\<^sub>p\<^bsup>Suc n\<^esup>). g (take n as) = as!n}"
-  using assms graph_memI fun_graph_def[of Q\<^sub>p n g]  
+  using assms graph_memI fun_graph_def[of Q\<^sub>p n g]
   by (smt Collect_cong Suc_eq_plus1 graph_memE(1) graph_mem_closed mem_Collect_eq)
 
 definition restricted_graph where
 "restricted_graph n g S = {as \<in> carrier (Q\<^sub>p\<^bsup>Suc n\<^esup>). take n as \<in> S \<and> g (take n as) = as!n }"
 
 lemma restricted_graph_closed:
  "restricted_graph n g S \<subseteq> carrier (Q\<^sub>p\<^bsup>Suc n\<^esup>)"
   by (metis (no_types, lifting) mem_Collect_eq restricted_graph_def subsetI)
 
 lemma restricted_graph_memE:
   assumes "a \<in> restricted_graph n g S"
-  shows "a \<in> carrier (Q\<^sub>p\<^bsup>Suc n\<^esup>)" "take n a \<in> S" "g (take n a) = a!n" 
-  using assms 
+  shows "a \<in> carrier (Q\<^sub>p\<^bsup>Suc n\<^esup>)" "take n a \<in> S" "g (take n a) = a!n"
+  using assms
   using restricted_graph_closed apply blast
   apply (metis (no_types, lifting) assms mem_Collect_eq restricted_graph_def)
-  using assms unfolding restricted_graph_def 
+  using assms unfolding restricted_graph_def
   by blast
 
 lemma restricted_graph_mem_formula:
   assumes "a \<in> restricted_graph n g S"
   shows "a = (take n a)@[g (take n a)]"
 proof-
   have "length a = Suc n"
-    using assms 
+    using assms
     by (metis (no_types, lifting) cartesian_power_car_memE mem_Collect_eq restricted_graph_def)
   then have "a = (take n a)@[a!n]"
     by (metis append_eq_append_conv_if hd_drop_conv_nth lessI take_hd_drop)
-  then show ?thesis 
+  then show ?thesis
     by (metis assms restricted_graph_memE(3))
 qed
 
-lemma restricted_graph_memI: 
+lemma restricted_graph_memI:
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>Suc n\<^esup>)"
   assumes "take n a \<in> S"
   assumes "g (take n a) = a!n"
   shows "a \<in> restricted_graph n g S"
   using assms restricted_graph_def
   by blast
 
-lemma restricted_graph_memI': 
+lemma restricted_graph_memI':
   assumes "a \<in> S"
   assumes "g \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<rightarrow> carrier Q\<^sub>p"
   assumes "S \<subseteq> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "(a@[g a]) \<in> restricted_graph n g S"
 proof-
   have "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
     using assms(1) assms(3) by blast
   then have "g a \<in> carrier Q\<^sub>p"
-    using assms by blast 
+    using assms by blast
   then have 0: "a @ [g a] \<in> carrier (Q\<^sub>p\<^bsup>Suc n\<^esup>)"
-    using assms 
+    using assms
     by (metis (no_types, lifting) add.commute cartesian_power_append plus_1_eq_Suc subsetD)
   have 1: "take n (a @ [g a]) \<in> S"
-    using assms 
+    using assms
     by (metis (no_types, lifting) append_eq_conv_conj cartesian_power_car_memE subsetD)
-  show ?thesis 
-    using assms restricted_graph_memI[of "a@[g a]" n S g] 
+  show ?thesis
+    using assms restricted_graph_memI[of "a@[g a]" n S g]
     by (metis "0" \<open>a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)\<close> append_eq_conv_conj cartesian_power_car_memE nth_append_length)
 qed
 
 lemma restricted_graph_subset:
   assumes "g \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<rightarrow> carrier Q\<^sub>p"
   assumes "S \<subseteq> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "restricted_graph n g S \<subseteq> graph n g"
 proof fix x  assume A: "x \<in> restricted_graph n g S"
   show "x \<in> graph n g"
     apply(rule graph_memI)
   using assms(1) apply blast
   using A restricted_graph_memE(3) apply blast
   by (metis A add.commute plus_1_eq_Suc restricted_graph_memE(1))
 qed
 
 lemma restricted_graph_subset':
   assumes "g \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<rightarrow> carrier Q\<^sub>p"
   assumes "S \<subseteq> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "restricted_graph n g S \<subseteq> cartesian_product S (carrier (Q\<^sub>p\<^bsup>1\<^esup>))"
 proof fix a assume A: "a \<in> restricted_graph n g S"
   then have "a = (take n a)@[g (take n a)]"
     using restricted_graph_mem_formula by blast
   then show "a \<in> cartesian_product S (carrier (Q\<^sub>p\<^bsup>1\<^esup>))"
-    using cartesian_product_memI' A unfolding restricted_graph_def 
-    by (metis (mono_tags, lifting) assms(2) last_closed' mem_Collect_eq subsetI Qp.to_R1_closed)    
+    using cartesian_product_memI' A unfolding restricted_graph_def
+    by (metis (mono_tags, lifting) assms(2) last_closed' mem_Collect_eq subsetI Qp.to_R1_closed)
 qed
 
 lemma restricted_graph_intersection:
   assumes "g \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<rightarrow> carrier Q\<^sub>p"
   assumes "S \<subseteq> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "restricted_graph n g S = graph n g \<inter> (cartesian_product S (carrier (Q\<^sub>p\<^bsup>1\<^esup>)))"
 proof
   show "restricted_graph n g S \<subseteq> graph n g \<inter> cartesian_product S (carrier (Q\<^sub>p\<^bsup>1\<^esup>))"
-    using assms restricted_graph_subset restricted_graph_subset' 
+    using assms restricted_graph_subset restricted_graph_subset'
     by (meson Int_subset_iff)
   show "graph n g \<inter> cartesian_product S (carrier (Q\<^sub>p\<^bsup>1\<^esup>)) \<subseteq> restricted_graph n g S"
   proof fix x assume A: " x \<in> graph n g \<inter> cartesian_product S (carrier (Q\<^sub>p\<^bsup>1\<^esup>))"
     show "x \<in> restricted_graph n g S"
       apply(rule restricted_graph_memI)
-      using A graph_memE[of g n x]  
+      using A graph_memE[of g n x]
       apply (metis (no_types, lifting) Int_iff add.commute assms(1) graph_mem_closed plus_1_eq_Suc)
-          using A graph_memE[of g n x] cartesian_product_memE[of x S "carrier (Q\<^sub>p\<^bsup>1\<^esup>)" Q\<^sub>p n] 
+          using A graph_memE[of g n x] cartesian_product_memE[of x S "carrier (Q\<^sub>p\<^bsup>1\<^esup>)" Q\<^sub>p n]
           using assms(2) apply blast
-          using A graph_memE[of g n x] cartesian_product_memE[of x S "carrier (Q\<^sub>p\<^bsup>1\<^esup>)" Q\<^sub>p n] 
+          using A graph_memE[of g n x] cartesian_product_memE[of x S "carrier (Q\<^sub>p\<^bsup>1\<^esup>)" Q\<^sub>p n]
           using assms(1) by blast
    qed
  qed
 
 lemma restricted_graph_is_semialgebraic:
   assumes "is_semialg_function n g"
   assumes "is_semialgebraic n S"
   shows "is_semialgebraic (n+1) (restricted_graph n g S)"
 proof-
   have 0: "restricted_graph n g S = graph n g \<inter> (cartesian_product S (carrier (Q\<^sub>p\<^bsup>1\<^esup>)))"
-    using assms is_semialg_function_closed is_semialgebraic_closed 
+    using assms is_semialg_function_closed is_semialgebraic_closed
       restricted_graph_intersection by presburger
   have 1: "is_semialgebraic (n + 1) (graph n g)"
     using assms semialg_graph
     by blast
   have 2: "is_semialgebraic (n + 1) (cartesian_product S (carrier (Q\<^sub>p\<^bsup>1\<^esup>)))"
     using cartesian_product_is_semialgebraic[of n S 1 "carrier (Q\<^sub>p\<^bsup>1\<^esup>)"]  assms
       carrier_is_semialgebraic less_one
     by presburger
   then show ?thesis
-    using 0 1 2 intersection_is_semialg[of "n+1" "graph n g" "cartesian_product S (carrier (Q\<^sub>p\<^bsup>1\<^esup>))"]  
+    using 0 1 2 intersection_is_semialg[of "n+1" "graph n g" "cartesian_product S (carrier (Q\<^sub>p\<^bsup>1\<^esup>))"]
     by presburger
 qed
 
-lemma take_closed: 
+lemma take_closed:
   assumes "n \<le> k"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>)"
   shows "take n x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
-  using assms take_closed 
+  using assms take_closed
   by blast
 
 lemma take_compose_closed:
   assumes "g \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<rightarrow> carrier Q\<^sub>p"
   assumes "n < k"
   shows "g \<circ> take n \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>) \<rightarrow> carrier Q\<^sub>p"
 proof fix x assume A: "x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>)"
   then have "(take n x) \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
-    using assms  less_imp_le_nat take_closed 
+    using assms  less_imp_le_nat take_closed
     by blast
   then have "g (take n x) \<in> carrier Q\<^sub>p"
-    using assms(1) by blast 
+    using assms(1) by blast
   then show "(g \<circ> take n) x \<in> carrier Q\<^sub>p"
-    using comp_apply[of g "take n" x] 
+    using comp_apply[of g "take n" x]
     by presburger
 qed
 
 lemma take_graph_formula:
   assumes "g \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<rightarrow> carrier Q\<^sub>p"
   assumes "n < k"
   assumes "0 < n"
   shows "graph k (g \<circ> (take n)) = {as \<in> carrier (Q\<^sub>p\<^bsup>k+1\<^esup>). g (take n as) = as!k}"
 proof-
   have "\<And>as. as \<in> carrier (Q\<^sub>p\<^bsup>k+1\<^esup>) \<Longrightarrow> (g \<circ> take n) (take k as) = g (take n as) "
-    using assms comp_apply take_take[of n k] 
+    using assms comp_apply take_take[of n k]
   proof -
     fix as :: "((nat \<Rightarrow> int) \<times> (nat \<Rightarrow> int)) set list"
     show "(g \<circ> take n) (take k as) = g (take n as)"
       by (metis (no_types) \<open>n < k\<close> comp_eq_dest_lhs min.strict_order_iff take_take)
   qed
   then show ?thesis
     using take_compose_closed[of g n k] assms comp_apply[of g "take n"] graph_formula[of "g \<circ> (take n)" k]
-   by (smt Collect_cong Suc_eq_plus1) 
-qed
-
-lemma graph_memI': 
+   by (smt Collect_cong Suc_eq_plus1)
+qed
+
+lemma graph_memI':
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>Suc n\<^esup>)"
   assumes "take n a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "g (take n a) = a!n"
   shows "a \<in> graph n g"
   using assms fun_graph_def[of Q\<^sub>p n g]
   by (smt cartesian_power_car_memE eq_imp_le lessI mem_Collect_eq take_Suc_conv_app_nth take_all)
 
-lemma graph_memI'': 
+lemma graph_memI'':
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "g \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<rightarrow> carrier Q\<^sub>p"
   shows "(a@[g a]) \<in> graph n g "
-  using assms fun_graph_def 
+  using assms fun_graph_def
   by blast
-  
+
 lemma graph_as_restricted_graph:
   assumes "f \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<rightarrow> carrier Q\<^sub>p"
   shows  "graph n f = restricted_graph n f (carrier (Q\<^sub>p\<^bsup>n\<^esup>))"
   apply(rule equalityI)
   apply (metis Suc_eq_plus1 assms graph_memE(1) graph_memE(3) graph_mem_closed restricted_graph_memI subsetI)
   by (simp add: assms restricted_graph_subset)
 
 definition double_graph where
 "double_graph n f g = {as \<in> carrier (Q\<^sub>p\<^bsup>n+2\<^esup>). f (take n as) = as!n \<and> g (take n as) = as!(n + 1)}"
 
 lemma double_graph_rep:
   assumes "g \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<rightarrow> carrier Q\<^sub>p"
   assumes "f \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<rightarrow> carrier Q\<^sub>p"
   shows "double_graph n f g = restricted_graph (n + 1) (g \<circ> take n) (graph n f)"
 proof
   show "double_graph n f g \<subseteq> restricted_graph (n + 1) (g \<circ> take n) (graph n f)"
   proof fix x assume A: "x \<in> double_graph n f g"
     then have 0: "x \<in> carrier (Q\<^sub>p\<^bsup>n+2\<^esup>) \<and> f (take n x) = x!n \<and> g (take n x) = x!(n + 1)"
       using double_graph_def by blast
      have 1: "take (n+1) x \<in> graph n f"
       apply(rule graph_memI)
        using assms(2) apply blast
         apply (metis "0" append_eq_conv_conj cartesian_power_car_memE le_add1 length_take
       less_add_same_cancel1 less_numeral_extra(1) min.absorb2 nth_take take_add)
        by (metis (no_types, opaque_lifting) "0" Suc_eq_plus1 Suc_n_not_le_n add_cancel_right_right
            dual_order.antisym le_iff_add not_less_eq_eq one_add_one plus_1_eq_Suc take_closed)
      show " x \<in> restricted_graph (n + 1) (g \<circ> take n) (graph n f)"
        apply(rule restricted_graph_memI)
          apply (metis "0" One_nat_def add_Suc_right numeral_2_eq_2)
        using "1" apply blast
-       using 0 take_take[of n "n + 1" x] comp_apply 
+       using 0 take_take[of n "n + 1" x] comp_apply
        by (metis le_add1 min.absorb1)
   qed
   show "restricted_graph (n + 1) (g \<circ> take n) (graph n f) \<subseteq> double_graph n f g"
   proof fix x
     assume A: "x \<in> restricted_graph (n + 1) (g \<circ> take n) (graph n f)"
     then have 0: "x \<in> carrier (Q\<^sub>p\<^bsup>Suc (n + 1)\<^esup>) \<and> take (n + 1) x \<in> graph n f \<and> (g \<circ> take n) (take (n + 1) x) = x ! (n + 1)"
       using restricted_graph_memE[of x "n+1" "(g \<circ> take n)" "graph n f" ]
-      by blast 
+      by blast
     then have 1: "x \<in> carrier (Q\<^sub>p\<^bsup>n+2\<^esup>)"
-      using 0 
+      using 0
       by (metis Suc_1 add_Suc_right)
     have 2: " f (take n x) = x ! n"
-      using 0 take_take[of n "n + 1" x] graph_memE[of f n "take (n + 1) x"] 
+      using 0 take_take[of n "n + 1" x] graph_memE[of f n "take (n + 1) x"]
       by (metis assms(2) le_add1 less_add_same_cancel1 less_numeral_extra(1) min.absorb1 nth_take)
     have 3: "g (take n x) = x ! (n + 1)"
       using 0 comp_apply take_take[of n "n + 1" x]
       by (metis le_add1 min.absorb1)
-    then show "x \<in> double_graph n f g" 
-      unfolding double_graph_def using 1 2 3 
+    then show "x \<in> double_graph n f g"
+      unfolding double_graph_def using 1 2 3
       by blast
   qed
 qed
 
 lemma double_graph_is_semialg:
   assumes "n > 0"
   assumes "is_semialg_function n f"
   assumes "is_semialg_function n g"
   shows "is_semialgebraic (n+2) (double_graph n f g)"
-  using double_graph_rep[of g n f] assms restricted_graph_is_semialgebraic[of n "g \<circ> take n" "graph n f"] 
-  by (metis (no_types, lifting) Suc_eq_plus1 add_Suc_right is_semialg_function_closed 
+  using double_graph_rep[of g n f] assms restricted_graph_is_semialgebraic[of n "g \<circ> take n" "graph n f"]
+  by (metis (no_types, lifting) Suc_eq_plus1 add_Suc_right is_semialg_function_closed
       less_add_same_cancel1 less_numeral_extra(1) one_add_one restricted_graph_is_semialgebraic
       comp_take_is_semialg semialg_graph)
-  
+
 definition add_vars :: "nat \<Rightarrow> nat  \<Rightarrow>  padic_tuple \<Rightarrow> padic_number" where
 "add_vars i j as = as!i \<oplus>\<^bsub>Q\<^sub>p\<^esub> as!j"
 
-lemma add_vars_rep: 
+lemma add_vars_rep:
   assumes "as \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "i < n"
   assumes "j < n"
   shows "add_vars i j as = Qp_ev ((pvar Q\<^sub>p i) \<oplus>\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> (pvar Q\<^sub>p j)) as"
   unfolding add_vars_def
-  using assms eval_at_point_add[of as n "pvar Q\<^sub>p i" "pvar Q\<^sub>p j"] 
+  using assms eval_at_point_add[of as n "pvar Q\<^sub>p i" "pvar Q\<^sub>p j"]
         eval_pvar  by (metis  pvar_closed)
 
 lemma add_vars_is_semialg:
   assumes "i < n"
   assumes "j < n"
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "is_semialg_function n (add_vars i j)"
 proof-
   have "pvar Q\<^sub>p i \<oplus>\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> pvar Q\<^sub>p j \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
-    using  assms pvar_closed[of ]  
-    by blast    
+    using  assms pvar_closed[of ]
+    by blast
   then have "is_semialg_function n (Qp_ev (pvar Q\<^sub>p i \<oplus>\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> pvar Q\<^sub>p j))"
-    using assms poly_is_semialg[of "(pvar Q\<^sub>p i) \<oplus>\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> (pvar Q\<^sub>p j)"]  
-    by blast
-  then show ?thesis 
-    using assms add_vars_rep 
-      semialg_function_on_carrier[of n  "Qp_ev ((pvar Q\<^sub>p i) \<oplus>\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> (pvar Q\<^sub>p j))" "add_vars i j"  ] 
+    using assms poly_is_semialg[of "(pvar Q\<^sub>p i) \<oplus>\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> (pvar Q\<^sub>p j)"]
+    by blast
+  then show ?thesis
+    using assms add_vars_rep
+      semialg_function_on_carrier[of n  "Qp_ev ((pvar Q\<^sub>p i) \<oplus>\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> (pvar Q\<^sub>p j))" "add_vars i j"  ]
   by (metis (no_types, lifting) restrict_ext)
 qed
 
 definition mult_vars :: "nat \<Rightarrow> nat  \<Rightarrow>  padic_tuple \<Rightarrow> padic_number" where
 "mult_vars i j as = as!i \<otimes> as!j"
 
-lemma mult_vars_rep: 
+lemma mult_vars_rep:
   assumes "as \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "i < n"
   assumes "j < n"
   shows "mult_vars i j as = Qp_ev ((pvar Q\<^sub>p i) \<otimes>\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> (pvar Q\<^sub>p j)) as"
   unfolding mult_vars_def
-  using assms eval_at_point_mult[of as n "pvar Q\<^sub>p i" "pvar Q\<^sub>p j"] 
-        eval_pvar[of i n as] eval_pvar[of j n as ] 
+  using assms eval_at_point_mult[of as n "pvar Q\<^sub>p i" "pvar Q\<^sub>p j"]
+        eval_pvar[of i n as] eval_pvar[of j n as ]
   by (metis pvar_closed)
 
 lemma mult_vars_is_semialg:
   assumes "i < n"
   assumes "j < n"
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "is_semialg_function n (mult_vars i j)"
 proof-
   have "pvar Q\<^sub>p i \<otimes>\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> pvar Q\<^sub>p j \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
-    using  assms pvar_closed[of ] 
-    by blast             
+    using  assms pvar_closed[of ]
+    by blast
   then have "is_semialg_function n (Qp_ev (pvar Q\<^sub>p i \<otimes>\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> pvar Q\<^sub>p j))"
-    using assms poly_is_semialg[of "(pvar Q\<^sub>p i) \<otimes>\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> (pvar Q\<^sub>p j)"]  
-    by blast
-  then show ?thesis 
-    using assms mult_vars_rep 
-      semialg_function_on_carrier[of n  "Qp_ev ((pvar Q\<^sub>p i) \<otimes>\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> (pvar Q\<^sub>p j))" "mult_vars i j"  ] 
+    using assms poly_is_semialg[of "(pvar Q\<^sub>p i) \<otimes>\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> (pvar Q\<^sub>p j)"]
+    by blast
+  then show ?thesis
+    using assms mult_vars_rep
+      semialg_function_on_carrier[of n  "Qp_ev ((pvar Q\<^sub>p i) \<otimes>\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> (pvar Q\<^sub>p j))" "mult_vars i j"  ]
     by (metis (no_types, lifting) restrict_ext)
 qed
 
 definition minus_vars :: "nat  \<Rightarrow>  padic_tuple \<Rightarrow> padic_number" where
 "minus_vars i as =  \<ominus>\<^bsub>Q\<^sub>p\<^esub> as!i"
 
-lemma minus_vars_rep: 
+lemma minus_vars_rep:
   assumes "as \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "i < n"
   shows "minus_vars i as = Qp_ev (\<ominus>\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub>(pvar Q\<^sub>p i)) as"
   unfolding minus_vars_def
-  using assms eval_pvar[of i n as] eval_at_point_a_inv[of as n "pvar Q\<^sub>p i"] 
-  by (metis pvar_closed)  
+  using assms eval_pvar[of i n as] eval_at_point_a_inv[of as n "pvar Q\<^sub>p i"]
+  by (metis pvar_closed)
 
 lemma minus_vars_is_semialg:
   assumes "i < n"
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "is_semialg_function n (minus_vars i)"
 proof-
   have 0: "pvar Q\<^sub>p i  \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
-    using  assms pvar_closed[of ] Qp.cring_axioms by presburger                      
+    using  assms pvar_closed[of ] Qp.cring_axioms by presburger
   have "is_semialg_function n  (Qp_ev (\<ominus>\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub>(pvar Q\<^sub>p i)))"
-    apply(rule poly_is_semialg ) 
+    apply(rule poly_is_semialg )
     using "0" by blast
-  then show ?thesis 
-    using assms minus_vars_rep[of a i n] 
-      semialg_function_on_carrier[of n  _ "minus_vars i"  ] 
+  then show ?thesis
+    using assms minus_vars_rep[of a i n]
+      semialg_function_on_carrier[of n  _ "minus_vars i"  ]
     by (metis (no_types, lifting) minus_vars_rep restrict_ext)
 qed
 
 definition extended_graph where
-"extended_graph n f g h = {as \<in> carrier (Q\<^sub>p\<^bsup>n+3\<^esup>). 
+"extended_graph n f g h = {as \<in> carrier (Q\<^sub>p\<^bsup>n+3\<^esup>).
                       f (take n as) = as!n \<and> g (take n as) = as! (n + 1) \<and> h [(f (take n as)),(g (take n as))] = as! (n + 2) }"
 
 lemma extended_graph_rep:
 "extended_graph n f g h = restricted_graph (n + 2) (h \<circ> (drop n)) (double_graph n f g)"
 proof
   show "extended_graph n f g h \<subseteq> restricted_graph (n + 2) (h \<circ> drop n) (double_graph n f g)"
-  proof fix x 
+  proof fix x
     assume "x \<in> extended_graph n f g h"
     then have A: "x \<in> carrier (Q\<^sub>p\<^bsup>n+3\<^esup>) \<and>f (take n x) = x!n \<and> g (take n x) = x! (n + 1) \<and>
                  h [(f (take n x)),(g (take n x))] = x! (n + 2)"
-      unfolding extended_graph_def by blast 
+      unfolding extended_graph_def by blast
     then have 0: "take (n + 2) x  \<in> carrier (Q\<^sub>p\<^bsup>n+2\<^esup>)"
     proof -
       have "Suc (Suc n) \<le> n + numeral (num.One + num.Bit0 num.One)"
         by simp
       then show ?thesis
-        by (metis (no_types) \<open>x \<in> carrier (Q\<^sub>p\<^bsup>n+3\<^esup>) \<and> f (take n x) = x ! n \<and> g (take n x) = x ! (n + 1) \<and> h [f (take n x), g (take n x)] = x ! (n + 2)\<close> add_2_eq_Suc' add_One_commute semiring_norm(5) take_closed)  
-    qed
-    have 1: "f (take n (take (n + 2) x)) = (take (n + 2) x) ! n" 
-      using A 
+        by (metis (no_types) \<open>x \<in> carrier (Q\<^sub>p\<^bsup>n+3\<^esup>) \<and> f (take n x) = x ! n \<and> g (take n x) = x ! (n + 1) \<and> h [f (take n x), g (take n x)] = x ! (n + 2)\<close> add_2_eq_Suc' add_One_commute semiring_norm(5) take_closed)
+    qed
+    have 1: "f (take n (take (n + 2) x)) = (take (n + 2) x) ! n"
+      using A
       by (metis Suc_1 add.commute append_same_eq append_take_drop_id
           less_add_same_cancel1 nth_take take_add take_drop zero_less_Suc)
     have 2: " g (take n (take (n + 2) x)) = (take (n + 2) x) ! (n + 1)"
-      using A 
-      by (smt add.assoc add.commute append_same_eq append_take_drop_id less_add_same_cancel1 
+      using A
+      by (smt add.assoc add.commute append_same_eq append_take_drop_id less_add_same_cancel1
           less_numeral_extra(1) nth_take one_add_one take_add take_drop)
     then have 3: "take (n + 2) x \<in> double_graph n f g"
-      unfolding double_graph_def 
-      using 0 1 2 
-      by blast 
+      unfolding double_graph_def
+      using 0 1 2
+      by blast
     have 4: "drop n (take (n + 2) x) = [(f (take n x)),(g (take n x))]"
     proof-
       have 40: "take (n + 2) x ! (n + 1) = x! (n + 1)"
         by (metis add.commute add_2_eq_Suc' lessI nth_take plus_1_eq_Suc)
       have 41: "take (n + 2) x ! n = x! n"
         by (metis Suc_1 less_SucI less_add_same_cancel1 less_numeral_extra(1) nth_take)
       have 42: "take (n + 2) x ! (n + 1) = g (take n x)"
-        using 40 A 
+        using 40 A
         by blast
       have 43: "take (n + 2) x ! n = f (take n x)"
-        using 41 A 
+        using 41 A
         by blast
-      show ?thesis using A 42 43 
-        by (metis "0" add_cancel_right_right cartesian_power_car_memE cartesian_power_drop 
+      show ?thesis using A 42 43
+        by (metis "0" add_cancel_right_right cartesian_power_car_memE cartesian_power_drop
             le_add_same_cancel1 nth_drop pair_id zero_le_numeral)
     qed
     then have 5: "(h \<circ> drop n) (take (n + 2) x) = x ! (n + 2)"
-      using 3 A 
+      using 3 A
       by (metis add_2_eq_Suc' comp_eq_dest_lhs)
     show "x \<in> restricted_graph (n + 2) (h \<circ> drop n) (double_graph n f g)"
-      using restricted_graph_def A 3 5 
+      using restricted_graph_def A 3 5
       by (metis (no_types, lifting) One_nat_def Suc_1
           add_Suc_right numeral_3_eq_3 restricted_graph_memI)
   qed
   show "restricted_graph (n + 2) (h \<circ> drop n) (double_graph n f g) \<subseteq> extended_graph n f g h"
   proof fix x assume A: "x \<in> restricted_graph (n + 2) (h \<circ> drop n) (double_graph n f g)"
     then have 0: "take (n+2) x \<in> double_graph n f g"
       using restricted_graph_memE(2) by blast
     have 1: "(h \<circ> drop n) (take (n+2) x) = x! (n+2) "
       by (meson A restricted_graph_memE(3) padic_fields_axioms)
     have 2: "x \<in> carrier (Q\<^sub>p\<^bsup>n+3\<^esup>)"
-      using A 
+      using A
       by (metis (no_types, opaque_lifting) Suc3_eq_add_3 add.commute add_2_eq_Suc'
           restricted_graph_closed subsetD)
     have 3: "length x = n + 3"
       using "2" cartesian_power_car_memE by blast
     have 4: "drop n (take (n+2) x) = [x!n, x!(n+1)]"
     proof-
       have "length (take (n+2) x) = n+2"
         by (simp add: "3")
       then have 40:"length (drop n (take (n+2) x)) = 2"
         by (metis add_2_eq_Suc' add_diff_cancel_left' length_drop)
       have 41: "(drop n (take (n+2) x))!0 = x!n"
-        using 3 
-        by (metis Nat.add_0_right \<open>length (take (n + 2) x) = n + 2\<close> add_gr_0 le_add1 less_add_same_cancel1 less_numeral_extra(1) nth_drop nth_take one_add_one)        
+        using 3
+        by (metis Nat.add_0_right \<open>length (take (n + 2) x) = n + 2\<close> add_gr_0 le_add1 less_add_same_cancel1 less_numeral_extra(1) nth_drop nth_take one_add_one)
       have 42: "(drop n (take (n+2) x))!1 = x!(n+1)"
-        using 3 nth_take nth_drop A 
+        using 3 nth_take nth_drop A
         by (metis add.commute le_add1 less_add_same_cancel1 less_numeral_extra(1) one_add_one take_drop)
-      show ?thesis 
-        using 40 41 42 
+      show ?thesis
+        using 40 41 42
         by (metis pair_id)
     qed
     have "(take n x) = take n (take (n+2) x)"
-      using take_take 3 
-      by (metis le_add1 min.absorb1)      
+      using take_take 3
+      by (metis le_add1 min.absorb1)
     then have 5: "f (take n x) = x ! n"
-      using 0 double_graph_def[of n f g] 3 
+      using 0 double_graph_def[of n f g] 3
       by (smt Suc_1 less_add_same_cancel1 mem_Collect_eq nth_take zero_less_Suc)
-    have 6: "g (take n x) = x ! (n + 1) " 
+    have 6: "g (take n x) = x ! (n + 1) "
       using 0 double_graph_def[of n f g] 3 take_take[of n "n+2" x]
       by (smt Suc_1 \<open>take n x = take n (take (n + 2) x)\<close> add_Suc_right lessI mem_Collect_eq nth_take)
     have 7: " h [f (take n x), g (take n x)] = x ! (n + 2)"
-      using 4 A comp_apply 
+      using 4 A comp_apply
       by (metis "1" "5" "6")
     show " x \<in> extended_graph n f g h"
-      unfolding extended_graph_def 
-      using 2  5 6 7 A 
+      unfolding extended_graph_def
+      using 2  5 6 7 A
       by blast
   qed
 qed
 
 lemma function_tuple_eval_closed:
-  assumes "is_function_tuple Q\<^sub>p n fs" 
+  assumes "is_function_tuple Q\<^sub>p n fs"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "function_tuple_eval Q\<^sub>p n fs x \<in> carrier (Q\<^sub>p\<^bsup>length fs\<^esup>)"
-  using function_tuple_eval_closed[of Q\<^sub>p n fs x] assms by blast 
+  using function_tuple_eval_closed[of Q\<^sub>p n fs x] assms by blast
 
 definition k_graph where
 "k_graph n fs = {x \<in> carrier (Q\<^sub>p\<^bsup>n + length fs\<^esup>). x = (take n x)@ (function_tuple_eval Q\<^sub>p n fs (take n x)) }"
 
 lemma k_graph_memI:
   assumes "is_function_tuple Q\<^sub>p n fs"
   assumes "x = as@function_tuple_eval Q\<^sub>p n fs as"
   assumes "as \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "x \<in> k_graph n fs"
 proof-
   have "take n x = as"
-    using assms 
+    using assms
     by (metis append_eq_conv_conj cartesian_power_car_memE)
-  then show ?thesis unfolding k_graph_def using assms 
-    by (smt append_eq_conv_conj cartesian_power_car_memE cartesian_power_car_memI'' length_append local.function_tuple_eval_closed mem_Collect_eq)    
+  then show ?thesis unfolding k_graph_def using assms
+    by (smt append_eq_conv_conj cartesian_power_car_memE cartesian_power_car_memI'' length_append local.function_tuple_eval_closed mem_Collect_eq)
 qed
 
 text\<open>composing a function with a function tuple\<close>
 
 lemma Qp_function_tuple_comp_closed:
   assumes "f \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<rightarrow> carrier Q\<^sub>p"
   assumes "length fs = n"
   assumes "is_function_tuple Q\<^sub>p m fs"
   shows "function_tuple_comp Q\<^sub>p fs f \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<rightarrow> carrier Q\<^sub>p"
-  using assms function_tuple_comp_closed 
-  by blast   
+  using assms function_tuple_comp_closed
+  by blast
 
       (********************************************************************************************)
       (********************************************************************************************)
       subsubsection\<open>Tuples of Semialgebraic Functions\<close>
       (********************************************************************************************)
       (********************************************************************************************)
 
 text\<open>Predicate for a tuple of semialgebraic functions\<close>
 
 definition is_semialg_function_tuple where
 "is_semialg_function_tuple n fs = (\<forall> f \<in> set fs. is_semialg_function n f)"
 
 lemma is_semialg_function_tupleI:
   assumes "\<And> f. f \<in> set fs \<Longrightarrow> is_semialg_function n f"
   shows "is_semialg_function_tuple n fs"
-  using assms is_semialg_function_tuple_def 
+  using assms is_semialg_function_tuple_def
   by blast
 
 lemma is_semialg_function_tupleE:
   assumes "is_semialg_function_tuple n fs"
   assumes "i < length fs"
   shows "is_semialg_function n (fs ! i)"
   by (meson assms(1) assms(2) in_set_conv_nth is_semialg_function_tuple_def padic_fields_axioms)
 
 lemma is_semialg_function_tupleE':
   assumes "is_semialg_function_tuple n fs"
   assumes "f \<in> set fs"
   shows "is_semialg_function n f"
-  using assms(1) assms(2) is_semialg_function_tuple_def 
+  using assms(1) assms(2) is_semialg_function_tuple_def
   by blast
 
 lemma semialg_function_tuple_is_function_tuple:
   assumes "is_semialg_function_tuple n fs"
   shows "is_function_tuple Q\<^sub>p n fs"
   apply(rule is_function_tupleI)
   using assms is_semialg_function_closed is_semialg_function_tupleE' by blast
 
 lemma const_poly_function_tuple_comp_is_semialg:
   assumes "n > 0"
   assumes "is_semialg_function_tuple n fs"
   assumes "a \<in> carrier Q\<^sub>p"
   shows "is_semialg_function n (poly_function_tuple_comp Q\<^sub>p n fs (Qp_to_IP a))"
   apply(rule semialg_function_on_carrier[of n "Qp_ev (Qp_to_IP a)"])
-  using poly_is_semialg[of  "(Qp_to_IP a)"] 
+  using poly_is_semialg[of  "(Qp_to_IP a)"]
   using assms(1) assms(3) Qp_to_IP_car apply blast
-  using poly_function_tuple_comp_eq[of n fs "(Qp_to_IP a)"]  assms unfolding restrict_def 
+  using poly_function_tuple_comp_eq[of n fs "(Qp_to_IP a)"]  assms unfolding restrict_def
   by (metis (no_types, opaque_lifting) eval_at_point_const poly_function_tuple_comp_constant semialg_function_tuple_is_function_tuple)
-  
+
 lemma pvar_poly_function_tuple_comp_is_semialg:
   assumes "n > 0"
   assumes "is_semialg_function_tuple n fs"
   assumes "i < length fs"
   shows "is_semialg_function n (poly_function_tuple_comp Q\<^sub>p n fs (pvar Q\<^sub>p i))"
   apply(rule semialg_function_on_carrier[of n "fs!i"])
   using assms(2) assms(3) is_semialg_function_tupleE apply blast
-  by (metis assms(2) assms(3) poly_function_tuple_comp_pvar 
+  by (metis assms(2) assms(3) poly_function_tuple_comp_pvar
       restrict_ext semialg_function_tuple_is_function_tuple)
 
 text\<open>Polynomial functions with semialgebraic coefficients\<close>
 
 definition point_to_univ_poly :: "nat \<Rightarrow> padic_tuple \<Rightarrow> padic_univ_poly" where
-"point_to_univ_poly n a = ring_cfs_to_univ_poly n a"   
+"point_to_univ_poly n a = ring_cfs_to_univ_poly n a"
 
 definition tuple_partial_image where
 "tuple_partial_image m fs x = (function_tuple_eval Q\<^sub>p m fs (take m x))@(drop m x)"
 
 lemma tuple_partial_image_closed:
   assumes "length fs > 0"
   assumes "is_function_tuple Q\<^sub>p n fs"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>n+l\<^esup>)"
   shows "tuple_partial_image n fs x \<in> carrier (Q\<^sub>p\<^bsup>length fs + l\<^esup>)"
-  using assms unfolding tuple_partial_image_def 
+  using assms unfolding tuple_partial_image_def
   by (meson cartesian_power_concat(1) cartesian_power_drop
       function_tuple_eval_closed le_add1 take_closed)
 
 lemma tuple_partial_image_indices:
   assumes "length fs > 0"
   assumes "is_function_tuple Q\<^sub>p n fs"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>n+l\<^esup>)"
   assumes "i < length fs"
   shows "(tuple_partial_image n fs x) ! i = (fs!i) (take n x)"
 proof-
   have 0: "(function_tuple_eval Q\<^sub>p n fs (take n x)) ! i = (fs!i) (take n x)"
-    using assms unfolding function_tuple_eval_def  
+    using assms unfolding function_tuple_eval_def
     by (meson nth_map)
   have 1: "length (function_tuple_eval Q\<^sub>p n fs (take n x)) > i"
     by (metis assms(4) function_tuple_eval_def length_map)
-  show ?thesis 
+  show ?thesis
     using 0 1 assms unfolding tuple_partial_image_def
     by (metis nth_append)
 qed
 
 lemma tuple_partial_image_indices':
   assumes "length fs > 0"
   assumes "is_function_tuple Q\<^sub>p n fs"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>n+l\<^esup>)"
   assumes "i < l"
   shows "(tuple_partial_image n fs x) ! (length fs + i) = x!(n + i)"
-  using assms unfolding tuple_partial_image_def 
-  by (metis (no_types, lifting) cartesian_power_car_memE function_tuple_eval_closed le_add1 
+  using assms unfolding tuple_partial_image_def
+  by (metis (no_types, lifting) cartesian_power_car_memE function_tuple_eval_closed le_add1
       nth_append_length_plus nth_drop take_closed)
 
 definition tuple_partial_pullback where
 "tuple_partial_pullback n fs l S = ((tuple_partial_image n fs)-`S)  \<inter> carrier (Q\<^sub>p\<^bsup>n+l\<^esup>)"
 
-lemma tuple_partial_pullback_memE: 
+lemma tuple_partial_pullback_memE:
   assumes "as \<in> tuple_partial_pullback m fs l S"
   shows "as \<in> carrier (Q\<^sub>p\<^bsup>m + l\<^esup>)" "tuple_partial_image m fs as \<in> S"
-  using assms 
+  using assms
   apply (metis (no_types, opaque_lifting) Int_iff add.commute tuple_partial_pullback_def)
-  using assms unfolding tuple_partial_pullback_def 
+  using assms unfolding tuple_partial_pullback_def
   by blast
-  
-lemma tuple_partial_pullback_closed: 
-"tuple_partial_pullback m fs l S \<subseteq> carrier (Q\<^sub>p\<^bsup>m + l\<^esup>)" 
+
+lemma tuple_partial_pullback_closed:
+"tuple_partial_pullback m fs l S \<subseteq> carrier (Q\<^sub>p\<^bsup>m + l\<^esup>)"
   using tuple_partial_pullback_memE by blast
 
-lemma tuple_partial_pullback_memI: 
+lemma tuple_partial_pullback_memI:
   assumes "as \<in> carrier (Q\<^sub>p\<^bsup>m + k\<^esup>)"
   assumes "is_function_tuple Q\<^sub>p m fs"
-  assumes "((function_tuple_eval Q\<^sub>p m fs) (take m as))@(drop m as) \<in> S" 
+  assumes "((function_tuple_eval Q\<^sub>p m fs) (take m as))@(drop m as) \<in> S"
   shows "as \<in> tuple_partial_pullback m fs k S"
-  using assms unfolding tuple_partial_pullback_def tuple_partial_image_def 
-  by blast 
-  
+  using assms unfolding tuple_partial_pullback_def tuple_partial_image_def
+  by blast
+
 lemma tuple_partial_image_eq:
   assumes "as \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "bs \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>)"
   assumes "x = as @ bs"
   shows "tuple_partial_image n fs x = ((function_tuple_eval Q\<^sub>p n fs) as)@bs"
 proof-
   have 0: "(take n x) = as"
     by (metis append_eq_conv_conj assms(1) assms(3) cartesian_power_car_memE)
   have 1: "drop n x = bs"
     by (metis "0" append_take_drop_id assms(3) same_append_eq)
-  show ?thesis using assms 0 1 unfolding tuple_partial_image_def 
-    by presburger    
-qed
-  
+  show ?thesis using assms 0 1 unfolding tuple_partial_image_def
+    by presburger
+qed
+
 lemma tuple_partial_pullback_memE':
   assumes "as \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "bs \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>)"
   assumes "x = as @ bs"
   assumes "x \<in> tuple_partial_pullback n fs k S"
-  shows "(function_tuple_eval Q\<^sub>p n fs as)@bs \<in> S" 
-  using tuple_partial_pullback_memE[of x n fs k S] tuple_partial_image_def[of n fs x] 
+  shows "(function_tuple_eval Q\<^sub>p n fs as)@bs \<in> S"
+  using tuple_partial_pullback_memE[of x n fs k S] tuple_partial_image_def[of n fs x]
   by (metis assms(1) assms(2) assms(3) assms(4) tuple_partial_image_eq)
 
 text\<open>tuple partial pullbacks have the same algebraic properties as pullbacks\<close>
 
 lemma tuple_partial_pullback_intersect:
 "tuple_partial_pullback m f l (S1 \<inter> S2) = (tuple_partial_pullback m f l S1) \<inter> (tuple_partial_pullback m f l S2)"
-  unfolding tuple_partial_pullback_def 
+  unfolding tuple_partial_pullback_def
   by blast
 
 lemma tuple_partial_pullback_union:
 "tuple_partial_pullback m f l (S1 \<union> S2) = (tuple_partial_pullback m f l S1) \<union> (tuple_partial_pullback m f l S2)"
-  unfolding tuple_partial_pullback_def 
+  unfolding tuple_partial_pullback_def
   by blast
 
 lemma tuple_partial_pullback_complement:
   assumes "is_function_tuple Q\<^sub>p m fs"
   shows "tuple_partial_pullback m fs l ((carrier (Q\<^sub>p\<^bsup>length fs + l\<^esup>)) - S) = carrier (Q\<^sub>p\<^bsup>m + l\<^esup>) - (tuple_partial_pullback m fs l S) "
   apply(rule equalityI)
-  using tuple_partial_pullback_def[of m fs l "((carrier (Q\<^sub>p\<^bsup>length fs + l\<^esup>)) - S)"] 
-        tuple_partial_pullback_def[of m fs l S] 
+  using tuple_partial_pullback_def[of m fs l "((carrier (Q\<^sub>p\<^bsup>length fs + l\<^esup>)) - S)"]
+        tuple_partial_pullback_def[of m fs l S]
    apply blast
 proof fix x assume A: " x \<in> carrier (Q\<^sub>p\<^bsup>m + l\<^esup>) - tuple_partial_pullback m fs l S"
   show " x \<in> tuple_partial_pullback m fs l (carrier (Q\<^sub>p\<^bsup>length fs + l\<^esup>) - S) "
     apply(rule tuple_partial_pullback_memI)
-    using A 
+    using A
       apply blast
-    using assms 
+    using assms
     apply blast
   proof
     have 0: "drop m x \<in> carrier (Q\<^sub>p\<^bsup>l\<^esup>)"
       by (meson A DiffD1 cartesian_power_drop)
     have 1: "take m x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
-      using A 
+      using A
       by (meson DiffD1 le_add1 take_closed)
     show "function_tuple_eval Q\<^sub>p m fs (take m x) @ drop m x
     \<in> carrier (Q\<^sub>p\<^bsup>length fs + l\<^esup>)"
-      using 0 1 assms 
+      using 0 1 assms
       using cartesian_power_concat(1) function_tuple_eval_closed by blast
     show "function_tuple_eval Q\<^sub>p m fs (take m x) @ drop m x \<notin> S"
-      using A unfolding tuple_partial_pullback_def tuple_partial_image_def 
-      by blast
-  qed
-qed     
+      using A unfolding tuple_partial_pullback_def tuple_partial_image_def
+      by blast
+  qed
+qed
 
 lemma tuple_partial_pullback_carrier:
   assumes "is_function_tuple Q\<^sub>p m fs"
   shows "tuple_partial_pullback m fs l (carrier (Q\<^sub>p\<^bsup>length fs + l\<^esup>)) = carrier (Q\<^sub>p\<^bsup>m + l\<^esup>)"
   apply(rule equalityI)
   using tuple_partial_pullback_memE(1) apply blast
 proof fix x assume A: "x \<in> carrier (Q\<^sub>p\<^bsup>m + l\<^esup>)"
   show "x \<in> tuple_partial_pullback m fs l (carrier (Q\<^sub>p\<^bsup>length fs + l\<^esup>))"
     apply(rule tuple_partial_pullback_memI)
   using A cartesian_power_drop[of x m l] take_closed assms
-    apply blast  
-  using assms apply blast 
+    apply blast
+  using assms apply blast
   proof-
     have "function_tuple_eval Q\<^sub>p m fs (take m x) \<in> carrier (Q\<^sub>p\<^bsup>length fs\<^esup>)"
-      using A take_closed assms 
+      using A take_closed assms
           function_tuple_eval_closed le_add1
       by blast
     then show "function_tuple_eval Q\<^sub>p m fs (take m x) @ drop m x
-    \<in> carrier (Q\<^sub>p\<^bsup>length fs + l\<^esup>)" 
-    using cartesian_power_drop[of x m l] A cartesian_power_concat(1) 
+    \<in> carrier (Q\<^sub>p\<^bsup>length fs + l\<^esup>)"
+    using cartesian_power_drop[of x m l] A cartesian_power_concat(1)
     by blast
   qed
 qed
 
 definition is_semialg_map_tuple where
-"is_semialg_map_tuple m fs = (is_function_tuple Q\<^sub>p m fs \<and> 
+"is_semialg_map_tuple m fs = (is_function_tuple Q\<^sub>p m fs \<and>
                           (\<forall>l \<ge> 0. \<forall>S \<in> semialg_sets ((length fs) + l). is_semialgebraic (m + l) (tuple_partial_pullback m fs l S)))"
 
 lemma is_semialg_map_tuple_closed:
   assumes "is_semialg_map_tuple m fs"
   shows "is_function_tuple Q\<^sub>p m fs"
-  using is_semialg_map_tuple_def assms by blast 
+  using is_semialg_map_tuple_def assms by blast
 
 lemma is_semialg_map_tupleE:
   assumes "is_semialg_map_tuple m fs"
   assumes "is_semialgebraic ((length fs) + l) S"
   shows " is_semialgebraic (m + l) (tuple_partial_pullback m fs l S)"
   using is_semialg_map_tuple_def[of m fs] assms is_semialgebraicE[of "((length fs) + l)" S]
   by blast
-  
+
 lemma is_semialg_map_tupleI:
   assumes "is_function_tuple Q\<^sub>p m fs"
   assumes "\<And>k  S. S \<in> semialg_sets ((length fs) + k) \<Longrightarrow> is_semialgebraic (m + k) (tuple_partial_pullback m fs k S)"
   shows "is_semialg_map_tuple m fs"
-  using assms unfolding is_semialg_map_tuple_def 
+  using assms unfolding is_semialg_map_tuple_def
   by blast
 
 text\<open>Semialgebraicity for maps can be verified on basic semialgebraic sets\<close>
 
 lemma is_semialg_map_tupleI':
   assumes "is_function_tuple Q\<^sub>p m fs"
   assumes "\<And>k  S. S \<in> basic_semialgs ((length fs) + k) \<Longrightarrow> is_semialgebraic (m + k) (tuple_partial_pullback m fs k S)"
   shows "is_semialg_map_tuple m fs"
   apply(rule is_semialg_map_tupleI)
   using assms(1) apply blast
 proof-
   show "\<And>k S. S \<in> semialg_sets ((length fs) + k) \<Longrightarrow> is_semialgebraic (m + k) (tuple_partial_pullback m fs k S)"
   proof- fix k S assume A: "S \<in> semialg_sets ((length fs) + k)"
     show "is_semialgebraic (m + k) (tuple_partial_pullback m fs k S)"
       apply(rule gen_boolean_algebra.induct[of S "carrier (Q\<^sub>p\<^bsup>length fs + k\<^esup>)" "basic_semialgs ((length fs) + k)"])
-        using A unfolding semialg_sets_def 
+        using A unfolding semialg_sets_def
             apply blast
         using tuple_partial_pullback_carrier assms carrier_is_semialgebraic plus_1_eq_Suc apply presburger
         using assms(1) assms(2) carrier_is_semialgebraic intersection_is_semialg tuple_partial_pullback_carrier tuple_partial_pullback_intersect apply presburger
                 using tuple_partial_pullback_union union_is_semialgebraic apply presburger
         using assms(1) complement_is_semialg tuple_partial_pullback_complement plus_1_eq_Suc by presburger
   qed
-qed      
+qed
 
 text\<open>
   The goal of this section is to show that tuples of semialgebraic functions are semialgebraic maps.
 \<close>
 
 text\<open>The function $(x_0, x, y) \mapsto (x_0, f(x), y)$\<close>
 
 definition twisted_partial_image where
 "twisted_partial_image n m f xs = (take n xs)@ partial_image m f (drop n xs)"
 
 text\<open>The set ${(x_0, x, y) \mid (x_0, f(x), y) \in S}$\<close>
 
-text\<open>Convention: a function which produces a subset of (Qp (i + j +k)) will receive the 3 arity 
+text\<open>Convention: a function which produces a subset of (Qp (i + j +k)) will receive the 3 arity
 parameters in sequence, at the very beginning of the function\<close>
 
 definition twisted_partial_pullback where
 "twisted_partial_pullback n m l f S = ((twisted_partial_image n m f)-`S)  \<inter> carrier (Q\<^sub>p\<^bsup>n+m+l\<^esup>)"
 
-lemma twisted_partial_pullback_memE: 
+lemma twisted_partial_pullback_memE:
   assumes "as \<in> twisted_partial_pullback n m l f S"
   shows "as \<in> carrier (Q\<^sub>p\<^bsup>n+m+l\<^esup>)" "twisted_partial_image n m f as \<in> S"
-  using assms 
+  using assms
   apply (metis (no_types, opaque_lifting) Int_iff add.commute twisted_partial_pullback_def subset_iff)
-  using assms unfolding twisted_partial_pullback_def   
+  using assms unfolding twisted_partial_pullback_def
   by blast
 
-lemma twisted_partial_pullback_closed: 
-"twisted_partial_pullback n m l f S \<subseteq> carrier (Q\<^sub>p\<^bsup>n+m+l\<^esup>)" 
+lemma twisted_partial_pullback_closed:
+"twisted_partial_pullback n m l f S \<subseteq> carrier (Q\<^sub>p\<^bsup>n+m+l\<^esup>)"
   using twisted_partial_pullback_memE(1) by blast
 
-lemma twisted_partial_pullback_memI: 
+lemma twisted_partial_pullback_memI:
   assumes "as \<in> carrier (Q\<^sub>p\<^bsup>n+m+l\<^esup>)"
-  assumes "(take n as)@((f (take m (drop n as)))#(drop (n + m) as)) \<in> S" 
+  assumes "(take n as)@((f (take m (drop n as)))#(drop (n + m) as)) \<in> S"
   shows "as \<in> twisted_partial_pullback n m l f S"
-  using assms unfolding twisted_partial_pullback_def twisted_partial_image_def 
+  using assms unfolding twisted_partial_pullback_def twisted_partial_image_def
   by (metis (no_types, lifting) IntI add.commute drop_drop local.partial_image_def vimageI)
-  
+
 lemma twisted_partial_image_eq:
   assumes "as \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "bs \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
   assumes "cs \<in> carrier (Q\<^sub>p\<^bsup>l\<^esup>)"
   assumes "x = as @ bs @ cs"
   shows "twisted_partial_image n m f x = as@((f bs)#cs)"
 proof-
   have 0: "(take n x) = as"
     by (metis append_eq_conv_conj assms(1) assms(4)
-        cartesian_power_car_memE) 
+        cartesian_power_car_memE)
   have 1: "twisted_partial_image n m f x = as@(partial_image m f (bs@cs))"
-    using 0 assms twisted_partial_image_def 
+    using 0 assms twisted_partial_image_def
     by (metis append_eq_conv_conj cartesian_power_car_memE)
   have 2: "(partial_image m f (bs@cs)) = (f bs)#cs"
-    using partial_image_eq[of bs m cs l "bs@cs" f] assms 
-    by blast    
-  show ?thesis using assms 0 1 2  
+    using partial_image_eq[of bs m cs l "bs@cs" f] assms
+    by blast
+  show ?thesis using assms 0 1 2
     by (metis )
 qed
-  
+
 lemma twisted_partial_pullback_memE':
   assumes "as \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "bs \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
   assumes "cs \<in> carrier (Q\<^sub>p\<^bsup>l\<^esup>)"
   assumes "x = as @ bs @ cs"
   assumes "x \<in> twisted_partial_pullback n m l f S"
-  shows "as@((f bs)#cs) \<in> S" 
-  by (metis (no_types, lifting) assms(1) assms(2) assms(3) assms(4) assms(5) 
+  shows "as@((f bs)#cs) \<in> S"
+  by (metis (no_types, lifting) assms(1) assms(2) assms(3) assms(4) assms(5)
       twisted_partial_image_eq twisted_partial_pullback_memE(2))
 
 text\<open>partial pullbacks have the same algebraic properties as pullbacks\<close>
 
 text\<open>permutation which moves the entry at index \<open>i\<close> to 0\<close>
 
 definition twisting_permutation where
-"twisting_permutation (i::nat) = (\<lambda>j. if j < i then j + 1 else 
+"twisting_permutation (i::nat) = (\<lambda>j. if j < i then j + 1 else
                                  (if j = i then 0 else j))"
 
 lemma twisting_permutation_permutes:
   assumes "i < n"
   shows "twisting_permutation i permutes {..<n}"
 proof-
   have 0: "\<And>x. x > i \<Longrightarrow> twisting_permutation i x = x"
-    unfolding twisting_permutation_def 
+    unfolding twisting_permutation_def
     by auto
   have 1: "(\<forall>x. x \<notin> {..<n} \<longrightarrow> twisting_permutation i x = x)"
-    using 0 assms 
+    using 0 assms
     by auto
   have 2: "(\<forall>y. \<exists>!x. twisting_permutation i x = y)"
   proof fix y
     show " \<exists>!x. twisting_permutation i x = y"
     proof(cases "y = 0")
       case True
-      show "\<exists>!x. twisting_permutation i x = y" 
+      show "\<exists>!x. twisting_permutation i x = y"
         by (metis  Suc_eq_plus1 True  add_eq_0_iff_both_eq_0 less_nat_zero_code
             nat_neq_iff twisting_permutation_def zero_neq_one)
     next
       case False
       show ?thesis
       proof(cases "y \<le>i")
         case True
         show ?thesis
         proof
           show "twisting_permutation i (y - 1) = y"
-            using True 
-            by (metis False add.commute add_diff_inverse_nat diff_less gr_zeroI le_eq_less_or_eq 
+            using True
+            by (metis False add.commute add_diff_inverse_nat diff_less gr_zeroI le_eq_less_or_eq
                   less_imp_diff_less less_one twisting_permutation_def)
           show "\<And>x. twisting_permutation i x = y \<Longrightarrow> x = y - 1"
             using True False twisting_permutation_def by force
         qed
       next
         case F: False
         then show ?thesis using False
-          unfolding twisting_permutation_def 
-          by (metis  add_leD1 add_leD2 add_le_same_cancel2 discrete le_numeral_extra(3) 
+          unfolding twisting_permutation_def
+          by (metis  add_leD1 add_leD2 add_le_same_cancel2 discrete le_numeral_extra(3)
               less_imp_not_less )
       qed
     qed
   qed
-  show ?thesis 
-    using 1 2 
+  show ?thesis
+    using 1 2
     by (simp add: permutes_def)
 qed
 
 lemma twisting_permutation_action:
   assumes "length as = i"
   shows "permute_list (twisting_permutation i) (b#(as@bs)) =  as@(b#bs)"
 proof-
   have 0: "length (permute_list (twisting_permutation i) (b#(as@bs))) = length  (as@(b#bs))"
     by (metis add.assoc length_append length_permute_list list.size(4))
-  have "\<And>j. j < length (as@(b#bs)) 
+  have "\<And>j. j < length (as@(b#bs))
           \<Longrightarrow> (permute_list (twisting_permutation i) (b#(as@bs))) ! j =  (as@(b#bs)) ! j"
   proof-
     fix j assume A: "j < length (as@(b#bs))"
     show "(permute_list (twisting_permutation i) (b#(as@bs))) ! j =  (as@(b#bs)) ! j"
     proof(cases "j < i")
       case True
       then have T0: "twisting_permutation i j = j + 1"
         using twisting_permutation_def by auto
       then have T1: "(b # as @ bs) ! twisting_permutation i j = as!j"
-        using assms 
+        using assms
         by (simp add: assms True nth_append)
-      have T2: "(permute_list (twisting_permutation i) (b # as @ bs)) ! j = as!j" 
+      have T2: "(permute_list (twisting_permutation i) (b # as @ bs)) ! j = as!j"
       proof-
         have "twisting_permutation i permutes {..<length (b # as @ bs)}"
-          by (metis (full_types) assms length_Cons length_append 
+          by (metis (full_types) assms length_Cons length_append
               not_add_less1 not_less_eq twisting_permutation_permutes)
-        then show ?thesis 
-          using True permute_list_nth[of "twisting_permutation i" "b#(as@bs)" j ] 
+        then show ?thesis
+          using True permute_list_nth[of "twisting_permutation i" "b#(as@bs)" j ]
               twisting_permutation_permutes[of i "length (b#(as@bs))"] assms
-          by (metis T0 T1 add_cancel_right_right lessThan_iff 
+          by (metis T0 T1 add_cancel_right_right lessThan_iff
               permutes_not_in zero_neq_one)
       qed
       have T3: "(as @ b # bs) ! j = as!j"
-        using assms True 
+        using assms True
         by (simp add: assms nth_append)
       show "(permute_list (twisting_permutation i) (b #( as @ bs))) ! j = (as @ b # bs) ! j"
-        using T3 T2 
+        using T3 T2
         by simp
     next
-      case False 
-      show ?thesis 
+      case False
+      show ?thesis
       proof(cases "j = i")
         case True
       then have T0: "twisting_permutation i j = 0"
         using twisting_permutation_def by auto
       then have T1: "(b # as @ bs) ! twisting_permutation i j = b"
-        using assms 
+        using assms
         by (simp add: assms True nth_append)
-      have T2: "(permute_list (twisting_permutation i) (b # as @ bs)) ! j = b" 
+      have T2: "(permute_list (twisting_permutation i) (b # as @ bs)) ! j = b"
       proof-
         have "twisting_permutation i permutes {..<length (b # as @ bs)}"
-          by (metis (full_types) assms length_Cons length_append 
+          by (metis (full_types) assms length_Cons length_append
               not_add_less1 not_less_eq twisting_permutation_permutes)
-        then show ?thesis 
-          using True permute_list_nth[of "twisting_permutation i" "b#(as@bs)" j ] 
+        then show ?thesis
+          using True permute_list_nth[of "twisting_permutation i" "b#(as@bs)" j ]
               twisting_permutation_permutes[of i "length (b#(as@bs))"] assms
-          by (metis "0" A T1 length_permute_list)         
+          by (metis "0" A T1 length_permute_list)
       qed
       have T3: "(as @ b # bs) ! j = b"
         by (metis True assms nth_append_length)
-      show ?thesis 
+      show ?thesis
         by (simp add: T2 T3)
       next
         case F: False
         then have F0: "twisting_permutation i j = j"
           by (simp add: False twisting_permutation_def)
         then have F1: "(b # as @ bs) ! twisting_permutation i j = bs! (j - i - 1)"
-          using assms 
+          using assms
           by (metis (mono_tags, lifting) F False Suc_diff_1
-              cancel_ab_semigroup_add_class.diff_right_commute linorder_neqE_nat not_gr_zero 
+              cancel_ab_semigroup_add_class.diff_right_commute linorder_neqE_nat not_gr_zero
               not_less_eq nth_Cons' nth_append)
-        have F2: "(permute_list (twisting_permutation i) (b # as @ bs)) ! j = bs ! (j - i - 1)" 
-          using F1 assms permute_list_nth 
+        have F2: "(permute_list (twisting_permutation i) (b # as @ bs)) ! j = bs ! (j - i - 1)"
+          using F1 assms permute_list_nth
           by (metis A add_cancel_right_right append.assoc last_to_first_eq le_add1
               le_eq_less_or_eq length_0_conv length_append length_permute_list list.distinct(1)
               twisting_permutation_permutes)
         have F3: "(as @ b # bs) ! j = bs!(j - i - 1)"
           by (metis F False assms linorder_neqE_nat nth_Cons_pos nth_append zero_less_diff)
-        then show ?thesis 
-          using F2 F3 
-          by presburger 
+        then show ?thesis
+          using F2 F3
+          by presburger
       qed
     qed
   qed
-  then show ?thesis 
-    using 0 
+  then show ?thesis
+    using 0
     by (metis nth_equalityI)
-qed        
+qed
 
 lemma twisting_permutation_action':
   assumes "length as = i"
   shows "permute_list (fun_inv (twisting_permutation i)) (as@(b#bs)) = (b#(as@bs)) "
 proof-
   obtain TI where TI_def: "TI = twisting_permutation i"
-    by blast 
+    by blast
   have 0: "TI permutes {..<length (as@(b#bs))}"
-    using assms TI_def twisting_permutation_permutes[of i "length (as@(b#bs))"] 
+    using assms TI_def twisting_permutation_permutes[of i "length (as@(b#bs))"]
     by (metis add_diff_cancel_left' gr0I length_0_conv length_append list.distinct(1) zero_less_diff)
   have 1: "(fun_inv TI) permutes {..<length (as@(b#bs))}"
     by (metis "0" Nil_is_append_conv fun_inv_permute(1) length_greater_0_conv list.distinct(1))
-  have "permute_list (fun_inv (twisting_permutation i)) (as@(b#bs)) = 
+  have "permute_list (fun_inv (twisting_permutation i)) (as@(b#bs)) =
         permute_list (fun_inv (twisting_permutation i)) (permute_list (twisting_permutation i) (b#(as@bs)))"
-    using twisting_permutation_action[of as i b bs] assms  
+    using twisting_permutation_action[of as i b bs] assms
     by (simp add: \<open>length as = i\<close>)
   then have "permute_list (fun_inv TI) (as@(b#bs)) =
   permute_list ((fun_inv TI) \<circ> TI) (b#(as@bs))"
-    using 0 1 
-    by (metis TI_def fun_inv_permute(2) fun_inv_permute(3) length_greater_0_conv 
+    using 0 1
+    by (metis TI_def fun_inv_permute(2) fun_inv_permute(3) length_greater_0_conv
         length_permute_list permute_list_compose)
-  then show ?thesis  
-    by (metis "0" Nil_is_append_conv TI_def fun_inv_permute(3) 
+  then show ?thesis
+    by (metis "0" Nil_is_append_conv TI_def fun_inv_permute(3)
         length_greater_0_conv list.distinct(1) permute_list_id)
 qed
 
 lemma twisting_semialg:
   assumes "is_semialgebraic n S"
   assumes "n > i"
   shows "is_semialgebraic n ((permute_list ((twisting_permutation i)) ` S))"
 proof-
   obtain TI where TI_def: "TI = twisting_permutation i"
-    by blast 
+    by blast
   have 0: "TI permutes {..<(n::nat)}"
-    using assms TI_def twisting_permutation_permutes[of i n]  
-    by blast 
-  have "(TI) permutes {..<n}" 
-    using TI_def  "0" 
-    by auto  
-  then show ?thesis 
-    using assms permutation_is_semialgebraic[of n S "TI"] TI_def 
-    by blast   
+    using assms TI_def twisting_permutation_permutes[of i n]
+    by blast
+  have "(TI) permutes {..<n}"
+    using TI_def  "0"
+    by auto
+  then show ?thesis
+    using assms permutation_is_semialgebraic[of n S "TI"] TI_def
+    by blast
 qed
 
 lemma twisting_semialg':
   assumes "is_semialgebraic n S"
   assumes "n > i"
   shows "is_semialgebraic n ((permute_list (fun_inv (twisting_permutation i)) ` S))"
 proof-
   obtain TI where TI_def: "TI = twisting_permutation i"
-    by blast 
+    by blast
   have 0: "TI permutes {..<(n::nat)}"
-    using assms TI_def twisting_permutation_permutes[of i n]  
-    by blast 
+    using assms TI_def twisting_permutation_permutes[of i n]
+    by blast
   have "(fun_inv TI) permutes {..<n}" using 0 permutes_inv[of TI "{..<n}"]
-    unfolding fun_inv_def 
-    by blast 
-  then show ?thesis 
-    using assms permutation_is_semialgebraic[of n S "fun_inv TI"] TI_def 
-    by blast   
+    unfolding fun_inv_def
+    by blast
+  then show ?thesis
+    using assms permutation_is_semialgebraic[of n S "fun_inv TI"] TI_def
+    by blast
 qed
 
 text\<open>Defining a permutation that does: $(x0, x1, y) \mapsto (x_1, x0, y)$\<close>
 
 definition tp_1 where
-"tp_1 i j = (\<lambda> n. (if n<i then j + n else 
-                  (if i \<le> n \<and> n < i + j then n - i else 
+"tp_1 i j = (\<lambda> n. (if n<i then j + n else
+                  (if i \<le> n \<and> n < i + j then n - i else
                    n)))"
 lemma permutes_I:
   assumes "\<And>x. x \<notin> S \<Longrightarrow> f x = x"
   assumes "\<And>y. y \<in> S \<Longrightarrow> \<exists>!x \<in> S. f x = y"
   assumes "\<And>x. x \<in> S \<Longrightarrow> f x \<in> S"
   shows "f permutes S"
 proof-
   have 0 : "(\<forall>x. x \<notin> S \<longrightarrow> f x = x) "
     using assms(1) by blast
   have 1: "(\<forall>y. \<exists>!x. f x = y)"
   proof fix y
     show "\<exists>!x. f x = y"
       apply(cases "y \<in> S")
        apply (metis "0" assms(2))
     proof-
       assume "y \<notin> S"
       then show "\<exists>!x. f x = y"
         by (metis assms(1) assms(3))
     qed
   qed
-  show ?thesis 
+  show ?thesis
     using assms 1
-    unfolding permutes_def 
+    unfolding permutes_def
     by blast
 qed
 
 lemma tp_1_permutes:
 "(tp_1 (i::nat) j) permutes {..< i + j}"
 proof(rule permutes_I)
   show "\<And>x. x \<notin> {..<i + j} \<Longrightarrow> tp_1 i j x = x"
   proof- fix x assume A: "x \<notin> {..<i + j}"
     then show "tp_1 i j x = x"
-      unfolding tp_1_def 
+      unfolding tp_1_def
       by auto
   qed
   show "\<And>y. y \<in> {..<i + j} \<Longrightarrow> \<exists>!x. x \<in> {..<i + j} \<and> tp_1 i j x = y"
   proof- fix y assume A: "y \<in> {..<i + j}"
     show "\<exists>!x. x \<in> {..<i + j} \<and> tp_1 i j x = y"
     proof(cases "y < j")
       case True
       then have 0:"tp_1 i j (y + i) = y"
         by (simp add: tp_1_def)
       have 1: "\<And>x. x \<noteq> y + i \<Longrightarrow> tp_1 i j x \<noteq> y"
       proof- fix x assume A: " x \<noteq> y + i"
         show "tp_1 i j x \<noteq> y"
           apply(cases "x < j")
            apply (metis A True add.commute le_add_diff_inverse le_eq_less_or_eq nat_neq_iff not_add_less1 tp_1_def trans_less_add2)
           by (metis A True add.commute le_add_diff_inverse less_not_refl tp_1_def trans_less_add1)
       qed
-      show ?thesis using 0 1 
+      show ?thesis using 0 1
         by (metis A \<open>\<And>x. x \<notin> {..<i + j} \<Longrightarrow> tp_1 i j x = x\<close>)
     next
       case False
       then have "y - j < i"
         using A by auto
       then have "tp_1 i j (y - j) = y"
-        using False tp_1_def 
-        by (simp add: tp_1_def)      
-      then show ?thesis 
+        using False tp_1_def
+        by (simp add: tp_1_def)
+      then show ?thesis
         by (smt A False \<open>\<And>x. x \<notin> {..<i + j} \<Longrightarrow> tp_1 i j x = x\<close>
-            add.commute add_diff_inverse_nat add_left_imp_eq 
-            less_diff_conv2 not_less tp_1_def 
+            add.commute add_diff_inverse_nat add_left_imp_eq
+            less_diff_conv2 not_less tp_1_def
             padic_fields_axioms)
     qed
   qed
   show "\<And>x. x \<in> {..<i + j} \<Longrightarrow> tp_1 i j x \<in> {..<i + j}"
   proof fix x assume A: "x \<in> {..<i + j}"
     show "tp_1 i j x < i + j"
-      unfolding tp_1_def using A 
+      unfolding tp_1_def using A
       by (simp add: trans_less_add2)
   qed
 qed
 
 lemma tp_1_permutes':
 "(tp_1 (i::nat) j) permutes {..< i + j + k}"
-  using tp_1_permutes 
+  using tp_1_permutes
   by (simp add: permutes_def)
-  
+
 lemma tp_1_permutation_action:
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>i\<^esup>)"
   assumes "b \<in> carrier (Q\<^sub>p\<^bsup>j\<^esup>)"
   assumes "c \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "permute_list (tp_1 i j) (b@a@c)= a@b@c"
 proof-
   have 0:"length (permute_list (tp_1 i j) (b@a@c))= length (a@b@c)"
     by (metis add.commute append.assoc length_append length_permute_list)
   have "\<And>x. x < length (a@b@c) \<Longrightarrow> (permute_list (tp_1 i j) (b@a@c)) ! x= (a@b@c) ! x"
   proof- fix x assume A: "x < length (a@b@c)"
     have B: "length (a @ b @ c) = i + j + length c"
-      using add.assoc assms(1) assms(2) cartesian_power_car_memE length_append 
-      by metis        
+      using add.assoc assms(1) assms(2) cartesian_power_car_memE length_append
+      by metis
     have C: "tp_1 i j permutes {..<length (a @ b @ c)}"
-      using B assms tp_1_permutes'[of i j "length b"] tp_1_permutes' 
+      using B assms tp_1_permutes'[of i j "length b"] tp_1_permutes'
       by presburger
     have D: "length a = i"
       using assms(1) cartesian_power_car_memE by blast
     have E: "length b = j"
       using assms(2) cartesian_power_car_memE by blast
     show "(permute_list (tp_1 i j) (b@a@c)) ! x= (a@b@c) ! x"
     proof(cases "x < i")
       case True
       have T0: "(tp_1 i j x) = j + x"
-        using tp_1_def[of i j ] True 
+        using tp_1_def[of i j ] True
         by auto
       then have "(b@ a @ c) ! (tp_1 i j x) = a!x"
-        using D E assms(1) assms(2) assms(3) True nth_append 
+        using D E assms(1) assms(2) assms(3) True nth_append
         by (metis nth_append_length_plus)
-      then show ?thesis 
+      then show ?thesis
         using A B C assms permute_list_nth[of "tp_1 i j" "a@b@c"]
         by (metis D True \<open>length (permute_list (tp_1 i j) (b @ a @ c)) =
-             length (a @ b @ c)\<close> length_permute_list nth_append permute_list_nth)      
+             length (a @ b @ c)\<close> length_permute_list nth_append permute_list_nth)
     next
       case False
-      show ?thesis 
+      show ?thesis
       proof(cases "x < i + j")
         case True
         then have T0: "(tp_1 i j x) = x - i"
           by (meson False not_less tp_1_def)
         have "x - i < length b"
           using E False True by linarith
         then have T1: "permute_list (tp_1 i j) (b@ a @ c) ! x = b!(x-i)"
-          using nth_append 
+          using nth_append
           by (metis A C T0 \<open>length (permute_list (tp_1 i j) (b @ a @ c)) = length (a @ b @ c)\<close>
               length_permute_list permute_list_nth)
-        then show ?thesis 
+        then show ?thesis
           by (metis D False \<open>x - i < length b\<close> nth_append)
       next
         case False
         then have "(tp_1 i j x) = x"
           by (meson tp_1_def trans_less_add1)
-        then show ?thesis 
+        then show ?thesis
           by (smt A C D E False add.commute add_diff_inverse_nat append.assoc
               length_append nth_append_length_plus permute_list_nth)
       qed
     qed
 qed
-  then show ?thesis 
-    using 0 
+  then show ?thesis
+    using 0
     by (metis list_eq_iff_nth_eq)
-qed    
-
-definition tw where 
+qed
+
+definition tw where
 "tw i j = permute_list (tp_1 j i)"
 
 lemma tw_is_semialg:
   assumes "n > 0"
   assumes "is_semialgebraic n S"
   assumes "n \<ge> i + j"
   shows "is_semialgebraic n ((tw i j)`S)"
-  unfolding tw_def 
+  unfolding tw_def
   using assms tp_1_permutes'[of j i "n - (j + i)"]
         permutation_is_semialgebraic[of n S]
   by (metis add.commute le_add_diff_inverse)
 
 lemma twisted_partial_pullback_factored:
   assumes "f \<in> (carrier (Q\<^sub>p\<^bsup>m\<^esup>)) \<rightarrow> carrier Q\<^sub>p"
   assumes "S \<subseteq> carrier (Q\<^sub>p\<^bsup>n+1+ l\<^esup>)"
   assumes "Y = partial_pullback m f (n + l) (permute_list (fun_inv (twisting_permutation n)) ` S)"
   shows "twisted_partial_pullback n m l f S = (tw m n) ` Y"
 proof
   show "twisted_partial_pullback n m l f S \<subseteq> tw m n ` Y"
   proof fix x
     assume A: "x \<in> twisted_partial_pullback n m l f S"
     then have x_closed: "x \<in> carrier (Q\<^sub>p\<^bsup>n+m+l\<^esup>)"
       using twisted_partial_pullback_memE(1) by blast
     obtain a where a_def: "a = take n x"
       by blast
     obtain b where b_def: "b = take m (drop n x)"
       by blast
     obtain c where c_def: "c = (drop (n + m) x)"
       by blast
     have x_eq:"x = a@(b@c)"
       by (metis a_def append.assoc append_take_drop_id b_def c_def take_add)
     have a_closed: "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
       by (metis (no_types, lifting) a_def dual_order.trans le_add1 take_closed x_closed)
     have b_closed: "b \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
     proof-
       have "drop n x \<in> carrier (Q\<^sub>p\<^bsup>m + l\<^esup>)"
         by (metis (no_types, lifting) add.assoc cartesian_power_drop x_closed)
-      then show ?thesis 
+      then show ?thesis
         using b_def le_add1 take_closed by blast
     qed
     have c_closed: "c \<in> carrier (Q\<^sub>p\<^bsup>l\<^esup>)"
       using c_def cartesian_power_drop x_closed by blast
     have B: "a@((f b)#c) \<in> S"
       using A twisted_partial_pullback_memE'
-      by (smt a_closed a_def add.commute append_take_drop_id b_closed 
+      by (smt a_closed a_def add.commute append_take_drop_id b_closed
           b_def c_closed c_def drop_drop)
     have  "permute_list (fun_inv (twisting_permutation n)) (a@((f b)#c)) = (f b)#(a@c)"
-       using assms twisting_permutation_action'[of a n "f b" c] 
-               a_closed cartesian_power_car_memE 
+       using assms twisting_permutation_action'[of a n "f b" c]
+               a_closed cartesian_power_car_memE
        by blast
     then have  C: "(f b)#(a@c) \<in> (permute_list (fun_inv (twisting_permutation n)) ` S)"
        by (metis B image_eqI)
     have C: "b@(a@c) \<in>  partial_pullback m f (n + l) (permute_list (fun_inv (twisting_permutation n)) ` S)"
-    proof(rule partial_pullback_memI)  
+    proof(rule partial_pullback_memI)
       show "b @ a @ c \<in> carrier (Q\<^sub>p\<^bsup>m + (n + l)\<^esup>)"
-        using a_closed b_closed c_closed cartesian_power_concat(1) 
+        using a_closed b_closed c_closed cartesian_power_concat(1)
         by blast
       have 0: "(take m (b @ a @ c)) = b"
-        by (metis append.right_neutral b_closed  cartesian_power_car_memE 
+        by (metis append.right_neutral b_closed  cartesian_power_car_memE
             diff_is_0_eq diff_self_eq_0 take0 take_all take_append)
       have 1: "drop m (b @ a @ c) = a@c"
-        by (metis "0" append_take_drop_id same_append_eq)            
-      show "f (take m (b @ a @ c)) # drop m (b @ a @ c) \<in> permute_list (fun_inv (twisting_permutation n)) ` S" 
-        using 0 1 C 
+        by (metis "0" append_take_drop_id same_append_eq)
+      show "f (take m (b @ a @ c)) # drop m (b @ a @ c) \<in> permute_list (fun_inv (twisting_permutation n)) ` S"
+        using 0 1 C
         by presburger
     qed
     have D: "tw m n (b@(a@c)) = a@(b@c)"
-      using assms tw_def a_closed b_closed c_closed 
+      using assms tw_def a_closed b_closed c_closed
       by (metis tp_1_permutation_action x_eq)
     then show "x \<in> tw m n ` Y"
-      using x_eq C assms 
+      using x_eq C assms
       by (metis image_eqI)
   qed
   show "tw m n ` Y \<subseteq> twisted_partial_pullback n m l f S"
   proof fix x
     assume A: "x \<in> tw m n ` Y"
     then obtain y where y_def: "x = tw m n y \<and> y \<in> Y"
-      by blast 
+      by blast
     obtain as where as_def: "as \<in> (permute_list (fun_inv (twisting_permutation n)) ` S) \<and>
                             as = partial_image m f y"
       using partial_pullback_memE
       by (metis assms(3) y_def)
     obtain s where s_def: "s \<in> S \<and> permute_list (fun_inv (twisting_permutation n)) s = as"
       using as_def by blast
     obtain b where b_def: "b = take m y"
       by blast
     obtain a where a_def: "a = take n (drop m y)"
       by blast
     obtain c where c_def: "c = drop (n + m) y"
       by blast
     have y_closed: "y \<in> carrier (Q\<^sub>p\<^bsup>m  + n + l\<^esup>)"
       by (metis add.assoc assms(3) partial_pullback_memE(1) y_def)
     then have y_eq: "y = b@a@c"
-      using a_def b_def c_def 
+      using a_def b_def c_def
       by (metis append_take_drop_id drop_drop)
     have a_closed: "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
       by (metis a_def add.commute cartesian_power_drop le_add1 take_closed take_drop y_closed)
     have b_closed: "b \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
-      using add_leD2 b_def le_add1 take_closed y_closed 
+      using add_leD2 b_def le_add1 take_closed y_closed
       by (meson trans_le_add1)
     have c_closed: "c \<in> carrier (Q\<^sub>p\<^bsup>l\<^esup>)"
-      using c_def cartesian_power_drop y_closed 
+      using c_def cartesian_power_drop y_closed
       by (metis add.commute)
     have ac_closed: "a@c \<in> carrier (Q\<^sub>p\<^bsup>n+l\<^esup>)"
       using a_closed c_closed cartesian_power_concat(1) by blast
     then have C: " local.partial_image m f y = f b # a @ c"
-      using b_closed y_eq partial_image_eq[of b m "a@c" "n + l" y f]  
-      by blast      
+      using b_closed y_eq partial_image_eq[of b m "a@c" "n + l" y f]
+      by blast
     then have as_eq: "as = (f b)#(a@c)"
-      using  as_def 
+      using  as_def
       by force
-    have B: "(tw m n) y = a@b@c" using y_eq tw_def[of n m] tp_1_permutation_action  
+    have B: "(tw m n) y = a@b@c" using y_eq tw_def[of n m] tp_1_permutation_action
       by (smt a_closed b_closed c_closed tw_def)
     then have "x = a@(b@c)"
       by (simp add: y_def)
     then have "twisted_partial_image n m f x =  a@((f b)# c)"
-      using a_closed b_closed c_closed twisted_partial_image_eq 
+      using a_closed b_closed c_closed twisted_partial_image_eq
       by blast
     then have D: "permute_list (twisting_permutation n) as = twisted_partial_image n m f x"
       using as_eq twisting_permutation_action[of a n "f b" c ]
       by (metis a_closed cartesian_power_car_memE)
     have "permute_list (twisting_permutation n) as \<in> S"
     proof-
       have S: "length s > n"
-        using s_def assms cartesian_power_car_memE le_add1 le_neq_implies_less 
-            le_trans less_add_same_cancel1 less_one not_add_less1 
+        using s_def assms cartesian_power_car_memE le_add1 le_neq_implies_less
+            le_trans less_add_same_cancel1 less_one not_add_less1
         by (metis (no_types, lifting) subset_iff)
       have "permute_list (twisting_permutation n) as = permute_list (twisting_permutation n) (permute_list (fun_inv (twisting_permutation n)) s)"
         using fun_inv_def  s_def by blast
-      then have "permute_list (twisting_permutation n) as = 
+      then have "permute_list (twisting_permutation n) as =
           permute_list ((twisting_permutation n) \<circ> (fun_inv (twisting_permutation n))) s"
-        using fun_inv_permute(2) fun_inv_permute(3) length_greater_0_conv 
-              length_permute_list twisting_permutation_permutes[of n "length s"]  
+        using fun_inv_permute(2) fun_inv_permute(3) length_greater_0_conv
+              length_permute_list twisting_permutation_permutes[of n "length s"]
               permute_list_compose[of "fun_inv (twisting_permutation n)" s "twisting_permutation n"]
         by (metis S permute_list_compose)
-      then have "permute_list (twisting_permutation n) as = 
+      then have "permute_list (twisting_permutation n) as =
           permute_list (id) s"
-        by (metis S \<open>permute_list (twisting_permutation n) as = permute_list 
-            (twisting_permutation n) (permute_list (fun_inv (twisting_permutation n)) s)\<close> 
-            fun_inv_permute(3) length_greater_0_conv length_permute_list permute_list_compose 
+        by (metis S \<open>permute_list (twisting_permutation n) as = permute_list
+            (twisting_permutation n) (permute_list (fun_inv (twisting_permutation n)) s)\<close>
+            fun_inv_permute(3) length_greater_0_conv length_permute_list permute_list_compose
             twisting_permutation_permutes)
       then have "permute_list (twisting_permutation n) as = s"
         by simp
       then show ?thesis
-        using s_def 
+        using s_def
         by (simp add: \<open>s \<in> S \<and> permute_list (fun_inv (twisting_permutation n)) s = as\<close>)
     qed
-    then show "x \<in> twisted_partial_pullback n m l f S" 
-      unfolding twisted_partial_pullback_def using D 
+    then show "x \<in> twisted_partial_pullback n m l f S"
+      unfolding twisted_partial_pullback_def using D
       by (smt \<open>x = a @ b @ c\<close> a_closed append.assoc append_eq_conv_conj b_closed
           c_closed cartesian_power_car_memE cartesian_power_concat(1) length_append
-          list.inject local.partial_image_def twisted_partial_image_def 
+          list.inject local.partial_image_def twisted_partial_image_def
           twisted_partial_pullback_def twisted_partial_pullback_memI)
   qed
 qed
 
 lemma twisted_partial_pullback_is_semialgebraic:
   assumes "is_semialg_function m f"
   assumes "is_semialgebraic (n + 1 + l) S"
   shows "is_semialgebraic (n + m + l)(twisted_partial_pullback n m l f S)"
 proof-
   have "(fun_inv (twisting_permutation n)) permutes {..<n + 1 + l}"
     by (simp add: fun_inv_permute(1) twisting_permutation_permutes)
   then have "is_semialgebraic (1 + n + l) (permute_list (fun_inv (twisting_permutation n)) ` S)"
-    using add_gr_0 assms(2) permutation_is_semialgebraic 
+    using add_gr_0 assms(2) permutation_is_semialgebraic
     by (metis add.commute)
-  then have "is_semialgebraic (n + m + l) 
-              (partial_pullback m f (n + l) (permute_list (fun_inv (twisting_permutation n)) ` S))"   
-    using assms is_semialg_functionE[of m f "n + l" "(permute_list (fun_inv (twisting_permutation n)) ` S)"]  
+  then have "is_semialgebraic (n + m + l)
+              (partial_pullback m f (n + l) (permute_list (fun_inv (twisting_permutation n)) ` S))"
+    using assms is_semialg_functionE[of m f "n + l" "(permute_list (fun_inv (twisting_permutation n)) ` S)"]
     by (metis (no_types, lifting) add.assoc add.commute)
-  then have "is_semialgebraic (n + m + l) 
-              ((tw m n) `(partial_pullback m f (n + l) (permute_list (fun_inv (twisting_permutation n)) ` S)))"   
-    unfolding tw_def 
-    using  tp_1_permutes'[of n m l] assms permutation_is_semialgebraic[of "n + m + l" 
-          "partial_pullback m f (n + l) (permute_list (fun_inv (twisting_permutation n)) ` S)" 
-          "tp_1 n m" ]  
-    by blast 
-  then show ?thesis 
-    using twisted_partial_pullback_factored assms(1) assms(2) 
-      is_semialg_function_closed is_semialgebraic_closed 
+  then have "is_semialgebraic (n + m + l)
+              ((tw m n) `(partial_pullback m f (n + l) (permute_list (fun_inv (twisting_permutation n)) ` S)))"
+    unfolding tw_def
+    using  tp_1_permutes'[of n m l] assms permutation_is_semialgebraic[of "n + m + l"
+          "partial_pullback m f (n + l) (permute_list (fun_inv (twisting_permutation n)) ` S)"
+          "tp_1 n m" ]
+    by blast
+  then show ?thesis
+    using twisted_partial_pullback_factored assms(1) assms(2)
+      is_semialg_function_closed is_semialgebraic_closed
     by presburger
 qed
 
 definition augment where
 "augment n x = take n x @ take n x @ drop n x"
 
 lemma augment_closed:
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>n+l\<^esup>)"
   shows "augment n x \<in> carrier (Q\<^sub>p\<^bsup>n+n+ l\<^esup>)"
   apply(rule cartesian_power_car_memI)
-  apply (smt ab_semigroup_add_class.add_ac(1) add.commute append_take_drop_id 
+  apply (smt ab_semigroup_add_class.add_ac(1) add.commute append_take_drop_id
     assms augment_def cartesian_power_car_memE cartesian_power_drop length_append)
-  using assms cartesian_power_car_memE'' unfolding augment_def   
+  using assms cartesian_power_car_memE'' unfolding augment_def
   by (metis (no_types, opaque_lifting) append_take_drop_id cartesian_power_concat(2) nat_le_iff_add take_closed)
-  
+
 lemma tuple_partial_image_factor:
   assumes "is_function_tuple Q\<^sub>p m fs"
   assumes "f \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<rightarrow> carrier Q\<^sub>p"
   assumes "length fs = n"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>m + l\<^esup>)"
   shows "tuple_partial_image m (fs@[f]) x = twisted_partial_image n m f (tuple_partial_image m fs (augment m x))"
 proof-
   obtain a where a_def: "a = take m x"
     by blast
   obtain b where b_def: "b = drop m x"
     by blast
   have a_closed: "a \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
-    using a_def assms(4) le_add1 take_closed 
-    by (meson dual_order.trans)    
+    using a_def assms(4) le_add1 take_closed
+    by (meson dual_order.trans)
   have b_closed: "b \<in> carrier (Q\<^sub>p\<^bsup>l\<^esup>)"
-    using assms(4) b_def cartesian_power_drop 
-    by (metis (no_types, lifting))    
+    using assms(4) b_def cartesian_power_drop
+    by (metis (no_types, lifting))
   have A: "(augment m x) = a @ (a @ b)"
-    using a_def augment_def b_def 
+    using a_def augment_def b_def
     by blast
   have 0: "tuple_partial_image m fs (augment m x) = ((function_tuple_eval Q\<^sub>p m fs) a) @ a @ b"
-    using A a_closed b_closed tuple_partial_image_eq[of a m "a@b" "m + l" "augment m x" fs] 
-      cartesian_power_concat(1) 
-    by blast   
+    using A a_closed b_closed tuple_partial_image_eq[of a m "a@b" "m + l" "augment m x" fs]
+      cartesian_power_concat(1)
+    by blast
   have 1: "tuple_partial_image m (fs@[f]) x = ((function_tuple_eval Q\<^sub>p m fs a) @ [f a])@b"
-    using 0 tuple_partial_image_eq[of a m b l x "fs@[f]"] unfolding function_tuple_eval_def 
+    using 0 tuple_partial_image_eq[of a m b l x "fs@[f]"] unfolding function_tuple_eval_def
     by (metis (no_types, lifting) a_closed a_def append_take_drop_id b_closed b_def
         list.simps(8) list.simps(9) map_append)
   have 2: "tuple_partial_image m (fs@[f]) x = (function_tuple_eval Q\<^sub>p m fs a) @ ((f a)#b)"
-    using 1 
-    by (metis (no_types, lifting) append_Cons append_Nil2 append_eq_append_conv2 same_append_eq)    
+    using 1
+    by (metis (no_types, lifting) append_Cons append_Nil2 append_eq_append_conv2 same_append_eq)
   have 3: "tuple_partial_image m fs x = (function_tuple_eval Q\<^sub>p m fs a) @ b"
     using a_def b_def 2 tuple_partial_image_eq[of a m b l x fs ] assms  tuple_partial_image_def
     by blast
   have 4: "twisted_partial_image n m f (tuple_partial_image m fs (augment m x)) =
                   (function_tuple_eval Q\<^sub>p m fs a) @ ((f a)#b)"
-    using twisted_partial_image_eq[of _ n _ m _ l] 0  assms(1) assms(3) b_closed 
-      local.a_closed local.function_tuple_eval_closed by blast    
-  show ?thesis using 2 4 
-    by presburger  
+    using twisted_partial_image_eq[of _ n _ m _ l] 0  assms(1) assms(3) b_closed
+      local.a_closed local.function_tuple_eval_closed by blast
+  show ?thesis using 2 4
+    by presburger
 qed
 
 definition diagonalize where
 "diagonalize n m S = S \<inter> cartesian_product (\<Delta> n) (carrier (Q\<^sub>p\<^bsup>m\<^esup>))"
 
 lemma diagaonlize_is_semiaglebraic:
   assumes "is_semialgebraic (n + n + m) S"
   shows "is_semialgebraic (n + n + m) (diagonalize n m S)"
 proof(cases "m = 0")
   case True
   then have 0: "carrier (Q\<^sub>p\<^bsup>m\<^esup>) = {[]}"
-    unfolding  cartesian_power_def 
+    unfolding  cartesian_power_def
     by simp
   have 1: "\<Delta> n \<subseteq> carrier (Q\<^sub>p\<^bsup>n+n\<^esup>)"
-    using Qp.cring_axioms assms diagonalE(2) 
+    using Qp.cring_axioms assms diagonalE(2)
     by blast
   then have "cartesian_product (\<Delta> n) (carrier (Q\<^sub>p\<^bsup>m\<^esup>)) = \<Delta> n"
-    using 0 cartesian_product_empty_right[of "\<Delta> n" Q\<^sub>p "n + n"  "carrier (Q\<^sub>p\<^bsup>m\<^esup>)"] 
+    using 0 cartesian_product_empty_right[of "\<Delta> n" Q\<^sub>p "n + n"  "carrier (Q\<^sub>p\<^bsup>m\<^esup>)"]
     by linarith
   then have "diagonalize n m S = S \<inter> (\<Delta> n)"
-    using diagonalize_def 
+    using diagonalize_def
     by presburger
-  then show ?thesis 
-    using intersection_is_semialg True assms diag_is_semialgebraic 
+  then show ?thesis
+    using intersection_is_semialg True assms diag_is_semialgebraic
     by auto
 next
   case False
   have "is_semialgebraic (n + n + m) (cartesian_product (\<Delta> n) (carrier (Q\<^sub>p\<^bsup>m\<^esup>)))"
-    using carrier_is_semialgebraic[of m]  
-          cartesian_product_is_semialgebraic[of "n + n" "\<Delta> n" m "carrier (Q\<^sub>p\<^bsup>m\<^esup>)"] 
-          diag_is_semialgebraic[of n]  False 
-    by blast
-  then show ?thesis 
-    using intersection_is_semialg assms(1) diagonalize_def 
+    using carrier_is_semialgebraic[of m]
+          cartesian_product_is_semialgebraic[of "n + n" "\<Delta> n" m "carrier (Q\<^sub>p\<^bsup>m\<^esup>)"]
+          diag_is_semialgebraic[of n]  False
+    by blast
+  then show ?thesis
+    using intersection_is_semialg assms(1) diagonalize_def
     by presburger
 qed
 
 lemma list_segment_take:
   assumes "length a \<ge>n"
   shows "list_segment 0 n a = take n a"
 proof-
   have 0: "length (list_segment 0 n a) = length (take n a)"
-    using assms unfolding list_segment_def 
-    by (metis (no_types, lifting) Groups.add_ac(2) add_diff_cancel_left' 
+    using assms unfolding list_segment_def
+    by (metis (no_types, lifting) Groups.add_ac(2) add_diff_cancel_left'
         append_take_drop_id le_Suc_ex length_append length_drop length_map map_nth)
   have "\<And>i. i < n \<Longrightarrow> list_segment 0 n a !i = take n a ! i"
-    unfolding list_segment_def using assms 
+    unfolding list_segment_def using assms
     by (metis add.left_neutral diff_zero nth_map_upt nth_take)
-  then show ?thesis using 0 
+  then show ?thesis using 0
     by (metis assms diff_zero le0 list_segment_length nth_equalityI)
 qed
 
 lemma augment_inverse_is_semialgebraic:
   assumes "is_semialgebraic (n+n+l) S"
   shows "is_semialgebraic (n+l) ((augment n -` S) \<inter> carrier (Q\<^sub>p\<^bsup>n+l\<^esup>))"
 proof-
   obtain Ps where Ps_def: "Ps = (var_list_segment 0 n)"
-    by blast 
+    by blast
   obtain Qs where Qs_def: "Qs = (var_list_segment n (n+l))"
     by blast
   obtain Fs where Fs_def: "Fs = Ps@Ps@Qs"
-    by blast 
+    by blast
   have 0: "is_poly_tuple (n+l) Ps"
     by (simp add: Ps_def var_list_segment_is_poly_tuple)
   have 1: "is_poly_tuple (n+l) Qs"
     by (simp add: Qs_def var_list_segment_is_poly_tuple)
   have 2: "is_poly_tuple (n+l) (Ps@Qs)"
     using Qp_is_poly_tuple_append[of "n+l" Ps Qs]
     by (metis (no_types, opaque_lifting) "0" "1" add.commute)
   have "is_poly_tuple (n+l) Fs"
     using 0 2 Qp_is_poly_tuple_append[of "n + l" Ps "Ps@Qs"] Fs_def assms
     by blast
   have 3: "\<And>x. x \<in> carrier (Q\<^sub>p\<^bsup>n+l\<^esup>) \<Longrightarrow> augment n x = poly_map (n + l) Fs x"
   proof- fix x assume A: "x \<in> carrier (Q\<^sub>p\<^bsup>n+l\<^esup>)"
     have 30: "poly_map (n+l) Ps x = take n x"
-      using Ps_def  map_by_var_list_segment[of x "n + l" n 0] 
-            list_segment_take[of n x] cartesian_power_car_memE[of x Q\<^sub>p "n+l"] 
-      by (simp add: A)      
+      using Ps_def  map_by_var_list_segment[of x "n + l" n 0]
+            list_segment_take[of n x] cartesian_power_car_memE[of x Q\<^sub>p "n+l"]
+      by (simp add: A)
     have 31: "poly_map (n + l) Qs x = drop n x"
       using Qs_def map_by_var_list_segment_to_length[of x "n + l" n] A le_add1
       by blast
     have 32: "poly_map (n + l) (Ps@Qs) x = take n x @ drop n x"
       using poly_map_append[of x "n+l" Ps Qs ]
       by (metis "30" "31" A append_take_drop_id)
     show "augment n x = poly_map (n + l) Fs x"
       using 30 32 poly_map_append
       by (metis A Fs_def poly_map_append augment_def)
   qed
   have 4: "(augment n -` S) \<inter> carrier (Q\<^sub>p\<^bsup>n+l\<^esup>) = poly_tuple_pullback (n + l) S Fs"
-  proof    
+  proof
     show "augment n -` S \<inter> carrier (Q\<^sub>p\<^bsup>n+l\<^esup>) \<subseteq> poly_tuple_pullback (n + l) S Fs"
     proof fix x assume A: "x \<in> augment n -` S \<inter> carrier (Q\<^sub>p\<^bsup>n+l\<^esup>)"
       then have 40: "augment n x \<in> S"
         by blast
       have 41: "augment n x \<in> carrier (Q\<^sub>p\<^bsup>n+n+ l\<^esup>)"
-        using 40 assms unfolding augment_def 
-        using is_semialgebraic_closed 
+        using 40 assms unfolding augment_def
+        using is_semialgebraic_closed
         by blast
       have "x \<in> carrier (Q\<^sub>p\<^bsup>n+l\<^esup>)"
       proof-
         have "take n x @ x \<in> carrier (Q\<^sub>p\<^bsup>n+n+ l\<^esup>)"
-          using augment_def A 
+          using augment_def A
           by (metis "41" append_take_drop_id)
         then have 0: "drop n (take n x @ x) \<in> carrier (Q\<^sub>p\<^bsup>n+l\<^esup>)"
           by (metis (no_types, lifting) add.assoc cartesian_power_drop)
         have "drop n (take n x @ x) = x"
         proof-
-          have "length x \<ge> n" 
-            using A 
+          have "length x \<ge> n"
+            using A
             by (metis IntD2 cartesian_power_car_memE le_add1)
           then have "length (take n x) = n"
-            by (metis add_right_cancel append_take_drop_id 
+            by (metis add_right_cancel append_take_drop_id
                 le_add_diff_inverse length_append length_drop)
-          then show ?thesis 
+          then show ?thesis
             by (metis append_eq_conv_conj)
         qed
-        then show ?thesis 
-          using 0 
+        then show ?thesis
+          using 0
           by presburger
       qed
       then show "x \<in> poly_tuple_pullback (n + l) S Fs"
-        using  41 3 unfolding poly_tuple_pullback_def 
-        by (metis (no_types, opaque_lifting) "40" add.commute cartesian_power_car_memE evimageI evimage_def poly_map_apply)       
+        using  41 3 unfolding poly_tuple_pullback_def
+        by (metis (no_types, opaque_lifting) "40" add.commute cartesian_power_car_memE evimageI evimage_def poly_map_apply)
     qed
     show "poly_tuple_pullback (n + l) S Fs \<subseteq> augment n -` S \<inter> carrier (Q\<^sub>p\<^bsup>n+l\<^esup>)"
     proof fix x assume A: "x \<in> poly_tuple_pullback (n + l) S Fs"
       have "x \<in> carrier (Q\<^sub>p\<^bsup>n+l\<^esup>)"
-        using A unfolding poly_tuple_pullback_def by blast         
+        using A unfolding poly_tuple_pullback_def by blast
       then show "x \<in> augment n -` S \<inter> carrier (Q\<^sub>p\<^bsup>n+l\<^esup>)"
-        using 3 
-        by (metis (no_types, lifting) A poly_map_apply poly_tuple_pullback_def vimage_inter_cong)        
+        using 3
+        by (metis (no_types, lifting) A poly_map_apply poly_tuple_pullback_def vimage_inter_cong)
     qed
   qed
   then show ?thesis using assms pullback_is_semialg[of "n + l" Fs]
     using poly_tuple_pullback_eq_poly_map_vimage
-    unfolding restrict_def evimage_def Fs_def  
+    unfolding restrict_def evimage_def Fs_def
     by (smt "4" Ex_list_of_length Fs_def Ps_def Qs_def \<open>is_poly_tuple (n + l) Fs\<close> add.commute
         add_diff_cancel_left' append_assoc diff_zero is_semialgebraic_closed le_add2 length_append
-        not_add_less1 not_gr_zero padic_fields.is_semialgebraicE padic_fields_axioms var_list_segment_length zero_le)        
+        not_add_less1 not_gr_zero padic_fields.is_semialgebraicE padic_fields_axioms var_list_segment_length zero_le)
 qed
 
 lemma tuple_partial_pullback_is_semialg_map_tuple_induct:
   assumes "is_semialg_map_tuple m fs"
   assumes "is_semialg_function m f"
   assumes "length fs = n"
   shows "is_semialg_map_tuple m (fs@[f])"
 proof(rule is_semialg_map_tupleI)
   have 0: "is_function_tuple Q\<^sub>p m fs"
-    using assms is_semialg_map_tuple_def 
+    using assms is_semialg_map_tuple_def
     by blast
   show "is_function_tuple Q\<^sub>p m (fs @ [f])"
   proof(rule is_function_tupleI)
     have A0: "set (fs @ [f]) = insert f (set fs)"
       by simp
     have A1: "set fs \<subseteq> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<rightarrow> carrier Q\<^sub>p"
-      using 0 is_function_tuple_def  
+      using 0 is_function_tuple_def
       by blast
     then show "set (fs @ [f]) \<subseteq> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<rightarrow> carrier Q\<^sub>p"
-      using assms 0 
-      by (metis (no_types, lifting) A0 is_semialg_function_closed list.simps(15) set_ConsD subset_code(1))      
+      using assms 0
+      by (metis (no_types, lifting) A0 is_semialg_function_closed list.simps(15) set_ConsD subset_code(1))
   qed
   show "\<And>k S. S \<in> semialg_sets (length (fs @ [f]) + k) \<Longrightarrow>
                  is_semialgebraic (m + k) (tuple_partial_pullback m (fs @ [f]) k S)"
   proof- fix l S
     assume A: "S \<in> semialg_sets (length (fs @ [f]) + l)"
     then have B: "S \<in> semialg_sets (n + l + 1)"
-      using assms 
-      by (metis (no_types, lifting) add.commute add_Suc_right add_diff_cancel_left' 
+      using assms
+      by (metis (no_types, lifting) add.commute add_Suc_right add_diff_cancel_left'
           append_Nil2 diff_Suc_1 length_Suc_conv length_append)
     show "is_semialgebraic (m + l) (tuple_partial_pullback m (fs @ [f]) l S)"
     proof-
       obtain S0 where S0_def: "S0 = tuple_partial_pullback m fs (l+1) S"
-        by blast 
+        by blast
       have 0: "is_semialgebraic (m + l + 1) S0"
         using B assms is_semialg_map_tupleE[of m fs "l + 1" S]
         by (metis S0_def add.assoc is_semialgebraicI)
       obtain S1 where S1_def: "S1 = twisted_partial_pullback m m l f S0"
-        by blast 
+        by blast
       then have "is_semialgebraic (m + m + l) S1"
         using S1_def assms(1) 0 twisted_partial_pullback_is_semialgebraic[of m f m l S0]
-        by (simp add: assms(2))        
+        by (simp add: assms(2))
       then have L: "is_semialgebraic (m + m + l) (diagonalize m l S1)"
         using assms  diagaonlize_is_semiaglebraic
         by blast
-      have 1: "(tuple_partial_pullback m (fs @ [f]) l S) 
+      have 1: "(tuple_partial_pullback m (fs @ [f]) l S)
                 = (augment m -` (diagonalize m l S1)) \<inter> carrier (Q\<^sub>p\<^bsup>m + l\<^esup>)"
       proof
-        show "tuple_partial_pullback m (fs @ [f]) l S \<subseteq> 
+        show "tuple_partial_pullback m (fs @ [f]) l S \<subseteq>
                       augment m -` diagonalize m l S1 \<inter> carrier (Q\<^sub>p\<^bsup>m + l\<^esup>)"
 
         proof fix x assume P0: "x \<in> tuple_partial_pullback m (fs @ [f]) l S "
           show "x \<in> augment m -` diagonalize m l S1 \<inter> carrier (Q\<^sub>p\<^bsup>m + l\<^esup>)"
           proof
             show "x \<in> carrier (Q\<^sub>p\<^bsup>m + l\<^esup>)"
-              using tuple_partial_pullback_closed P0 
+              using tuple_partial_pullback_closed P0
               by blast
             show "x \<in> augment m -` diagonalize m l S1"
             proof-
               obtain a where a_def: "a = take m x"
                 by blast
               then have a_closed: "a \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
-                using \<open>x \<in> carrier (Q\<^sub>p\<^bsup>m + l\<^esup>)\<close> le_add1 take_closed 
+                using \<open>x \<in> carrier (Q\<^sub>p\<^bsup>m + l\<^esup>)\<close> le_add1 take_closed
                 by blast
               obtain b where b_def: "b = drop m x"
                 by blast
               then have b_closed: "b \<in> carrier (Q\<^sub>p\<^bsup>l\<^esup>)"
-                using \<open>x \<in> carrier (Q\<^sub>p\<^bsup>m + l\<^esup>)\<close> cartesian_power_drop 
+                using \<open>x \<in> carrier (Q\<^sub>p\<^bsup>m + l\<^esup>)\<close> cartesian_power_drop
                 by blast
               have x_eq: "x = a@b"
-                using a_def b_def 
+                using a_def b_def
                 by (metis append_take_drop_id)
               have X0: "a @ a @ b = augment m x"
                 by (metis a_def augment_def b_def)
               have "a @ a @ b \<in>  diagonalize m l S1"
               proof-
                 have "length (a@a) = m + m"
-                  using a_closed cartesian_power_car_memE length_append 
+                  using a_closed cartesian_power_car_memE length_append
                   by blast
                 then have "take (m + m) (a @ a @ b) = a@a"
                   by (metis append.assoc append_eq_conv_conj)
                 then have X00: "take (m + m) (a @ a @ b) \<in> \<Delta>  m"
-                  using diagonalI[of  "a@a"] a_def a_closed  
-                  by (metis append_eq_conv_conj cartesian_power_car_memE)                  
+                  using diagonalI[of  "a@a"] a_def a_closed
+                  by (metis append_eq_conv_conj cartesian_power_car_memE)
                 then have X01: "a @ a @ b \<in> cartesian_product (\<Delta> m) (carrier (Q\<^sub>p\<^bsup>l\<^esup>))"
                   using a_closed b_closed cartesian_product_memI[of "\<Delta> m" Q\<^sub>p "m+m" "carrier (Q\<^sub>p\<^bsup>l\<^esup>)" l "a @ a @ b"]
-                  unfolding diagonal_def 
-                  by (metis (no_types, lifting) X0 \<open>x \<in> carrier (Q\<^sub>p\<^bsup>m + l\<^esup>)\<close> augment_closed cartesian_power_drop mem_Collect_eq subsetI)  
+                  unfolding diagonal_def
+                  by (metis (no_types, lifting) X0 \<open>x \<in> carrier (Q\<^sub>p\<^bsup>m + l\<^esup>)\<close> augment_closed cartesian_power_drop mem_Collect_eq subsetI)
                 have X02: "twisted_partial_image m m f (a @ a @ b) = a @ ((f a)# b)"
-                  using twisted_partial_image_eq[of a m a m b l _ f] a_closed b_closed 
+                  using twisted_partial_image_eq[of a m a m b l _ f] a_closed b_closed
                   by blast
                 have "a @ a @ b \<in> S1"
                 proof-
                   have "twisted_partial_image m m f (a @ a @ b) \<in> S0"
                   proof-
-                    have X020:"tuple_partial_image m fs (a @ ((f a)# b)) 
+                    have X020:"tuple_partial_image m fs (a @ ((f a)# b))
                               = (function_tuple_eval Q\<^sub>p m fs a)@[f a]@b"
                       using tuple_partial_image_eq[of a m "(f a)# b" "l + 1" _ fs]
                       by (metis (no_types, lifting) a_closed append_Cons append_eq_conv_conj
                           cartesian_power_car_memE self_append_conv2 tuple_partial_image_def)
                     have X021: "(function_tuple_eval Q\<^sub>p m fs a)@[f a]@b \<in> S"
                     proof-
-                      have X0210: "(function_tuple_eval Q\<^sub>p m fs a)@[f a]@b = 
+                      have X0210: "(function_tuple_eval Q\<^sub>p m fs a)@[f a]@b =
                                 (function_tuple_eval Q\<^sub>p m (fs@[f]) a)@b"
-                        unfolding function_tuple_eval_def 
-                        by (metis (mono_tags, lifting) append.assoc list.simps(8) list.simps(9) map_append)                        
-                      have X0211: "(function_tuple_eval Q\<^sub>p m (fs@[f]) a)@b = 
+                        unfolding function_tuple_eval_def
+                        by (metis (mono_tags, lifting) append.assoc list.simps(8) list.simps(9) map_append)
+                      have X0211: "(function_tuple_eval Q\<^sub>p m (fs@[f]) a)@b =
                                 tuple_partial_image m (fs @ [f]) x"
                         using x_eq tuple_partial_image_eq[of a m b l x "fs@[f]"]
                         by (simp add: a_closed b_closed)
                       have "tuple_partial_image m (fs @ [f]) x \<in> S"
-                        using P0 tuple_partial_pullback_memE(2) 
+                        using P0 tuple_partial_pullback_memE(2)
                         by blast
                       then show ?thesis using X0211 X0210 by presburger
                     qed
                     have X022: "tuple_partial_image m fs (twisted_partial_image m m f (a @ a @ b))
                                  = (function_tuple_eval Q\<^sub>p m fs a)@[f a]@b"
                     proof-
-                      have X0220: "tuple_partial_image m fs (twisted_partial_image m m f (a @ a @ b)) = 
+                      have X0220: "tuple_partial_image m fs (twisted_partial_image m m f (a @ a @ b)) =
                                     tuple_partial_image m fs (a @ ((f a)# b))"
-                        using X02 by presburger                        
-                      have X0221: "tuple_partial_image m fs (twisted_partial_image m m f (a @ a @ b)) = 
+                        using X02 by presburger
+                      have X0221: "tuple_partial_image m fs (twisted_partial_image m m f (a @ a @ b)) =
                                     (function_tuple_eval Q\<^sub>p m fs a) @ ((f a)# b)"
-                        using tuple_partial_image_eq 
+                        using tuple_partial_image_eq
                         by (metis X02 X020 append_Cons self_append_conv2)
-                      then show ?thesis 
-                        unfolding function_tuple_eval_def 
-                        by (metis X02 X020 X0221 append_same_eq)               
+                      then show ?thesis
+                        unfolding function_tuple_eval_def
+                        by (metis X02 X020 X0221 append_same_eq)
                     qed
                     have X023: "tuple_partial_image m fs (twisted_partial_image m m f (a @ a @ b)) \<in> S"
                       using X02 X020 X021 by presburger
                     have "twisted_partial_image m m f (a @ a @ b) \<in> carrier (Q\<^sub>p\<^bsup>m + (l+1)\<^esup>)"
                     proof-
                       have "a @ ((f a)# b) \<in> carrier (Q\<^sub>p\<^bsup>m + (l+1)\<^esup>)"
                         apply(rule cartesian_power_car_memI)
-                         apply (metis a_closed add.commute b_closed cartesian_power_car_memE 
+                         apply (metis a_closed add.commute b_closed cartesian_power_car_memE
                             length_Cons length_append plus_1_eq_Suc)
                       proof-
                         have "f \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<rightarrow> carrier Q\<^sub>p"
                           using assms(2) is_semialg_function_closed by blast
-                        then have "f a \<in> carrier Q\<^sub>p" 
-                          using a_closed assms 
+                        then have "f a \<in> carrier Q\<^sub>p"
+                          using a_closed assms
                           by blast
                         then show "set (a @ f a # b) \<subseteq> carrier Q\<^sub>p"
-                          using  assms a_closed b_closed  
+                          using  assms a_closed b_closed
                           by (meson cartesian_power_car_memE'' cartesian_power_concat(1) cartesian_power_cons)
                       qed
-                      then show ?thesis 
-                        using X02 
+                      then show ?thesis
+                        using X02
                         by presburger
-                    qed  
-                    then show ?thesis 
-                      using S0_def X023 tuple_partial_pullback_def[of m fs "l+1" S ]             
+                    qed
+                    then show ?thesis
+                      using S0_def X023 tuple_partial_pullback_def[of m fs "l+1" S ]
                       by blast
                   qed
-                  then show ?thesis using X02 S1_def twisted_partial_pullback_def   
-                    by (metis (no_types, lifting) X0 \<open>x \<in> carrier (Q\<^sub>p\<^bsup>m + l\<^esup>)\<close> augment_closed 
-                        drop_drop local.partial_image_def twisted_partial_image_def 
+                  then show ?thesis using X02 S1_def twisted_partial_pullback_def
+                    by (metis (no_types, lifting) X0 \<open>x \<in> carrier (Q\<^sub>p\<^bsup>m + l\<^esup>)\<close> augment_closed
+                        drop_drop local.partial_image_def twisted_partial_image_def
                         twisted_partial_pullback_memI)
                 qed
-                then show ?thesis using X01 diagonalize_def[of m l S1]  
+                then show ?thesis using X01 diagonalize_def[of m l S1]
                   by blast
               qed
-              then show ?thesis 
+              then show ?thesis
                 by (metis X0 vimageI)
             qed
           qed
         qed
         show "augment m -` diagonalize m l S1 \<inter> carrier (Q\<^sub>p\<^bsup>m + l\<^esup>) \<subseteq> tuple_partial_pullback m (fs @ [f]) l S"
         proof
-          fix x 
+          fix x
           assume A: "x \<in> augment m -` diagonalize m l S1 \<inter> carrier (Q\<^sub>p\<^bsup>m + l\<^esup>)"
           then have X0: "x \<in> carrier (Q\<^sub>p\<^bsup>m + l\<^esup>)"
             by blast
           obtain a where a_def: "a = take m x"
-            by blast 
+            by blast
           then have a_closed: "a \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
             using X0 le_add1 take_closed by blast
           obtain b where b_def: "b = drop m x"
-            by blast 
+            by blast
           then have a_closed: "b \<in> carrier (Q\<^sub>p\<^bsup>l\<^esup>)"
-            using X0 cartesian_power_drop 
+            using X0 cartesian_power_drop
             by blast
           have X1: "augment m x = a@a@b"
-            using a_def augment_def b_def 
+            using a_def augment_def b_def
             by blast
           have X2: "a@a@b \<in> diagonalize m l S1"
-            using A X1 
+            using A X1
             by (metis Int_iff vimage_eq)
           have X3: "a@a@b \<in>  S1"
-            using X2 diagonalize_def 
+            using X2 diagonalize_def
             by blast
           have X4: "twisted_partial_image m m f (a@a@b) \<in> S0"
-            using X3 S1_def twisted_partial_pullback_memE(2) 
+            using X3 S1_def twisted_partial_pullback_memE(2)
             by blast
           have X5: "a@((f a)#b) \<in> S0"
             using X4 twisted_partial_image_eq[of a m a m b l _ f]
             by (metis X0 a_closed a_def le_add1 take_closed)
           have X6: "tuple_partial_image m fs (a@((f a)#b)) \<in> S"
-            using S0_def X5 tuple_partial_pullback_memE(2) 
+            using S0_def X5 tuple_partial_pullback_memE(2)
             by blast
           have X7: "((function_tuple_eval Q\<^sub>p m fs a)@((f a)#b)) \<in> S"
-            using X6 using tuple_partial_image_eq  
-            by (metis X0 a_def append_eq_conv_conj cartesian_power_car_memE 
+            using X6 using tuple_partial_image_eq
+            by (metis X0 a_def append_eq_conv_conj cartesian_power_car_memE
                 le_add1 take_closed tuple_partial_image_def)
-          have X8: "((function_tuple_eval Q\<^sub>p m fs a)@((f a)#b)) = 
+          have X8: "((function_tuple_eval Q\<^sub>p m fs a)@((f a)#b)) =
                         tuple_partial_image m (fs @ [f]) x"
           proof-
             have X80: "tuple_partial_image m (fs @ [f]) x = (function_tuple_eval Q\<^sub>p m (fs@[f]) a)@b"
-              using tuple_partial_image_def a_def b_def 
+              using tuple_partial_image_def a_def b_def
               by blast
-            then show ?thesis unfolding function_tuple_eval_def 
-              by (metis (no_types, lifting) append_Cons append_eq_append_conv2 list.simps(9) map_append self_append_conv2)              
+            then show ?thesis unfolding function_tuple_eval_def
+              by (metis (no_types, lifting) append_Cons append_eq_append_conv2 list.simps(9) map_append self_append_conv2)
           qed
           show "x \<in> tuple_partial_pullback m (fs @ [f]) l S"
-            using X8 X7 tuple_partial_pullback_def 
-            by (metis X0 \<open>is_function_tuple Q\<^sub>p m (fs @ [f])\<close> 
+            using X8 X7 tuple_partial_pullback_def
+            by (metis X0 \<open>is_function_tuple Q\<^sub>p m (fs @ [f])\<close>
                 tuple_partial_image_def tuple_partial_pullback_memI)
         qed
       qed
-      then show ?thesis 
-        using augment_inverse_is_semialgebraic 
-        by (simp add: L)        
+      then show ?thesis
+        using augment_inverse_is_semialgebraic
+        by (simp add: L)
     qed
   qed
 qed
 
 lemma singleton_tuple_partial_pullback_is_semialg_map_tuple:
   assumes "is_semialg_function_tuple m fs"
   assumes "length fs = 1"
   shows "is_semialg_map_tuple m fs"
 proof(rule is_semialg_map_tupleI)
   show "is_function_tuple Q\<^sub>p m fs"
-    by (simp add: assms(1) semialg_function_tuple_is_function_tuple)    
+    by (simp add: assms(1) semialg_function_tuple_is_function_tuple)
   show "\<And>k S. S \<in> semialg_sets (length fs + k) \<Longrightarrow> is_semialgebraic (m + k) (tuple_partial_pullback m fs k S)"
   proof- fix k S assume A: "S \<in> semialg_sets (length fs + k)"
     show "is_semialgebraic (m + k) (tuple_partial_pullback m fs k S)"
     proof-
       obtain f where f_def: "fs = [f]"
-        using assms 
+        using assms
         by (metis One_nat_def length_0_conv length_Suc_conv)
       have 0: "is_semialg_function m f"
-        using f_def assms is_semialg_function_tupleE'[of m fs f] 
+        using f_def assms is_semialg_function_tupleE'[of m fs f]
         by simp
       have 1: "\<And>x. tuple_partial_image m fs x = partial_image m f x"
-        unfolding function_tuple_eval_def tuple_partial_image_def partial_image_def 
+        unfolding function_tuple_eval_def tuple_partial_image_def partial_image_def
         by (metis (no_types, lifting) append_Cons append_Nil2 append_eq_append_conv_if
-            f_def list.simps(8) list.simps(9))        
+            f_def list.simps(8) list.simps(9))
       have 2: "tuple_partial_pullback m fs k S = partial_pullback m f k S"
       proof
         show "tuple_partial_pullback m fs k S \<subseteq> partial_pullback m f k S"
-            using 1 unfolding tuple_partial_pullback_def  partial_pullback_def evimage_def 
-            by (metis (no_types, lifting) set_eq_subset vimage_inter_cong)            
+            using 1 unfolding tuple_partial_pullback_def  partial_pullback_def evimage_def
+            by (metis (no_types, lifting) set_eq_subset vimage_inter_cong)
         show "partial_pullback m f k S \<subseteq> tuple_partial_pullback m fs k S"
-          using 1 unfolding tuple_partial_pullback_def  partial_pullback_def evimage_def 
+          using 1 unfolding tuple_partial_pullback_def  partial_pullback_def evimage_def
             by blast
       qed
-      then show ?thesis 
+      then show ?thesis
         by (metis "0" A assms(2) is_semialg_functionE is_semialgebraicI)
     qed
   qed
 qed
 
 lemma empty_tuple_partial_pullback_is_semialg_map_tuple:
   assumes "is_semialg_function_tuple m fs"
   assumes "length fs = 0"
   shows "is_semialg_map_tuple m fs"
-  apply(rule is_semialg_map_tupleI)  
+  apply(rule is_semialg_map_tupleI)
   using assms(1) semialg_function_tuple_is_function_tuple apply blast
 proof-
   fix k S assume A: "S \<in> semialg_sets (length fs + k)"
   then have 0: "is_semialgebraic k S"
     by (metis add.left_neutral assms(2) is_semialgebraicI)
   have 1: "tuple_partial_pullback m fs k S = cartesian_product (carrier (Q\<^sub>p\<^bsup>m\<^esup>)) S"
   proof
     have 1: "\<And>x. function_tuple_eval Q\<^sub>p m fs (take m x) = []"
-      using assms unfolding function_tuple_eval_def 
-      by blast      
+      using assms unfolding function_tuple_eval_def
+      by blast
     show "tuple_partial_pullback m fs k S \<subseteq> cartesian_product (carrier (Q\<^sub>p\<^bsup>m\<^esup>)) S"
       apply(rule subsetI) apply(rule cartesian_product_memI[of "carrier (Q\<^sub>p\<^bsup>m\<^esup>)" Q\<^sub>p m S k])
       apply blast using 0 is_semialgebraic_closed apply blast
-      using  0 assms unfolding 1 tuple_partial_pullback_def tuple_partial_image_def 
+      using  0 assms unfolding 1 tuple_partial_pullback_def tuple_partial_image_def
       apply (meson IntD2 le_add1 take_closed)
      by (metis append_Nil evimageD evimage_def)
    have 2: "cartesian_product (carrier (Q\<^sub>p\<^bsup>m\<^esup>)) S \<subseteq> carrier (Q\<^sub>p\<^bsup>m + k\<^esup>)"
-     using is_semialgebraic_closed[of k S] 0 assms cartesian_product_closed[of "carrier (Q\<^sub>p\<^bsup>m\<^esup>)" Q\<^sub>p m S k]  by blast 
+     using is_semialgebraic_closed[of k S] 0 assms cartesian_product_closed[of "carrier (Q\<^sub>p\<^bsup>m\<^esup>)" Q\<^sub>p m S k]  by blast
    show "cartesian_product (carrier (Q\<^sub>p\<^bsup>m\<^esup>)) S \<subseteq> tuple_partial_pullback m fs k S"
      apply(rule subsetI)  apply(rule tuple_partial_pullback_memI)
-     using 2 apply blast 
+     using 2 apply blast
      using assms semialg_function_tuple_is_function_tuple apply blast
-     unfolding 1 
+     unfolding 1
      by (metis carrier_is_semialgebraic cartesian_product_memE(2) is_semialgebraic_closed self_append_conv2)
   qed
   show "is_semialgebraic (m + k) (tuple_partial_pullback m fs k S)"
-    unfolding 1 
+    unfolding 1
     using "0" car_times_semialg_is_semialg by blast
 qed
 
 lemma tuple_partial_pullback_is_semialg_map_tuple:
   assumes "is_semialg_function_tuple m fs"
   shows "is_semialg_map_tuple m fs"
 proof-
   have "\<And>n fs. is_semialg_function_tuple m fs \<and> length fs = n \<Longrightarrow> is_semialg_map_tuple m fs"
-  proof- fix n 
+  proof- fix n
     show " \<And> fs. is_semialg_function_tuple m fs \<and> length fs = n \<Longrightarrow> is_semialg_map_tuple m fs"
       apply(induction n)
-      using singleton_tuple_partial_pullback_is_semialg_map_tuple empty_tuple_partial_pullback_is_semialg_map_tuple apply blast      
+      using singleton_tuple_partial_pullback_is_semialg_map_tuple empty_tuple_partial_pullback_is_semialg_map_tuple apply blast
     proof-
       fix n fs
       assume IH: "(\<And>fs. is_semialg_function_tuple m fs \<and> length fs = n \<Longrightarrow> is_semialg_map_tuple m fs)"
       assume A: "is_semialg_function_tuple m fs \<and> length fs = Suc n"
       then obtain gs f where gs_f_def: "fs = gs@[f]"
         by (metis length_Suc_conv list.discI rev_exhaust)
       have gs_length: "length gs = n"
-        using gs_f_def 
+        using gs_f_def
         by (metis A length_append_singleton nat.inject)
       have 0: "set gs \<subseteq> set fs"
         by (simp add: gs_f_def subsetI)
       have 1: "is_semialg_function_tuple m gs"
         apply(rule is_semialg_function_tupleI)
-        using 0 A gs_f_def is_semialg_function_tupleE'[of m fs] 
+        using 0 A gs_f_def is_semialg_function_tupleE'[of m fs]
         by blast
-      then have 2: "is_semialg_map_tuple m gs" 
-        using IH gs_length 
+      then have 2: "is_semialg_map_tuple m gs"
+        using IH gs_length
         by blast
       have 3: "is_semialg_function m f"
-        using gs_f_def A 
+        using gs_f_def A
         by (metis gs_length is_semialg_function_tupleE lessI nth_append_length)
       then show "is_semialg_map_tuple m fs"
-        using assms 2 gs_f_def tuple_partial_pullback_is_semialg_map_tuple_induct 
-        by blast        
-    qed
-  qed
-  then show ?thesis 
-    using assms by blast 
+        using assms 2 gs_f_def tuple_partial_pullback_is_semialg_map_tuple_induct
+        by blast
+    qed
+  qed
+  then show ?thesis
+    using assms by blast
 qed
 
 
       (********************************************************************************************)
       (********************************************************************************************)
       subsubsection\<open>Semialgebraic Functions are Closed under Composition with Semialgebraic Tuples\<close>
       (********************************************************************************************)
       (********************************************************************************************)
 
 lemma function_tuple_comp_partial_pullback:
   assumes "is_semialg_function_tuple m fs"
   assumes "length fs = n"
   assumes "is_semialg_function n f"
   assumes "S \<subseteq> carrier (Q\<^sub>p\<^bsup>1+k\<^esup>)"
-  shows "partial_pullback m (function_tuple_comp Q\<^sub>p fs f) k S = 
+  shows "partial_pullback m (function_tuple_comp Q\<^sub>p fs f) k S =
              tuple_partial_pullback m fs k (partial_pullback n f k S)"
-proof- 
-  have 0: "\<And>x. partial_image m (function_tuple_comp Q\<^sub>p fs f) x = 
+proof-
+  have 0: "\<And>x. partial_image m (function_tuple_comp Q\<^sub>p fs f) x =
         partial_image n f (tuple_partial_image m fs x)"
-    unfolding partial_image_def function_tuple_comp_def tuple_partial_image_def 
+    unfolding partial_image_def function_tuple_comp_def tuple_partial_image_def
     using comp_apply[of f "function_tuple_eval Q\<^sub>p 0 fs"]
-    unfolding function_tuple_eval_def 
+    unfolding function_tuple_eval_def
   proof -
     fix x :: "((nat \<Rightarrow> int) \<times> (nat \<Rightarrow> int)) set list"
     assume a1: "\<And>x. (f \<circ> (\<lambda>x. map (\<lambda>f. f x) fs)) x = f (map (\<lambda>f. f x) fs)"
     have f2: "\<forall>f rs. drop n (map f fs @ (rs::((nat \<Rightarrow> int) \<times> (nat \<Rightarrow> int)) set list)) = rs"
       by (simp add: assms(2))
     have "\<forall>f rs. take n (map f fs @ (rs::((nat \<Rightarrow> int) \<times> (nat \<Rightarrow> int)) set list)) = map f fs"
       by (simp add: assms(2))
-    then show "(f \<circ> (\<lambda>rs. map (\<lambda>f. f rs) fs)) (take m x) # drop m x = 
+    then show "(f \<circ> (\<lambda>rs. map (\<lambda>f. f rs) fs)) (take m x) # drop m x =
               f (take n (map (\<lambda>f. f (take m x)) fs @ drop m x)) # drop n (map (\<lambda>f. f (take m x)) fs @ drop m x)"
       using f2 a1 by presburger
   qed
-  show "partial_pullback m (function_tuple_comp Q\<^sub>p fs f) k S = 
+  show "partial_pullback m (function_tuple_comp Q\<^sub>p fs f) k S =
               tuple_partial_pullback m fs k (partial_pullback n f k S)"
   proof
     show "partial_pullback m (function_tuple_comp Q\<^sub>p fs f) k S \<subseteq> tuple_partial_pullback m fs k (partial_pullback n f k S)"
     proof fix x assume A: "x \<in> partial_pullback m (function_tuple_comp Q\<^sub>p fs f) k S"
       then have 1: "partial_image m (function_tuple_comp Q\<^sub>p fs f) x \<in> S"
         using partial_pullback_memE(2) by blast
       have 2: "  partial_image n f (tuple_partial_image m fs x) \<in> S"
-        using 0 1 
+        using 0 1
         by presburger
       have 3: "x \<in> carrier (Q\<^sub>p\<^bsup>m + k\<^esup>)"
-        using A assms 
-        by (metis partial_pullback_memE(1))  
+        using A assms
+        by (metis partial_pullback_memE(1))
       have 4: "tuple_partial_image m fs x \<in> partial_pullback n f k S"
         apply(rule partial_pullback_memI)
          apply (metis "0" "3" add_cancel_left_left assms(1) assms(2) cartesian_power_drop drop0
-            list.inject local.partial_image_def not_gr_zero semialg_function_tuple_is_function_tuple 
+            list.inject local.partial_image_def not_gr_zero semialg_function_tuple_is_function_tuple
             tuple_partial_image_closed)
         by (metis "2" local.partial_image_def)
       show " x \<in> tuple_partial_pullback m fs k (partial_pullback n f k S)"
         apply(rule tuple_partial_pullback_memI)
         apply (simp add: "3")
         using assms(1) semialg_function_tuple_is_function_tuple apply blast
         by (metis "4" tuple_partial_image_def)
     qed
     show " tuple_partial_pullback m fs k (partial_pullback n f k S) \<subseteq> partial_pullback m (function_tuple_comp Q\<^sub>p fs f) k S"
     proof fix x assume A: "x \<in> tuple_partial_pullback m fs k (partial_pullback n f k S)"
       show "x \<in> partial_pullback m (function_tuple_comp Q\<^sub>p fs f) k S "
       proof-
         have "partial_image n f (tuple_partial_image m fs x) \<in> S"
-          using A partial_pullback_memE(2) tuple_partial_pullback_memE(2) 
+          using A partial_pullback_memE(2) tuple_partial_pullback_memE(2)
           by blast
-        show ?thesis 
+        show ?thesis
           apply(rule partial_pullback_memI)
            apply (meson A subset_eq tuple_partial_pullback_closed)
-          by (metis "0" \<open>local.partial_image n f (tuple_partial_image m fs x) \<in> S\<close> 
+          by (metis "0" \<open>local.partial_image n f (tuple_partial_image m fs x) \<in> S\<close>
               local.partial_image_def)
       qed
     qed
   qed
 qed
 
 lemma semialg_function_tuple_comp:
   assumes "is_semialg_function_tuple m fs"
   assumes "length fs = n"
   assumes "is_semialg_function n f"
   shows "is_semialg_function m (function_tuple_comp Q\<^sub>p fs f)"
 proof(rule is_semialg_functionI)
   show "function_tuple_comp Q\<^sub>p fs f \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<rightarrow> carrier Q\<^sub>p"
-    using function_tuple_comp_closed[of f Q\<^sub>p n fs] assms(1) assms(2) 
+    using function_tuple_comp_closed[of f Q\<^sub>p n fs] assms(1) assms(2)
       assms(3) is_semialg_function_closed semialg_function_tuple_is_function_tuple
     by blast
   show "\<And>k S. S \<in> semialg_sets (1 + k) \<Longrightarrow> is_semialgebraic (m + k) (partial_pullback m (function_tuple_comp Q\<^sub>p fs f) k S)"
   proof- fix k S
     assume A0: "S \<in> semialg_sets (1 + k)"
     show "is_semialgebraic (m + k) (partial_pullback m (function_tuple_comp Q\<^sub>p fs f) k S)"
     proof-
-      have 0: "partial_pullback m (function_tuple_comp Q\<^sub>p fs f) k S = 
+      have 0: "partial_pullback m (function_tuple_comp Q\<^sub>p fs f) k S =
              tuple_partial_pullback m fs k (partial_pullback n f k S)"
-        using  function_tuple_comp_partial_pullback[of m fs n f S k] assms 
-            \<open>S \<in> semialg_sets (1 + k)\<close> is_semialgebraicI is_semialgebraic_closed 
+        using  function_tuple_comp_partial_pullback[of m fs n f S k] assms
+            \<open>S \<in> semialg_sets (1 + k)\<close> is_semialgebraicI is_semialgebraic_closed
         by blast
       have 1: "is_semialgebraic (n + k) (partial_pullback n f k S)"
-        using assms A0 is_semialg_functionE is_semialgebraicI 
+        using assms A0 is_semialg_functionE is_semialgebraicI
         by blast
       have 2: "is_semialgebraic (m + k) (tuple_partial_pullback m fs k (partial_pullback n f k S))"
-        using 1 0 assms tuple_partial_pullback_is_semialg_map_tuple[of m fs] 
+        using 1 0 assms tuple_partial_pullback_is_semialg_map_tuple[of m fs]
           is_semialg_map_tupleE[of m fs k "partial_pullback n f k S"]
         by blast
       then show ?thesis
-        using 0 
+        using 0
         by simp
     qed
   qed
 qed
 
 
       (********************************************************************************************)
       (********************************************************************************************)
       subsubsection\<open>Algebraic Operations on Semialgebraic Functions\<close>
       (********************************************************************************************)
       (********************************************************************************************)
 
 
 text\<open>Defining the set of extensional semialgebraic functions\<close>
 
 definition Qp_add_fun where
 "Qp_add_fun xs = xs!0 \<oplus>\<^bsub>Q\<^sub>p\<^esub> xs!1"
 
 definition Qp_mult_fun where
 "Qp_mult_fun xs = xs!0 \<otimes> xs!1"
 
 text\<open>Inversion function on first coordinates of Qp tuples. Arbitrarily redefined at 0 to map to 0\<close>
 
 definition Qp_invert where
 "Qp_invert xs = (if ((xs!0) = \<zero>) then \<zero> else (inv (xs!0)))"
 
 text\<open>Addition is semialgebraic\<close>
 
 lemma addition_is_semialg:
 "is_semialg_function 2 Qp_add_fun"
 proof-
   have 0: "\<And>x. x \<in> carrier (Q\<^sub>p\<^bsup>2\<^esup>) \<Longrightarrow> Qp_add_fun x = Qp_ev (pvar Q\<^sub>p 0 \<oplus>\<^bsub>Q\<^sub>p[\<X>\<^bsub>2\<^esub>]\<^esub> pvar Q\<^sub>p 1) x"
   proof- fix x assume A: "x \<in> carrier (Q\<^sub>p\<^bsup>2\<^esup>)"
     have "Qp_ev (pvar Q\<^sub>p 0 \<oplus>\<^bsub>Q\<^sub>p[\<X>\<^bsub>2\<^esub>]\<^esub> pvar Q\<^sub>p 1) x = (Qp_ev (pvar Q\<^sub>p 0) x) \<oplus>\<^bsub>Q\<^sub>p\<^esub> (Qp_ev (pvar Q\<^sub>p 1) x)"
       by (metis A One_nat_def eval_at_point_add  pvar_closed less_Suc_eq numeral_2_eq_2)
     then show " Qp_add_fun x = Qp_ev (pvar Q\<^sub>p 0 \<oplus>\<^bsub>Q\<^sub>p[\<X>\<^bsub>2\<^esub>]\<^esub> pvar Q\<^sub>p 1) x"
       by (metis A Qp_add_fun_def add_vars_def add_vars_rep one_less_numeral_iff
           pos2 semiring_norm(76))
   qed
-  then have 1: "restrict Qp_add_fun (carrier (Q\<^sub>p\<^bsup>2\<^esup>)) = 
+  then have 1: "restrict Qp_add_fun (carrier (Q\<^sub>p\<^bsup>2\<^esup>)) =
       restrict (Qp_ev (pvar Q\<^sub>p 0 \<oplus>\<^bsub>Q\<^sub>p[\<X>\<^bsub>2\<^esub>]\<^esub> pvar Q\<^sub>p 1)) (carrier (Q\<^sub>p\<^bsup>2\<^esup>))"
     by (meson restrict_ext)
   have "is_semialg_function 2 (Qp_ev (pvar Q\<^sub>p 0 \<oplus>\<^bsub>Q\<^sub>p[\<X>\<^bsub>2\<^esub>]\<^esub> pvar Q\<^sub>p 1))"
     using poly_is_semialg[of "pvar Q\<^sub>p 0 \<oplus>\<^bsub>Q\<^sub>p[\<X>\<^bsub>2\<^esub>]\<^esub> pvar Q\<^sub>p 1"]
-    by (meson MP.add.m_closed local.pvar_closed one_less_numeral_iff pos2 semiring_norm(76))    
-  then show ?thesis 
-    using 1 semialg_function_on_carrier[of 2 "Qp_add_fun" "Qp_ev (pvar Q\<^sub>p 0 \<oplus>\<^bsub>Q\<^sub>p[\<X>\<^bsub>2\<^esub>]\<^esub> pvar Q\<^sub>p 1)"] 
-          semialg_function_on_carrier 
+    by (meson MP.add.m_closed local.pvar_closed one_less_numeral_iff pos2 semiring_norm(76))
+  then show ?thesis
+    using 1 semialg_function_on_carrier[of 2 "Qp_add_fun" "Qp_ev (pvar Q\<^sub>p 0 \<oplus>\<^bsub>Q\<^sub>p[\<X>\<^bsub>2\<^esub>]\<^esub> pvar Q\<^sub>p 1)"]
+          semialg_function_on_carrier
     by presburger
 qed
 
 text\<open>Multiplication is semialgebraic:\<close>
 
 lemma multiplication_is_semialg:
 "is_semialg_function 2 Qp_mult_fun"
 proof-
   have 0: "\<And>x. x \<in> carrier (Q\<^sub>p\<^bsup>2\<^esup>) \<Longrightarrow> Qp_mult_fun x = Qp_ev (pvar Q\<^sub>p 0 \<otimes>\<^bsub>Q\<^sub>p[\<X>\<^bsub>2\<^esub>]\<^esub> pvar Q\<^sub>p 1) x"
   proof- fix x assume A: "x \<in> carrier (Q\<^sub>p\<^bsup>2\<^esup>)"
-    have "Qp_ev (pvar Q\<^sub>p 0 \<otimes>\<^bsub>Q\<^sub>p[\<X>\<^bsub>2\<^esub>]\<^esub> pvar Q\<^sub>p 1) x = 
+    have "Qp_ev (pvar Q\<^sub>p 0 \<otimes>\<^bsub>Q\<^sub>p[\<X>\<^bsub>2\<^esub>]\<^esub> pvar Q\<^sub>p 1) x =
                 (Qp_ev (pvar Q\<^sub>p 0) x) \<otimes> (Qp_ev (pvar Q\<^sub>p 1) x)"
-      by (metis A One_nat_def eval_at_point_mult pvar_closed less_Suc_eq numeral_2_eq_2)      
+      by (metis A One_nat_def eval_at_point_mult pvar_closed less_Suc_eq numeral_2_eq_2)
     then show " Qp_mult_fun x = Qp_ev (pvar Q\<^sub>p 0 \<otimes>\<^bsub>Q\<^sub>p[\<X>\<^bsub>2\<^esub>]\<^esub> pvar Q\<^sub>p 1) x"
-      by (metis A Qp_mult_fun_def mult_vars_def mult_vars_rep 
-          one_less_numeral_iff pos2 semiring_norm(76))     
-  qed
-  then have 1: "restrict Qp_mult_fun (carrier (Q\<^sub>p\<^bsup>2\<^esup>)) = 
+      by (metis A Qp_mult_fun_def mult_vars_def mult_vars_rep
+          one_less_numeral_iff pos2 semiring_norm(76))
+  qed
+  then have 1: "restrict Qp_mult_fun (carrier (Q\<^sub>p\<^bsup>2\<^esup>)) =
       restrict (Qp_ev (pvar Q\<^sub>p 0 \<otimes>\<^bsub>Q\<^sub>p[\<X>\<^bsub>2\<^esub>]\<^esub> pvar Q\<^sub>p 1)) (carrier (Q\<^sub>p\<^bsup>2\<^esup>))"
     by (meson restrict_ext)
   have "is_semialg_function 2 (Qp_ev (pvar Q\<^sub>p 0 \<otimes>\<^bsub>Q\<^sub>p[\<X>\<^bsub>2\<^esub>]\<^esub> pvar Q\<^sub>p 1))"
     using poly_is_semialg[of "pvar Q\<^sub>p 0 \<otimes>\<^bsub>Q\<^sub>p[\<X>\<^bsub>2\<^esub>]\<^esub> pvar Q\<^sub>p 1"]
     by (meson MP.m_closed local.pvar_closed one_less_numeral_iff pos2 semiring_norm(76))
-  thus ?thesis     
-    using 1 semialg_function_on_carrier[of 2 "Qp_mult_fun" "Qp_ev (pvar Q\<^sub>p 0 \<otimes>\<^bsub>Q\<^sub>p[\<X>\<^bsub>2\<^esub>]\<^esub> pvar Q\<^sub>p 1)"] 
-          semialg_function_on_carrier 
+  thus ?thesis
+    using 1 semialg_function_on_carrier[of 2 "Qp_mult_fun" "Qp_ev (pvar Q\<^sub>p 0 \<otimes>\<^bsub>Q\<^sub>p[\<X>\<^bsub>2\<^esub>]\<^esub> pvar Q\<^sub>p 1)"]
+          semialg_function_on_carrier
     by presburger
 qed
 
 text\<open>Inversion is semialgebraic:\<close>
 
 lemma(in field) field_nat_pow_inv:
   assumes "a \<in> carrier R"
   assumes "a \<noteq> \<zero>"
   shows "inv (a [^] (n::nat)) = (inv a) [^] (n :: nat)"
   apply(induction n)
   using inv_one local.nat_pow_0 apply presburger
-  using assms nat_pow_of_inv 
+  using assms nat_pow_of_inv
   by (metis Units_one_closed field_inv(2) field_inv(3) unit_factor)
 
 lemma Qp_invert_basic_semialg:
   assumes "is_basic_semialg (1 + k)  S"
   shows "is_semialgebraic (1 + k) (partial_pullback 1 Qp_invert k S)"
 proof-
   obtain P n where P_n_def: "(n::nat) \<noteq> 0 \<and> P \<in> carrier (Q\<^sub>p[\<X>\<^bsub>1+k\<^esub>]) \<and> S = basic_semialg_set (1+k) n P \<and> P \<in> carrier (Q\<^sub>p[\<X>\<^bsub>1+k\<^esub>])"
-    using assms is_basic_semialg_def  
+    using assms is_basic_semialg_def
     by meson
   obtain d::nat where d_def: "d =  deg (coord_ring Q\<^sub>p k) (to_univ_poly (Suc k) 0 P)"
     by auto
   obtain l where l_def: "l =  ((- d) mod n)"
     by blast
   have 1: "(l + d) mod n = 0"
     by (metis add_eq_0_iff equation_minus_iff l_def mod_0 mod_add_cong mod_minus_eq zmod_int)
   then  obtain m::int where m_def:  " (l + d) = m*n "
-    using d_def l_def   
+    using d_def l_def
     by (metis mult_of_nat_commute zmod_eq_0D)
-  have 2: "m \<ge>0" 
-  proof- 
+  have 2: "m \<ge>0"
+  proof-
     have 10: "n > 0"
-      using P_n_def 
+      using P_n_def
       by blast
     have 11: "l \<ge> 0"
-      using l_def 10 Euclidean_Division.pos_mod_sign of_nat_0_less_iff 
+      using l_def 10 Euclidean_Division.pos_mod_sign of_nat_0_less_iff
       by blast
     then show ?thesis
-      using m_def 
+      using m_def
       by (metis "10" le_add_same_cancel1 minus_add_cancel negative_zle
         neq0_conv  of_nat_le_0_iff zero_le_imp_eq_int zero_le_mult_iff)
   qed
   obtain N where N_def: "N = m*n"
     by blast
   have 3: "N \<ge> d"
   proof-
     have "l \<ge> 0"
-      using l_def d_def m_def  Euclidean_Division.pos_mod_sign[of n "-d"] P_n_def 
+      using l_def d_def m_def  Euclidean_Division.pos_mod_sign[of n "-d"] P_n_def
       by linarith
-    then show ?thesis 
+    then show ?thesis
       using d_def N_def m_def
       by linarith
   qed
   have 4: "deg (coord_ring Q\<^sub>p k) (to_univ_poly (Suc k) 0 P) \<le> nat N"
-    using d_def N_def 3 
+    using d_def N_def 3
     by linarith
   have 5: " P \<in> carrier (coord_ring Q\<^sub>p (Suc k))"
     by (metis P_n_def  plus_1_eq_Suc)
   have 6: " \<exists>q\<in>carrier (coord_ring Q\<^sub>p (Suc k)).
        \<forall>x\<in>carrier Q\<^sub>p - {\<zero>}. \<forall>a\<in>carrier (Q\<^sub>p\<^bsup>k\<^esup>). Qp_ev q (insert_at_index a x 0) = (x[^]nat N) \<otimes> Qp_ev P (insert_at_index a (inv x) 0)"
-    using 3 4 d_def to_univ_poly_one_over_poly''[of 0 k P "nat N"] "5"  Qp.field_axioms 
+    using 3 4 d_def to_univ_poly_one_over_poly''[of 0 k P "nat N"] "5"  Qp.field_axioms
     by blast
   obtain q where q_def: "q \<in> carrier (coord_ring Q\<^sub>p (Suc k)) \<and> ( \<forall> x \<in> carrier Q\<^sub>p - {\<zero>}. ( \<forall> a \<in>  carrier (Q\<^sub>p\<^bsup>k\<^esup>).
         eval_at_point Q\<^sub>p (insert_at_index a x 0) q =  (x[^] (nat N)) \<otimes> (eval_at_point Q\<^sub>p (insert_at_index a (inv x) 0) P)))"
-    using 6 
+    using 6
     by blast
   obtain T where T_def: "T = basic_semialg_set (1+k) n q"
     by auto
   have "is_basic_semialg (1 + k) T"
   proof-
     have "q \<in> carrier ( Q\<^sub>p[\<X>\<^bsub>Suc k\<^esub>])"
-      using q_def  
+      using q_def
       by presburger
-    then show ?thesis 
-      using T_def is_basic_semialg_def  
+    then show ?thesis
+      using T_def is_basic_semialg_def
       by (metis P_n_def plus_1_eq_Suc)
   qed
   then have T_semialg: "is_semialgebraic (1+k) T"
-    using T_def basic_semialg_is_semialg[of "1+k" T] is_semialgebraicI 
+    using T_def basic_semialg_is_semialg[of "1+k" T] is_semialgebraicI
     by blast
   obtain Nz where Nz_def: "Nz = {xs \<in> carrier (Q\<^sub>p\<^bsup>Suc k\<^esup>). xs!0 \<noteq> \<zero>}"
-    by blast 
+    by blast
   have Nz_semialg: "is_semialgebraic (1+k) Nz"
   proof-
     obtain Nzc where Nzc_def: "Nzc = {xs \<in> carrier (Q\<^sub>p\<^bsup>Suc k\<^esup>). xs!0 = \<zero>}"
-      by blast 
+      by blast
     have 0: "Nzc = zero_set Q\<^sub>p (Suc k) (pvar Q\<^sub>p 0)"
-      unfolding zero_set_def 
-      using Nzc_def 
+      unfolding zero_set_def
+      using Nzc_def
       by (metis (no_types, lifting) Collect_cong eval_pvar zero_less_Suc)
     have 1: "is_algebraic Q\<^sub>p (1+k)  Nzc"
-      using 0 pvar_closed[of ]  
+      using 0 pvar_closed[of ]
       by (metis is_algebraicI' plus_1_eq_Suc zero_less_Suc)
     then have 2: "is_semialgebraic  (1+k) Nzc"
       using is_algebraic_imp_is_semialg by blast
     have 3: "Nz = carrier (Q\<^sub>p\<^bsup>Suc k\<^esup>) - Nzc"
-      using Nz_def Nzc_def 
+      using Nz_def Nzc_def
       by blast
     then show ?thesis
-      using 2 
+      using 2
       by (simp add: complement_is_semialg)
   qed
   have 7: "(partial_pullback 1 Qp_invert k S) \<inter> Nz = T \<inter> Nz"
   proof
     show "partial_pullback 1 Qp_invert k S \<inter> Nz \<subseteq> T \<inter> Nz"
     proof fix c assume A: "c \<in> partial_pullback 1 Qp_invert k S \<inter> Nz"
       show "c  \<in> T \<inter> Nz"
       proof-
         have c_closed: "c \<in> carrier (Q\<^sub>p\<^bsup>1+k\<^esup>)"
-          using A partial_pullback_closed[of 1  Qp_invert  k S] 
+          using A partial_pullback_closed[of 1  Qp_invert  k S]
           by blast
         obtain x a where xa_def: "c = (x#a)"
-          using c_closed 
+          using c_closed
           by (metis Suc_eq_plus1 add.commute cartesian_power_car_memE length_Suc_conv)
         have x_closed: "x \<in> carrier Q\<^sub>p"
-          using xa_def  c_closed 
-          by (metis (no_types, lifting) append_Cons cartesian_power_decomp 
+          using xa_def  c_closed
+          by (metis (no_types, lifting) append_Cons cartesian_power_decomp
               list.inject Qp.to_R1_to_R Qp.to_R_pow_closed)
         have a_closed: "a \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>)"
-          using  xa_def c_closed 
+          using  xa_def c_closed
           by (metis One_nat_def cartesian_power_drop drop0 drop_Suc_Cons)
         have 0: "c \<in> Nz"
           using A by blast
         then have "x \<noteq> \<zero>"
-          using Nz_def xa_def 
+          using Nz_def xa_def
           by (metis (mono_tags, lifting) mem_Collect_eq nth_Cons_0)
         have 1: "Qp_invert [x] = inv x"
-          unfolding Qp_invert_def 
-          by (metis \<open>x \<noteq> \<zero>\<close> nth_Cons_0)          
+          unfolding Qp_invert_def
+          by (metis \<open>x \<noteq> \<zero>\<close> nth_Cons_0)
         have 2: "partial_image 1 Qp_invert c \<in> S"
-          using A partial_pullback_memE[of c 1 "Qp_invert" k S] 
+          using A partial_pullback_memE[of c 1 "Qp_invert" k S]
           by blast
         have 3: "inv x # a  \<in> S"
         proof-
           have 30: "[x] = take 1 c"
             by (simp add: xa_def)
           have 31: "a = drop 1 c"
             by (simp add: xa_def)
-          show ?thesis 
+          show ?thesis
             using 1 30 31 partial_image_def[of 1 "Qp_invert" c] xa_def  "2"
-            by metis 
+            by metis
         qed
         obtain y where y_def: "y \<in> carrier Q\<^sub>p \<and> eval_at_point Q\<^sub>p (inv x # a) P = y [^] n"
           using 3 P_n_def basic_semialg_set_memE(2)
           by blast
-        then have 4: "x [^] (nat N) \<otimes> eval_at_point Q\<^sub>p (inv x # a) P 
+        then have 4: "x [^] (nat N) \<otimes> eval_at_point Q\<^sub>p (inv x # a) P
                       = x [^] (nat N) \<otimes> y [^] n"
-          by presburger          
+          by presburger
         have 5: "x [^] (nat N) \<otimes> y [^] n = ((x[^]m)\<otimes>y)[^]n"
         proof-
           have 50: "x [^] (N) \<otimes> y [^] n = x [^] (m*n) \<otimes> y [^] n"
             using N_def by blast
           have 51: "x [^] (m*n) = (x[^]m)[^]n"
-            using Qp_int_nat_pow_pow \<open>x \<noteq> \<zero>\<close> not_nonzero_Qp x_closed 
-            by metis          
+            using Qp_int_nat_pow_pow \<open>x \<noteq> \<zero>\<close> not_nonzero_Qp x_closed
+            by metis
           have 52: "x [^] (m*n)\<otimes> y [^] n = ((x[^]m) \<otimes> y) [^] n"
           proof-
             have 0: "x [^] (m*n)\<otimes> y [^] n= (x[^]m)[^]n \<otimes> (y[^] n)"
               using "51" by presburger
             have 1: "(x[^]m)[^]n \<otimes> (y[^] n) = ((x[^]m) \<otimes> y) [^] n"
-              apply(induction n) 
+              apply(induction n)
               using Qp.nat_pow_0 Qp.one_closed Qp.r_one apply presburger
-              using x_closed y_def 
+              using x_closed y_def
               by (metis Qp.nat_pow_distrib Qp.nonzero_closed Qp_int_pow_nonzero \<open>x \<noteq> \<zero>\<close> not_nonzero_Qp)
-            then show ?thesis 
+            then show ?thesis
               using "0" by blast
           qed
           have 53: "x [^] N = x [^] (nat N)"
           proof-
             have "N \<ge> 0"
               by (metis (full_types) Euclidean_Division.pos_mod_sign N_def P_n_def
                   add_increasing2 int.lless_eq l_def  m_def  of_nat_0_le_iff of_nat_le_0_iff )
-            then show ?thesis 
+            then show ?thesis
                by (metis pow_nat)
           qed
           then show ?thesis
-            using 50 52 
-            by presburger            
-        qed  
+            using 50 52
+            by presburger
+        qed
         have 6: "x [^] (nat N) \<otimes> eval_at_point Q\<^sub>p (inv x # a) P = ((x[^]m)\<otimes>y)[^]n"
-          using "4" "5" 
+          using "4" "5"
           by blast
         have 7: "eval_at_point Q\<^sub>p c q = ((x[^]m)\<otimes>y)[^]n"
         proof-
           have 70: "(insert_at_index a (inv x) 0) = inv x # a"
-            using insert_at_index.simps 
-            by (metis (no_types, lifting) append_eq_append_conv2 append_same_eq append_take_drop_id drop0 same_append_eq)       
-          have 71: "(insert_at_index a x) 0 =  x # a"            
-            by simp 
-          then show ?thesis using 6 q_def 
+            using insert_at_index.simps
+            by (metis (no_types, lifting) append_eq_append_conv2 append_same_eq append_take_drop_id drop0 same_append_eq)
+          have 71: "(insert_at_index a x) 0 =  x # a"
+            by simp
+          then show ?thesis using 6 q_def
             by (metis "70" DiffI \<open>x \<noteq> \<zero>\<close> a_closed empty_iff insert_iff x_closed xa_def)
         qed
         have 8: "(x[^]m)\<otimes>y \<in> carrier Q\<^sub>p"
         proof-
           have 80: "x[^]m \<in> carrier Q\<^sub>p"
-            using  \<open>x \<noteq> \<zero>\<close> x_closed Qp_int_pow_nonzero[of x m] unfolding nonzero_def 
-            by blast   
-          then show ?thesis 
-            using y_def by blast       
-        qed 
-        then have "c \<in> T" 
+            using  \<open>x \<noteq> \<zero>\<close> x_closed Qp_int_pow_nonzero[of x m] unfolding nonzero_def
+            by blast
+          then show ?thesis
+            using y_def by blast
+        qed
+        then have "c \<in> T"
           using T_def basic_semialg_set_def "7" c_closed by blast
-        then show ?thesis 
+        then show ?thesis
           by (simp add: \<open>c \<in> T\<close> "0")
       qed
     qed
     show "T \<inter> Nz \<subseteq> partial_pullback 1 Qp_invert k S \<inter> Nz"
     proof fix x assume A: "x \<in> T \<inter> Nz"
       show " x \<in> partial_pullback 1 Qp_invert k S \<inter> Nz "
       proof-
         have " x \<in> partial_pullback 1 Qp_invert k S"
         proof(rule partial_pullback_memI)
           show x_closed: "x \<in> carrier (Q\<^sub>p\<^bsup>1+k\<^esup>)"
-            using T_def A 
+            using T_def A
             by (meson IntD1 basic_semialg_set_memE(1))
           show "Qp_invert (take 1 x) # drop 1 x \<in> S"
           proof-
             have 00: "x!0  \<noteq> \<zero>"
-              using A Nz_def  
+              using A Nz_def
               by blast
             then have 0: "Qp_invert (take 1 x) # drop 1 x = inv (x!0) # drop 1 x"
-              unfolding Qp_invert_def  
+              unfolding Qp_invert_def
               by (smt One_nat_def lessI nth_take)
             have "drop 1 x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>)"
               using \<open>x \<in> carrier (Q\<^sub>p\<^bsup>1+k\<^esup>)\<close> cartesian_power_drop by blast
             obtain a where a_def: "a = (x!0)"
-              by blast 
+              by blast
             have a_closed: "a \<in> carrier Q\<^sub>p"
-              using 00 a_def A Nz_def cartesian_power_car_memE'[of x Q\<^sub>p "Suc k" 0] inv_in_frac(1) 
+              using 00 a_def A Nz_def cartesian_power_car_memE'[of x Q\<^sub>p "Suc k" 0] inv_in_frac(1)
               by blast
             have a_nz: "a \<noteq> \<zero>"
-              using a_def Nz_def  A 
-              by blast                
+              using a_def Nz_def  A
+              by blast
             obtain b where b_def: "b = drop 1 x"
-              by blast 
+              by blast
             have b_closed: "b \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>)"
-              using b_def A Nz_def  \<open>drop 1 x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>)\<close> 
+              using b_def A Nz_def  \<open>drop 1 x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>)\<close>
               by blast
             have abx: "x = a#b"
-              using a_def b_def x_closed 
-              by (metis (no_types, lifting) One_nat_def append_Cons append_Nil 
-                  append_eq_conv_conj cartesian_power_car_memE cartesian_power_decomp 
-                  lessI nth_take Qp.to_R1_to_R)              
+              using a_def b_def x_closed
+              by (metis (no_types, lifting) One_nat_def append_Cons append_Nil
+                  append_eq_conv_conj cartesian_power_car_memE cartesian_power_decomp
+                  lessI nth_take Qp.to_R1_to_R)
             have 1: "Qp_invert (take 1 x) # drop 1 x = (inv a)#b"
-              using "0" a_def b_def 
+              using "0" a_def b_def
               by blast
-            have 22: "eval_at_point Q\<^sub>p (insert_at_index b a 0) q =  
+            have 22: "eval_at_point Q\<^sub>p (insert_at_index b a 0) q =
                 (a[^] (nat N)) \<otimes> (eval_at_point Q\<^sub>p (insert_at_index b (inv a) 0) P)"
-              using q_def a_closed a_nz b_closed 
+              using q_def a_closed a_nz b_closed
               by blast
             obtain c where c_def: "c \<in> carrier Q\<^sub>p \<and> Qp_ev q x = (c[^]n)"
-              using  A T_def unfolding basic_semialg_set_def  
+              using  A T_def unfolding basic_semialg_set_def
               by blast
             obtain c' where c'_def: "c' = (inv a)[^]m  \<otimes>  c"
-              by blast 
+              by blast
             have c'_closed: "c' \<in> carrier Q\<^sub>p"
-              using c_def a_def a_closed a_nz Qp_int_pow_nonzero nonzero_def 
-                 c'_def inv_in_frac(3) Qp.m_closed Qp.nonzero_closed by presburger                 
+              using c_def a_def a_closed a_nz Qp_int_pow_nonzero nonzero_def
+                 c'_def inv_in_frac(3) Qp.m_closed Qp.nonzero_closed by presburger
             have 3: "(eval_at_point Q\<^sub>p ((inv a) # b) P) = (c'[^]n)"
             proof-
               have 30: "x = insert_at_index b a 0"
-                using abx 
+                using abx
                 by simp
-              have 31: "(c[^]n) =  
+              have 31: "(c[^]n) =
                 (a[^] (nat N)) \<otimes> (eval_at_point Q\<^sub>p (insert_at_index b (inv a) 0) P)"
-                using 22 30 c_def 
+                using 22 30 c_def
                 by blast
               have 32: "insert_at_index b (inv a) 0 = (inv a) # b"
-                using insert_at_index.simps 
-                by (metis drop0 self_append_conv2 take0)      
-              have 33: "(c[^]n) =  
+                using insert_at_index.simps
+                by (metis drop0 self_append_conv2 take0)
+              have 33: "(c[^]n) =
                 (a[^] (nat N)) \<otimes> (eval_at_point Q\<^sub>p ((inv a) # b) P)"
-               using "31" "32" by presburger               
+               using "31" "32" by presburger
               have 34: "(inv a) # b \<in> carrier (Q\<^sub>p\<^bsup>1+k\<^esup>)"
                 apply(rule cartesian_power_car_memI'')
-                 apply (metis b_closed cartesian_power_car_memE length_Suc_conv plus_1_eq_Suc)             
-                using a_closed a_nz b_closed 
+                 apply (metis b_closed cartesian_power_car_memE length_Suc_conv plus_1_eq_Suc)
+                using a_closed a_nz b_closed
                 apply (metis One_nat_def inv_in_frac(1) take0 take_Suc_Cons Qp.to_R1_closed)
-                by (metis abx b_closed b_def drop_Cons' not_Cons_self2)                
+                by (metis abx b_closed b_def drop_Cons' not_Cons_self2)
               have 35: "(eval_at_point Q\<^sub>p ((inv a) # b) P) \<in> carrier Q\<^sub>p"
-                using 34 P_n_def  eval_at_point_closed 
+                using 34 P_n_def  eval_at_point_closed
                 by blast
-              have  "inv(a[^] (nat N)) \<otimes> (c[^]n) =  
+              have  "inv(a[^] (nat N)) \<otimes> (c[^]n) =
                inv(a[^] (nat N)) \<otimes> ((a[^] (nat N)) \<otimes> (eval_at_point Q\<^sub>p ((inv a) # b) P))"
-                using 31 "33"  by presburger                
-              then have 6: "inv(a[^] (nat N)) \<otimes> (c[^]n) =  
-               inv(a[^] (nat N)) \<otimes> (a[^] (nat N)) \<otimes> (eval_at_point Q\<^sub>p ((inv a) # b) P)"        
+                using 31 "33"  by presburger
+              then have 6: "inv(a[^] (nat N)) \<otimes> (c[^]n) =
+               inv(a[^] (nat N)) \<otimes> (a[^] (nat N)) \<otimes> (eval_at_point Q\<^sub>p ((inv a) # b) P)"
                 using 35 monoid.m_assoc[of Q\<^sub>p] Qp.monoid_axioms Qp.nat_pow_closed
-                  Qp.nonzero_pow_nonzero a_nz inv_in_frac(1) local.a_closed by presburger                
-              have 37:"inv(a[^] (nat N)) \<otimes> (c[^]n) = (eval_at_point Q\<^sub>p ((inv a) # b) P)"    
+                  Qp.nonzero_pow_nonzero a_nz inv_in_frac(1) local.a_closed by presburger
+              have 37:"inv(a[^] (nat N)) \<otimes> (c[^]n) = (eval_at_point Q\<^sub>p ((inv a) # b) P)"
               proof-
                 have "inv(a[^] (nat N)) \<otimes> (a[^] (nat N)) = \<one> "
-                  using  a_closed  a_nz Qp.nat_pow_closed Qp.nonzero_pow_nonzero field_inv(1) 
-                  by blast     
-                then have "inv(a[^] (nat N)) \<otimes> (c[^]n) =  
+                  using  a_closed  a_nz Qp.nat_pow_closed Qp.nonzero_pow_nonzero field_inv(1)
+                  by blast
+                then have "inv(a[^] (nat N)) \<otimes> (c[^]n) =
                         \<one> \<otimes> (eval_at_point Q\<^sub>p ((inv a) # b) P)"
-                  using 6 by presburger                  
-                then show ?thesis using 35 Qp.l_one by blast                
+                  using 6 by presburger
+                then show ?thesis using 35 Qp.l_one by blast
               qed
-              have 38:"(inv a)[^] (nat N) \<otimes> (c[^]n) = (eval_at_point Q\<^sub>p ((inv a) # b) P)"    
+              have 38:"(inv a)[^] (nat N) \<otimes> (c[^]n) = (eval_at_point Q\<^sub>p ((inv a) # b) P)"
                 using 37 group.nat_pow_inv[of Q\<^sub>p a "nat N"] a_closed Qp.field_axioms field.field_nat_pow_inv[of Q\<^sub>p]
                 by (metis a_nz)
-              have 39:"((inv a)[^]m) [^] \<^bsub>Q\<^sub>p\<^esub> n \<otimes> (c[^]n) = (eval_at_point Q\<^sub>p ((inv a) # b) P)"    
-                using 2  38 monoid.nat_pow_pow[of Q\<^sub>p "inv a" ] N_def 
+              have 39:"((inv a)[^]m) [^] \<^bsub>Q\<^sub>p\<^esub> n \<otimes> (c[^]n) = (eval_at_point Q\<^sub>p ((inv a) # b) P)"
+                using 2  38 monoid.nat_pow_pow[of Q\<^sub>p "inv a" ] N_def
                 by (smt "3" Qp_int_nat_pow_pow a_closed a_nz inv_in_frac(3) of_nat_0_le_iff pow_nat)
-              have 310:"((((inv a)[^]m) \<otimes> c)[^]n) = (eval_at_point Q\<^sub>p ((inv a) # b) P)"    
+              have 310:"((((inv a)[^]m) \<otimes> c)[^]n) = (eval_at_point Q\<^sub>p ((inv a) # b) P)"
               proof-
                 have AA: "(inv a)[^]m \<in> carrier Q\<^sub>p"
                   using Qp_int_pow_nonzero nonzero_def a_closed a_nz inv_in_frac(3) Qp.nonzero_closed
                   by presburger
                 have "((inv a)[^]m) [^] \<^bsub>Q\<^sub>p\<^esub> n \<otimes> (c[^]n) = ((((inv a)[^]m) \<otimes> c)[^]n)"
                   using Qp.nat_pow_distrib[of  "(inv a)[^]m" c n]  a_closed a_def c_def AA by blast
-                then show ?thesis 
+                then show ?thesis
                   using "39" by blast
               qed
-              then show ?thesis using c'_def   
+              then show ?thesis using c'_def
                 by blast
             qed
             have 4: "inv a # b \<in> carrier (Q\<^sub>p\<^bsup>1+k\<^esup>)"
               by (metis a_closed a_nz add.commute b_closed cartesian_power_cons inv_in_frac(1))
             then have 5: "((inv a) # b) \<in> S"
-              using 3 P_n_def c'_closed  basic_semialg_set_memI[of "(inv a) # b" "1 + k" c' P n] 
+              using 3 P_n_def c'_closed  basic_semialg_set_memI[of "(inv a) # b" "1 + k" c' P n]
               by blast
             have 6: "Qp_invert (take 1 x) # drop 1 x = inv a # b"
-              using a_def b_def unfolding Qp_invert_def using "1" Qp_invert_def 
+              using a_def b_def unfolding Qp_invert_def using "1" Qp_invert_def
               by blast
-            show ?thesis using 5 6 
+            show ?thesis using 5 6
               by presburger
           qed
         qed
-        then show ?thesis 
+        then show ?thesis
           using A by blast
       qed
     qed
   qed
   have 8: "is_semialgebraic (1+k) ((partial_pullback 1 Qp_invert k S) \<inter> Nz)"
-    using "7" Nz_semialg T_semialg intersection_is_semialg 
+    using "7" Nz_semialg T_semialg intersection_is_semialg
     by auto
   have 9: "(partial_pullback 1 Qp_invert k S) - Nz = {xs \<in> carrier (Q\<^sub>p\<^bsup>Suc k\<^esup>). xs!0 = \<zero>} \<inter>S"
   proof
     show "partial_pullback 1 Qp_invert k S - Nz \<subseteq> {xs \<in> carrier (Q\<^sub>p\<^bsup>Suc k\<^esup>). xs ! 0 = \<zero>} \<inter>  S"
     proof fix x assume A: " x \<in> partial_pullback 1 Qp_invert k S - Nz"
       have 0: "x \<in> carrier (Q\<^sub>p\<^bsup>Suc k\<^esup>)"
-        using A 
+        using A
         by (metis DiffD1 partial_pullback_memE(1) plus_1_eq_Suc)
       have 1: "take 1 x \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>)"
         by (metis "0" le_add1 plus_1_eq_Suc take_closed)
       have 2: "drop 1 x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>)"
-        using "0" cartesian_power_drop plus_1_eq_Suc 
+        using "0" cartesian_power_drop plus_1_eq_Suc
         by presburger
       have 3: " x = take 1 x @ drop 1 x "
-        using 0 
+        using 0
         by (metis append_take_drop_id)
       have 4: "Qp_invert (take 1 x) # drop 1 x \<in> S"
         using A partial_pullback_memE'[of "take 1 x" 1 "drop 1 x" k x Qp_invert S] 1 2 3
-        by blast 
+        by blast
       have 5: "x!0 = \<zero>"
-        using A 0 Nz_def by blast 
+        using A 0 Nz_def by blast
       have 6: "Qp_invert (take 1 x) # drop 1 x = x"
       proof-
         have "(take 1 x) =[x!0]"
-          using 0 
+          using 0
           by (metis "1" "3" append_Cons nth_Cons_0 Qp.to_R1_to_R)
         then have "Qp_invert (take 1 x) = \<zero>"
-          unfolding Qp_invert_def using 5 
+          unfolding Qp_invert_def using 5
           by (metis less_one nth_take)
-        then show ?thesis using 0 5 
+        then show ?thesis using 0 5
           by (metis "3" Cons_eq_append_conv \<open>take 1 x = [x ! 0]\<close> self_append_conv2)
       qed
       have "x \<in> S"
-        using 6 4 
+        using 6 4
         by presburger
       then show "x \<in> {xs \<in> carrier (Q\<^sub>p\<^bsup>Suc k\<^esup>). xs ! 0 = \<zero>} \<inter>  S"
-        using Nz_def A 0 
-        by blast        
+        using Nz_def A 0
+        by blast
     qed
     show "{xs \<in> carrier (Q\<^sub>p\<^bsup>Suc k\<^esup>). xs ! 0 = \<zero>} \<inter> S \<subseteq> partial_pullback 1 Qp_invert k S - Nz"
     proof fix x assume A: "x \<in> {xs \<in> carrier (Q\<^sub>p\<^bsup>Suc k\<^esup>). xs ! 0 = \<zero>} \<inter> S"
       have A0: "x \<in> carrier (Q\<^sub>p\<^bsup>Suc k\<^esup>)"
-        using A by blast 
+        using A by blast
       have A1: "x!0  = \<zero>"
-        using A by blast 
+        using A by blast
       have A2: "x \<in> S"
-        using A by blast 
+        using A by blast
       show " x \<in> partial_pullback 1 Qp_invert k S - Nz"
       proof
         show "x \<notin> Nz"
-          using Nz_def A1 by blast 
+          using Nz_def A1 by blast
         show " x \<in> partial_pullback 1 Qp_invert k S"
         proof(rule partial_pullback_memI)
           show "x \<in> carrier (Q\<^sub>p\<^bsup>1+k\<^esup>)"
-            using A0 
+            using A0
             by (simp add: A0)
           show "Qp_invert (take 1 x) # drop 1 x \<in> S"
           proof-
             have "Qp_invert (take 1 x) = \<zero>"
-              unfolding Qp_invert_def  using A0 A1 
+              unfolding Qp_invert_def  using A0 A1
               by (metis less_numeral_extra(1) nth_take)
             then have "Qp_invert (take 1 x) # drop 1 x = x"
-              using A0 A1 A2 
+              using A0 A1 A2
               by (metis (no_types, lifting) Cons_eq_append_conv Qp_invert_def \<open>x \<in> carrier (Q\<^sub>p\<^bsup>1+k\<^esup>)\<close>
-                  append_take_drop_id inv_in_frac(2) le_add_same_cancel1 self_append_conv2 
-                  take_closed Qp.to_R1_to_R Qp.to_R_pow_closed zero_le)              
-            then show ?thesis 
+                  append_take_drop_id inv_in_frac(2) le_add_same_cancel1 self_append_conv2
+                  take_closed Qp.to_R1_to_R Qp.to_R_pow_closed zero_le)
+            then show ?thesis
               using A2 by presburger
           qed
         qed
       qed
     qed
   qed
   have 10: "is_semialgebraic (1+k) {xs \<in> carrier (Q\<^sub>p\<^bsup>Suc k\<^esup>). xs!0 = \<zero>}"
   proof-
     have "{xs \<in> carrier (Q\<^sub>p\<^bsup>Suc k\<^esup>). xs!0 = \<zero>} = V\<^bsub>Q\<^sub>p\<^esub> (Suc k) (pvar Q\<^sub>p 0)"
-      unfolding zero_set_def using eval_pvar[of  0 "Suc k"] Qp.cring_axioms 
-      by blast
-    then show ?thesis using          
-        is_zero_set_imp_basic_semialg pvar_closed[of  0 "Suc k"] Qp.cring_axioms 
-        is_zero_set_imp_semialg plus_1_eq_Suc zero_less_Suc 
+      unfolding zero_set_def using eval_pvar[of  0 "Suc k"] Qp.cring_axioms
+      by blast
+    then show ?thesis using
+        is_zero_set_imp_basic_semialg pvar_closed[of  0 "Suc k"] Qp.cring_axioms
+        is_zero_set_imp_semialg plus_1_eq_Suc zero_less_Suc
       by presburger
   qed
   have 11: "is_semialgebraic (1+k) ({xs \<in> carrier (Q\<^sub>p\<^bsup>Suc k\<^esup>). xs!0 = \<zero>} \<inter>S)"
-    using 10 assms basic_semialg_is_semialgebraic intersection_is_semialg 
+    using 10 assms basic_semialg_is_semialgebraic intersection_is_semialg
     by blast
   have 12: "(partial_pullback 1 Qp_invert k S) = ((partial_pullback 1 Qp_invert k S) \<inter> Nz) \<union>
                                                   ((partial_pullback 1 Qp_invert k S) - Nz)"
     by blast
   have 13: "is_semialgebraic (1+k) ((partial_pullback 1 Qp_invert k S) - Nz)"
-    using 11  9 by metis  
-  show ?thesis 
-    using 8  12 13 
+    using 11  9 by metis
+  show ?thesis
+    using 8  12 13
     by (metis "7" Int_Diff_Un Int_commute plus_1_eq_Suc union_is_semialgebraic)
 qed
-   
+
 lemma Qp_invert_is_semialg:
 "is_semialg_function 1 Qp_invert"
 proof(rule is_semialg_functionI')
   show 0: "Qp_invert \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>) \<rightarrow> carrier Q\<^sub>p"
   proof fix x
     assume A: "x \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>)"
     then obtain a where a_def: "x = [a]"
       by (metis Qp.to_R1_to_R)
     have a_closed: "a \<in> carrier Q\<^sub>p"
-      using a_def A cartesian_power_concat(1) last_closed' 
+      using a_def A cartesian_power_concat(1) last_closed'
       by blast
     show " Qp_invert x \<in> carrier Q\<^sub>p"
     apply(cases "a = \<zero>")
-    unfolding Qp_invert_def using a_def a_closed 
+    unfolding Qp_invert_def using a_def a_closed
     apply (meson Qp.to_R_to_R1)
     by (metis a_closed a_def inv_in_frac(1) Qp.to_R_to_R1)
   qed
   show "\<And>k S. S \<in> basic_semialgs (1 + k) \<Longrightarrow> is_semialgebraic (1 + k) (partial_pullback 1 Qp_invert k S)"
-    using Qp_invert_basic_semialg 
+    using Qp_invert_basic_semialg
     by blast
 qed
 
 lemma Taylor_deg_1_expansion'':
   assumes "f \<in> carrier Q\<^sub>p_x"
   assumes "\<And>n. f n \<in> \<O>\<^sub>p"
   assumes "a \<in> \<O>\<^sub>p "
   assumes "b \<in> \<O>\<^sub>p"
   shows "\<exists>c c' c''. c = to_fun f a \<and> c' = deriv f a \<and> c \<in> \<O>\<^sub>p \<and> c' \<in> \<O>\<^sub>p \<and>c'' \<in> \<O>\<^sub>p \<and>
                   to_fun f (b) = c \<oplus> c' \<otimes> (b \<ominus> a) \<oplus> (c'' \<otimes> (b \<ominus> a)[^](2::nat))"
 proof-
   obtain S where S_def: "S = (Q\<^sub>p  \<lparr> carrier := \<O>\<^sub>p \<rparr>)"
-    by blast 
+    by blast
   have 1: "f \<in> carrier (UP S)"
-    unfolding S_def using val_ring_subring UPQ.poly_cfs_subring[of \<O>\<^sub>p f] assms  
+    unfolding S_def using val_ring_subring UPQ.poly_cfs_subring[of \<O>\<^sub>p f] assms
     by blast
   have 2: " f \<in> carrier (UP (Q\<^sub>p\<lparr>carrier := \<O>\<^sub>p\<rparr>))"
-    using val_ring_subring 1 assms poly_cfs_subring[of \<O>\<^sub>p] 
+    using val_ring_subring 1 assms poly_cfs_subring[of \<O>\<^sub>p]
     by blast
   have 3: "\<exists>c\<in>\<O>\<^sub>p. f \<bullet> b = f \<bullet> a \<oplus> UPQ.deriv f a \<otimes> (b \<ominus> a) \<oplus> c \<otimes> (b \<ominus> a) [^] (2::nat)"
     using UP_subring_taylor_appr'[of \<O>\<^sub>p f b a] UP_subring_taylor_appr[of \<O>\<^sub>p f b a] val_ring_subring 1 2 assms
     by blast
   then show ?thesis
-    using UP_subring_taylor_appr[of \<O>\<^sub>p f b a] assms UP_subring_deriv_closed[of \<O>\<^sub>p f a] 
+    using UP_subring_taylor_appr[of \<O>\<^sub>p f b a] assms UP_subring_deriv_closed[of \<O>\<^sub>p f a]
       UP_subring_eval_closed[of \<O>\<^sub>p f a] 2 val_ring_subring by blast
 qed
 
 end
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsection\<open>Sets Defined by Residues of Valuation Ring Elements\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
-sublocale padic_fields < Res: cring "Zp_res_ring (Suc n)" 
-  using p_residues residues.cring 
+sublocale padic_fields < Res: cring "Zp_res_ring (Suc n)"
+  using p_residues residues.cring
   by blast
 
 context padic_fields
 begin
 
-definition Qp_res where 
+definition Qp_res where
 "Qp_res x n =  to_Zp x n "
 
-lemma Qp_res_closed: 
+lemma Qp_res_closed:
   assumes "x \<in> \<O>\<^sub>p"
   shows "Qp_res  x n \<in> carrier (Zp_res_ring n)"
   unfolding Qp_res_def using assms val_ring_memE residue_closed to_Zp_closed by blast
 
-lemma Qp_res_add: 
+lemma Qp_res_add:
   assumes "x \<in> \<O>\<^sub>p"
   assumes "y \<in> \<O>\<^sub>p"
   shows "Qp_res (x \<oplus> y) n = Qp_res x n \<oplus>\<^bsub>Zp_res_ring n\<^esub> Qp_res y n"
   unfolding Qp_res_def
   using assms residue_of_sum to_Zp_add by presburger
 
-lemma Qp_res_mult: 
+lemma Qp_res_mult:
   assumes "x \<in> \<O>\<^sub>p"
   assumes "y \<in> \<O>\<^sub>p"
   shows "Qp_res (x \<otimes> y) n = Qp_res x n \<otimes>\<^bsub>Zp_res_ring n\<^esub> Qp_res y n"
   unfolding Qp_res_def
   using assms residue_of_prod to_Zp_mult by presburger
 
-lemma Qp_res_diff: 
+lemma Qp_res_diff:
   assumes "x \<in> \<O>\<^sub>p"
   assumes "y \<in> \<O>\<^sub>p"
   shows "Qp_res (x \<ominus> y) n = Qp_res x n \<ominus>\<^bsub>Zp_res_ring n\<^esub> Qp_res y n"
   unfolding Qp_res_def
-  using assms residue_of_diff to_Zp_minus 
+  using assms residue_of_diff to_Zp_minus
   by (meson val_ring_res)
 
 lemma Qp_res_zero:
   shows  "Qp_res \<zero> n = 0"
   unfolding Qp_res_def to_Zp_zero
-  using residue_of_zero(2) by blast 
+  using residue_of_zero(2) by blast
 
 lemma Qp_res_one:
   assumes "n > 0"
   shows  "Qp_res \<one> n = (1::int)"
-  using assms 
+  using assms
   unfolding Qp_res_def to_Zp_one
   using residue_of_one(2) by blast
 
 lemma Qp_res_nat_inc:
   shows  "Qp_res ([(n::nat)]\<cdot>\<one>) n = n mod p^n"
   unfolding Qp_res_def unfolding to_Zp_nat_inc
   using Zp_nat_inc_res by blast
 
 lemma Qp_res_int_inc:
   shows  "Qp_res ([(k::int)]\<cdot>\<one>) n = k mod p^n"
   unfolding Qp_res_def unfolding to_Zp_int_inc
   using Zp_int_inc_res by blast
 
-lemma Qp_poly_res_monom: 
+lemma Qp_poly_res_monom:
 assumes "a \<in> \<O>\<^sub>p"
 assumes "x \<in> \<O>\<^sub>p"
 assumes "Qp_res a n = 0"
 assumes "k > 0"
 shows "Qp_res (up_ring.monom (UP Q\<^sub>p) a k \<bullet> x) n = 0"
-proof- 
+proof-
   have 0: "up_ring.monom (UP Q\<^sub>p) a k \<bullet> x = a \<otimes> x [^] k"
     apply(rule UPQ.to_fun_monom[of a x k])
     using assms val_ring_memE apply blast
     using assms val_ring_memE by blast
   have 1: "x[^]k \<in> \<O>\<^sub>p"
     using assms val_ring_nat_pow_closed by blast
-  show ?thesis unfolding 0 
-    using Qp_res_mult[of a "x[^]k" n] assms 
+  show ?thesis unfolding 0
+    using Qp_res_mult[of a "x[^]k" n] assms
     using "1" residue_times_zero_r by presburger
 qed
 
-lemma Qp_poly_res_zero: 
+lemma Qp_poly_res_zero:
   assumes "q \<in> carrier (UP Q\<^sub>p)"
   assumes "\<And>i. q i \<in> \<O>\<^sub>p"
   assumes "\<And>i. Qp_res (q i) n = 0"
   assumes "x \<in> \<O>\<^sub>p"
   shows "Qp_res (q \<bullet> x) n = 0"
-proof- 
+proof-
   have "(\<forall>i. q i \<in> \<O>\<^sub>p \<and> Qp_res (q i) n = 0) \<longrightarrow> Qp_res (q \<bullet> x) n = 0"
   proof(rule UPQ.poly_induct[of q], rule assms, rule )
     fix p assume A: "p \<in> carrier (UP Q\<^sub>p)" " deg Q\<^sub>p p = 0" " \<forall>i. p i \<in> \<O>\<^sub>p \<and> Qp_res (p i) n = 0"
     have 0: "p \<bullet> x = p 0"
-      using assms 
+      using assms
       by (metis A(1) A(2) val_ring_memE UPQ.ltrm_deg_0 UPQ.to_fun_ctrm)
     show "Qp_res (p \<bullet> x) n = 0"
-      unfolding 0 using A by blast 
+      unfolding 0 using A by blast
   next
-    fix p 
+    fix p
     assume A0: "(\<And>q. q \<in> carrier (UP Q\<^sub>p) \<Longrightarrow> deg Q\<^sub>p q < deg Q\<^sub>p p \<Longrightarrow> (\<forall>i. q i \<in> \<O>\<^sub>p \<and> Qp_res (q i) n = 0) \<longrightarrow> Qp_res (q \<bullet> x) n = 0)"
       "p \<in> carrier (UP Q\<^sub>p)" "0 < deg Q\<^sub>p p"
     show "(\<forall>i. p i \<in> \<O>\<^sub>p \<and> Qp_res (p i) n = 0) \<longrightarrow> Qp_res (p \<bullet> x) n = 0"
     proof assume A1: " \<forall>i. p i \<in> \<O>\<^sub>p \<and> Qp_res (p i) n = 0"
       obtain k where k_def: "k = deg Q\<^sub>p p"
-        by blast 
+        by blast
       obtain q where q_def: "q = UPQ.trunc p"
         by blast
       have q_closed: "q \<in> carrier (UP Q\<^sub>p)"
-        unfolding q_def 
+        unfolding q_def
         using A0(2) UPQ.trunc_closed by blast
       have q_deg: "deg Q\<^sub>p q < deg Q\<^sub>p p"
-        unfolding q_def 
+        unfolding q_def
         using A0(2) A0(3) UPQ.trunc_degree by blast
       have 9: "\<And>i. i < deg Q\<^sub>p p \<Longrightarrow> q i = p i"
-        unfolding q_def 
+        unfolding q_def
         using A0(2) UPQ.trunc_cfs by blast
       have 90:  "\<And>i. \<not> i < deg Q\<^sub>p p \<Longrightarrow> q i = \<zero>"
-        unfolding q_def 
+        unfolding q_def
       proof -
         fix i :: nat
         assume "\<not> i < deg Q\<^sub>p p"
         then have "deg Q\<^sub>p q < i"
           using q_deg by linarith
         then show "Cring_Poly.truncate Q\<^sub>p p i = \<zero>"
           using UPQ.deg_gtE q_closed q_def by blast
       qed
       have 10: "(\<forall>i. q i \<in> \<O>\<^sub>p \<and> Qp_res (q i) n = 0)"
-      proof fix i 
+      proof fix i
         show "q i \<in> \<O>\<^sub>p \<and> Qp_res (q i) n = 0"
           apply(cases "i < deg Q\<^sub>p p")
            using A1  9[of i] apply presburger
-           unfolding q_def using Qp_res_zero  90 
+           unfolding q_def using Qp_res_zero  90
            by (metis q_def zero_in_val_ring)
       qed
       have 11: "Qp_res (q \<bullet> x) n = 0"
         using 10 A1 A0 q_closed q_deg by blast
       have 12: "p = q \<oplus>\<^bsub>UP Q\<^sub>p\<^esub> up_ring.monom (UP Q\<^sub>p) (p k) k"
-        unfolding k_def q_def 
+        unfolding k_def q_def
         using A0(2) UPQ.trunc_simps(1) by blast
       have 13: "p \<bullet> x = q \<bullet> x \<oplus> (up_ring.monom (UP Q\<^sub>p) (p k) k) \<bullet> x"
       proof-
         have 0: " (q \<oplus>\<^bsub>UP Q\<^sub>p\<^esub> up_ring.monom (UP Q\<^sub>p) (p k) k) \<bullet> x = q \<bullet> x \<oplus> up_ring.monom (UP Q\<^sub>p) (p k) k \<bullet> x"
-          apply(rule UPQ.to_fun_plus) 
+          apply(rule UPQ.to_fun_plus)
           using A0(2) UPQ.ltrm_closed k_def apply blast
           unfolding q_def apply(rule UPQ.trunc_closed, rule A0)
           using assms val_ring_memE by blast
-        show ?thesis 
-          using 0 12 by metis 
+        show ?thesis
+          using 0 12 by metis
       qed
       have 14: "(up_ring.monom (UP Q\<^sub>p) (p k) k) \<bullet> x \<in> \<O>\<^sub>p"
         apply(rule val_ring_poly_eval)
           using A0(2) UPQ.ltrm_closed k_def apply blast
-          using UPQ.cfs_monom[of "p k" k ] A1 zero_in_val_ring 
+          using UPQ.cfs_monom[of "p k" k ] A1 zero_in_val_ring
           using A0(2) UPQ.ltrm_cfs k_def apply presburger
           using assms(4) by blast
       have 15: "Qp_res ((up_ring.monom (UP Q\<^sub>p) (p k) k) \<bullet> x) n = 0"
         apply(rule Qp_poly_res_monom)
         using A1 apply blast using assms apply blast
-        using A1 apply blast unfolding k_def using A0 by blast 
+        using A1 apply blast unfolding k_def using A0 by blast
       have 16: "Qp_res (q \<bullet> x) n = 0"
         using A0 10 11 by blast
       have 17: "q \<bullet> x \<in> \<O>\<^sub>p"
         apply(rule val_ring_poly_eval, rule q_closed)
         using 10 apply blast by(rule assms)
       have 18: "Qp_res (q \<bullet> x \<oplus> (up_ring.monom (UP Q\<^sub>p) (p k) k) \<bullet> x) n = 0"
-        using Qp_res_add[of "q \<bullet> x" "up_ring.monom (UP Q\<^sub>p) (p k) k \<bullet> x" n] 14  17 
-        unfolding 15 16 
+        using Qp_res_add[of "q \<bullet> x" "up_ring.monom (UP Q\<^sub>p) (p k) k \<bullet> x" n] 14  17
+        unfolding 15 16
         by (metis "10" Qp_res_add UPQ.cfs_add UPQ.coeff_of_sum_diff_degree0 q_closed q_deg)
       show "Qp_res (p \<bullet> x) n = 0"
-        using 13 18 by metis 
-    qed
-  qed
-  thus ?thesis using assms by blast 
-qed
-
-lemma Qp_poly_res_eval_0: 
+        using 13 18 by metis
+    qed
+  qed
+  thus ?thesis using assms by blast
+qed
+
+lemma Qp_poly_res_eval_0:
   assumes "f \<in> carrier (UP Q\<^sub>p)"
   assumes "g \<in> carrier (UP Q\<^sub>p)"
   assumes "x \<in> \<O>\<^sub>p"
   assumes "\<And>i. f i \<in> \<O>\<^sub>p"
   assumes "\<And>i. g i \<in> \<O>\<^sub>p"
   assumes "\<And>i. Qp_res (f i) n = Qp_res (g i) n"
   shows "Qp_res (f \<bullet> x) n = Qp_res (g \<bullet> x) n"
 proof-
-    obtain F where F_def: "F = f \<ominus>\<^bsub>UP Q\<^sub>p\<^esub>g" 
-      by blast 
+    obtain F where F_def: "F = f \<ominus>\<^bsub>UP Q\<^sub>p\<^esub>g"
+      by blast
     have F_closed: "F \<in> carrier (UP Q\<^sub>p)"
-      unfolding F_def 
+      unfolding F_def
       using assms by blast
     have F_cfs: "\<And>i. F i = (f i) \<ominus> (g i)"
-      unfolding F_def 
+      unfolding F_def
       using assms UPQ.cfs_minus by blast
     have F_cfs_res: "\<And>i. Qp_res (F i) n = Qp_res (f i) n \<ominus>\<^bsub>Zp_res_ring n\<^esub> Qp_res (g i) n"
       unfolding F_cfs apply(rule Qp_res_diff)
       using assms apply blast using assms by blast
     have 0: "\<And>i. Qp_res (f i) n = Qp_res (g i) n"
-      using assms by blast 
+      using assms by blast
     have F_cfs_res': "\<And>i. Qp_res (F i) n = 0"
-      unfolding F_cfs_res 0 
+      unfolding F_cfs_res 0
       by (metis diff_self mod_0 residue_minus)
     have 1: "\<And>i. F i \<in> \<O>\<^sub>p"
-      unfolding F_cfs using assms  
+      unfolding F_cfs using assms
       using val_ring_minus_closed by blast
     have 2: "Qp_res (F \<bullet> x) n = 0"
       by(rule Qp_poly_res_zero, rule F_closed, rule 1, rule F_cfs_res', rule assms)
     have 3: "F \<bullet> x = f \<bullet> x \<ominus> g \<bullet> x"
-      unfolding F_def using assms 
+      unfolding F_def using assms
       by (meson assms UPQ.to_fun_diff val_ring_memE)
-    have 4: "Qp_res (F \<bullet> x) n = Qp_res (f \<bullet> x) n \<ominus>\<^bsub>Zp_res_ring n\<^esub> Qp_res (g \<bullet> x) n" 
+    have 4: "Qp_res (F \<bullet> x) n = Qp_res (f \<bullet> x) n \<ominus>\<^bsub>Zp_res_ring n\<^esub> Qp_res (g \<bullet> x) n"
       unfolding 3 apply(rule Qp_res_diff, rule val_ring_poly_eval, rule assms)
       using assms apply blast using assms apply blast
       apply(rule val_ring_poly_eval, rule assms)
       using assms apply blast by(rule assms)
     have 5: "f \<bullet> x \<in> \<O>\<^sub>p"
       apply(rule val_ring_poly_eval, rule assms)
       using assms apply blast using assms by blast
     have 6: "g \<bullet> x \<in> \<O>\<^sub>p"
       apply(rule val_ring_poly_eval, rule assms)
       using assms apply blast by(rule assms)
     show "Qp_res (f \<bullet> x) n = Qp_res (g \<bullet> x) n"
-      using 5 6 2 Qp_res_closed[of "f \<bullet> x" n] Qp_res_closed[of "g \<bullet> x" n] 
-      unfolding 4 
+      using 5 6 2 Qp_res_closed[of "f \<bullet> x" n] Qp_res_closed[of "g \<bullet> x" n]
+      unfolding 4
     proof -
       assume "Qp_res (f \<bullet> x) n \<ominus>\<^bsub>Zp_res_ring n\<^esub> Qp_res (g \<bullet> x) n = 0"
       then show ?thesis
         by (metis (no_types) Qp_res_def 5 6 res_diff_zero_fact(1) residue_of_diff to_Zp_closed val_ring_memE)
     qed
 qed
 
 lemma Qp_poly_res_eval_1:
   assumes "f \<in> carrier (UP Q\<^sub>p)"
   assumes "x \<in> \<O>\<^sub>p"
   assumes "y \<in> \<O>\<^sub>p"
   assumes "\<And>i. f i \<in> \<O>\<^sub>p"
   assumes "Qp_res x n = Qp_res y n"
   shows "Qp_res (f \<bullet> x) n = Qp_res (f \<bullet> y) n"
-proof- 
+proof-
   have "(\<forall>i. f i \<in> \<O>\<^sub>p) \<longrightarrow> Qp_res (f \<bullet> x) n = Qp_res (f \<bullet> y) n"
     apply(rule UPQ.poly_induct[of f], rule assms)
   proof fix f assume A: "f \<in> carrier (UP Q\<^sub>p)" "deg Q\<^sub>p f = 0" "\<forall>i. f i \<in> \<O>\<^sub>p"
     show "Qp_res (f \<bullet> x) n = Qp_res (f \<bullet> y) n"
-    proof- 
+    proof-
       obtain a where a_def: "a \<in> carrier Q\<^sub>p \<and> f = to_polynomial Q\<^sub>p a"
-        using assms 
+        using assms
         by (metis A(1) A(2) UPQ.lcf_closed UPQ.to_poly_inverse)
       have a_eq: "f = to_polynomial Q\<^sub>p a"
-        using a_def  by blast 
+        using a_def  by blast
       have 0: "f \<bullet> x = a"
-        using a_def assms unfolding a_eq  
+        using a_def assms unfolding a_eq
         by (meson UPQ.to_fun_to_poly val_ring_memE)
       have 1: "f \<bullet> y = a"
-        using a_def assms unfolding a_eq 
+        using a_def assms unfolding a_eq
         by (meson UPQ.to_fun_to_poly val_ring_memE)
       show " Qp_res (f \<bullet> x) n = Qp_res (f \<bullet> y) n"
-        unfolding 0 1 by blast 
-    qed
-  next 
-    fix f 
+        unfolding 0 1 by blast
+    qed
+  next
+    fix f
     assume A: " (\<And>q. q \<in> carrier (UP Q\<^sub>p) \<Longrightarrow> deg Q\<^sub>p q < deg Q\<^sub>p f \<Longrightarrow> (\<forall>i. q i \<in> \<O>\<^sub>p) \<longrightarrow> Qp_res (q \<bullet> x) n = Qp_res (q \<bullet> y) n)"
         "f \<in> carrier (UP Q\<^sub>p)" " 0 < deg Q\<^sub>p f"
     show "(\<forall>i. f i \<in> \<O>\<^sub>p) \<longrightarrow> Qp_res (f \<bullet> x) n = Qp_res (f \<bullet> y) n"
     proof assume A1: "\<forall>i. f i \<in> \<O>\<^sub>p"
       obtain q where q_def: "q = UPQ.trunc f"
-        by blast 
+        by blast
       have q_closed: "q \<in> carrier (UP Q\<^sub>p)"
         using q_def A UPQ.trunc_closed by presburger
       have q_deg: "deg Q\<^sub>p q < deg Q\<^sub>p f"
         using q_def A UPQ.trunc_degree by blast
       have q_cfs: "\<forall>i. q i \<in> \<O>\<^sub>p"
       proof fix i show "q i \<in> \<O>\<^sub>p"
         apply(cases "i < deg Q\<^sub>p f")
         unfolding q_def using A A1 UPQ.trunc_cfs
          apply presburger
-        using q_deg q_closed 
+        using q_deg q_closed
         proof -
         assume "\<not> i < deg Q\<^sub>p f"
         then have "deg Q\<^sub>p f \<le> i"
           by (meson diff_is_0_eq neq0_conv zero_less_diff)
         then show "Cring_Poly.truncate Q\<^sub>p f i \<in> \<O>\<^sub>p"
           by (metis (no_types) UPQ.deg_eqI diff_is_0_eq' le_trans nat_le_linear neq0_conv q_closed q_def q_deg zero_in_val_ring zero_less_diff)
         qed
       qed
       hence 0: "Qp_res (q \<bullet> x) n = Qp_res (q \<bullet> y) n"
-        using A q_closed q_deg by blast 
+        using A q_closed q_deg by blast
       have 1: "Qp_res (UPQ.ltrm f \<bullet> x) n = Qp_res (UPQ.ltrm f \<bullet> y) n"
-      proof- 
+      proof-
         have 10: "UPQ.ltrm f \<bullet> x =  (f (deg Q\<^sub>p f)) \<otimes> x[^](deg Q\<^sub>p f)"
           using A assms  A1 UPQ.to_fun_monom val_ring_memE by presburger
         have 11: "UPQ.ltrm f \<bullet> y =  (f (deg Q\<^sub>p f)) \<otimes> y[^](deg Q\<^sub>p f)"
           using A assms  A1 UPQ.to_fun_monom val_ring_memE by presburger
         obtain d where d_def: "d = deg Q\<^sub>p f"
-          by blast 
+          by blast
         have 12: "Qp_res (x[^]d) n = Qp_res (y[^]d) n"
-          apply(induction d) 
+          apply(induction d)
           using Qp.nat_pow_0 apply presburger
           using Qp_res_mult assms Qp.nat_pow_Suc val_ring_nat_pow_closed by presburger
         hence 13: "Qp_res (x [^] deg Q\<^sub>p f) n = Qp_res (y [^] deg Q\<^sub>p f) n"
-          unfolding d_def by blast 
+          unfolding d_def by blast
         have 14: "x [^] deg Q\<^sub>p f \<in> \<O>\<^sub>p"
           using assms val_ring_nat_pow_closed by blast
         have 15: "y [^] deg Q\<^sub>p f \<in> \<O>\<^sub>p"
           using assms val_ring_nat_pow_closed by blast
         have 16: "Qp_res (f (deg Q\<^sub>p f) \<otimes> x [^] deg Q\<^sub>p f) n = Qp_res (f (deg Q\<^sub>p f)) n \<otimes>\<^bsub>residue_ring (p ^ n)\<^esub> Qp_res (x [^] deg Q\<^sub>p f) n"
           apply(rule Qp_res_mult[of "f (deg Q\<^sub>p f)" " x[^](deg Q\<^sub>p f)" n])
           using A1 apply blast by(rule 14)
         have 17: "Qp_res (f (deg Q\<^sub>p f) \<otimes> y [^] deg Q\<^sub>p f) n = Qp_res (f (deg Q\<^sub>p f)) n \<otimes>\<^bsub>residue_ring (p ^ n)\<^esub> Qp_res (y [^] deg Q\<^sub>p f) n"
           apply(rule Qp_res_mult[of "f (deg Q\<^sub>p f)" " y[^](deg Q\<^sub>p f)" n])
           using A1 apply blast by(rule 15)
         show ?thesis
-          unfolding 10 11 16 17 13 by blast 
+          unfolding 10 11 16 17 13 by blast
       qed
       have f_decomp: "f = q \<oplus>\<^bsub>UP Q\<^sub>p\<^esub> UPQ.ltrm f"
-        using A unfolding q_def 
+        using A unfolding q_def
         using UPQ.trunc_simps(1) by blast
       have 2: "f \<bullet> x = q \<bullet> x \<oplus> (UPQ.ltrm f \<bullet> x)"
         using A f_decomp q_closed q_cfs
         by (metis val_ring_memE UPQ.ltrm_closed UPQ.to_fun_plus assms(2))
       have 3: "f \<bullet> y = q \<bullet> y \<oplus> (UPQ.ltrm f \<bullet> y)"
         using A f_decomp q_closed q_cfs
         by (metis val_ring_memE UPQ.ltrm_closed UPQ.to_fun_plus assms(3))
       show 4: " Qp_res (f \<bullet> x) n = Qp_res (f \<bullet> y) n "
-        unfolding 2 3 using assms q_cfs Qp_res_add 0 1 
+        unfolding 2 3 using assms q_cfs Qp_res_add 0 1
         by (metis (no_types, opaque_lifting) "2" "3" A(2) A1 Qp_res_def poly_eval_cong)
     qed
   qed
-  thus ?thesis using assms by blast 
+  thus ?thesis using assms by blast
 qed
 
 lemma Qp_poly_res_eval_2:
   assumes "f \<in> carrier (UP Q\<^sub>p)"
   assumes "g \<in> carrier (UP Q\<^sub>p)"
   assumes "x \<in> \<O>\<^sub>p"
   assumes "y \<in> \<O>\<^sub>p"
   assumes "\<And>i. f i \<in> \<O>\<^sub>p"
   assumes "\<And>i. g i \<in> \<O>\<^sub>p"
   assumes "\<And>i. Qp_res (f i) n = Qp_res (g i) n"
   assumes "Qp_res x n = Qp_res y n"
   shows "Qp_res (f \<bullet> x) n = Qp_res (g \<bullet> y) n"
-proof- 
+proof-
   have 0: "Qp_res (f \<bullet> x) n = Qp_res (g \<bullet> x) n"
-    using Qp_poly_res_eval_0 assms by blast 
+    using Qp_poly_res_eval_0 assms by blast
   have 1: "Qp_res (g \<bullet> x) n = Qp_res (g \<bullet> y) n"
-    using Qp_poly_res_eval_1 assms by blast 
-  show ?thesis unfolding 0 1 by blast 
+    using Qp_poly_res_eval_1 assms by blast
+  show ?thesis unfolding 0 1 by blast
 qed
 
 definition poly_res_class where
 "poly_res_class n d f = {q \<in> carrier (UP Q\<^sub>p). deg Q\<^sub>p q \<le> d \<and> (\<forall>i. q i \<in> \<O>\<^sub>p \<and> Qp_res (f i) n = Qp_res (q i) n) }"
 
-lemma poly_res_class_closed: 
+lemma poly_res_class_closed:
   assumes "f \<in> carrier (UP Q\<^sub>p)"
   assumes "g \<in> carrier (UP Q\<^sub>p)"
   assumes "deg Q\<^sub>p f \<le> d"
   assumes "deg Q\<^sub>p g \<le> d"
   assumes "g \<in> poly_res_class n d f"
   shows "poly_res_class n d f = poly_res_class n d g"
-  unfolding poly_res_class_def 
+  unfolding poly_res_class_def
   apply(rule equalityI)
 apply(rule subsetI)
-  unfolding mem_Collect_eq apply(rule conjI, blast, rule conjI, blast) 
-  using assms unfolding poly_res_class_def mem_Collect_eq 
+  unfolding mem_Collect_eq apply(rule conjI, blast, rule conjI, blast)
+  using assms unfolding poly_res_class_def mem_Collect_eq
    apply presburger
 apply(rule subsetI) unfolding mem_Collect_eq
-  apply(rule conjI, blast, rule conjI, blast) 
-  using assms unfolding poly_res_class_def mem_Collect_eq 
+  apply(rule conjI, blast, rule conjI, blast)
+  using assms unfolding poly_res_class_def mem_Collect_eq
   by presburger
 
 lemma poly_res_class_memE:
   assumes "f \<in> poly_res_class n d g"
   shows "f \<in> carrier (UP Q\<^sub>p)"
         "deg Q\<^sub>p f \<le> d"
         "f i \<in> \<O>\<^sub>p"
         "Qp_res (g i) n = Qp_res (f i) n"
   using assms unfolding poly_res_class_def mem_Collect_eq apply blast
   using assms unfolding poly_res_class_def mem_Collect_eq apply blast
   using assms unfolding poly_res_class_def mem_Collect_eq apply blast
   using assms unfolding poly_res_class_def mem_Collect_eq by blast
 
-definition val_ring_polys where 
+definition val_ring_polys where
 "val_ring_polys = {f \<in> carrier (UP Q\<^sub>p). (\<forall>i. f i \<in> \<O>\<^sub>p)} "
 
-lemma val_ring_polys_closed: 
+lemma val_ring_polys_closed:
 "val_ring_polys \<subseteq> carrier (UP Q\<^sub>p)"
-  unfolding val_ring_polys_def by blast 
-
-lemma val_ring_polys_memI: 
+  unfolding val_ring_polys_def by blast
+
+lemma val_ring_polys_memI:
   assumes "f \<in> carrier (UP Q\<^sub>p)"
   assumes "\<And>i. f i \<in> \<O>\<^sub>p"
   shows "f \<in> val_ring_polys"
-  using assms   unfolding val_ring_polys_def by blast 
-
-lemma val_ring_polys_memE: 
+  using assms   unfolding val_ring_polys_def by blast
+
+lemma val_ring_polys_memE:
   assumes "f \<in> val_ring_polys"
   shows "f \<in> carrier (UP Q\<^sub>p)"
         "f i \<in> \<O>\<^sub>p"
   using assms unfolding val_ring_polys_def apply blast
   using assms unfolding val_ring_polys_def by blast
 
 definition val_ring_polys_grad where
 "val_ring_polys_grad d = {f \<in> val_ring_polys. deg Q\<^sub>p f \<le> d}"
 
-lemma val_ring_polys_grad_closed: 
+lemma val_ring_polys_grad_closed:
 "val_ring_polys_grad d \<subseteq> val_ring_polys"
-  unfolding val_ring_polys_grad_def by blast 
-
-lemma val_ring_polys_grad_closed': 
+  unfolding val_ring_polys_grad_def by blast
+
+lemma val_ring_polys_grad_closed':
 "val_ring_polys_grad d \<subseteq> carrier (UP Q\<^sub>p)"
-  unfolding val_ring_polys_grad_def val_ring_polys_def by blast 
-
-lemma val_ring_polys_grad_memI: 
+  unfolding val_ring_polys_grad_def val_ring_polys_def by blast
+
+lemma val_ring_polys_grad_memI:
   assumes "f \<in> carrier (UP Q\<^sub>p)"
   assumes "\<And>i. f i \<in> \<O>\<^sub>p"
   assumes "deg Q\<^sub>p f \<le> d"
   shows "f \<in> val_ring_polys_grad d"
-  using assms unfolding val_ring_polys_grad_def val_ring_polys_def by blast 
-
-lemma val_ring_polys_grad_memE: 
+  using assms unfolding val_ring_polys_grad_def val_ring_polys_def by blast
+
+lemma val_ring_polys_grad_memE:
   assumes "f \<in> val_ring_polys_grad d"
   shows "f \<in> carrier (UP Q\<^sub>p)"
         "deg Q\<^sub>p f \<le> d"
         "f i \<in> \<O>\<^sub>p"
   using assms unfolding val_ring_polys_grad_def val_ring_polys_def apply blast
   using assms unfolding val_ring_polys_grad_def val_ring_polys_def apply blast
   using assms unfolding val_ring_polys_grad_def val_ring_polys_def by blast
 
 lemma poly_res_classes_in_val_ring_polys_grad:
   assumes "f \<in> val_ring_polys_grad d"
   shows "poly_res_class n d f \<subseteq> val_ring_polys_grad d"
   apply(rule subsetI, rule val_ring_polys_grad_memI)
     apply(rule poly_res_class_memE[of _ n d f], blast)
    apply(rule poly_res_class_memE[of _ n d f], blast)
   by(rule poly_res_class_memE[of _ n d f], blast)
 
-lemma poly_res_class_disjoint: 
+lemma poly_res_class_disjoint:
   assumes "f \<in> val_ring_polys_grad d"
   assumes "f \<notin> poly_res_class n d g"
   shows "poly_res_class n d f \<inter> poly_res_class n d g = {}"
   apply(rule equalityI)
   apply(rule subsetI)
-  using assms 
-  unfolding poly_res_class_def mem_Collect_eq Int_iff 
+  using assms
+  unfolding poly_res_class_def mem_Collect_eq Int_iff
   apply (metis val_ring_polys_grad_memE(1) val_ring_polys_grad_memE(2) val_ring_polys_grad_memE(3))
   by blast
 
-lemma poly_res_class_refl: 
+lemma poly_res_class_refl:
   assumes "f \<in> val_ring_polys_grad d"
   shows "f \<in> poly_res_class n d f"
-  unfolding poly_res_class_def mem_Collect_eq 
+  unfolding poly_res_class_def mem_Collect_eq
   using assms val_ring_polys_grad_memE(1) val_ring_polys_grad_memE(2) val_ring_polys_grad_memE(3) by blast
 
 lemma poly_res_class_memI:
   assumes "f \<in> carrier (UP Q\<^sub>p)"
-  assumes "deg Q\<^sub>p f \<le> d" 
+  assumes "deg Q\<^sub>p f \<le> d"
   assumes "\<And>i. f i \<in> \<O>\<^sub>p"
   assumes "\<And>i. Qp_res (f i) n = Qp_res (g i) n"
   shows "f \<in> poly_res_class n d g"
-  unfolding poly_res_class_def mem_Collect_eq using assms 
-  by metis 
-
-definition poly_res_classes where 
+  unfolding poly_res_class_def mem_Collect_eq using assms
+  by metis
+
+definition poly_res_classes where
 "poly_res_classes n d = poly_res_class n d ` val_ring_polys_grad d"
 
-lemma poly_res_classes_disjoint: 
+lemma poly_res_classes_disjoint:
   assumes "A \<in> poly_res_classes n d"
   assumes "B \<in> poly_res_classes n d"
   assumes "g \<in> A - B"
   shows "A \<inter> B = {}"
-proof- 
+proof-
   obtain a where a_def: "a \<in> val_ring_polys_grad d \<and> A = poly_res_class n d a"
-    using assms unfolding poly_res_classes_def by blast 
+    using assms unfolding poly_res_classes_def by blast
   obtain b where b_def: "b \<in> val_ring_polys_grad d \<and> B = poly_res_class n d b"
-    using assms unfolding poly_res_classes_def by blast 
+    using assms unfolding poly_res_classes_def by blast
   have 0: "\<And>f. f \<in> A \<inter> B \<Longrightarrow> False"
-  proof- 
+  proof-
     fix f assume A: "f \<in> A \<inter> B"
     have 1: "\<exists>i. Qp_res (g i) n \<noteq> Qp_res (f i) n"
     proof(rule ccontr)
       assume B: "\<nexists>i. Qp_res (g i) n \<noteq> Qp_res (f i) n"
       then have 2: "\<And>i.  Qp_res (g i) n = Qp_res (f i) n"
-        by blast 
+        by blast
       have 3: "g \<in> poly_res_class n d a"
         using a_def assms by blast
       have 4: "\<And>i. Qp_res (b i) n = Qp_res (f i) n"
         apply(rule poly_res_class_memE[of _ n d])
         using assms A b_def by blast
       have 5: "\<And>i. Qp_res (a i) n = Qp_res (g i) n"
         apply(rule poly_res_class_memE[of _ n d])
         using assms A a_def by blast
       have 6: "g \<in> poly_res_class n d b"
-        apply(rule poly_res_class_memI, rule poly_res_class_memE[of _ n d a], rule 3, 
+        apply(rule poly_res_class_memI, rule poly_res_class_memE[of _ n d a], rule 3,
             rule poly_res_class_memE[of _ n d a], rule 3,  rule poly_res_class_memE[of _ n d a], rule 3)
-        unfolding 2 4 by blast 
-      show False using 6 b_def assms by blast 
+        unfolding 2 4 by blast
+      show False using 6 b_def assms by blast
     qed
     then obtain i where i_def: "Qp_res (g i) n \<noteq> Qp_res (f i) n"
-      by blast 
+      by blast
     have 2: "\<And>i. Qp_res (a i) n = Qp_res (f i) n"
-      apply(rule poly_res_class_memE[of _ n d]) 
-      using A a_def by blast 
+      apply(rule poly_res_class_memE[of _ n d])
+      using A a_def by blast
     have 3: "\<And>i. Qp_res (b i) n = Qp_res (f i) n"
-      apply(rule poly_res_class_memE[of _ n d]) 
-      using A b_def by blast 
+      apply(rule poly_res_class_memE[of _ n d])
+      using A b_def by blast
     have 4: "\<And>i. Qp_res (a i) n = Qp_res (g i) n"
       apply(rule poly_res_class_memE[of _ n d])
-      using assms a_def by blast 
-    show False using i_def 2 unfolding 4 2 by blast 
-  qed
-  show ?thesis using 0 by blast 
+      using assms a_def by blast
+    show False using i_def 2 unfolding 4 2 by blast
+  qed
+  show ?thesis using 0 by blast
 qed
 
 definition int_fun_to_poly where
 "int_fun_to_poly (f::nat \<Rightarrow> int) i = [(f i)]\<cdot>\<one>"
 
-lemma int_fun_to_poly_closed: 
+lemma int_fun_to_poly_closed:
   assumes "\<And>i. i > d \<Longrightarrow> f i = 0"
   shows "int_fun_to_poly f \<in> carrier (UP Q\<^sub>p)"
-  apply(rule UPQ.UP_car_memI[of d]) 
-  using assms unfolding int_fun_to_poly_def 
+  apply(rule UPQ.UP_car_memI[of d])
+  using assms unfolding int_fun_to_poly_def
   using Qp.int_inc_zero apply presburger
   by(rule Qp.int_inc_closed)
 
-lemma int_fun_to_poly_deg: 
+lemma int_fun_to_poly_deg:
   assumes "\<And>i. i > d \<Longrightarrow> f i = 0"
   shows "deg Q\<^sub>p (int_fun_to_poly f) \<le> d"
   apply(rule UPQ.deg_leqI, rule int_fun_to_poly_closed, rule assms, blast)
-  unfolding int_fun_to_poly_def using assms 
+  unfolding int_fun_to_poly_def using assms
   using Qp.int_inc_zero by presburger
 
 lemma Qp_res_mod_triv:
   assumes "a \<in> \<O>\<^sub>p"
   shows "Qp_res a n mod p ^ n = Qp_res a n"
-  using assms Qp_res_closed[of a n] 
+  using assms Qp_res_closed[of a n]
   by (meson mod_pos_pos_trivial p_residue_ring_car_memE(1) p_residue_ring_car_memE(2))
 
 lemma int_fun_to_poly_is_class_wit:
   assumes "f \<in> poly_res_class n d g"
   shows "(int_fun_to_poly (\<lambda>i::nat. Qp_res (f i) n)) \<in> poly_res_class n d g"
 proof(rule poly_res_class_memI[of   ], rule int_fun_to_poly_closed[of d])
   show 0: "\<And>i. d < i \<Longrightarrow> Qp_res (f i) n = 0"
   proof- fix i assume A: "d < i"
     hence 0: "deg Q\<^sub>p f < i"
-      using A  assms poly_res_class_memE(2)[of f n d g] 
+      using A  assms poly_res_class_memE(2)[of f n d g]
       by linarith
     have 1: "f i = \<zero>"
-      using 0 assms poly_res_class_memE[of f n d g] 
+      using 0 assms poly_res_class_memE[of f n d g]
       using UPQ.UP_car_memE(2) by blast
     show "Qp_res (f i) n = 0"
-      unfolding 1 Qp_res_zero by blast 
+      unfolding 1 Qp_res_zero by blast
   qed
   show "deg Q\<^sub>p (int_fun_to_poly (\<lambda>i. Qp_res (f i) n)) \<le> d"
     by(rule int_fun_to_poly_deg, rule 0, blast)
   show "\<And>i. int_fun_to_poly (\<lambda>i. Qp_res (f i) n) i \<in> \<O>\<^sub>p"
-    unfolding int_fun_to_poly_def 
+    unfolding int_fun_to_poly_def
     using Qp.int_mult_closed Qp_val_ringI val_of_int_inc by blast
   show "\<And>i. Qp_res (int_fun_to_poly (\<lambda>i. Qp_res (f i) n) i) n = Qp_res (g i) n"
     unfolding int_fun_to_poly_def Qp_res_int_inc
-    using Qp_res_mod_triv assms poly_res_class_memE(4) Qp_res_closed UPQ.cfs_closed 
+    using Qp_res_mod_triv assms poly_res_class_memE(4) Qp_res_closed UPQ.cfs_closed
     by (metis poly_res_class_memE(3))
 qed
 
-lemma finite_support_funs_finite: 
+lemma finite_support_funs_finite:
 "finite (({..d} \<rightarrow> carrier (Zp_res_ring n)) \<inter> {(f::nat \<Rightarrow> int). \<forall>i > d. f i = 0})"
-proof- 
+proof-
   have 0: "finite (\<Pi>\<^sub>E i \<in> {..d}.carrier (Zp_res_ring n))"
     apply(rule finite_PiE, blast)
-    using residue_ring_card[of n] by blast 
+    using residue_ring_card[of n] by blast
   obtain g where g_def: "g = (\<lambda>f. (\<lambda>i::nat. if i \<in> {..d} then f i else (0::int)))"
-    by blast 
+    by blast
   have 1: "g `  (\<Pi>\<^sub>E i \<in> {..d}.carrier (Zp_res_ring n)) = (({..d} \<rightarrow> carrier (Zp_res_ring n)) \<inter> {(f::nat \<Rightarrow> int). \<forall>i > d. f i = 0})"
   proof(rule equalityI, rule subsetI)
     fix x assume A: "x \<in> g ` ({..d} \<rightarrow>\<^sub>E carrier (residue_ring (p ^ n)))"
     obtain f where f_def: "f \<in> (\<Pi>\<^sub>E i \<in> {..d}.carrier (Zp_res_ring n)) \<and> x = g f"
-      using A by blast 
+      using A by blast
     have x_eq: "x = g f"
-      using f_def by blast 
+      using f_def by blast
     show "x \<in> ({..d} \<rightarrow> carrier (residue_ring (p ^ n))) \<inter> {f. \<forall>i>d. f i = 0}"
     proof(rule, rule)
       fix i assume A: "i \<in> {..d}"
       show "x i \<in> carrier (Zp_res_ring n)"
       proof(cases "i \<in> {..d}")
         case True
         then have T0: "f i \<in> carrier (Zp_res_ring n)"
-          using f_def by blast 
+          using f_def by blast
         have "x i = f i"
-          unfolding x_eq g_def 
-          using True by metis 
-        thus ?thesis using T0 by metis 
+          unfolding x_eq g_def
+          using True by metis
+        thus ?thesis using T0 by metis
       next
         case False
         then have F0: "x i = 0"
-          unfolding x_eq g_def by metis 
+          unfolding x_eq g_def by metis
         show ?thesis
-          unfolding F0 
+          unfolding F0
           by (metis residue_mult_closed residue_times_zero_r)
       qed
     next
       show "x \<in> {f. \<forall>i>d. f i = 0}"
       proof(rule, rule, rule)
         fix i assume A: "d < i"
         then have 0: "i \<notin> {..d}"
           by simp
         thus "x i = 0"
-          unfolding x_eq g_def 
-          by metis 
+          unfolding x_eq g_def
+          by metis
       qed
     qed
-  next 
+  next
     show "({..d} \<rightarrow> carrier (residue_ring (p ^ n))) \<inter> {f. \<forall>i>d. f i = 0}
     \<subseteq> g ` ({..d} \<rightarrow>\<^sub>E carrier (residue_ring (p ^ n)))"
     proof(rule subsetI)
-    fix x 
+    fix x
     assume A: " x \<in> ({..d} \<rightarrow> carrier (residue_ring (p ^ n))) \<inter> {f. \<forall>i>d. f i = 0}"
     show " x \<in> g ` ({..d} \<rightarrow>\<^sub>E carrier (residue_ring (p ^ n)))"
-    proof- 
+    proof-
       obtain h where h_def: "h = restrict x {..d}"
-        by blast 
+        by blast
       have 0: "\<And>i. i \<in> {..d} \<Longrightarrow> h i = x i"
-        unfolding h_def restrict_apply by metis 
+        unfolding h_def restrict_apply by metis
       have 1: "\<And>i. i \<notin> {..d} \<Longrightarrow> h i = undefined"
-        unfolding h_def restrict_apply by metis 
+        unfolding h_def restrict_apply by metis
       have 2: "\<And>i. i \<in> {..d} \<Longrightarrow> h i \<in> carrier (Zp_res_ring n)"
         using A 0 unfolding 0 by blast
       have 3: "h \<in> {..d} \<rightarrow>\<^sub>E carrier (residue_ring (p ^ n))"
         by(rule, rule 2, blast, rule 1, blast)
       have 4: "\<And>i. i \<notin> {..d} \<Longrightarrow> x i = 0"
-        using A unfolding Int_iff mem_Collect_eq 
+        using A unfolding Int_iff mem_Collect_eq
         by (metis atMost_iff eq_imp_le le_simps(1) linorder_neqE_nat)
       have 5: "x = g h"
-      proof fix i 
+      proof fix i
         show "x i = g h i"
-          unfolding g_def 
+          unfolding g_def
           apply(cases "i \<in> {..d}")
-          using 0 apply metis unfolding 4 
-          by metis 
+          using 0 apply metis unfolding 4
+          by metis
       qed
-      show ?thesis unfolding 5 using 3 by blast 
+      show ?thesis unfolding 5 using 3 by blast
      qed
    qed
   qed
   have 2: "finite (g `  (\<Pi>\<^sub>E i \<in> {..d}.carrier (Zp_res_ring n)))"
-    using 0 by blast 
-  show ?thesis using 2 unfolding 1 by blast 
-qed
-
-lemma poly_res_classes_finite: 
+    using 0 by blast
+  show ?thesis using 2 unfolding 1 by blast
+qed
+
+lemma poly_res_classes_finite:
 "finite (poly_res_classes n d)"
 proof-
   have 0: "poly_res_class n d ` int_fun_to_poly ` (({..d} \<rightarrow> carrier (Zp_res_ring n)) \<inter> {(f::nat \<Rightarrow> int). \<forall>i > d. f i = 0}) = poly_res_classes n d"
   proof(rule equalityI, rule subsetI)
     fix x assume A: " x \<in> poly_res_class n d ` int_fun_to_poly ` (({..d} \<rightarrow> carrier (residue_ring (p ^ n))) \<inter> {f. \<forall>i>d. f i = 0})"
-    then obtain f where f_def: "f \<in> ({..d} \<rightarrow> carrier (residue_ring (p ^ n))) \<inter> {f. \<forall>i>d. f i = 0} \<and> 
+    then obtain f where f_def: "f \<in> ({..d} \<rightarrow> carrier (residue_ring (p ^ n))) \<inter> {f. \<forall>i>d. f i = 0} \<and>
                                 x = poly_res_class n d (int_fun_to_poly f)"
-      by blast 
+      by blast
     have x_eq: "x = poly_res_class n d (int_fun_to_poly f)"
-      using f_def by blast 
+      using f_def by blast
     show "x \<in> poly_res_classes n d"
-    proof- 
+    proof-
       have 0: "int_fun_to_poly f \<in> val_ring_polys_grad d"
         apply(rule val_ring_polys_grad_memI, rule int_fun_to_poly_closed[of d])
         using f_def apply blast
-        using int_fun_to_poly_def 
+        using int_fun_to_poly_def
          apply (metis Qp.int_inc_closed padic_fields.int_fun_to_poly_def padic_fields.val_of_int_inc padic_fields_axioms val_ring_memI)
         apply(rule int_fun_to_poly_deg)
         using f_def by blast
-      show ?thesis unfolding poly_res_classes_def x_eq 
-        using 0 by blast 
-    qed
-  next 
+      show ?thesis unfolding poly_res_classes_def x_eq
+        using 0 by blast
+    qed
+  next
     show "poly_res_classes n d
     \<subseteq> poly_res_class n d `
        int_fun_to_poly `
        (({..d} \<rightarrow> carrier (residue_ring (p ^ n))) \<inter>
         {f. \<forall>i>d. f i = 0})"
     proof(rule subsetI)
     fix x assume A: " x \<in> poly_res_classes n d"
     show "x \<in> poly_res_class n d ` int_fun_to_poly ` (({..d} \<rightarrow> carrier (residue_ring (p ^ n))) \<inter> {f. \<forall>i>d. f i = 0})"
-    proof- 
+    proof-
       obtain f where f_def: "f \<in> val_ring_polys_grad d \<and> x = poly_res_class n d f"
-        using A unfolding poly_res_classes_def by blast 
+        using A unfolding poly_res_classes_def by blast
       have x_eq: "x = poly_res_class n d f"
-        using f_def by blast 
+        using f_def by blast
       obtain h where h_def: "h = (\<lambda>i::nat. Qp_res (f i) n)"
-        by blast 
+        by blast
       have 0: "\<And>i. i > d \<Longrightarrow> f i = \<zero>"
       proof- fix i assume A: "i > d"
          have "i > deg Q\<^sub>p f"
            apply(rule le_less_trans[of _ d])
-           using f_def unfolding val_ring_polys_grad_def val_ring_polys_def mem_Collect_eq 
+           using f_def unfolding val_ring_polys_grad_def val_ring_polys_def mem_Collect_eq
             apply blast
            by(rule A)
         then show "f i = \<zero>"
-          using f_def unfolding val_ring_polys_grad_def val_ring_polys_def mem_Collect_eq 
+          using f_def unfolding val_ring_polys_grad_def val_ring_polys_def mem_Collect_eq
           using UPQ.deg_leE by blast
       qed
       have 1: "\<And>i. i > d \<Longrightarrow> h i = 0"
-        unfolding h_def 0 Qp_res_zero by blast 
+        unfolding h_def 0 Qp_res_zero by blast
       have 2: "x = poly_res_class n d (int_fun_to_poly h)"
-        unfolding x_eq 
+        unfolding x_eq
         apply(rule poly_res_class_closed)
         using f_def unfolding val_ring_polys_grad_def val_ring_polys_def mem_Collect_eq apply blast
            apply(rule int_fun_to_poly_closed[of d], rule 1, blast)
         using f_def unfolding val_ring_polys_grad_def val_ring_polys_def mem_Collect_eq apply blast
          apply(rule int_fun_to_poly_deg, rule 1, blast)
         unfolding h_def
         apply(rule int_fun_to_poly_is_class_wit, rule poly_res_class_refl)
-        using f_def by blast 
+        using f_def by blast
       have 3: "h \<in> ({..d} \<rightarrow> carrier (residue_ring (p ^ n))) \<inter> {f. \<forall>i>d. f i = 0}"
         apply(rule , rule )
         unfolding h_def apply(rule Qp_res_closed, rule val_ring_polys_grad_memE[of _ d])
         using f_def apply blast
         unfolding mem_Collect_eq apply(rule, rule)
-        unfolding 0 Qp_res_zero by blast 
-      show ?thesis 
-        unfolding 2 using 3 by blast 
+        unfolding 0 Qp_res_zero by blast
+      show ?thesis
+        unfolding 2 using 3 by blast
       qed
     qed
   qed
   have 1: "finite (poly_res_class n d ` int_fun_to_poly ` (({..d} \<rightarrow> carrier (Zp_res_ring n)) \<inter> {(f::nat \<Rightarrow> int). \<forall>i > d. f i = 0}))"
-    using finite_support_funs_finite by blast 
-  show ?thesis using 1 unfolding 0 by blast 
-qed
-
-lemma Qp_res_eq_zeroI: 
+    using finite_support_funs_finite by blast
+  show ?thesis using 1 unfolding 0 by blast
+qed
+
+lemma Qp_res_eq_zeroI:
   assumes "a \<in> \<O>\<^sub>p"
   assumes "val a \<ge> n"
   shows "Qp_res a n = 0"
-proof- 
+proof-
   have 0: "val_Zp (to_Zp a) \<ge> n"
     using assms to_Zp_val by presburger
   have 1: "to_Zp a n = 0"
     apply(rule zero_below_val_Zp, rule to_Zp_closed)
     using val_ring_closed assms apply blast
     by(rule 0)
-  thus ?thesis unfolding Qp_res_def by blast 
-qed
-
-lemma Qp_res_eqI: 
+  thus ?thesis unfolding Qp_res_def by blast
+qed
+
+lemma Qp_res_eqI:
   assumes "a \<in> \<O>\<^sub>p"
   assumes "b \<in> \<O>\<^sub>p"
   assumes "Qp_res (a \<ominus> b) n = 0"
   shows "Qp_res a n = Qp_res b n"
   using assms by (metis Qp_res_def val_ring_memE res_diff_zero_fact(1) to_Zp_closed to_Zp_minus)
 
-lemma Qp_res_eqI': 
+lemma Qp_res_eqI':
   assumes "a \<in> \<O>\<^sub>p"
   assumes "b \<in> \<O>\<^sub>p"
   assumes "val (a \<ominus> b) \<ge> n"
   shows "Qp_res a n = Qp_res b n"
   apply(rule Qp_res_eqI, rule assms, rule assms, rule Qp_res_eq_zeroI)
   using assms Q\<^sub>p_def Zp_def \<iota>_def padic_fields.val_ring_minus_closed padic_fields_axioms apply blast
   by(rule assms)
 
-lemma Qp_res_eqE: 
+lemma Qp_res_eqE:
   assumes "a \<in> \<O>\<^sub>p"
   assumes "b \<in> \<O>\<^sub>p"
   assumes "Qp_res a n = Qp_res b n"
   shows "val (a \<ominus> b) \<ge> n"
-proof- 
+proof-
   have 0: "val (a \<ominus> b) = val_Zp (to_Zp a \<ominus>\<^bsub>Z\<^sub>p\<^esub> to_Zp b)"
-    using assms  
-    by (metis to_Zp_minus to_Zp_val val_ring_minus_closed)    
+    using assms
+    by (metis to_Zp_minus to_Zp_val val_ring_minus_closed)
   have 1: "(to_Zp a \<ominus>\<^bsub>Z\<^sub>p\<^esub> to_Zp b) n = 0"
-    using assms unfolding Qp_res_def 
+    using assms unfolding Qp_res_def
     by (meson val_ring_memE res_diff_zero_fact'' to_Zp_closed)
   have 2: "val_Zp (to_Zp a \<ominus>\<^bsub>Z\<^sub>p\<^esub> to_Zp b) \<ge> n"
     apply(cases "to_Zp a \<ominus>\<^bsub>Z\<^sub>p\<^esub> to_Zp b = \<zero>\<^bsub>Z\<^sub>p\<^esub>")
   proof -
     assume a1: "to_Zp a \<ominus>\<^bsub>Z\<^sub>p\<^esub> to_Zp b = \<zero>\<^bsub>Z\<^sub>p\<^esub>"
     have "\<forall>n. eint (int n) \<le> val_Zp \<zero>\<^bsub>Z\<^sub>p\<^esub>"
       by (metis (no_types) Zp.r_right_minus_eq Zp.zero_closed val_Zp_dist_def val_Zp_dist_res_eq2)
        then show ?thesis
       using a1 by presburger
   next
     assume a1: "to_Zp a \<ominus>\<^bsub>Z\<^sub>p\<^esub> to_Zp b \<noteq> \<zero>\<^bsub>Z\<^sub>p\<^esub>"
     have 00: "to_Zp a \<ominus>\<^bsub>Z\<^sub>p\<^esub> to_Zp b \<in> carrier Z\<^sub>p"
-      using assms 
+      using assms
       by (meson val_ring_memE Zp.cring_simprules(4) to_Zp_closed)
-    show ?thesis 
+    show ?thesis
       using 1 a1 ord_Zp_geq[of "to_Zp a \<ominus>\<^bsub>Z\<^sub>p\<^esub> to_Zp b" n] 00
-            val_ord_Zp[of "to_Zp a \<ominus>\<^bsub>Z\<^sub>p\<^esub> to_Zp b"] eint_ord_code by metis 
-  qed
-  thus ?thesis unfolding 0 by blast 
+            val_ord_Zp[of "to_Zp a \<ominus>\<^bsub>Z\<^sub>p\<^esub> to_Zp b"] eint_ord_code by metis
+  qed
+  thus ?thesis unfolding 0 by blast
 qed
 
 lemma notin_closed:
-"(\<not> ((c::eint) \<le> x \<and> x \<le> d)) = (x < c \<or> d < x)" 
+"(\<not> ((c::eint) \<le> x \<and> x \<le> d)) = (x < c \<or> d < x)"
   by auto
 
-lemma Qp_res_neqI: 
+lemma Qp_res_neqI:
   assumes "a \<in> \<O>\<^sub>p"
   assumes "b \<in> \<O>\<^sub>p"
-  assumes "val (a \<ominus> b) < n" 
+  assumes "val (a \<ominus> b) < n"
   shows "Qp_res a n \<noteq> Qp_res b n"
   apply(rule ccontr)
-  using Qp_res_eqE[of a b n] assms 
+  using Qp_res_eqE[of a b n] assms
   using notin_closed by blast
 
-lemma Qp_res_equal: 
+lemma Qp_res_equal:
   assumes "a \<in> \<O>\<^sub>p"
   assumes "l = Qp_res a n"
   shows "Qp_res a n = Qp_res ([l]\<cdot>\<one>) n "
   unfolding Qp_res_int_inc assms using assms Qp_res_mod_triv by presburger
 
 definition Qp_res_class where
 "Qp_res_class n b = {a \<in> \<O>\<^sub>p. Qp_res a n = Qp_res b n}"
 
-definition Qp_res_classes where 
+definition Qp_res_classes where
 "Qp_res_classes n = Qp_res_class n ` \<O>\<^sub>p"
 
-lemma val_ring_int_inc_closed: 
-"[(k::int)]\<cdot>\<one> \<in> \<O>\<^sub>p" 
-proof- 
+lemma val_ring_int_inc_closed:
+"[(k::int)]\<cdot>\<one> \<in> \<O>\<^sub>p"
+proof-
   have 0: "[(k::int)]\<cdot>\<one> = \<iota> ([(k::int)]\<cdot>\<^bsub>Z\<^sub>p\<^esub>\<one>\<^bsub>Z\<^sub>p\<^esub>)"
     using inc_of_int by blast
-  thus ?thesis 
-    by blast 
-qed
-
-lemma val_ring_nat_inc_closed: 
+  thus ?thesis
+    by blast
+qed
+
+lemma val_ring_nat_inc_closed:
 "[(k::nat)]\<cdot>\<one> \<in> \<O>\<^sub>p"
 proof-
  have 0: "[k]\<cdot>\<one> = \<iota> ([k]\<cdot>\<^bsub>Z\<^sub>p\<^esub>\<one>\<^bsub>Z\<^sub>p\<^esub>)"
    using inc_of_nat by blast
-  thus ?thesis 
-    by blast
-qed
-
-lemma Qp_res_classes_wits: 
+  thus ?thesis
+    by blast
+qed
+
+lemma Qp_res_classes_wits:
 "Qp_res_classes n  =  (\<lambda>l::int. Qp_res_class n ([l]\<cdot>\<one>)) ` (carrier (Zp_res_ring n))"
 proof-
  obtain F where F_def: "F = (\<lambda>l::int. Qp_res_class n ([l]\<cdot>\<one>))"
-    by blast 
+    by blast
   have 0: "Qp_res_classes n  = F ` (carrier (Zp_res_ring n))"
   proof(rule equalityI, rule subsetI)
     fix x assume A: "x \<in> Qp_res_classes n"
     then obtain a where a_def: "a \<in> \<O>\<^sub>p \<and> x = Qp_res_class n a"
-      unfolding Qp_res_classes_def by blast 
+      unfolding Qp_res_classes_def by blast
     have 1: "Qp_res a n = Qp_res ([(Qp_res a n)]\<cdot>\<one>) n "
       apply(rule Qp_res_equal)
       using a_def apply blast by blast
     have 2: "Qp_res_class n a = Qp_res_class n ([(Qp_res a n)]\<cdot>\<one>)"
-      unfolding Qp_res_class_def using 1 by metis 
+      unfolding Qp_res_class_def using 1 by metis
     have 3: "x =  Qp_res_class n ([(Qp_res a n)]\<cdot>\<one>)"
-      using a_def unfolding 2 by blast 
+      using a_def unfolding 2 by blast
     have 4: "a \<in> \<O>\<^sub>p"
-      using a_def by blast 
+      using a_def by blast
     show " x \<in> F ` carrier (Zp_res_ring n)"
-      unfolding F_def  3 
+      unfolding F_def  3
       using Qp_res_closed[of a n] 4 by blast
   next
     show "F ` carrier (residue_ring (p ^ n)) \<subseteq> Qp_res_classes n"
     proof(rule subsetI)
     fix x assume A: "x \<in> F ` (carrier (Zp_res_ring n))"
     then obtain l where l_def: "l \<in> carrier (Zp_res_ring n) \<and> x = F l"
-      using A by blast 
+      using A by blast
     have 0: "x = F l"
       using l_def by blast
     show "x \<in> Qp_res_classes n"
-      unfolding 0 F_def Qp_res_classes_def using val_ring_int_inc_closed by blast 
-    qed
-  qed
-  then show ?thesis unfolding F_def by blast 
-qed
-
-lemma Qp_res_classes_finite: 
+      unfolding 0 F_def Qp_res_classes_def using val_ring_int_inc_closed by blast
+    qed
+  qed
+  then show ?thesis unfolding F_def by blast
+qed
+
+lemma Qp_res_classes_finite:
 "finite (Qp_res_classes n)"
 by (metis Qp_res_classes_wits finite_atLeastLessThan_int finite_imageI p_res_ring_car)
 
 definition Qp_cong_set where
-"Qp_cong_set \<alpha> a =  {x \<in> \<O>\<^sub>p. to_Zp x \<alpha> = a \<alpha>}" 
+"Qp_cong_set \<alpha> a =  {x \<in> \<O>\<^sub>p. to_Zp x \<alpha> = a \<alpha>}"
 
 lemma Qp_cong_set_as_ball:
   assumes "a \<in> carrier Z\<^sub>p"
   assumes "a \<alpha> = 0"
-  shows "Qp_cong_set \<alpha> a = B\<^bsub>\<alpha>\<^esub>[\<zero>]" 
+  shows "Qp_cong_set \<alpha> a = B\<^bsub>\<alpha>\<^esub>[\<zero>]"
 proof-
   have 0: "\<iota> a \<in> carrier Q\<^sub>p"
-    using assms inc_closed[of a] by blast   
-  show ?thesis 
-  proof 
+    using assms inc_closed[of a] by blast
+  show ?thesis
+  proof
     show "Qp_cong_set \<alpha> a \<subseteq> B\<^bsub>\<alpha>\<^esub>[\<zero>]"
     proof fix x assume A: "x \<in> Qp_cong_set \<alpha> a"
       show "x \<in> B\<^bsub>\<alpha> \<^esub>[\<zero>]"
       proof(rule c_ballI)
         show t0: "x \<in> carrier Q\<^sub>p"
-          using A unfolding Qp_cong_set_def 
+          using A unfolding Qp_cong_set_def
           using val_ring_memE by blast
         show "eint (int \<alpha>) \<le> val (x \<ominus> \<zero>)"
         proof-
           have t1: "to_Zp x \<alpha> = 0"
-            using A unfolding Qp_cong_set_def  
+            using A unfolding Qp_cong_set_def
             by (metis (mono_tags, lifting) assms(2) mem_Collect_eq)
           have t2: "val_Zp (to_Zp x) \<ge> \<alpha>"
             apply(cases "to_Zp x = \<zero>\<^bsub>Z\<^sub>p\<^esub>")
              apply (metis Zp.r_right_minus_eq Zp.zero_closed val_Zp_dist_def val_Zp_dist_res_eq2)
-              using ord_Zp_geq[of "to_Zp x" \<alpha>] A unfolding Qp_cong_set_def  
+              using ord_Zp_geq[of "to_Zp x" \<alpha>] A unfolding Qp_cong_set_def
               by (metis (no_types, lifting) val_ring_memE eint_ord_simps(1) t1 to_Zp_closed to_Zp_def val_ord_Zp)
           then show ?thesis using A unfolding Qp_cong_set_def mem_Collect_eq
-            using val_ring_memE  
+            using val_ring_memE
              by (metis Qp_res_eqE Qp_res_eq_zeroI Qp_res_zero to_Zp_val zero_in_val_ring)
          qed
       qed
     qed
     show "B\<^bsub>int \<alpha>\<^esub>[\<zero>] \<subseteq> Qp_cong_set \<alpha> a"
-    proof fix x assume A: "x \<in> B\<^bsub>int \<alpha>\<^esub>[\<zero>]" 
+    proof fix x assume A: "x \<in> B\<^bsub>int \<alpha>\<^esub>[\<zero>]"
       then have 0: "val x \<ge> \<alpha>"
-        using assms c_ballE[of x \<alpha> \<zero>] 
+        using assms c_ballE[of x \<alpha> \<zero>]
         by (smt Qp.minus_closed Qp.r_right_minus_eq Qp_diff_diff)
       have 1: "to_Zp x \<in> carrier Z\<^sub>p"
         using A 0 assms c_ballE(1) to_Zp_closed by blast
       have 2: "x \<in> \<O>\<^sub>p"
-        using 0 A val_ringI c_ballE 
+        using 0 A val_ringI c_ballE
         by (smt Q\<^sub>p_def Zp_def \<iota>_def eint_ord_simps(1) of_nat_0 of_nat_le_0_iff val_ring_ord_criterion padic_fields_axioms val_ord' zero_in_val_ring)
       then have "val_Zp (to_Zp  x) \<ge> \<alpha>"
         using 0 1 A assms c_ballE[of x \<alpha> \<zero>] to_Zp_val by presburger
-      then have "to_Zp x \<alpha> = 0" 
-        using 1 zero_below_val_Zp by blast 
-      then show " x \<in> Qp_cong_set \<alpha> a" 
-        unfolding Qp_cong_set_def using assms(2) 2 
+      then have "to_Zp x \<alpha> = 0"
+        using 1 zero_below_val_Zp by blast
+      then show " x \<in> Qp_cong_set \<alpha> a"
+        unfolding Qp_cong_set_def using assms(2) 2
         by (metis (mono_tags, lifting) mem_Collect_eq)
     qed
   qed
 qed
 
 lemma Qp_cong_set_as_ball':
   assumes "a \<in> carrier Z\<^sub>p"
   assumes "val_Zp a < eint (int \<alpha>)"
-  shows "Qp_cong_set \<alpha> a = B\<^bsub>\<alpha>\<^esub>[(\<iota> a)]" 
+  shows "Qp_cong_set \<alpha> a = B\<^bsub>\<alpha>\<^esub>[(\<iota> a)]"
 proof
   show "Qp_cong_set \<alpha> a \<subseteq> B\<^bsub>\<alpha>\<^esub>[\<iota> a]"
-  proof fix x 
+  proof fix x
     assume A:  "x \<in> Qp_cong_set \<alpha> a"
     then have 0: "to_Zp x \<alpha> = a \<alpha>"
-      unfolding Qp_cong_set_def  by blast 
+      unfolding Qp_cong_set_def  by blast
     have 1:  "x \<in> \<O>\<^sub>p"
-      using A unfolding Qp_cong_set_def by blast 
+      using A unfolding Qp_cong_set_def by blast
     have 2: "to_Zp x \<in> carrier Z\<^sub>p"
       using 1 val_ring_memE to_Zp_closed by blast
-    have 3: "val_Zp (to_Zp x \<ominus>\<^bsub>Z\<^sub>p\<^esub> a) \<ge> \<alpha>" 
+    have 3: "val_Zp (to_Zp x \<ominus>\<^bsub>Z\<^sub>p\<^esub> a) \<ge> \<alpha>"
       using 0 assms 2 val_Zp_dist_def val_Zp_dist_res_eq2 by presburger
     have 4: "val_Zp (to_Zp x \<ominus>\<^bsub>Z\<^sub>p\<^esub> a) > val_Zp a"
       using 3 assms(2) less_le_trans[of "val_Zp a" "eint (int \<alpha>)" "val_Zp (to_Zp x \<ominus>\<^bsub>Z\<^sub>p\<^esub> a)" ]
-      by blast 
+      by blast
     then have 5: "val_Zp (to_Zp x) = val_Zp a"
       using assms 2 equal_val_Zp by blast
     have 7: "val (x \<ominus> (\<iota> a)) \<ge> \<alpha>"
       using 3 5 1 by (metis "2" Zp.minus_closed assms(1) inc_of_diff to_Zp_inc val_of_inc)
     then show "x \<in> B\<^bsub>int \<alpha>\<^esub>[\<iota> a]"
       using c_ballI[of x \<alpha> "\<iota> a"]  1 assms val_ring_memE by blast
   qed
   show "B\<^bsub>int \<alpha>\<^esub>[\<iota> a] \<subseteq> Qp_cong_set \<alpha> a"
   proof fix x
     assume A: "x \<in> B\<^bsub>int \<alpha>\<^esub>[\<iota> a]"
     then have 0: "val (x \<ominus> \<iota> a) \<ge> \<alpha>"
-      using c_ballE by blast 
+      using c_ballE by blast
     have 1: "val (\<iota> a) = val_Zp a"
-      using assms  Zp_def \<iota>_def padic_fields.val_of_inc padic_fields_axioms 
+      using assms  Zp_def \<iota>_def padic_fields.val_of_inc padic_fields_axioms
       by metis
     then have 2: "val (x \<ominus> \<iota> a) > val (\<iota> a)"
       using 0 assms less_le_trans[of "val (\<iota> a)" "eint (int \<alpha>)" "val (x \<ominus> \<iota> a)"]
-      by metis 
+      by metis
     have "\<iota> a \<in> carrier Q\<^sub>p"
       using assms(1) inc_closed by blast
     then have B: "val x = val (\<iota> a)"
-      using 2  A assms c_ballE(1)[of x \<alpha> "\<iota> a"] 
+      using 2  A assms c_ballE(1)[of x \<alpha> "\<iota> a"]
       by (metis ultrametric_equal_eq)
     have 3: "val_Zp (to_Zp x) = val_Zp a"
       by (metis "1" A \<open>val x = val (\<iota> a)\<close> assms(1) c_ballE(1) to_Zp_val val_pos val_ringI)
     have 4: "val_Zp (to_Zp x \<ominus>\<^bsub>Z\<^sub>p\<^esub> a) \<ge> \<alpha>"
-      using 0 A 3 
-      by (metis B Zp.minus_closed assms(1) c_ballE(1) inc_of_diff to_Zp_closed to_Zp_inc val_of_inc val_pos val_ring_val_criterion)      
+      using 0 A 3
+      by (metis B Zp.minus_closed assms(1) c_ballE(1) inc_of_diff to_Zp_closed to_Zp_inc val_of_inc val_pos val_ring_val_criterion)
     then have 5: "to_Zp x \<alpha> = a \<alpha>"
       by (meson A Zp.minus_closed assms(1) c_ballE(1) res_diff_zero_fact(1) to_Zp_closed zero_below_val_Zp)
     have 6: "x \<in> \<O>\<^sub>p"
     proof-
       have "val x \<ge> 0"
         using B assms 1 val_pos by presburger
-      then show ?thesis 
+      then show ?thesis
         using A c_ballE(1) val_ringI by blast
     qed
-    then show "x \<in> Qp_cong_set \<alpha> a" unfolding Qp_cong_set_def  
+    then show "x \<in> Qp_cong_set \<alpha> a" unfolding Qp_cong_set_def
       using "5" by blast
   qed
 qed
 
 lemma Qp_cong_set_is_univ_semialgebraic:
   assumes "a \<in> carrier Z\<^sub>p"
   shows "is_univ_semialgebraic (Qp_cong_set \<alpha> a)"
 proof(cases "a \<alpha> = 0")
   case True
-  then show ?thesis 
-    using ball_is_univ_semialgebraic[of \<zero> \<alpha>] Qp.zero_closed  Qp_cong_set_as_ball assms 
-    by metis  
+  then show ?thesis
+    using ball_is_univ_semialgebraic[of \<zero> \<alpha>] Qp.zero_closed  Qp_cong_set_as_ball assms
+    by metis
 next
   case False
   then have "\<alpha> \<noteq> 0"
-    using assms residues_closed[of a 0] 
+    using assms residues_closed[of a 0]
     by (meson p_res_ring_0')
   then obtain n where n_def: "Suc n = \<alpha>"
     by (metis lessI less_Suc_eq_0_disj)
   then have "val_Zp a < eint (int \<alpha>)"
     using below_val_Zp_zero[of a n]
     by (smt False assms eint_ile eint_ord_simps(1) eint_ord_simps(2) zero_below_val_Zp)
   then show ?thesis
-    using ball_is_univ_semialgebraic[of "\<iota> a" \<alpha>] Qp.zero_closed  Qp_cong_set_as_ball'[of a \<alpha>] assms 
+    using ball_is_univ_semialgebraic[of "\<iota> a" \<alpha>] Qp.zero_closed  Qp_cong_set_as_ball'[of a \<alpha>] assms
          inc_closed by presburger
 qed
 
-lemma constant_res_set_semialg: 
+lemma constant_res_set_semialg:
   assumes "l \<in> carrier (Zp_res_ring n)"
   shows "is_univ_semialgebraic {x \<in> \<O>\<^sub>p. Qp_res x n = l}"
-proof- 
+proof-
   have 0: "{x \<in> \<O>\<^sub>p. Qp_res x n = l} = Qp_cong_set n ([l]\<cdot>\<^bsub>Z\<^sub>p\<^esub>\<one>\<^bsub>Z\<^sub>p\<^esub>)"
     apply(rule equalityI')
-    unfolding mem_Collect_eq Qp_cong_set_def Qp_res_def 
+    unfolding mem_Collect_eq Qp_cong_set_def Qp_res_def
     apply (metis val_ring_memE Zp_int_inc_rep nat_le_linear p_residue_padic_int to_Zp_closed)
-    using assms 
+    using assms
     by (metis Zp_int_inc_res mod_pos_pos_trivial p_residue_ring_car_memE(1) p_residue_ring_car_memE(2))
-  show ?thesis unfolding 0 
+  show ?thesis unfolding 0
     apply(rule Qp_cong_set_is_univ_semialgebraic)
     by(rule Zp.int_inc_closed)
 qed
- 
+
 end
 
 end
diff --git a/thys/Padic_Field/Padic_Semialgebraic_Function_Ring.thy b/thys/Padic_Field/Padic_Semialgebraic_Function_Ring.thy
--- a/thys/Padic_Field/Padic_Semialgebraic_Function_Ring.thy
+++ b/thys/Padic_Field/Padic_Semialgebraic_Function_Ring.thy
@@ -1,6503 +1,6503 @@
 theory Padic_Semialgebraic_Function_Ring
   imports Padic_Field_Powers
 begin
 
 section\<open>Rings of Semialgebraic Functions\<close>
 
 text\<open>
-  In order to efficiently formalize Denef's proof of Macintyre's theorem, it is necessary to be 
-  able to reason about semialgebraic functions algebraically. For example, we need to consider 
-  polynomials in one variable whose coefficients are semialgebraic functions, and take their 
+  In order to efficiently formalize Denef's proof of Macintyre's theorem, it is necessary to be
+  able to reason about semialgebraic functions algebraically. For example, we need to consider
+  polynomials in one variable whose coefficients are semialgebraic functions, and take their
   Taylor expansions centered at a semialgebraic function. To facilitate this kind of reasoning, it
   is necessary to construct, for each arity $m$, a ring \texttt{SA(m)} of semialgebraic functions in
-  $m$ variables. These functions must be extensional functions which are undefined outside of the 
-  carrier set of $\mathbb{Q}_p^m$. 
-
-  The developments in this theory are mainly lemmas and defintitions which build the necessary 
-  theory to prove the cell decomposition theorems of \cite{denef1986}, and finally Macintyre's 
-  Theorem, which says that semi-algebraic sets are closed under projections. 
+  $m$ variables. These functions must be extensional functions which are undefined outside of the
+  carrier set of $\mathbb{Q}_p^m$.
+
+  The developments in this theory are mainly lemmas and defintitions which build the necessary
+  theory to prove the cell decomposition theorems of \cite{denef1986}, and finally Macintyre's
+  Theorem, which says that semi-algebraic sets are closed under projections.
 \<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsection\<open>Some eint Arithmetic\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 context padic_fields
 begin
 
 lemma eint_minus_ineq':
   assumes "a \<le> eint N"
   assumes "b - a \<le> c"
   shows "b - eint N \<le> c"
   using assms by(induction c, induction b, induction a, auto )
 
 lemma eint_minus_plus:
 "a - (eint b + eint c) = a - eint b - eint c"
   apply(induction a)
   apply (metis diff_add_eq_diff_diff_swap idiff_eint_eint plus_eint_simps(1) semiring_normalization_rules(24))
   using idiff_infinity by presburger
 
 lemma eint_minus_plus':
 "a - (eint b + eint c) = a - eint c - eint b"
-  by (metis add.commute eint_minus_plus)  
+  by (metis add.commute eint_minus_plus)
 
 lemma eint_minus_plus'':
   assumes "a - eint c - eint b = eint f"
   shows "a - eint c - eint f =  eint b"
-  using assms  apply(induction a ) 
+  using assms  apply(induction a )
   apply (metis add.commute add_diff_cancel_eint eint.distinct(2) eint_add_cancel_fact)
   by simp
 
 lemma uminus_involutive[simp]:
 "-(-x::eint) = x"
   apply(induction x)
   unfolding uminus_eint_def by auto
 
 lemma eint_minus:
 "(a::eint) - (b::eint) = a + (-b)"
   apply(induction a)
    apply(induction b)
 proof -
   fix int :: int and inta :: int
   have "\<forall>e ea. (ea::eint) + (e + - ea) = ea - ea + e"
     by (simp add: eint_uminus_eq)
   then have "\<forall>i ia. eint (ia + i) + - eint ia = eint i"
     by (metis ab_group_add_class.ab_diff_conv_add_uminus add.assoc add_minus_cancel idiff_eint_eint plus_eint_simps(1))
   then show "eint inta - eint int = eint inta + - eint int"
     by (metis ab_group_add_class.ab_diff_conv_add_uminus add.commute add_minus_cancel idiff_eint_eint)
 next
   show "\<And>int. eint int - \<infinity> = eint int + - \<infinity>"
   by (metis eint_uminus_eq i0_ne_infinity idiff_infinity_right idiff_self plus_eq_infty_iff_eint uminus_involutive)
   show " \<infinity> - b = \<infinity> + - b"
     apply(induction b)
   apply simp
-    by auto  
+    by auto
 qed
 
 lemma eint_mult_Suc:
    "eint (Suc k) * a = eint k * a + a"
   apply(induction a)
   apply (metis add.commute eSuc_eint mult_eSuc' of_nat_Suc)
-  using plus_eint_simps(3) times_eint_simps(4) 
+  using plus_eint_simps(3) times_eint_simps(4)
   by presburger
 
 lemma eint_mult_Suc_mono:
 assumes "a \<le> eint b \<longrightarrow> eint (int k) * a \<le> eint (int k) * eint b"
 shows "a \<le> eint b \<longrightarrow> eint (int (Suc k)) * a \<le> eint (int (Suc k)) * eint b"
-  using assms eint_mult_Suc 
+  using assms eint_mult_Suc
   by (metis add_mono_thms_linordered_semiring(1))
 
 lemma eint_nat_mult_mono:
   assumes "(a::eint) \<le> b"
   shows "eint (k::nat)*a \<le> eint k*b"
 proof-
   have "(a::eint) \<le> b \<longrightarrow> eint (k::nat)*a \<le> eint k*b"
     apply(induction k) apply(induction b)
   apply (metis eint_ile eq_iff mult_not_zero of_nat_0 times_eint_simps(1))
   apply simp
- apply(induction b) 
+ apply(induction b)
     using eint_mult_Suc_mono apply blast
   using eint_ord_simps(3) times_eint_simps(4) by presburger
-  thus ?thesis using assms by blast 
+  thus ?thesis using assms by blast
 qed
 
 lemma eint_Suc_zero:
 "eint (int (Suc 0)) * a = a"
   apply(induction a)
   apply simp
   by simp
 
 lemma eint_add_mono:
   assumes "(a::eint) \<le> b"
   assumes "(c::eint) \<le> d"
   shows "a + c \<le> b + d"
-  using assms 
+  using assms
   by (simp add: add_mono)
 
 lemma eint_nat_mult_mono_rev:
   assumes "k > 0"
-  assumes "eint (k::nat)*a \<le> eint k*b" 
+  assumes "eint (k::nat)*a \<le> eint k*b"
   shows "(a::eint) \<le> b"
 proof(rule ccontr)
   assume "\<not> a \<le> b"
   then have A: "b < a"
     using leI by blast
   have "b < a \<longrightarrow> eint (k::nat)*b < eint k*a"
     apply(induction b) apply(induction a)
       using A assms eint_ord_simps(2) times_eint_simps(1) zmult_zless_mono2_lemma apply presburger
     using eint_ord_simps(4) nat_mult_not_infty times_eint_simps(4) apply presburger
     using eint_ord_simps(6) by blast
-  then have "eint (k::nat)*b < eint k*a" 
-    using A by blast 
+  then have "eint (k::nat)*b < eint k*a"
+    using A by blast
   hence "\<not> eint (k::nat)*a \<le> eint k*b"
-    by (metis \<open>\<not> a \<le> b\<close> antisym eint_nat_mult_mono linear neq_iff)   
-  then show False using assms by blast 
+    by (metis \<open>\<not> a \<le> b\<close> antisym eint_nat_mult_mono linear neq_iff)
+  then show False using assms by blast
 qed
 
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsection\<open>Lemmas on Function Ring Operations\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 lemma Qp_funs_is_cring:
 "cring (Fun\<^bsub>n\<^esub> Q\<^sub>p)"
-  using F.cring by blast 
+  using F.M_cring by blast
 
 lemma Qp_funs_is_monoid:
 "monoid (Fun\<^bsub>n\<^esub> Q\<^sub>p)"
   using F.is_monoid by blast
-  
+
 lemma Qp_funs_car_memE:
   assumes "f \<in> carrier (Fun\<^bsub>n\<^esub> Q\<^sub>p)"
   shows "f \<in> (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) \<rightarrow> (carrier Q\<^sub>p)"
   by (simp add: assms ring_pow_function_ring_car_memE(2))
-  
+
 lemma Qp_funs_car_memI:
   assumes "g \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<rightarrow> carrier Q\<^sub>p"
   assumes "\<And>x. x \<notin> (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) \<Longrightarrow> g x = undefined"
   shows "g \<in> carrier (Fun\<^bsub>n\<^esub> Q\<^sub>p)"
 apply(rule Qp.function_ring_car_memI)
   using assms  apply blast
-  using assms by blast  
+  using assms by blast
 
 lemma Qp_funs_car_memI':
   assumes "g \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<rightarrow> carrier Q\<^sub>p"
   assumes "restrict g (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) = g"
   shows "g \<in> carrier (Fun\<^bsub>n\<^esub> Q\<^sub>p)"
   apply(intro Qp_funs_car_memI assms)
-  using assms  unfolding restrict_def 
-  by (metis (mono_tags, lifting)) 
+  using assms  unfolding restrict_def
+  by (metis (mono_tags, lifting))
 
 lemma Qp_funs_car_memI'':
   assumes "f \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<rightarrow> carrier Q\<^sub>p"
   assumes "g = (\<lambda> x \<in> (carrier (Q\<^sub>p\<^bsup>n\<^esup>)). f x)"
   shows "g \<in> carrier (Fun\<^bsub>n\<^esub> Q\<^sub>p)"
   apply(rule Qp_funs_car_memI)
-  using assms 
+  using assms
   apply (meson restrict_Pi_cancel)
   by (metis assms(2) restrict_def)
 
 lemma Qp_funs_one:
 "\<one>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> = (\<lambda> x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). \<one>)"
-  unfolding function_ring_def function_one_def 
+  unfolding function_ring_def function_one_def
   by (meson monoid.select_convs(2))
 
 lemma Qp_funs_zero:
 "\<zero>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> = (\<lambda> x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). \<zero>\<^bsub>Q\<^sub>p\<^esub>)"
-  unfolding function_ring_def function_zero_def 
+  unfolding function_ring_def function_zero_def
   by (meson ring_record_simps(11))
 
 lemma Qp_funs_add:
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "f \<in> (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) \<rightarrow> carrier Q\<^sub>p"
   assumes "g \<in> (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) \<rightarrow> carrier Q\<^sub>p"
   shows "(f \<oplus>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> g) x = f x \<oplus>\<^bsub>Q\<^sub>p\<^esub> g x"
-  using assms function_ring_def[of "carrier (Q\<^sub>p\<^bsup>n\<^esup>)" "Q\<^sub>p"] 
-  unfolding function_add_def  
+  using assms function_ring_def[of "carrier (Q\<^sub>p\<^bsup>n\<^esup>)" "Q\<^sub>p"]
+  unfolding function_add_def
   by (metis (mono_tags, lifting) restrict_apply' ring_record_simps(12))
 
 lemma Qp_funs_add':
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "f \<in> (carrier (Fun\<^bsub>n\<^esub> Q\<^sub>p))"
   assumes "g \<in> (carrier (Fun\<^bsub>n\<^esub> Q\<^sub>p))"
   shows "(f \<oplus>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> g) x = f x \<oplus>\<^bsub>Q\<^sub>p\<^esub> g x"
   using assms Qp_funs_add Qp_funs_car_memE
-  by blast 
+  by blast
 
 lemma Qp_funs_add'':
   assumes "f \<in> (carrier (Fun\<^bsub>n\<^esub> Q\<^sub>p))"
   assumes "g \<in> (carrier (Fun\<^bsub>n\<^esub> Q\<^sub>p))"
   shows "(f \<oplus>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> g) = (\<lambda> x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). f x \<oplus>\<^bsub>Q\<^sub>p\<^esub> g x)"
   unfolding function_ring_def function_add_def using ring_record_simps(12)
-  by metis 
+  by metis
 
 lemma Qp_funs_add''':
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "(f \<oplus>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> g) x = f x \<oplus>\<^bsub>Q\<^sub>p\<^esub> g x"
-  using assms function_ring_def[of "carrier (Q\<^sub>p\<^bsup>n\<^esup>)" "Q\<^sub>p"] 
-  unfolding function_add_def  
+  using assms function_ring_def[of "carrier (Q\<^sub>p\<^bsup>n\<^esup>)" "Q\<^sub>p"]
+  unfolding function_add_def
   by (metis (mono_tags, lifting) restrict_apply' ring_record_simps(12))
 
 lemma Qp_funs_mult:
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "f \<in> (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) \<rightarrow> carrier Q\<^sub>p"
   assumes "g \<in> (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) \<rightarrow> carrier Q\<^sub>p"
   shows "(f \<otimes>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> g) x = f x \<otimes> g x"
-  using assms function_ring_def[of "carrier (Q\<^sub>p\<^bsup>n\<^esup>)" "Q\<^sub>p"] 
-  unfolding function_mult_def  
+  using assms function_ring_def[of "carrier (Q\<^sub>p\<^bsup>n\<^esup>)" "Q\<^sub>p"]
+  unfolding function_mult_def
   by (metis (no_types, lifting) monoid.select_convs(1) restrict_apply')
 
 lemma Qp_funs_mult':
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "f \<in> (carrier (Fun\<^bsub>n\<^esub> Q\<^sub>p))"
   assumes "g \<in> (carrier (Fun\<^bsub>n\<^esub> Q\<^sub>p))"
   shows "(f \<otimes>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> g) x = f x \<otimes> g x"
   using assms Qp_funs_mult Qp_funs_car_memE
   by blast
 
 lemma Qp_funs_mult'':
   assumes "f \<in> (carrier (Fun\<^bsub>n\<^esub> Q\<^sub>p))"
   assumes "g \<in> (carrier (Fun\<^bsub>n\<^esub> Q\<^sub>p))"
   shows "(f \<otimes>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> g) = (\<lambda> x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). f x \<otimes> g x)"
-  unfolding function_ring_def function_mult_def using ring_record_simps(5) 
-  by metis 
+  unfolding function_ring_def function_mult_def using ring_record_simps(5)
+  by metis
 
 lemma Qp_funs_mult''':
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "(f \<otimes>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> g) x = f x \<otimes> g x"
-  using assms function_ring_def[of "carrier (Q\<^sub>p\<^bsup>n\<^esup>)" "Q\<^sub>p"] 
-  unfolding function_mult_def  
+  using assms function_ring_def[of "carrier (Q\<^sub>p\<^bsup>n\<^esup>)" "Q\<^sub>p"]
+  unfolding function_mult_def
   by (metis (mono_tags, lifting) monoid.select_convs(1) restrict_apply')
 
 lemma Qp_funs_a_inv:
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "f \<in> (carrier (Fun\<^bsub>n\<^esub> Q\<^sub>p))"
   shows "(\<ominus>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> f) x = \<ominus> (f x)"
-  using assms local.function_uminus_eval 
+  using assms local.function_uminus_eval
   by (simp add: local.function_uminus_eval'')
-    
+
 lemma Qp_funs_a_inv':
   assumes "f \<in> (carrier (Fun\<^bsub>n\<^esub> Q\<^sub>p))"
   shows "(\<ominus>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> f) = (\<lambda> x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). \<ominus> (f x))"
-proof fix x 
+proof fix x
   show "(\<ominus>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> f) x = (\<lambda>x\<in>carrier (Q\<^sub>p\<^bsup>n\<^esup>). \<ominus> f x) x"
     apply(cases "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)")
      apply (metis (no_types, lifting) Qp_funs_a_inv assms restrict_apply')
   by (simp add: assms local.function_ring_not_car)
 qed
 
 abbreviation(input) Qp_const ("\<cc>\<^bsub>_\<^esub>_") where
 "Qp_const n c \<equiv> constant_function (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) c"
 
-lemma Qp_constE: 
+lemma Qp_constE:
   assumes "c \<in> carrier Q\<^sub>p"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "Qp_const n c x = c"
-  using assms unfolding constant_function_def 
+  using assms unfolding constant_function_def
   by (meson restrict_apply')
 
 lemma Qp_funs_Units_memI:
   assumes "f \<in> (carrier (Fun\<^bsub>n\<^esub> Q\<^sub>p))"
   assumes "\<And> x. x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<Longrightarrow>  f x \<noteq> \<zero>\<^bsub>Q\<^sub>p\<^esub>"
   shows "f \<in> (Units (Fun\<^bsub>n\<^esub> Q\<^sub>p))"
         "inv\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> f = (\<lambda> x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). inv\<^bsub>Q\<^sub>p\<^esub> (f x))"
 proof-
   obtain g where g_def: "g = (\<lambda> x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). inv\<^bsub>Q\<^sub>p\<^esub> (f x))"
-    by blast 
+    by blast
   have g_closed: "g \<in> (carrier (Fun\<^bsub>n\<^esub> Q\<^sub>p))"
-    by(rule Qp_funs_car_memI, unfold  g_def, auto,  
-            intro field_inv(3) assms Qp.function_ring_car_memE[of _ n], auto ) 
+    by(rule Qp_funs_car_memI, unfold  g_def, auto,
+            intro field_inv(3) assms Qp.function_ring_car_memE[of _ n], auto )
   have "\<And>x. x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<Longrightarrow> f x \<otimes> g x = \<one>"
-    using assms g_def 
-    by (metis (no_types, lifting) Qp.function_ring_car_memE field_inv(2) restrict_apply)   
+    using assms g_def
+    by (metis (no_types, lifting) Qp.function_ring_car_memE field_inv(2) restrict_apply)
   then have 0: "f \<otimes>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> g = \<one>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub>"
-    using assms g_def Qp_funs_mult''[of f n g] Qp_funs_one[of n] g_closed 
+    using assms g_def Qp_funs_mult''[of f n g] Qp_funs_one[of n] g_closed
     by (metis (no_types, lifting) restrict_ext)
   then show "f \<in> (Units (Fun\<^bsub>n\<^esub> Q\<^sub>p))"
-    using comm_monoid.UnitsI[of "Fun\<^bsub>n\<^esub> Q\<^sub>p"] assms(1) g_closed local.F.comm_monoid_axioms by presburger  
+    using comm_monoid.UnitsI[of "Fun\<^bsub>n\<^esub> Q\<^sub>p"] assms(1) g_closed local.F.comm_monoid_axioms by presburger
   have "inv\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> f = g"
-    using g_def g_closed 0 cring.invI[of "Fun\<^bsub>n\<^esub> Q\<^sub>p"] Qp_funs_is_cring assms(1) 
+    using g_def g_closed 0 cring.invI[of "Fun\<^bsub>n\<^esub> Q\<^sub>p"] Qp_funs_is_cring assms(1)
     by presburger
   show "inv\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> f = (\<lambda> x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). inv\<^bsub>Q\<^sub>p\<^esub> (f x))"
-    using assms g_def 0  \<open>inv\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> f = g\<close> 
+    using assms g_def 0  \<open>inv\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> f = g\<close>
     by blast
 qed
 
 lemma Qp_funs_Units_memE:
   assumes "f \<in> (Units (Fun\<^bsub>n\<^esub> Q\<^sub>p))"
   shows "f \<otimes>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> inv\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> f = \<one>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub>"
         "inv\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> f \<otimes>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> f = \<one>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub>"
         "\<And> x. x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<Longrightarrow>  f x \<noteq> \<zero>\<^bsub>Q\<^sub>p\<^esub>"
-  using monoid.Units_r_inv[of "Fun\<^bsub>n\<^esub> Q\<^sub>p" f ] assms  Qp_funs_is_monoid 
+  using monoid.Units_r_inv[of "Fun\<^bsub>n\<^esub> Q\<^sub>p" f ] assms  Qp_funs_is_monoid
     apply blast
-  using monoid.Units_l_inv[of "Fun\<^bsub>n\<^esub> Q\<^sub>p" f ] assms  Qp_funs_is_monoid 
+  using monoid.Units_l_inv[of "Fun\<^bsub>n\<^esub> Q\<^sub>p" f ] assms  Qp_funs_is_monoid
     apply blast
 proof-
   obtain g where g_def: "g = inv\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> f"
-    by blast 
+    by blast
   show "\<And> x. x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<Longrightarrow>  f x \<noteq> \<zero>\<^bsub>Q\<^sub>p\<^esub>"
   proof- fix x assume A: "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
     have "f \<otimes>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> g = \<one>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub>"
-      using assms g_def Qp_funs_is_monoid 
-        \<open>\<lbrakk>Group.monoid (Fun\<^bsub>n\<^esub> Q\<^sub>p); f \<in> Units (Fun\<^bsub>n\<^esub> Q\<^sub>p)\<rbrakk> \<Longrightarrow> f \<otimes>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> inv\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> f = \<one>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub>\<close> 
+      using assms g_def Qp_funs_is_monoid
+        \<open>\<lbrakk>Group.monoid (Fun\<^bsub>n\<^esub> Q\<^sub>p); f \<in> Units (Fun\<^bsub>n\<^esub> Q\<^sub>p)\<rbrakk> \<Longrightarrow> f \<otimes>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> inv\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> f = \<one>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub>\<close>
       by blast
     then have 0: "f x \<otimes> g x = \<one>"
-      using A assms g_def Qp_funs_mult'[of x n f g] Qp_funs_one[of n] 
+      using A assms g_def Qp_funs_mult'[of x n f g] Qp_funs_one[of n]
       by (metis Qp_funs_is_monoid monoid.Units_closed monoid.Units_inv_closed restrict_apply)
     have 1: "g x \<in> carrier Q\<^sub>p"
-      using g_def A assms local.function_ring_car_closed by auto      
+      using g_def A assms local.function_ring_car_closed by auto
     then show "f x \<noteq> \<zero>\<^bsub>Q\<^sub>p\<^esub>"
-      using 0 
-      by (metis Qp.l_null local.one_neq_zero)      
+      using 0
+      by (metis Qp.l_null local.one_neq_zero)
   qed
 qed
-  
+
 lemma Qp_funs_m_inv:
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "f \<in> (Units (Fun\<^bsub>n\<^esub> Q\<^sub>p))"
   shows "(inv\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> f) x = inv\<^bsub>Q\<^sub>p\<^esub> (f x)"
-  using Qp_funs_Units_memI(2)  Qp_funs_Units_memE(3) assms 
+  using Qp_funs_Units_memI(2)  Qp_funs_Units_memE(3) assms
   by (metis (no_types, lifting) Qp_funs_is_monoid monoid.Units_closed restrict_apply)
 
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsection\<open>Defining the Rings of Semialgebraic Functions\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 
 definition semialg_functions  where
 "semialg_functions n  = {f \<in> (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) \<rightarrow> carrier Q\<^sub>p. is_semialg_function n f \<and> f = restrict f (carrier (Q\<^sub>p\<^bsup>n\<^esup>))}"
 
-lemma semialg_functions_memE: 
+lemma semialg_functions_memE:
   assumes "f \<in> semialg_functions n"
   shows "is_semialg_function n f"
         "f \<in> carrier (Fun\<^bsub>n\<^esub> Q\<^sub>p)"
         "f \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<rightarrow> carrier Q\<^sub>p"
-  using semialg_functions_def assms apply blast 
+  using semialg_functions_def assms apply blast
   apply(rule Qp_funs_car_memI')
-  using assms 
+  using assms
   apply (metis (no_types, lifting) mem_Collect_eq semialg_functions_def)
-  using assms unfolding semialg_functions_def 
+  using assms unfolding semialg_functions_def
    apply (metis (mono_tags, lifting) mem_Collect_eq)
   by (metis (no_types, lifting) assms mem_Collect_eq semialg_functions_def)
 
 lemma semialg_functions_in_Qp_funs:
 "semialg_functions n \<subseteq> carrier (Fun\<^bsub>n\<^esub> Q\<^sub>p)"
-  using semialg_functions_memE 
+  using semialg_functions_memE
   by blast
 
 lemma semialg_functions_memI:
   assumes "f \<in> carrier (Fun\<^bsub>n\<^esub> Q\<^sub>p)"
   assumes "is_semialg_function n f"
   shows "f \<in> semialg_functions n"
-  using assms unfolding semialg_functions_def 
-  by (metis (mono_tags, lifting) Qp_funs_car_memI function_ring_car_eqI 
+  using assms unfolding semialg_functions_def
+  by (metis (mono_tags, lifting) Qp_funs_car_memI function_ring_car_eqI
       is_semialg_function_closed mem_Collect_eq restrict_Pi_cancel restrict_apply)
 
 lemma restrict_is_semialg:
   assumes "is_semialg_function n f"
   shows "is_semialg_function n (restrict f (carrier (Q\<^sub>p\<^bsup>n\<^esup>)))"
 proof(rule is_semialg_functionI)
   show 0: "restrict f (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<rightarrow> carrier Q\<^sub>p"
     using assms is_semialg_function_closed by blast
   show "\<And>k S. S \<in> semialg_sets (1 + k) \<Longrightarrow> is_semialgebraic (n + k) (partial_pullback n (restrict f (carrier (Q\<^sub>p\<^bsup>n\<^esup>))) k S)"
   proof- fix k S assume A: "S \<in> semialg_sets (1 + k)"
     have "(partial_pullback n (restrict f (carrier (Q\<^sub>p\<^bsup>n\<^esup>))) k S) = partial_pullback n f k S"
       apply(intro equalityI'  partial_pullback_memI, meson partial_pullback_memE)
       unfolding partial_pullback_def partial_image_def evimage_eq restrict_def
         apply (metis (mono_tags, lifting) le_add1 local.take_closed)
-       apply blast 
+       apply blast
       by (metis (mono_tags, lifting) le_add1 local.take_closed)
     then show " is_semialgebraic (n + k) (partial_pullback n (restrict f (carrier (Q\<^sub>p\<^bsup>n\<^esup>))) k S)"
       using assms A is_semialg_functionE is_semialgebraicI
       by presburger
   qed
 qed
 
 lemma restrict_in_semialg_functions:
   assumes "is_semialg_function n f"
   shows "(restrict f (carrier (Q\<^sub>p\<^bsup>n\<^esup>))) \<in> semialg_functions n"
-  using assms restrict_is_semialg 
-  unfolding semialg_functions_def  
+  using assms restrict_is_semialg
+  unfolding semialg_functions_def
   by (metis (mono_tags, lifting) is_semialg_function_closed mem_Collect_eq restrict_apply' restrict_ext)
 
 lemma constant_function_is_semialg:
   assumes "a \<in> carrier Q\<^sub>p"
   shows "is_semialg_function n (constant_function (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) a)"
 proof-
   have "(constant_function (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) a) = restrict (Qp_ev (Qp.indexed_const a)) (carrier (Q\<^sub>p\<^bsup>n\<^esup>))"
     apply(rule ext)
-    unfolding constant_function_def 
-    using eval_at_point_const assms by simp   
+    unfolding constant_function_def
+    using eval_at_point_const assms by simp
   then show ?thesis using restrict_in_semialg_functions poly_is_semialg[of  "Qp.indexed_const a"]
-    using assms(1)  Qp_to_IP_car restrict_is_semialg by presburger    
+    using assms(1)  Qp_to_IP_car restrict_is_semialg by presburger
 qed
 
 lemma constant_function_in_semialg_functions:
   assumes "a \<in> carrier Q\<^sub>p"
   shows "Qp_const n a \<in> semialg_functions n"
   apply(unfold semialg_functions_def constant_function_def mem_Collect_eq, intro conjI, auto simp: assms)
-  using assms constant_function_is_semialg[of a n] unfolding constant_function_def by auto 
+  using assms constant_function_is_semialg[of a n] unfolding constant_function_def by auto
 
 lemma function_one_as_constant:
 "\<one>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> = Qp_const n \<one>"
-  unfolding constant_function_def  function_ring_def[of "carrier (Q\<^sub>p\<^bsup>n\<^esup>)" Q\<^sub>p] function_one_def 
-  by simp 
+  unfolding constant_function_def  function_ring_def[of "carrier (Q\<^sub>p\<^bsup>n\<^esup>)" Q\<^sub>p] function_one_def
+  by simp
 
 lemma function_zero_as_constant:
 "\<zero>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> = Qp_const n \<zero>\<^bsub>Q\<^sub>p\<^esub>"
-  unfolding constant_function_def function_ring_def[of "carrier (Q\<^sub>p\<^bsup>n\<^esup>)" Q\<^sub>p] function_zero_def 
-  by simp 
+  unfolding constant_function_def function_ring_def[of "carrier (Q\<^sub>p\<^bsup>n\<^esup>)" Q\<^sub>p] function_zero_def
+  by simp
 
 lemma sum_in_semialg_functions:
   assumes "f \<in> semialg_functions n"
   assumes "g \<in> semialg_functions n"
   shows "f \<oplus>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> g \<in> semialg_functions n"
 proof-
   have 0:"f \<oplus>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> g = restrict (function_tuple_comp Q\<^sub>p [f,g] Qp_add_fun) (carrier (Q\<^sub>p\<^bsup>n\<^esup>))"
   proof fix x
     show "(f \<oplus>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> g) x = restrict (function_tuple_comp Q\<^sub>p [f, g] Qp_add_fun) (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) x"
     proof(cases "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)")
       case True
       have " restrict (function_tuple_comp Q\<^sub>p [f, g] Qp_add_fun) (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) x = Qp_add_fun [f x, g x]"
         unfolding function_tuple_comp_def function_tuple_eval_def restrict_def
-        using comp_apply[of "Qp_add_fun" "(\<lambda>x. map (\<lambda>f. f x) [f, g])" x]   
-        by (metis (no_types, lifting) True list.simps(8) list.simps(9))       
+        using comp_apply[of "Qp_add_fun" "(\<lambda>x. map (\<lambda>f. f x) [f, g])" x]
+        by (metis (no_types, lifting) True list.simps(8) list.simps(9))
       then have "restrict (function_tuple_comp Q\<^sub>p [f, g] Qp_add_fun) (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) x = f x \<oplus>\<^bsub>Q\<^sub>p\<^esub> g x"
-        unfolding Qp_add_fun_def 
+        unfolding Qp_add_fun_def
         by (metis One_nat_def nth_Cons_0 nth_Cons_Suc)
-      then show ?thesis  using True function_ring_def[of "carrier (Q\<^sub>p\<^bsup>n\<^esup>)" Q\<^sub>p] 
-        unfolding function_add_def 
+      then show ?thesis  using True function_ring_def[of "carrier (Q\<^sub>p\<^bsup>n\<^esup>)" Q\<^sub>p]
+        unfolding function_add_def
         by (metis (no_types, lifting) Qp_funs_add assms(1) assms(2) mem_Collect_eq semialg_functions_def)
     next
       case False
       have "(f \<oplus>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> g) x = undefined"
-        using function_ring_def[of "carrier (Q\<^sub>p\<^bsup>n\<^esup>)" Q\<^sub>p] unfolding function_add_def 
+        using function_ring_def[of "carrier (Q\<^sub>p\<^bsup>n\<^esup>)" Q\<^sub>p] unfolding function_add_def
         by (metis (mono_tags, lifting) False restrict_apply ring_record_simps(12))
-      then show ?thesis 
+      then show ?thesis
         by (metis False restrict_def)
     qed
   qed
   have 1: "is_semialg_function_tuple n [f, g]"
-    using assms is_semialg_function_tupleI[of "[f, g]" n] semialg_functions_memE 
+    using assms is_semialg_function_tupleI[of "[f, g]" n] semialg_functions_memE
     by (metis list.distinct(1) list.set_cases set_ConsD)
   have 2: "is_semialg_function n (function_tuple_comp Q\<^sub>p [f,g] Qp_add_fun)"
     apply(rule semialg_function_tuple_comp[of _ _  2])
         apply (simp add: "1")
        apply simp
     by (simp add: addition_is_semialg)
   show ?thesis
     apply(rule semialg_functions_memI)
   apply (meson Qp_funs_is_cring assms(1) assms(2) cring.cring_simprules(1) semialg_functions_memE(2))
     using 0 2 restrict_is_semialg by presburger
 qed
 
 lemma prod_in_semialg_functions:
   assumes "f \<in> semialg_functions n"
   assumes "g \<in> semialg_functions n"
   shows "f \<otimes>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> g \<in> semialg_functions n"
 proof-
   have 0:"f \<otimes>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> g = restrict (function_tuple_comp Q\<^sub>p [f,g] Qp_mult_fun) (carrier (Q\<^sub>p\<^bsup>n\<^esup>))"
   proof fix x
     show "(f \<otimes>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> g) x = restrict (function_tuple_comp Q\<^sub>p [f, g] Qp_mult_fun) (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) x"
     proof(cases "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)")
       case True
       have " restrict (function_tuple_comp Q\<^sub>p [f, g] Qp_mult_fun) (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) x = Qp_mult_fun [f x, g x]"
         unfolding function_tuple_comp_def function_tuple_eval_def restrict_def
-        using comp_apply[of "Qp_mult_fun" "(\<lambda>x. map (\<lambda>f. f x) [f, g])" x] 
-        by (metis (no_types, lifting) True list.simps(8) list.simps(9))        
+        using comp_apply[of "Qp_mult_fun" "(\<lambda>x. map (\<lambda>f. f x) [f, g])" x]
+        by (metis (no_types, lifting) True list.simps(8) list.simps(9))
       then have "restrict (function_tuple_comp Q\<^sub>p [f, g] Qp_mult_fun) (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) x = f x \<otimes> g x"
-        unfolding Qp_mult_fun_def 
+        unfolding Qp_mult_fun_def
         by (metis One_nat_def nth_Cons_0 nth_Cons_Suc)
-      then show ?thesis  using True function_ring_def[of "carrier (Q\<^sub>p\<^bsup>n\<^esup>)" Q\<^sub>p] 
-        unfolding function_mult_def 
+      then show ?thesis  using True function_ring_def[of "carrier (Q\<^sub>p\<^bsup>n\<^esup>)" Q\<^sub>p]
+        unfolding function_mult_def
         by (metis (no_types, lifting) Qp_funs_mult  assms(1) assms(2) mem_Collect_eq semialg_functions_def)
     next
       case False
       have "(f \<otimes>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> g) x = undefined"
-        using function_ring_def[of "carrier (Q\<^sub>p\<^bsup>n\<^esup>)" Q\<^sub>p] unfolding function_mult_def 
+        using function_ring_def[of "carrier (Q\<^sub>p\<^bsup>n\<^esup>)" Q\<^sub>p] unfolding function_mult_def
         by (metis (mono_tags, lifting) False restrict_apply ring_record_simps(5))
-      then show ?thesis 
+      then show ?thesis
         by (metis False restrict_def)
     qed
   qed
   have 1: "is_semialg_function_tuple n [f, g]"
-    using assms is_semialg_function_tupleI[of "[f, g]" n] semialg_functions_memE 
+    using assms is_semialg_function_tupleI[of "[f, g]" n] semialg_functions_memE
     by (metis list.distinct(1) list.set_cases set_ConsD)
   have 2: "is_semialg_function n (function_tuple_comp Q\<^sub>p [f,g] Qp_mult_fun)"
     apply(rule semialg_function_tuple_comp[of _ _  2])
         apply (simp add: "1")
        apply simp
     by (simp add: multiplication_is_semialg)
   show ?thesis
     apply(rule semialg_functions_memI)
     apply (meson Qp_funs_is_cring assms(1) assms(2) cring.cring_simprules(5) semialg_functions_memE(2))
     using 0 2 restrict_is_semialg by presburger
 qed
-    
+
 lemma inv_in_semialg_functions:
   assumes "f \<in> semialg_functions n"
   assumes "\<And> x. x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<Longrightarrow> f x \<noteq> \<zero>\<^bsub>Q\<^sub>p\<^esub>"
   shows "inv\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> f \<in> semialg_functions n "
 proof-
   have 0: "inv\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> f = restrict (function_tuple_comp Q\<^sub>p [f] Qp_invert) (carrier (Q\<^sub>p\<^bsup>n\<^esup>))"
-  proof fix x 
+  proof fix x
     show "(inv\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> f) x = restrict (function_tuple_comp Q\<^sub>p [f] Qp_invert) (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) x"
     proof(cases "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)")
       case True
       have "(function_tuple_comp Q\<^sub>p [f] Qp_invert) x = Qp_invert [f x]"
-        unfolding function_tuple_comp_def function_tuple_eval_def 
+        unfolding function_tuple_comp_def function_tuple_eval_def
         using comp_apply  by (metis (no_types, lifting) list.simps(8) list.simps(9))
       then have "(function_tuple_comp Q\<^sub>p [f] Qp_invert) x = inv\<^bsub>Q\<^sub>p\<^esub> (f x)"
-        unfolding Qp_invert_def 
-        using True assms(2) Qp.to_R_to_R1 by presburger        
-      then show ?thesis 
-        using True restrict_apply 
-        by (metis (mono_tags, opaque_lifting) Qp_funs_Units_memI(1) 
-            Qp_funs_m_inv assms(1) assms(2) semialg_functions_memE(2))      
+        unfolding Qp_invert_def
+        using True assms(2) Qp.to_R_to_R1 by presburger
+      then show ?thesis
+        using True restrict_apply
+        by (metis (mono_tags, opaque_lifting) Qp_funs_Units_memI(1)
+            Qp_funs_m_inv assms(1) assms(2) semialg_functions_memE(2))
     next
       case False
       have "inv\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> f \<in> carrier (Fun\<^bsub>n\<^esub> Q\<^sub>p)"
-        using assms 
+        using assms
         by (meson Qp_funs_Units_memI(1) Qp_funs_is_monoid monoid.Units_inv_closed semialg_functions_memE(2))
       then show ?thesis using False restrict_apply function_ring_not_car
-        by auto 
+        by auto
     qed
   qed
   have "is_semialg_function n (function_tuple_comp Q\<^sub>p [f] Qp_invert)"
     apply(rule semialg_function_tuple_comp[of _ _  1])
         apply (simp add: assms(1) is_semialg_function_tuple_def semialg_functions_memE(1))
-        apply simp 
+        apply simp
         using Qp_invert_is_semialg by blast
-  then show ?thesis 
-    using "0" restrict_in_semialg_functions 
+  then show ?thesis
+    using "0" restrict_in_semialg_functions
     by presburger
 qed
 
 lemma a_inv_in_semialg_functions:
   assumes "f \<in> semialg_functions n"
   shows "\<ominus>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> f \<in> semialg_functions n"
 proof-
   have "\<ominus>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> f = (Qp_const n (\<ominus> \<one>)) \<otimes>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> f"
-  proof fix x 
+  proof fix x
     show "(\<ominus>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> f) x = (Qp_const n (\<ominus> \<one>) \<otimes>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> f) x"
     proof(cases "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)")
       case True
       have 0: "(\<ominus>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> f) x = \<ominus> (f x)"
-        using Qp_funs_a_inv semialg_functions_memE True assms(1) by blast 
+        using Qp_funs_a_inv semialg_functions_memE True assms(1) by blast
       have 1: "(Qp_const n (\<ominus> \<one>) \<otimes>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> f) x = (\<ominus> \<one>)\<otimes>(f x)"
-        using Qp_funs_mult[of x n "Qp_const n (\<ominus> \<one>)" f] assms Qp_constE[of "\<ominus> \<one>" x n] 
+        using Qp_funs_mult[of x n "Qp_const n (\<ominus> \<one>)" f] assms Qp_constE[of "\<ominus> \<one>" x n]
           Qp_funs_mult'  Qp.add.inv_closed Qp.one_closed Qp_funs_mult''' True by presburger
       have 2: "f x \<in> carrier Q\<^sub>p"
         using True semialg_functions_memE[of f n]  assms by blast
-      show ?thesis 
-        using True assms 0 1 2 Qp.l_minus Qp.l_one Qp.one_closed by presburger        
+      show ?thesis
+        using True assms 0 1 2 Qp.l_minus Qp.l_one Qp.one_closed by presburger
     next
       case False
       have "(\<ominus>\<^bsub>function_ring (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) Q\<^sub>p\<^esub> f) x = undefined"
-        using  Qp_funs_a_inv'[of f n] False assms semialg_functions_memE  
+        using  Qp_funs_a_inv'[of f n] False assms semialg_functions_memE
         by (metis (no_types, lifting) restrict_apply)
-      then show ?thesis 
+      then show ?thesis
         using False function_ring_defs(2)[of n] Qp_funs_a_inv'[of f n]
         unfolding function_mult_def restrict_def
         by presburger
     qed
   qed
-  then show ?thesis 
+  then show ?thesis
     using prod_in_semialg_functions[of "Qp_const n (\<ominus> \<one>)" n f] assms
-      constant_function_in_semialg_functions[of "\<ominus> \<one>" n]  Qp.add.inv_closed Qp.one_closed 
-    by presburger     
-qed
-    
+      constant_function_in_semialg_functions[of "\<ominus> \<one>" n]  Qp.add.inv_closed Qp.one_closed
+    by presburger
+qed
+
 lemma semialg_functions_subring:
   shows "subring (semialg_functions n) (Fun\<^bsub>n\<^esub> Q\<^sub>p)"
   apply(rule ring.subringI)
   using Qp_funs_is_cring cring.axioms(1) apply blast
   apply (simp add: semialg_functions_in_Qp_funs)
   using Qp.one_closed  constant_function_in_semialg_functions function_one_as_constant apply presburger
   using a_inv_in_semialg_functions  apply blast
   using  prod_in_semialg_functions apply blast
   using  sum_in_semialg_functions by blast
 
 lemma semialg_functions_subcring:
   shows "subcring (semialg_functions n) (Fun\<^bsub>n\<^esub> Q\<^sub>p)"
   using semialg_functions_subring cring.subcringI'
   using Qp_funs_is_cring by blast
 
 definition SA where
 "SA n = (Fun\<^bsub>n\<^esub> Q\<^sub>p)\<lparr> carrier := semialg_functions n\<rparr>"
 
 lemma SA_is_ring:
   shows "ring (SA n)"
 proof-
   have "ring (Fun\<^bsub>n\<^esub> Q\<^sub>p)"
     by (simp add: Qp_funs_is_cring cring.axioms(1))
-  then show ?thesis 
+  then show ?thesis
     unfolding SA_def
   using  ring.subring_is_ring[of "Fun\<^bsub>n\<^esub> Q\<^sub>p" "semialg_functions n"] semialg_functions_subring[of n]
-  by blast 
+  by blast
 qed
 
 lemma SA_is_cring:
   shows "cring (SA n)"
   using ring.subcring_iff[of "Fun\<^bsub>n\<^esub> Q\<^sub>p" "semialg_functions n"] semialg_functions_subcring[of n]
-         Qp_funs_is_cring cring.axioms(1) semialg_functions_in_Qp_funs 
+         Qp_funs_is_cring cring.axioms(1) semialg_functions_in_Qp_funs
     unfolding SA_def
   by blast
 
 lemma SA_is_monoid:
   shows "monoid (SA n)"
-  using SA_is_ring[of n]  unfolding ring_def 
+  using SA_is_ring[of n]  unfolding ring_def
   by linarith
 
 lemma SA_is_abelian_monoid:
   shows "abelian_monoid (SA n)"
-  using SA_is_ring[of n]  unfolding ring_def abelian_group_def by blast 
-
-lemma SA_car: 
+  using SA_is_ring[of n]  unfolding ring_def abelian_group_def by blast
+
+lemma SA_car:
 "carrier (SA n) = semialg_functions n"
     unfolding SA_def
-  by simp 
-
-lemma SA_car_in_Qp_funs_car: 
+  by simp
+
+lemma SA_car_in_Qp_funs_car:
 "carrier (SA n) \<subseteq> carrier (Fun\<^bsub>n\<^esub> Q\<^sub>p)"
   by (simp add: SA_car semialg_functions_in_Qp_funs)
 
 lemma SA_car_memI:
   assumes "f \<in> carrier (Fun\<^bsub>n\<^esub> Q\<^sub>p)"
   assumes "is_semialg_function n f"
   shows "f \<in> carrier (SA n)"
-  using assms semialg_functions_memI[of f n] SA_car 
+  using assms semialg_functions_memI[of f n] SA_car
   by blast
 
 lemma SA_car_memE:
   assumes "f \<in> carrier (SA n)"
   shows "is_semialg_function n f"
         "f \<in> carrier (Fun\<^bsub>n\<^esub> Q\<^sub>p)"
         "f \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<rightarrow> carrier Q\<^sub>p"
   using SA_car assms semialg_functions_memE(1) apply blast
   using SA_car assms semialg_functions_memE(2) apply blast
   using SA_car assms semialg_functions_memE(3) by blast
 
-lemma SA_plus: 
+lemma SA_plus:
 "(\<oplus>\<^bsub>SA n\<^esub>) = (\<oplus>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub>)"
   unfolding SA_def
-  by simp 
-
-lemma SA_times: 
+  by simp
+
+lemma SA_times:
 "(\<otimes>\<^bsub>SA n\<^esub>) = (\<otimes>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub>)"
   unfolding SA_def
-  by simp 
-
-lemma SA_one: 
+  by simp
+
+lemma SA_one:
 "(\<one>\<^bsub>SA n\<^esub>) = (\<one>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub>)"
   unfolding SA_def
-  by simp 
-
-lemma SA_zero: 
+  by simp
+
+lemma SA_zero:
 "(\<zero>\<^bsub>SA n\<^esub>) = (\<zero>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub>)"
   unfolding SA_def
-  by simp 
+  by simp
 
 lemma SA_zero_is_function_ring:
 "(Fun\<^bsub>0\<^esub> Q\<^sub>p) = SA 0"
 proof-
   have 0: "carrier (Fun\<^bsub>0\<^esub> Q\<^sub>p) = carrier (SA 0)"
   proof
     show "carrier (function_ring (carrier (Q\<^sub>p\<^bsup>0\<^esup>)) Q\<^sub>p) \<subseteq> carrier (SA 0)"
     proof fix f assume A0: "f \<in> carrier (function_ring (carrier (Q\<^sub>p\<^bsup>0\<^esup>)) Q\<^sub>p)"
       show "f \<in> carrier (SA 0)"
       proof(rule SA_car_memI)
         show "f \<in> carrier (function_ring (carrier (Q\<^sub>p\<^bsup>0\<^esup>)) Q\<^sub>p)"
-          using A0 by blast 
+          using A0 by blast
         show "is_semialg_function 0 f"
         proof(rule is_semialg_functionI)
           show "f \<in> carrier (Q\<^sub>p\<^bsup>0\<^esup>) \<rightarrow> carrier Q\<^sub>p"
-            using A0 Qp.function_ring_car_memE by blast 
+            using A0 Qp.function_ring_car_memE by blast
           show "\<And>k S. S \<in> semialg_sets (1 + k) \<Longrightarrow> is_semialgebraic (0 + k) (partial_pullback 0 f k S)"
           proof- fix k S assume A: "S \<in> semialg_sets (1+k)"
             obtain a where a_def: "a = f []"
-              by blast 
+              by blast
             have 0: "carrier (Q\<^sub>p\<^bsup>0\<^esup>) = {[]}"
               using Qp_zero_carrier by blast
             have 1: "(partial_pullback 0 f k S) = {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). a#x \<in> S}"
             proof
               show "partial_pullback 0 f k S \<subseteq> {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). a # x \<in> S}"
                 apply(rule subsetI)
-              unfolding partial_pullback_def partial_image_def using a_def 
+              unfolding partial_pullback_def partial_image_def using a_def
               by (metis (no_types, lifting) add.left_neutral drop0 evimage_eq mem_Collect_eq take_eq_Nil)
               show "{x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). a # x \<in> S} \<subseteq> partial_pullback 0 f k S"
                 apply(rule subsetI)
-                unfolding partial_pullback_def partial_image_def  a_def 
+                unfolding partial_pullback_def partial_image_def  a_def
                 by (metis (no_types, lifting) add.left_neutral drop0 evimageI2 mem_Collect_eq take0)
             qed
             have 2: "cartesian_product {[a]} (partial_pullback 0 f k S) = (cartesian_product {[a]} (carrier (Q\<^sub>p\<^bsup>k\<^esup>))) \<inter> S"
             proof
               show "cartesian_product {[a]} (partial_pullback 0 f k S) \<subseteq> cartesian_product {[a]} (carrier (Q\<^sub>p\<^bsup>k\<^esup>)) \<inter> S"
               proof(rule subsetI) fix x assume A: "x \<in> cartesian_product {[a]} (partial_pullback 0 f k S)"
                 then obtain y where y_def: "y \<in> (partial_pullback 0 f k S) \<and>x = a#y"
-                  using cartesian_product_memE' 
+                  using cartesian_product_memE'
                   by (metis (no_types, lifting) Cons_eq_appendI self_append_conv2 singletonD)
                 hence 20: "x \<in> S"
-                  unfolding 1 by blast  
+                  unfolding 1 by blast
                 have 21: "x = [a]@y"
                   using y_def by (simp add: y_def)
                 have "x \<in> cartesian_product {[a]} (carrier (Q\<^sub>p\<^bsup>k\<^esup>))"
                   unfolding 21 apply(rule cartesian_product_memI'[of _ Q\<^sub>p 1 _ k])
                   using a_def  apply (metis function_ring_car_closed Qp_zero_carrier \<open>f \<in> carrier (function_ring (carrier (Q\<^sub>p\<^bsup>0\<^esup>)) Q\<^sub>p)\<close> empty_subsetI insert_subset singletonI Qp.to_R1_closed)
-                    apply blast 
-                   apply blast 
-                  using y_def unfolding partial_pullback_def  evimage_def 
+                    apply blast
+                   apply blast
+                  using y_def unfolding partial_pullback_def  evimage_def
                   by (metis IntD2 add_cancel_right_left)
                 thus "x \<in> cartesian_product {[a]} (carrier (Q\<^sub>p\<^bsup>k\<^esup>)) \<inter> S"
-                  using 20 by blast 
+                  using 20 by blast
               qed
               show " cartesian_product {[a]} (carrier (Q\<^sub>p\<^bsup>k\<^esup>)) \<inter> S \<subseteq> cartesian_product {[a]} (partial_pullback 0 f k S)"
               proof fix x assume A: "x \<in> cartesian_product {[a]} (carrier (Q\<^sub>p\<^bsup>k\<^esup>)) \<inter> S"
                 then obtain y where y_def: "x = a#y \<and> y \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>)"
-                  using cartesian_product_memE'   
+                  using cartesian_product_memE'
                   by (metis (no_types, lifting) Cons_eq_appendI IntD1 append_Nil singletonD)
                 have 00: "y \<in> partial_pullback 0 f k S"
                   using y_def unfolding 1  using A by blast
                 have 01: "x = [a]@y"
                   using y_def by (simp add: y_def)
                 have 02: "partial_pullback 0 f k S \<subseteq> carrier (Q\<^sub>p\<^bsup>k\<^esup>)"
-                  unfolding partial_pullback_def 
-                  by (simp add: extensional_vimage_closed)                 
+                  unfolding partial_pullback_def
+                  by (simp add: extensional_vimage_closed)
                 show "x \<in> cartesian_product {[a]} (partial_pullback 0 f k S)"
-                  unfolding 01 apply(rule cartesian_product_memI'[of "{[a]}" Q\<^sub>p 1 "partial_pullback 0 f k S" k "[a]" y ])    
+                  unfolding 01 apply(rule cartesian_product_memI'[of "{[a]}" Q\<^sub>p 1 "partial_pullback 0 f k S" k "[a]" y ])
                   apply (metis function_ring_car_closed Qp_zero_carrier \<open>f \<in> carrier (function_ring (carrier (Q\<^sub>p\<^bsup>0\<^esup>)) Q\<^sub>p)\<close> a_def empty_subsetI insert_subset singletonI Qp.to_R1_closed)
                   using "02" apply blast
-                   apply blast 
-                  using 00 by blast 
+                   apply blast
+                  using 00 by blast
               qed
-            qed 
+            qed
             have 3:"is_semialgebraic 1 {[a]}"
             proof-
               have "a \<in> carrier Q\<^sub>p"
                 using a_def Qp.function_ring_car_memE 0  \<open>f \<in> carrier (function_ring (carrier (Q\<^sub>p\<^bsup>0\<^esup>)) Q\<^sub>p)\<close> by blast
               hence "[a] \<in> carrier (Q\<^sub>p\<^bsup>1\<^esup>)"
                 using Qp.to_R1_closed by blast
-              thus ?thesis  
+              thus ?thesis
                 using is_algebraic_imp_is_semialg singleton_is_algebraic by blast
             qed
             have  4: "is_semialgebraic (1+k)  (cartesian_product {[a]} (carrier (Q\<^sub>p\<^bsup>k\<^esup>)))"
               using 3 carrier_is_semialgebraic cartesian_product_is_semialgebraic less_one by blast
             have  5: "is_semialgebraic (1+k) (cartesian_product {[a]} (partial_pullback 0 f k S))"
               unfolding 2 using 3 4 A intersection_is_semialg padic_fields.is_semialgebraicI padic_fields_axioms by blast
             have 6: "{[a]} \<subseteq> carrier (Q\<^sub>p\<^bsup>1\<^esup>)"
               using a_def A0 0 by (metis Qp.function_ring_car_memE empty_subsetI insert_subset singletonI Qp.to_R1_closed)
             have  7: "is_semialgebraic (k+1) (cartesian_product (partial_pullback 0 f k S) {[a]})"
               apply(rule cartesian_product_swap)
                 using "6" apply blast
                apply (metis add_cancel_right_left partial_pullback_closed)
               using "5" by auto
             have 8: "is_semialgebraic k (partial_pullback 0 f k S)"
               apply(rule cartesian_product_singleton_factor_projection_is_semialg'[of _ _ "[a]" 1])
                   apply (metis add_cancel_right_left partial_pullback_closed)
                  apply (metis A0 Qp.function_ring_car_memE Qp_zero_carrier a_def singletonI Qp.to_R1_closed)
                 using "7" by blast
             thus "is_semialgebraic (0 + k) (partial_pullback 0 f k S)"
-                by simp 
+                by simp
           qed
         qed
       qed
     qed
     show "carrier (SA 0) \<subseteq> carrier (function_ring (carrier (Q\<^sub>p\<^bsup>0\<^esup>)) Q\<^sub>p)"
       using SA_car_in_Qp_funs_car by blast
   qed
   then have 1: "semialg_functions 0 = carrier (Fun\<^bsub>0\<^esub> Q\<^sub>p)"
-    unfolding 0 SA_def by auto 
-  show ?thesis unfolding SA_def 1 by auto 
-qed
-
-lemma constant_fun_closed: 
+    unfolding 0 SA_def by auto
+  show ?thesis unfolding SA_def 1 by auto
+qed
+
+lemma constant_fun_closed:
   assumes "c \<in> carrier Q\<^sub>p"
   shows "constant_function (carrier (Q\<^sub>p\<^bsup>m\<^esup>)) c \<in> carrier (SA m)"
   using constant_function_in_semialg_functions SA_car assms by blast
 
 lemma SA_0_car_memI:
   assumes "\<xi> \<in> carrier (Q\<^sub>p\<^bsup>0\<^esup>) \<rightarrow> carrier Q\<^sub>p"
   assumes "\<And>x. x \<notin> carrier (Q\<^sub>p\<^bsup>0\<^esup>) \<Longrightarrow> \<xi> x = undefined"
   shows "\<xi> \<in> carrier (SA 0)"
-proof- 
+proof-
   have 0: "carrier (Q\<^sub>p\<^bsup>0\<^esup>) = {[]}"
     by (simp add: Qp_zero_carrier)
   obtain c where c_def: "\<xi> [] = c"
-    by blast 
+    by blast
   have 1: "\<xi> = constant_function (carrier (Q\<^sub>p\<^bsup>0\<^esup>)) c"
-    unfolding constant_function_def restrict_def 
+    unfolding constant_function_def restrict_def
     using assms c_def  unfolding 0
     by (metis empty_iff insert_iff)
   have 2: "c \<in> carrier Q\<^sub>p"
-    using assms(1) c_def unfolding 0 by blast 
+    using assms(1) c_def unfolding 0 by blast
   show ?thesis unfolding 1
     using 2 constant_fun_closed by blast
 qed
 
 lemma car_SA_0_mem_imp_const:
   assumes "a \<in> carrier (SA 0)"
   shows "\<exists> c \<in> carrier Q\<^sub>p. a = Qp_const 0 c"
 proof-
   obtain c where c_def: "c =  a []"
-    by blast 
+    by blast
   have car_zero: "carrier (Q\<^sub>p\<^bsup>0\<^esup>) = {[]}"
     using Qp_zero_carrier by blast
   have 0: "a = constant_function (carrier (Q\<^sub>p\<^bsup>0\<^esup>)) c"
-  proof fix x 
+  proof fix x
     show " a x = constant_function (carrier (Q\<^sub>p\<^bsup>0\<^esup>)) c x"
       apply(cases "x \<in> carrier (Q\<^sub>p\<^bsup>0\<^esup>)")
       using assms SA_car_memE[of a 0] c_def
-      unfolding constant_function_def restrict_def  car_zero 
+      unfolding constant_function_def restrict_def  car_zero
        apply (metis empty_iff insert_iff)
-      using assms SA_car_memE(2)[of a 0] c_def 
+      using assms SA_car_memE(2)[of a 0] c_def
       unfolding constant_function_def restrict_def car_zero
       by (metis car_zero function_ring_not_car)
   qed
   have c_closed: "c \<in> carrier Q\<^sub>p"
     using assms  SA_car_memE(3)[of  a 0] unfolding c_def car_zero
-    by blast  
-  thus ?thesis using 0 by blast   
+    by blast
+  thus ?thesis using 0 by blast
 qed
 
 lemma SA_zeroE:
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "\<zero> \<^bsub>SA n\<^esub> a = \<zero>"
   using function_zero_eval SA_zero assms by presburger
 
 lemma SA_oneE:
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "\<one> \<^bsub>SA n\<^esub> a = \<one>"
-  using function_one_eval  SA_one assms by presburger  
-end 
+  using function_one_eval  SA_one assms by presburger
+end
 
 sublocale padic_fields < UPSA?: UP_cring "SA m" "UP (SA m)"
-  unfolding UP_cring_def using SA_is_cring[of m] by auto 
+  unfolding UP_cring_def using SA_is_cring[of m] by auto
 
 context padic_fields
 begin
 
-lemma SA_add: 
+lemma SA_add:
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "(f \<oplus>\<^bsub>SA n\<^esub> g) x = f x \<oplus>\<^bsub>Q\<^sub>p\<^esub> g x"
   using Qp_funs_add''' SA_plus assms by presburger
-  
-lemma SA_add': 
+
+lemma SA_add':
   assumes "x \<notin> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "(f \<oplus>\<^bsub>SA n\<^esub> g) x = undefined"
 proof-
   have "(f \<oplus>\<^bsub>SA n\<^esub> g) x = function_add (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) Q\<^sub>p f g x"
     using SA_plus[of n] unfolding function_ring_def
     by (metis ring_record_simps(12))
-  then show ?thesis 
-    unfolding function_add_def using restrict_apply assms  
+  then show ?thesis
+    unfolding function_add_def using restrict_apply assms
     by (metis (no_types, lifting))
 qed
 
 lemma SA_mult:
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "(f \<otimes>\<^bsub>SA n\<^esub> g) x = f x \<otimes> g x"
   using Qp_funs_mult''' SA_times assms by presburger
-  
-lemma SA_mult': 
+
+lemma SA_mult':
   assumes "x \<notin> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "(f \<otimes>\<^bsub>SA n\<^esub> g) x = undefined"
 proof-
   have "(f \<otimes>\<^bsub>SA n\<^esub> g) x = function_mult (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) Q\<^sub>p f g x"
     using SA_times[of n] unfolding function_ring_def
     by (metis ring_record_simps(5))
-  then show ?thesis 
-    unfolding function_mult_def using restrict_apply assms  
+  then show ?thesis
+    unfolding function_mult_def using restrict_apply assms
     by (metis (no_types, lifting))
 qed
 
 lemma SA_u_minus_eval:
   assumes "f \<in> carrier (SA n)"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
-  shows "(\<ominus>\<^bsub>SA n\<^esub> f) x = \<ominus> (f x)" 
+  shows "(\<ominus>\<^bsub>SA n\<^esub> f) x = \<ominus> (f x)"
 proof-
   have  "f \<oplus>\<^bsub>SA n\<^esub> (\<ominus>\<^bsub>SA n\<^esub> f) = \<zero> \<^bsub>SA n\<^esub>"
-    using assms SA_is_cring cring.cring_simprules(17) by metis 
+    using assms SA_is_cring cring.cring_simprules(17) by metis
   have  "(f \<oplus>\<^bsub>SA n\<^esub> (\<ominus>\<^bsub>SA n\<^esub> f)) x = \<zero> \<^bsub>SA n\<^esub> x"
     using assms \<open>f \<oplus>\<^bsub>SA n\<^esub> \<ominus>\<^bsub>SA n\<^esub> f = \<zero>\<^bsub>SA n\<^esub>\<close> by presburger
   then have "(f x) \<oplus> (\<ominus>\<^bsub>SA n\<^esub> f) x = \<zero>"
-    using assms function_zero_eval SA_add unfolding   SA_zero by blast  
-  then show ?thesis 
-    using assms SA_is_ring 
+    using assms function_zero_eval SA_add unfolding   SA_zero by blast
+  then show ?thesis
+    using assms SA_is_ring
     by (meson Qp.add.inv_closed Qp.add.inv_comm Qp.function_ring_car_memE Qp.minus_unique Qp.r_neg SA_car_memE(2) ring.ring_simprules(3))
 qed
 
 lemma SA_a_inv_eval:
   assumes "f \<in> carrier (SA n)"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
-  shows "(\<ominus>\<^bsub>SA n\<^esub> f) x = \<ominus> (f x)" 
+  shows "(\<ominus>\<^bsub>SA n\<^esub> f) x = \<ominus> (f x)"
 proof-
   have  "f \<oplus>\<^bsub>SA n\<^esub> (\<ominus>\<^bsub>SA n\<^esub> f) = \<zero> \<^bsub>SA n\<^esub>"
-    using assms SA_is_cring cring.cring_simprules(17) by metis   
+    using assms SA_is_cring cring.cring_simprules(17) by metis
   have  "(f \<oplus>\<^bsub>SA n\<^esub> (\<ominus>\<^bsub>SA n\<^esub> f)) x = \<zero> \<^bsub>SA n\<^esub> x"
     using assms \<open>f \<oplus>\<^bsub>SA n\<^esub> \<ominus>\<^bsub>SA n\<^esub> f = \<zero>\<^bsub>SA n\<^esub>\<close> by presburger
   then have "(f x) \<oplus> (\<ominus>\<^bsub>SA n\<^esub> f) x = \<zero>"
     by (metis function_zero_eval SA_add SA_zero assms)
-  then show ?thesis 
+  then show ?thesis
     by (metis (no_types, lifting) PiE Q\<^sub>p_def Qp.add.m_comm Qp.minus_equality SA_is_cring Zp_def assms(1) assms(2) cring.cring_simprules(3) padic_fields.SA_car_memE(3) padic_fields_axioms)
 qed
 
 lemma SA_nat_pow:
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "(f [^]\<^bsub>SA n\<^esub> (k::nat)) x = (f x) [^]\<^bsub>Q\<^sub>p\<^esub> k"
   apply(induction k)
-  using assms  nat_pow_def   
+  using assms  nat_pow_def
   apply (metis function_one_eval SA_one old.nat.simps(6))
-  using assms SA_mult 
+  using assms SA_mult
   by (metis Group.nat_pow_Suc)
 
 lemma SA_nat_pow':
   assumes "x \<notin> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "(f [^]\<^bsub>SA n\<^esub> (k::nat)) x = undefined"
   apply(induction k)
-  using assms nat_pow_def[of "SA n" f] 
+  using assms nat_pow_def[of "SA n" f]
   apply (metis (no_types, lifting) Group.nat_pow_0 Qp_funs_one SA_one restrict_apply)
   by (metis Group.nat_pow_Suc SA_mult' assms)
-  
-lemma SA_add_closed_id: 
+
+lemma SA_add_closed_id:
   assumes "is_semialg_function n f"
   assumes "is_semialg_function n g"
   shows "restrict f (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) \<oplus>\<^bsub>SA n\<^esub> restrict g (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) = f \<oplus>\<^bsub>SA n\<^esub> g "
-proof fix x 
+proof fix x
   show "(restrict f (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) \<oplus>\<^bsub>SA n\<^esub> restrict g (carrier (Q\<^sub>p\<^bsup>n\<^esup>))) x = (f \<oplus>\<^bsub>SA n\<^esub> g) x"
     apply(cases "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)")
-    using assms restrict_apply  
+    using assms restrict_apply
     apply (metis SA_add)
-    using assms  
+    using assms
     by (metis SA_add')
 qed
 
-lemma SA_mult_closed_id: 
+lemma SA_mult_closed_id:
   assumes "is_semialg_function n f"
   assumes "is_semialg_function n g"
   shows "restrict f (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) \<otimes>\<^bsub>SA n\<^esub> restrict g (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) = f \<otimes>\<^bsub>SA n\<^esub> g "
-proof fix x 
+proof fix x
   show "(restrict f (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) \<otimes>\<^bsub>SA n\<^esub> restrict g (carrier (Q\<^sub>p\<^bsup>n\<^esup>))) x = (f \<otimes>\<^bsub>SA n\<^esub> g) x"
     apply(cases "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)")
-    using assms restrict_apply  
+    using assms restrict_apply
     apply (metis SA_mult)
-    using assms  
+    using assms
     by (metis SA_mult')
 qed
 
-lemma SA_add_closed: 
+lemma SA_add_closed:
   assumes "is_semialg_function n f"
   assumes "is_semialg_function n g"
   shows "f \<oplus>\<^bsub>SA n\<^esub> g \<in> carrier (SA n)"
-    using assms SA_add_closed_id 
+    using assms SA_add_closed_id
     by (metis SA_car SA_plus restrict_in_semialg_functions sum_in_semialg_functions)
 
-lemma SA_mult_closed: 
+lemma SA_mult_closed:
   assumes "is_semialg_function n f"
   assumes "is_semialg_function n g"
   shows "f \<otimes>\<^bsub>SA n\<^esub> g \<in> carrier (SA n)"
-    using assms SA_mult_closed_id 
+    using assms SA_mult_closed_id
   by (metis SA_car SA_is_cring cring.cring_simprules(5) restrict_in_semialg_functions)
 
-lemma SA_add_closed_right: 
+lemma SA_add_closed_right:
   assumes "is_semialg_function n f"
   assumes "g \<in> carrier (SA n)"
   shows "f \<oplus>\<^bsub>SA n\<^esub> g \<in> carrier (SA n)"
   using SA_add_closed SA_car_memE(1) assms(1) assms(2)   by blast
-  
-lemma SA_mult_closed_right: 
+
+lemma SA_mult_closed_right:
   assumes "is_semialg_function n f"
   assumes "g \<in> carrier (SA n)"
   shows "f \<otimes>\<^bsub>SA n\<^esub> g \<in> carrier (SA n)"
   using SA_car_memE(1) SA_mult_closed assms(1) assms(2)   by blast
-  
-lemma SA_add_closed_left: 
+
+lemma SA_add_closed_left:
   assumes "f \<in> carrier (SA n)"
   assumes "is_semialg_function n g"
   shows "f \<oplus>\<^bsub>SA n\<^esub> g \<in> carrier (SA n)"
   using SA_add_closed SA_car_memE(1) assms(1) assms(2)   by blast
-  
-lemma SA_mult_closed_left: 
+
+lemma SA_mult_closed_left:
   assumes "f \<in> carrier (SA n)"
   assumes "is_semialg_function n g"
   shows "f \<otimes>\<^bsub>SA n\<^esub> g \<in> carrier (SA n)"
   using SA_car_memE(1) SA_mult_closed assms(1) assms(2)  by blast
 
 lemma SA_nat_pow_closed:
   assumes "is_semialg_function n f"
   shows "f [^]\<^bsub>SA n\<^esub> (k::nat) \<in> carrier (SA n)"
   apply(induction k)
-  using nat_pow_def[of "SA n" f ] 
+  using nat_pow_def[of "SA n" f ]
   apply (metis Group.nat_pow_0  monoid.one_closed SA_is_monoid)
   by (metis Group.nat_pow_Suc SA_car assms(1) assms SA_mult_closed semialg_functions_memE(1))
 
 lemma SA_imp_semialg:
   assumes "f \<in> carrier (SA n)"
   shows "is_semialg_function n f"
   using SA_car assms semialg_functions_memE(1) by blast
 
 lemma SA_minus_closed:
   assumes "f \<in> carrier (SA n)"
   assumes "g \<in> carrier (SA n)"
   shows "(f  \<ominus>\<^bsub>SA n\<^esub> g) \<in> carrier (SA n)"
-  using assms unfolding a_minus_def 
+  using assms unfolding a_minus_def
   by (meson SA_add_closed_left SA_imp_semialg SA_is_ring ring.ring_simprules(3))
 
-lemma(in ring) add_pow_closed : 
+lemma(in ring) add_pow_closed :
   assumes "b \<in> carrier R"
-  shows "([(m::nat)]\<cdot>\<^bsub>R\<^esub>b) \<in> carrier R" 
+  shows "([(m::nat)]\<cdot>\<^bsub>R\<^esub>b) \<in> carrier R"
   by(rule add.nat_pow_closed, rule assms)
-   
-lemma(in ring) add_pow_Suc: 
+
+lemma(in ring) add_pow_Suc:
   assumes "b \<in> carrier R"
   shows "[(Suc m)]\<cdot>b = [m]\<cdot>b \<oplus> b"
   using assms add.nat_pow_Suc by blast
 
-lemma(in ring) add_pow_zero: 
+lemma(in ring) add_pow_zero:
   assumes "b \<in> carrier R"
   shows "[(0::nat)]\<cdot>b = \<zero>"
-  using assms nat_mult_zero 
+  using assms nat_mult_zero
   by blast
-  
+
 lemma Fun_add_pow_apply:
   assumes "b \<in> carrier (Fun\<^bsub>n\<^esub> Q\<^sub>p)"
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "([(m::nat)]\<cdot>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> b) a = [m]\<cdot>( b a)"
 proof-
   have 0: "b a \<in> carrier Q\<^sub>p"
-    using Qp.function_ring_car_mem_closed assms by fastforce    
+    using Qp.function_ring_car_mem_closed assms by fastforce
   have 1: "ring (Fun\<^bsub>n\<^esub> Q\<^sub>p)"
     using function_ring_is_ring by blast
-  show ?thesis 
-  proof(induction m) 
+  show ?thesis
+  proof(induction m)
     case 0
     have "([(0::nat)] \<cdot>\<^bsub>function_ring (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) Q\<^sub>p\<^esub> b) = \<zero>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub>"
-      using 1 ring.add_pow_zero[of "Fun\<^bsub>n\<^esub> Q\<^sub>p" b ] assms  by blast  
-    then show ?case 
-      using  function_zero_eval Qp.nat_mult_zero assms  by presburger   
+      using 1 ring.add_pow_zero[of "Fun\<^bsub>n\<^esub> Q\<^sub>p" b ] assms  by blast
+    then show ?case
+      using  function_zero_eval Qp.nat_mult_zero assms  by presburger
   next
     case (Suc m)
-    then show ?case using Suc ring.add_pow_Suc[of "SA n" b m] assms 
-      by (metis (no_types, lifting) "0" "1" Qp.ring_axioms SA_add SA_plus ring.add_pow_Suc)   
+    then show ?case using Suc ring.add_pow_Suc[of "SA n" b m] assms
+      by (metis (no_types, lifting) "0" "1" Qp.ring_axioms SA_add SA_plus ring.add_pow_Suc)
   qed
 qed
 
 lemma SA_add_pow_apply:
   assumes "b \<in> carrier (SA n)"
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "([(m::nat)]\<cdot>\<^bsub>SA n\<^esub> b) a = [m]\<cdot>( b a)"
   apply(induction m)
-  using assms SA_is_ring[of n] Fun_add_pow_apply 
-  apply (metis function_zero_eval Qp.nat_mult_zero SA_zero ring.add_pow_zero) 
+  using assms SA_is_ring[of n] Fun_add_pow_apply
+  apply (metis function_zero_eval Qp.nat_mult_zero SA_zero ring.add_pow_zero)
   using assms SA_is_ring[of n] ring.add_pow_Suc[of "Fun\<^bsub>n\<^esub> Q\<^sub>p" b]  ring.add_pow_Suc[of "SA n" b] SA_plus[of n]
-  using Fun_add_pow_apply 
+  using Fun_add_pow_apply
   by (metis Qp.add.nat_pow_Suc SA_add)
 
 lemma Qp_funs_Units_SA_Units:
   assumes "f \<in> Units (Fun\<^bsub>n\<^esub> Q\<^sub>p)"
   assumes "is_semialg_function n f"
   shows "f \<in> Units (SA n)"
 proof-
   have 0: "f \<in> carrier (Fun\<^bsub>n\<^esub> Q\<^sub>p)"
     by (meson Qp_funs_is_monoid assms(1) monoid.Units_closed)
   have 1: "inv\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> f \<in> semialg_functions n"
     using monoid.Units_closed[of "Fun\<^bsub>n\<^esub> Q\<^sub>p" f]
-      assms inv_in_semialg_functions[of f n] Qp_funs_Units_memE(3)[of f n] 
+      assms inv_in_semialg_functions[of f n] Qp_funs_Units_memE(3)[of f n]
       semialg_functions_memI[of f n] Qp_funs_is_monoid by blast
   then have 2: "f \<otimes>\<^bsub>SA n\<^esub> (inv\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> f ) = \<one>\<^bsub>SA n\<^esub>"
-    using Qp_funs_Units_memE(1)[of f n] SA_one SA_times assms(1) 
+    using Qp_funs_Units_memE(1)[of f n] SA_one SA_times assms(1)
     by presburger
   then have 3: "(inv\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> f ) \<otimes>\<^bsub>SA n\<^esub> f  = \<one>\<^bsub>SA n\<^esub>"
-    using Qp_funs_Units_memE(2)[of f n] SA_one SA_times assms(1) 
+    using Qp_funs_Units_memE(2)[of f n] SA_one SA_times assms(1)
     by presburger
   have 4: "f \<in> carrier (SA n)"
     using "0" SA_car assms(2) semialg_functions_memI by blast
   have 5: "inv\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> f \<in> carrier (SA n)"
     using SA_car_memI "1" SA_car by blast
-  show ?thesis 
-    using 5 4 3 2 unfolding Units_def 
+  show ?thesis
+    using 5 4 3 2 unfolding Units_def
     by blast
 qed
 
 lemma SA_Units_memE:
   assumes "f \<in> (Units (SA n))"
   shows "f \<otimes>\<^bsub>SA n\<^esub> inv\<^bsub>SA n\<^esub> f = \<one>\<^bsub>SA n\<^esub>"
         "inv\<^bsub>SA n\<^esub> f \<otimes>\<^bsub>SA n\<^esub> f = \<one>\<^bsub>SA n\<^esub>"
-  using assms SA_is_monoid[of n] monoid.Units_r_inv[of "SA n" f] 
+  using assms SA_is_monoid[of n] monoid.Units_r_inv[of "SA n" f]
   apply blast
-  using assms SA_is_monoid[of n] monoid.Units_l_inv[of "SA n" f] 
-  by blast 
+  using assms SA_is_monoid[of n] monoid.Units_l_inv[of "SA n" f]
+  by blast
 
 lemma SA_Units_closed:
   assumes "f \<in> (Units (SA n))"
   shows "f \<in> carrier (SA n)"
-  using assms unfolding Units_def by blast 
+  using assms unfolding Units_def by blast
 
 lemma SA_Units_inv_closed:
   assumes "f \<in> (Units (SA n))"
   shows "inv\<^bsub>SA n\<^esub> f \<in> carrier (SA n)"
   using assms SA_is_monoid[of n] monoid.Units_inv_closed[of "SA n" f]
   by blast
 
 lemma SA_Units_Qp_funs_Units:
   assumes "f \<in> (Units (SA n))"
   shows "f \<in> (Units (Fun\<^bsub>n\<^esub> Q\<^sub>p))"
 proof-
   have 0:  "f \<otimes>\<^bsub>SA n\<^esub> inv\<^bsub>SA n\<^esub> f = \<one>\<^bsub>SA n\<^esub>"
         "inv\<^bsub>SA n\<^esub> f \<otimes>\<^bsub>SA n\<^esub> f = \<one>\<^bsub>SA n\<^esub>"
-     using R.Units_r_inv assms apply blast    
-     using R.Units_l_inv assms by blast     
+     using R.Units_r_inv assms apply blast
+     using R.Units_l_inv assms by blast
   have 1: "f \<in> carrier (Fun\<^bsub>n\<^esub> Q\<^sub>p)"
-    using assms 
+    using assms
     by (metis SA_car SA_is_monoid monoid.Units_closed semialg_functions_memE(2))
   have 2: "inv\<^bsub>SA n\<^esub> f \<in> carrier (Fun\<^bsub>n\<^esub> Q\<^sub>p)"
     using SA_Units_inv_closed SA_car assms semialg_functions_memE(2) by blast
-  then show ?thesis 
+  then show ?thesis
    using 0 1 2  SA_one SA_times local.F.UnitsI(1) by auto
 qed
 
 lemma SA_Units_Qp_funs_inv:
   assumes "f \<in> (Units (SA n))"
   shows "inv\<^bsub>SA n\<^esub> f = inv\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub> f"
   using assms SA_Units_Qp_funs_Units
-  by (metis (no_types, opaque_lifting) Qp_funs_is_cring Qp_funs_is_monoid SA_Units_memE(1) 
+  by (metis (no_types, opaque_lifting) Qp_funs_is_cring Qp_funs_is_monoid SA_Units_memE(1)
       SA_is_monoid SA_one SA_times cring.invI(1) monoid.Units_closed monoid.Units_inv_Units)
 
 lemma SA_Units_memI:
   assumes "f \<in> (carrier (SA n))"
   assumes "\<And> x. x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<Longrightarrow>  f x \<noteq> \<zero>\<^bsub>Q\<^sub>p\<^esub>"
   shows "f \<in> (Units (SA n))"
-  using assms Qp_funs_Units_memI[of f n] Qp_funs_Units_SA_Units SA_car SA_imp_semialg 
+  using assms Qp_funs_Units_memI[of f n] Qp_funs_Units_SA_Units SA_car SA_imp_semialg
       semialg_functions_memE(2) by blast
-  
+
 lemma SA_Units_memE':
   assumes "f \<in> (Units (SA n))"
-  shows  "\<And> x. x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<Longrightarrow>  f x \<noteq> \<zero>\<^bsub>Q\<^sub>p\<^esub>" 
-  using assms Qp_funs_Units_memE[of f n] SA_Units_Qp_funs_Units 
+  shows  "\<And> x. x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<Longrightarrow>  f x \<noteq> \<zero>\<^bsub>Q\<^sub>p\<^esub>"
+  using assms Qp_funs_Units_memE[of f n] SA_Units_Qp_funs_Units
   by blast
 
 lemma Qp_n_nonempty:
   shows "carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<noteq> {}"
   apply(induction n)
   apply (simp add: Qp_zero_carrier)
-  using cartesian_power_cons[of _ Q\<^sub>p _ \<one>] Qp.one_closed  
+  using cartesian_power_cons[of _ Q\<^sub>p _ \<one>] Qp.one_closed
   by (metis Suc_eq_plus1 all_not_in_conv cartesian_power_cons empty_iff)
 
 lemma SA_one_not_zero:
   shows "\<one>\<^bsub>SA n\<^esub> \<noteq> \<zero>\<^bsub> SA n\<^esub>"
 proof-
   obtain a where a_def: "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
-    using Qp_n_nonempty by blast 
+    using Qp_n_nonempty by blast
   have "\<one>\<^bsub>SA n\<^esub> a  \<noteq> \<zero>\<^bsub> SA n\<^esub> a"
     using function_one_eval function_zero_eval SA_one SA_zero a_def local.one_neq_zero by presburger
-  then show ?thesis 
-    by metis 
-qed
-  
+  then show ?thesis
+    by metis
+qed
+
 lemma SA_units_not_zero:
   assumes "f \<in> Units (SA n)"
-  shows "f \<noteq> \<zero>\<^bsub> SA n\<^esub>"  
-  using SA_one_not_zero 
+  shows "f \<noteq> \<zero>\<^bsub> SA n\<^esub>"
+  using SA_one_not_zero
   by (metis assms padic_fields.SA_is_ring padic_fields_axioms ring.ring_in_Units_imp_not_zero)
 
 lemma SA_Units_nonzero:
   assumes "f \<in> Units (SA m)"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
   shows "f x \<in> nonzero Q\<^sub>p"
-  unfolding nonzero_def  mem_Collect_eq 
+  unfolding nonzero_def  mem_Collect_eq
   apply(rule conjI)
-  using assms  SA_Units_closed SA_car_memE(3)[of f m]  apply blast 
-  using assms SA_Units_memE' by blast 
+  using assms  SA_Units_closed SA_car_memE(3)[of f m]  apply blast
+  using assms SA_Units_memE' by blast
 
 lemma SA_car_closed:
   assumes "f \<in> carrier (SA m)"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
   shows "f x \<in> carrier Q\<^sub>p"
-  using assms SA_car_memE(3) by blast 
+  using assms SA_car_memE(3) by blast
 
 lemma SA_Units_closed_fun:
   assumes "f \<in> Units (SA m)"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
   shows "f x \<in> carrier Q\<^sub>p"
-  using SA_Units_closed SA_car_closed assms by blast 
+  using SA_Units_closed SA_car_closed assms by blast
 
 lemma SA_inv_eval:
   assumes "f \<in> Units (SA n)"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
-  shows "(inv\<^bsub>SA n\<^esub> f) x = inv (f x)" 
+  shows "(inv\<^bsub>SA n\<^esub> f) x = inv (f x)"
 proof-
   have  "f \<otimes>\<^bsub>SA n\<^esub> (inv\<^bsub>SA n\<^esub> f) = \<one> \<^bsub>SA n\<^esub>"
     using assms SA_is_cring  SA_Units_memE(1) by blast
   hence  "(f \<otimes>\<^bsub>SA n\<^esub> (inv\<^bsub>SA n\<^esub> f)) x = \<one>  \<^bsub>SA n\<^esub> x"
     using assms by presburger
   then have "(f x) \<otimes> (inv\<^bsub>SA n\<^esub> f) x = \<one>"
     by (metis function_one_eval SA_mult SA_one assms)
-  then show ?thesis 
+  then show ?thesis
     by (metis Q\<^sub>p_def Qp_funs_m_inv Zp_def assms(1) assms(2) padic_fields.SA_Units_Qp_funs_Units padic_fields.SA_Units_Qp_funs_inv padic_fields_axioms)
 qed
 
 lemma SA_div_eval:
   assumes "f \<in> Units (SA n)"
   assumes "h \<in> carrier (SA n)"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
-  shows "(h \<otimes>\<^bsub>SA n\<^esub> (inv\<^bsub>SA n\<^esub> f)) x = h x \<otimes> inv (f x)" 
+  shows "(h \<otimes>\<^bsub>SA n\<^esub> (inv\<^bsub>SA n\<^esub> f)) x = h x \<otimes> inv (f x)"
   using assms SA_inv_eval  SA_mult by presburger
 
 lemma SA_unit_int_pow:
   assumes "f \<in> Units (SA m)"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
   shows "(f[^]\<^bsub>SA m\<^esub>(i::int)) x = (f x)[^]i"
 proof(induction i)
   case (nonneg n)
   have  0: "(f [^]\<^bsub>SA m\<^esub> int n) = (f [^]\<^bsub>SA m\<^esub> n)"
     using assms by (meson int_pow_int)
-  then show ?case using SA_Units_closed[of f m] assms 
+  then show ?case using SA_Units_closed[of f m] assms
     by (metis SA_nat_pow int_pow_int)
 next
   case (neg n)
   have 0: "(f [^]\<^bsub>SA m\<^esub> - int (Suc n)) = inv \<^bsub>SA m\<^esub>(f [^]\<^bsub>SA m\<^esub> (Suc n))"
     using assms by (metis R.int_pow_inv' int_pow_int)
-  then show ?case unfolding 0 using assms 
+  then show ?case unfolding 0 using assms
     by (metis Qp.int_pow_inv' R.Units_pow_closed SA_Units_nonzero SA_inv_eval SA_nat_pow Units_eq_nonzero int_pow_int)
 qed
 
 lemma restrict_in_SA_car:
   assumes "is_semialg_function n f"
   shows "restrict f (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) \<in> carrier (SA n)"
-  using assms  SA_car restrict_in_semialg_functions 
+  using assms  SA_car restrict_in_semialg_functions
   by blast
 
 lemma SA_smult:
 "(\<odot>\<^bsub>SA n\<^esub>) = (\<odot>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub>)"
-  unfolding SA_def by auto 
-
-lemma SA_smult_formula: 
+  unfolding SA_def by auto
+
+lemma SA_smult_formula:
   assumes "h \<in> carrier (SA n)"
   assumes "q \<in> carrier Q\<^sub>p"
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "(q \<odot>\<^bsub>SA n\<^esub> h) a = q \<otimes>(h a)"
   using SA_smult assms function_smult_eval SA_car_memE(2) by presburger
 
-lemma SA_smult_closed: 
+lemma SA_smult_closed:
   assumes "h \<in> carrier (SA n)"
   assumes "q \<in> carrier Q\<^sub>p"
   shows "q \<odot>\<^bsub>SA n\<^esub> h \<in> carrier (SA n)"
 proof-
   obtain g where g_def: "g = \<cc>\<^bsub>n\<^esub> q"
-    by blast 
+    by blast
   have g_closed: "g \<in> carrier (SA n)"
-    using g_def assms constant_function_is_semialg[of q n] constant_function_closed SA_car_memI 
+    using g_def assms constant_function_is_semialg[of q n] constant_function_closed SA_car_memI
     by blast
   have "q \<odot>\<^bsub>SA n\<^esub> h = g \<otimes>\<^bsub>SA n\<^esub> h"
     apply(rule function_ring_car_eqI[of _ n])
-    using function_smult_closed SA_car_memE(2) SA_smult assms apply presburger    
+    using function_smult_closed SA_car_memE(2) SA_smult assms apply presburger
     using SA_car_memE(2) assms(1) assms(2) g_closed padic_fields.SA_imp_semialg padic_fields.SA_mult_closed_right padic_fields_axioms apply blast
     using Qp_constE SA_mult SA_smult_formula assms   g_def by presburger
-  thus ?thesis 
+  thus ?thesis
     using SA_imp_semialg SA_mult_closed_right assms(1) assms(2) g_closed by presburger
 qed
 
 lemma p_mult_function_val:
   assumes "f \<in> carrier (SA m)"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
   shows "val ((\<pp> \<odot>\<^bsub>SA m\<^esub>f) x) = val (f x) + 1"
 proof-
   have 0: "(\<pp>\<odot>\<^bsub>SA m\<^esub>f) x = \<pp>\<otimes>(f x)"
     using Qp.int_inc_closed SA_smult_formula assms(1) assms(2) by blast
-  show ?thesis  unfolding 0 using assms 
+  show ?thesis  unfolding 0 using assms
     by (metis Qp.function_ring_car_memE Qp.int_inc_closed Qp.m_comm SA_car semialg_functions_memE(2) val_mult val_p)
 qed
 
 lemma Qp_char_0'':
   assumes "a \<in> carrier Q\<^sub>p"
   assumes "a \<noteq> \<zero>"
   assumes "(k::nat) > 0"
   shows "[k]\<cdot>a \<noteq> \<zero>"
 proof-
   have 0: "[k]\<cdot>\<one> \<noteq>\<zero>"
     using Qp_char_0 assms(3) by blast
   have "[k]\<cdot>a = [k]\<cdot>\<one> \<otimes> a"
     using Qp.add_pow_ldistr Qp.l_one Qp.one_closed assms(1) by presburger
-  thus ?thesis using 0 assms 
+  thus ?thesis using 0 assms
     using Qp.integral by blast
 qed
 
 lemma SA_char_zero:
   assumes "f \<in> carrier (SA m)"
   assumes "f \<noteq> \<zero>\<^bsub>SA m\<^esub>"
   assumes "n > 0"
   shows "[(n::nat)]\<cdot>\<^bsub>SA m\<^esub>f \<noteq> \<zero>\<^bsub>SA m\<^esub>"
 proof assume A: "[n] \<cdot>\<^bsub>SA m\<^esub> f = \<zero>\<^bsub>SA m\<^esub>"
   obtain x where x_def: "x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<and> f x \<noteq> \<zero>"
-    using assms 
+    using assms
     by (metis function_ring_car_eqI R.cring_simprules(2) SA_car_memE(2) SA_zeroE)
   have 0: "([(n::nat)]\<cdot>\<^bsub>SA m\<^esub>f) x = [n]\<cdot>(f x)"
     using SA_add_pow_apply assms(1) x_def by blast
   have 1: "[n]\<cdot>(f x) = \<zero>"
     using 0 unfolding A  using SA_zeroE x_def by blast
   have 2: "f x \<in> nonzero Q\<^sub>p"
-    using x_def assms 
+    using x_def assms
     by (metis Qp.function_ring_car_memE SA_car not_nonzero_Qp semialg_functions_memE(2))
-  then show False using x_def 
+  then show False using x_def
     using "1" Qp.nonzero_memE(1) Qp_char_0'' assms(3) by blast
 qed
 
 
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsection\<open>Defining Semialgebraic Maps\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 text\<open>
-  We can define a semialgebraic map in essentially the same way that Denef defines 
-  semialgebraic functions. As for functions, we can define the partial pullback of a set 
+  We can define a semialgebraic map in essentially the same way that Denef defines
+  semialgebraic functions. As for functions, we can define the partial pullback of a set
   $S \subseteq \mathbb{Q}_p^{n+l}$ by a map $g: \mathbb{Q}_p^m \to \mathbb{Q}_p^n$ to be the set
   \[
   \{(x,y) \in \mathbb{Q}_p^m \times \mathbb{Q}_p^l \mid (f(x), y) \in S \}
   \]
-  and say that $g$ is a semialgebraic map if for every $l$, and every semialgebraic 
-  $S \subseteq \mathbb{Q}_p^{n+l}$, the partial pullback of $S$ by $g$ is also semialgebraic. 
-  On this definition, it is immediate that the composition $f \circ g$ of a semialgebraic 
-  function $f: \mathbb{Q}_p^n \to \mathbb{Q}$ and a semialgebraic map 
-  $g: \mathbb{Q}_p^m \to \mathbb{Q}_p^n$ is semialgebraic. It is also not hard to show that a map 
-  is semialgebraic if and only if all of its coordinate functions are semialgebraic functions. 
+  and say that $g$ is a semialgebraic map if for every $l$, and every semialgebraic
+  $S \subseteq \mathbb{Q}_p^{n+l}$, the partial pullback of $S$ by $g$ is also semialgebraic.
+  On this definition, it is immediate that the composition $f \circ g$ of a semialgebraic
+  function $f: \mathbb{Q}_p^n \to \mathbb{Q}$ and a semialgebraic map
+  $g: \mathbb{Q}_p^m \to \mathbb{Q}_p^n$ is semialgebraic. It is also not hard to show that a map
+  is semialgebraic if and only if all of its coordinate functions are semialgebraic functions.
   This allows us to build new semialgebraic functions out of old ones via composition.
 \<close>
 
 
 text\<open>Generalizing the notion of partial image partial pullbacks from functions to maps:\<close>
 
 definition map_partial_image where
 "map_partial_image m f xs = (f (take m xs))@(drop m xs)"
 
 definition map_partial_pullback where
 "map_partial_pullback m f l S = (map_partial_image m f)  \<inverse>\<^bsub>m+l\<^esub> S"
 
-lemma map_partial_pullback_memE: 
+lemma map_partial_pullback_memE:
   assumes "as \<in> map_partial_pullback m f l S"
   shows "as \<in> carrier (Q\<^sub>p\<^bsup>m+l\<^esup>)" "map_partial_image m f as \<in> S"
   using assms unfolding map_partial_pullback_def  evimage_def
   apply (metis (no_types, opaque_lifting) Int_iff add.commute)
-  using assms unfolding map_partial_pullback_def   
+  using assms unfolding map_partial_pullback_def
   by blast
 
-lemma map_partial_pullback_closed: 
-"map_partial_pullback m f l S \<subseteq> carrier (Q\<^sub>p\<^bsup>m+l\<^esup>)" 
+lemma map_partial_pullback_closed:
+"map_partial_pullback m f l S \<subseteq> carrier (Q\<^sub>p\<^bsup>m+l\<^esup>)"
   using map_partial_pullback_memE(1) by blast
 
-lemma map_partial_pullback_memI: 
+lemma map_partial_pullback_memI:
   assumes "as \<in> carrier (Q\<^sub>p\<^bsup>m+k\<^esup>)"
-  assumes "(f (take m as))@(drop m as) \<in> S" 
+  assumes "(f (take m as))@(drop m as) \<in> S"
   shows "as \<in> map_partial_pullback m f k S"
-  using assms unfolding map_partial_pullback_def map_partial_image_def 
-  by blast 
+  using assms unfolding map_partial_pullback_def map_partial_image_def
+  by blast
 
 lemma map_partial_image_eq:
   assumes "as \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "bs \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>)"
   assumes "x = as @ bs"
   shows "map_partial_image n f x = (f as)@bs"
 proof-
   have 0: "(take n x) = as"
     by (metis append_eq_conv_conj assms(1) assms(3) cartesian_power_car_memE)
   have 1: "drop n x = bs"
     by (metis "0" append_take_drop_id assms(3) same_append_eq)
-  show ?thesis using 0 1 unfolding map_partial_image_def 
-    by blast 
-qed
-  
+  show ?thesis using 0 1 unfolding map_partial_image_def
+    by blast
+qed
+
 lemma map_partial_pullback_memE':
   assumes "as \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "bs \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>)"
   assumes "x = as @ bs"
   assumes "x \<in> map_partial_pullback n f k S"
-  shows "(f as)@bs \<in> S" 
-  using map_partial_pullback_memE[of x n f k S] map_partial_image_def[of n f x] 
+  shows "(f as)@bs \<in> S"
+  using map_partial_pullback_memE[of x n f k S] map_partial_image_def[of n f x]
   by (metis assms(1) assms(2) assms(3) assms(4) map_partial_image_eq)
 
 text\<open>Partial pullbacks have the same algebraic properties as pullbacks.\<close>
 
 lemma map_partial_pullback_intersect:
 "map_partial_pullback m f l (S1 \<inter> S2) = (map_partial_pullback m f l S1) \<inter> (map_partial_pullback m f l S2)"
-  unfolding map_partial_pullback_def 
+  unfolding map_partial_pullback_def
   by simp
-  
+
 lemma map_partial_pullback_union:
 "map_partial_pullback m f l (S1 \<union> S2) = (map_partial_pullback m f l S1) \<union> (map_partial_pullback m f l S2)"
-  unfolding map_partial_pullback_def 
+  unfolding map_partial_pullback_def
   by simp
 
 lemma map_partial_pullback_complement:
   assumes "f \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<rightarrow> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "map_partial_pullback m f l (carrier (Q\<^sub>p\<^bsup>n+l\<^esup>) - S) = carrier (Q\<^sub>p\<^bsup>m+l\<^esup>) - (map_partial_pullback m f l S) "
   apply(rule equalityI)
-  using map_partial_pullback_def[of m f l "(carrier (Q\<^sub>p\<^bsup>n+l\<^esup>) - S)"] 
-        map_partial_pullback_def[of m f l S] 
-   apply (metis (no_types, lifting) DiffD2 DiffI evimage_Diff map_partial_pullback_memE(1) subsetI)   
+  using map_partial_pullback_def[of m f l "(carrier (Q\<^sub>p\<^bsup>n+l\<^esup>) - S)"]
+        map_partial_pullback_def[of m f l S]
+   apply (metis (no_types, lifting) DiffD2 DiffI evimage_Diff map_partial_pullback_memE(1) subsetI)
 proof fix x assume A: " x \<in> carrier (Q\<^sub>p\<^bsup>m+l\<^esup>) - map_partial_pullback m f l S"
   show " x \<in> map_partial_pullback m f l (carrier (Q\<^sub>p\<^bsup>n+l\<^esup>) - S) "
     apply(rule map_partial_pullback_memI)
-    using A 
+    using A
      apply blast
   proof
     have 0: "drop m x \<in> carrier (Q\<^sub>p\<^bsup>l\<^esup>)"
       by (meson A DiffD1 cartesian_power_drop)
     have 1: "take m x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
-      using A 
-      by (meson DiffD1 take_closed le_add1)     
+      using A
+      by (meson DiffD1 take_closed le_add1)
     show "f (take m x) @ drop m x \<in> carrier (Q\<^sub>p\<^bsup>n+l\<^esup>) "
-      using 0 1 assms 
-      by (meson Pi_iff cartesian_power_concat(1))      
+      using 0 1 assms
+      by (meson Pi_iff cartesian_power_concat(1))
     show "f (take m x) @ drop m x \<notin> S"
-      using A unfolding map_partial_pullback_def map_partial_image_def 
+      using A unfolding map_partial_pullback_def map_partial_image_def
       by blast
   qed
-qed     
+qed
 
 lemma map_partial_pullback_carrier:
   assumes "f \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<rightarrow> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "map_partial_pullback m f l (carrier (Q\<^sub>p\<^bsup>n+l\<^esup>)) = carrier (Q\<^sub>p\<^bsup>m+l\<^esup>)"
   apply(rule equalityI)
   using map_partial_pullback_memE(1) apply blast
 proof fix x assume A: "x \<in> carrier (Q\<^sub>p\<^bsup>m+l\<^esup>)"
   show "x \<in> map_partial_pullback m f l (carrier (Q\<^sub>p\<^bsup>n+l\<^esup>))"
     apply(rule map_partial_pullback_memI)
   using A cartesian_power_drop[of x m l] take_closed assms
-   apply blast  
+   apply blast
 proof-
   have "f (take m x) \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
     using A take_closed assms
     by (meson Pi_mem le_add1)
-  then show "f (take m x) @ drop m x \<in> carrier (Q\<^sub>p\<^bsup>n+l\<^esup>)" 
-    using cartesian_power_drop[of x m l] A cartesian_power_concat(1)[of _ Q\<^sub>p n _ l] 
-    by blast    
+  then show "f (take m x) @ drop m x \<in> carrier (Q\<^sub>p\<^bsup>n+l\<^esup>)"
+    using cartesian_power_drop[of x m l] A cartesian_power_concat(1)[of _ Q\<^sub>p n _ l]
+    by blast
 qed
 qed
 
 definition is_semialg_map where
-"is_semialg_map m n f = (f \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<rightarrow> carrier (Q\<^sub>p\<^bsup>n\<^esup>)  \<and> 
+"is_semialg_map m n f = (f \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<rightarrow> carrier (Q\<^sub>p\<^bsup>n\<^esup>)  \<and>
                           (\<forall>l \<ge> 0. \<forall>S \<in> semialg_sets (n + l). is_semialgebraic (m + l) (map_partial_pullback m f l S)))"
 
 lemma is_semialg_map_closed:
   assumes "is_semialg_map m n f"
   shows "f \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<rightarrow> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
-  using is_semialg_map_def assms by blast 
+  using is_semialg_map_def assms by blast
 
 lemma is_semialg_map_closed':
   assumes "is_semialg_map m n f" "x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
   shows "f x \<in>  carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
-  using is_semialg_map_def assms by blast 
+  using is_semialg_map_def assms by blast
 
 lemma is_semialg_mapE:
   assumes "is_semialg_map m n f"
   assumes "is_semialgebraic (n + k) S"
   shows " is_semialgebraic (m + k) (map_partial_pullback m f k S)"
-  using is_semialg_map_def assms 
+  using is_semialg_map_def assms
   by (meson is_semialgebraicE le0)
 
 lemma is_semialg_mapE':
   assumes "is_semialg_map m n f"
   assumes "is_semialgebraic (n + k) S"
   shows " is_semialgebraic (m + k) (map_partial_image m f \<inverse>\<^bsub>m+k\<^esub> S)"
   using assms is_semialg_mapE unfolding map_partial_pullback_def
-  by blast 
+  by blast
 
 lemma is_semialg_mapI:
   assumes "f \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<rightarrow> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "\<And>k  S. S \<in> semialg_sets (n + k) \<Longrightarrow> is_semialgebraic (m + k) (map_partial_pullback m f k S)"
   shows "is_semialg_map m n f"
-  using assms unfolding is_semialg_map_def 
+  using assms unfolding is_semialg_map_def
   by blast
 
 lemma is_semialg_mapI':
   assumes "f \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<rightarrow> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "\<And>k  S. S \<in> semialg_sets (n + k) \<Longrightarrow> is_semialgebraic (m + k) (map_partial_image m f \<inverse>\<^bsub>m+k\<^esub> S)"
   shows "is_semialg_map m n f"
-  using assms is_semialg_mapI unfolding map_partial_pullback_def 
+  using assms is_semialg_mapI unfolding map_partial_pullback_def
   by blast
 
 text\<open>Semialgebraicity for functions can be verified on basic semialgebraic sets.\<close>
 
 lemma is_semialg_mapI'':
   assumes "f \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<rightarrow> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "\<And>k  S. S \<in> basic_semialgs (n + k) \<Longrightarrow> is_semialgebraic (m + k) (map_partial_pullback m f k S)"
   shows "is_semialg_map m n f"
   apply(rule is_semialg_mapI)
   using assms(1) apply blast
 proof-
   show "\<And>k S. S \<in> semialg_sets (n + k) \<Longrightarrow> is_semialgebraic (m + k) (map_partial_pullback m f k S)"
   proof- fix k S assume A: "S \<in> semialg_sets (n + k)"
     show "is_semialgebraic (m + k) (map_partial_pullback m f k S)"
       apply(rule gen_boolean_algebra.induct[of S "carrier (Q\<^sub>p\<^bsup>n+k\<^esup>)" "basic_semialgs (n + k)"])
-        using A unfolding semialg_sets_def 
+        using A unfolding semialg_sets_def
             apply blast
         using map_partial_pullback_carrier assms carrier_is_semialgebraic plus_1_eq_Suc apply presburger
         apply (simp add: assms(1) assms(2) carrier_is_semialgebraic intersection_is_semialg map_partial_pullback_carrier map_partial_pullback_intersect)
         using map_partial_pullback_union union_is_semialgebraic apply presburger
         using assms(1) complement_is_semialg map_partial_pullback_complement plus_1_eq_Suc by presburger
   qed
-qed      
+qed
 
 lemma is_semialg_mapI''':
   assumes "f \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<rightarrow> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "\<And>k  S. S \<in> basic_semialgs (n + k) \<Longrightarrow> is_semialgebraic (m + k) (map_partial_image m f \<inverse>\<^bsub>m+k\<^esub> S)"
   shows "is_semialg_map m n f"
-  using is_semialg_mapI'' assms unfolding map_partial_pullback_def 
+  using is_semialg_mapI'' assms unfolding map_partial_pullback_def
   by blast
 
-lemma id_is_semialg_map: 
+lemma id_is_semialg_map:
 "is_semialg_map n n (\<lambda> x. x)"
 proof-
-  have 0: "\<And>k S. S \<in> semialg_sets (n + k) \<Longrightarrow> (\<lambda>xs. take n xs @ drop n xs) -` S \<inter> carrier (Q\<^sub>p\<^bsup>n + k\<^esup>) = 
+  have 0: "\<And>k S. S \<in> semialg_sets (n + k) \<Longrightarrow> (\<lambda>xs. take n xs @ drop n xs) -` S \<inter> carrier (Q\<^sub>p\<^bsup>n + k\<^esup>) =
             S"
-    apply(rule equalityI') 
+    apply(rule equalityI')
      apply (metis (no_types, lifting) Int_iff append_take_drop_id vimage_eq)
     by (metis (no_types, lifting) IntI append_take_drop_id in_mono is_semialgebraicI is_semialgebraic_closed vimageI)
-  show ?thesis 
+  show ?thesis
     by(intro is_semialg_mapI,
-          unfold  map_partial_pullback_def map_partial_image_def evimage_def is_semialgebraic_def  0, 
+          unfold  map_partial_pullback_def map_partial_image_def evimage_def is_semialgebraic_def  0,
           auto)
 qed
 
 lemma map_partial_pullback_comp:
   assumes "is_semialg_map m n f"
   assumes "is_semialg_map k m g"
   shows  "(map_partial_pullback k (f \<circ> g) l S) = (map_partial_pullback k g l (map_partial_pullback m f l S))"
 proof
   show "map_partial_pullback k (f \<circ> g) l S \<subseteq> map_partial_pullback k g l (map_partial_pullback m f l S)"
   proof fix x assume A: " x \<in> map_partial_pullback k (f \<circ> g) l S"
     show " x \<in> map_partial_pullback k g l (map_partial_pullback m f l S)"
-    proof(rule map_partial_pullback_memI) 
+    proof(rule map_partial_pullback_memI)
       show 0: "x \<in> carrier (Q\<^sub>p\<^bsup>k+l\<^esup>)"
         using A map_partial_pullback_memE(1) by blast
-      show "g (take k x) @ drop k x \<in> map_partial_pullback m f l S"  
-      proof(rule map_partial_pullback_memI) 
-        show "g (take k x) @ drop k x \<in> carrier (Q\<^sub>p\<^bsup>m+l\<^esup>)" 
+      show "g (take k x) @ drop k x \<in> map_partial_pullback m f l S"
+      proof(rule map_partial_pullback_memI)
+        show "g (take k x) @ drop k x \<in> carrier (Q\<^sub>p\<^bsup>m+l\<^esup>)"
         proof-
           have 1: "g (take k x) \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
-            using 0 assms(2) is_semialg_map_closed[of k m g] 
+            using 0 assms(2) is_semialg_map_closed[of k m g]
             by (meson Pi_iff le_add1 take_closed)
-          then show ?thesis 
+          then show ?thesis
             by (metis "0" add.commute cartesian_power_concat(2) cartesian_power_drop)
         qed
         show "f (take m (g (take k x) @ drop k x)) @ drop m (g (take k x) @ drop k x) \<in> S"
-        using map_partial_pullback_memE[of x k "f \<circ> g" l S] 
+        using map_partial_pullback_memE[of x k "f \<circ> g" l S]
           comp_apply[of f g] map_partial_image_eq[of "take k x" k "drop k x" l x "f \<circ> g"]
-        by (metis (no_types, lifting) A \<open>g (take k x) @ drop k x \<in> carrier (Q\<^sub>p\<^bsup>m + l\<^esup>)\<close> 
-            append_eq_append_conv append_take_drop_id cartesian_power_car_memE 
+        by (metis (no_types, lifting) A \<open>g (take k x) @ drop k x \<in> carrier (Q\<^sub>p\<^bsup>m + l\<^esup>)\<close>
+            append_eq_append_conv append_take_drop_id cartesian_power_car_memE
             cartesian_power_drop map_partial_image_def)
       qed
     qed
   qed
   show "map_partial_pullback k g l (map_partial_pullback m f l S) \<subseteq> map_partial_pullback k (f \<circ> g) l S"
   proof fix x assume A: "x \<in> map_partial_pullback k g l (map_partial_pullback m f l S)"
     have 0: "(take m (map_partial_image k g x)) = g (take k x)"
     proof-
       have "take k x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>)"
-        using map_partial_pullback_memE[of x k g l] A le_add1 take_closed 
+        using map_partial_pullback_memE[of x k g l] A le_add1 take_closed
         by blast
       then have "length (g (take k x)) = m"
-        using assms is_semialg_map_closed[of k m g] cartesian_power_car_memE 
+        using assms is_semialg_map_closed[of k m g] cartesian_power_car_memE
         by blast
       then show ?thesis
-        using assms unfolding map_partial_image_def 
+        using assms unfolding map_partial_image_def
         by (metis append_eq_conv_conj)
     qed
     show " x \<in> map_partial_pullback k (f \<circ> g) l S"
       apply(rule map_partial_pullback_memI)
-      using A map_partial_pullback_memE 
+      using A map_partial_pullback_memE
        apply blast
       using 0 assms A comp_apply map_partial_pullback_memE[of x k g l "map_partial_pullback m f l S"]
             map_partial_pullback_memE[of "map_partial_image k g x" m f l S]
   map_partial_image_eq[of "take k x" k "drop k x" l x g]
    map_partial_image_eq[of "take m (map_partial_image k g x)" m "drop m (map_partial_image k g x)" l "(map_partial_image k g x)" f ]
   by (metis (no_types, lifting) cartesian_power_drop le_add1 map_partial_image_def map_partial_pullback_memE' take_closed)
 qed
 qed
 
 lemma semialg_map_comp_closed:
   assumes "is_semialg_map m n f"
   assumes "is_semialg_map k m g"
   shows "is_semialg_map k n (f \<circ> g)"
   apply(intro is_semialg_mapI , unfold Pi_iff comp_def, intro ballI,
-        intro is_semialg_map_closed'[of m n f] is_semialg_map_closed'[of k m g]  assms, blast) 
-proof- fix l S assume A: "S \<in> semialg_sets (n + l)"  
+        intro is_semialg_map_closed'[of m n f] is_semialg_map_closed'[of k m g]  assms, blast)
+proof- fix l S assume A: "S \<in> semialg_sets (n + l)"
   have " is_semialgebraic (k + l) (map_partial_pullback k (f \<circ> g) l S)"
-    using map_partial_pullback_comp is_semialg_mapE A assms(1) assms(2) is_semialgebraicI 
+    using map_partial_pullback_comp is_semialg_mapE A assms(1) assms(2) is_semialgebraicI
     by presburger
   thus "is_semialgebraic (k + l) (map_partial_pullback k (\<lambda>x. f (g x)) l S)"
-    unfolding comp_def by auto 
+    unfolding comp_def by auto
 qed
 
 lemma partial_pullback_comp:
   assumes "is_semialg_function m f"
   assumes "is_semialg_map k m g"
   shows  "(partial_pullback k (f \<circ> g) l S) = (map_partial_pullback k g l (partial_pullback m f l S))"
 proof
   show "partial_pullback k (f \<circ> g) l S \<subseteq> map_partial_pullback k g l (partial_pullback m f l S)"
   proof fix x assume A: "x \<in> partial_pullback k (f \<circ> g) l S"
     show "x \<in> map_partial_pullback k g l (partial_pullback m f l S)"
-    proof(rule map_partial_pullback_memI) 
+    proof(rule map_partial_pullback_memI)
       show 0: "x \<in> carrier (Q\<^sub>p\<^bsup>k+l\<^esup>)"
         using A partial_pullback_memE(1) by blast
-      show "g (take k x) @ drop k x \<in> partial_pullback m f l S"  
-      proof(rule partial_pullback_memI) 
-        show "g (take k x) @ drop k x \<in> carrier (Q\<^sub>p\<^bsup>m+l\<^esup>)" 
+      show "g (take k x) @ drop k x \<in> partial_pullback m f l S"
+      proof(rule partial_pullback_memI)
+        show "g (take k x) @ drop k x \<in> carrier (Q\<^sub>p\<^bsup>m+l\<^esup>)"
         proof-
           have 1: "g (take k x) \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
-            using 0 assms(2) is_semialg_map_closed[of k m g] 
+            using 0 assms(2) is_semialg_map_closed[of k m g]
             by (meson Pi_iff le_add1 take_closed)
-          then show ?thesis 
+          then show ?thesis
             by (metis "0" add.commute cartesian_power_concat(2) cartesian_power_drop)
         qed
         show "f (take m (g (take k x) @ drop k x)) # drop m (g (take k x) @ drop k x) \<in> S"
-        using partial_pullback_memE[of x k "f \<circ> g" l S] 
+        using partial_pullback_memE[of x k "f \<circ> g" l S]
           comp_apply[of f g] partial_image_eq[of "take k x" k "drop k x" l x "f \<circ> g"]
-        by (metis (no_types, lifting) A \<open>g (take k x) @ drop k x \<in> carrier (Q\<^sub>p\<^bsup>m+l\<^esup>)\<close> 
-            add_diff_cancel_left' append_eq_append_conv append_take_drop_id 
-            cartesian_power_car_memE length_drop local.partial_image_def)        
+        by (metis (no_types, lifting) A \<open>g (take k x) @ drop k x \<in> carrier (Q\<^sub>p\<^bsup>m+l\<^esup>)\<close>
+            add_diff_cancel_left' append_eq_append_conv append_take_drop_id
+            cartesian_power_car_memE length_drop local.partial_image_def)
       qed
     qed
   qed
   show "map_partial_pullback k g l (partial_pullback m f l S) \<subseteq> partial_pullback k (f \<circ> g) l S"
   proof fix x assume A: "x \<in> map_partial_pullback k g l (partial_pullback m f l S)"
     have 0: "(take m (map_partial_image k g x)) = g (take k x)"
     proof-
       have "take k x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>)"
-        using map_partial_pullback_memE[of x k g l] A le_add1 take_closed 
+        using map_partial_pullback_memE[of x k g l] A le_add1 take_closed
         by blast
       then have "length (g (take k x)) = m"
-        using assms is_semialg_map_closed[of k m g] cartesian_power_car_memE 
+        using assms is_semialg_map_closed[of k m g] cartesian_power_car_memE
         by blast
       then show ?thesis
-        using assms unfolding map_partial_image_def 
+        using assms unfolding map_partial_image_def
         by (metis append_eq_conv_conj)
     qed
     show "x \<in> partial_pullback k (f \<circ> g) l S"
       apply(rule partial_pullback_memI)
-      using A map_partial_pullback_memE 
+      using A map_partial_pullback_memE
        apply blast
       using 0 assms A comp_apply map_partial_pullback_memE[of x k g l "partial_pullback m f l S"]
             partial_pullback_memE[of "map_partial_image k g x" m f l S]
             map_partial_image_eq[of "take k x" k "drop k x" l x g]
    partial_image_eq[of "take m (map_partial_image k g x)" m "drop m (map_partial_image k g x)" l "(map_partial_image k g x)" f ]
   by (metis (no_types, lifting) cartesian_power_drop le_add1 map_partial_image_def partial_pullback_memE' take_closed)
 
 qed
 qed
 
 lemma semialg_function_comp_closed:
   assumes "is_semialg_function m f"
   assumes "is_semialg_map k m g"
   shows "is_semialg_function k (f \<circ> g)"
 proof(rule is_semialg_functionI)
   show "f \<circ> g \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>) \<rightarrow> carrier Q\<^sub>p"
   proof fix x assume A: "x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>)"
     show " (f \<circ> g) x \<in> carrier Q\<^sub>p"
       using A assms is_semialg_map_closed[of k m g ] is_semialg_function_closed[of m f] comp_apply[of f g x]
       by (metis (no_types, lifting) Pi_mem)
   qed
   show " \<And>ka S. S \<in> semialg_sets (1 + ka) \<Longrightarrow> is_semialgebraic (k + ka) (partial_pullback k (f \<circ> g) ka S)"
   proof- fix n S assume A: "S \<in> semialg_sets (1 + n)"
     show "is_semialgebraic (k + n) (partial_pullback k (f \<circ> g) n S)"
       using A assms partial_pullback_comp is_semialg_functionE is_semialg_mapE
         is_semialgebraicI  by presburger
   qed
 qed
 
 lemma semialg_map_evimage_is_semialg:
   assumes "is_semialg_map k m g"
   assumes "is_semialgebraic m S"
   shows "is_semialgebraic k (g  \<inverse>\<^bsub>k\<^esub> S)"
 proof-
   have "g  \<inverse>\<^bsub>k\<^esub> S = map_partial_pullback k g 0 S"
   proof
     show "g \<inverse>\<^bsub>k\<^esub> S \<subseteq> map_partial_pullback k g 0 S"
     proof fix x assume A: "x \<in> g  \<inverse>\<^bsub>k\<^esub> S"
       then have 0: "x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>) \<and> g x \<in> S"
         by (meson evimage_eq)
       have "x = take k x @ drop k x"
         using 0  by (simp add: "0")
       then show "x \<in> map_partial_pullback k g 0 S"
         unfolding map_partial_pullback_def map_partial_image_def
-        by (metis (no_types, lifting) "0" Nat.add_0_right add.commute append_Nil2 append_same_eq 
+        by (metis (no_types, lifting) "0" Nat.add_0_right add.commute append_Nil2 append_same_eq
         append_take_drop_id evimageI2 map_partial_image_def map_partial_image_eq take0 take_drop)
     qed
     show "map_partial_pullback k g 0 S \<subseteq> g \<inverse>\<^bsub>k\<^esub> S"
-    proof fix x assume A: "x \<in> map_partial_pullback k g 0 S " 
+    proof fix x assume A: "x \<in> map_partial_pullback k g 0 S "
       then have 0: " g (take k x) @ (drop k x) \<in> S"
         unfolding map_partial_pullback_def map_partial_image_def
         by blast
       have 1: "x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>)"
-        using A unfolding map_partial_pullback_def map_partial_image_def 
+        using A unfolding map_partial_pullback_def map_partial_image_def
         by (metis (no_types, lifting) Nat.add_0_right evimage_eq)
       hence "take k x = x"
         by (metis cartesian_power_car_memE le_eq_less_or_eq take_all)
       then show " x \<in> g \<inverse>\<^bsub>k\<^esub> S"
-        using 0 1 unfolding evimage_def  
+        using 0 1 unfolding evimage_def
         by (metis (no_types, lifting) IntI append.assoc append_same_eq append_take_drop_id same_append_eq vimageI)
     qed
   qed
-  then show ?thesis using assms 
+  then show ?thesis using assms
     by (metis add.right_neutral is_semialg_mapE' map_partial_pullback_def)
 qed
 
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsection \<open>Examples of Semialgebraic Maps\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 
 lemma semialg_map_on_carrier:
   assumes "is_semialg_map n m f"
   assumes "restrict f (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) = restrict g (carrier (Q\<^sub>p\<^bsup>n\<^esup>))"
   shows "is_semialg_map n m g"
 proof(rule is_semialg_mapI)
   have 0: "f \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<rightarrow> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
-    using assms(1) is_semialg_map_closed 
+    using assms(1) is_semialg_map_closed
     by blast
   show "g \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<rightarrow> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
   proof fix x assume A: "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)" then show " g x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
-      using assms(2) 0 
+      using assms(2) 0
       by (metis (no_types, lifting) PiE restrict_Pi_cancel)
-  qed    
+  qed
   show "\<And>k S. S \<in> semialg_sets (m + k) \<Longrightarrow> is_semialgebraic (n + k) (map_partial_pullback n g k S)"
-  proof- fix k S 
+  proof- fix k S
     assume A: "S \<in> semialg_sets (m + k)"
     have 1: "is_semialgebraic (n + k) (map_partial_pullback n f k S)"
-      using A assms(1) is_semialg_mapE is_semialgebraicI 
+      using A assms(1) is_semialg_mapE is_semialgebraicI
       by blast
     have 2: "(map_partial_pullback n f k S) = (map_partial_pullback n g k S)"
       unfolding map_partial_pullback_def map_partial_image_def evimage_def
     proof
       show "(\<lambda>xs. f (take n xs) @ drop n xs) -` S \<inter> carrier (Q\<^sub>p\<^bsup>n+k\<^esup>) \<subseteq> (\<lambda>xs. g (take n xs) @ drop n xs) -` S \<inter> carrier (Q\<^sub>p\<^bsup>n+k\<^esup>)"
       proof fix x assume A: "x \<in> (\<lambda>xs. f (take n xs) @ drop n xs) -` S \<inter> carrier (Q\<^sub>p\<^bsup>n+k\<^esup>)"
         have "(take n x) \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
           using assms A
           by (meson Int_iff le_add1 take_closed)
         then have "f (take n x) = g (take n x)"
           using assms unfolding  restrict_def
-          by meson 
+          by meson
         then show "x \<in> (\<lambda>xs. g (take n xs) @ drop n xs) -` S \<inter> carrier (Q\<^sub>p\<^bsup>n+k\<^esup>)"
-          using assms 
-          by (metis (no_types, lifting) A Int_iff vimageE vimageI)         
+          using assms
+          by (metis (no_types, lifting) A Int_iff vimageE vimageI)
       qed
       show " (\<lambda>xs. g (take n xs) @ drop n xs) -` S \<inter> carrier (Q\<^sub>p\<^bsup>n+k\<^esup>) \<subseteq> (\<lambda>xs. f (take n xs) @ drop n xs) -` S \<inter> carrier (Q\<^sub>p\<^bsup>n+k\<^esup>)"
       proof fix x assume A: "x \<in> (\<lambda>xs. g (take n xs) @ drop n xs) -` S \<inter> carrier (Q\<^sub>p\<^bsup>n+k\<^esup>)"
         have "(take n x) \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
-          using assms 
+          using assms
           by (meson A  inf_le2 le_add1 subset_iff take_closed)
         then have "f (take n x) = g (take n x)"
           using assms unfolding  restrict_def
-          by meson 
+          by meson
         then show " x \<in> (\<lambda>xs. f (take n xs) @ drop n xs) -` S \<inter> carrier (Q\<^sub>p\<^bsup>n+k\<^esup>)"
           using A by blast
       qed
     qed
-    then show "is_semialgebraic (n + k) (map_partial_pullback n g k S)" 
-      using 1 by auto 
+    then show "is_semialgebraic (n + k) (map_partial_pullback n g k S)"
+      using 1 by auto
   qed
 qed
 
 lemma semialg_function_tuple_is_semialg_map:
   assumes "is_semialg_function_tuple m fs"
   assumes "length fs = n"
   shows "is_semialg_map m n (function_tuple_eval Q\<^sub>p m fs)"
   apply(rule is_semialg_mapI)
-  using function_tuple_eval_closed[of m fs] 
+  using function_tuple_eval_closed[of m fs]
   apply (metis Pi_I assms(1) assms(2) semialg_function_tuple_is_function_tuple)
 proof- fix k S assume A: "S \<in> semialg_sets (n + k)"
   show "is_semialgebraic (m + k) (map_partial_pullback m (function_tuple_eval Q\<^sub>p m fs) k S)"
   using is_semialg_map_tupleE[of m fs k S] assms A tuple_partial_pullback_is_semialg_map_tuple[of m fs]
   unfolding tuple_partial_pullback_def map_partial_pullback_def
-        map_partial_image_def tuple_partial_image_def is_semialgebraic_def  
-  by (metis evimage_def)  
+        map_partial_image_def tuple_partial_image_def is_semialgebraic_def
+  by (metis evimage_def)
 qed
 
 lemma index_is_semialg_function:
   assumes "n > k"
   shows "is_semialg_function n (\<lambda>as. as!k)"
 proof-
   have 0: "restrict (\<lambda>as. as!k) (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) = restrict (Qp_ev (pvar Q\<^sub>p k)) (carrier (Q\<^sub>p\<^bsup>n\<^esup>))"
-    using assms by (metis (no_types, lifting) eval_pvar restrict_ext)    
+    using assms by (metis (no_types, lifting) eval_pvar restrict_ext)
   have 1: "is_semialg_function n (Qp_ev (pvar Q\<^sub>p k))"
     using pvar_closed   assms poly_is_semialg[of  "pvar Q\<^sub>p k"] by blast
-  show ?thesis 
-    using 0 1 semialg_function_on_carrier[of n "Qp_ev (pvar Q\<^sub>p k)" "(\<lambda>as. as!k)"] 
+  show ?thesis
+    using 0 1 semialg_function_on_carrier[of n "Qp_ev (pvar Q\<^sub>p k)" "(\<lambda>as. as!k)"]
     by presburger
 qed
 
-definition Qp_ith where 
+definition Qp_ith where
 "Qp_ith m i = (\<lambda> x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). x!i)"
 
-lemma Qp_ith_closed: 
+lemma Qp_ith_closed:
   assumes "i < m"
   shows "Qp_ith m i \<in> carrier (SA m)"
 proof(rule SA_car_memI)
   show "Qp_ith m i \<in> carrier (function_ring (carrier (Q\<^sub>p\<^bsup>m\<^esup>)) Q\<^sub>p)"
     apply(rule Qp.function_ring_car_memI[of "carrier(Q\<^sub>p\<^bsup>m\<^esup>)"])
-    using assms cartesian_power_car_memE'[of _ Q\<^sub>p m i] unfolding Qp_ith_def 
+    using assms cartesian_power_car_memE'[of _ Q\<^sub>p m i] unfolding Qp_ith_def
     apply (metis restrict_apply)
     unfolding restrict_def by meson
   have 0: "\<And>x. x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<Longrightarrow> Qp_ith m i x = eval_at_point Q\<^sub>p  x (pvar Q\<^sub>p i)"
     using assms eval_pvar[of i m] unfolding Qp_ith_def restrict_def by presburger
   have 1: "is_semialg_function m (eval_at_poly Q\<^sub>p  (pvar Q\<^sub>p i))"
     using assms pvar_closed[of i m] poly_is_semialg by blast
-  show "is_semialg_function m (local.Qp_ith m i)" 
+  show "is_semialg_function m (local.Qp_ith m i)"
     apply(rule semialg_function_on_carrier'[of m "eval_at_poly Q\<^sub>p  (pvar Q\<^sub>p i)"])
-    using 1 apply blast 
-    using 0 by blast  
+    using 1 apply blast
+    using 0 by blast
 qed
 
 lemma take_is_semialg_map:
   assumes "n \<ge> k"
   shows "is_semialg_map n k (take k)"
 proof-
   obtain fs where fs_def: "fs = map (\<lambda>i::nat. (\<lambda>as \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). as!i)) [0::nat..< k]"
-    by blast 
+    by blast
   have 0: "is_semialg_function_tuple n fs"
   proof(rule is_semialg_function_tupleI)
     fix f assume A: "f \<in> set fs"
     then obtain i where i_def: "i \<in> set [0::nat..< k] \<and> f = (\<lambda>as \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). as!i)"
-      using A fs_def 
+      using A fs_def
       by (metis (no_types, lifting) in_set_conv_nth length_map nth_map)
     have i_less: "i < k"
     proof-
       have "set [0::nat..< k] = {0..<k}"
         using atLeastLessThan_upt by blast
-      then show ?thesis using i_def 
+      then show ?thesis using i_def
         using atLeastLessThan_iff by blast
     qed
     show "is_semialg_function n f"
-      apply(rule semialg_function_on_carrier[of n "(\<lambda>as. as ! i)"], 
+      apply(rule semialg_function_on_carrier[of n "(\<lambda>as. as ! i)"],
             rule index_is_semialg_function[of i n ]  )
-      using A i_def assms by auto               
+      using A i_def assms by auto
   qed
   have 1: "restrict (function_tuple_eval Q\<^sub>p n fs) (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) = restrict (take k) (carrier (Q\<^sub>p\<^bsup>n\<^esup>))"
-  unfolding function_tuple_eval_def 
-  proof fix x 
+  unfolding function_tuple_eval_def
+  proof fix x
     show " (\<lambda>x\<in>carrier (Q\<^sub>p\<^bsup>n\<^esup>). map (\<lambda>f. f x) fs) x = restrict (take k) (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) x"
   proof(cases "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)")
     case True
     have "(map (\<lambda>f. f x) fs) = take k x"
     proof-
       have "\<And>i. i < k \<Longrightarrow> (map (\<lambda>f. f x) fs) ! i = x ! i"
       proof- fix i assume A: "i < k"
         have "length [0::nat..< k] = k"
           using assms by simp
         then have "length fs = k"
-          using fs_def  
+          using fs_def
           by (metis length_map)
         then have 0: "(map (\<lambda>f. f x) fs) ! i = (fs!i) x"
-          using A by (meson nth_map) 
+          using A by (meson nth_map)
         have 1: "(fs!i) x = x!i"
-          using A nth_map[of i "[0..<k]" "(\<lambda>i. \<lambda>as\<in>carrier (Q\<^sub>p\<^bsup>n\<^esup>). as ! i)"] True 
-          unfolding fs_def  restrict_def  by auto                  
-        then show "map (\<lambda>f. f x) fs ! i = x ! i" 
+          using A nth_map[of i "[0..<k]" "(\<lambda>i. \<lambda>as\<in>carrier (Q\<^sub>p\<^bsup>n\<^esup>). as ! i)"] True
+          unfolding fs_def  restrict_def  by auto
+        then show "map (\<lambda>f. f x) fs ! i = x ! i"
           using 0 assms A True fs_def nth_map[of i fs] cartesian_power_car_memE[of x Q\<^sub>p n]
           by blast
       qed
       then have 0: "\<And>i. i < k \<Longrightarrow> (map (\<lambda>f. f x) fs) ! i = (take k x) ! i"
         using assms True  nth_take by blast
       have 1: "length (map (\<lambda>f. f x) fs) = length (take k x)"
-        using fs_def True assms 
+        using fs_def True assms
         by (metis cartesian_power_car_memE length_map map_nth take_closed)
       have 2: "length (take k x) = k"
-        using assms True cartesian_power_car_memE take_closed 
+        using assms True cartesian_power_car_memE take_closed
         by blast
-      show ?thesis using 0 1 2 
+      show ?thesis using 0 1 2
         by (metis nth_equalityI)
     qed
-    then show ?thesis using True unfolding restrict_def  
-      by presburger   
+    then show ?thesis using True unfolding restrict_def
+      by presburger
   next
     case False
-    then show ?thesis unfolding restrict_def  
+    then show ?thesis unfolding restrict_def
       by (simp add: False)
   qed
   qed
   have 2: " is_semialg_map n k (function_tuple_eval Q\<^sub>p n fs)"
     using 0 semialg_function_tuple_is_semialg_map[of n fs k] assms fs_def  length_map
     by (metis (no_types, lifting) diff_zero length_upt)
-  show ?thesis using 1 2 
+  show ?thesis using 1 2
     using semialg_map_on_carrier by blast
 qed
 
 lemma drop_is_semialg_map:
   shows "is_semialg_map (k + n) n (drop k)"
 proof-
   obtain fs where fs_def: "fs = map (\<lambda>i::nat. (\<lambda>as \<in> carrier (Q\<^sub>p\<^bsup>n+k\<^esup>). as!i)) [k..<n+k]"
-    by blast 
+    by blast
   have 0: "is_semialg_function_tuple (k+n) fs"
   proof(rule is_semialg_function_tupleI)
     fix f assume A: "f \<in> set fs"
     then obtain i where i_def: "i \<in> set [k..<n+k] \<and> f = (\<lambda>as \<in> carrier (Q\<^sub>p\<^bsup>n+k\<^esup>). as!i)"
-      using A fs_def 
+      using A fs_def
       by (metis (no_types, lifting) in_set_conv_nth length_map nth_map)
     have i_less: "i \<ge> k \<and> i < n + k"
     proof-
       have "set [k..<n+k] = {k..<n+k}"
         using atLeastLessThan_upt by blast
-      then show ?thesis using i_def 
+      then show ?thesis using i_def
         using atLeastLessThan_iff by blast
     qed
     have "is_semialg_function (n + k) f"
-      using A index_is_semialg_function[of i "n+k" ] 
+      using A index_is_semialg_function[of i "n+k" ]
             i_less semialg_function_on_carrier[of "n+k" "(\<lambda>as. as ! i)" f] i_def
-            restrict_is_semialg 
-      by blast           
+            restrict_is_semialg
+      by blast
     then show "is_semialg_function (k + n) f"
       by (simp add: add.commute)
   qed
   have 1: "restrict (function_tuple_eval Q\<^sub>p (n+k) fs) (carrier (Q\<^sub>p\<^bsup>n+k\<^esup>)) = restrict (drop k) (carrier (Q\<^sub>p\<^bsup>n+k\<^esup>))"
-  unfolding function_tuple_eval_def 
-  proof fix x 
+  unfolding function_tuple_eval_def
+  proof fix x
     show " (\<lambda>x\<in>carrier (Q\<^sub>p\<^bsup>n+k\<^esup>). map (\<lambda>f. f x) fs) x = restrict (drop k) (carrier (Q\<^sub>p\<^bsup>n+k\<^esup>)) x"
   proof(cases "x\<in>carrier (Q\<^sub>p\<^bsup>n+k\<^esup>)")
     case True
     have "(map (\<lambda>f. f x) fs) = drop k x"
     proof-
       have "\<And>i. i < n \<Longrightarrow> (map (\<lambda>f. f x) fs) ! i = x ! (i+k)"
       proof- fix i assume A: "i < n"
         have 00: "length [k..<n+k] = n"
            by simp
         then have "length fs = n"
-          using fs_def  
+          using fs_def
           by (metis length_map)
         then have 0:"(map (\<lambda>f. f x) fs) ! i = (fs!i) x"
-          using A  by (meson nth_map)          
+          using A  by (meson nth_map)
         have "[k..<n+k]!i = i + k"
           by (simp add: A)
         have "( map (\<lambda>i. \<lambda>as\<in>carrier (Q\<^sub>p\<^bsup>n+k\<^esup>). as ! i) [k..<n + k]) ! i = ((\<lambda>i. \<lambda>as\<in>carrier (Q\<^sub>p\<^bsup>n+k\<^esup>). as ! i) ([k..<n + k] ! i))"
-          using A 00 nth_map[of i "[k..< n + k]" "(\<lambda>i. \<lambda>as\<in>carrier (Q\<^sub>p\<^bsup>n+k\<^esup>). as ! i)"] 
+          using A 00 nth_map[of i "[k..< n + k]" "(\<lambda>i. \<lambda>as\<in>carrier (Q\<^sub>p\<^bsup>n+k\<^esup>). as ! i)"]
           by linarith
         then have 1: "fs!i =  (\<lambda>as \<in> carrier (Q\<^sub>p\<^bsup>n+k\<^esup>). as!(i+k))"
           using fs_def A  \<open>[k..<n + k] ! i = i + k\<close> by presburger
-        then show "map (\<lambda>f. f x) fs ! i = x ! (i + k)" 
+        then show "map (\<lambda>f. f x) fs ! i = x ! (i + k)"
           using True 0
-          by (metis (no_types, lifting) restrict_apply)         
+          by (metis (no_types, lifting) restrict_apply)
       qed
       then have 0: "\<And>i. i < n \<Longrightarrow> (map (\<lambda>f. f x) fs) ! i = (drop k x) ! i"
-        using True  nth_drop 
+        using True  nth_drop
         by (metis add.commute cartesian_power_car_memE le_add2)
       have 1: "length (map (\<lambda>f. f x) fs) = length (drop k x)"
         using fs_def True length_drop[of k x]
         by (metis cartesian_power_car_memE length_map length_upt)
       have 2: "length (drop k x) = n"
-        using True cartesian_power_car_memE  
-        by (metis add_diff_cancel_right' length_drop)    
-      show ?thesis using 0 1 2 
+        using True cartesian_power_car_memE
+        by (metis add_diff_cancel_right' length_drop)
+      show ?thesis using 0 1 2
         by (metis nth_equalityI)
     qed
-    then show ?thesis using True unfolding restrict_def  
-      by presburger   
+    then show ?thesis using True unfolding restrict_def
+      by presburger
   next
     case False
-    then show ?thesis unfolding restrict_def  
+    then show ?thesis unfolding restrict_def
       by (simp add: False)
   qed
   qed
   have 00: "length [k..<n+k] = n"
     by simp
   then have "length fs = n"
-    using fs_def  
+    using fs_def
     by (metis length_map)
   then have 2: " is_semialg_map (k + n) n (function_tuple_eval Q\<^sub>p (k + n) fs)"
-    using 0 semialg_function_tuple_is_semialg_map[of "k + n" fs n]   
+    using 0 semialg_function_tuple_is_semialg_map[of "k + n" fs n]
     by blast
   show ?thesis using 1 2
-    using semialg_map_on_carrier[of "k + n" n "function_tuple_eval Q\<^sub>p (k + n) fs" "drop k"] 
-    by (metis add.commute)    
+    using semialg_map_on_carrier[of "k + n" n "function_tuple_eval Q\<^sub>p (k + n) fs" "drop k"]
+    by (metis add.commute)
 qed
 
 lemma project_at_indices_is_semialg_map:
   assumes "S \<subseteq> {..<n}"
   shows "is_semialg_map n (card S) \<pi>\<^bsub>S\<^esub>"
 proof-
   obtain k where k_def: "k = card S"
     by blast
   have 0: "card {..<n} = n"
     by simp
   have 1: "finite S"
-    using assms finite_subset 
+    using assms finite_subset
     by blast
   have 2: "card S \<le> n"
-    using assms 0 1 
+    using assms 0 1
     by (metis card_mono finite_lessThan)
   then have k_size: " k \<le> n"
-    using k_def assms 0 1 2 
+    using k_def assms 0 1 2
     by blast
   obtain fs where fs_def: "fs = map (\<lambda>i::nat. (\<lambda>as \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). as!(nth_elem S i))) [0..<k]"
-    by blast 
+    by blast
   have 4: "length fs = k"
-    using fs_def assms "1" k_def  
-    by (metis Ex_list_of_length length_map map_nth)   
+    using fs_def assms "1" k_def
+    by (metis Ex_list_of_length length_map map_nth)
   have 5: "is_semialg_function_tuple n fs"
   proof(rule is_semialg_function_tupleI)
     fix f assume A: "f \<in> set fs"
     then obtain i where i_def: "i \<in> set [0..<k] \<and> f = (\<lambda>as \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). as!(nth_elem S i))"
-      using A fs_def atLeast_upt "4"  in_set_conv_nth  map_eq_conv map_nth 
+      using A fs_def atLeast_upt "4"  in_set_conv_nth  map_eq_conv map_nth
       by auto
     have i_le_k:"i < k"
     proof-
       have "set [0..<k] = {..<k}"
         using atLeast_upt by blast
       then show ?thesis
-        using i_def 
+        using i_def
         by blast
     qed
     then have i_in_S: "nth_elem S i \<in> S"
-      using "1" k_def nth_elem_closed by blast      
+      using "1" k_def nth_elem_closed by blast
     then have "nth_elem S i < n"
-      using assms 
+      using assms
       by blast
     show "is_semialg_function n f"
-      using A index_is_semialg_function[of "nth_elem S i" n]  
-            semialg_function_on_carrier[of n "(\<lambda>as. as ! nth_elem S i)"] i_def restrict_is_semialg 
-            \<open>nth_elem S i < n\<close> by blast 
+      using A index_is_semialg_function[of "nth_elem S i" n]
+            semialg_function_on_carrier[of n "(\<lambda>as. as ! nth_elem S i)"] i_def restrict_is_semialg
+            \<open>nth_elem S i < n\<close> by blast
   qed
   have 6: "restrict (function_tuple_eval Q\<^sub>p n fs) (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) = restrict (\<pi>\<^bsub>S\<^esub>) (carrier (Q\<^sub>p\<^bsup>n\<^esup>))"
-  unfolding function_tuple_eval_def 
-  proof fix x 
+  unfolding function_tuple_eval_def
+  proof fix x
     show " (\<lambda>x\<in>carrier (Q\<^sub>p\<^bsup>n\<^esup>). map (\<lambda>f. f x) fs) x = restrict (\<pi>\<^bsub>S\<^esub>) (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) x"
     proof(cases "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)")
     case True
     have "(map (\<lambda>f. f x) fs) = \<pi>\<^bsub>S\<^esub> x"
     proof-
       have "\<And>i. i < k \<Longrightarrow> (map (\<lambda>f. f x) fs) ! i = (\<pi>\<^bsub>S\<^esub> x) ! i"
       proof- fix i assume A: "i < k"
         have T0:"(map (\<lambda>f. f x) fs) ! i = (fs!i) x"
-          using A nth_map by (metis "4")          
+          using A nth_map by (metis "4")
         have T1: "indices_of x = {..< n}"
-          using True cartesian_power_car_memE indices_of_def 
+          using True cartesian_power_car_memE indices_of_def
           by blast
         have T2: "set (set_to_list S) \<subseteq> indices_of x"
-          using assms True  by (simp add: True "1" T1)          
+          using assms True  by (simp add: True "1" T1)
         have T3: "length x = n"
           using True cartesian_power_car_memE by blast
         have T4: "([0..<k] ! i) =   i"
-          using A by simp 
+          using A by simp
         have T5: "nth_elem S i < n"
-          using assms 0 1 2 A k_def 
-          by (meson lessThan_iff nth_elem_closed subsetD)          
+          using assms 0 1 2 A k_def
+          by (meson lessThan_iff nth_elem_closed subsetD)
         have T6: "nth_elem S ([0..<k] ! i) = nth_elem S i"
           by (simp add: T4)
         have T6: "(\<lambda>as\<in>carrier (Q\<^sub>p\<^bsup>n\<^esup>). as ! nth_elem S ([0..<k] ! i)) x = x! (nth_elem S i)"
         proof-
           have "(\<lambda>as\<in>carrier (Q\<^sub>p\<^bsup>n\<^esup>). as ! nth_elem S ([0..<k] ! i)) x = x! nth_elem S ([0..<k] ! i)"
-            using True restrict_def by metis 
-          then show ?thesis 
-            using A T3 T4 0 1 2 T5 T6  True 
-            by metis 
+            using True restrict_def by metis
+          then show ?thesis
+            using A T3 T4 0 1 2 T5 T6  True
+            by metis
         qed
         have T7: " map (\<lambda>f. f x) fs ! i = x! (nth_elem S i)"
-          using fs_def T0 A nth_map[of i "[0..<k]" "(\<lambda>i. \<lambda>as\<in>carrier (Q\<^sub>p\<^bsup>n\<^esup>). as ! nth_elem S i)"] 
+          using fs_def T0 A nth_map[of i "[0..<k]" "(\<lambda>i. \<lambda>as\<in>carrier (Q\<^sub>p\<^bsup>n\<^esup>). as ! nth_elem S i)"]
           by (metis "4" T6 length_map)
         show "map (\<lambda>f. f x) fs ! i = \<pi>\<^bsub>S\<^esub> x ! i"
-          using True T0 T1 T2 fs_def 5 unfolding T7 
-          by (metis A assms(1) k_def project_at_indices_nth)                                    
+          using True T0 T1 T2 fs_def 5 unfolding T7
+          by (metis A assms(1) k_def project_at_indices_nth)
       qed
       have 1: "length (map (\<lambda>f. f x) fs) = length (\<pi>\<^bsub>S\<^esub> x)"
-        using fs_def True assms proj_at_index_list_length[of S x] k_def  
-        by (metis "4" cartesian_power_car_memE indices_of_def length_map)         
+        using fs_def True assms proj_at_index_list_length[of S x] k_def
+        by (metis "4" cartesian_power_car_memE indices_of_def length_map)
       have 2: "length (\<pi>\<^bsub>S\<^esub> x) = k"
         using assms True 0 1 2
-        by (metis "4" length_map)        
-      show ?thesis using  1 2 
-        by (metis \<open>\<And>i. i < k \<Longrightarrow> map (\<lambda>f. f x) fs ! i = \<pi>\<^bsub>S\<^esub> x ! i\<close> nth_equalityI)        
+        by (metis "4" length_map)
+      show ?thesis using  1 2
+        by (metis \<open>\<And>i. i < k \<Longrightarrow> map (\<lambda>f. f x) fs ! i = \<pi>\<^bsub>S\<^esub> x ! i\<close> nth_equalityI)
     qed
-    then show ?thesis using True unfolding restrict_def  
-      by presburger   
+    then show ?thesis using True unfolding restrict_def
+      by presburger
     next
     case False
-    then show ?thesis unfolding restrict_def  
+    then show ?thesis unfolding restrict_def
       by (simp add: False)
     qed
   qed
   have 7: " is_semialg_map n k (function_tuple_eval Q\<^sub>p n fs)"
-    using 0 semialg_function_tuple_is_semialg_map[of n fs k] assms fs_def  length_map "4" "5" k_size 
+    using 0 semialg_function_tuple_is_semialg_map[of n fs k] assms fs_def  length_map "4" "5" k_size
     by blast
-  show ?thesis using 6 7 
-    using semialg_map_on_carrier k_def 
-    by blast    
+  show ?thesis using 6 7
+    using semialg_map_on_carrier k_def
+    by blast
 qed
 
 lemma tl_is_semialg_map:
   shows "is_semialg_map (Suc n) n tl"
 proof-
   have 0: "(card {1..<Suc n}) = n"
   proof-
     have "Suc n - 1 = n"
       using diff_Suc_1 by blast
-    then show ?thesis 
+    then show ?thesis
       by simp
-  qed   
+  qed
   have 3: "{1..<Suc n} \<subseteq> {..<Suc n}"
     using atLeastLessThan_iff by blast
   have 4: " is_semialg_map (Suc n) n (project_at_indices {1..<Suc n})"
-    using 0 project_at_indices_is_semialg_map 
-    by (metis "3")   
+    using 0 project_at_indices_is_semialg_map
+    by (metis "3")
   show ?thesis
-    using 0 3 4 
-        semialg_map_on_carrier[of "Suc n" n "(project_at_indices {1..<Suc n})" tl] 
+    using 0 3 4
+        semialg_map_on_carrier[of "Suc n" n "(project_at_indices {1..<Suc n})" tl]
     unfolding restrict_def
     by (metis (no_types, lifting) tl_as_projection)
 qed
 
 text\<open>Coordinate functions are semialgebraic maps.\<close>
 
 lemma coord_fun_is_SA:
   assumes "is_semialg_map n m g"
   assumes "i < m"
   shows "coord_fun Q\<^sub>p n g i \<in> carrier (SA n)"
 proof(rule SA_car_memI)
   show "coord_fun Q\<^sub>p n g i \<in> carrier (function_ring (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) Q\<^sub>p)"
     apply(rule Qp.function_ring_car_memI)
-    unfolding coord_fun_def using assms 
+    unfolding coord_fun_def using assms
      apply (metis (no_types, lifting) Pi_iff cartesian_power_car_memE' is_semialg_map_closed restrict_apply')
       by (meson restrict_apply)
   show "is_semialg_function n (coord_fun Q\<^sub>p n g i)"
   proof-
     have 0: "is_semialg_function m (\<lambda> x. x ! i)"
       using assms gr_implies_not0 index_is_semialg_function by blast
     have 1: "(coord_fun Q\<^sub>p n g i) = restrict ((\<lambda>x.  x ! i) \<circ> g) (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) "
-      unfolding coord_fun_def using comp_apply 
-      by metis 
-    show ?thesis 
+      unfolding coord_fun_def using comp_apply
+      by metis
+    show ?thesis
       using semialg_function_on_carrier[of n "((\<lambda>x. x ! i) \<circ> g)" "coord_fun Q\<^sub>p n g i"]
              assms semialg_function_comp_closed[of m "\<lambda>x. x ! i" n g] assms 0 1
-      unfolding coord_fun_def 
+      unfolding coord_fun_def
       using restrict_is_semialg by presburger
   qed
 qed
 
 lemma coord_fun_map_is_semialg_tuple:
   assumes "is_semialg_map n m g"
   shows "is_semialg_function_tuple n (map (coord_fun Q\<^sub>p n g) [0..<m])"
 proof(rule is_semialg_function_tupleI)
   have 0: "set [0..<m] = {..<m}"
     using atLeast_upt by blast
   fix f assume A: "f \<in> set (map (coord_fun Q\<^sub>p n g) [0..<m])"
   then obtain i where i_def: "i < m \<and> f = coord_fun Q\<^sub>p n g i"
-    using set_map[of "coord_fun Q\<^sub>p n g" "[0..<m]"] 0 
+    using set_map[of "coord_fun Q\<^sub>p n g" "[0..<m]"] 0
     by (metis image_iff lessThan_iff)
   show " is_semialg_function n f"
   using i_def A assms coord_fun_is_SA[of n m g i] SA_imp_semialg by blast
 qed
 
 lemma semialg_map_cons:
   assumes "is_semialg_map n m g"
   assumes "f \<in> carrier (SA n)"
   shows "is_semialg_map n (Suc m) (\<lambda> x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). f x # g x)"
 proof-
   obtain Fs where Fs_def: "Fs = f # (map (coord_fun Q\<^sub>p n g) [0..<m])"
-    by blast 
+    by blast
   have 0: "is_semialg_function_tuple n Fs"
     apply(rule  is_semialg_function_tupleI)
-    using is_semialg_function_tupleE'[of n "map (coord_fun Q\<^sub>p n g) [0..<m]"] 
+    using is_semialg_function_tupleE'[of n "map (coord_fun Q\<^sub>p n g) [0..<m]"]
           coord_fun_map_is_semialg_tuple[of n m g] assms SA_car_memE(1)[of f n]
           set_ConsD[of _ f "map (coord_fun Q\<^sub>p n g) [0..<m]"] assms
     unfolding Fs_def by blast
   have 1: "(\<lambda> x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). f x # g x) = (\<lambda> x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). function_tuple_eval Q\<^sub>p n Fs x)"
   proof(rule ext) fix x show "(\<lambda>x\<in>carrier (Q\<^sub>p\<^bsup>n\<^esup>). f x # g x) x = restrict (function_tuple_eval Q\<^sub>p n Fs) (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) x"
     proof(cases "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)")
       case True then have T0: "(\<lambda>x\<in>carrier (Q\<^sub>p\<^bsup>n\<^esup>). f x # g x) x = f x # g x"
         by (meson restrict_apply')
       have T1: "restrict (function_tuple_eval Q\<^sub>p n Fs) (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) x = function_tuple_eval Q\<^sub>p n Fs x"
         using True by (meson restrict_apply')
       hence T2: "restrict (function_tuple_eval Q\<^sub>p n Fs) (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) x = f x # (function_tuple_eval Q\<^sub>p n (map (coord_fun Q\<^sub>p n g) [0..<m]) x)"
         unfolding function_tuple_eval_def Fs_def by (metis (no_types, lifting) list.simps(9))
       have T3: "g x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
-        using assms True is_semialg_map_closed by blast 
+        using assms True is_semialg_map_closed by blast
       have "length [0..<m] = m"
-        by auto 
+        by auto
       hence T4: "length (map (coord_fun Q\<^sub>p n g) [0..<m]) = m"
-        using length_map[of "(coord_fun Q\<^sub>p n g)" "[0..<m]"] by metis  
+        using length_map[of "(coord_fun Q\<^sub>p n g)" "[0..<m]"] by metis
       hence T5: "length (function_tuple_eval Q\<^sub>p n (map (coord_fun Q\<^sub>p n g) [0..<m]) x) = m"
         unfolding function_tuple_eval_def using length_map by metis
       have T6: "(function_tuple_eval Q\<^sub>p n (map (coord_fun Q\<^sub>p n g) [0..<m]) x) = g x"
-        apply(rule nth_equalityI) using T3 T5 
+        apply(rule nth_equalityI) using T3 T5
           using cartesian_power_car_memE apply blast
-          using cartesian_power_car_memE[of "g x" Q\<^sub>p m]  T5 T4  T3 True  
+          using cartesian_power_car_memE[of "g x" Q\<^sub>p m]  T5 T4  T3 True
                 nth_map[of _ "(map (\<lambda>i. \<lambda>x\<in>carrier (Q\<^sub>p\<^bsup>n\<^esup>). g x ! i) [0..<m])" "(\<lambda>f. f x)"]
                 nth_map[of _ "[0..<m]" "(\<lambda>i. \<lambda>x\<in>carrier (Q\<^sub>p\<^bsup>n\<^esup>). g x ! i)"]
-          unfolding function_tuple_eval_def  coord_fun_def restrict_def 
+          unfolding function_tuple_eval_def  coord_fun_def restrict_def
           by (metis (no_types, lifting) \<open>length [0..<m] = m\<close> map_nth nth_map)
       hence T7: "restrict (function_tuple_eval Q\<^sub>p n Fs) (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) x = f x # g x"
         using T4 T2 by presburger
       thus ?thesis using T0
         by presburger
     next
       case False
-      then show ?thesis unfolding restrict_def 
-        by metis  
+      then show ?thesis unfolding restrict_def
+        by metis
     qed
   qed
   have 2: "length Fs = Suc m"
     unfolding Fs_def using length_map[of "(coord_fun Q\<^sub>p n g)" "[0..<m]"] length_Cons[of f "map (coord_fun Q\<^sub>p n g) [0..<m]"]
-    using length_upt by presburger    
+    using length_upt by presburger
   have 3: "is_semialg_map n (Suc m) (\<lambda> x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). function_tuple_eval Q\<^sub>p n Fs x)"
-    apply(rule semialg_map_on_carrier[of _ _  "function_tuple_eval Q\<^sub>p n Fs"], 
+    apply(rule semialg_map_on_carrier[of _ _  "function_tuple_eval Q\<^sub>p n Fs"],
           intro semialg_function_tuple_is_semialg_map[of n Fs "Suc m"] 0 2)
-    by auto 
+    by auto
   show ?thesis using 1 3
-    by presburger  
+    by presburger
 qed
 
 text\<open>Extensional Semialgebraic Maps:\<close>
 
 definition semialg_maps where
 "semialg_maps n m \<equiv> {f. is_semialg_map n m f \<and> f \<in> struct_maps (Q\<^sub>p\<^bsup>n\<^esup>) (Q\<^sub>p\<^bsup>m\<^esup>)}"
 
 lemma semialg_mapsE:
   assumes "f \<in> (semialg_maps n m)"
   shows "is_semialg_map n m f"
         "f \<in> struct_maps (Q\<^sub>p\<^bsup>n\<^esup>) (Q\<^sub>p\<^bsup>m\<^esup>)"
         "f \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<rightarrow> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
-  using assms unfolding semialg_maps_def apply blast 
-  using assms unfolding semialg_maps_def apply blast 
+  using assms unfolding semialg_maps_def apply blast
+  using assms unfolding semialg_maps_def apply blast
   apply(rule is_semialg_map_closed)
-  using assms unfolding semialg_maps_def by blast 
-
-definition to_semialg_map where 
+  using assms unfolding semialg_maps_def by blast
+
+definition to_semialg_map where
 "to_semialg_map n m f = restrict f (carrier (Q\<^sub>p\<^bsup>n\<^esup>))"
 
 lemma to_semialg_map_is_semialg_map:
   assumes "is_semialg_map n m f"
   shows "to_semialg_map n m f \<in> semialg_maps n m"
   using assms unfolding to_semialg_map_def semialg_maps_def struct_maps_def mem_Collect_eq
   using is_semialg_map_closed' semialg_map_on_carrier by force
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsection\<open>Application of Functions to Segments of Tuples\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 
 definition take_apply where
 "take_apply m n f = restrict (f \<circ> take n) (carrier (Q\<^sub>p\<^bsup>m\<^esup>))"
 
 definition drop_apply where
 "drop_apply m n f = restrict (f \<circ> drop n) (carrier (Q\<^sub>p\<^bsup>m\<^esup>))"
 
 lemma take_apply_closed:
   assumes "f \<in> carrier (Fun\<^bsub>n\<^esub> Q\<^sub>p)"
   assumes "k \<ge> n"
   shows "take_apply k n f \<in> carrier (Fun\<^bsub>k\<^esub> Q\<^sub>p)"
 proof(rule Qp.function_ring_car_memI)
   show  "\<And>a. a \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>) \<Longrightarrow> take_apply k n f a \<in> carrier Q\<^sub>p"
   proof-  fix a assume A: "a \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>)" show "take_apply k n f a \<in> carrier Q\<^sub>p"
       using A assms comp_apply[of f "take n" a] Qp.function_ring_car_memE[of f n] take_closed[of n k a]
-      unfolding take_apply_def restrict_def 
+      unfolding take_apply_def restrict_def
       by metis
   qed
   show " \<And>a. a \<notin> carrier (Q\<^sub>p\<^bsup>k\<^esup>) \<Longrightarrow> take_apply k n f a = undefined"
-    unfolding take_apply_def restrict_def 
+    unfolding take_apply_def restrict_def
     by meson
-qed 
+qed
 
 lemma take_apply_SA_closed:
   assumes "f \<in> carrier (SA n)"
   assumes "k \<ge> n"
   shows "take_apply k n f \<in> carrier (SA k)"
   apply(rule SA_car_memI)
   using SA_car_memE(2) assms(1) assms(2) take_apply_closed apply blast
-  using assms take_is_semialg_map[of n k] unfolding take_apply_def 
-  by (metis padic_fields.SA_imp_semialg 
+  using assms take_is_semialg_map[of n k] unfolding take_apply_def
+  by (metis padic_fields.SA_imp_semialg
       padic_fields_axioms restrict_is_semialg semialg_function_comp_closed)
-  
+
 lemma drop_apply_closed:
   assumes "f \<in> carrier (Fun\<^bsub>k - n\<^esub> Q\<^sub>p)"
   assumes "k \<ge> n"
   shows "drop_apply k n f \<in> carrier (Fun\<^bsub>k\<^esub> Q\<^sub>p)"
 proof(rule Qp.function_ring_car_memI)
   show  " \<And>a. a \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>) \<Longrightarrow> drop_apply k n f a \<in> carrier Q\<^sub>p"
   proof-  fix a assume A: "a \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>)" show "drop_apply k n f a \<in> carrier Q\<^sub>p"
       using A assms comp_apply[of f "drop n" a] Qp.function_ring_car_memE[of f ] drop_closed[of n k a Q\<^sub>p]
-      unfolding drop_apply_def restrict_def 
-      by (metis (no_types, opaque_lifting) Qp.function_ring_car_memE add_diff_cancel_right' 
-          cartesian_power_drop dec_induct diff_diff_cancel diff_less_Suc diff_less_mono2 
-          infinite_descent le_neq_implies_less less_antisym linorder_neqE_nat not_less0 not_less_eq_eq)      
+      unfolding drop_apply_def restrict_def
+      by (metis (no_types, opaque_lifting) Qp.function_ring_car_memE add_diff_cancel_right'
+          cartesian_power_drop dec_induct diff_diff_cancel diff_less_Suc diff_less_mono2
+          infinite_descent le_neq_implies_less less_antisym linorder_neqE_nat not_less0 not_less_eq_eq)
   qed
   show " \<And>a. a \<notin> carrier (Q\<^sub>p\<^bsup>k\<^esup>) \<Longrightarrow> drop_apply k n f a = undefined"
-    unfolding drop_apply_def restrict_def 
+    unfolding drop_apply_def restrict_def
     by meson
-qed 
-  
+qed
+
 lemma drop_apply_SA_closed:
   assumes "f \<in> carrier (SA (k-n))"
   assumes "k \<ge> n"
   shows "drop_apply k n f \<in> carrier (SA k)"
   apply(rule SA_car_memI)
-  using SA_car_memE(2) assms(1) assms(2) drop_apply_closed less_imp_le_nat apply blast 
-  using assms drop_is_semialg_map[of n "k - n" ] semialg_function_comp_closed[of "k - n" f k "drop n"] unfolding drop_apply_def 
+  using SA_car_memE(2) assms(1) assms(2) drop_apply_closed less_imp_le_nat apply blast
+  using assms drop_is_semialg_map[of n "k - n" ] semialg_function_comp_closed[of "k - n" f k "drop n"] unfolding drop_apply_def
   by (metis (no_types, lifting) SA_imp_semialg le_add_diff_inverse  restrict_is_semialg)
 
 lemma take_apply_apply:
   assumes "f \<in> carrier (SA n)"
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "b \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>)"
   shows "take_apply (n+k) n f (a@b) = f a"
 proof-
   have "a@b \<in> carrier (Q\<^sub>p\<^bsup>n+k\<^esup>)"
     using assms cartesian_power_concat(1) by blast
-  thus ?thesis 
-    unfolding take_apply_def restrict_def 
-    using assms cartesian_power_car_memE comp_apply[of f "take n"]  
+  thus ?thesis
+    unfolding take_apply_def restrict_def
+    using assms cartesian_power_car_memE comp_apply[of f "take n"]
     by (metis append_eq_conv_conj)
 qed
 
 lemma drop_apply_apply:
   assumes "f \<in> carrier (SA k)"
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "b \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>)"
   shows "drop_apply (n+k) n f (a@b) = f b"
 proof-
   have "a@b \<in> carrier (Q\<^sub>p\<^bsup>n+k\<^esup>)"
     using assms cartesian_power_concat(1) by blast
-  thus ?thesis 
-    unfolding drop_apply_def  restrict_def 
-    using assms cartesian_power_car_memE comp_apply[of f "drop n"]  
+  thus ?thesis
+    unfolding drop_apply_def  restrict_def
+    using assms cartesian_power_car_memE comp_apply[of f "drop n"]
     by (metis append_eq_conv_conj)
 qed
 
 lemma drop_apply_add:
   assumes "f \<in> carrier (SA n)"
   assumes "g \<in> carrier (SA n)"
   shows "drop_apply (n+k) k (f \<oplus>\<^bsub>SA n\<^esub> g) = drop_apply (n+k) k f \<oplus>\<^bsub>SA (n + k)\<^esub> drop_apply (n+k) k g"
   apply(rule function_ring_car_eqI[of _ "n + k"])
   using drop_apply_SA_closed  assms fun_add_closed SA_car_memE(2) SA_plus diff_add_inverse2 drop_apply_closed le_add2 apply presburger
   using drop_apply_SA_closed assms fun_add_closed SA_car_memE(2) SA_plus diff_add_inverse2 drop_apply_closed le_add2 apply presburger
 proof-
   fix a  assume A: "a \<in> carrier (Q\<^sub>p\<^bsup>n+k\<^esup>)"
   then obtain  b c where bc_def: "b \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>) \<and> c \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<and> a  =  b@c"
     by (metis (no_types, lifting) add.commute cartesian_power_decomp)
   have 0: "drop_apply (n + k) k (f \<oplus>\<^bsub>SA n\<^esub> g) a = f c \<oplus> g c"
-    using assms  bc_def drop_apply_apply[of "f \<oplus>\<^bsub>SA n\<^esub> g" n b k c ] 
+    using assms  bc_def drop_apply_apply[of "f \<oplus>\<^bsub>SA n\<^esub> g" n b k c ]
     by (metis SA_add SA_imp_semialg add.commute padic_fields.SA_add_closed_left padic_fields_axioms)
   then show " drop_apply (n + k) k (f \<oplus>\<^bsub>SA n\<^esub> g) a = (drop_apply (n + k) k f \<oplus>\<^bsub>SA (n + k)\<^esub> drop_apply (n + k) k g) a"
-    using bc_def drop_apply_apply assms 
+    using bc_def drop_apply_apply assms
     by (metis A SA_add add.commute)
 qed
-    
+
 lemma drop_apply_mult:
   assumes "f \<in> carrier (SA n)"
   assumes "g \<in> carrier (SA n)"
   shows "drop_apply (n+k) k (f \<otimes> \<^bsub>SA n\<^esub> g) = drop_apply (n+k) k f \<otimes>\<^bsub>SA (n + k)\<^esub> drop_apply (n+k) k g"
   apply(rule function_ring_car_eqI[of _ "n + k"])
   using drop_apply_SA_closed  assms fun_mult_closed SA_car_memE(2) SA_times diff_add_inverse2 drop_apply_closed le_add2 apply presburger
   using drop_apply_SA_closed assms fun_mult_closed SA_car_memE(2)  SA_times diff_add_inverse2 drop_apply_closed le_add2 apply presburger
 proof-
   fix a  assume A: "a \<in> carrier (Q\<^sub>p\<^bsup>n+k\<^esup>)"
   then obtain  b c where bc_def: "b \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>) \<and> c \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<and> a  =  b@c"
     by (metis (no_types, lifting) add.commute cartesian_power_decomp)
   have 0: "drop_apply (n + k) k (f \<otimes>\<^bsub>SA n\<^esub> g) a = f c \<otimes> g c"
-    using assms  bc_def drop_apply_apply[of "f \<otimes>\<^bsub>SA n\<^esub> g" n b k c ] 
-    by (metis SA_imp_semialg SA_mult SA_mult_closed_right add.commute)    
+    using assms  bc_def drop_apply_apply[of "f \<otimes>\<^bsub>SA n\<^esub> g" n b k c ]
+    by (metis SA_imp_semialg SA_mult SA_mult_closed_right add.commute)
   then show " drop_apply (n + k) k (f \<otimes>\<^bsub>SA n\<^esub> g) a = (drop_apply (n + k) k f \<otimes>\<^bsub>SA (n + k)\<^esub> drop_apply (n + k) k g) a"
     using bc_def drop_apply_apply assms by (metis A SA_mult add.commute)
 qed
 
 lemma drop_apply_one:
   shows "drop_apply (n+k) k \<one>\<^bsub>SA n\<^esub> = \<one>\<^bsub>SA (n+k)\<^esub>"
   apply(rule function_ring_car_eqI[of _  "n + k"])
   apply (metis function_one_closed SA_one add_diff_cancel_right' drop_apply_closed le_add2)
-  using function_one_closed SA_one apply presburger  
+  using function_one_closed SA_one apply presburger
 proof-
   fix a assume A: "a \<in> carrier (Q\<^sub>p\<^bsup>n+k\<^esup>)"
   show "drop_apply (n + k) k \<one>\<^bsub>SA n\<^esub> a = \<one>\<^bsub>SA (n + k)\<^esub> a"
-  unfolding drop_apply_def restrict_def 
-  using  SA_one[of "n+k"] SA_one[of n] comp_apply[of "\<one>\<^bsub>SA n\<^esub>" "drop k" a] drop_closed[of k "n+k" a Q\<^sub>p]  
+  unfolding drop_apply_def restrict_def
+  using  SA_one[of "n+k"] SA_one[of n] comp_apply[of "\<one>\<^bsub>SA n\<^esub>" "drop k" a] drop_closed[of k "n+k" a Q\<^sub>p]
         function_ring_defs(4)
-  unfolding function_one_def 
+  unfolding function_one_def
   by (metis A function_one_eval add.commute cartesian_power_drop)
 qed
 
 lemma drop_apply_is_hom:
   shows "drop_apply (n + k) k \<in> ring_hom (SA n) (SA (n + k))"
   apply(rule ring_hom_memI)
-  using drop_apply_SA_closed[of _ "k+n" k]  
-  apply (metis add.commute add_diff_cancel_left' le_add1)    
+  using drop_apply_SA_closed[of _ "k+n" k]
+  apply (metis add.commute add_diff_cancel_left' le_add1)
   using drop_apply_mult apply blast
    using drop_apply_add apply blast
      using drop_apply_one by blast
 
 lemma take_apply_add:
   assumes "f \<in> carrier (SA k)"
   assumes "g \<in> carrier (SA k)"
   shows "take_apply (n+k) k (f \<oplus>\<^bsub>SA k\<^esub> g) = take_apply (n+k) k f \<oplus>\<^bsub>SA (n + k)\<^esub> take_apply (n+k) k g"
   apply(rule function_ring_car_eqI[of _ "n + k"])
   using take_apply_SA_closed  assms fun_add_closed SA_car_memE(2) SA_plus diff_add_inverse2 take_apply_closed le_add2 apply presburger
   using take_apply_SA_closed assms fun_add_closed SA_car_memE(2) SA_plus diff_add_inverse2 take_apply_closed le_add2 apply presburger
 proof-
   fix a  assume A: "a \<in> carrier (Q\<^sub>p\<^bsup>n+k\<^esup>)"
   then obtain  b c where bc_def: "b \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>) \<and> c \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<and> a  =  b@c"
     by (metis (no_types, lifting) add.commute cartesian_power_decomp)
   hence 0: "take_apply (n + k) k (f \<oplus>\<^bsub>SA k\<^esub> g) a =  f b \<oplus> g b"
-    using assms bc_def take_apply_apply[of "f \<oplus>\<^bsub>SA k\<^esub> g" k b c ] 
+    using assms bc_def take_apply_apply[of "f \<oplus>\<^bsub>SA k\<^esub> g" k b c ]
     by (metis SA_add SA_imp_semialg add.commute padic_fields.SA_add_closed_left padic_fields_axioms)
   then show "take_apply (n + k) k (f \<oplus>\<^bsub>SA k\<^esub> g) a = (take_apply (n + k) k f \<oplus>\<^bsub>SA (n + k)\<^esub> take_apply (n + k) k g) a"
-    using bc_def take_apply_apply assms 
+    using bc_def take_apply_apply assms
     by (metis A SA_add add.commute)
 qed
-    
+
 lemma take_apply_mult:
   assumes "f \<in> carrier (SA k)"
   assumes "g \<in> carrier (SA k)"
   shows "take_apply (n+k) k (f \<otimes>\<^bsub>SA k\<^esub> g) = take_apply (n+k) k f \<otimes>\<^bsub>SA (n + k)\<^esub> take_apply (n+k) k g"
   apply(rule function_ring_car_eqI[of _ "n + k"])
   using take_apply_SA_closed  assms fun_mult_closed SA_car_memE(2) SA_times diff_add_inverse2 take_apply_closed le_add2 apply presburger
   using take_apply_SA_closed assms fun_mult_closed SA_car_memE(2) SA_times diff_add_inverse2 take_apply_closed le_add2 apply presburger
 proof-
   fix a  assume A: "a \<in> carrier (Q\<^sub>p\<^bsup>n+k\<^esup>)"
   then obtain  b c where bc_def: "b \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>) \<and> c \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<and> a  =  b@c"
     by (metis (no_types, lifting) add.commute cartesian_power_decomp)
   hence 0: "take_apply (n + k) k (f \<otimes>\<^bsub>SA k\<^esub> g) a =  f b \<otimes> g b"
-    using assms bc_def take_apply_apply[of "f \<otimes>\<^bsub>SA k\<^esub> g" k b c ] 
+    using assms bc_def take_apply_apply[of "f \<otimes>\<^bsub>SA k\<^esub> g" k b c ]
     by (metis SA_mult SA_imp_semialg add.commute padic_fields.SA_mult_closed_left padic_fields_axioms)
   then show "take_apply (n + k) k (f \<otimes>\<^bsub>SA k\<^esub> g) a = (take_apply (n + k) k f \<otimes>\<^bsub>SA (n + k)\<^esub> take_apply (n + k) k g) a"
-    using bc_def take_apply_apply assms 
+    using bc_def take_apply_apply assms
     by (metis A SA_mult add.commute)
 qed
 
 lemma take_apply_one:
   shows "take_apply (n+k) k \<one>\<^bsub>SA k\<^esub> = \<one>\<^bsub>SA (n+k)\<^esub>"
   apply(rule function_ring_car_eqI[of _  "n + k"])
   using function_one_closed SA_one le_add2 take_apply_closed apply presburger
   using function_one_closed SA_one apply presburger
 proof-
   fix a assume A: "a \<in> carrier (Q\<^sub>p\<^bsup>n+k\<^esup>)"
   show "take_apply (n + k) k \<one>\<^bsub>SA k\<^esub> a = \<one>\<^bsub>SA (n + k)\<^esub> a"
-  unfolding take_apply_def restrict_def 
-  using  SA_one[of "n+k"] SA_one[of k] comp_apply[of "\<one>\<^bsub>SA k\<^esub>" "take k" a] take_closed[of k "n + k" a] 
+  unfolding take_apply_def restrict_def
+  using  SA_one[of "n+k"] SA_one[of k] comp_apply[of "\<one>\<^bsub>SA k\<^esub>" "take k" a] take_closed[of k "n + k" a]
         function_ring_defs(4)
-  unfolding function_one_def 
-  using A function_one_eval le_add2 by metis 
+  unfolding function_one_def
+  using A function_one_eval le_add2 by metis
 qed
 
 lemma take_apply_is_hom:
   shows "take_apply (n + k) k \<in> ring_hom (SA k) (SA (n + k))"
   apply(rule ring_hom_memI)
-  using take_apply_SA_closed[of _ k "n+k"]  le_add2 apply blast  
+  using take_apply_SA_closed[of _ k "n+k"]  le_add2 apply blast
   using   take_apply_mult apply blast
    using   take_apply_add apply blast
      using   take_apply_one by blast
 
 lemma drop_apply_units:
   assumes "f \<in> Units (SA n)"
   shows "drop_apply (n+k) k f \<in> Units (SA (n+k))"
   apply(rule SA_Units_memI)
-  using drop_apply_SA_closed[of f "n+k" k ] assms SA_Units_closed 
+  using drop_apply_SA_closed[of f "n+k" k ] assms SA_Units_closed
   apply (metis add_diff_cancel_right' le_add2)
 proof-
   show "\<And>x. x \<in> carrier (Q\<^sub>p\<^bsup>n + k\<^esup>) \<Longrightarrow> drop_apply (n + k) k f x \<noteq> \<zero>"
   proof- fix x assume A: "x \<in> carrier (Q\<^sub>p\<^bsup>n+k\<^esup>)"
     then have "drop_apply (n + k) k f x = f (drop k x)"
       unfolding drop_apply_def restrict_def by (meson comp_def)
     then show "drop_apply (n + k) k f x \<noteq> \<zero>"
-      using SA_Units_memE'[of f n "drop k x"] 
+      using SA_Units_memE'[of f n "drop k x"]
       by (metis A add.commute assms  cartesian_power_drop)
   qed
 qed
 
 lemma take_apply_units:
   assumes "f \<in> Units (SA k)"
   shows "take_apply (n+k) k f \<in> Units (SA (n+k))"
   apply(rule SA_Units_memI)
   using take_apply_SA_closed[of f k "n+k" ] assms SA_Units_closed le_add2 apply blast
 proof-
   show "\<And>x. x \<in> carrier (Q\<^sub>p\<^bsup>n + k\<^esup>) \<Longrightarrow> take_apply (n + k) k f x \<noteq> \<zero>"
   proof- fix x assume A: "x \<in> carrier (Q\<^sub>p\<^bsup>n+k\<^esup>)"
     then have "take_apply (n + k) k f x = f (take k x)"
       unfolding take_apply_def restrict_def by (meson comp_def)
     then show "take_apply (n + k) k f x \<noteq> \<zero>"
-      using SA_Units_memE'[of f k "take k x"]  A assms le_add2 local.take_closed by blast      
+      using SA_Units_memE'[of f k "take k x"]  A assms le_add2 local.take_closed by blast
   qed
 qed
 
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsection\<open>Level Sets of Semialgebraic Functions\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 
 lemma evimage_is_semialg:
-  assumes "h \<in> carrier (SA n)" 
+  assumes "h \<in> carrier (SA n)"
   assumes "is_univ_semialgebraic S"
   shows "is_semialgebraic n (h  \<inverse>\<^bsub>n\<^esub> S)"
 proof-
   have 0: "is_semialgebraic 1 (to_R1 ` S)"
     using assms is_univ_semialgebraicE by blast
   have 1: "h  \<inverse>\<^bsub>n\<^esub> S = partial_pullback n h (0::nat) (to_R1 ` S)"
   proof show "h \<inverse>\<^bsub>n\<^esub> S \<subseteq> partial_pullback n h 0 ((\<lambda>a. [a]) ` S)"
     proof fix x assume A: "x \<in> h \<inverse>\<^bsub>n\<^esub> S"
       then have 0: "h x \<in> S" by blast
       have x_closed: "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
         by (meson A evimage_eq)
       have 1: "drop n x = []"
-        using cartesian_power_car_memE[of x Q\<^sub>p n] drop_all x_closed 
+        using cartesian_power_car_memE[of x Q\<^sub>p n] drop_all x_closed
         by blast
       have 2: "take n x = x"
-        using cartesian_power_car_memE[of x Q\<^sub>p n] x_closed take_all 
+        using cartesian_power_car_memE[of x Q\<^sub>p n] x_closed take_all
         by blast
       then show "x \<in> partial_pullback n h 0 ((\<lambda>a. [a]) ` S)"
-        unfolding partial_pullback_def partial_image_def evimage_def 
-        using 0 1 2 x_closed 
+        unfolding partial_pullback_def partial_image_def evimage_def
+        using 0 1 2 x_closed
         by (metis (no_types, lifting) IntI Nat.add_0_right image_iff vimageI)
     qed
     show "partial_pullback n h 0 ((\<lambda>a. [a]) ` S) \<subseteq> h \<inverse>\<^bsub>n\<^esub> S"
     proof fix x assume A: "x \<in> partial_pullback n h 0 ((\<lambda>a. [a]) ` S)"
       have x_closed: "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
-        using A unfolding partial_pullback_def evimage_def 
+        using A unfolding partial_pullback_def evimage_def
         by (metis A Nat.add_0_right partial_pullback_memE(1))
-      then have "(partial_image n h) x = [h x]" 
-        unfolding partial_image_def 
-        by (metis (no_types, opaque_lifting) One_nat_def Qp.zero_closed append.right_neutral append_Nil 
+      then have "(partial_image n h) x = [h x]"
+        unfolding partial_image_def
+        by (metis (no_types, opaque_lifting) One_nat_def Qp.zero_closed append.right_neutral append_Nil
             local.partial_image_def partial_image_eq segment_in_car' Qp.to_R1_closed)
       then have "h x \<in> S"
-        using A unfolding partial_pullback_def 
+        using A unfolding partial_pullback_def
         by (metis (no_types, lifting) A image_iff partial_pullback_memE(2) Qp.to_R_to_R1)
       thus "x \<in> h \<inverse>\<^bsub>n\<^esub> S"
         using x_closed by blast
     qed
   qed
-  then show ?thesis 
+  then show ?thesis
     using 0 is_semialg_functionE[of n h 0 "((\<lambda>a. [a]) ` S)"] assms SA_car_memE(1)[of h n]
     by (metis Nat.add_0_right SA_car)
 qed
 
 lemma semialg_val_ineq_set_is_semialg:
   assumes "g \<in> carrier (SA k)"
   assumes "f \<in> carrier (SA k)"
   shows "is_semialgebraic k {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (g x) \<le> val (f x)}"
 proof-
   obtain F where F_def: "F = function_tuple_eval Q\<^sub>p  k [f, g]"
-    by blast 
+    by blast
   have P0: "is_semialg_function_tuple k [f, g] "
     using is_semialg_function_tupleI[of "[f, g]" k]
     by (metis assms  list.distinct(1) list.set_cases padic_fields.SA_imp_semialg padic_fields_axioms set_ConsD)
   hence P1: "is_semialg_map k 2 F"
     using  assms semialg_function_tuple_is_semialg_map[of k "[f, g]" 2]
     unfolding F_def by (simp add: \<open>f \<in> carrier (SA k)\<close> \<open>g \<in> carrier (SA k)\<close>)
   have "{x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (g x) \<le> val (f x)} = F  \<inverse>\<^bsub>k\<^esub> val_relation_set"
   proof
     show "{x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (g x) \<le> val (f x)} \<subseteq> F \<inverse>\<^bsub>k\<^esub> val_relation_set"
     proof fix x assume A: "x \<in> {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (g x) \<le> val (f x)}"
       then have 0: "x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>) \<and> val (g x) \<le> val (f x)" by blast
       have 1: "F x = [f x, g x]"
-        unfolding F_def using A unfolding function_tuple_eval_def  
+        unfolding F_def using A unfolding function_tuple_eval_def
         by (metis (no_types, lifting) list.simps(8) map_eq_Cons_conv)
       have 2: "val (g x) \<le> val (f x)"
-        using A 
+        using A
         by blast
       have 3: "F x \<in> carrier (Q\<^sub>p\<^bsup>2\<^esup>)"
-        using assms A 1 
+        using assms A 1
         by (metis (no_types, lifting) "0" Qp.function_ring_car_mem_closed Qp_2I SA_car_memE(2))
       then have 4: "x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>) \<and> F x \<in> val_relation_set"
-        unfolding val_relation_set_def F_def using 0 1 2 3 
-        by (metis (no_types, lifting) cartesian_power_car_memE cartesian_power_car_memE' 
-            list.inject local.F_def one_less_numeral_iff pair_id semiring_norm(76) val_relation_setI 
+        unfolding val_relation_set_def F_def using 0 1 2 3
+        by (metis (no_types, lifting) cartesian_power_car_memE cartesian_power_car_memE'
+            list.inject local.F_def one_less_numeral_iff pair_id semiring_norm(76) val_relation_setI
             val_relation_set_def zero_less_numeral)
       then show "x \<in> F \<inverse>\<^bsub>k\<^esub> val_relation_set"
         by blast
     qed
     show "F \<inverse>\<^bsub>k\<^esub> val_relation_set \<subseteq> {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (g x) \<le> val (f x)}"
     proof fix x assume A: "x \<in> F \<inverse>\<^bsub>k\<^esub> val_relation_set"
       then have 0: "x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>) \<and> F x \<in> val_relation_set"
         by (meson evimage_eq)
       then have 0: "x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>) \<and> [f x, g x] \<in> val_relation_set"
-        unfolding F_def function_tuple_eval_def 
+        unfolding F_def function_tuple_eval_def
         by (metis (no_types, lifting) list.simps(8) list.simps(9))
       then have "x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>) \<and> val (g x) \<le> val (f x)"
-        unfolding F_def val_relation_set_def 
+        unfolding F_def val_relation_set_def
         by (metis (no_types, lifting) "0" list.inject val_relation_setE)
       then show "x \<in> {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (g x) \<le> val (f x)}"
         by blast
     qed
   qed
-  then show ?thesis 
-    using assms P0 P1 val_relation_is_semialgebraic semialg_map_evimage_is_semialg[of k 2 F val_relation_set] pos2 
+  then show ?thesis
+    using assms P0 P1 val_relation_is_semialgebraic semialg_map_evimage_is_semialg[of k 2 F val_relation_set] pos2
     by presburger
 qed
 
 lemma semialg_val_ineq_set_is_semialg':
   assumes "f \<in> carrier (SA k)"
   shows "is_semialgebraic k {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (f x) \<le> C}"
 proof-
   obtain a where a_def: "a \<in> carrier Q\<^sub>p \<and> val a = C"
     by (meson Qp.carrier_not_empty Qp.minus_closed dist_nonempty' equals0I)
   then obtain g where g_def: "g = constant_function (carrier (Q\<^sub>p\<^bsup>k\<^esup>)) a"
-    by blast 
+    by blast
   have 0: "g \<in> carrier (SA k)"
-    using g_def a_def SA_car assms(1) constant_function_in_semialg_functions by blast  
+    using g_def a_def SA_car assms(1) constant_function_in_semialg_functions by blast
   have 1: "{x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (f x) \<le> C} = {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (f x) \<le> val (g x)}"
     using g_def by (metis (no_types, lifting) Qp_constE a_def)
-  then show ?thesis using assms 0 semialg_val_ineq_set_is_semialg[of f k g] 
+  then show ?thesis using assms 0 semialg_val_ineq_set_is_semialg[of f k g]
     by presburger
 qed
 
 lemma semialg_val_ineq_set_is_semialg'':
   assumes "f \<in> carrier (SA k)"
   shows "is_semialgebraic k {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (f x) \<ge> C}"
 proof-
   obtain a where a_def: "a \<in> carrier Q\<^sub>p \<and> val a = C"
     by (meson Qp.carrier_not_empty Qp.minus_closed dist_nonempty' equals0I)
   then obtain g where g_def: "g = constant_function (carrier (Q\<^sub>p\<^bsup>k\<^esup>)) a"
-    by blast 
+    by blast
   have 0: "g \<in> carrier (SA k)"
-    using g_def a_def SA_car assms(1) constant_function_in_semialg_functions by blast  
+    using g_def a_def SA_car assms(1) constant_function_in_semialg_functions by blast
   have 1: "{x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (f x) \<ge> C} = {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (f x) \<ge> val (g x)}"
     using g_def by (metis (no_types, lifting) Qp_constE a_def)
-  then show ?thesis using assms 0 semialg_val_ineq_set_is_semialg[of g k f] 
+  then show ?thesis using assms 0 semialg_val_ineq_set_is_semialg[of g k f]
     by presburger
 qed
 
 lemma semialg_level_set_is_semialg:
   assumes "f \<in> carrier (SA k)"
   assumes "c \<in> carrier Q\<^sub>p"
   shows "is_semialgebraic k {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). f x = c}"
 proof-
   have 0: "is_univ_semialgebraic {c}"
-    apply(rule finite_is_univ_semialgebraic) using assms apply blast by auto 
+    apply(rule finite_is_univ_semialgebraic) using assms apply blast by auto
   have 1: "{x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). f x = c} = f  \<inverse>\<^bsub>k\<^esub> {c}"
-    unfolding evimage_def by auto 
+    unfolding evimage_def by auto
   then show ?thesis using 0 assms evimage_is_semialg by presburger
 qed
 
 lemma semialg_val_eq_set_is_semialg':
   assumes "f \<in> carrier (SA k)"
   shows "is_semialgebraic k {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (f x) = C}"
 proof(cases "C = \<infinity>")
   case True
   then have "{x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (f x) = C} = {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). f x = \<zero>}"
     using assms unfolding val_def by (meson eint.distinct(1))
   then have "{x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (f x) = C} = f  \<inverse>\<^bsub>k\<^esub> {\<zero>}"
-    unfolding evimage_def by blast 
-  then show ?thesis 
+    unfolding evimage_def by blast
+  then show ?thesis
     using assms semialg_level_set_is_semialg[of f k \<zero>]  Qp.zero_closed \<open>{x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (f x) = C} = {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). f x = \<zero>}\<close> by presburger
 next
   case False
   then obtain N::int where N_def: "C = eint N"
     by blast
   have 0: "{x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (f x) = C} = {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (f x) \<le> N} - {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (f x) \<le> (eint (N-1))}"
-  proof 
+  proof
     show "{x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (f x) = C} \<subseteq> {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (f x) \<le> N} - {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (f x) \<le> (eint (N-1))}"
     proof
       fix x assume A: "x \<in> {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (f x) = C}"
       then have 0: "x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>) \<and> val (f x) = C"
-        by blast 
+        by blast
       have 1: "\<not> val (f x) \<le> (eint (N-1))"
         using A N_def assms 0 eint_ord_simps(1) by presburger
       have 2: "val (f x) \<le> (eint N)"
         using 0 N_def eint_ord_simps(1) by presburger
       show "x \<in> {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (f x) \<le> N} - {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (f x) \<le> (eint (N-1))}"
         using 0 1 2
-        by blast 
+        by blast
     qed
     show  "{x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (f x) \<le> N} - {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (f x) \<le> (eint (N-1))} \<subseteq>  {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (f x) = C}"
     proof fix x assume A: "x \<in> {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (f x) \<le> N} - {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (f x) \<le> (eint (N-1))}"
-      have 0:  "x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>) \<and> val (f x) \<le> C" 
-        using A N_def by blast      
+      have 0:  "x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>) \<and> val (f x) \<le> C"
+        using A N_def by blast
       have 1: "\<not> val (f x) \<le> (eint (N-1))"
         using A 0  by blast
       have 2: "val (f x) = C"
       proof(rule ccontr)
         assume "val (f x) \<noteq> C"
         then have "val (f x) < C"
-          using 0 by auto 
+          using 0 by auto
         then obtain M where M_def: "val (f x) = eint M"
           using N_def eint_iless by blast
         then have "M < N"
           by (metis N_def \<open>val (f x) < C\<close> eint_ord_simps(2))
         then have "val (f x) \<le> eint (N - 1)"
           using M_def eint_ord_simps(1) by presburger
-        then show False using 1 by blast 
+        then show False using 1 by blast
       qed
       show "x \<in> {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (f x) = C} "
-        using 0 2  by blast        
+        using 0 2  by blast
     qed
   qed
   have 1: "is_semialgebraic k {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (f x) \<le> N}"
-    using assms semialg_val_ineq_set_is_semialg'[of f k N] by blast 
+    using assms semialg_val_ineq_set_is_semialg'[of f k N] by blast
   have 2: "is_semialgebraic k {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (f x) \<le> (eint (N-1))}"
-    using assms semialg_val_ineq_set_is_semialg' by blast  
-  show ?thesis using 0 1 2 
+    using assms semialg_val_ineq_set_is_semialg' by blast
+  show ?thesis using 0 1 2
     using diff_is_semialgebraic by presburger
 qed
 
 lemma semialg_val_eq_set_is_semialg:
   assumes "g \<in> carrier (SA k)"
   assumes "f \<in> carrier (SA k)"
   shows "is_semialgebraic k {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (g x) = val (f x)}"
 proof-
   have 0: "is_semialgebraic k {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (g x) \<le> val (f x)}"
-    using assms semialg_val_ineq_set_is_semialg[of g k f] by blast 
+    using assms semialg_val_ineq_set_is_semialg[of g k f] by blast
   have 1: "is_semialgebraic k {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (g x) \<ge> val (f x)}"
-    using assms semialg_val_ineq_set_is_semialg[of f k g] by blast 
+    using assms semialg_val_ineq_set_is_semialg[of f k g] by blast
   have 2: " {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (g x) = val (f x)} =  {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (g x) \<le> val (f x)} \<inter> {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (g x) \<ge> val (f x)}"
-    using eq_iff by blast 
+    using eq_iff by blast
   show ?thesis using 0 1 2 intersection_is_semialg by presburger
 qed
 
 lemma semialg_val_strict_ineq_set_formula:
 "{x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (g x) < val (f x)} = {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (g x) \<le> val (f x)} - {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (g x) = val (f x)}"
-  using neq_iff le_less by blast 
+  using neq_iff le_less by blast
 
 lemma semialg_val_ineq_set_complement:
 "carrier (Q\<^sub>p\<^bsup>k\<^esup>) - {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (g x) \<le> val (f x)} = {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (f x) < val (g x)}"
-  using not_le by blast 
+  using not_le by blast
 
 lemma semialg_val_strict_ineq_set_complement:
 "carrier (Q\<^sub>p\<^bsup>k\<^esup>) - {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (g x) < val (f x)} = {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (f x) \<le> val (g x)}"
   using not_le by blast
 
 lemma semialg_val_strict_ineq_set_is_semialg:
   assumes "g \<in> carrier (SA k)"
   assumes "f \<in> carrier (SA k)"
   shows "is_semialgebraic k {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (g x) < val (f x)}"
-  using semialg_val_ineq_set_complement[of k f g] assms diff_is_semialgebraic 
-        semialg_val_ineq_set_is_semialg[of f ]  
+  using semialg_val_ineq_set_complement[of k f g] assms diff_is_semialgebraic
+        semialg_val_ineq_set_is_semialg[of f ]
   by (metis (no_types, lifting) complement_is_semialg)
 
 lemma semialg_val_strict_ineq_set_is_semialg':
   assumes "f \<in> carrier (SA k)"
   shows "is_semialgebraic k {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (f x) < C}"
 proof-
   obtain a where a_def: "a \<in> carrier Q\<^sub>p \<and> val a = C"
     by (meson Qp.carrier_not_empty Qp.minus_closed dist_nonempty' equals0I)
   then obtain g where g_def: "g = constant_function (carrier (Q\<^sub>p\<^bsup>k\<^esup>)) a"
-    by blast 
+    by blast
   have 0: "g \<in> carrier (SA k)"
-    using g_def a_def SA_car assms(1) constant_function_in_semialg_functions by blast  
+    using g_def a_def SA_car assms(1) constant_function_in_semialg_functions by blast
   have 1: "{x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (f x) < C} = {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (f x) < val (g x)}"
     using g_def by (metis (no_types, lifting) Qp_constE a_def)
-  then show ?thesis using assms 0 semialg_val_strict_ineq_set_is_semialg[of f k g] 
+  then show ?thesis using assms 0 semialg_val_strict_ineq_set_is_semialg[of f k g]
     by presburger
 qed
 
 lemma semialg_val_strict_ineq_set_is_semialg'':
   assumes "f \<in> carrier (SA k)"
   shows "is_semialgebraic k {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (f x) > C}"
 proof-
   obtain a where a_def: "a \<in> carrier Q\<^sub>p \<and> val a = C"
     by (meson Qp.carrier_not_empty Qp.minus_closed dist_nonempty' equals0I)
   then obtain g where g_def: "g = constant_function (carrier (Q\<^sub>p\<^bsup>k\<^esup>)) a"
-    by blast 
+    by blast
   have 0: "g \<in> carrier (SA k)"
-    using g_def a_def SA_car assms(1) constant_function_in_semialg_functions by blast  
+    using g_def a_def SA_car assms(1) constant_function_in_semialg_functions by blast
   have 1: "{x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (f x) > C} = {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). val (f x) > val (g x)}"
     using g_def by (metis (no_types, lifting) Qp_constE a_def)
-  then show ?thesis using assms 0 semialg_val_strict_ineq_set_is_semialg[of g k f] 
+  then show ?thesis using assms 0 semialg_val_strict_ineq_set_is_semialg[of g k f]
     by presburger
 qed
 
 lemma semialg_val_ineq_set_plus:
   assumes "N > 0"
   assumes "c \<in> carrier (SA N)"
   assumes "a \<in> carrier (SA N)"
   shows "is_semialgebraic N  {x \<in> carrier (Q\<^sub>p\<^bsup>N\<^esup>). val (c x) \<le> val (a x) + eint n}"
 proof-
   obtain b where b_def: "b = \<pp>[^]n \<odot>\<^bsub>SA N\<^esub> a"
-    by blast 
+    by blast
   have b_closed: "b \<in> carrier (SA N)"
     unfolding b_def using assms SA_smult_closed p_intpow_closed(1) by blast
   have "\<And>x. x \<in> carrier (Q\<^sub>p\<^bsup>N\<^esup>) \<Longrightarrow> val (b x) = val (a x) + eint n"
     unfolding b_def by (metis Qp.function_ring_car_memE SA_car_memE(2) SA_smult_formula assms(3) p_intpow_closed(1) times_p_pow_val)
   hence 0: "{x \<in> carrier (Q\<^sub>p\<^bsup>N\<^esup>). val (c x) \<le> val (a x) + eint n} = {x \<in> carrier (Q\<^sub>p\<^bsup>N\<^esup>). val (c x) \<le> val (b x)}"
     by (metis (no_types, opaque_lifting) add.commute)
-  show ?thesis unfolding 0 using assms b_def b_closed semialg_val_ineq_set_is_semialg[of c N b]  by blast 
+  show ?thesis unfolding 0 using assms b_def b_closed semialg_val_ineq_set_is_semialg[of c N b]  by blast
 qed
 
 lemma semialg_val_eq_set_plus:
   assumes "N > 0"
   assumes "c \<in> carrier (SA N)"
   assumes "a \<in> carrier (SA N)"
   shows "is_semialgebraic N  {x \<in> carrier (Q\<^sub>p\<^bsup>N\<^esup>). val (c x) = val (a x) + eint n}"
 proof-
   obtain b where b_def: "b = \<pp>[^]n \<odot>\<^bsub>SA N\<^esub> a"
-    by blast 
+    by blast
   have b_closed: "b \<in> carrier (SA N)"
     unfolding b_def using assms SA_smult_closed p_intpow_closed(1) by blast
   have "\<And>x. x \<in> carrier (Q\<^sub>p\<^bsup>N\<^esup>) \<Longrightarrow> val (b x) = val (a x) + eint n"
     unfolding b_def by (metis Qp.function_ring_car_memE SA_car_memE(2) SA_smult_formula assms(3) p_intpow_closed(1) times_p_pow_val)
   hence 0: "{x \<in> carrier (Q\<^sub>p\<^bsup>N\<^esup>). val (c x) = val (a x) + eint n} = {x \<in> carrier (Q\<^sub>p\<^bsup>N\<^esup>). val (c x) = val (b x)}"
     by (metis (no_types, opaque_lifting) add.commute)
-  show ?thesis unfolding 0 using assms b_def b_closed semialg_val_eq_set_is_semialg[of c N b]  by blast 
-qed
-
-definition SA_zero_set where 
+  show ?thesis unfolding 0 using assms b_def b_closed semialg_val_eq_set_is_semialg[of c N b]  by blast
+qed
+
+definition SA_zero_set where
 "SA_zero_set n f = {x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). f x = \<zero>}"
 
 lemma SA_zero_set_is_semialgebraic:
   assumes "f \<in> carrier (SA n)"
   shows "is_semialgebraic n (SA_zero_set n f)"
-  using assms semialg_level_set_is_semialg[of f n \<zero>] unfolding SA_zero_set_def 
+  using assms semialg_level_set_is_semialg[of f n \<zero>] unfolding SA_zero_set_def
   by blast
 
 lemma SA_zero_set_memE:
   assumes "f \<in> carrier (SA n)"
   assumes "x \<in> SA_zero_set n f"
   shows "f x = \<zero>"
-  using assms unfolding SA_zero_set_def by blast 
+  using assms unfolding SA_zero_set_def by blast
 
 lemma SA_zero_set_memI:
   assumes "f \<in> carrier (SA n)"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "f x = \<zero>"
   shows  "x \<in> SA_zero_set n f"
-  using assms unfolding SA_zero_set_def by blast 
+  using assms unfolding SA_zero_set_def by blast
 
 lemma SA_zero_set_of_zero:
 "SA_zero_set m (\<zero>\<^bsub>SA m\<^esub>) = carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
   apply(rule equalityI')
-  unfolding SA_zero_set_def mem_Collect_eq 
-  apply blast 
+  unfolding SA_zero_set_def mem_Collect_eq
+  apply blast
   using SA_zeroE by blast
-  
-definition SA_nonzero_set where 
+
+definition SA_nonzero_set where
 "SA_nonzero_set n f = {x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). f x \<noteq> \<zero>}"
 
 lemma nonzero_evimage_closed:
   assumes "f \<in> carrier (SA n)"
   shows "is_semialgebraic n {x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). f x \<noteq> \<zero>}"
 proof-
   have "{x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). f x \<noteq> \<zero>} = f  \<inverse>\<^bsub>n\<^esub> nonzero Q\<^sub>p"
     unfolding nonzero_def evimage_def using SA_car_memE[of f n] assms by blast
-  thus ?thesis   using assms evimage_is_semialg[of f n "nonzero Q\<^sub>p"] nonzero_is_univ_semialgebraic 
+  thus ?thesis   using assms evimage_is_semialg[of f n "nonzero Q\<^sub>p"] nonzero_is_univ_semialgebraic
     by presburger
 qed
 
 lemma SA_nonzero_set_is_semialgebraic:
   assumes "f \<in> carrier (SA n)"
   shows "is_semialgebraic n (SA_nonzero_set n f)"
-  using assms semialg_level_set_is_semialg[of f n \<zero>] unfolding SA_nonzero_set_def 
+  using assms semialg_level_set_is_semialg[of f n \<zero>] unfolding SA_nonzero_set_def
   using nonzero_evimage_closed by blast
-  
+
 lemma SA_nonzero_set_memE:
   assumes "f \<in> carrier (SA n)"
   assumes "x \<in> SA_nonzero_set n f"
   shows "f x \<noteq> \<zero>"
-  using assms unfolding SA_nonzero_set_def by blast 
+  using assms unfolding SA_nonzero_set_def by blast
 
 lemma SA_nonzero_set_memI:
   assumes "f \<in> carrier (SA n)"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "f x \<noteq>  \<zero>"
   shows  "x \<in> SA_nonzero_set n f"
   using assms unfolding SA_nonzero_set_def
-  by blast 
+  by blast
 
 lemma SA_nonzero_set_of_zero:
 "SA_nonzero_set m (\<zero>\<^bsub>SA m\<^esub>) = {}"
   apply(rule equalityI')
-  unfolding SA_nonzero_set_def mem_Collect_eq 
+  unfolding SA_nonzero_set_def mem_Collect_eq
   using SA_zeroE apply blast
-  by blast 
+  by blast
 
 lemma SA_car_memI':
   assumes "\<And>x. x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<Longrightarrow> f x \<in> carrier Q\<^sub>p"
   assumes "\<And>x. x \<notin> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<Longrightarrow> f x = undefined"
   assumes "\<And>k n P. n > 0 \<Longrightarrow> P \<in> carrier (Q\<^sub>p [\<X>\<^bsub>1 + k\<^esub>]) \<Longrightarrow> is_semialgebraic (m + k) (partial_pullback m f k (basic_semialg_set (1 + k) n P))"
   shows "f \<in> carrier (SA m)"
   apply(rule SA_car_memI)
    apply(rule Qp.function_ring_car_memI)
-  using assms(1) apply blast using assms(2) apply blast 
+  using assms(1) apply blast using assms(2) apply blast
   apply(rule is_semialg_functionI')
   using assms(1) apply blast
-  using assms(3) unfolding is_basic_semialg_def 
-  by blast 
+  using assms(3) unfolding is_basic_semialg_def
+  by blast
 
 lemma(in padic_fields) SA_zero_set_is_semialg:
   assumes "a \<in> carrier (SA m)"
   shows "is_semialgebraic m {x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). a x = \<zero>}"
-  using assms semialg_level_set_is_semialg[of a m \<zero>] Qp.zero_closed by blast 
+  using assms semialg_level_set_is_semialg[of a m \<zero>] Qp.zero_closed by blast
 
 lemma(in padic_fields) SA_nonzero_set_is_semialg:
   assumes "a \<in> carrier (SA m)"
   shows "is_semialgebraic m {x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). a x \<noteq> \<zero>}"
 proof-
   have 0: "{x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). a x \<noteq> \<zero>} = carrier (Q\<^sub>p\<^bsup>m\<^esup>) - {x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). a x = \<zero>}"
     by blast
   show ?thesis using assms  SA_zero_set_is_semialg[of a m] complement_is_semialg[of m "{x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). a x = \<zero>}"]
-    unfolding 0  by blast 
+    unfolding 0  by blast
 qed
 
 lemma zero_set_nonzero_set_covers:
 "carrier (Q\<^sub>p\<^bsup>n\<^esup>) = SA_zero_set n f \<union> SA_nonzero_set n f"
-  unfolding SA_zero_set_def SA_nonzero_set_def  
+  unfolding SA_zero_set_def SA_nonzero_set_def
   apply(rule equalityI')
-  unfolding mem_Collect_eq 
+  unfolding mem_Collect_eq
   apply blast
   by blast
 
 lemma zero_set_nonzero_set_covers':
   assumes "S \<subseteq> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "S = (S \<inter> SA_zero_set n f) \<union> (S \<inter> SA_nonzero_set n f)"
-  using assms zero_set_nonzero_set_covers by blast 
+  using assms zero_set_nonzero_set_covers by blast
 
 lemma zero_set_nonzero_set_covers_semialg_set:
   assumes "is_semialgebraic n S"
   shows "S = (S \<inter> SA_zero_set n f) \<union> (S \<inter> SA_nonzero_set n f)"
-  using assms is_semialgebraic_closed zero_set_nonzero_set_covers' by blast 
+  using assms is_semialgebraic_closed zero_set_nonzero_set_covers' by blast
 
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsection\<open>Partitioning Semialgebraic Sets According to Valuations of Functions\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 
 text\<open>
-  Given a semialgebraic set $A$ and a finite set of semialgebraic functions $Fs$, a common 
+  Given a semialgebraic set $A$ and a finite set of semialgebraic functions $Fs$, a common
   construction is to simplify one's understanding of the behaviour of the functions $\mathit{Fs}$ on
-  $A$ by finitely paritioning $A$ into subsets where the element $f \in F$ for which $val (f x)$ is 
-  minimal is constant as $x$ ranges over each piece of the parititon. The function 
+  $A$ by finitely paritioning $A$ into subsets where the element $f \in F$ for which $val (f x)$ is
+  minimal is constant as $x$ ranges over each piece of the parititon. The function
   \texttt{Min\_set} helps construct this by picking out the subset of a set $A$ where the valuation
   of a particular element of $\mathit{Fs}$ is minimal. Such a set will always be semialgebraic.
 \<close>
 
 lemma disjointify_semialg:
   assumes "finite As"
   assumes "As \<subseteq> semialg_sets n"
   shows "disjointify As \<subseteq> semialg_sets n"
-  using assms unfolding semialg_sets_def 
+  using assms unfolding semialg_sets_def
   by (simp add: disjointify_gen_boolean_algebra)
 
 lemma semialgebraic_subalgebra:
   assumes "finite Xs"
   assumes "Xs \<subseteq> semialg_sets n"
   shows "atoms_of Xs \<subseteq> semialg_sets n"
-  using assms unfolding semialg_sets_def  
+  using assms unfolding semialg_sets_def
   by (simp add: atoms_of_gen_boolean_algebra)
 
 lemma(in padic_fields) finite_intersection_is_semialg:
   assumes "finite  Xs"
   assumes "Xs \<noteq> {}"
   assumes "F ` Xs \<subseteq> semialg_sets m"
   shows "is_semialgebraic m (\<Inter> i \<in> Xs. F i)"
 proof-
   have 0: "F ` Xs \<subseteq> semialg_sets m \<and> F ` Xs \<noteq> {} "
     using assms by blast
   thus ?thesis
     using assms finite_intersection_is_semialgebraic[of "F ` Xs" m]
-    by blast 
+    by blast
 qed
 
 
 definition Min_set where
 "Min_set m As a = carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<inter> (\<Inter> f \<in> As. {x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). val (a x) \<le> val (f x) })"
 
-lemma Min_set_memE: 
+lemma Min_set_memE:
   assumes "x \<in> Min_set m As a"
   assumes "f \<in> As"
   shows "val (a x) \<le> val (f x)"
-  using assms unfolding Min_set_def by blast 
+  using assms unfolding Min_set_def by blast
 
 lemma Min_set_closed:
 "Min_set m As a \<subseteq> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
-  unfolding Min_set_def by blast 
-
-lemma Min_set_semialg0: 
+  unfolding Min_set_def by blast
+
+lemma Min_set_semialg0:
   assumes "As \<subseteq> carrier (SA m)"
   assumes "finite As"
   assumes "a \<in> As"
   assumes "As \<noteq> {}"
   shows "is_semialgebraic m (Min_set m As a)"
   unfolding Min_set_def apply(rule intersection_is_semialg)
   using carrier_is_semialgebraic apply blast
-  apply(rule finite_intersection_is_semialg) 
+  apply(rule finite_intersection_is_semialg)
   using assms apply blast
   using assms apply blast
 proof(rule subsetI) fix x assume A: " x \<in> (\<lambda>i. {x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). val (a x) \<le> val (i x)}) ` As"
   then obtain f where f_def: "f \<in> As \<and> x = {x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). val (a x) \<le> val (f x)}"
-    by blast 
+    by blast
   have f_closed: "f \<in> carrier (SA m)"
-    using f_def assms by blast 
+    using f_def assms by blast
   have a_closed: "a \<in> carrier (SA m)"
-    using assms by blast 
+    using assms by blast
   have "is_semialgebraic m  {x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). val (a x) \<le> val (f x)}"
     using a_closed f_closed semialg_val_ineq_set_is_semialg by blast
   thus " x \<in> semialg_sets m"
-    using f_def unfolding is_semialgebraic_def by blast 
-qed
-
-lemma Min_set_semialg: 
+    using f_def unfolding is_semialgebraic_def by blast
+qed
+
+lemma Min_set_semialg:
   assumes "As \<subseteq> carrier (SA m)"
   assumes "finite As"
   assumes "a \<in> As"
   shows "is_semialgebraic m (Min_set m As a)"
   apply(cases "As = {}")
   using Min_set_def  assms(3) apply blast
-  using assms Min_set_semialg0 by blast 
+  using assms Min_set_semialg0 by blast
 
 lemma Min_sets_cover:
   assumes "As \<noteq> {}"
   assumes "finite As"
   shows "carrier (Q\<^sub>p\<^bsup>m\<^esup>) = (\<Union> a \<in> As. Min_set m As a)"
 proof
   show "carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<subseteq> \<Union> (Min_set m As ` As)"
   proof fix x assume A: "x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) "
     obtain v where v_def: "v = Min ((\<lambda>f. val (f x)) ` As)"
-      by blast 
+      by blast
     obtain f where f_def: "f \<in> As \<and> v = val (f x)"
-      unfolding v_def using assms Min_in[of "((\<lambda>f. val (f x)) ` As)"] 
+      unfolding v_def using assms Min_in[of "((\<lambda>f. val (f x)) ` As)"]
       by blast
     have v_def': "v = val (f x)"
-      using f_def by blast 
+      using f_def by blast
     have 0: "x \<in> Min_set m As f"
-      unfolding Min_set_def 
+      unfolding Min_set_def
       apply(rule IntI)
-      using A apply blast 
+      using A apply blast
     proof(rule InterI) fix s assume s: "s \<in> (\<lambda>fa. {x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). val (f x) \<le> val (fa x)}) ` As"
       then obtain g where g_def: "g \<in> As \<and> s=  {x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). val (f x) \<le> val (g x)}"
-        by blast 
+        by blast
       have s_def: "s=  {x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). val (f x) \<le> val (g x)}"
-        using g_def by blast 
+        using g_def by blast
       have 00: " val (g x) \<in> ((\<lambda>f. val (f x)) ` As)"
-        using g_def by blast 
+        using g_def by blast
       show "x \<in> s"
-        unfolding s_def mem_Collect_eq using 00 A assms  MinE[of "((\<lambda>f. val (f x)) ` As)" v "val (g x)"] 
+        unfolding s_def mem_Collect_eq using 00 A assms  MinE[of "((\<lambda>f. val (f x)) ` As)" v "val (g x)"]
         unfolding v_def  by (metis f_def finite_imageI v_def)
     qed
     thus "x \<in> \<Union> (Min_set m As ` As)"
       using f_def by blast
   qed
   show "\<Union> (Min_set m As ` As) \<subseteq> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
-    unfolding Min_set_def by blast 
+    unfolding Min_set_def by blast
 qed
 
 text\<open>
   The sets defined by the function \texttt{Min\_set} for a fixed set of functions may not all be
-   disjoint, but we can easily refine then to obtain a finite partition via the function 
+   disjoint, but we can easily refine then to obtain a finite partition via the function
   "disjointify".
 \<close>
 
-definition Min_set_partition where 
+definition Min_set_partition where
 "Min_set_partition m As B = disjointify ((\<inter>)B  ` (Min_set m As ` As))"
 
 lemma Min_set_partition_finite:
   assumes "finite As"
   shows "finite (Min_set_partition m As B)"
-  unfolding Min_set_partition_def 
+  unfolding Min_set_partition_def
   by (meson assms disjointify_finite finite_imageI)
 
 lemma Min_set_partition_semialg0:
   assumes "finite As"
   assumes "As \<subseteq> carrier (SA m)"
   assumes "is_semialgebraic m B"
   assumes "S \<in> ((\<inter>)B  ` (Min_set m As ` As))"
   shows "is_semialgebraic m S"
-  using Min_set_semialg[of As m] assms intersection_is_semialg[of m B] 
+  using Min_set_semialg[of As m] assms intersection_is_semialg[of m B]
   by blast
-  
+
 lemma Min_set_partition_semialg:
   assumes "finite As"
   assumes "As \<subseteq> carrier (SA m)"
   assumes "is_semialgebraic m B"
   assumes "S \<in> (Min_set_partition m As B)"
   shows "is_semialgebraic m S"
 proof-
   have 0: "(\<inter>) B ` Min_set m As ` As \<subseteq> semialg_sets m "
     apply(rule subsetI)
-    using Min_set_partition_semialg0[of As m B ] assms unfolding is_semialgebraic_def 
+    using Min_set_partition_semialg0[of As m B ] assms unfolding is_semialgebraic_def
     by blast
   thus ?thesis
   unfolding is_semialgebraic_def
-  using assms  Min_set_partition_semialg0[of As m B] disjointify_semialg[of "((\<inter>) B ` Min_set m As ` As)" m] 
-  unfolding Min_set_partition_def  is_semialgebraic_def by blast 
-qed
-
-lemma Min_set_partition_covers0: 
+  using assms  Min_set_partition_semialg0[of As m B] disjointify_semialg[of "((\<inter>) B ` Min_set m As ` As)" m]
+  unfolding Min_set_partition_def  is_semialgebraic_def by blast
+qed
+
+lemma Min_set_partition_covers0:
   assumes "finite As"
   assumes "As \<noteq> {}"
   assumes "As \<subseteq> carrier (SA m)"
   assumes "is_semialgebraic m B"
   shows "\<Union> ((\<inter>)B  ` (Min_set m As ` As)) = B"
-proof- 
+proof-
   have 0: "\<Union> ((\<inter>)B  ` (Min_set m As ` As)) = B \<inter> \<Union> (Min_set m As ` As)"
-    by blast 
+    by blast
   have 1: "B \<subseteq> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
     using assms is_semialgebraic_closed by blast
-  show ?thesis unfolding 0 using 1 assms Min_sets_cover[of As m]  by blast 
-qed
-
-lemma Min_set_partition_covers: 
+  show ?thesis unfolding 0 using 1 assms Min_sets_cover[of As m]  by blast
+qed
+
+lemma Min_set_partition_covers:
   assumes "finite As"
   assumes "As \<subseteq> carrier (SA m)"
   assumes "As \<noteq> {}"
   assumes "is_semialgebraic m B"
   shows "\<Union> (Min_set_partition m As B)  = B"
-  unfolding Min_set_partition_def 
+  unfolding Min_set_partition_def
   using Min_set_partition_covers0[of As m B] assms disjointify_union[of "((\<inter>) B ` Min_set m As ` As)"]
   by (metis finite_imageI)
 
-lemma Min_set_partition_disjoint: 
+lemma Min_set_partition_disjoint:
   assumes "finite As"
   assumes "As \<subseteq> carrier (SA m)"
   assumes "As \<noteq> {}"
   assumes "is_semialgebraic m B"
   assumes "s \<in> Min_set_partition m As B"
   assumes "s' \<in> Min_set_partition m As B"
   assumes "s \<noteq>  s'"
   shows  "s \<inter> s' = {}"
   apply(rule  disjointify_is_disjoint[of "((\<inter>) B ` Min_set m As ` As)" s s'])
-  using assms finite_imageI apply blast 
-  using assms unfolding Min_set_partition_def apply blast 
+  using assms finite_imageI apply blast
   using assms unfolding Min_set_partition_def apply blast
-  using assms by blast 
+  using assms unfolding Min_set_partition_def apply blast
+  using assms by blast
 
 lemma Min_set_partition_memE:
   assumes "finite As"
   assumes "As \<subseteq> carrier (SA m)"
   assumes "As \<noteq> {}"
   assumes "is_semialgebraic m B"
   assumes "s \<in> Min_set_partition m As B"
   shows "\<exists>f \<in> As. (\<forall>x \<in> s. (\<forall>g \<in> As. val (f x) \<le> val (g x)))"
 proof-
   obtain s' where s'_def: "s' \<in> ((\<inter>) B ` Min_set m As ` As) \<and> s \<subseteq>  s'"
-    using finite_imageI assms disjointify_subset[of "((\<inter>) B ` Min_set m As ` As)" s] unfolding Min_set_partition_def  by blast 
+    using finite_imageI assms disjointify_subset[of "((\<inter>) B ` Min_set m As ` As)" s] unfolding Min_set_partition_def  by blast
   obtain f where f_def: "f \<in> As \<and> s' = B \<inter> Min_set m As f"
-    using s'_def by blast 
+    using s'_def by blast
   have 0: "(\<forall>x \<in> s'. (\<forall>g \<in> As. val (f x) \<le> val (g x)))"
-    using f_def Min_set_memE[of _ m As f] by blast 
-  thus ?thesis 
+    using f_def Min_set_memE[of _ m As f] by blast
+  thus ?thesis
     using s'_def  by (meson f_def subset_iff)
 qed
 
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsection\<open>Valuative Congruence Sets for Semialgebraic Functions\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 text\<open>
   The set of points $x$ where the values $\mathit{ord}\ f(x)$ satisfy a congruence are important
   basic examples of semialgebraic sets, and will be vital in the proof of Macintyre's Theorem. The
   lemma below is essentially the content of Denef's Lemma 2.1.3 from his cell decomposition paper.
 \<close>
 
 lemma pre_SA_unit_cong_set_is_semialg:
   assumes "k \<ge> 0"
   assumes "f \<in> Units (SA n)"
   shows "is_semialgebraic n {x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). ord (f x) mod k  = a }"
 proof-
   have 0: "{x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). ord (f x) mod k  = a } = f  \<inverse>\<^bsub>n\<^esub> ord_congruence_set k a"
-    unfolding ord_congruence_set_def 
-    apply(rule equalityI')  
-    using assms unfolding evimage_def vimage_def mem_Collect_eq 
+    unfolding ord_congruence_set_def
+    apply(rule equalityI')
+    using assms unfolding evimage_def vimage_def mem_Collect_eq
     apply (metis (mono_tags, lifting) IntI Qp.function_ring_car_memE SA_Units_closed SA_Units_memE' SA_car_memE(2) mem_Collect_eq not_nonzero_Qp)
     using assms by blast
-  show ?thesis unfolding 0 
-    apply(rule evimage_is_semialg) 
+  show ?thesis unfolding 0
+    apply(rule evimage_is_semialg)
     using assms  apply blast
-    using  assms ord_congruence_set_univ_semialg[of k a]   
+    using  assms ord_congruence_set_univ_semialg[of k a]
     by blast
 qed
 
 lemma SA_unit_cong_set_is_semialg:
   assumes "f \<in> Units (SA n)"
   shows "is_semialgebraic n {x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). ord (f x) mod k = a}"
 proof(cases "k \<ge> 0")
   case True
-  then show ?thesis 
+  then show ?thesis
     using assms pre_SA_unit_cong_set_is_semialg by presburger
 next
   case False
-  show ?thesis 
+  show ?thesis
   proof(cases "a = 0")
     case True
-    have T0: "{x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). ord (f x) mod k  = a } = {x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). ord (f x) mod (-k)  = a }" 
+    have T0: "{x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). ord (f x) mod k  = a } = {x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). ord (f x) mod (-k)  = a }"
       apply(rule equalityI')
       unfolding mem_Collect_eq using True zmod_zminus2_not_zero  apply meson
-      using True zmod_zminus2_not_zero  
-      by (metis equation_minus_iff)     
+      using True zmod_zminus2_not_zero
+      by (metis equation_minus_iff)
     show ?thesis unfolding T0 apply(rule pre_SA_unit_cong_set_is_semialg[of "-k" f n a])
-      using False apply presburger using assms by blast 
+      using False apply presburger using assms by blast
   next
     case F: False
-    show ?thesis 
+    show ?thesis
     proof(cases "a = k")
       case True
       have 0: "{x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). ord (f x) mod k = a} = {x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). ord (f x) mod k = k}"
         using True by blast
       have 1: "{x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). ord (f x) mod k = a} = {} \<or> k = 0"
       proof(cases "{x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). ord (f x) mod k = a} \<noteq> {}")
         case T: True
         then obtain x where "ord (f x) mod k = k"
           unfolding True by blast
-        then have "k = 0"  
+        then have "k = 0"
           by (metis mod_mod_trivial mod_self)
-        thus ?thesis by blast 
+        thus ?thesis by blast
       next
         case False
-        then show ?thesis by blast 
+        then show ?thesis by blast
       qed
       show ?thesis apply(cases "{x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). ord (f x) mod k = a} = {}")
         using empty_is_semialgebraic apply presburger
-        using 1 pre_SA_unit_cong_set_is_semialg assms by blast 
+        using 1 pre_SA_unit_cong_set_is_semialg assms by blast
     next
       case F': False
       have 0: "{x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). ord (f x) mod k = a} = {x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). ord (f x) mod (-k) = a - k}"
         apply(rule equalityI')
         unfolding mem_Collect_eq using zmod_zminus2_eq_if assms apply (metis F)
         unfolding mem_Collect_eq zmod_zminus2_eq_if using False F F' assms
-        by (metis (no_types, opaque_lifting) cancel_ab_semigroup_add_class.diff_right_commute group_add_class.right_minus_eq) 
+        by (metis (no_types, opaque_lifting) cancel_ab_semigroup_add_class.diff_right_commute group_add_class.right_minus_eq)
       show ?thesis unfolding 0 apply(rule pre_SA_unit_cong_set_is_semialg)
-        using False apply presburger using assms by blast 
+        using False apply presburger using assms by blast
     qed
   qed
 qed
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsection\<open>Gluing Functions Along Semialgebraic Sets\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 text\<open>
   Semialgebraic functions have the useful property that they are closed under piecewise definitions.
-  That is, if $f, g$ are semialgebraic and $C \subseteq \mathbb{Q}_p^m$ is a semialgebraic set, 
+  That is, if $f, g$ are semialgebraic and $C \subseteq \mathbb{Q}_p^m$ is a semialgebraic set,
   then the function:
   \[
    h(x) =
     \begin{cases}
                                      f(x) & \text{if $x \in C$} \\
                                      g(x) & \text{if $x \in \mathbb{Q}_p^m - C$} \\
     undefined & \text{otherwise}
     \end{cases}
   \]
-  is again semialgebraic. The function $h$ can be obtained by the definition 
+  is again semialgebraic. The function $h$ can be obtained by the definition
   \[\texttt{h = fun\_glue m C f g}\] which is defined below. This closure property means that we
-  can avoid having to define partial semialgebraic functions which are undefined outside of some 
-  proper subset of $\mathbb{Q}_p^m$, since it usually suffices to just define the function as some 
-  arbitrary constant outside of the desired domain. This is useful for defining partial 
-  multiplicative inverses of arbitrary functions. If $f$ is semialgebraic, then its nonzero set 
-  $\{x \in \mathbb{Q}_p^m \mid f x \neq 0\}$ is semialgebraic. By gluing $f$ to the constant 
-  function $1$ outside of its nonzero set, we obtain an invertible element in the ring 
-  \texttt{SA(m)} which evaluates to a multiplicative inverse of $f(x)$ on the largest domain 
+  can avoid having to define partial semialgebraic functions which are undefined outside of some
+  proper subset of $\mathbb{Q}_p^m$, since it usually suffices to just define the function as some
+  arbitrary constant outside of the desired domain. This is useful for defining partial
+  multiplicative inverses of arbitrary functions. If $f$ is semialgebraic, then its nonzero set
+  $\{x \in \mathbb{Q}_p^m \mid f x \neq 0\}$ is semialgebraic. By gluing $f$ to the constant
+  function $1$ outside of its nonzero set, we obtain an invertible element in the ring
+  \texttt{SA(m)} which evaluates to a multiplicative inverse of $f(x)$ on the largest domain
   possible.
 \<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsubsection\<open>Defining Piecewise Semialgebraic Functions\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 
 text\<open>
   An important property that will be repeatedly used is that we can define piecewise semialgebraic
   functions, which will themselves be semialgebraic as long as the pieces are semialgebraic sets.
-  An important application of this principle will be that a function $f$ which is always nonzero 
-  on some semialgebraic set $A$ can be replaced with a global unit in the ring of semialgebraic 
-  functions. This global unit admits a global multiplicative inverse that inverts $f$ pointwise on 
-  $A$, and allows us to avoid having to consider localizations of function rings to locally invert 
+  An important application of this principle will be that a function $f$ which is always nonzero
+  on some semialgebraic set $A$ can be replaced with a global unit in the ring of semialgebraic
+  functions. This global unit admits a global multiplicative inverse that inverts $f$ pointwise on
+  $A$, and allows us to avoid having to consider localizations of function rings to locally invert
   such functions.
 \<close>
 
 definition fun_glue where
 "fun_glue n S f g = (\<lambda>x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). if x \<in> S then f x else g x)"
 
 lemma fun_glueE:
   assumes "f \<in> carrier (SA n)"
   assumes "g \<in> carrier (SA n)"
   assumes "S \<subseteq> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "x \<in> S"
   shows "fun_glue n S f g x = f x"
 proof-
   have "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
-    using assms by blast 
-  thus ?thesis 
-    unfolding fun_glue_def using assms 
+    using assms by blast
+  thus ?thesis
+    unfolding fun_glue_def using assms
     by (metis (mono_tags, lifting) restrict_apply')
 qed
 
 lemma fun_glueE':
   assumes "f \<in> carrier (SA n)"
   assumes "g \<in> carrier (SA n)"
   assumes "S \<subseteq> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) - S"
   shows "fun_glue n S f g x = g x"
 proof-
   have 0: "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
     using assms by blast
   have 1: "x \<notin> S"
-    using assms by blast 
-  show ?thesis 
+    using assms by blast
+  show ?thesis
     unfolding fun_glue_def using assms 0 1
     by (metis (mono_tags, lifting) restrict_apply')
 qed
 
 lemma fun_glue_evimage:
   assumes "f \<in> carrier (SA n)"
   assumes "g \<in> carrier (SA n)"
   assumes "S \<subseteq> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "fun_glue n S f g \<inverse>\<^bsub>n\<^esub> T = ((f  \<inverse>\<^bsub>n\<^esub> T) \<inter> S) \<union> ((g  \<inverse>\<^bsub>n\<^esub> T) - S)"
 proof
   show "fun_glue n S f g \<inverse>\<^bsub>n\<^esub> T \<subseteq> ((f  \<inverse>\<^bsub>n\<^esub> T) \<inter> S) \<union> ((g  \<inverse>\<^bsub>n\<^esub> T) - S)"
   proof fix x assume A: "x \<in> fun_glue n S f g \<inverse>\<^bsub>n\<^esub> T "
     then have 0: "fun_glue n S f g x \<in> T"
       by blast
     have 1: "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
       using A  by (meson evimage_eq)
     show "x \<in> ((f  \<inverse>\<^bsub>n\<^esub> T) \<inter> S) \<union> ((g  \<inverse>\<^bsub>n\<^esub> T) - S)"
       apply(cases "x \<in> S")
-      apply auto[1] 
+      apply auto[1]
       using "1" apply force
       using "0" assms(1) assms(2) assms(3) fun_glueE apply force
-      apply auto[1] using 1 apply blast 
-      using A 1  unfolding fun_glue_def evimage_def Int_iff  by auto 
+      apply auto[1] using 1 apply blast
+      using A 1  unfolding fun_glue_def evimage_def Int_iff  by auto
   qed
   show " f \<inverse>\<^bsub>n\<^esub> T \<inter> S \<union> (g \<inverse>\<^bsub>n\<^esub> T - S) \<subseteq> fun_glue n S f g \<inverse>\<^bsub>n\<^esub> T"
   proof fix x assume A: "x \<in> f \<inverse>\<^bsub>n\<^esub> T \<inter> S \<union> (g \<inverse>\<^bsub>n\<^esub> T - S)"
     then have x_closed: "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
       by (metis (no_types, opaque_lifting) Diff_iff Int_iff UnE extensional_vimage_closed subsetD)
     show "x \<in> fun_glue n S f g \<inverse>\<^bsub>n\<^esub> T"
       apply(cases "x \<in> S")
-      using x_closed fun_glueE assms 
+      using x_closed fun_glueE assms
        apply (metis A DiffD2 IntD1 UnE evimage_eq)
-      using x_closed fun_glueE' assms 
+      using x_closed fun_glueE' assms
       by (metis A Diff_iff Int_iff Un_iff evimageD evimageI2)
   qed
 qed
 
 lemma fun_glue_partial_pullback:
   assumes "f \<in> carrier (SA k)"
   assumes "g \<in> carrier (SA k)"
   assumes "S \<subseteq> carrier (Q\<^sub>p\<^bsup>k\<^esup>)"
-  shows "partial_pullback k (fun_glue k S f g) n T = 
+  shows "partial_pullback k (fun_glue k S f g) n T =
           ((cartesian_product S (carrier (Q\<^sub>p\<^bsup>n\<^esup>))) \<inter> partial_pullback k f n T) \<union> ((partial_pullback k g n T)- (cartesian_product S (carrier (Q\<^sub>p\<^bsup>n\<^esup>))))"
 proof
   show "partial_pullback k (fun_glue k S f g) n T \<subseteq> (cartesian_product S (carrier (Q\<^sub>p\<^bsup>n\<^esup>))) \<inter> partial_pullback k f n T \<union> (partial_pullback k g n T - (cartesian_product S (carrier (Q\<^sub>p\<^bsup>n\<^esup>))))"
   proof fix x assume A: "x \<in> partial_pullback k (fun_glue k S f g) n T "
     then have x_closed: "x \<in> carrier (Q\<^sub>p\<^bsup>k+n\<^esup>)" unfolding partial_pullback_def partial_image_def
-      by (meson evimage_eq) 
+      by (meson evimage_eq)
     show " x \<in> cartesian_product S (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) \<inter> partial_pullback k f n T \<union> (partial_pullback k g n T - (cartesian_product S (carrier (Q\<^sub>p\<^bsup>n\<^esup>))))"
     proof(cases "x \<in> cartesian_product S (carrier (Q\<^sub>p\<^bsup>n\<^esup>))")
       case True
       then have T0: "take k x \<in> S"
         using assms cartesian_product_memE(1) by blast
       then have "(fun_glue k S f g) (take k x) = f (take k x)"
-        using assms fun_glueE[of f k g S "take k x"] 
+        using assms fun_glueE[of f k g S "take k x"]
         by blast
       then have "partial_image k (fun_glue k S f g) x = partial_image k f x"
-        using A x_closed unfolding partial_pullback_def partial_image_def 
-        by blast        
-      then show ?thesis using T0 A unfolding partial_pullback_def evimage_def  
+        using A x_closed unfolding partial_pullback_def partial_image_def
+        by blast
+      then show ?thesis using T0 A unfolding partial_pullback_def evimage_def
         by (metis IntI Int_iff True Un_iff vimageI vimage_eq x_closed)
     next
       case False
       then have F0: "take k x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>) - S"
-        using A x_closed assms cartesian_product_memI 
+        using A x_closed assms cartesian_product_memI
         by (metis (no_types, lifting) DiffI carrier_is_semialgebraic cartesian_power_drop is_semialgebraic_closed le_add1 local.take_closed)
       then have "(fun_glue k S f g) (take k x) = g (take k x)"
-        using assms fun_glueE'[of f k g S "take k x"] 
+        using assms fun_glueE'[of f k g S "take k x"]
         by blast
       then have "partial_image k (fun_glue k S f g) x = partial_image k g x"
-        using A x_closed unfolding partial_pullback_def partial_image_def 
-        by blast        
+        using A x_closed unfolding partial_pullback_def partial_image_def
+        by blast
       then have "x \<in> partial_pullback k g n T "
-        using F0 x_closed A unfolding partial_pullback_def partial_image_def evimage_def 
+        using F0 x_closed A unfolding partial_pullback_def partial_image_def evimage_def
         by (metis (no_types, lifting) A IntI local.partial_image_def partial_pullback_memE(2) vimageI)
       then have "x \<in> (partial_pullback k g n T - cartesian_product S (carrier (Q\<^sub>p\<^bsup>n\<^esup>)))"
-        using False by blast 
-      then show ?thesis by blast 
+        using False by blast
+      then show ?thesis by blast
     qed
-  qed 
+  qed
   show "cartesian_product S (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) \<inter> partial_pullback k f n T \<union> (partial_pullback k g n T - cartesian_product S (carrier (Q\<^sub>p\<^bsup>n\<^esup>)))
     \<subseteq> partial_pullback k (fun_glue k S f g) n T"
   proof fix x assume A: "x \<in> cartesian_product S (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) \<inter> partial_pullback k f n T \<union> (partial_pullback k g n T - cartesian_product S (carrier (Q\<^sub>p\<^bsup>n\<^esup>)))"
     then have x_closed: "x \<in> carrier (Q\<^sub>p\<^bsup>n+k\<^esup>)"
       by (metis DiffD1 Int_iff Un_iff add.commute partial_pullback_memE(1))
     show "x \<in> partial_pullback k (fun_glue k S f g) n T"
     proof(cases "x \<in> cartesian_product S (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) \<inter> partial_pullback k f n T")
       case True
       show ?thesis apply(rule partial_pullback_memI)
-        using x_closed apply (metis add.commute)      
+        using x_closed apply (metis add.commute)
         using x_closed True assms fun_glueE[of f k g S "take k x"] partial_pullback_memE[of x k f n T]
         unfolding partial_image_def by (metis Int_iff cartesian_product_memE(1))
     next
       case False
       show ?thesis apply(rule partial_pullback_memI)
-        using x_closed apply (metis add.commute)   
+        using x_closed apply (metis add.commute)
         using A x_closed False assms fun_glueE'[of f k g S "take k x"] partial_pullback_memE[of x k g n T]
-        unfolding partial_image_def 
+        unfolding partial_image_def
         by (metis (no_types, lifting) Diff_iff Un_iff carrier_is_semialgebraic cartesian_power_drop cartesian_product_memI is_semialgebraic_closed le_add2 local.take_closed)
     qed
   qed
 qed
 
 lemma fun_glue_eval_closed:
   assumes "f \<in> carrier (SA n)"
   assumes "g \<in> carrier (SA n)"
   assumes "is_semialgebraic n S"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "fun_glue n S f g x \<in> carrier Q\<^sub>p"
   apply(cases "x \<in> S")
-  using assms fun_glueE SA_car_memE 
+  using assms fun_glueE SA_car_memE
    apply (metis Qp.function_ring_car_mem_closed is_semialgebraic_closed)
 proof- assume A: "x \<notin> S"
   then have 0: "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) - S"
-    using assms by auto 
+    using assms by auto
   hence 1: "fun_glue n S f g x = g x"
-    using assms fun_glueE' is_semialgebraic_closed by auto  
+    using assms fun_glueE' is_semialgebraic_closed by auto
   show "fun_glue n S f g x \<in> carrier Q\<^sub>p"
-    unfolding 1 using assms SA_car_memE by blast 
+    unfolding 1 using assms SA_car_memE by blast
 qed
 
 lemma fun_glue_closed:
   assumes "f \<in> carrier (SA n)"
   assumes "g \<in> carrier (SA n)"
   assumes "is_semialgebraic n S"
   shows "fun_glue n S f g \<in> carrier (SA n)"
   apply(rule SA_car_memI)
    apply(rule Qp.function_ring_car_memI)
-  using fun_glue_eval_closed assms apply blast   
-  using fun_glue_def unfolding restrict_apply apply metis 
-  apply(rule is_semialg_functionI, intro Pi_I fun_glue_eval_closed assms, blast) 
+  using fun_glue_eval_closed assms apply blast
+  using fun_glue_def unfolding restrict_apply apply metis
+  apply(rule is_semialg_functionI, intro Pi_I fun_glue_eval_closed assms, blast)
 proof-
   fix k T assume A: "T \<in> semialg_sets (1 + k)"
   have 0: "is_semialgebraic (n+k) (partial_pullback n f k T)"
     using assms A SA_car_memE[of f n]  is_semialg_functionE[of n f k T]  padic_fields.is_semialgebraicI padic_fields_axioms by blast
   have 1: "is_semialgebraic (n+k) (partial_pullback n g k T)"
     using assms A SA_car_memE[of g n]  is_semialg_functionE[of n g k T] padic_fields.is_semialgebraicI padic_fields_axioms by blast
   have 2: "partial_pullback n (fun_glue n S f g) k T =
     cartesian_product S (carrier (Q\<^sub>p\<^bsup>k\<^esup>)) \<inter> partial_pullback n f k T \<union> (partial_pullback n g k T - cartesian_product S (carrier (Q\<^sub>p\<^bsup>k\<^esup>)))"
-    using assms fun_glue_partial_pullback[of f n g S k T] \<open>f \<in> carrier (SA n)\<close> \<open>g \<in> carrier (SA n)\<close> is_semialgebraic_closed 
+    using assms fun_glue_partial_pullback[of f n g S k T] \<open>f \<in> carrier (SA n)\<close> \<open>g \<in> carrier (SA n)\<close> is_semialgebraic_closed
     by blast
   show "is_semialgebraic (n + k) (partial_pullback n (fun_glue n S f g) k T)"
-    using assms 0 1 2 cartesian_product_is_semialgebraic carrier_is_semialgebraic 
-      diff_is_semialgebraic intersection_is_semialg union_is_semialgebraic by presburger 
-qed  
+    using assms 0 1 2 cartesian_product_is_semialgebraic carrier_is_semialgebraic
+      diff_is_semialgebraic intersection_is_semialg union_is_semialgebraic by presburger
+qed
 
 lemma fun_glue_unit:
   assumes "f \<in> carrier (SA n)"
   assumes "is_semialgebraic n S"
   assumes "\<And>x. x \<in> S \<Longrightarrow> f x \<noteq> \<zero>"
   shows "fun_glue n S f \<one>\<^bsub>SA n\<^esub> \<in> Units (SA n)"
 proof(rule SA_Units_memI)
   show "fun_glue n S f \<one>\<^bsub>SA n\<^esub> \<in> carrier (SA n)"
     using fun_glue_closed assms SA_is_cring cring.cring_simprules(6) by blast
   show "\<And>x. x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<Longrightarrow> fun_glue n S f \<one>\<^bsub>SA n\<^esub> x \<noteq> \<zero>"
   proof- fix x assume A: "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
     show "fun_glue n S f \<one>\<^bsub>SA n\<^esub> x \<noteq> \<zero>"
       apply(cases "x \<in> S")
-      using assms SA_is_cring cring.cring_simprules(6) assms(3)[of x] fun_glueE[of f n "\<one>\<^bsub>SA n\<^esub>" S x] 
+      using assms SA_is_cring cring.cring_simprules(6) assms(3)[of x] fun_glueE[of f n "\<one>\<^bsub>SA n\<^esub>" S x]
       apply (metis is_semialgebraic_closed)
-      using assms SA_is_cring[of n] cring.cring_simprules(6)[of "SA n"] 
-            A  fun_glueE'[of f n "\<one>\<^bsub>SA n\<^esub>" S x] is_semialgebraic_closed[of n S] 
+      using assms SA_is_cring[of n] cring.cring_simprules(6)[of "SA n"]
+            A  fun_glueE'[of f n "\<one>\<^bsub>SA n\<^esub>" S x] is_semialgebraic_closed[of n S]
       unfolding SA_one[of n] function_ring_defs(4)[of n] function_one_def
       by (metis Diff_iff function_one_eval Qp_funs_one local.one_neq_zero)
   qed
 qed
 
 definition parametric_fun_glue where
 "parametric_fun_glue n Xs fs = (\<lambda>x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). let S = (THE S. S \<in> Xs \<and> x \<in> S) in (fs S x))"
 
 lemma parametric_fun_glue_formula:
   assumes "Xs partitions (carrier (Q\<^sub>p\<^bsup>n\<^esup>))"
   assumes "x \<in> S"
   assumes "S \<in> Xs"
   shows "parametric_fun_glue n Xs fs x = fs S x"
 proof-
   have 0: "(THE S. S \<in> Xs \<and> x \<in> S) = S"
     apply(rule the_equality)
-    using assms apply blast 
+    using assms apply blast
     using assms unfolding is_partition_def by (metis Int_iff empty_iff disjointE)
   have 1: "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
-    using assms unfolding is_partition_def by blast 
-  then show ?thesis using 0 unfolding parametric_fun_glue_def  restrict_def by metis 
+    using assms unfolding is_partition_def by blast
+  then show ?thesis using 0 unfolding parametric_fun_glue_def  restrict_def by metis
 qed
 
 definition char_fun where
 "char_fun n S = (\<lambda> x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). if x \<in> S then \<one> else \<zero>)"
 
 lemma char_fun_is_semialg:
   assumes "is_semialgebraic n S"
   shows "char_fun n S \<in> carrier (SA n)"
 proof-
   have "char_fun n S = fun_glue n S \<one>\<^bsub>SA n\<^esub> \<zero>\<^bsub>SA n\<^esub>"
-    unfolding char_fun_def fun_glue_def 
+    unfolding char_fun_def fun_glue_def
     by (metis (no_types, lifting) function_one_eval function_zero_eval SA_one SA_zero restrict_ext)
-  then show ?thesis 
+  then show ?thesis
     using assms fun_glue_closed
-    by (metis SA_is_cring cring.cring_simprules(2) cring.cring_simprules(6))    
+    by (metis SA_is_cring cring.cring_simprules(2) cring.cring_simprules(6))
 qed
 
 lemma SA_finsum_apply:
   assumes "finite S"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "F \<in> S \<rightarrow> carrier (SA n) \<longrightarrow> finsum (SA n) F S x = (\<Oplus>s\<in>S. F s x)"
 proof(rule finite.induct[of S])
   show "finite S"
-    using assms by blast 
+    using assms by blast
   show " F \<in> {} \<rightarrow> carrier (SA n) \<longrightarrow> finsum (SA n) F {} x = (\<Oplus>s\<in>{}. F s x)"
     using assms abelian_monoid.finsum_empty[of "SA n"]   Qp.abelian_monoid_axioms SA_is_abelian_monoid
     by (simp add: SA_zeroE)
   show "\<And>A a. finite A \<Longrightarrow>
            F \<in> A \<rightarrow> carrier (SA n) \<longrightarrow> finsum (SA n) F A x = (\<Oplus>s\<in>A. F s x) \<Longrightarrow>
            F \<in> insert a A \<rightarrow> carrier (SA n) \<longrightarrow> finsum (SA n) F (insert a A) x = (\<Oplus>s\<in>insert a A. F s x)"
   proof- fix A a assume A: "finite A" "F \<in> A \<rightarrow> carrier (SA n) \<longrightarrow> finsum (SA n) F A x = (\<Oplus>s\<in>A. F s x)"
     show " F \<in> insert a A \<rightarrow> carrier (SA n) \<longrightarrow> finsum (SA n) F (insert a A) x = (\<Oplus>s\<in>insert a A. F s x)"
     proof assume A': "F \<in> insert a A \<rightarrow> carrier (SA n)"
       then have 0: "F \<in> A \<rightarrow> carrier (SA n)"
-        by blast 
+        by blast
       hence 1: "finsum (SA n) F A x = (\<Oplus>s\<in>A. F s x)"
-        using A by blast 
+        using A by blast
       show "finsum (SA n) F (insert a A) x = (\<Oplus>s\<in>insert a A. F s x)"
       proof(cases "a \<in> A")
         case True
-        then show ?thesis        
+        then show ?thesis
           using 1 by (metis insert_absorb)
       next
         case False
         have F00: "(\<lambda>s. F s x) \<in> A \<rightarrow> carrier Q\<^sub>p"
-          apply(rule Pi_I, rule SA_car_closed[of _ n] )  
-          using "0" assms by auto 
+          apply(rule Pi_I, rule SA_car_closed[of _ n] )
+          using "0" assms by auto
         have F01: "F a x \<in> carrier Q\<^sub>p"
-          using A' assms 
+          using A' assms
           by (metis (no_types, lifting) Qp.function_ring_car_mem_closed Pi_split_insert_domain SA_car_in_Qp_funs_car subsetD)
         have F0: "(\<Oplus>s\<in>insert a A. F s x) = F a x \<oplus> (\<Oplus>s\<in>A. F s x)"
-          using F00 F01  A' False A(1) Qp.finsum_insert[of A a "\<lambda>s. F s x"] by blast 
+          using F00 F01  A' False A(1) Qp.finsum_insert[of A a "\<lambda>s. F s x"] by blast
         have F1: "finsum (SA n) F (insert a A) = F a \<oplus>\<^bsub>SA n\<^esub> finsum (SA n) F A"
           using abelian_monoid.finsum_insert[of "SA n" A a F]
           by (metis (no_types, lifting) A' A(1) False Pi_split_insert_domain SA_is_abelian_monoid assms(1))
-        show ?thesis 
+        show ?thesis
           using Qp.finsum_closed[of "\<lambda>s. F s x" A] abelian_monoid.finsum_closed[of "SA n" F A]
                 SA_is_abelian_monoid[of n] assms F0 F1 "0" A(2) SA_add by presburger
       qed
     qed
   qed
 qed
 
 lemma SA_finsum_apply_zero:
   assumes "finite S"
   assumes "F \<in> S \<rightarrow> carrier (SA n)"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "\<And>s. s \<in> S \<Longrightarrow> F s x = \<zero>"
   shows "finsum (SA n) F S x = \<zero>"
 proof-
   have "finsum (SA n) F S x = (\<Oplus>s\<in>S. F s x)"
-    using SA_finsum_apply assms by blast   
-  then show ?thesis using assms 
+    using SA_finsum_apply assms by blast
+  then show ?thesis using assms
     by (metis Qp.add.finprod_one_eqI)
 qed
 
 lemma parametric_fun_glue_is_SA:
   assumes "finite Xs"
   assumes "Xs partitions (carrier (Q\<^sub>p\<^bsup>n\<^esup>))"
   assumes "fs \<in> Xs \<rightarrow> carrier (SA n)"
   assumes "\<forall> S \<in> Xs. is_semialgebraic n S"
   shows "parametric_fun_glue n Xs fs \<in> carrier (SA n)"
 proof-
   obtain F where F_def: "F = (\<lambda>S. fs S \<otimes>\<^bsub>SA n\<^esub> char_fun n S)"
-    by blast 
+    by blast
   have 0: "F \<in> Xs \<rightarrow> carrier (SA n)" proof fix S assume "S \<in> Xs" then show "F S \<in> carrier (SA n)"
       using  SA_mult_closed[of n "fs S" "char_fun n S"] char_fun_is_semialg[of n S] assms SA_car_memE
       unfolding F_def by blast qed
   have 1: "\<And>S x. S \<in> Xs \<Longrightarrow> x \<in> S \<Longrightarrow> F S x = fs S x"
   proof- fix S x assume A: "S \<in> Xs" "x \<in> S"
     then have x_closed: "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
-      using assms unfolding is_partition_def by blast 
+      using assms unfolding is_partition_def by blast
     then have 0: "F S x = fs S x \<otimes> char_fun n S x"
       unfolding F_def using SA_mult by blast
     have 1: "char_fun n S x = \<one>"
-      using char_fun_def A x_closed by auto 
+      using char_fun_def A x_closed by auto
     have 2: "fs S x \<in> carrier Q\<^sub>p"
       apply(intro SA_car_closed[of _ n] x_closed )
-      using assms A by auto 
+      using assms A by auto
     show "F S x = fs S x"
-      unfolding 0  1 using 2 Qp.cring_simprules(12) by auto             
+      unfolding 0  1 using 2 Qp.cring_simprules(12) by auto
   qed
   have 2: "\<And>S x. S \<in> Xs \<Longrightarrow> x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<Longrightarrow> x \<notin> S \<Longrightarrow> F S x = \<zero>"
   proof- fix S x assume A: "S \<in> Xs" "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)" "x \<notin> S"
     then have x_closed: "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
-      using assms unfolding is_partition_def by blast 
+      using assms unfolding is_partition_def by blast
     hence 20: "F S x = fs S x \<otimes> char_fun n S x"
       unfolding F_def using SA_mult by blast
     have 21: "char_fun n S x = \<zero>"
-      unfolding char_fun_def using A x_closed by auto 
+      unfolding char_fun_def using A x_closed by auto
     have 22: "fs S x \<in> carrier Q\<^sub>p"
       apply(intro SA_car_closed[of _ n] x_closed )
-      using assms A by auto 
+      using assms A by auto
     show "F S x = \<zero>"
-      using 22 unfolding 20 21 by auto       
+      using 22 unfolding 20 21 by auto
   qed
   obtain g where g_def: "g = finsum (SA n) F Xs"
-    by blast 
+    by blast
   have g_closed: "g \<in> carrier (SA n)"
     using abelian_monoid.finsum_closed[of "SA n" F Xs] assms SA_is_ring 0
-    unfolding g_def ring_def abelian_group_def by blast 
+    unfolding g_def ring_def abelian_group_def by blast
   have "g = parametric_fun_glue n Xs fs"
   proof fix x show "g x = parametric_fun_glue n Xs fs x"
     proof(cases "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)")
       case True
       then obtain S where S_def: "S \<in> Xs \<and> x \<in> S"
-        using assms is_partitionE by blast 
+        using assms is_partitionE by blast
       then have T0: "parametric_fun_glue n Xs fs x = F S x"
         using 1 assms parametric_fun_glue_formula by blast
       have T1: "g x = F S x"
       proof-
         have 00: " F S \<oplus>\<^bsub>SA n\<^esub> finsum (SA n) F (Xs - {S}) =  finsum (SA n) F (insert S (Xs - {S}))"
-          using abelian_monoid.finsum_insert[of "SA n" "Xs - {S}" S F ] 
+          using abelian_monoid.finsum_insert[of "SA n" "Xs - {S}" S F ]
           by (metis (no_types, lifting) "0" Diff_iff Pi_anti_mono Pi_split_insert_domain SA_is_abelian_monoid S_def Set.basic_monos(7) assms(1) finite_Diff insert_iff subsetI)
         have T10: "g  = F S \<oplus>\<^bsub>SA n\<^esub> finsum (SA n) F (Xs - {S})"
-          using S_def unfolding 00 g_def 
-          by (simp add: insert_absorb)         
+          using S_def unfolding 00 g_def
+          by (simp add: insert_absorb)
         have T11: "finsum (SA n) F (Xs - {S}) \<in> carrier (SA n)"
           using abelian_monoid.finsum_closed[of "SA n" F "Xs - {S}"] assms SA_is_ring 0
-          unfolding g_def ring_def abelian_group_def by blast 
+          unfolding g_def ring_def abelian_group_def by blast
         hence T12: "g x = F S x \<oplus> finsum (SA n) F (Xs - {S}) x"
           using SA_add S_def T10 assms is_semialgebraic_closed by blast
         have T13: "finsum (SA n) F (Xs - {S}) x = \<zero>"
-          apply(rule SA_finsum_apply_zero[of "Xs - {S}" F n x]) 
+          apply(rule SA_finsum_apply_zero[of "Xs - {S}" F n x])
               using assms apply blast
                 using "0" apply blast
-                using True apply blast 
+                using True apply blast
         proof-
            fix s assume AA: "s \<in> Xs - {S}"
            then have "x \<notin> s"
              using True assms S_def is_partitionE[of Xs "carrier (Q\<^sub>p\<^bsup>n\<^esup>)"]  disjointE[of Xs S s]
-             by blast              
+             by blast
            then show "F s x = \<zero>"
              using AA 2[of s x] True by blast
         qed
         have T14: "F S x \<in> carrier Q\<^sub>p"
           using assms True S_def  by (metis (no_types, lifting) "0" Qp.function_ring_car_mem_closed PiE SA_car_memE(2))
         then show ?thesis using T12 T13 assms True Qp.add.l_cancel_one Qp.zero_closed by presburger
       qed
-      show ?thesis using T0 T1 by blast 
+      show ?thesis using T0 T1 by blast
     next
       case False
-      then show ?thesis 
-        using g_closed unfolding parametric_fun_glue_def 
+      then show ?thesis
+        using g_closed unfolding parametric_fun_glue_def
         by (metis (mono_tags, lifting) function_ring_not_car SA_car_memE(2) restrict_def)
     qed
   qed
-  then show ?thesis using g_closed by blast 
+  then show ?thesis using g_closed by blast
 qed
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsubsection\<open>Turning Functions into Units Via Gluing\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 text\<open>
   By gluing a function to the multiplicative unit on its zero set, we can get a canonical choice of
   local multiplicative inverse of a function $f$. Denef's proof frequently reasons about functions
-  of the form $\frac{f(x)}{g(x)}$ with the tacit understanding that they are meant to be defined 
-  on the largest domain of definition possible. This technical tool allows us to replicate this 
+  of the form $\frac{f(x)}{g(x)}$ with the tacit understanding that they are meant to be defined
+  on the largest domain of definition possible. This technical tool allows us to replicate this
   kind of reasoning in our formal proofs.
 \<close>
 
 definition to_fun_unit where
 "to_fun_unit n f = fun_glue n {x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). f x \<noteq> \<zero>} f \<one>\<^bsub>SA n\<^esub>"
 
 lemma to_fun_unit_is_unit:
   assumes "f \<in> carrier (SA n)"
   shows "to_fun_unit n f \<in> Units (SA n)"
-  unfolding to_fun_unit_def 
-  apply(rule fun_glue_unit) 
+  unfolding to_fun_unit_def
+  apply(rule fun_glue_unit)
   apply (simp add: assms)
-  using assms nonzero_evimage_closed[of f] apply blast 
-  by blast 
+  using assms nonzero_evimage_closed[of f] apply blast
+  by blast
 
 lemma to_fun_unit_closed:
   assumes "f \<in> carrier (SA n)"
   shows "to_fun_unit n f \<in> carrier (SA n)"
   using assms to_fun_unit_is_unit SA_is_ring SA_Units_closed by blast
 
 lemma to_fun_unit_eq:
   assumes "f \<in> carrier (SA n)"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "f x \<noteq> \<zero>"
   shows "to_fun_unit n f x = f x"
-  unfolding to_fun_unit_def fun_glue_def using assms 
+  unfolding to_fun_unit_def fun_glue_def using assms
   by simp
-  
+
 lemma to_fun_unit_eq':
   assumes "f \<in> carrier (SA n)"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "f x = \<zero>"
   shows "to_fun_unit n f x = \<one>"
-  unfolding to_fun_unit_def fun_glue_def using assms 
+  unfolding to_fun_unit_def fun_glue_def using assms
   by (simp add: SA_oneE)
 
 definition one_over_fun where
 "one_over_fun n f = inv\<^bsub>SA n\<^esub> (to_fun_unit n f)"
 
-lemma one_over_fun_closed: 
+lemma one_over_fun_closed:
   assumes "f \<in> carrier (SA n)"
   shows "one_over_fun n f \<in> carrier (SA n)"
-  using assms SA_is_ring[of n] to_fun_unit_is_unit[of f n] 
+  using assms SA_is_ring[of n] to_fun_unit_is_unit[of f n]
   by (metis SA_Units_closed one_over_fun_def ring.Units_inverse)
 
-lemma one_over_fun_eq: 
+lemma one_over_fun_eq:
   assumes "f \<in> carrier (SA n)"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
-  assumes "f x \<noteq> \<zero>" 
+  assumes "f x \<noteq> \<zero>"
   shows "one_over_fun n f x = inv (f x)"
-  using assms to_fun_unit_eq unfolding one_over_fun_def 
+  using assms to_fun_unit_eq unfolding one_over_fun_def
   using Qp_funs_m_inv SA_Units_Qp_funs_Units SA_Units_Qp_funs_inv to_fun_unit_is_unit by presburger
 
-lemma one_over_fun_smult_eval: 
+lemma one_over_fun_smult_eval:
   assumes "f \<in> carrier (SA n)"
   assumes "a \<noteq> \<zero>"
   assumes "a \<in> carrier Q\<^sub>p"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "f x \<noteq> \<zero>"
   shows "one_over_fun n (a \<odot>\<^bsub>SA n\<^esub>f) x = inv (a \<otimes> (f x))"
-  using one_over_fun_eq[of "a \<odot>\<^bsub>SA n\<^esub> f" n x] assms 
+  using one_over_fun_eq[of "a \<odot>\<^bsub>SA n\<^esub> f" n x] assms
   by (metis Qp.function_ring_car_memE Qp.integral SA_car_memE(2) SA_smult_closed SA_smult_formula)
 
 lemma one_over_fun_smult_eval':
   assumes "f \<in> carrier (SA n)"
   assumes "a \<noteq> \<zero>"
   assumes "a \<in> carrier Q\<^sub>p"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "f x \<noteq> \<zero>"
   shows "one_over_fun n (a \<odot>\<^bsub>SA n\<^esub>f) x = inv a \<otimes> inv (f x)"
-proof- 
+proof-
   have 0: "one_over_fun n (a \<odot>\<^bsub>SA n\<^esub> f) x = inv (a \<otimes> f x)"
     using assms one_over_fun_smult_eval[of f n a x]
     by fastforce
   have 1: "f x \<in> nonzero Q\<^sub>p"
     by(intro nonzero_memI SA_car_closed[ of _ n] assms)
   show ?thesis
-    unfolding 0 using 1 assms  
+    unfolding 0 using 1 assms
     using Qp.comm_inv_char Qp.cring_simprules(11) Qp.cring_simprules(5) SA_car_closed field_inv(2) field_inv(3) local.fract_cancel_right by presburger
 qed
 
 
 
 lemma SA_add_pow_closed:
   assumes "f \<in> carrier (SA n)"
   shows "([(k::nat)]\<cdot>\<^bsub>SA n\<^esub>f) \<in> carrier (SA n)"
   using assms SA_is_ring[of n]
   by (meson ring.nat_mult_closed)
 
-lemma one_over_fun_add_pow_eval: 
+lemma one_over_fun_add_pow_eval:
   assumes "f \<in> carrier (SA n)"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "f x \<noteq> \<zero>"
   assumes "(k::nat) > 0"
   shows "one_over_fun n ([k]\<cdot>\<^bsub>SA n\<^esub>f) x = inv ([k] \<cdot>f x)"
 proof-
   have 0: "([k] \<cdot>\<^bsub>SA n\<^esub> f) x = [k] \<cdot> f x"
     using assms SA_add_pow_apply[of f n x k] by linarith
   hence 1: "([k] \<cdot>\<^bsub>SA n\<^esub> f) x \<noteq> \<zero>"
-    using assms Qp_char_0'' Qp.function_ring_car_mem_closed SA_car_memE(2) 
+    using assms Qp_char_0'' Qp.function_ring_car_mem_closed SA_car_memE(2)
     by metis
   have 2: "one_over_fun n ([k] \<cdot>\<^bsub>SA n\<^esub> f) x = inv ([k] \<cdot>\<^bsub>SA n\<^esub> f) x"
     using assms one_over_fun_eq[of "[k]\<cdot>\<^bsub>SA n\<^esub>f" n x] 1 SA_add_pow_closed by blast
   thus ?thesis using 1 0 by presburger
 qed
 
-lemma one_over_fun_pow_closed: 
+lemma one_over_fun_pow_closed:
   assumes "f \<in> carrier (SA n)"
   shows "one_over_fun n (f[^]\<^bsub>SA n\<^esub>(k::nat)) \<in> carrier (SA n)"
-  using assms 
+  using assms
   by (meson SA_nat_pow_closed one_over_fun_closed padic_fields.SA_imp_semialg padic_fields_axioms)
 
-lemma one_over_fun_pow_eval: 
+lemma one_over_fun_pow_eval:
   assumes "f \<in> carrier (SA n)"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "f x \<noteq> \<zero>"
   shows "one_over_fun n (f[^]\<^bsub>SA n\<^esub>(k::nat)) x = inv ((f x) [^] k)"
-  using one_over_fun_eq[of  "f[^]\<^bsub>SA n\<^esub>k" n x] assms 
+  using one_over_fun_eq[of  "f[^]\<^bsub>SA n\<^esub>k" n x] assms
   by (metis Qp.function_ring_car_memE Qp.nonzero_pow_nonzero SA_car_memE(2) SA_nat_pow SA_nat_pow_closed padic_fields.SA_car_memE(1) padic_fields_axioms)
- 
+
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsection\<open>Inclusions of Lower Dimensional Function Rings\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 
 definition fun_inc where
 "fun_inc m n f = (\<lambda> x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). f (take n x))"
 
 lemma fun_inc_closed:
   assumes "f \<in> carrier (SA n)"
   assumes "m \<ge> n"
   shows  "fun_inc m n f \<in> carrier (SA m)"
 proof-
   have 0: "\<And>x. x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<Longrightarrow> fun_inc m n f x = (f \<circ> take n) x"
     unfolding fun_inc_def by (metis comp_apply restrict_apply')
   have 1: "is_semialg_function m (f \<circ> take n)"
     using assms comp_take_is_semialg
     by (metis SA_imp_semialg le_neq_implies_less padic_fields.semialg_function_comp_closed padic_fields_axioms take_is_semialg_map)
   have 2: "is_semialg_function m (fun_inc m n f)"
     using 0 1 semialg_function_on_carrier' by blast
   show ?thesis apply(rule SA_car_memI) apply(rule  Qp.function_ring_car_memI)
     using "2" is_semialg_function_closed apply blast
     using fun_inc_def[of m n f] unfolding restrict_def apply presburger
-    using 2 by blast 
+    using 2 by blast
 qed
 
 lemma fun_inc_eval:
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
   shows "fun_inc m n f x = f (take n x)"
-  unfolding fun_inc_def using assms 
+  unfolding fun_inc_def using assms
   by (meson restrict_apply')
 
-lemma ord_congruence_set_univ_semialg_fixed: 
+lemma ord_congruence_set_univ_semialg_fixed:
   assumes "n \<ge> 0"
-  shows "is_univ_semialgebraic (ord_congruence_set n a)" 
-  using ord_congruence_set_univ_semialg assms 
+  shows "is_univ_semialgebraic (ord_congruence_set n a)"
+  using ord_congruence_set_univ_semialg assms
   by auto
-  
-lemma ord_congruence_set_SA_function: 
+
+lemma ord_congruence_set_SA_function:
   assumes "n \<ge> 0"
   assumes "c \<in> carrier (SA (m+l))"
   shows "is_semialgebraic (m+l) {x \<in> carrier (Q\<^sub>p\<^bsup>m+l\<^esup>). c x \<in> nonzero Q\<^sub>p \<and> ord (c x) mod n = a}"
 proof-
   have 0: "{x \<in> carrier (Q\<^sub>p\<^bsup>m+l\<^esup>). c x \<in> nonzero Q\<^sub>p \<and> ord (c x) mod n = a} = c  \<inverse>\<^bsub>m+l\<^esub> (ord_congruence_set n a)"
     unfolding ord_congruence_set_def evimage_def using assms by blast
   show ?thesis unfolding 0 apply(rule evimage_is_semialg)
-    using assms apply blast using assms  ord_congruence_set_univ_semialg_fixed[of n a]  
-    by blast  
+    using assms apply blast using assms  ord_congruence_set_univ_semialg_fixed[of n a]
+    by blast
 qed
 
 lemma ac_cong_set_SA:
   assumes "n > 0"
   assumes "k \<in> Units (Zp_res_ring n)"
   assumes "c \<in> carrier (SA (m+l))"
   shows "is_semialgebraic (m+l) {x \<in> carrier (Q\<^sub>p\<^bsup>m+l\<^esup>). c x \<in> nonzero Q\<^sub>p \<and> ac n (c x) = k}"
 proof-
   have  "{x \<in> carrier (Q\<^sub>p\<^bsup>m+l\<^esup>). c x \<in> nonzero Q\<^sub>p \<and> ac n (c x) = k}= (c \<inverse>\<^bsub>m + l\<^esub> ac_cong_set n k)"
     unfolding ac_cong_set_def evimage_def nonzero_def mem_Collect_eq using assms by blast
-  thus ?thesis   
-    using assms ac_cong_set_is_univ_semialg[of n k] evimage_is_semialg[of c "m+l" "ac_cong_set n k"] 
+  thus ?thesis
+    using assms ac_cong_set_is_univ_semialg[of n k] evimage_is_semialg[of c "m+l" "ac_cong_set n k"]
     by presburger
 qed
 
 lemma ac_cong_set_SA':
   assumes "n >0 "
   assumes "k \<in> Units (Zp_res_ring n)"
   assumes "c \<in> carrier (SA m)"
   shows "is_semialgebraic m {x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). c x \<in> nonzero Q\<^sub>p \<and> ac n (c x) = k}"
-  using assms ac_cong_set_SA[of n k c m 0] unfolding Nat.add_0_right by blast 
+  using assms ac_cong_set_SA[of n k c m 0] unfolding Nat.add_0_right by blast
 
 lemma ac_cong_set_SA'':
   assumes "n >0 "
   assumes "m > 0"
   assumes "k \<in> Units (Zp_res_ring n)"
   assumes "c \<in> carrier (SA m)"
   assumes "\<And>x. x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<Longrightarrow> c x \<noteq> \<zero>"
   shows "is_semialgebraic m {x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). ac n (c x) = k}"
 proof-
   have "{x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). c x \<in> nonzero Q\<^sub>p \<and> ac n (c x) = k} = {x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). ac n (c x) = k}"
-    apply(rule subset_antisym) apply blast 
-    apply(rule subsetI) using assms unfolding nonzero_def mem_Collect_eq 
-    using Qp.function_ring_car_memE SA_car_memE(2) by blast 
-  thus ?thesis using assms ac_cong_set_SA'[of n k c m] by metis 
+    apply(rule subset_antisym) apply blast
+    apply(rule subsetI) using assms unfolding nonzero_def mem_Collect_eq
+    using Qp.function_ring_car_memE SA_car_memE(2) by blast
+  thus ?thesis using assms ac_cong_set_SA'[of n k c m] by metis
 qed
 
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsection\<open>Miscellaneous\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 
 lemma nth_pow_wits_SA_fun_prep:
   assumes "n > 0"
   assumes "h \<in> carrier (SA m)"
   assumes "\<rho> \<in> nth_pow_wits n"
   shows "is_semialgebraic m (h \<inverse>\<^bsub>m\<^esub>pow_res n \<rho>)"
   by(intro evimage_is_semialg assms  pow_res_is_univ_semialgebraic nth_pow_wits_closed(1)[of n])
 
 definition kth_rt where
 "kth_rt m (k::nat) f x = (if x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) then (THE b. b \<in> carrier Q\<^sub>p \<and> b[^]k = (f x) \<and> ac (nat (ord ([k]\<cdot>\<one>)) + 1) b = 1)
                                                            else undefined )"
 
 text\<open>Normalizing a semialgebraic function to have a constant angular component\<close>
 
 lemma ac_res_Unit_inc:
   assumes "n > 0"
   assumes "t \<in> Units (Zp_res_ring n)"
   shows "ac n ([t]\<cdot>\<one>) = t"
 proof-
   have 0: "[t]\<cdot>\<one> \<noteq>\<zero>"
     using assms by (metis Qp_char_0_int less_one less_or_eq_imp_le nat_neq_iff zero_not_in_residue_units)
-  have 1: "[t]\<cdot>\<one> \<in> \<O>\<^sub>p" 
+  have 1: "[t]\<cdot>\<one> \<in> \<O>\<^sub>p"
     by (metis Zp.one_closed Zp_int_mult_closed image_eqI inc_of_int)
   hence 2: "angular_component ([t]\<cdot>\<one>) = ac_Zp ([t]\<cdot>\<^bsub>Z\<^sub>p\<^esub>\<one>\<^bsub>Z\<^sub>p\<^esub>)"
-    using angular_component_of_inclusion[of "[t]\<cdot>\<^bsub>Z\<^sub>p\<^esub>\<one>\<^bsub>Z\<^sub>p\<^esub>"] 
+    using angular_component_of_inclusion[of "[t]\<cdot>\<^bsub>Z\<^sub>p\<^esub>\<one>\<^bsub>Z\<^sub>p\<^esub>"]
     by (metis "0" Qp.int_inc_zero Zp.int_inc_zero Zp_int_inc_closed inc_of_int not_nonzero_Qp)
   hence 3: "ac n ([t]\<cdot>\<one>) = ac_Zp ([t]\<cdot>\<^bsub>Z\<^sub>p\<^esub>\<one>\<^bsub>Z\<^sub>p\<^esub>) n"
     unfolding ac_def using 0 by presburger
   hence "val_Zp ([t]\<cdot>\<^bsub>Z\<^sub>p\<^esub>\<one>\<^bsub>Z\<^sub>p\<^esub>) = 0"
   proof-
     have "coprime p t"
-      using assms 
+      using assms
       by (metis coprime_commute less_one less_or_eq_imp_le nat_neq_iff padic_integers.residue_UnitsE padic_integers_axioms)
-    then show ?thesis 
+    then show ?thesis
       by (metis Zp_int_inc_closed Zp_int_inc_res coprime_mod_right_iff coprime_power_right_iff mod_by_0 order_refl p_residues residues.m_gt_one residues.mod_in_res_units val_Zp_0_criterion val_Zp_p val_Zp_p_int_unit zero_less_one zero_neq_one_class.one_neq_zero zero_not_in_residue_units)
   qed
   hence 4: "[t]\<cdot>\<^bsub>Z\<^sub>p\<^esub>\<one>\<^bsub>Z\<^sub>p\<^esub> \<in> Units Z\<^sub>p"
     using val_Zp_0_imp_unit by blast
   hence 5: "ac_Zp ([t]\<cdot>\<^bsub>Z\<^sub>p\<^esub>\<one>\<^bsub>Z\<^sub>p\<^esub>)  = [t]\<cdot>\<^bsub>Z\<^sub>p\<^esub>\<one>\<^bsub>Z\<^sub>p\<^esub>"  using
       ac_Zp_of_Unit  \<open>val_Zp ([t] \<cdot>\<^bsub>Z\<^sub>p\<^esub> \<one>\<^bsub>Z\<^sub>p\<^esub>) = 0\<close> by blast
   have 6: "ac_Zp ([t]\<cdot>\<^bsub>Z\<^sub>p\<^esub>\<one>\<^bsub>Z\<^sub>p\<^esub>) n  = t"
   proof-
     have "t \<in> carrier (Zp_res_ring n)"
       using assms monoid.Units_closed[of "Zp_res_ring n" t] cring_def padic_integers.R_cring padic_integers_axioms ring_def by blast
     hence "t < p^n \<and> t \<ge> 0 "
-      using p_residue_ring_car_memE by auto 
+      using p_residue_ring_car_memE by auto
     thus ?thesis
-      unfolding 5 unfolding Zp_int_inc_rep p_residue_def residue_def by auto 
+      unfolding 5 unfolding Zp_int_inc_rep p_residue_def residue_def by auto
   qed
-  show ?thesis 
-    unfolding 3 using 6 by blast 
+  show ?thesis
+    unfolding 3 using 6 by blast
 qed
 
 lemma val_of_res_Unit:
   assumes "n > 0"
   assumes "t \<in> Units (Zp_res_ring n)"
   shows "val ([t]\<cdot>\<one>) = 0"
 proof-
   have 0: "[t]\<cdot>\<one> \<noteq>\<zero>"
     using assms by (metis Qp_char_0_int less_one less_or_eq_imp_le nat_neq_iff zero_not_in_residue_units)
-  have 1: "[t]\<cdot>\<one> \<in> \<O>\<^sub>p" 
+  have 1: "[t]\<cdot>\<one> \<in> \<O>\<^sub>p"
     by (metis Zp.one_closed Zp_int_mult_closed image_eqI inc_of_int)
   hence 2: "angular_component ([t]\<cdot>\<one>) = ac_Zp ([t]\<cdot>\<^bsub>Z\<^sub>p\<^esub>\<one>\<^bsub>Z\<^sub>p\<^esub>)"
-    using angular_component_of_inclusion[of "[t]\<cdot>\<^bsub>Z\<^sub>p\<^esub>\<one>\<^bsub>Z\<^sub>p\<^esub>"] 
+    using angular_component_of_inclusion[of "[t]\<cdot>\<^bsub>Z\<^sub>p\<^esub>\<one>\<^bsub>Z\<^sub>p\<^esub>"]
     by (metis "0" Qp.int_inc_zero Zp.int_inc_zero Zp_int_inc_closed inc_of_int not_nonzero_Qp)
   hence 3: "ac n ([t]\<cdot>\<one>) = ac_Zp ([t]\<cdot>\<^bsub>Z\<^sub>p\<^esub>\<one>\<^bsub>Z\<^sub>p\<^esub>) n"
     unfolding ac_def using 0 by presburger
   hence "val_Zp ([t]\<cdot>\<^bsub>Z\<^sub>p\<^esub>\<one>\<^bsub>Z\<^sub>p\<^esub>) = 0"
   proof-
     have "coprime p t"
-      using assms 
+      using assms
       by (metis coprime_commute less_one less_or_eq_imp_le nat_neq_iff padic_integers.residue_UnitsE padic_integers_axioms)
-    then show ?thesis 
+    then show ?thesis
       by (metis Zp_int_inc_closed Zp_int_inc_res coprime_mod_right_iff coprime_power_right_iff mod_by_0 order_refl p_residues residues.m_gt_one residues.mod_in_res_units val_Zp_0_criterion val_Zp_p val_Zp_p_int_unit zero_less_one zero_neq_one_class.one_neq_zero zero_not_in_residue_units)
   qed
-  then show ?thesis using assms 
+  then show ?thesis using assms
     by (metis Zp_int_inc_closed inc_of_int val_of_inc)
 qed
 
 
 lemma(in padic_integers) res_map_is_hom:
   assumes "N > 0"
   shows "ring_hom_ring Zp (Zp_res_ring N) (\<lambda> x. x N)"
   apply(rule ring_hom_ringI)
   apply (simp add: R.ring_axioms)
   using assms cring.axioms(1) local.R_cring apply blast
   using residue_closed apply blast
   using residue_of_prod apply blast
   using residue_of_sum apply blast
   using assms residue_of_one(1) by blast
 
 lemma ac_of_fraction:
   assumes "N > 0"
   assumes "a \<in> nonzero Q\<^sub>p"
   assumes "b \<in> nonzero Q\<^sub>p"
   shows "ac N (a \<div> b) = ac N a \<otimes>\<^bsub>Zp_res_ring N\<^esub> inv \<^bsub>Zp_res_ring N\<^esub> ac N b"
-  using ac_mult[of a "inv b" N] ac_inv assms Qp.nonzero_closed nonzero_inverse_Qp by presburger 
+  using ac_mult[of a "inv b" N] ac_inv assms Qp.nonzero_closed nonzero_inverse_Qp by presburger
 
 lemma pow_res_eq_rel:
   assumes "n > 0"
   assumes "b \<in> carrier Q\<^sub>p"
   shows "{x \<in> carrier Q\<^sub>p. pow_res n x = pow_res n b} = pow_res n b"
-  apply(rule equalityI',  unfold mem_Collect_eq, metis pow_res_refl, 
+  apply(rule equalityI',  unfold mem_Collect_eq, metis pow_res_refl,
         intro conjI)
-    using pow_res_def apply auto[1] 
+    using pow_res_def apply auto[1]
     apply(rule equal_pow_resI)
-    using pow_res_def apply auto[1] 
+    using pow_res_def apply auto[1]
     using  pow_res_refl assms  by (metis equal_pow_resI)
 
 lemma pow_res_is_univ_semialgebraic':
   assumes "n > 0"
   assumes "b \<in> carrier Q\<^sub>p"
   shows "is_univ_semialgebraic {x \<in> carrier Q\<^sub>p. pow_res n x = pow_res n b}"
   using assms pow_res_eq_rel pow_res_is_univ_semialgebraic by presburger
 
 lemma evimage_eqI:
   assumes "c \<in> carrier (SA n)"
-  shows "c  \<inverse>\<^bsub>n\<^esub> {x \<in> carrier Q\<^sub>p. P x} = {x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). P (c x)}"  
+  shows "c  \<inverse>\<^bsub>n\<^esub> {x \<in> carrier Q\<^sub>p. P x} = {x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). P (c x)}"
   by(rule equalityI', unfold evimage_def mem_Collect_eq Int_iff,  intro conjI, auto
         , rule SA_car_closed[of _ n], auto simp: assms)
 
 lemma SA_pow_res_is_semialgebraic:
   assumes "n > 0"
   assumes "b \<in> carrier Q\<^sub>p"
   assumes "c \<in> carrier (SA N)"
   shows "is_semialgebraic N  {x \<in> carrier (Q\<^sub>p\<^bsup>N\<^esup>). pow_res n (c x) = pow_res n b}"
 proof-
   have " c  \<inverse>\<^bsub>N\<^esub> {x \<in> carrier Q\<^sub>p. pow_res n x = pow_res n b} = {x \<in> carrier (Q\<^sub>p\<^bsup>N\<^esup>). pow_res n (c x) = pow_res n b}"
-    apply(rule evimage_eqI) using assms by blast 
-  thus ?thesis 
+    apply(rule evimage_eqI) using assms by blast
+  thus ?thesis
     using pow_res_is_univ_semialgebraic' evimage_is_semialg assms
-    by (metis (no_types, lifting))      
+    by (metis (no_types, lifting))
 qed
 
 lemma eint_diff_imp_eint:
   assumes "a \<in> nonzero Q\<^sub>p"
   assumes "b \<in> carrier Q\<^sub>p"
   assumes "val a = val b + eint i"
   shows "b \<in> nonzero Q\<^sub>p"
-  using assms val_zero 
+  using assms val_zero
   by (metis Qp.nonzero_closed Qp.not_nonzero_memE not_eint_eq plus_eint_simps(2) val_ord')
 
 lemma SA_minus_eval:
   assumes "f \<in> carrier (SA n)"
   assumes "g \<in> carrier (SA n)"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "(f  \<ominus>\<^bsub>SA n\<^esub> g) x = f x \<ominus> g x"
-  using assms unfolding a_minus_def 
-  using SA_a_inv_eval SA_add  by metis  
+  using assms unfolding a_minus_def
+  using SA_a_inv_eval SA_add  by metis
 
 lemma Qp_cong_set_evimage:
   assumes "f \<in> carrier (SA n)"
   assumes "a \<in> carrier Z\<^sub>p"
   shows  "is_semialgebraic n (f \<inverse>\<^bsub>n\<^esub> (Qp_cong_set \<alpha> a))"
   using assms Qp_cong_set_is_univ_semialgebraic evimage_is_semialg by blast
 
-lemma SA_constant_res_set_semialg: 
+lemma SA_constant_res_set_semialg:
   assumes "l \<in> carrier (Zp_res_ring n)"
   assumes "f \<in> carrier (SA m)"
   shows "is_semialgebraic m {x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). f x \<in> \<O>\<^sub>p \<and> Qp_res (f x) n = l}"
-proof- 
+proof-
   have 0: "{x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). f x \<in> \<O>\<^sub>p \<and> Qp_res (f x) n = l} = f  \<inverse>\<^bsub>m\<^esub> {x \<in> \<O>\<^sub>p. Qp_res x n = l}"
-    unfolding evimage_def  by blast 
-  show ?thesis unfolding 0 
+    unfolding evimage_def  by blast
+  show ?thesis unfolding 0
     by(rule evimage_is_semialg, rule assms, rule constant_res_set_semialg, rule assms)
 qed
 
 lemma val_ring_cong_set:
   assumes "f \<in> carrier (SA k)"
   assumes "\<And>x. x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>) \<Longrightarrow> f x \<in> \<O>\<^sub>p"
   assumes "t \<in> carrier (Zp_res_ring n)"
   shows "is_semialgebraic k {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). to_Zp (f x) n = t}"
 proof-
   have 0: "[t] \<cdot>\<^bsub>Z\<^sub>p\<^esub> \<one>\<^bsub>Z\<^sub>p\<^esub> \<in> carrier Z\<^sub>p "
-    by blast 
+    by blast
   have 1: "([t] \<cdot>\<^bsub>Z\<^sub>p\<^esub> \<one>\<^bsub>Z\<^sub>p\<^esub>) n = t"
-    using assms 
-    unfolding Zp_int_inc_rep p_residue_def residue_def residue_ring_def by simp 
+    using assms
+    unfolding Zp_int_inc_rep p_residue_def residue_def residue_ring_def by simp
   have "{x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). to_Zp (f x) n = t} = f  \<inverse>\<^bsub>k\<^esub> {x \<in> \<O>\<^sub>p. (to_Zp x) n = t}"
     unfolding evimage_def using assms by auto
-  then show ?thesis using 0 1 assms  Qp_cong_set_evimage[of f k "[t]\<cdot>\<^bsub>Z\<^sub>p\<^esub> \<one>\<^bsub>Z\<^sub>p\<^esub>" n] unfolding Qp_cong_set_def 
+  then show ?thesis using 0 1 assms  Qp_cong_set_evimage[of f k "[t]\<cdot>\<^bsub>Z\<^sub>p\<^esub> \<one>\<^bsub>Z\<^sub>p\<^esub>" n] unfolding Qp_cong_set_def
     by presburger
 qed
 
 lemma val_ring_pullback_SA:
   assumes "N > 0"
   assumes "c \<in> carrier (SA N)"
   shows "is_semialgebraic N {x \<in> carrier (Q\<^sub>p\<^bsup>N\<^esup>). c x \<in> \<O>\<^sub>p}"
 proof-
   have 0: "{x \<in> carrier (Q\<^sub>p\<^bsup>N\<^esup>). c x \<in> \<O>\<^sub>p} = c  \<inverse>\<^bsub>N\<^esub> \<O>\<^sub>p"
     unfolding evimage_def by blast
   have 1: "is_univ_semialgebraic \<O>\<^sub>p"
     using Qp_val_ring_is_univ_semialgebraic by blast
   show ?thesis using assms 0 1 evimage_is_semialg by presburger
 qed
 
 lemma(in padic_fields) res_eq_set_is_semialg:
   assumes "k > 0"
   assumes "c \<in> carrier (SA k)"
   assumes "s \<in> carrier (Zp_res_ring n)"
   shows "is_semialgebraic k {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). c x \<in> \<O>\<^sub>p \<and> to_Zp (c x) n = s}"
 proof-
   obtain a where a_def: "a = [s]\<cdot>\<one>"
-    by blast 
+    by blast
   have 0: "a \<in> \<O>\<^sub>p"
-    using a_def 
+    using a_def
     by (metis Zp.one_closed Zp_int_mult_closed image_iff inc_of_int)
   have 1: "to_Zp a = [s]\<cdot>\<^bsub>Z\<^sub>p\<^esub>\<one>\<^bsub>Z\<^sub>p\<^esub>"
-    using 0 unfolding a_def 
+    using 0 unfolding a_def
     by (metis Q\<^sub>p_def Zp_def Zp_int_inc_closed \<iota>_def inc_to_Zp padic_fields.inc_of_int padic_fields_axioms)
   have 2: "([s]\<cdot>\<^bsub>Z\<^sub>p\<^esub>\<one>\<^bsub>Z\<^sub>p\<^esub>) n = s"
-    using assms 
+    using assms
     by (metis Zp_int_inc_res mod_pos_pos_trivial p_residue_ring_car_memE(1) p_residue_ring_car_memE(2))
   have 3: "{x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). c x \<in> \<O>\<^sub>p \<and> to_Zp (c x) n = s} = c  \<inverse>\<^bsub>k\<^esub> B\<^bsub>n\<^esub>[a]"
   proof
     show "{x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). c x \<in> \<O>\<^sub>p \<and> to_Zp (c x) n = s} \<subseteq> c \<inverse>\<^bsub>k\<^esub> B\<^bsub>int n\<^esub>[a]"
     proof fix x assume A: "x \<in> {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). c x \<in> \<O>\<^sub>p \<and> to_Zp (c x) n = s}"
-      then have  30: "x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>)" by blast 
-      have 31: "c x \<in> \<O>\<^sub>p" using A by blast 
-      have 32: "to_Zp (c x) n = s" using A by blast 
+      then have  30: "x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>)" by blast
+      have 31: "c x \<in> \<O>\<^sub>p" using A by blast
+      have 32: "to_Zp (c x) n = s" using A by blast
       have 33: "to_Zp (c x) \<in> carrier Z\<^sub>p"
         using 31 val_ring_memE to_Zp_closed by blast
       have 34: "to_Zp (c x) n = (to_Zp a) n"
         using "1" "2" "32" by presburger
       hence "((to_Zp (c x)) \<ominus>\<^bsub>Z\<^sub>p\<^esub> (to_Zp a)) n = 0"
         using "1" "33" Zp_int_inc_closed res_diff_zero_fact'' by presburger
       hence 35: "val_Zp ((to_Zp (c x)) \<ominus>\<^bsub>Z\<^sub>p\<^esub> (to_Zp a)) \<ge> n"
         using "1" "33" "34" Zp.one_closed Zp_int_mult_closed val_Zp_dist_def val_Zp_dist_res_eq2 by presburger
       have 36: "val (c x \<ominus> a) = val_Zp ((to_Zp (c x)) \<ominus>\<^bsub>Z\<^sub>p\<^esub> (to_Zp a))"
-        using 31 0 
+        using 31 0
         by (metis to_Zp_minus to_Zp_val val_ring_minus_closed)
       hence "val (c x \<ominus> a) \<ge> n"
         using 35 by presburger
       hence "c x \<in>  B\<^bsub>int n\<^esub>[a]"
         using 31 c_ballI  val_ring_memE by blast
       thus "x \<in> c \<inverse>\<^bsub>k\<^esub> B\<^bsub>int n\<^esub>[a]"
-        using 30  by blast 
+        using 30  by blast
     qed
     show "c \<inverse>\<^bsub>k\<^esub> B\<^bsub>int n\<^esub>[a] \<subseteq> {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). c x \<in> \<O>\<^sub>p \<and> to_Zp (c x) n = s}"
     proof fix x assume A: "x \<in> c \<inverse>\<^bsub>k\<^esub> B\<^bsub>int n\<^esub>[a]"
       have x_closed: "x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>)"
         using A  by (meson evimage_eq)
       have 00: "val (c x \<ominus> a) \<ge> n"
         using A c_ballE(2) by blast
       have cx_closed: "c x \<in> carrier Q\<^sub>p"
         using x_closed assms function_ring_car_closed SA_car_memE(2) by blast
       hence 11: "c x \<ominus> a \<in> \<O>\<^sub>p"
       proof-
         have "(0::eint) \<le> n"
           by (metis eint_ord_simps(1) of_nat_0_le_iff zero_eint_def)
-        thus ?thesis using 00 order_trans[of "0::eint" n] Qp_val_ringI 
+        thus ?thesis using 00 order_trans[of "0::eint" n] Qp_val_ringI
           by (meson "0" Qp.minus_closed val_ring_memE cx_closed)
       qed
       hence 22: "c x \<in> \<O>\<^sub>p"
       proof-
         have 00: "c x = (c x \<ominus> a) \<oplus> a"
-          using cx_closed 0 
+          using cx_closed 0
           by (metis "11" Qp.add.inv_solve_right' Qp.minus_eq val_ring_memE(2))
         have 01: "(c x \<ominus> a) \<oplus> a \<in> \<O>\<^sub>p"
           by(intro val_ring_add_closed 0 11)
-        then show ?thesis 
-          using 0 11 image_iff  "00" by auto 
+        then show ?thesis
+          using 0 11 image_iff  "00" by auto
       qed
       have 33: "val (c x \<ominus> a) = val_Zp (to_Zp (c x) \<ominus>\<^bsub>Z\<^sub>p\<^esub> to_Zp a)"
-        using 11 22 0 
-        by (metis to_Zp_minus to_Zp_val)       
+        using 11 22 0
+        by (metis to_Zp_minus to_Zp_val)
       have 44: "val_Zp (to_Zp (c x) \<ominus>\<^bsub>Z\<^sub>p\<^esub> to_Zp a) \<ge> n"
         using 33 00 by presburger
       have tzpcx: "to_Zp (c x) \<in> carrier Z\<^sub>p"
         using 22 by (metis image_iff inc_to_Zp)
       have tzpa: "to_Zp a \<in> carrier Z\<^sub>p"
         using 0 val_ring_memE to_Zp_closed by blast
       have 55: "(to_Zp (c x) \<ominus>\<^bsub>Z\<^sub>p\<^esub> to_Zp a) n = 0"
         using 44 tzpcx tzpa Zp.minus_closed zero_below_val_Zp by blast
       hence 66: "to_Zp (c x) n = s"
-        using 0 1 2 tzpa tzpcx 
+        using 0 1 2 tzpa tzpcx
         by (metis res_diff_zero_fact(1))
       then show "x \<in> {x \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>). c x \<in> \<O>\<^sub>p \<and> to_Zp (c x) n = s}"
         using "22" x_closed by blast
     qed
   qed
-  thus ?thesis 
+  thus ?thesis
     using evimage_is_semialg[of c k] 0  val_ring_memE assms(2) ball_is_univ_semialgebraic by presburger
 qed
 
-lemma SA_constant_res_set_semialg': 
+lemma SA_constant_res_set_semialg':
   assumes "f \<in> carrier (SA m)"
   assumes "C \<in> Qp_res_classes n"
   shows "is_semialgebraic m (f  \<inverse>\<^bsub>m\<^esub> C)"
-proof- 
+proof-
   obtain l where l_def: "l \<in> carrier (Zp_res_ring n) \<and> C = Qp_res_class n ([l]\<cdot>\<one>)"
-    using Qp_res_classes_wits assms by blast 
+    using Qp_res_classes_wits assms by blast
   have C_eq: "C = Qp_res_class n ([l]\<cdot>\<one>)"
-    using l_def by blast 
+    using l_def by blast
   have 0: "Qp_res ([l] \<cdot> \<one>) n = l"
-    using l_def 
+    using l_def
     by (metis Qp_res_int_inc mod_pos_pos_trivial p_residue_ring_car_memE(1) p_residue_ring_car_memE(2))
   have 1: "f  \<inverse>\<^bsub>m\<^esub> C = {x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). f x \<in> \<O>\<^sub>p \<and> Qp_res (f x) n = l}"
-    apply(rule equalityI') 
+    apply(rule equalityI')
     unfolding evimage_def C_eq Qp_res_class_def mem_Collect_eq unfolding 0 apply blast
-    by blast 
-  show ?thesis 
+    by blast
+  show ?thesis
     unfolding 1 apply(rule SA_constant_res_set_semialg )
     using  l_def apply blast by(rule assms)
 qed
 
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsection\<open>Semialgebraic Polynomials\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 
 lemma UP_SA_n_is_ring:
   shows "ring (UP (SA n))"
-  using SA_is_ring  
+  using SA_is_ring
   by (simp add: UP_ring.UP_ring UP_ring.intro)
-  
+
 lemma UP_SA_n_is_cring:
   shows "cring (UP (SA n))"
-  using SA_is_cring  
+  using SA_is_cring
   by (simp add: UP_cring.UP_cring UP_cring.intro)
 
 text\<open>The evaluation homomorphism from \texttt{Qp\_funs} to \texttt{Qp}\<close>
 
-definition eval_hom where 
+definition eval_hom where
 "eval_hom a = (\<lambda>f. f a)"
 
 lemma eval_hom_is_hom:
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "ring_hom_ring (Fun\<^bsub>n\<^esub> Q\<^sub>p) Q\<^sub>p (eval_hom a)"
   apply(rule ring_hom_ringI)
   using Qp_funs_is_cring cring.axioms(1) apply blast
-  apply (simp add: Qp.ring_axioms)  
-  apply (simp add: Qp.function_ring_car_mem_closed assms eval_hom_def) 
+  apply (simp add: Qp.ring_axioms)
+  apply (simp add: Qp.function_ring_car_mem_closed assms eval_hom_def)
   apply (metis Qp_funs_mult' assms eval_hom_def)
   apply (metis Qp_funs_add' assms eval_hom_def)
   by (metis function_one_eval assms eval_hom_def)
 
 text\<open>the homomorphism from \texttt{Fun n Qp [x]} to \texttt{Qp [x]} induced by evaluation of coefficients\<close>
 
 definition Qp_fpoly_to_Qp_poly where
 "Qp_fpoly_to_Qp_poly n a = poly_lift_hom (Fun\<^bsub>n\<^esub> Q\<^sub>p) Q\<^sub>p (eval_hom a)"
 
 lemma Qp_fpoly_to_Qp_poly_is_hom:
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "(Qp_fpoly_to_Qp_poly n a) \<in> ring_hom (UP (Fun\<^bsub>n\<^esub> Q\<^sub>p)) (Q\<^sub>p_x) "
-  unfolding Qp_fpoly_to_Qp_poly_def 
+  unfolding Qp_fpoly_to_Qp_poly_def
   apply(rule UP_cring.poly_lift_hom_is_hom)
-  unfolding UP_cring_def 
+  unfolding UP_cring_def
   apply (simp add: Qp_funs_is_cring)
   apply (simp add: UPQ.R_cring)
   using assms eval_hom_is_hom[of a] ring_hom_ring.homh by blast
 
 lemma Qp_fpoly_to_Qp_poly_extends_apply:
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "f \<in> carrier (Fun\<^bsub>n\<^esub> Q\<^sub>p)"
   shows "Qp_fpoly_to_Qp_poly n a (to_polynomial (Fun\<^bsub>n\<^esub> Q\<^sub>p) f) = to_polynomial Q\<^sub>p (f a)"
-  unfolding Qp_fpoly_to_Qp_poly_def 
+  unfolding Qp_fpoly_to_Qp_poly_def
   using assms eval_hom_is_hom[of a] UP_cring.poly_lift_hom_extends_hom[of "Fun\<^bsub>n\<^esub> Q\<^sub>p" Q\<^sub>p "eval_hom a" ]
         Qp.function_ring_car_memE[of f n] ring_hom_ring.homh
-  unfolding eval_hom_def UP_cring_def 
+  unfolding eval_hom_def UP_cring_def
   using Qp_funs_is_cring UPQ.R_cring by blast
 
 lemma Qp_fpoly_to_Qp_poly_X_var:
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "Qp_fpoly_to_Qp_poly n a (X_poly (Fun\<^bsub>n\<^esub> Q\<^sub>p)) = X_poly Q\<^sub>p"
   unfolding X_poly_def Qp_fpoly_to_Qp_poly_def
-  apply(rule UP_cring.poly_lift_hom_X_var) unfolding UP_cring_def 
-    apply (simp add: Qp_funs_is_cring)    
-      apply (simp add: UPQ.R_cring)      
-        using assms(1) eval_hom_is_hom ring_hom_ring.homh 
+  apply(rule UP_cring.poly_lift_hom_X_var) unfolding UP_cring_def
+    apply (simp add: Qp_funs_is_cring)
+      apply (simp add: UPQ.R_cring)
+        using assms(1) eval_hom_is_hom ring_hom_ring.homh
   by blast
-  
+
 lemma Qp_fpoly_to_Qp_poly_monom:
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "f \<in> carrier (Fun\<^bsub>n\<^esub> Q\<^sub>p)"
   shows "Qp_fpoly_to_Qp_poly n a (up_ring.monom (UP (Fun\<^bsub>n\<^esub> Q\<^sub>p)) f m) = up_ring.monom Q\<^sub>p_x (f a) m"
-  unfolding Qp_fpoly_to_Qp_poly_def 
+  unfolding Qp_fpoly_to_Qp_poly_def
   using UP_cring.poly_lift_hom_monom[of "Fun\<^bsub>n\<^esub> Q\<^sub>p" Q\<^sub>p "eval_hom a" f m] assms ring_hom_ring.homh
-        eval_hom_is_hom[of a] unfolding eval_hom_def UP_cring_def 
+        eval_hom_is_hom[of a] unfolding eval_hom_def UP_cring_def
   using Qp_funs_is_cring UPQ.R_cring by blast
-  
+
 lemma Qp_fpoly_to_Qp_poly_coeff:
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "f \<in> carrier (UP (Fun\<^bsub>n\<^esub> Q\<^sub>p))"
   shows "Qp_fpoly_to_Qp_poly n a f k = (f k) a"
-  using assms UP_cring.poly_lift_hom_cf[of "Fun\<^bsub>n\<^esub> Q\<^sub>p" Q\<^sub>p "eval_hom a" f k] eval_hom_is_hom[of a] 
-  unfolding  Qp_fpoly_to_Qp_poly_def eval_hom_def 
+  using assms UP_cring.poly_lift_hom_cf[of "Fun\<^bsub>n\<^esub> Q\<^sub>p" Q\<^sub>p "eval_hom a" f k] eval_hom_is_hom[of a]
+  unfolding  Qp_fpoly_to_Qp_poly_def eval_hom_def
   using Qp_funs_is_cring ring_hom_ring.homh  ring_hom_ring.homh
-  unfolding eval_hom_def UP_cring_def 
-  using UPQ.R_cring by blast  
+  unfolding eval_hom_def UP_cring_def
+  using UPQ.R_cring by blast
 
 lemma Qp_fpoly_to_Qp_poly_eval:
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "P \<in> carrier (UP (Fun\<^bsub>n\<^esub> Q\<^sub>p))"
   assumes "f \<in> carrier (Fun\<^bsub>n\<^esub> Q\<^sub>p)"
   shows "(UP_cring.to_fun (Fun\<^bsub>n\<^esub> Q\<^sub>p) P f) a  = UP_cring.to_fun Q\<^sub>p (Qp_fpoly_to_Qp_poly n a P) (f a)"
-  unfolding Qp_fpoly_to_Qp_poly_def 
-  using UP_cring.poly_lift_hom_eval[of "Fun\<^bsub>n\<^esub> Q\<^sub>p" Q\<^sub>p "eval_hom a" P f] 
+  unfolding Qp_fpoly_to_Qp_poly_def
+  using UP_cring.poly_lift_hom_eval[of "Fun\<^bsub>n\<^esub> Q\<^sub>p" Q\<^sub>p "eval_hom a" P f]
         eval_hom_is_hom[of a] eval_hom_def assms ring_hom_ring.homh Qp_funs_is_cring
-  unfolding eval_hom_def UP_cring_def 
+  unfolding eval_hom_def UP_cring_def
   using UPQ.R_cring by blast
- 
+
 lemma Qp_fpoly_to_Qp_poly_sub:
   assumes "f \<in> carrier (UP (Fun\<^bsub>n\<^esub> Q\<^sub>p))"
   assumes "g \<in> carrier (UP (Fun\<^bsub>n\<^esub> Q\<^sub>p))"
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "Qp_fpoly_to_Qp_poly n a (compose (Fun\<^bsub>n\<^esub> Q\<^sub>p) f g) = compose Q\<^sub>p (Qp_fpoly_to_Qp_poly n a f) (Qp_fpoly_to_Qp_poly n a g)"
-  unfolding Qp_fpoly_to_Qp_poly_def 
+  unfolding Qp_fpoly_to_Qp_poly_def
   using assms UP_cring.poly_lift_hom_sub[of "Fun\<^bsub>n\<^esub> Q\<^sub>p" Q\<^sub>p "eval_hom a" f g]
         eval_hom_is_hom[of a] ring_hom_ring.homh[of "Fun\<^bsub>n\<^esub> Q\<^sub>p" Q\<^sub>p "eval_hom a"]
-        Qp_funs_is_cring  
-  unfolding eval_hom_def UP_cring_def 
+        Qp_funs_is_cring
+  unfolding eval_hom_def UP_cring_def
   using UPQ.R_cring by blast
 
 lemma Qp_fpoly_to_Qp_poly_taylor_poly:
   assumes "F \<in> carrier (UP (Fun\<^bsub>n\<^esub> Q\<^sub>p))"
   assumes "c \<in> carrier (Fun\<^bsub>n\<^esub> Q\<^sub>p)"
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
-  shows "Qp_fpoly_to_Qp_poly n a (taylor_expansion (Fun\<^bsub>n\<^esub> Q\<^sub>p) c F) = 
+  shows "Qp_fpoly_to_Qp_poly n a (taylor_expansion (Fun\<^bsub>n\<^esub> Q\<^sub>p) c F) =
         taylor_expansion Q\<^sub>p (c a) (Qp_fpoly_to_Qp_poly n a F)"
 proof-
   have 0: "X_poly (Fun\<^bsub>n\<^esub> Q\<^sub>p) \<oplus>\<^bsub>UP (Fun\<^bsub>n\<^esub> Q\<^sub>p)\<^esub> to_polynomial (Fun\<^bsub>n\<^esub> Q\<^sub>p) c \<in> carrier (UP (Fun\<^bsub>n\<^esub> Q\<^sub>p))"
     by (metis Qp_funs_is_cring UP_cring.X_plus_closed UP_cring_def X_poly_plus_def assms(2))
   have 1: "poly_lift_hom (Fun\<^bsub>n\<^esub> Q\<^sub>p) Q\<^sub>p (eval_hom a) (X_poly (Fun\<^bsub>n\<^esub> Q\<^sub>p) \<oplus>\<^bsub>UP (Fun\<^bsub>n\<^esub> Q\<^sub>p)\<^esub> to_polynomial (Fun\<^bsub>n\<^esub> Q\<^sub>p) c) = X_poly Q\<^sub>p \<oplus>\<^bsub>Q\<^sub>p_x\<^esub> UPQ.to_poly (c a)"
   proof-
     have 10: "poly_lift_hom (Fun\<^bsub>n\<^esub> Q\<^sub>p) Q\<^sub>p (eval_hom a) \<in> ring_hom (UP (Fun\<^bsub>n\<^esub> Q\<^sub>p)) Q\<^sub>p_x"
       using Qp_fpoly_to_Qp_poly_def Qp_fpoly_to_Qp_poly_is_hom assms
       by presburger
     have 11: " to_polynomial (Fun\<^bsub>n\<^esub> Q\<^sub>p) c \<in> carrier (UP (Fun\<^bsub>n\<^esub> Q\<^sub>p))"
       by (meson Qp_funs_is_cring UP_cring.intro UP_cring.to_poly_closed assms)
     have 12: "X_poly (Fun\<^bsub>n\<^esub> Q\<^sub>p) \<in> carrier (UP (Fun\<^bsub>n\<^esub> Q\<^sub>p)) "
-      using UP_cring.X_closed[of "Fun\<^bsub>n\<^esub> Q\<^sub>p"] unfolding UP_cring_def 
-      using Qp_funs_is_cring 
+      using UP_cring.X_closed[of "Fun\<^bsub>n\<^esub> Q\<^sub>p"] unfolding UP_cring_def
+      using Qp_funs_is_cring
       by blast
     have "Qp_fpoly_to_Qp_poly n a (X_poly (Fun\<^bsub>n\<^esub> Q\<^sub>p) \<oplus>\<^bsub>UP (Fun\<^bsub>n\<^esub> Q\<^sub>p)\<^esub> to_polynomial (Fun\<^bsub>n\<^esub> Q\<^sub>p) c) =
-    Qp_fpoly_to_Qp_poly n a (X_poly (Fun\<^bsub>n\<^esub> Q\<^sub>p)) \<oplus>\<^bsub>Q\<^sub>p_x\<^esub> Qp_fpoly_to_Qp_poly n a (to_polynomial (Fun\<^bsub>n\<^esub> Q\<^sub>p) c)"    
+    Qp_fpoly_to_Qp_poly n a (X_poly (Fun\<^bsub>n\<^esub> Q\<^sub>p)) \<oplus>\<^bsub>Q\<^sub>p_x\<^esub> Qp_fpoly_to_Qp_poly n a (to_polynomial (Fun\<^bsub>n\<^esub> Q\<^sub>p) c)"
       using assms 0 10 11 12 Qp_fpoly_to_Qp_poly_extends_apply[of a n c] Qp_fpoly_to_Qp_poly_is_hom[of a] Qp_fpoly_to_Qp_poly_X_var[of a]
       using ring_hom_add[of "Qp_fpoly_to_Qp_poly n a"  "UP (Fun\<^bsub>n\<^esub> Q\<^sub>p)" Q\<^sub>p_x "X_poly (Fun\<^bsub>n\<^esub> Q\<^sub>p)" "to_polynomial (Fun\<^bsub>n\<^esub> Q\<^sub>p) c" ]
-      unfolding Qp_fpoly_to_Qp_poly_def 
+      unfolding Qp_fpoly_to_Qp_poly_def
       by blast
-     then show ?thesis 
-       using Qp_fpoly_to_Qp_poly_X_var Qp_fpoly_to_Qp_poly_def Qp_fpoly_to_Qp_poly_extends_apply assms 
-       by metis 
+     then show ?thesis
+       using Qp_fpoly_to_Qp_poly_X_var Qp_fpoly_to_Qp_poly_def Qp_fpoly_to_Qp_poly_extends_apply assms
+       by metis
   qed
   have 2: "poly_lift_hom (Fun\<^bsub>n\<^esub> Q\<^sub>p) Q\<^sub>p (eval_hom a) (compose (Fun\<^bsub>n\<^esub> Q\<^sub>p) F (X_poly (Fun\<^bsub>n\<^esub> Q\<^sub>p) \<oplus>\<^bsub>UP (Fun\<^bsub>n\<^esub> Q\<^sub>p)\<^esub> to_polynomial (Fun\<^bsub>n\<^esub> Q\<^sub>p) c)) =
-   UPQ.sub (poly_lift_hom (Fun\<^bsub>n\<^esub> Q\<^sub>p) Q\<^sub>p (eval_hom a) F) 
+   UPQ.sub (poly_lift_hom (Fun\<^bsub>n\<^esub> Q\<^sub>p) Q\<^sub>p (eval_hom a) F)
     (poly_lift_hom (Fun\<^bsub>n\<^esub> Q\<^sub>p) Q\<^sub>p (eval_hom a) (X_poly (Fun\<^bsub>n\<^esub> Q\<^sub>p) \<oplus>\<^bsub>UP (Fun\<^bsub>n\<^esub> Q\<^sub>p)\<^esub> to_polynomial (Fun\<^bsub>n\<^esub> Q\<^sub>p) c))"
     using 0 1 Qp_fpoly_to_Qp_poly_sub[of F n "X_poly_plus (Fun\<^bsub>n\<^esub> Q\<^sub>p) c" a] assms
-    unfolding Qp_fpoly_to_Qp_poly_def X_poly_plus_def 
+    unfolding Qp_fpoly_to_Qp_poly_def X_poly_plus_def
     by blast
-  show ?thesis  
+  show ?thesis
     using assms 0 1
     unfolding Qp_fpoly_to_Qp_poly_def  taylor_expansion_def X_poly_plus_def
-    using "2" by presburger    
-qed 
+    using "2" by presburger
+qed
 
 lemma SA_is_UP_cring:
   shows "UP_cring (SA n)"
-  unfolding UP_cring_def 
+  unfolding UP_cring_def
   by (simp add: SA_is_cring)
 
 lemma eval_hom_is_SA_hom:
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "ring_hom_ring (SA n) Q\<^sub>p (eval_hom a)"
   apply(rule ring_hom_ringI)
-  using SA_is_cring cring.axioms(1) assms(1) apply blast 
-  using Qp.ring_axioms apply blast 
+  using SA_is_cring cring.axioms(1) assms(1) apply blast
+  using Qp.ring_axioms apply blast
   apply (metis (no_types, lifting) SA_car assms eval_hom_def Qp.function_ring_car_mem_closed semialg_functions_memE(2))
   apply (metis (mono_tags, lifting) Qp_funs_mult' SA_car SA_times assms eval_hom_def semialg_functions_memE(2))
   apply (metis (mono_tags, lifting) Qp_funs_add' SA_car SA_plus assms eval_hom_def semialg_functions_memE(2))
   using Qp_constE Qp.one_closed SA_one assms eval_hom_def function_one_as_constant
   by (metis function_one_eval)
 
 text\<open>the homomorphism from \texttt{(SA n)[x]} to \texttt{Qp [x]} induced by evaluation of coefficients\<close>
 
 definition SA_poly_to_Qp_poly where
 "SA_poly_to_Qp_poly n a = poly_lift_hom (SA n) Q\<^sub>p (eval_hom a)"
 
 lemma SA_poly_to_Qp_poly_is_hom:
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "(SA_poly_to_Qp_poly n a) \<in> ring_hom (UP (SA n)) (Q\<^sub>p_x) "
-  unfolding SA_poly_to_Qp_poly_def 
+  unfolding SA_poly_to_Qp_poly_def
   apply(rule UP_cring.poly_lift_hom_is_hom)
-  using SA_is_cring assms(1) UP_cring.intro apply blast  
+  using SA_is_cring assms(1) UP_cring.intro apply blast
   apply (simp add: UPQ.R_cring)
   using assms eval_hom_is_SA_hom ring_hom_ring.homh by blast
 
 lemma SA_poly_to_Qp_poly_closed:
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "P \<in> carrier (UP (SA n))"
   shows "SA_poly_to_Qp_poly n a P \<in> carrier Q\<^sub>p_x"
-  using assms SA_poly_to_Qp_poly_is_hom[of a] ring_hom_closed[of "SA_poly_to_Qp_poly n a" "UP (SA n)" Q\<^sub>p_x P] 
+  using assms SA_poly_to_Qp_poly_is_hom[of a] ring_hom_closed[of "SA_poly_to_Qp_poly n a" "UP (SA n)" Q\<^sub>p_x P]
   by blast
 
 lemma SA_poly_to_Qp_poly_add:
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "f \<in> carrier (UP (SA n))"
   assumes "g \<in> carrier (UP (SA n))"
   shows "SA_poly_to_Qp_poly n a (f \<oplus>\<^bsub>UP (SA n)\<^esub> g) = SA_poly_to_Qp_poly n a f \<oplus>\<^bsub>Q\<^sub>p_x\<^esub> SA_poly_to_Qp_poly n a g"
-  using SA_poly_to_Qp_poly_is_hom ring_hom_add assms 
+  using SA_poly_to_Qp_poly_is_hom ring_hom_add assms
   by (metis (no_types, opaque_lifting))
 
 lemma SA_poly_to_Qp_poly_minus:
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "f \<in> carrier (UP (SA n))"
   assumes "g \<in> carrier (UP (SA n))"
    shows "SA_poly_to_Qp_poly n a (f \<ominus>\<^bsub>UP (SA n)\<^esub> g) = SA_poly_to_Qp_poly n a f \<ominus>\<^bsub>Q\<^sub>p_x\<^esub> SA_poly_to_Qp_poly n a g"
-  using SA_poly_to_Qp_poly_is_hom[of a] assms SA_is_ring[of n] 
-        ring.ring_hom_minus[of "UP (SA n)" Q\<^sub>p_x "SA_poly_to_Qp_poly n a" f g] UP_SA_n_is_ring 
-        UPQ.UP_ring 
+  using SA_poly_to_Qp_poly_is_hom[of a] assms SA_is_ring[of n]
+        ring.ring_hom_minus[of "UP (SA n)" Q\<^sub>p_x "SA_poly_to_Qp_poly n a" f g] UP_SA_n_is_ring
+        UPQ.UP_ring
   by blast
 
 lemma SA_poly_to_Qp_poly_mult:
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "f \<in> carrier (UP (SA n))"
   assumes "g \<in> carrier (UP (SA n))"
   shows "SA_poly_to_Qp_poly n a (f \<otimes>\<^bsub>UP (SA n)\<^esub> g) = SA_poly_to_Qp_poly n a f \<otimes>\<^bsub>Q\<^sub>p_x\<^esub> SA_poly_to_Qp_poly n a g"
-  using SA_poly_to_Qp_poly_is_hom ring_hom_mult assms 
+  using SA_poly_to_Qp_poly_is_hom ring_hom_mult assms
   by (metis (no_types, opaque_lifting))
 
 lemma SA_poly_to_Qp_poly_extends_apply:
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "f \<in> carrier (SA n)"
   shows "SA_poly_to_Qp_poly n a (to_polynomial (SA n) f) = to_polynomial Q\<^sub>p (f a)"
-  unfolding SA_poly_to_Qp_poly_def 
+  unfolding SA_poly_to_Qp_poly_def
   using assms eval_hom_is_SA_hom[of a] UP_cring.poly_lift_hom_extends_hom[of "SA n" Q\<^sub>p "eval_hom a" f]
          eval_hom_def   SA_is_cring Qp.cring_axioms ring_hom_ring.homh
-  unfolding eval_hom_def UP_cring_def 
+  unfolding eval_hom_def UP_cring_def
   by blast
 
 lemma SA_poly_to_Qp_poly_X_var:
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "SA_poly_to_Qp_poly n a (X_poly (SA n)) = X_poly Q\<^sub>p"
   unfolding X_poly_def SA_poly_to_Qp_poly_def
   apply(rule UP_cring.poly_lift_hom_X_var)
-    using SA_is_cring assms(1) 
+    using SA_is_cring assms(1)
     using UP_cring.intro apply blast
     apply (simp add: Qp.cring_axioms)
-    using assms eval_hom_is_SA_hom ring_hom_ring.homh by blast    
+    using assms eval_hom_is_SA_hom ring_hom_ring.homh by blast
 
 lemma SA_poly_to_Qp_poly_X_plus:
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "c \<in> carrier (SA n)"
   shows "SA_poly_to_Qp_poly n a (X_poly_plus (SA n) c) = UPQ.X_plus (c a)"
-  unfolding X_poly_plus_def 
+  unfolding X_poly_plus_def
   using assms SA_poly_to_Qp_poly_add[of a n "X_poly (SA n)" "to_polynomial (SA n) c"]
         SA_poly_to_Qp_poly_extends_apply[of a n c] UP_cring.X_closed[of "SA n"] SA_is_cring[of n]
-        SA_poly_to_Qp_poly_X_var[of a] UP_cring.to_poly_closed[of "SA n" c] 
-  unfolding UP_cring_def 
-  by metis 
+        SA_poly_to_Qp_poly_X_var[of a] UP_cring.to_poly_closed[of "SA n" c]
+  unfolding UP_cring_def
+  by metis
 
 lemma SA_poly_to_Qp_poly_X_minus:
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "c \<in> carrier (SA n)"
   shows "SA_poly_to_Qp_poly n a (X_poly_minus (SA n) c) = UPQ.X_minus (c a)"
-  unfolding X_poly_minus_def 
+  unfolding X_poly_minus_def
   using assms SA_poly_to_Qp_poly_minus[of a n "X_poly (SA n)" "to_polynomial (SA n) c"]
         SA_poly_to_Qp_poly_extends_apply[of a n c] UP_cring.X_closed[of "SA n"] SA_is_cring[of n]
-        SA_poly_to_Qp_poly_X_var[of a n] UP_cring.to_poly_closed[of "SA n" c] 
-  unfolding UP_cring_def 
-  by metis 
+        SA_poly_to_Qp_poly_X_var[of a n] UP_cring.to_poly_closed[of "SA n" c]
+  unfolding UP_cring_def
+  by metis
 
 lemma SA_poly_to_Qp_poly_monom:
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "f \<in> carrier (SA n)"
   shows "SA_poly_to_Qp_poly n a (up_ring.monom (UP (SA n)) f m) = up_ring.monom Q\<^sub>p_x (f a) m"
-  unfolding SA_poly_to_Qp_poly_def 
-  using UP_cring.poly_lift_hom_monom[of "SA n" Q\<^sub>p "eval_hom a" f n] assms eval_hom_is_SA_hom eval_hom_def 
+  unfolding SA_poly_to_Qp_poly_def
+  using UP_cring.poly_lift_hom_monom[of "SA n" Q\<^sub>p "eval_hom a" f n] assms eval_hom_is_SA_hom eval_hom_def
    SA_is_cring Qp.cring_axioms UP_cring.poly_lift_hom_monom ring_hom_ring.homh
   by (metis UP_cring.intro)
 
 lemma SA_poly_to_Qp_poly_coeff:
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "f \<in> carrier (UP (SA n))"
   shows "SA_poly_to_Qp_poly n a f k = (f k) a"
-  using assms UP_cring.poly_lift_hom_cf[of "SA n" Q\<^sub>p "eval_hom a" f k] eval_hom_is_SA_hom[of a] 
+  using assms UP_cring.poly_lift_hom_cf[of "SA n" Q\<^sub>p "eval_hom a" f k] eval_hom_is_SA_hom[of a]
   using SA_is_cring Qp.cring_axioms ring_hom_ring.homh
   unfolding  SA_poly_to_Qp_poly_def eval_hom_def UP_cring_def
   by blast
 
 lemma SA_poly_to_Qp_poly_eval:
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "P \<in> carrier (UP (SA n))"
   assumes "f \<in> carrier (SA n)"
   shows "(UP_cring.to_fun (SA n) P f) a  = UP_cring.to_fun Q\<^sub>p (SA_poly_to_Qp_poly n a P) (f a)"
-  unfolding SA_poly_to_Qp_poly_def 
-  using UP_cring.poly_lift_hom_eval[of "SA n" Q\<^sub>p "eval_hom a" P f] 
+  unfolding SA_poly_to_Qp_poly_def
+  using UP_cring.poly_lift_hom_eval[of "SA n" Q\<^sub>p "eval_hom a" P f]
         eval_hom_is_SA_hom[of a] eval_hom_def assms SA_is_cring Qp.cring_axioms ring_hom_ring.homh
   unfolding  SA_poly_to_Qp_poly_def eval_hom_def UP_cring_def
   by blast
 
 lemma SA_poly_to_Qp_poly_sub:
   assumes "f \<in> carrier (UP (SA n))"
   assumes "g \<in> carrier (UP (SA n))"
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "SA_poly_to_Qp_poly n a (compose (SA n) f g) = compose Q\<^sub>p (SA_poly_to_Qp_poly n a f) (SA_poly_to_Qp_poly n a g)"
-  unfolding SA_poly_to_Qp_poly_def 
+  unfolding SA_poly_to_Qp_poly_def
   using assms UP_cring.poly_lift_hom_sub[of "SA n" Q\<^sub>p "eval_hom a" f g]
         eval_hom_is_SA_hom[of a] ring_hom_ring.homh[of "SA n" Q\<^sub>p "eval_hom a"]
-        SA_is_cring Qp.cring_axioms 
+        SA_is_cring Qp.cring_axioms
   unfolding  SA_poly_to_Qp_poly_def eval_hom_def UP_cring_def
   by blast
 
-lemma SA_poly_to_Qp_poly_deg_bound: 
+lemma SA_poly_to_Qp_poly_deg_bound:
   assumes "g \<in> carrier (UP (SA m))"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
   shows "deg Q\<^sub>p (SA_poly_to_Qp_poly m x g) \<le> deg (SA m) g"
-  apply(rule  UPQ.deg_leqI) 
-    using assms SA_poly_to_Qp_poly_closed[of x m g] apply blast 
+  apply(rule  UPQ.deg_leqI)
+    using assms SA_poly_to_Qp_poly_closed[of x m g] apply blast
   proof- fix n assume A: "deg (SA m) g < n"
     then have "g n = \<zero>\<^bsub>SA m\<^esub>"
       using assms SA_is_UP_cring[of m] UP_cring.UP_car_memE(2) by blast
     thus "SA_poly_to_Qp_poly m x g n = \<zero>"
       using assms SA_poly_to_Qp_poly_coeff[of x m g n] function_zero_eval SA_zero by presburger
   qed
 
 lemma SA_poly_to_Qp_poly_taylor_poly:
   assumes "F \<in> carrier (UP (SA n))"
   assumes "c \<in> carrier (SA n)"
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
-  shows "SA_poly_to_Qp_poly n a (taylor_expansion (SA n) c F) = 
+  shows "SA_poly_to_Qp_poly n a (taylor_expansion (SA n) c F) =
         taylor_expansion Q\<^sub>p (c a) (SA_poly_to_Qp_poly n a F)"
-  unfolding SA_poly_to_Qp_poly_def using assms Qp.cring_axioms SA_is_cring eval_hom_def 
+  unfolding SA_poly_to_Qp_poly_def using assms Qp.cring_axioms SA_is_cring eval_hom_def
       eval_hom_is_SA_hom UP_cring.poly_lift_hom_comm_taylor_expansion[of "SA n" Q\<^sub>p "eval_hom a" F c] ring_hom_ring.homh
-  unfolding  SA_poly_to_Qp_poly_def eval_hom_def UP_cring_def 
+  unfolding  SA_poly_to_Qp_poly_def eval_hom_def UP_cring_def
   by metis
 
 lemma SA_poly_to_Qp_poly_comm_taylor_term:
   assumes "F \<in> carrier (UP (SA n))"
   assumes "c \<in> carrier (SA n)"
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
-  shows "SA_poly_to_Qp_poly n a (UP_cring.taylor_term (SA n) c F i) = 
+  shows "SA_poly_to_Qp_poly n a (UP_cring.taylor_term (SA n) c F i) =
         UP_cring.taylor_term Q\<^sub>p (c a) (SA_poly_to_Qp_poly n a F) i"
   unfolding SA_poly_to_Qp_poly_def using assms  Qp.cring_axioms SA_is_cring eval_hom_def
       eval_hom_is_SA_hom UP_cring.poly_lift_hom_comm_taylor_term[of "SA n" Q\<^sub>p "eval_hom a" F c i] ring_hom_ring.homh
-  unfolding  SA_poly_to_Qp_poly_def eval_hom_def UP_cring_def 
+  unfolding  SA_poly_to_Qp_poly_def eval_hom_def UP_cring_def
   by metis
 
 lemma SA_poly_to_Qp_poly_comm_pderiv:
   assumes "F \<in> carrier (UP (SA n))"
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
-  shows "SA_poly_to_Qp_poly n a (UP_cring.pderiv (SA n) F) = 
+  shows "SA_poly_to_Qp_poly n a (UP_cring.pderiv (SA n) F) =
         UP_cring.pderiv Q\<^sub>p (SA_poly_to_Qp_poly n a F)"
-  apply(rule UP_ring.poly_induct3[of "SA n" F]) unfolding UP_ring_def 
+  apply(rule UP_ring.poly_induct3[of "SA n" F]) unfolding UP_ring_def
   apply (simp add: SA_is_ring assms(1))
   using assms apply blast
 proof-
   show "\<And>p q. q \<in> carrier (UP (SA n)) \<Longrightarrow>
            p \<in> carrier (UP (SA n)) \<Longrightarrow>
            SA_poly_to_Qp_poly n a (UP_cring.pderiv (SA n) p) = UPQ.pderiv (SA_poly_to_Qp_poly n a p) \<Longrightarrow>
            SA_poly_to_Qp_poly n a (UP_cring.pderiv (SA n) q) = UPQ.pderiv (SA_poly_to_Qp_poly n a q) \<Longrightarrow>
            SA_poly_to_Qp_poly n a (UP_cring.pderiv (SA n) (p \<oplus>\<^bsub>UP (SA n)\<^esub> q)) = UPQ.pderiv (SA_poly_to_Qp_poly n a (p \<oplus>\<^bsub>UP (SA n)\<^esub> q))"
   proof- fix p q assume A: "q \<in> carrier (UP (SA n))"
            "p \<in> carrier (UP (SA n))"
            "SA_poly_to_Qp_poly n a (UP_cring.pderiv (SA n) p) = UPQ.pderiv (SA_poly_to_Qp_poly n a p)"
            "SA_poly_to_Qp_poly n a (UP_cring.pderiv (SA n) q) = UPQ.pderiv (SA_poly_to_Qp_poly n a q)"
     show "SA_poly_to_Qp_poly n a (UP_cring.pderiv (SA n) (p \<oplus>\<^bsub>UP (SA n)\<^esub> q)) = UPQ.pderiv (SA_poly_to_Qp_poly n a (p \<oplus>\<^bsub>UP (SA n)\<^esub> q))"
     proof-
-      have 0: "SA_poly_to_Qp_poly n a p \<in> carrier (UP Q\<^sub>p)" 
-        using A assms SA_poly_to_Qp_poly_closed[of a n p] 
+      have 0: "SA_poly_to_Qp_poly n a p \<in> carrier (UP Q\<^sub>p)"
+        using A assms SA_poly_to_Qp_poly_closed[of a n p]
         by blast
       have 1: "SA_poly_to_Qp_poly n a q \<in> carrier (UP Q\<^sub>p)"
-        using A SA_poly_to_Qp_poly_closed[of a n q] assms by blast 
-      have 2: "UPQ.pderiv (SA_poly_to_Qp_poly n a p) \<in> carrier (UP Q\<^sub>p)" 
+        using A SA_poly_to_Qp_poly_closed[of a n q] assms by blast
+      have 2: "UPQ.pderiv (SA_poly_to_Qp_poly n a p) \<in> carrier (UP Q\<^sub>p)"
         using  UPQ.pderiv_closed[of "SA_poly_to_Qp_poly n a p"] 0  by blast
       have 3: "UPQ.pderiv (SA_poly_to_Qp_poly n a q) \<in> carrier (UP Q\<^sub>p)"
-        using A assms  UPQ.pderiv_closed[of "SA_poly_to_Qp_poly n a q"] 1 by blast 
+        using A assms  UPQ.pderiv_closed[of "SA_poly_to_Qp_poly n a q"] 1 by blast
       have 4: "UP_cring.pderiv (SA n) p \<in> carrier (UP (SA n))"
-        using A UP_cring.pderiv_closed[of "SA n" p] unfolding UP_cring_def 
+        using A UP_cring.pderiv_closed[of "SA n" p] unfolding UP_cring_def
         using SA_is_cring assms(1) by blast
       have 5: "UP_cring.pderiv (SA n) q \<in> carrier (UP (SA n))"
-        using A UP_cring.pderiv_closed[of "SA n" q] unfolding UP_cring_def  
+        using A UP_cring.pderiv_closed[of "SA n" q] unfolding UP_cring_def
         using SA_is_cring assms(1) by blast
       have 6: "SA_poly_to_Qp_poly n a (UP_cring.pderiv (SA n) p \<oplus>\<^bsub>UP (SA n)\<^esub> UP_cring.pderiv (SA n) q) =
     SA_poly_to_Qp_poly n a (UP_cring.pderiv (SA n) p) \<oplus>\<^bsub>UP Q\<^sub>p\<^esub> SA_poly_to_Qp_poly n a (UP_cring.pderiv (SA n) q)"
         using A 4 5 SA_poly_to_Qp_poly_add assms  by blast
       have 7: "UPQ.pderiv (SA_poly_to_Qp_poly n a p \<oplus>\<^bsub>UP Q\<^sub>p\<^esub> SA_poly_to_Qp_poly n a q) =
     UPQ.pderiv (SA_poly_to_Qp_poly n a p) \<oplus>\<^bsub>UP Q\<^sub>p\<^esub> UPQ.pderiv (SA_poly_to_Qp_poly n a q)"
         using "0" "1" UPQ.pderiv_add by blast
-      have 8: "UP_cring.pderiv (SA n) (p \<oplus>\<^bsub>UP (SA n)\<^esub> q) = UP_cring.pderiv (SA n) p \<oplus>\<^bsub>UP (SA n)\<^esub> UP_cring.pderiv (SA n) q"  
+      have 8: "UP_cring.pderiv (SA n) (p \<oplus>\<^bsub>UP (SA n)\<^esub> q) = UP_cring.pderiv (SA n) p \<oplus>\<^bsub>UP (SA n)\<^esub> UP_cring.pderiv (SA n) q"
         using A assms UP_cring.pderiv_add[of "SA n" p q]
         unfolding UP_cring_def  using SA_is_cring by blast
       have 9: "SA_poly_to_Qp_poly n a (UP_cring.pderiv (SA n) (p \<oplus>\<^bsub>UP (SA n)\<^esub> q)) =
     UPQ.pderiv (SA_poly_to_Qp_poly n a p) \<oplus>\<^bsub>UP Q\<^sub>p\<^esub> UPQ.pderiv (SA_poly_to_Qp_poly n a q)"
         using A 6 8 by presburger
-      have 10: "UPQ.pderiv (SA_poly_to_Qp_poly n a (p \<oplus>\<^bsub>UP (SA n)\<^esub> q)) = 
+      have 10: "UPQ.pderiv (SA_poly_to_Qp_poly n a (p \<oplus>\<^bsub>UP (SA n)\<^esub> q)) =
     UPQ.pderiv (SA_poly_to_Qp_poly n a p) \<oplus> \<^bsub>UP Q\<^sub>p\<^esub> UPQ.pderiv (SA_poly_to_Qp_poly n a q)"
         using "7" A(1) A(2) SA_poly_to_Qp_poly_add assms  by presburger
-      show ?thesis using 9 10 
+      show ?thesis using 9 10
         by presburger
     qed
   qed
   show "\<And>aa na.
        aa \<in> carrier (SA n) \<Longrightarrow>
        SA_poly_to_Qp_poly n a (UP_cring.pderiv (SA n) (up_ring.monom (UP (SA n)) aa na)) =
        UPQ.pderiv (SA_poly_to_Qp_poly n a (up_ring.monom (UP (SA n)) aa na))"
   proof- fix b m assume A: "b \<in> carrier (SA n)"
     show "SA_poly_to_Qp_poly n a (UP_cring.pderiv (SA n) (up_ring.monom (UP (SA n)) b m)) =
        UPQ.pderiv (SA_poly_to_Qp_poly n a (up_ring.monom (UP (SA n)) b m))"
     proof-
-      have 0: "(UP_cring.pderiv (SA n) (up_ring.monom (UP (SA n)) b m)) = 
+      have 0: "(UP_cring.pderiv (SA n) (up_ring.monom (UP (SA n)) b m)) =
               (up_ring.monom (UP (SA n)) ([m]\<cdot>\<^bsub>SA n\<^esub>b) (m-1))"
-        using UP_cring.pderiv_monom[of "SA n" b m] unfolding UP_cring_def 
+        using UP_cring.pderiv_monom[of "SA n" b m] unfolding UP_cring_def
         using SA_is_cring \<open>b \<in> carrier (SA n)\<close> assms(1) by blast
-      have 1: "(SA_poly_to_Qp_poly n a (up_ring.monom (UP (SA n)) b m)) =up_ring.monom (UP Q\<^sub>p) (b a) m" 
+      have 1: "(SA_poly_to_Qp_poly n a (up_ring.monom (UP (SA n)) b m)) =up_ring.monom (UP Q\<^sub>p) (b a) m"
         using SA_poly_to_Qp_poly_monom \<open>b \<in> carrier (SA n)\<close> assms  by blast
       have 2: "b a \<in> carrier Q\<^sub>p"
-        using A assms Qp.function_ring_car_mem_closed SA_car_memE(2) by metis        
+        using A assms Qp.function_ring_car_mem_closed SA_car_memE(2) by metis
       hence 3: "UPQ.pderiv (SA_poly_to_Qp_poly n a (up_ring.monom (UP (SA n)) b m)) = up_ring.monom (UP Q\<^sub>p) ([m]\<cdot>(b a)) (m-1)"
-        using 1 2 A UPQ.pderiv_monom[of "b a" m] 
+        using 1 2 A UPQ.pderiv_monom[of "b a" m]
         by presburger
       have 4: "[m] \<cdot>\<^bsub>SA n\<^esub> b \<in> carrier (SA n)"
-        using A assms  SA_is_cring[of n] ring.add_pow_closed[of "SA n" b m] SA_is_ring 
+        using A assms  SA_is_cring[of n] ring.add_pow_closed[of "SA n" b m] SA_is_ring
         by blast
       have 5: "SA_poly_to_Qp_poly n a (up_ring.monom (UP (SA n)) ([m] \<cdot>\<^bsub>SA n\<^esub> b) (m-1)) = up_ring.monom (UP Q\<^sub>p) (([m] \<cdot>\<^bsub>SA n\<^esub> b) a) (m-1)"
         using SA_poly_to_Qp_poly_monom[of a n "[m]\<cdot>\<^bsub>SA n\<^esub>b" "m-1"] assms 4 by blast
       have 6: "SA_poly_to_Qp_poly n a (UP_cring.pderiv (SA n) (up_ring.monom (UP (SA n)) b m)) = up_ring.monom (UP Q\<^sub>p) (([m] \<cdot>\<^bsub>SA n\<^esub> b) a) (m - 1)"
          using 5 0  by presburger
-      thus ?thesis using assms A 3 6 SA_add_pow_apply[of b n a] 
-        by auto 
+      thus ?thesis using assms A 3 6 SA_add_pow_apply[of b n a]
+        by auto
     qed
   qed
 qed
 
 lemma SA_poly_to_Qp_poly_pderiv:
   assumes "g \<in> carrier (UP (SA m))"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
   shows "UPQ.pderiv (SA_poly_to_Qp_poly m x g) = (SA_poly_to_Qp_poly m x (pderiv m g))"
 proof
-  fix n 
+  fix n
   have 0: "UPQ.pderiv (SA_poly_to_Qp_poly m x g) n = [Suc n] \<cdot> SA_poly_to_Qp_poly m x g (Suc n)"
     by(rule UPQ.pderiv_cfs[of "SA_poly_to_Qp_poly m x g"n], rule SA_poly_to_Qp_poly_closed, rule assms , rule assms)
   have 1: "SA_poly_to_Qp_poly m x (UPSA.pderiv m g) n = UPSA.pderiv m g n x"
     by(rule SA_poly_to_Qp_poly_coeff[of x m "UPSA.pderiv m g" n], rule assms, rule UPSA.pderiv_closed, rule assms)
   have 2: "SA_poly_to_Qp_poly m x (UPSA.pderiv m g) n = ([Suc n] \<cdot>\<^bsub>SA m\<^esub> g (Suc n)) x"
-    using UPSA.pderiv_cfs[of g m n] assms unfolding 1 by auto  
+    using UPSA.pderiv_cfs[of g m n] assms unfolding 1 by auto
   show "UPQ.pderiv (SA_poly_to_Qp_poly m x g) n = SA_poly_to_Qp_poly m x (UPSA.pderiv m g) n"
-    unfolding 0 2 using SA_poly_to_Qp_poly_coeff assms 
+    unfolding 0 2 using SA_poly_to_Qp_poly_coeff assms
     by (metis "0" "2" SA_poly_to_Qp_poly_comm_pderiv)
 qed
-    
+
 lemma(in UP_cring) pderiv_deg_lt:
   assumes "f \<in> carrier (UP R)"
   assumes "deg R f > 0"
   shows "deg R (pderiv f) < deg R f"
-proof- 
+proof-
   obtain n where n_def: "n = deg R f"
-    by blast 
+    by blast
   have 0: "\<And>k. k \<ge> n \<Longrightarrow> pderiv f k = \<zero>"
-    using pderiv_cfs assms unfolding n_def 
-    by (simp add: UP_car_memE(2)) 
+    using pderiv_cfs assms unfolding n_def
+    by (simp add: UP_car_memE(2))
   obtain k where k_def: "n = Suc k"
-    using n_def assms gr0_implies_Suc by presburger 
+    using n_def assms gr0_implies_Suc by presburger
   have "deg R (pderiv f) \<le> k"
     apply(rule deg_leqI)
-    using P_def assms(1) pderiv_closed apply presburger 
+    using P_def assms(1) pderiv_closed apply presburger
     apply(rule 0)
-    unfolding k_def by presburger 
-  thus ?thesis using k_def unfolding n_def by linarith 
+    unfolding k_def by presburger
+  thus ?thesis using k_def unfolding n_def by linarith
 qed
 
 lemma deg_pderiv:
   assumes "f \<in> carrier (UP (SA m))"
   assumes "deg (SA m) f > 0"
   shows "deg (SA m) (pderiv m f) = deg (SA m) f - 1"
-proof- 
+proof-
   obtain n where n_def: "n = deg (SA m) f"
     by blast
   have 0: "f n \<noteq> \<zero>\<^bsub>SA m\<^esub>"
     unfolding n_def using assms UPSA.deg_ltrm by fastforce
   have 1: "(pderiv m f) (n-1) = [n]\<cdot>\<^bsub>SA m\<^esub> (f n)"
     using assms unfolding n_def using  Suc_diff_1 UPSA.pderiv_cfs by presburger
   have 2: "deg (SA m) (pderiv m f) \<ge> (n-1)"
-    using 0 assms SA_char_zero 
-    by (metis "1" UPSA.deg_eqI UPSA.lcf_closed UPSA.pderiv_closed n_def nat_le_linear) 
+    using 0 assms SA_char_zero
+    by (metis "1" UPSA.deg_eqI UPSA.lcf_closed UPSA.pderiv_closed n_def nat_le_linear)
   have 3: "deg (SA m) (pderiv m f) < deg (SA m) f"
-    using assms  pderiv_deg_lt by auto 
+    using assms  pderiv_deg_lt by auto
   thus ?thesis using 2  unfolding  n_def by presburger
 qed
 
 lemma SA_poly_to_Qp_poly_smult:
   assumes "a \<in> carrier (SA m)"
   assumes "f \<in> carrier (UP (SA m))"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
   shows "SA_poly_to_Qp_poly m x (a \<odot>\<^bsub>UP (SA m)\<^esub> f) = a x \<odot>\<^bsub>UP Q\<^sub>p\<^esub> SA_poly_to_Qp_poly m x f"
 proof-
   have 0: "a \<odot>\<^bsub>UP (SA m)\<^esub> f = to_polynomial (SA m) a \<otimes>\<^bsub>UP (SA m)\<^esub> f"
     using assms  UPSA.to_poly_mult_simp(1) by presburger
   have 1: "SA_poly_to_Qp_poly m x (a \<odot>\<^bsub>UP (SA m)\<^esub> f) = SA_poly_to_Qp_poly m x (to_polynomial (SA m) a) \<otimes>\<^bsub>UP Q\<^sub>p\<^esub> SA_poly_to_Qp_poly m x f"
     unfolding 0 apply(rule SA_poly_to_Qp_poly_mult)
     using assms apply blast
     using assms to_poly_closed  apply blast
-    using assms by blast 
+    using assms by blast
   have 2: "SA_poly_to_Qp_poly m x (a \<odot>\<^bsub>UP (SA m)\<^esub> f) = (to_polynomial Q\<^sub>p (a x)) \<otimes>\<^bsub>UP Q\<^sub>p\<^esub> SA_poly_to_Qp_poly m x f"
     unfolding 1 using assms SA_poly_to_Qp_poly_monom unfolding to_polynomial_def
     by presburger
-  show ?thesis 
+  show ?thesis
     unfolding 2 apply(rule UP_cring.to_poly_mult_simp(1)[of Q\<^sub>p "a x" "SA_poly_to_Qp_poly m x f"])
-    unfolding UP_cring_def 
-    apply (simp add: Qp.cring)
+    unfolding UP_cring_def
+    using F.R_cring apply blast
     using assms SA_car_memE apply blast
-    using   assms SA_poly_to_Qp_poly_closed[of x m f] by blast 
+    using assms SA_poly_to_Qp_poly_closed[of x m f] by blast
 qed
 
 lemma SA_poly_constant_res_class_semialg:
   assumes "f \<in> carrier (UP (SA m))"
   assumes "\<And>i x. x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<Longrightarrow> f i x \<in> \<O>\<^sub>p"
   assumes "deg (SA m) f \<le> d"
   assumes "C \<in> poly_res_classes n d"
   shows "is_semialgebraic m {x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). SA_poly_to_Qp_poly m x f \<in> C}"
 proof-
   obtain Fs where Fs_def: "Fs = f ` {..d}"
-    by blast 
+    by blast
   obtain g where g_def: "g \<in> val_ring_polys_grad d \<and> C = poly_res_class n d g"
-    using assms unfolding poly_res_classes_def by blast 
+    using assms unfolding poly_res_classes_def by blast
   have C_eq: " C = poly_res_class n d g"
-    using g_def by blast 
-  have 0: "{x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). SA_poly_to_Qp_poly m x f \<in> C} = 
+    using g_def by blast
+  have 0: "{x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). SA_poly_to_Qp_poly m x f \<in> C} =
             (\<Inter> i \<in> {..d}. {x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). f i x \<in> \<O>\<^sub>p \<and> Qp_res (f i x) n = Qp_res (g i) n})"
     apply(rule equalityI')
-    unfolding mem_Collect_eq 
+    unfolding mem_Collect_eq
   proof(rule InterI)
     fix x S
     assume A: " x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<and> SA_poly_to_Qp_poly m x f \<in> C"
            "S \<in> (\<lambda>i. {x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). f i x \<in> \<O>\<^sub>p \<and> Qp_res (f i x) n = Qp_res (g i) n}) ` {..d}"
     have 0: "C = poly_res_class n d (SA_poly_to_Qp_poly m x f)"
-      unfolding C_eq 
+      unfolding C_eq
       apply(rule equalityI', rule poly_res_class_memI, rule poly_res_class_memE[of _ n d g], blast
           , rule poly_res_class_memE[of _ n d g], blast,  rule poly_res_class_memE[of _ n d g], blast)
-      using poly_res_class_memE[of _ n d ]A 
+      using poly_res_class_memE[of _ n d ]A
        apply (metis (no_types, lifting) C_eq)
-      apply(rule poly_res_class_memI, rule poly_res_class_memE[of _ n d "SA_poly_to_Qp_poly m x f"], blast, 
-            rule poly_res_class_memE[of _ n d "SA_poly_to_Qp_poly m x f"], blast, 
+      apply(rule poly_res_class_memI, rule poly_res_class_memE[of _ n d "SA_poly_to_Qp_poly m x f"], blast,
+            rule poly_res_class_memE[of _ n d "SA_poly_to_Qp_poly m x f"], blast,
             rule poly_res_class_memE[of _ n d "SA_poly_to_Qp_poly m x f"], blast)
-      using poly_res_class_memE[of _ n d ]A 
+      using poly_res_class_memE[of _ n d ]A
        by (metis (no_types, lifting) C_eq)
-    obtain i where i_def: "i \<in> {..d} \<and>  
+    obtain i where i_def: "i \<in> {..d} \<and>
                             S = {x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). f i x \<in> \<O>\<^sub>p \<and> Qp_res (f i x) n = Qp_res (g i) n}"
       using A by blast
     have S_eq: "S = {x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). f i x \<in> \<O>\<^sub>p \<and> Qp_res (f i x) n = Qp_res (g i) n}"
-      using i_def by blast 
+      using i_def by blast
     have 1: "\<And>i. SA_poly_to_Qp_poly m x f i = f i x"
       apply(rule SA_poly_to_Qp_poly_coeff)
       using A apply blast by(rule assms)
     have 2: "deg Q\<^sub>p (SA_poly_to_Qp_poly m x f) \<le> d"
       using assms SA_poly_to_Qp_poly_deg_bound[of f m x]
       using A(1) by linarith
     have 3: "Qp_res (SA_poly_to_Qp_poly m x f i) n = Qp_res (g i) n"
       apply(rule poly_res_class_memE[of _ _ d], rule poly_res_class_memI)
       using g_def val_ring_polys_grad_memE apply blast
       using g_def val_ring_polys_grad_memE apply blast
       using g_def val_ring_polys_grad_memE apply blast
       apply(rule poly_res_class_memE[of _ _ d],rule poly_res_class_memI)
          apply(rule SA_poly_to_Qp_poly_closed)
       using A apply blast
          apply(rule assms)
       apply(rule 2)
       unfolding 1 using assms A apply blast
-      using A unfolding C_eq 
+      using A unfolding C_eq
       using poly_res_class_memE(4)[of "SA_poly_to_Qp_poly m x f" n d g]
-      unfolding 1 by metis 
+      unfolding 1 by metis
     show "x \<in> S"
       using A 3 assms
-      unfolding S_eq mem_Collect_eq unfolding 1 
-      by blast 
+      unfolding S_eq mem_Collect_eq unfolding 1
+      by blast
   next
-    
+
     show "\<And>x. x \<in> (\<Inter>i\<le>d. {x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). f i x \<in> \<O>\<^sub>p \<and> Qp_res (f i x) n = Qp_res (g i) n}) \<Longrightarrow> x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<and> SA_poly_to_Qp_poly m x f \<in> C"
-    proof- 
+    proof-
       fix x assume A: "x \<in> (\<Inter>i\<le>d. {x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). f i x \<in> \<O>\<^sub>p \<and> Qp_res (f i x) n = Qp_res (g i) n})"
       have x_closed: "x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
-        using A by blast 
+        using A by blast
       have 0: "\<And>i. SA_poly_to_Qp_poly m x f i = f i x"
         apply(rule SA_poly_to_Qp_poly_coeff)
         using A apply blast by(rule assms)
       have 1: "deg Q\<^sub>p (SA_poly_to_Qp_poly m x f) \<le> d"
         using assms SA_poly_to_Qp_poly_deg_bound[of f m x]
         using x_closed by linarith
       have 2: "\<And>i. Qp_res (f i x) n = Qp_res (g i) n"
-      proof- fix i 
+      proof- fix i
         have 20: "i > d \<Longrightarrow> i > deg Q\<^sub>p g"
-          using g_def val_ring_polys_grad_memE(2) by fastforce          
+          using g_def val_ring_polys_grad_memE(2) by fastforce
         have 21: "i > d \<Longrightarrow> g i= \<zero>"
           using 20 g_def val_ring_polys_grad_memE UPQ.deg_leE by blast
         have 22: "i > d \<Longrightarrow> Qp_res (g i) n = 0"
-          unfolding 21 Qp_res_zero by blast 
+          unfolding 21 Qp_res_zero by blast
         have 23: "i > d \<Longrightarrow> SA_poly_to_Qp_poly m x f i = \<zero>"
           apply(rule UPQ.deg_leE, rule SA_poly_to_Qp_poly_closed, rule x_closed, rule assms)
-          by(rule le_less_trans[of _ d], rule 1, blast) 
+          by(rule le_less_trans[of _ d], rule 1, blast)
         show " Qp_res (f i x) n = Qp_res (g i) n"
           apply(cases "i \<le> d")
           using A apply blast
-          using 22 21 1 23 unfolding 0 
+          using 22 21 1 23 unfolding 0
           by (metis less_or_eq_imp_le linorder_neqE_nat)
       qed
       have 3: "SA_poly_to_Qp_poly m x f \<in> C"
-        unfolding C_eq 
+        unfolding C_eq
         apply(rule poly_res_class_memI, rule SA_poly_to_Qp_poly_closed, rule x_closed, rule assms, rule 1)
-        unfolding 0 
+        unfolding 0
         by(rule assms, rule x_closed, rule 2)
       show " x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<and> SA_poly_to_Qp_poly m x f \<in> C"
-        using x_closed 3 by blast 
+        using x_closed 3 by blast
     qed
   qed
   have 1: "\<And>i. is_semialgebraic m {x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). f i x \<in> \<O>\<^sub>p \<and> Qp_res (f i x) n = Qp_res (g i) n}"
     apply(rule SA_constant_res_set_semialg, rule Qp_res_closed, rule val_ring_polys_grad_memE[of _ d])
     using g_def apply blast
     using assms UPSA.UP_car_memE(1) by blast
   show "is_semialgebraic m {x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). SA_poly_to_Qp_poly m x f \<in> C}"
-    unfolding 0 
+    unfolding 0
     apply(rule finite_intersection_is_semialg, blast, blast, rule subsetI)
     using 1 unfolding is_semialgebraic_def by blast
 qed
 
 text\<open>Maps a polynomial $F(t) \in UP (SA n)$ to a function sending $(t, a) \in (Q_p (n + 1) \mapsto F(a)(t) \in Q_p$ \<close>
 
 definition SA_poly_to_SA_fun where
   "SA_poly_to_SA_fun n P = (\<lambda> a \<in> carrier (Q\<^sub>p\<^bsup>Suc n\<^esup>). UP_cring.to_fun Q\<^sub>p (SA_poly_to_Qp_poly n (tl a) P) (hd a))"
 
 lemma SA_poly_to_SA_fun_is_fun:
   assumes "P \<in> carrier (UP (SA n))"
   shows "SA_poly_to_SA_fun n P \<in> (carrier (Q\<^sub>p\<^bsup>Suc n\<^esup>) \<rightarrow> carrier Q\<^sub>p)"
 proof fix x assume A: "x \<in> carrier (Q\<^sub>p\<^bsup>Suc n\<^esup>)"
-  obtain t where t_def: "t = hd x" by blast 
-  obtain a where a_def: "a = tl x" by blast 
+  obtain t where t_def: "t = hd x" by blast
+  obtain a where a_def: "a = tl x" by blast
   have t_closed: "t \<in> carrier Q\<^sub>p"
-    using A t_def cartesian_power_head 
+    using A t_def cartesian_power_head
     by blast
   have a_closed: "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
-    using A a_def cartesian_power_tail 
+    using A a_def cartesian_power_tail
     by blast
   have 0: "SA_poly_to_SA_fun n P x = UP_cring.to_fun Q\<^sub>p (SA_poly_to_Qp_poly n a P) t"
-    unfolding SA_poly_to_SA_fun_def using t_def a_def 
+    unfolding SA_poly_to_SA_fun_def using t_def a_def
     by (meson A restrict_apply')
   show "SA_poly_to_SA_fun n P x \<in> carrier Q\<^sub>p"
-    using assms t_closed a_def 0 UP_cring.to_fun_closed[of Q\<^sub>p "SA_poly_to_Qp_poly n a P" ] 
-    unfolding SA_poly_to_SA_fun_def 
+    using assms t_closed a_def 0 UP_cring.to_fun_closed[of Q\<^sub>p "SA_poly_to_Qp_poly n a P" ]
+    unfolding SA_poly_to_SA_fun_def
     using SA_poly_to_Qp_poly_closed a_closed UPQ.to_fun_closed by presburger
 qed
 
 lemma SA_poly_to_SA_fun_formula:
   assumes "P \<in> carrier (UP (SA n))"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "t \<in> carrier Q\<^sub>p"
   shows "SA_poly_to_SA_fun n P (t#x) = (SA_poly_to_Qp_poly n x P)\<bullet>t"
 proof-
   have 0: "hd (t#x) = t"
     by simp
   have 1: "tl (t#x) = x"
     by auto
   have 2: "(t#x) \<in> carrier (Q\<^sub>p\<^bsup>Suc n\<^esup>)"
     by (metis add.commute assms cartesian_power_cons plus_1_eq_Suc)
-  show ?thesis 
-    unfolding SA_poly_to_SA_fun_def 
-    using 0 1 2 assms 
+  show ?thesis
+    unfolding SA_poly_to_SA_fun_def
+    using 0 1 2 assms
     by (metis (no_types, lifting) restrict_apply')
 qed
 
 lemma semialg_map_comp_in_SA:
   assumes "f \<in> carrier (SA n)"
   assumes "is_semialg_map m n g"
-  shows "(\<lambda> a \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). f (g a)) \<in> carrier (SA m)"  
+  shows "(\<lambda> a \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). f (g a)) \<in> carrier (SA m)"
 proof(rule SA_car_memI)
   show "(\<lambda>a\<in>carrier (Q\<^sub>p\<^bsup>m\<^esup>). f (g a)) \<in> carrier (Qp_funs m)"
   proof(rule Qp_funs_car_memI)
-    show " (\<lambda>a\<in>carrier (Q\<^sub>p\<^bsup>m\<^esup>). f (g a)) \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<rightarrow> carrier Q\<^sub>p"      
+    show " (\<lambda>a\<in>carrier (Q\<^sub>p\<^bsup>m\<^esup>). f (g a)) \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<rightarrow> carrier Q\<^sub>p"
     proof fix a assume A: "a \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
       then have "g a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
         using is_semialg_map_def[of m n g] assms
-        by blast 
-      then show  "f (g a) \<in> carrier Q\<^sub>p" 
-        using A assms  SA_car_memE(3)[of f n] 
-        by blast 
+        by blast
+      then show  "f (g a) \<in> carrier Q\<^sub>p"
+        using A assms  SA_car_memE(3)[of f n]
+        by blast
     qed
     show " \<And>x. x \<notin> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<Longrightarrow> (\<lambda>a\<in>carrier (Q\<^sub>p\<^bsup>m\<^esup>). f (g a)) x = undefined"
-      unfolding restrict_def by metis  
+      unfolding restrict_def by metis
   qed
   have 0: "is_semialg_function m (f \<circ> g)"
     using assms semialg_function_comp_closed[of n f m g] SA_car_memE(1)[of f n]
-    by blast 
+    by blast
   have 1: " (\<And>a. a \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<Longrightarrow> (f \<circ> g) a = (\<lambda>a\<in>carrier (Q\<^sub>p\<^bsup>m\<^esup>). f (g a)) a)"
     using assms  comp_apply[of f g] unfolding restrict_def
-    by metis 
+    by metis
   then show "is_semialg_function m (\<lambda>a\<in>carrier (Q\<^sub>p\<^bsup>m\<^esup>). f (g a))"
-    using 0 1 semialg_function_on_carrier'[of m "f \<circ> g" "\<lambda> a \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). f (g a)" ] 
-    by blast 
+    using 0 1 semialg_function_on_carrier'[of m "f \<circ> g" "\<lambda> a \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). f (g a)" ]
+    by blast
 qed
 
 lemma tl_comp_in_SA:
   assumes "f \<in> carrier (SA n)"
-  shows "(\<lambda> a \<in> carrier (Q\<^sub>p\<^bsup>Suc n\<^esup>). f (tl a)) \<in> carrier (SA (Suc n))"  
+  shows "(\<lambda> a \<in> carrier (Q\<^sub>p\<^bsup>Suc n\<^esup>). f (tl a)) \<in> carrier (SA (Suc n))"
   using assms semialg_map_comp_in_SA[of f _ _  tl] tl_is_semialg_map[of "n"]
-  by blast 
+  by blast
 
 lemma SA_poly_to_SA_fun_add_eval:
   assumes "f \<in> carrier (UP (SA n))"
   assumes "g \<in> carrier (UP (SA n))"
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>Suc n\<^esup>)"
   shows "SA_poly_to_SA_fun n (f \<oplus>\<^bsub>UP (SA n)\<^esub> g) a = SA_poly_to_SA_fun n f a \<oplus>\<^bsub>Q\<^sub>p\<^esub> SA_poly_to_SA_fun n g a"
-  unfolding SA_poly_to_SA_fun_def 
+  unfolding SA_poly_to_SA_fun_def
   using assms SA_poly_to_Qp_poly_add[of "tl a" n f g]
   by (metis (no_types, lifting) SA_poly_to_Qp_poly_closed UPQ.to_fun_plus cartesian_power_head cartesian_power_tail restrict_apply')
-  
+
 lemma SA_poly_to_SA_fun_add:
   assumes "f \<in> carrier (UP (SA n))"
   assumes "g \<in> carrier (UP (SA n))"
   shows "SA_poly_to_SA_fun n (f \<oplus>\<^bsub>UP (SA n)\<^esub> g) = SA_poly_to_SA_fun n f \<oplus>\<^bsub>SA (Suc n)\<^esub> SA_poly_to_SA_fun n g"
 proof fix x
   show " SA_poly_to_SA_fun n (f \<oplus>\<^bsub>UP (SA n)\<^esub> g) x = (SA_poly_to_SA_fun n f \<oplus>\<^bsub>SA (Suc n)\<^esub> SA_poly_to_SA_fun n g) x"
   proof(cases "x \<in> carrier (Q\<^sub>p\<^bsup>Suc n\<^esup>)")
     case True
     then show ?thesis using SA_poly_to_SA_fun_add_eval[of f n g x] SA_add[of x n]
-      using SA_mult assms(1) assms(2) SA_add 
-      by blast     
+      using SA_mult assms(1) assms(2) SA_add
+      by blast
   next
     case False
     have F0: "SA_poly_to_SA_fun n (f \<oplus>\<^bsub>UP (SA n)\<^esub> g) x = undefined"
-      unfolding SA_poly_to_SA_fun_def 
+      unfolding SA_poly_to_SA_fun_def
       by (meson False restrict_apply)
     have F1: "(SA_poly_to_SA_fun n f \<oplus>\<^bsub>SA (Suc n)\<^esub> SA_poly_to_SA_fun n g) x = undefined"
       using  False SA_add' by blast
-    then show ?thesis 
+    then show ?thesis
       using F0 by blast
   qed
 qed
 
 lemma SA_poly_to_SA_fun_monom:
   assumes "f \<in> carrier (SA n)"
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>Suc n\<^esup>)"
   shows "SA_poly_to_SA_fun n (up_ring.monom (UP (SA n)) f k) a = (f (tl a))\<otimes>(hd a)[^]\<^bsub>Q\<^sub>p\<^esub>k "
 proof-
   have "SA_poly_to_SA_fun n (up_ring.monom (UP (SA n)) f k) a = SA_poly_to_Qp_poly n (tl a) (up_ring.monom (UP (SA n)) f k) \<bullet> lead_coeff a"
-    unfolding SA_poly_to_SA_fun_def using assms 
+    unfolding SA_poly_to_SA_fun_def using assms
     by (meson restrict_apply)
   then have "SA_poly_to_SA_fun n (up_ring.monom (UP (SA n)) f k) a = up_ring.monom Q\<^sub>p_x (f (tl a)) k\<bullet> lead_coeff a"
-    using SA_poly_to_Qp_poly_monom[of "tl a" n f k] assms 
+    using SA_poly_to_Qp_poly_monom[of "tl a" n f k] assms
     by (metis cartesian_power_tail)
-  then show ?thesis using assms 
+  then show ?thesis using assms
     by (metis (no_types, lifting) SA_car cartesian_power_head cartesian_power_tail UPQ.to_fun_monom Qp.function_ring_car_mem_closed semialg_functions_memE(2))
 qed
 
 lemma SA_poly_to_SA_fun_monom':
   assumes "f \<in> carrier (SA n)"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "t \<in> carrier Q\<^sub>p"
   shows "SA_poly_to_SA_fun n (up_ring.monom (UP (SA n)) f k) (t#x) = (f x)\<otimes>t[^]\<^bsub>Q\<^sub>p\<^esub>k "
 proof-
   have 0: "hd (t#x) = t"
     by simp
   have 1: "tl (t#x) = x"
     by auto
   have 2: "(t#x) \<in> carrier (Q\<^sub>p\<^bsup>Suc n\<^esup>)"
     by (metis add.commute assms  cartesian_power_cons plus_1_eq_Suc)
-  show ?thesis  
-    using 0 1 2 SA_poly_to_SA_fun_monom[of f n "t#x" k] assms SA_poly_to_SA_fun_monom 
+  show ?thesis
+    using 0 1 2 SA_poly_to_SA_fun_monom[of f n "t#x" k] assms SA_poly_to_SA_fun_monom
     by presburger
 qed
 
-lemma hd_is_semialg_function: 
+lemma hd_is_semialg_function:
   assumes "n > 0"
   shows "is_semialg_function n hd"
 proof-
   have 0: "is_semialg_function n (\<lambda> a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). a!0)"
     using assms index_is_semialg_function restrict_is_semialg by blast
   have 1: "restrict (\<lambda>a\<in>carrier (Q\<^sub>p\<^bsup>n\<^esup>). a ! 0) (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) = restrict lead_coeff (carrier (Q\<^sub>p\<^bsup>n\<^esup>))"
-  proof fix x 
+  proof fix x
     show "restrict (\<lambda>a\<in>carrier (Q\<^sub>p\<^bsup>n\<^esup>). a ! 0) (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) x = restrict lead_coeff (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) x"
       unfolding restrict_def
       apply(cases "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)")
        apply (metis assms cartesian_power_car_memE drop_0 hd_drop_conv_nth)
       by presburger
   qed
-  show ?thesis 
+  show ?thesis
     using 0 1 assms semialg_function_on_carrier[of n "(\<lambda> a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>). a!0)" hd]
     by blast
 qed
 
 lemma SA_poly_to_SA_fun_monom_closed:
   assumes "f \<in> carrier (SA n)"
   shows "SA_poly_to_SA_fun n (up_ring.monom (UP (SA n)) f k) \<in> carrier (SA (Suc n))"
 proof-
   have 0: "is_semialg_function (Suc n) (f \<circ> tl)"
     using SA_imp_semialg assms(1)  semialg_function_comp_closed tl_is_semialg_map by blast
   obtain h where h_def: "h = restrict hd (carrier (Q\<^sub>p\<^bsup>Suc n\<^esup>))"
-    by blast 
+    by blast
   have h_closed: "h \<in> carrier (SA (Suc n))"
-    using hd_is_semialg_function SA_car h_def restrict_in_semialg_functions 
+    using hd_is_semialg_function SA_car h_def restrict_in_semialg_functions
     by blast
   have h_pow_closed: "h[^]\<^bsub>SA (Suc n)\<^esub> k \<in> carrier (SA (Suc n))"
-    using assms(1) h_closed monoid.nat_pow_closed[of "SA (Suc n)" h k] SA_is_monoid[of "Suc n"] 
-    by blast 
+    using assms(1) h_closed monoid.nat_pow_closed[of "SA (Suc n)" h k] SA_is_monoid[of "Suc n"]
+    by blast
   have monom_term_closed: "(f \<circ> tl) \<otimes>\<^bsub>SA (Suc n)\<^esub> h[^]\<^bsub>SA (Suc n)\<^esub> k \<in> carrier (SA (Suc n))"
     apply(rule SA_mult_closed_right)
     using "0" apply linarith
     using h_pow_closed by blast
-    
+
   have 0: "\<And>a. a \<in> carrier (Q\<^sub>p\<^bsup>Suc n\<^esup>) \<Longrightarrow> SA_poly_to_SA_fun n (up_ring.monom (UP (SA n)) f k) a = ((f \<circ> tl) \<otimes>\<^bsub>SA (Suc n)\<^esub> h[^]\<^bsub>SA (Suc n)\<^esub> k) a"
   proof- fix a assume "a \<in> carrier (Q\<^sub>p\<^bsup>Suc n\<^esup>)"
     then show "SA_poly_to_SA_fun n (up_ring.monom (UP (SA n)) f k) a = ((f \<circ> tl) \<otimes>\<^bsub>SA (Suc n)\<^esub> h[^]\<^bsub>SA (Suc n)\<^esub> k) a"
-      using assms SA_poly_to_SA_fun_monom[of f n a k] comp_apply[of f tl a] h_def restrict_apply 
-            SA_mult[of a "Suc n" "f \<circ> tl" "h [^]\<^bsub>SA (Suc n)\<^esub> k"] SA_nat_pow[of a "Suc n" h k]  
+      using assms SA_poly_to_SA_fun_monom[of f n a k] comp_apply[of f tl a] h_def restrict_apply
+            SA_mult[of a "Suc n" "f \<circ> tl" "h [^]\<^bsub>SA (Suc n)\<^esub> k"] SA_nat_pow[of a "Suc n" h k]
       by metis
   qed
   have 1: "SA_poly_to_SA_fun n (up_ring.monom (UP (SA n)) f k) = ((f \<circ> tl) \<otimes>\<^bsub>SA (Suc n)\<^esub> h[^]\<^bsub>SA (Suc n)\<^esub> k)"
-  proof fix x 
+  proof fix x
     show "SA_poly_to_SA_fun n (up_ring.monom (UP (SA n)) f k) x = ((f \<circ> tl) \<otimes>\<^bsub>SA (Suc n)\<^esub> h [^]\<^bsub>SA (Suc n)\<^esub> k) x"
       apply(cases "x \<in> carrier (Q\<^sub>p\<^bsup>Suc n\<^esup>)")
       using "0" apply blast
-      using monom_term_closed unfolding SA_poly_to_SA_fun_def 
-      using restrict_apply 
+      using monom_term_closed unfolding SA_poly_to_SA_fun_def
+      using restrict_apply
       by (metis (no_types, lifting) SA_mult')
   qed
-  show ?thesis 
-    using 1 monom_term_closed 
+  show ?thesis
+    using 1 monom_term_closed
     by presburger
 qed
 
 lemma SA_poly_to_SA_fun_is_SA:
   assumes "P \<in> carrier (UP (SA n))"
   shows "SA_poly_to_SA_fun n P \<in> carrier (SA (Suc n))"
   apply(rule UP_ring.poly_induct3[of "SA n" P])
-  unfolding UP_ring_def using assms SA_is_ring apply blast 
-  using assms apply blast 
+  unfolding UP_ring_def using assms SA_is_ring apply blast
+  using assms apply blast
   using assms SA_poly_to_SA_fun_add[of ]
   using SA_add_closed_right SA_imp_semialg  zero_less_Suc apply presburger
-  using SA_poly_to_SA_fun_monom_closed assms(1) 
+  using SA_poly_to_SA_fun_monom_closed assms(1)
   by blast
 
 lemma SA_poly_to_SA_fun_mult:
   assumes "f \<in> carrier (UP (SA n))"
   assumes "g \<in> carrier (UP (SA n))"
   shows "SA_poly_to_SA_fun n (f \<otimes>\<^bsub>UP (SA n)\<^esub> g) = SA_poly_to_SA_fun n f \<otimes>\<^bsub>SA (Suc n)\<^esub> SA_poly_to_SA_fun n g"
 proof(rule function_ring_car_eqI[of _ "Suc n"])
   show "SA_poly_to_SA_fun n (f \<otimes>\<^bsub>UP (SA n)\<^esub> g) \<in> carrier (function_ring (carrier (Q\<^sub>p\<^bsup>Suc n\<^esup>)) Q\<^sub>p)"
   proof-
     have "f \<otimes>\<^bsub>UP (SA n)\<^esub> g \<in> carrier (UP (SA n))"
-      using assms  SA_is_UP_cring 
+      using assms  SA_is_UP_cring
       by (meson cring.cring_simprules(5) padic_fields.UP_SA_n_is_cring padic_fields_axioms)
-    thus ?thesis 
-      using SA_is_UP_cring assms SA_poly_to_SA_fun_is_SA[of "f \<otimes>\<^bsub>UP (SA n)\<^esub> g"] SA_car_in_Qp_funs_car[of "Suc n"]  
+    thus ?thesis
+      using SA_is_UP_cring assms SA_poly_to_SA_fun_is_SA[of "f \<otimes>\<^bsub>UP (SA n)\<^esub> g"] SA_car_in_Qp_funs_car[of "Suc n"]
       by blast
   qed
   show "SA_poly_to_SA_fun n f \<otimes>\<^bsub>SA (Suc n)\<^esub> SA_poly_to_SA_fun n g \<in> carrier (function_ring (carrier (Q\<^sub>p\<^bsup>Suc n\<^esup>)) Q\<^sub>p)"
-    using SA_is_UP_cring SA_poly_to_SA_fun_is_SA assms 
+    using SA_is_UP_cring SA_poly_to_SA_fun_is_SA assms
     by (meson SA_car_memE(2) SA_mult_closed_left less_Suc_eq_0_disj padic_fields.SA_car_memE(1) padic_fields_axioms)
   show "\<And>a. a \<in> carrier (Q\<^sub>p\<^bsup>Suc n\<^esup>) \<Longrightarrow> SA_poly_to_SA_fun n (f \<otimes>\<^bsub>UP (SA n)\<^esub> g) a = (SA_poly_to_SA_fun n f \<otimes>\<^bsub>SA (Suc n)\<^esub> SA_poly_to_SA_fun n g) a"
   proof- fix a assume  A: "a \<in> carrier (Q\<^sub>p\<^bsup>Suc n\<^esup>)"
     then obtain t b where tb_def: "a = t#b"
       using cartesian_power_car_memE[of a Q\<^sub>p "Suc n"] by (meson length_Suc_conv)
     have t_closed: "t \<in> carrier Q\<^sub>p"
       using tb_def A cartesian_power_head[of a Q\<^sub>p n] by (metis list.sel(1))
     have b_closed: "b \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
       using tb_def A cartesian_power_tail[of a Q\<^sub>p n] by (metis list.sel(3))
     have 0: "f \<otimes>\<^bsub>UP (SA n)\<^esub> g \<in> carrier (UP (SA n))"
       using assms by (meson UP_SA_n_is_cring cring.cring_simprules(5))
     have 1: "SA_poly_to_SA_fun n (f \<otimes>\<^bsub>UP (SA n)\<^esub> g) a  = (SA_poly_to_Qp_poly n b (f \<otimes>\<^bsub>UP (SA n)\<^esub> g))\<bullet>t"
-      using SA_poly_to_SA_fun_formula[of "f \<otimes>\<^bsub>UP (SA n)\<^esub> g" n b t] t_closed b_closed tb_def  0  assms(1) 
+      using SA_poly_to_SA_fun_formula[of "f \<otimes>\<^bsub>UP (SA n)\<^esub> g" n b t] t_closed b_closed tb_def  0  assms(1)
       by blast
-    have 2: "(SA_poly_to_SA_fun n f \<otimes>\<^bsub>SA (Suc n)\<^esub> SA_poly_to_SA_fun n g) a = 
+    have 2: "(SA_poly_to_SA_fun n f \<otimes>\<^bsub>SA (Suc n)\<^esub> SA_poly_to_SA_fun n g) a =
             SA_poly_to_SA_fun n f a \<otimes> SA_poly_to_SA_fun n g a"
       using SA_poly_to_SA_fun_is_fun assms A SA_mult by blast
-    hence 3: "(SA_poly_to_SA_fun n f \<otimes>\<^bsub>SA (Suc n)\<^esub> SA_poly_to_SA_fun n g) a = 
+    hence 3: "(SA_poly_to_SA_fun n f \<otimes>\<^bsub>SA (Suc n)\<^esub> SA_poly_to_SA_fun n g) a =
             ((SA_poly_to_Qp_poly n b f) \<bullet> t) \<otimes> ((SA_poly_to_Qp_poly n b g) \<bullet> t)"
-      using SA_poly_to_SA_fun_formula assms 
+      using SA_poly_to_SA_fun_formula assms
       by (metis b_closed t_closed tb_def)
     have 4: "SA_poly_to_SA_fun n (f \<otimes>\<^bsub>UP (SA n)\<^esub> g) a  = ((SA_poly_to_Qp_poly n b f) \<bullet> t) \<otimes> ((SA_poly_to_Qp_poly n b g) \<bullet> t)"
-      using 1 assms SA_poly_to_Qp_poly_closed[of b] SA_poly_to_Qp_poly_mult UPQ.to_fun_mult b_closed t_closed 
+      using 1 assms SA_poly_to_Qp_poly_closed[of b] SA_poly_to_Qp_poly_mult UPQ.to_fun_mult b_closed t_closed
       by presburger
     show " SA_poly_to_SA_fun n (f \<otimes>\<^bsub>UP (SA n)\<^esub> g) a = (SA_poly_to_SA_fun n f \<otimes>\<^bsub>SA (Suc n)\<^esub> SA_poly_to_SA_fun n g) a"
       using "3" "4" by blast
   qed
 qed
 
 lemma SA_poly_to_SA_fun_one:
   shows "SA_poly_to_SA_fun n (\<one>\<^bsub>UP (SA n)\<^esub>) = \<one>\<^bsub>SA (Suc n)\<^esub>"
 proof(rule function_ring_car_eqI[of _ "Suc n"])
   have "\<one>\<^bsub>UP (SA n)\<^esub> \<in> carrier (UP (SA n))"
     using UP_SA_n_is_cring cring.cring_simprules(6) by blast
   thus " SA_poly_to_SA_fun n \<one>\<^bsub>UP (SA n)\<^esub> \<in> carrier (function_ring (carrier (Q\<^sub>p\<^bsup>Suc n\<^esup>)) Q\<^sub>p)"
-    using SA_poly_to_SA_fun_is_SA[of "\<one>\<^bsub>UP (SA n)\<^esub>"] SA_car_in_Qp_funs_car[of "Suc n"]  SA_is_UP_cring[of n]            
+    using SA_poly_to_SA_fun_is_SA[of "\<one>\<^bsub>UP (SA n)\<^esub>"] SA_car_in_Qp_funs_car[of "Suc n"]  SA_is_UP_cring[of n]
     by blast
   show "\<one>\<^bsub>SA (Suc n)\<^esub> \<in> carrier (function_ring (carrier (Q\<^sub>p\<^bsup>Suc n\<^esup>)) Q\<^sub>p)"
-    unfolding SA_one 
+    unfolding SA_one
     using function_one_closed by blast
   show "\<And>a. a \<in> carrier (Q\<^sub>p\<^bsup>Suc n\<^esup>) \<Longrightarrow> SA_poly_to_SA_fun n \<one>\<^bsub>UP (SA n)\<^esub> a = \<one>\<^bsub>SA (Suc n)\<^esub> a"
   proof- fix a assume A: "  a \<in> carrier (Q\<^sub>p\<^bsup>Suc n\<^esup>)"
     then obtain t b where tb_def: "a = t#b"
       using cartesian_power_car_memE[of a Q\<^sub>p "Suc n"] by (meson length_Suc_conv)
     have t_closed: "t \<in> carrier Q\<^sub>p"
       using tb_def A cartesian_power_head[of a Q\<^sub>p n] by (metis list.sel(1))
     have b_closed: "b \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
       using tb_def A cartesian_power_tail[of a Q\<^sub>p n] by (metis list.sel(3))
     have 0: "SA_poly_to_SA_fun n \<one>\<^bsub>UP (SA n)\<^esub> a  = (SA_poly_to_Qp_poly n b \<one>\<^bsub>UP (SA n)\<^esub>)\<bullet>t"
-      using SA_poly_to_SA_fun_formula \<open>\<one>\<^bsub>UP (SA n)\<^esub> \<in> carrier (UP (SA n))\<close>  b_closed t_closed tb_def 
+      using SA_poly_to_SA_fun_formula \<open>\<one>\<^bsub>UP (SA n)\<^esub> \<in> carrier (UP (SA n))\<close>  b_closed t_closed tb_def
       by blast
     have 1: "SA_poly_to_Qp_poly n b \<one>\<^bsub>UP (SA n)\<^esub> = \<one>\<^bsub>UP Q\<^sub>p\<^esub>"
-      using  SA_poly_to_Qp_poly_is_hom[of b] ring_hom_one[of _ "UP (SA n)" "UP Q\<^sub>p"] b_closed 
+      using  SA_poly_to_Qp_poly_is_hom[of b] ring_hom_one[of _ "UP (SA n)" "UP Q\<^sub>p"] b_closed
       by blast
     thus "SA_poly_to_SA_fun n \<one>\<^bsub>UP (SA n)\<^esub> a = \<one>\<^bsub>SA (Suc n)\<^esub> a"
       using "0" A function_one_eval SA_one UPQ.to_fun_one t_closed by presburger
   qed
 qed
 
 lemma SA_poly_to_SA_fun_ring_hom:
   shows "SA_poly_to_SA_fun n \<in> ring_hom (UP (SA n)) (SA (Suc n))"
   apply(rule ring_hom_memI)
-  using SA_poly_to_SA_fun_is_SA  apply blast 
+  using SA_poly_to_SA_fun_is_SA  apply blast
   apply (meson SA_poly_to_SA_fun_mult)
   apply (meson SA_poly_to_SA_fun_add)
   by (meson SA_poly_to_SA_fun_one)
 
 lemma SA_poly_to_SA_fun_taylor_term:
   assumes "F \<in> carrier (UP (SA n))"
   assumes "c \<in> carrier (SA n)"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "t \<in> carrier Q\<^sub>p"
   assumes "f = SA_poly_to_Qp_poly n x F"
   shows "SA_poly_to_SA_fun n (UP_cring.taylor_term (SA n) c F k) (t#x) = (taylor_expansion Q\<^sub>p (c x) f k) \<otimes>(t \<ominus> c x)[^]\<^bsub>Q\<^sub>p\<^esub> k"
 proof-
   have 0: "hd (t#x) = t"
     by simp
   have 1: "tl (t#x) = x"
     by auto
   have 2: "(t#x) \<in> carrier (Q\<^sub>p\<^bsup>Suc n\<^esup>)"
     by (metis Suc_eq_plus1 assms cartesian_power_cons)
-  show ?thesis 
+  show ?thesis
     using assms 0 1 2 Pi_iff  SA_car_memE(3)
           SA_poly_to_Qp_poly_closed SA_poly_to_Qp_poly_comm_taylor_term[of F n c x k] restrict_apply'
-    unfolding SA_poly_to_SA_fun_def 
-    by (metis (no_types, lifting) UPQ.taylor_def UPQ.to_fun_taylor_term)    
-qed
-    
-lemma SA_finsum_eval: 
+    unfolding SA_poly_to_SA_fun_def
+    by (metis (no_types, lifting) UPQ.taylor_def UPQ.to_fun_taylor_term)
+qed
+
+lemma SA_finsum_eval:
   assumes "finite I"
   assumes "F \<in> I \<rightarrow> carrier (SA m)"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
   shows "(\<Oplus>\<^bsub>SA m\<^esub>i\<in>I. F i) x = (\<Oplus>i\<in>I. F i x)"
 proof-
   have "F \<in> I \<rightarrow> carrier (SA m) \<longrightarrow> (\<Oplus>\<^bsub>SA m\<^esub>i\<in>I. F i) x = (\<Oplus>i\<in>I. F i x)"
     apply(rule finite.induct[of I])
     apply (simp add: assms(1))
-    using  abelian_monoid.finsum_empty[of "SA m" F] assms unfolding Qp.finsum_empty 
+    using  abelian_monoid.finsum_empty[of "SA m" F] assms unfolding Qp.finsum_empty
     using SA_is_abelian_monoid SA_zeroE apply presburger
   proof fix A a assume IH: "finite A" " F \<in> A \<rightarrow> carrier (SA m) \<longrightarrow> finsum (SA m) F A x = (\<Oplus>i\<in>A. F i x)"
                             "F \<in> insert a A \<rightarrow> carrier (SA m)"
     then have 0: "F \<in> A \<rightarrow> carrier (SA m)"
       by blast
     then have 1: "finsum (SA m) F A x = (\<Oplus>i\<in>A. F i x)"
       using IH by blast
     show "finsum (SA m) F (insert a A) x = (\<Oplus>i\<in>insert a A. F i x)"
     proof(cases "a \<in> A")
-      case True     
-      then show ?thesis using insert_absorb[of a A] IH 
+      case True
+      then show ?thesis using insert_absorb[of a A] IH
         by presburger
     next
       case False
       have F0: "(\<lambda>i. F i x) \<in> A \<rightarrow> carrier Q\<^sub>p" proof fix i assume "i \<in> A" thus "F i x \<in> carrier Q\<^sub>p"
         using 0 assms(3) SA_car_memE(3)[of "F i" m] by blast qed
       have F1: "F a x \<in> carrier Q\<^sub>p"
-        using IH assms(3) SA_car_memE(3)[of "F a" m] by blast  
+        using IH assms(3) SA_car_memE(3)[of "F a" m] by blast
       have F2: "finsum (SA m) F (insert a A) x = F a x \<oplus> finsum (SA m) F A x"
       proof-
         have " finsum (SA m) F (insert a A) = F a \<oplus>\<^bsub>SA m\<^esub> finsum (SA m) F A"
-          using False IH abelian_monoid.finsum_insert[of "SA m" A a F ] 0 
+          using False IH abelian_monoid.finsum_insert[of "SA m" A a F ] 0
           by (meson Pi_split_insert_domain SA_is_abelian_monoid)
         thus ?thesis using    abelian_monoid.finsum_closed[of "SA m" F A] 1 F0  F1
-            SA_add[of x m "F a" "finsum (SA m) F A"] 
+            SA_add[of x m "F a" "finsum (SA m) F A"]
           using assms(3) by presburger
       qed
-      show ?thesis using False 0 IH  Qp.finsum_insert[of A a "\<lambda>i. F i x"] unfolding F2 1 
-        using F0 F1 by blast     
+      show ?thesis using False 0 IH  Qp.finsum_insert[of A a "\<lambda>i. F i x"] unfolding F2 1
+        using F0 F1 by blast
     qed
   qed
-  thus ?thesis using assms by blast 
+  thus ?thesis using assms by blast
 qed
 
 lemma(in ring) finsum_ring_hom:
   assumes "ring S"
   assumes "h \<in> ring_hom R S"
   assumes "F \<in> I \<rightarrow> carrier R"
   assumes "finite I"
   shows "h (\<Oplus>i\<in>I. F i) = (\<Oplus>\<^bsub>S\<^esub>i\<in>I. h (F i))"
 proof-
   have "F \<in> I \<rightarrow> carrier R \<longrightarrow> h (\<Oplus>i\<in>I. F i) = (\<Oplus>\<^bsub>S\<^esub>i\<in>I. h (F i))"
     apply(rule finite.induct[of I])
       apply (simp add: assms(4))
     unfolding finsum_empty using assms abelian_monoid.finsum_empty[of S]
-    unfolding ring_def abelian_group_def 
-    apply (simp add: \<open>\<And>f. abelian_monoid S \<Longrightarrow> finsum S f {} = \<zero>\<^bsub>S\<^esub>\<close> assms(1) local.ring_axioms ring_hom_zero)    
+    unfolding ring_def abelian_group_def
+    apply (simp add: \<open>\<And>f. abelian_monoid S \<Longrightarrow> finsum S f {} = \<zero>\<^bsub>S\<^esub>\<close> assms(1) local.ring_axioms ring_hom_zero)
   proof fix A a assume A: "finite A" "  F \<in> A \<rightarrow> carrier R \<longrightarrow> h (finsum R F A) = (\<Oplus>\<^bsub>S\<^esub>i\<in>A. h (F i))"
        "   F \<in> insert a A \<rightarrow> carrier R"
     then have 0: "F \<in> A \<rightarrow> carrier R "
       by blast
     have 1: "h (finsum R F A) = (\<Oplus>\<^bsub>S\<^esub>i\<in>A. h (F i))"
       using "0" A(2) by linarith
     have 2: "(finsum R F A) \<in> carrier R"
-      using finsum_closed[of F A] 0 by blast  
+      using finsum_closed[of F A] 0 by blast
     have 3: "(\<Oplus>\<^bsub>S\<^esub>i\<in>A. h (F i)) \<in> carrier S"
-    using assms 1 2 ring_hom_closed 
+    using assms 1 2 ring_hom_closed
     by metis
-    have 4: "F a \<in> carrier R" using A by blast 
+    have 4: "F a \<in> carrier R" using A by blast
     have 5: "h (F a) \<in> carrier S"
-      using assms 4 
+      using assms 4
       by (meson ring_hom_closed)
     show "h (finsum R F (insert a A)) = (\<Oplus>\<^bsub>S\<^esub>i\<in>insert a A. h (F i))"
       apply(cases "a \<in> A")
-      using insert_absorb 1 
+      using insert_absorb 1
        apply metis
     proof- assume C: "a \<notin>A"
       have 6: "finsum R F (insert a A) = F a \<oplus> finsum R F A"
       using A finsum_insert[of A a F]  C by blast
     have 7: "(\<Oplus>\<^bsub>S\<^esub>i\<in>insert a A. h (F i)) = h (F a) \<oplus>\<^bsub>S\<^esub> (\<Oplus>\<^bsub>S\<^esub>i\<in>A. h (F i))"
     apply(rule abelian_monoid.finsum_insert[of S A a "\<lambda>i. h (F i)"])
           apply (simp add: abelian_group.axioms(1) assms(1) ring.is_abelian_group)
          apply (simp add: A(1))
         apply (simp add: C)
        apply(intro Pi_I ring_hom_closed[of h R S] assms)
-      using 0 A 5 assms by auto       
-    thus ?thesis 
-    using    0 1 2 3 4 5 6  7 assms ring_hom_add[of h R S "F a" "finsum R F A" ] 
-
-      unfolding 1 ring_def abelian_group_def 
+      using 0 A 5 assms by auto
+    thus ?thesis
+    using    0 1 2 3 4 5 6  7 assms ring_hom_add[of h R S "F a" "finsum R F A" ]
+
+      unfolding 1 ring_def abelian_group_def
       by presburger
   qed
   qed
-  thus ?thesis using assms by auto  
+  thus ?thesis using assms by auto
 qed
 
 lemma SA_poly_to_SA_fun_finsum:
   assumes "finite I"
   assumes "F \<in> I \<rightarrow> carrier (UP (SA m))"
   assumes "f = (\<Oplus>\<^bsub>UP (SA m)\<^esub>i\<in>I. F i)"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>Suc m\<^esup>)"
   shows "SA_poly_to_SA_fun m f x = (\<Oplus>i\<in>I. SA_poly_to_SA_fun m (F i) x)"
-proof- 
+proof-
   have "SA_poly_to_SA_fun m \<in> ring_hom (UP (SA m)) (SA (Suc m))"
     using SA_poly_to_SA_fun_ring_hom by blast
   have f_closed: "f \<in> carrier (UP (SA m))"
-    unfolding assms apply(rule finsum_closed)  using assms by blast 
+    unfolding assms apply(rule finsum_closed)  using assms by blast
   have 0: "SA_poly_to_SA_fun m f = (\<Oplus>\<^bsub>SA (Suc m)\<^esub>i\<in>I. SA_poly_to_SA_fun m (F i))"
-    unfolding assms 
+    unfolding assms
     apply(rule finsum_ring_hom)
-       apply (simp add: R.is_ring)
+       apply (simp add: SA_is_ring)
     using \<open>SA_poly_to_SA_fun m \<in> ring_hom (UP (SA m)) (SA (Suc m))\<close> apply blast
     using assms(2) apply blast
     by (simp add: assms(1))
-  show ?thesis unfolding 0 apply(rule SA_finsum_eval) 
-    using assms apply blast using assms 
+  show ?thesis unfolding 0 apply(rule SA_finsum_eval)
+    using assms apply blast using assms
     apply (meson Pi_I Pi_mem padic_fields.SA_poly_to_SA_fun_is_SA padic_fields_axioms)
-    using assms by blast     
+    using assms by blast
 qed
 
 lemma SA_poly_to_SA_fun_taylor_expansion:
   assumes "f \<in> carrier (UP (SA m))"
   assumes "c \<in> carrier (SA m)"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>Suc m\<^esup>)"
   shows "SA_poly_to_SA_fun m f x = (\<Oplus>i\<in>{..deg (SA m) f}. taylor_expansion (SA m) c f i (tl x) \<otimes> (hd x \<ominus> c (tl x)) [^] i)"
 proof-
   have 0: "f = (\<Oplus>\<^bsub>UP (SA m)\<^esub>i\<in>{..deg (SA m) f}. UP_cring.taylor_term (SA m) c f i)"
-    using taylor_sum[of f m "deg (SA m) f" c] assms unfolding UPSA.taylor_term_def UPSA.taylor_def  
-    by blast  
+    using taylor_sum[of f m "deg (SA m) f" c] assms unfolding UPSA.taylor_term_def UPSA.taylor_def
+    by blast
   have 1: "SA_poly_to_SA_fun m f x = (\<Oplus>i\<in>{..deg (SA m) f}. SA_poly_to_SA_fun m (UP_cring.taylor_term (SA m) c f i) x)"
     apply(rule SA_poly_to_SA_fun_finsum)
        apply simp
       apply (meson Pi_I UPSA.taylor_term_closed assms(1) assms(2))
-    using 0 apply blast 
+    using 0 apply blast
     using assms by blast
   have hd_closed: "hd x \<in> carrier Q\<^sub>p"
     using assms cartesian_power_head by blast
   have tl_closed: "tl x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
     using assms cartesian_power_tail by blast
   obtain t a where ta_def: " x = t#a"
-    using assms cartesian_power_car_memE[of x Q\<^sub>p "Suc m" ] 
+    using assms cartesian_power_car_memE[of x Q\<^sub>p "Suc m" ]
     by (metis Suc_length_conv)
   have 2: "t = hd x"
     by (simp add: ta_def)
   have 3: "a = tl x "
     by (simp add: ta_def)
   have 4: "\<And>i. SA_poly_to_SA_fun m (UPSA.taylor_term m c f i) x =
     taylor_expansion Q\<^sub>p (c (tl x)) (SA_poly_to_Qp_poly m (tl x) f) i \<otimes> (lead_coeff x \<ominus> c (tl x)) [^] i"
-    using tl_closed hd_closed assms SA_poly_to_SA_fun_taylor_term[of f m c a t  "SA_poly_to_Qp_poly m a f"  ] 
-    unfolding 2 3  by (metis "2" "3" ta_def) 
+    using tl_closed hd_closed assms SA_poly_to_SA_fun_taylor_term[of f m c a t  "SA_poly_to_Qp_poly m a f"  ]
+    unfolding 2 3  by (metis "2" "3" ta_def)
   have 5: "SA_poly_to_SA_fun m f x = (\<Oplus>i\<in>{..deg (SA m) f}. (taylor_expansion Q\<^sub>p (c (tl x)) (SA_poly_to_Qp_poly m (tl x) f) i) \<otimes>((hd x) \<ominus> c (tl x))[^]\<^bsub>Q\<^sub>p\<^esub> i)"
     using 1 2 unfolding 4 by blast
   have 6: "taylor_expansion Q\<^sub>p (c (tl x)) (SA_poly_to_Qp_poly m (tl x) f) = SA_poly_to_Qp_poly m (tl x) (taylor_expansion (SA m) c f)"
 
     using SA_poly_to_Qp_poly_taylor_poly[of f m c "tl x"] assms(1) assms(2) tl_closed by presburger
   have 7: "\<And>i. taylor_expansion Q\<^sub>p (c (tl x)) (SA_poly_to_Qp_poly m (tl x) f) i =
                 taylor_expansion (SA m) c f i (tl x)"
-    unfolding 6 using SA_poly_to_Qp_poly_coeff[of "tl x" m "taylor_expansion (SA m) c f"] 
+    unfolding 6 using SA_poly_to_Qp_poly_coeff[of "tl x" m "taylor_expansion (SA m) c f"]
     by (metis UPSA.taylor_closed UPSA.taylor_def assms(1) assms(2) tl_closed)
-  show ?thesis using 5 unfolding 7 by blast 
+  show ?thesis using 5 unfolding 7 by blast
 qed
 
 lemma SA_deg_one_eval:
   assumes "g \<in> carrier (UP (SA m))"
   assumes "deg (SA m) g = 1"
   assumes "\<xi> \<in> carrier (Fun\<^bsub>m\<^esub> Q\<^sub>p)"
   assumes "UP_ring.lcf (SA m) g \<in> Units (SA m)"
   assumes "\<forall>x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). (SA_poly_to_SA_fun m g) (\<xi> x#x) = \<zero>"
   shows "\<xi> = \<ominus>\<^bsub>SA m\<^esub>(g 0)\<otimes>\<^bsub>SA m\<^esub> (inv\<^bsub>SA m\<^esub> (g 1))"
 proof(rule ext)
   fix x show " \<xi> x = (\<ominus>\<^bsub>SA m\<^esub> g 0 \<otimes>\<^bsub>SA m\<^esub> inv\<^bsub>SA m\<^esub> g 1) x"
   proof(cases "x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)")
     case True
     then have "(SA_poly_to_SA_fun m g) (\<xi> x#x) = \<zero>"
-      using assms by blast 
+      using assms by blast
     then have T0: "SA_poly_to_Qp_poly m x g \<bullet> \<xi> x = \<zero>"
-      using SA_poly_to_SA_fun_formula[of g m x "\<xi> x"] assms 
+      using SA_poly_to_SA_fun_formula[of g m x "\<xi> x"] assms
             Qp.function_ring_car_mem_closed True by metis
     have "deg Q\<^sub>p (SA_poly_to_Qp_poly m x g) = 1"
     proof(rule UPQ.deg_eqI)
       show "SA_poly_to_Qp_poly m x g \<in> carrier (UP Q\<^sub>p)"
         using assms True SA_poly_to_Qp_poly_closed by blast
       show "deg Q\<^sub>p (SA_poly_to_Qp_poly m x g) \<le> 1"
         using SA_poly_to_Qp_poly_deg_bound by (metis True assms(1) assms(2) )
       show " SA_poly_to_Qp_poly m x g 1 \<noteq> \<zero>"
         using SA_poly_to_Qp_poly_coeff[of x m g 1] assms SA_Units_memE' True by presburger
     qed
     hence T1: "SA_poly_to_Qp_poly m x g \<bullet> \<xi> x = SA_poly_to_Qp_poly m x g 0 \<oplus> SA_poly_to_Qp_poly m x g 1 \<otimes> (\<xi> x)"
       using UP_cring.deg_one_eval[of Q\<^sub>p _ "\<xi> x"]
       by (meson Qp.function_ring_car_mem_closed SA_poly_to_Qp_poly_closed True UPQ.deg_one_eval assms)
     hence T2: "g 0 x \<oplus> g 1 x \<otimes> (\<xi> x) = \<zero>"
-      using True T0 assms SA_poly_to_Qp_poly_coeff[of x m g] 
-      by metis       
+      using True T0 assms SA_poly_to_Qp_poly_coeff[of x m g]
+      by metis
     have T3: "g 0 x \<in> carrier Q\<^sub>p"
-      using True assms UP_ring.cfs_closed 
+      using True assms UP_ring.cfs_closed
       by (metis SA_poly_to_Qp_poly_closed SA_poly_to_Qp_poly_coeff UPQ.is_UP_ring)
     have T4: "\<xi> x \<in> carrier Q\<^sub>p"
       using True assms Qp.function_ring_car_memE by blast
     have T5: "g 1 x \<in> nonzero Q\<^sub>p"
-      using True assms 
+      using True assms
       by (metis Qp.function_ring_car_memE SA_Units_closed SA_Units_memE' SA_car_memE(2) not_nonzero_Qp)
     have T6: "\<ominus> (g 0 x) = g 1 x \<otimes> (\<xi> x)"
       using T2 T3 T4 T5 by (metis Qp.m_closed Qp.minus_equality Qp.minus_minus Qp.nonzero_closed)
     hence T7: "\<ominus> (g 0 x) \<otimes> inv (g 1 x)= \<xi> x"
       using T5 by (metis Qp.inv_cancelR(2) Qp.m_closed Qp.nonzero_closed T4 Units_eq_nonzero)
     have T8: "(inv\<^bsub>SA m\<^esub> g 1) x = inv (g 1 x)"
       using assms True Qp_funs_m_inv SA_Units_Qp_funs_Units SA_Units_Qp_funs_inv by presburger
-    have T9: "(\<ominus>\<^bsub>SA m\<^esub> g 0) x = \<ominus> (g 0 x)" 
+    have T9: "(\<ominus>\<^bsub>SA m\<^esub> g 0) x = \<ominus> (g 0 x)"
       using SA_a_inv_eval[of "g 0" m x] UP_ring.cfs_closed[of "SA m" g 0] assms True SA_is_ring
-      unfolding UP_ring_def by blast 
+      unfolding UP_ring_def by blast
     have T11: "((\<ominus>\<^bsub>SA m\<^esub> g 0) \<otimes>\<^bsub>SA m\<^esub> inv\<^bsub>SA m\<^esub> g 1) x = \<ominus> (g 0 x) \<otimes> inv (g 1 x)"
       using assms UP_ring.cfs_closed T8 T9 T7 True SA_mult by presburger
     thus "\<xi> x = (\<ominus>\<^bsub>SA m\<^esub> g 0 \<otimes>\<^bsub>SA m\<^esub> inv\<^bsub>SA m\<^esub> g 1) x"
       using T7 by blast
   next
     case False
-    then show ?thesis 
+    then show ?thesis
       using SA_mult' SA_times assms function_ring_not_car by auto
   qed
 qed
 
 lemma SA_deg_one_eval':
   assumes "g \<in> carrier (UP (SA m))"
   assumes "deg (SA m) g = 1"
   assumes "\<xi> \<in> carrier (Fun\<^bsub>m\<^esub> Q\<^sub>p)"
   assumes "UP_ring.lcf (SA m) g \<in> Units (SA m)"
   assumes "\<forall>x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). (SA_poly_to_SA_fun m g) (\<xi> x#x) = \<zero>"
   shows "\<xi> \<in> carrier (SA m)"
 proof-
   have 0: "\<xi> = \<ominus>\<^bsub>SA m\<^esub>(g 0)\<otimes>\<^bsub>SA m\<^esub> (inv\<^bsub>SA m\<^esub> (g 1))"
     using assms SA_deg_one_eval by blast
   have 1: "(inv\<^bsub>SA m\<^esub> (g 1)) \<in> carrier (SA m)"
     using assms SA_Units_inv_closed by presburger
   have "(g 0) \<in> carrier (SA m)"
-    using assms(1) assms(2) UP_ring.cfs_closed[of "SA m" g 0] SA_is_ring[of m ] unfolding UP_ring_def 
-    by blast 
+    using assms(1) assms(2) UP_ring.cfs_closed[of "SA m" g 0] SA_is_ring[of m ] unfolding UP_ring_def
+    by blast
   hence 2: "\<ominus>\<^bsub>SA m\<^esub>(g 0) \<in> carrier (SA m)"
     by (meson SA_is_cring assms(1) cring.cring_simprules(3))
-  show ?thesis 
+  show ?thesis
     using 0 1 2  SA_imp_semialg SA_mult_closed_left assms(1) by blast
 qed
 
 lemma Qp_pow_ConsI:
   assumes "t \<in> carrier Q\<^sub>p"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
   shows "t#x \<in> carrier (Q\<^sub>p\<^bsup>Suc m\<^esup>)"
   using assms cartesian_power_cons[of x Q\<^sub>p m t] Suc_eq_plus1 by presburger
 
 lemma Qp_pow_ConsE:
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>Suc m\<^esup>)"
   shows "tl x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
         "hd x \<in> carrier Q\<^sub>p"
   using assms cartesian_power_tail apply blast
-  using assms cartesian_power_head by blast 
+  using assms cartesian_power_head by blast
 
 lemma(in ring) add_monoid_one:
 "\<one>\<^bsub>add_monoid R\<^esub> = \<zero>"
-  by (metis add.generate_empty add.group_l_invI add.l_inv_ex empty_iff group.generate_empty insert_iff)
+  by simp
 
 lemma(in ring) add_monoid_carrier:
 "carrier (add_monoid R) = carrier R"
-  unfolding Congruence.partial_object.simps(1) 
+  unfolding Congruence.partial_object.simps(1)
   by simp
-  
+
 lemma(in ring) finsum_mono_neutral_cong:
   assumes "F \<in> I \<rightarrow> carrier R"
   assumes "finite I"
   assumes "\<And>i. i \<notin> J \<Longrightarrow> F i = \<zero>"
   assumes "J \<subseteq> I"
   shows "finsum R F I = finsum R F J"
   unfolding finsum_def apply(rule comm_monoid.finprod_mono_neutral_cong)
   using local.add.comm_monoid_axioms apply blast
   using assms(2) assms(4) rev_finite_subset apply blast
   apply (simp add: assms(2))
   using assms(4) apply blast
   unfolding add_monoid_one using assms apply blast
-  apply blast 
+  apply blast
   using add_monoid_carrier assms(1) apply blast
   using add_monoid_carrier assms  by blast
 
 text\<open>
   This lemma helps to formalize statements like "by passing to a partition, we can assume the
   Taylor coefficients are either always zero or never zero"\<close>
 
 lemma SA_poly_to_SA_fun_taylor_on_refined_set:
   assumes "f \<in> carrier (UP (SA m))"
   assumes "c \<in> carrier (SA m)"
   assumes "is_semialgebraic m A"
   assumes "\<And>i. A \<subseteq> SA_zero_set m (taylor_expansion (SA m) c f i) \<or> A \<subseteq> SA_nonzero_set m (taylor_expansion (SA m) c f i)"
   assumes "a = to_fun_unit m \<circ> taylor_expansion (SA m) c f"
   assumes "inds = {i. i \<le> deg (SA m) f \<and> A \<subseteq> SA_nonzero_set m (taylor_expansion (SA m) c f i)}"
   assumes "x \<in> A"
   assumes "t \<in> carrier Q\<^sub>p"
   shows "SA_poly_to_SA_fun m f (t#x) = (\<Oplus>i\<in>inds. (a i x)\<otimes>(t \<ominus> c x)[^]i)"
 proof-
   have x_closed: "x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
-    using assms(3) assms(7)  
+    using assms(3) assms(7)
   by (metis (no_types, lifting) Diff_eq_empty_iff Diff_iff empty_iff is_semialgebraic_closed)
   have tx_closed: "t#x \<in> carrier (Q\<^sub>p\<^bsup>Suc m\<^esup>)"
     using x_closed assms(8) cartesian_power_cons[of x Q\<^sub>p m t] Qp_pow_ConsI by blast
   have "SA_poly_to_SA_fun m f (t # x) =
     (\<Oplus>i\<in>{..deg (SA m) f}. taylor_expansion (SA m) c f i (tl (t # x)) \<otimes> (hd (t # x) \<ominus> c (tl (t # x))) [^] i)"
-    using tx_closed assms SA_poly_to_SA_fun_taylor_expansion[of f m c "t#x"] 
+    using tx_closed assms SA_poly_to_SA_fun_taylor_expansion[of f m c "t#x"]
     by linarith
   then have 0: "SA_poly_to_SA_fun m f (t#x) = (\<Oplus>i\<in>{..deg (SA m) f}. taylor_expansion (SA m) c f i x \<otimes> (t \<ominus> c x) [^] i)"
-    unfolding list_tl list_hd by blast 
+    unfolding list_tl list_hd by blast
   have 1: "\<And>i. i \<notin> inds \<Longrightarrow> taylor_expansion (SA m) c f i x = \<zero>"
-  proof- fix i assume A: "i \<notin> inds" 
+  proof- fix i assume A: "i \<notin> inds"
     show "taylor_expansion (SA m) c f i x = \<zero>"
     proof(cases "i \<le> deg (SA m) f")
     case True
     then have "A \<subseteq> {x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). taylor_expansion (SA m) c f i x = \<zero>}"
-      using A assms(8) assms(4)[of i] unfolding assms mem_Collect_eq SA_zero_set_def 
+      using A assms(8) assms(4)[of i] unfolding assms mem_Collect_eq SA_zero_set_def
       by linarith
-    thus ?thesis using x_closed assms by blast 
+    thus ?thesis using x_closed assms by blast
    next
     case False
     then have "taylor_expansion (SA m) c f i = \<zero>\<^bsub>SA m\<^esub>"
-      using assms taylor_deg[of c m f] unfolding UPSA.taylor_def  
+      using assms taylor_deg[of c m f] unfolding UPSA.taylor_def
       by (metis (no_types, lifting) UPSA.taylor_closed UPSA.taylor_def UPSA.deg_eqI nat_le_linear)
-    then show ?thesis 
+    then show ?thesis
       using x_closed SA_car_memE SA_zeroE by presburger
     qed
   qed
   hence 2: "\<And>i. i \<notin> inds \<Longrightarrow> taylor_expansion (SA m) c f i x \<otimes> (t \<ominus> c x) [^] i = \<zero>"
-    using assms x_closed  SA_car_memE 
+    using assms x_closed  SA_car_memE
     by (metis (no_types, lifting) Qp.cring_simprules(26) Qp.function_ring_car_memE Qp.minus_closed Qp.nat_pow_closed)
   have 3: "(\<lambda>i. taylor_expansion (SA m) c f i x \<otimes> (t \<ominus> c x) [^] i) \<in> {..deg (SA m) f} \<rightarrow> carrier Q\<^sub>p"
   proof fix i assume A: "i \<in> {..deg (SA m) f}"
     have 30: "taylor_expansion (SA m) c f i \<in> carrier (SA m)"
-      using  assms UPSA.taylor_closed[of f m c] unfolding UPSA.taylor_def  
+      using  assms UPSA.taylor_closed[of f m c] unfolding UPSA.taylor_def
       using UPSA.UP_car_memE(1) by blast
     hence 31: "taylor_expansion (SA m) c f i x \<in> carrier Q\<^sub>p"
       using x_closed function_ring_car_closed SA_car_memE(2) by blast
     have 32: "c x \<in> carrier Q\<^sub>p"
-      using assms x_closed SA_car_memE(3) by blast 
+      using assms x_closed SA_car_memE(3) by blast
     hence 33: "(t \<ominus> c x)[^] i \<in> carrier Q\<^sub>p"
       using assms by blast
     show "taylor_expansion (SA m) c f i x \<otimes> (t \<ominus> c x) [^] i \<in> carrier Q\<^sub>p"
       using 30 31 32 33 by blast
   qed
   have 4: "SA_poly_to_SA_fun m f (t#x) = (\<Oplus>i\<in>inds. taylor_expansion (SA m) c f i x \<otimes> (t \<ominus> c x) [^] i)"
     unfolding 0 apply(rule Qp.finsum_mono_neutral_cong)
-    using assms UPSA.taylor_closed[of f m c] unfolding UPSA.taylor_def  
+    using assms UPSA.taylor_closed[of f m c] unfolding UPSA.taylor_def
     using "3" apply blast
-      apply blast 
-    using 2 apply blast 
-    unfolding assms by blast 
+      apply blast
+    using 2 apply blast
+    unfolding assms by blast
   have A: "\<And>i. i \<in> inds \<Longrightarrow> a i x = taylor_expansion (SA m) c f i x"
     unfolding assms mem_Collect_eq SA_nonzero_set_def comp_apply
     apply(rule to_fun_unit_eq[of _ m x])
-    using UPSA.taylor_closed[of f m c] assms unfolding UPSA.taylor_def 
+    using UPSA.taylor_closed[of f m c] assms unfolding UPSA.taylor_def
     using UPSA.UP_car_memE(1) apply blast
-    using x_closed apply blast 
-    using assms(7) by blast 
+    using x_closed apply blast
+    using assms(7) by blast
   have a_closed: "a \<in> UNIV \<rightarrow> carrier (SA m)"
     apply(rule Pi_I) unfolding assms comp_apply apply(rule to_fun_unit_closed[of _ m])
-    apply(rule cfs_closed) using assms UPSA.taylor_closed[of f m c] unfolding UPSA.taylor_def by blast 
+    apply(rule cfs_closed) using assms UPSA.taylor_closed[of f m c] unfolding UPSA.taylor_def by blast
   have 5: "(\<lambda>i. a i x \<otimes> (t \<ominus> c x) [^] i) \<in> inds \<rightarrow> carrier Q\<^sub>p"
   proof fix i assume A: "i \<in> inds"
     show "a i x \<otimes> (t \<ominus> c x) [^] i \<in> carrier Q\<^sub>p"
-    using assms(8) x_closed a_closed SA_car_memE(3)[of c m] SA_car_memE(3)[of "a i" m] 
+    using assms(8) x_closed a_closed SA_car_memE(3)[of c m] SA_car_memE(3)[of "a i" m]
          assms(2) by blast
   qed
   have 6: "\<And>i. i \<in> inds \<Longrightarrow> taylor_expansion (SA m) c f i x \<otimes> (t \<ominus> c x) [^] i = a i x \<otimes> (t \<ominus> c x) [^] i"
-    unfolding A by blast 
-  show ?thesis 
-    unfolding 4 apply(rule Qp.finsum_cong') apply blast 
-    using 5 apply blast 
-    using 6 by blast 
-qed
-
-lemma SA_poly_to_Qp_poly_taylor_cfs: 
+    unfolding A by blast
+  show ?thesis
+    unfolding 4 apply(rule Qp.finsum_cong') apply blast
+    using 5 apply blast
+    using 6 by blast
+qed
+
+lemma SA_poly_to_Qp_poly_taylor_cfs:
   assumes "f \<in> carrier (UP (SA m))"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
   assumes "c \<in> carrier (SA m)"
   shows "taylor_expansion (SA m) c f i x =
     taylor_expansion Q\<^sub>p (c x) (SA_poly_to_Qp_poly m x f) i"
 proof-
   have 0: "SA_poly_to_Qp_poly m x (taylor_expansion (SA m) c f) =
          taylor_expansion Q\<^sub>p (c x) (SA_poly_to_Qp_poly m x f)"
-    using SA_poly_to_Qp_poly_taylor_poly[of f m c x] assms by blast 
+    using SA_poly_to_Qp_poly_taylor_poly[of f m c x] assms by blast
   hence 1: "SA_poly_to_Qp_poly m x (taylor_expansion (SA m) c f) i =
   taylor_expansion Q\<^sub>p (c x) (SA_poly_to_Qp_poly m x f) i"
     by presburger
-  thus ?thesis 
-    using assms SA_poly_to_Qp_poly_coeff[of x m "taylor_expansion (SA m) c f" i]  
-      UPSA.taylor_closed[of f m c] unfolding UPSA.taylor_def assms by blast 
+  thus ?thesis
+    using assms SA_poly_to_Qp_poly_coeff[of x m "taylor_expansion (SA m) c f" i]
+      UPSA.taylor_closed[of f m c] unfolding UPSA.taylor_def assms by blast
 qed
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsubsection\<open>Common Morphisms on Polynomial Rings\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 
 text\<open>Evaluation homomorphism from multivariable polynomials to semialgebraic functions\<close>
 
 definition Qp_ev_hom where
 "Qp_ev_hom n P = restrict (Qp_ev P) (carrier (Q\<^sub>p\<^bsup>n\<^esup>))"
 
 lemma Qp_ev_hom_ev:
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   shows "Qp_ev_hom n P a = Qp_ev P a"
-  using assms unfolding Qp_ev_hom_def 
+  using assms unfolding Qp_ev_hom_def
   by (meson restrict_apply')
 
 lemma Qp_ev_hom_closed:
   assumes "f \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   shows "Qp_ev_hom n f \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>) \<rightarrow> carrier Q\<^sub>p"
   using Qp_ev_hom_ev assms by (metis Pi_I eval_at_point_closed)
 
 lemma Qp_ev_hom_is_semialg_function:
   assumes "f \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   shows "is_semialg_function n (Qp_ev_hom n f)"
-  unfolding Qp_ev_hom_def  
+  unfolding Qp_ev_hom_def
   using assms poly_is_semialg[of  f] restrict_is_semialg by blast
 
 lemma Qp_ev_hom_closed':
   assumes "f \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   shows "Qp_ev_hom n f \<in> carrier (Fun\<^bsub>n\<^esub> Q\<^sub>p)"
   apply(rule Qp.function_ring_car_memI)
   using  Qp_ev_hom_closed[of f n] assms apply blast
   unfolding Qp_ev_hom_def using assms by (meson restrict_apply)
 
 lemma Qp_ev_hom_in_SA:
   assumes "f \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   shows "Qp_ev_hom n f \<in> carrier (SA n)"
   apply(rule SA_car_memI)
   using Qp_ev_hom_closed' assms(1)  apply blast
-  using Qp_ev_hom_is_semialg_function assms(1) by blast 
+  using Qp_ev_hom_is_semialg_function assms(1) by blast
 
 lemma Qp_ev_hom_add:
   assumes "f \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   assumes "g \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   shows "Qp_ev_hom n (f \<oplus>\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> g) = (Qp_ev_hom n f) \<oplus>\<^bsub>SA n\<^esub> (Qp_ev_hom n g)"
   apply(rule function_ring_car_eqI[of _ n])
   using assms MP.add.m_closed Qp_ev_hom_closed' apply blast
   using assms Qp_ev_hom_closed' fun_add_closed SA_plus apply presburger
 proof- fix a assume A: "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   have " Qp_ev_hom n (f \<oplus>\<^bsub>Q\<^sub>p [\<X>\<^bsub>n\<^esub>]\<^esub> g) a =  Qp_ev_hom n f a \<oplus> Qp_ev_hom n g a"
     using A Qp_ev_hom_ev assms eval_at_point_add by presburger
   then show "Qp_ev_hom n (f \<oplus>\<^bsub>Q\<^sub>p [\<X>\<^bsub>n\<^esub>]\<^esub> g) a = (Qp_ev_hom n f \<oplus>\<^bsub>SA n\<^esub> Qp_ev_hom n g) a"
     using A SA_add by blast
 qed
-        
+
 lemma Qp_ev_hom_mult:
   assumes "f \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   assumes "g \<in> carrier (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])"
   shows "Qp_ev_hom n (f \<otimes>\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> g) = (Qp_ev_hom n f) \<otimes>\<^bsub>SA n\<^esub> (Qp_ev_hom n g)"
   apply(rule function_ring_car_eqI[of _ n])
-  using assms MP.m_closed Qp_ev_hom_closed' apply blast 
-  using assms Qp_ev_hom_closed' fun_mult_closed SA_mult 
-  apply (meson Qp_ev_hom_in_SA SA_car_memE(2) SA_imp_semialg SA_mult_closed)  
+  using assms MP.m_closed Qp_ev_hom_closed' apply blast
+  using assms Qp_ev_hom_closed' fun_mult_closed SA_mult
+  apply (meson Qp_ev_hom_in_SA SA_car_memE(2) SA_imp_semialg SA_mult_closed)
 proof- fix a assume A: "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   have " Qp_ev_hom n (f \<otimes>\<^bsub>Q\<^sub>p [\<X>\<^bsub>n\<^esub>]\<^esub> g) a =  Qp_ev_hom n f a \<otimes> Qp_ev_hom n g a"
     using A Qp_ev_hom_ev assms eval_at_point_mult by presburger
   then show "Qp_ev_hom n (f \<otimes>\<^bsub>Q\<^sub>p [\<X>\<^bsub>n\<^esub>]\<^esub> g) a = (Qp_ev_hom n f \<otimes>\<^bsub>SA n\<^esub> Qp_ev_hom n g) a"
     using A SA_mult by blast
 qed
 
 lemma Qp_ev_hom_one:
   shows "Qp_ev_hom n \<one>\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> = \<one>\<^bsub>SA n\<^esub>"
   apply(rule function_ring_car_eqI[of _ n])
   using Qp_ev_hom_closed' apply blast
-  using  function_one_closed SA_one apply presburger 
-  unfolding Qp_ev_hom_def 
+  using  function_one_closed SA_one apply presburger
+  unfolding Qp_ev_hom_def
   using function_one_eval Qp_ev_hom_def Qp_ev_hom_ev Qp_ev_one SA_one by presburger
 
 lemma Qp_ev_hom_is_hom:
   shows "Qp_ev_hom n \<in> ring_hom (Q\<^sub>p[\<X>\<^bsub>n\<^esub>]) (SA n)"
   apply(rule ring_hom_memI)
   using   Qp_ev_hom_in_SA apply blast
     using  Qp_ev_hom_mult apply blast
-      using  Qp_ev_hom_add apply blast  
-        using  Qp_ev_hom_one by blast 
+      using  Qp_ev_hom_add apply blast
+        using  Qp_ev_hom_one by blast
 
 lemma Qp_ev_hom_constant:
   assumes "c \<in> carrier Q\<^sub>p"
   shows "Qp_ev_hom n (Qp.indexed_const c) = \<cc>\<^bsub>n\<^esub> c"
   apply(rule function_ring_car_eqI[of _ n])
   using Qp_ev_hom_closed' Qp_to_IP_car assms(1)  apply blast
   using constant_function_closed assms  apply blast
   by (metis Qp_constE Qp_ev_hom_ev assms eval_at_point_const)
 
 notation  Qp.variable ("\<vv>\<^bsub>_, _\<^esub>")
 
 lemma Qp_ev_hom_pvar:
   assumes "i < n"
   shows "Qp_ev_hom n (pvar Q\<^sub>p i) = \<vv>\<^bsub>n, i\<^esub>"
-  apply(rule function_ring_car_eqI[of _ n]) 
+  apply(rule function_ring_car_eqI[of _ n])
   using assms Qp_ev_hom_closed' local.pvar_closed apply blast
   using Qp.var_in_car assms apply blast
-  unfolding Qp_ev_hom_def var_def using assms eval_pvar 
+  unfolding Qp_ev_hom_def var_def using assms eval_pvar
   by (metis (no_types, lifting) restrict_apply)
 
-definition ext_hd where 
+definition ext_hd where
 "ext_hd m = (\<lambda> x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). hd x)"
 
 lemma hd_zeroth:
 "length x > 0 \<Longrightarrow> x!0 = hd x"
   apply(induction x)
   apply simp
   by simp
 
 lemma ext_hd_pvar:
   assumes "m > 0"
   shows "ext_hd m = (\<lambda>x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). eval_at_point Q\<^sub>p x (pvar Q\<^sub>p 0) )"
   unfolding ext_hd_def restrict_def using assms eval_pvar[of 0 m]
-  using hd_zeroth 
+  using hd_zeroth
   by (metis (no_types, opaque_lifting) cartesian_power_car_memE)
 
-lemma ext_hd_closed: 
+lemma ext_hd_closed:
   assumes "m > 0"
   shows "ext_hd m \<in> carrier (SA m)"
-  using ext_hd_pvar[of m] assms pvar_closed[of 0 m] Qp_ev_hom_def Qp_ev_hom_in_SA by presburger 
-  
+  using ext_hd_pvar[of m] assms pvar_closed[of 0 m] Qp_ev_hom_def Qp_ev_hom_in_SA by presburger
+
 lemma UP_Qp_poly_to_UP_SA_is_hom:
   shows "poly_lift_hom (Q\<^sub>p[\<X>\<^bsub>n\<^esub>]) (SA n) (Qp_ev_hom n) \<in> ring_hom (UP (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])) (UP (SA n))"
   using UP_cring.poly_lift_hom_is_hom[of "Q\<^sub>p[\<X>\<^bsub>n\<^esub>]" "SA n" "Qp_ev_hom n"]
-  unfolding UP_cring_def 
+  unfolding UP_cring_def
   using Qp_ev_hom_is_hom SA_is_cring  coord_cring_cring by blast
 
 definition coord_ring_to_UP_SA where
 "coord_ring_to_UP_SA n = poly_lift_hom (Q\<^sub>p[\<X>\<^bsub>n\<^esub>]) (SA n) (Qp_ev_hom n) \<circ> to_univ_poly (Suc n) 0"
 
 lemma coord_ring_to_UP_SA_is_hom:
   shows "coord_ring_to_UP_SA n \<in> ring_hom (Q\<^sub>p[\<X>\<^bsub>Suc n\<^esub>]) (UP (SA n))"
-  unfolding coord_ring_to_UP_SA_def 
-  using UP_Qp_poly_to_UP_SA_is_hom[of n] to_univ_poly_is_hom[of 0 "n"] ring_hom_trans 
+  unfolding coord_ring_to_UP_SA_def
+  using UP_Qp_poly_to_UP_SA_is_hom[of n] to_univ_poly_is_hom[of 0 "n"] ring_hom_trans
   by blast
 
 lemma coord_ring_to_UP_SA_add:
   assumes "f \<in> carrier (Q\<^sub>p[\<X>\<^bsub>Suc n\<^esub>])"
   assumes "g \<in> carrier (Q\<^sub>p[\<X>\<^bsub>Suc n\<^esub>])"
   shows "coord_ring_to_UP_SA n (f \<oplus>\<^bsub>Q\<^sub>p[\<X>\<^bsub>Suc n\<^esub>]\<^esub>g) = coord_ring_to_UP_SA n f  \<oplus>\<^bsub>UP (SA n)\<^esub> coord_ring_to_UP_SA n g"
-  using assms coord_ring_to_UP_SA_is_hom ring_hom_add 
+  using assms coord_ring_to_UP_SA_is_hom ring_hom_add
   by (metis (mono_tags, opaque_lifting))
 
 lemma coord_ring_to_UP_SA_mult:
   assumes "f \<in> carrier (Q\<^sub>p[\<X>\<^bsub>Suc n\<^esub>])"
   assumes "g \<in> carrier (Q\<^sub>p[\<X>\<^bsub>Suc n\<^esub>])"
   shows "coord_ring_to_UP_SA n (f \<otimes>\<^bsub>Q\<^sub>p[\<X>\<^bsub>Suc n\<^esub>]\<^esub>g) = coord_ring_to_UP_SA n f  \<otimes>\<^bsub>UP (SA n)\<^esub> coord_ring_to_UP_SA n g"
-  using assms coord_ring_to_UP_SA_is_hom ring_hom_mult 
+  using assms coord_ring_to_UP_SA_is_hom ring_hom_mult
   by (metis (no_types, opaque_lifting))
 
 lemma coord_ring_to_UP_SA_one:
   shows "coord_ring_to_UP_SA n \<one>\<^bsub>Q\<^sub>p[\<X>\<^bsub>Suc n\<^esub>]\<^esub> = \<one>\<^bsub>UP (SA n)\<^esub>"
-  using coord_ring_to_UP_SA_is_hom ring_hom_one 
+  using coord_ring_to_UP_SA_is_hom ring_hom_one
   by blast
 
 lemma coord_ring_to_UP_SA_closed:
   assumes "f \<in> carrier (Q\<^sub>p[\<X>\<^bsub>Suc n\<^esub>])"
   shows "coord_ring_to_UP_SA n f \<in> carrier (UP (SA n))"
-  using assms coord_ring_to_UP_SA_is_hom ring_hom_closed 
+  using assms coord_ring_to_UP_SA_is_hom ring_hom_closed
   by (metis (no_types, opaque_lifting))
 
 lemma coord_ring_to_UP_SA_constant:
   assumes "c \<in> carrier Q\<^sub>p"
   shows "coord_ring_to_UP_SA n (Qp.indexed_const c) = to_polynomial (SA n) (\<cc>\<^bsub>n\<^esub> c)"
 proof-
   have 0: "pre_to_univ_poly (Suc n) 0 (Qp.indexed_const c) = ring.indexed_const (Q\<^sub>p[\<X>\<^bsub>n\<^esub>]) (Qp.indexed_const c)"
-    unfolding to_univ_poly_def 
-    using pre_to_univ_poly_is_hom(5)[of 0 "Suc n" "pre_to_univ_poly (Suc n) 0" c] assms unfolding coord_ring_def 
+    unfolding to_univ_poly_def
+    using pre_to_univ_poly_is_hom(5)[of 0 "Suc n" "pre_to_univ_poly (Suc n) 0" c] assms unfolding coord_ring_def
     using diff_Suc_1 zero_less_Suc by presburger
   hence 1: "to_univ_poly (Suc n) 0 (Qp.indexed_const c) = to_polynomial (Q\<^sub>p[\<X>\<^bsub>n\<^esub>]) (Qp.indexed_const c) "
-    unfolding to_univ_poly_def 
-    using UP_cring.IP_to_UP_indexed_const[of "Q\<^sub>p[\<X>\<^bsub>n\<^esub>]" "Qp.indexed_const c" "0::nat"] assms 
+    unfolding to_univ_poly_def
+    using UP_cring.IP_to_UP_indexed_const[of "Q\<^sub>p[\<X>\<^bsub>n\<^esub>]" "Qp.indexed_const c" "0::nat"] assms
           comp_apply[of "IP_to_UP (0::nat)" "pre_to_univ_poly (Suc n) (0::nat)" "Qp.indexed_const c"]
     unfolding UP_cring_def using Qp_to_IP_car coord_cring_cring by presburger
   have 2: "Qp_ev_hom n (Qp.indexed_const c) = \<cc>\<^bsub>n\<^esub> c"
-    using Qp_ev_hom_constant[of c n] assms by blast 
+    using Qp_ev_hom_constant[of c n] assms by blast
   have 3: "poly_lift_hom (Q\<^sub>p [\<X>\<^bsub>n\<^esub>]) (SA n) (Qp_ev_hom n) (to_polynomial (Q\<^sub>p [\<X>\<^bsub>n\<^esub>]) (Qp.indexed_const c)) = to_polynomial (SA n) (Qp_ev_hom n (Qp.indexed_const c))"
     using UP_cring.poly_lift_hom_extends_hom[of "Q\<^sub>p[\<X>\<^bsub>n\<^esub>]" "SA n" "Qp_ev_hom n" "Qp.indexed_const c"]
     unfolding UP_cring_def coord_ring_def coord_ring_to_UP_SA_def
     by (metis Qp_ev_hom_is_hom Qp_to_IP_car SA_is_cring assms(1) coord_cring_cring coord_ring_def)
   hence 4: "poly_lift_hom (Q\<^sub>p [\<X>\<^bsub>n\<^esub>]) (SA n) (Qp_ev_hom n) (to_univ_poly (Suc n) 0 (Qp.indexed_const c) ) = to_polynomial (SA n) (\<cc>\<^bsub>n\<^esub> c)"
     using "1" "2" by presburger
-  show ?thesis 
-    using assms 4 Qp.indexed_const_closed[of c "{..<n}"] 
-          comp_apply[of "poly_lift_hom (Pring Q\<^sub>p {..<n}) (SA n) (Qp_ev_hom n)" "to_univ_poly (Suc n) 0" "Qp.indexed_const c"] 
+  show ?thesis
+    using assms 4 Qp.indexed_const_closed[of c "{..<n}"]
+          comp_apply[of "poly_lift_hom (Pring Q\<^sub>p {..<n}) (SA n) (Qp_ev_hom n)" "to_univ_poly (Suc n) 0" "Qp.indexed_const c"]
     unfolding coord_ring_to_UP_SA_def
     by (metis coord_ring_def)
 qed
 
 lemma coord_ring_to_UP_SA_pvar_0:
   shows "coord_ring_to_UP_SA n (pvar Q\<^sub>p 0) = up_ring.monom (UP (SA n)) \<one>\<^bsub>SA n\<^esub> 1"
 proof-
   have 0: "pre_to_univ_poly (Suc n) 0 (pvar Q\<^sub>p 0) = pvar (Q\<^sub>p[\<X>\<^bsub>n\<^esub>]) 0"
-    using pre_to_univ_poly_is_hom(3)[of 0 "Suc n" "pre_to_univ_poly (Suc n) 0"] diff_Suc_1 zero_less_Suc 
+    using pre_to_univ_poly_is_hom(3)[of 0 "Suc n" "pre_to_univ_poly (Suc n) 0"] diff_Suc_1 zero_less_Suc
     by presburger
   have 1: "IP_to_UP (0::nat) (pvar (Q\<^sub>p[\<X>\<^bsub>n\<^esub>]) 0) = up_ring.monom (UP (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])) \<one>\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> 1"
     using cring.IP_to_UP_var[of "Q\<^sub>p[\<X>\<^bsub>n\<^esub>]" "0::nat"] unfolding X_poly_def var_to_IP_def
     using coord_cring_cring by blast
   have 2: "to_univ_poly (Suc n) 0 (pvar Q\<^sub>p 0) = up_ring.monom (UP (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])) \<one>\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> 1"
-    unfolding to_univ_poly_def using 0 1 comp_apply 
+    unfolding to_univ_poly_def using 0 1 comp_apply
     by metis
-  have 3: "poly_lift_hom (Q\<^sub>p[\<X>\<^bsub>n\<^esub>]) (SA n) (Qp_ev_hom n) (up_ring.monom (UP (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])) \<one>\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> 1) 
+  have 3: "poly_lift_hom (Q\<^sub>p[\<X>\<^bsub>n\<^esub>]) (SA n) (Qp_ev_hom n) (up_ring.monom (UP (Q\<^sub>p[\<X>\<^bsub>n\<^esub>])) \<one>\<^bsub>Q\<^sub>p[\<X>\<^bsub>n\<^esub>]\<^esub> 1)
              = up_ring.monom (UP (SA n)) \<one>\<^bsub>SA n\<^esub> 1"
     using UP_cring.poly_lift_hom_monom[of "Q\<^sub>p[\<X>\<^bsub>n\<^esub>]" "SA n" "Qp_ev_hom n"]
     unfolding UP_cring_def
     using MP.one_closed Qp_ev_hom_is_hom Qp_ev_hom_one SA_is_cring coord_cring_cring by presburger
-  thus ?thesis unfolding coord_ring_to_UP_SA_def  
-    using 2 3  comp_apply 
+  thus ?thesis unfolding coord_ring_to_UP_SA_def
+    using 2 3  comp_apply
     by metis
 qed
 
 lemma coord_ring_to_UP_SA_pvar_Suc:
   assumes "i > 0"
   assumes "i < Suc n"
   shows "coord_ring_to_UP_SA n (pvar Q\<^sub>p i) = to_polynomial (SA n) (\<vv>\<^bsub>n, i-1\<^esub>)"
 proof-
   have 0: "pre_to_univ_poly (Suc n) 0 (pvar Q\<^sub>p i) = ring.indexed_const (Q\<^sub>p[\<X>\<^bsub>n\<^esub>]) (pvar Q\<^sub>p (i-1))"
-    using pre_to_univ_poly_is_hom(4)[of 0 "Suc n" "pre_to_univ_poly (Suc n) 0" i] diff_Suc_1 zero_less_Suc 
+    using pre_to_univ_poly_is_hom(4)[of 0 "Suc n" "pre_to_univ_poly (Suc n) 0" i] diff_Suc_1 zero_less_Suc
           assms
-    unfolding coord_ring_def by presburger 
+    unfolding coord_ring_def by presburger
   have 1: "IP_to_UP (0::nat) (ring.indexed_const (Q\<^sub>p[\<X>\<^bsub>n\<^esub>]) (pvar Q\<^sub>p (i-1))) = to_polynomial (Q\<^sub>p[\<X>\<^bsub>n\<^esub>]) (pvar Q\<^sub>p (i-1))"
     using UP_cring.IP_to_UP_indexed_const[of "Q\<^sub>p[\<X>\<^bsub>n\<^esub>]" "pvar Q\<^sub>p (i-1)" "0::nat"] coord_cring_cring
-    unfolding UP_cring_def  
-    by (metis assms diff_less less_Suc_eq less_imp_diff_less less_numeral_extra(1) local.pvar_closed)    
+    unfolding UP_cring_def
+    by (metis assms diff_less less_Suc_eq less_imp_diff_less less_numeral_extra(1) local.pvar_closed)
   have 2: "to_univ_poly (Suc n) 0 (pvar Q\<^sub>p i) = to_polynomial (Q\<^sub>p[\<X>\<^bsub>n\<^esub>]) (pvar Q\<^sub>p (i-1))"
-    unfolding to_univ_poly_def using 0 1 comp_apply 
+    unfolding to_univ_poly_def using 0 1 comp_apply
     by metis
-  have 3: "poly_lift_hom (Q\<^sub>p[\<X>\<^bsub>n\<^esub>]) (SA n) (Qp_ev_hom n) (to_polynomial (Q\<^sub>p[\<X>\<^bsub>n\<^esub>]) (pvar Q\<^sub>p (i-1))) 
+  have 3: "poly_lift_hom (Q\<^sub>p[\<X>\<^bsub>n\<^esub>]) (SA n) (Qp_ev_hom n) (to_polynomial (Q\<^sub>p[\<X>\<^bsub>n\<^esub>]) (pvar Q\<^sub>p (i-1)))
              = to_polynomial (SA n) (\<vv>\<^bsub>n, (i-1)\<^esub>)"
     using UP_cring.poly_lift_hom_extends_hom[of "Q\<^sub>p[\<X>\<^bsub>n\<^esub>]" "SA n" "Qp_ev_hom n" "pvar Q\<^sub>p (i-1)"]
-    unfolding UP_cring_def 
-    by (metis (no_types, lifting) Qp_ev_hom_is_hom Qp_ev_hom_pvar SA_is_cring Suc_diff_1 
-        Suc_less_eq assms  coord_cring_cring local.pvar_closed) 
-  thus ?thesis unfolding coord_ring_to_UP_SA_def  
-    using 2 3 assms comp_apply 
+    unfolding UP_cring_def
+    by (metis (no_types, lifting) Qp_ev_hom_is_hom Qp_ev_hom_pvar SA_is_cring Suc_diff_1
+        Suc_less_eq assms  coord_cring_cring local.pvar_closed)
+  thus ?thesis unfolding coord_ring_to_UP_SA_def
+    using 2 3 assms comp_apply
     by metis
 qed
- 
+
 lemma coord_ring_to_UP_SA_eval:
   assumes "f \<in> carrier (Q\<^sub>p[\<X>\<^bsub>Suc n\<^esub>])"
   assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
   assumes "t \<in> carrier Q\<^sub>p"
   shows "Qp_ev f (t#a) = ((SA_poly_to_Qp_poly n a (coord_ring_to_UP_SA n f))) \<bullet> t"
 proof(rule coord_ring_car_induct[of f "Suc n"])
   have ta_closed: "t # a \<in> carrier (Q\<^sub>p\<^bsup>Suc n\<^esup>)"
     using assms cartesian_power_cons  by (metis Suc_eq_plus1)
   show "f \<in> carrier (Q\<^sub>p [\<X>\<^bsub>Suc n\<^esub>])"
     by (simp add: assms)
   show "\<And>c. c \<in> carrier Q\<^sub>p \<Longrightarrow> eval_at_point Q\<^sub>p (t # a) (Qp.indexed_const c) = SA_poly_to_Qp_poly n a (coord_ring_to_UP_SA n (Qp.indexed_const c)) \<bullet> t"
   proof- fix c assume A: "c \<in> carrier Q\<^sub>p"
     have 0: "eval_at_point Q\<^sub>p (t # a) (Qp.indexed_const c) = c"
       using A eval_at_point_const ta_closed by blast
     have 1: "SA_poly_to_Qp_poly n a (coord_ring_to_UP_SA n (Qp.indexed_const c)) = to_polynomial Q\<^sub>p c"
-      using coord_ring_to_UP_SA_constant[of c n]  A Qp_constE SA_car 
-            SA_poly_to_Qp_poly_extends_apply assms  constant_function_in_semialg_functions 
+      using coord_ring_to_UP_SA_constant[of c n]  A Qp_constE SA_car
+            SA_poly_to_Qp_poly_extends_apply assms  constant_function_in_semialg_functions
       by presburger
     show "eval_at_point Q\<^sub>p (t # a) (Qp.indexed_const c) = SA_poly_to_Qp_poly n a (coord_ring_to_UP_SA n (Qp.indexed_const c)) \<bullet> t"
       using 0 1 A UPQ.to_fun_to_poly assms by presburger
-  qed 
+  qed
   show " \<And>p q. p \<in> carrier (Q\<^sub>p [\<X>\<^bsub>Suc n\<^esub>]) \<Longrightarrow>
            q \<in> carrier (Q\<^sub>p [\<X>\<^bsub>Suc n\<^esub>]) \<Longrightarrow>
            eval_at_point Q\<^sub>p (t # a) p = SA_poly_to_Qp_poly n a (coord_ring_to_UP_SA n p) \<bullet> t \<Longrightarrow>
            eval_at_point Q\<^sub>p (t # a) q = SA_poly_to_Qp_poly n a (coord_ring_to_UP_SA n q) \<bullet> t \<Longrightarrow>
            eval_at_point Q\<^sub>p (t # a) (p \<oplus>\<^bsub>Q\<^sub>p [\<X>\<^bsub>Suc n\<^esub>]\<^esub> q) = SA_poly_to_Qp_poly n a (coord_ring_to_UP_SA n (p \<oplus>\<^bsub>Q\<^sub>p [\<X>\<^bsub>Suc n\<^esub>]\<^esub> q)) \<bullet> t"
   proof- fix p q assume A: "p \<in> carrier (Q\<^sub>p [\<X>\<^bsub>Suc n\<^esub>])" "q \<in> carrier (Q\<^sub>p [\<X>\<^bsub>Suc n\<^esub>])"
            "eval_at_point Q\<^sub>p (t # a) p = SA_poly_to_Qp_poly n a (coord_ring_to_UP_SA n p) \<bullet> t"
            "eval_at_point Q\<^sub>p (t # a) q = SA_poly_to_Qp_poly n a (coord_ring_to_UP_SA n q) \<bullet> t"
     have 0: "eval_at_point Q\<^sub>p (t # a) (p \<oplus>\<^bsub>Q\<^sub>p [\<X>\<^bsub>Suc n\<^esub>]\<^esub> q) = eval_at_point Q\<^sub>p (t # a) p \<oplus> eval_at_point Q\<^sub>p (t # a) q"
       using A(1) A(2) eval_at_point_add ta_closed by blast
-    have 1: "SA_poly_to_Qp_poly n a (coord_ring_to_UP_SA n (p \<oplus>\<^bsub>Q\<^sub>p [\<X>\<^bsub>Suc n\<^esub>]\<^esub> q))= 
+    have 1: "SA_poly_to_Qp_poly n a (coord_ring_to_UP_SA n (p \<oplus>\<^bsub>Q\<^sub>p [\<X>\<^bsub>Suc n\<^esub>]\<^esub> q))=
     SA_poly_to_Qp_poly n a (coord_ring_to_UP_SA n p)  \<oplus>\<^bsub>UP Q\<^sub>p\<^esub> SA_poly_to_Qp_poly n a (coord_ring_to_UP_SA n q)"
       using coord_ring_to_UP_SA_add[of ] assms coord_ring_to_UP_SA_closed A
-            SA_poly_to_Qp_poly_add[of  a n "coord_ring_to_UP_SA n p" "coord_ring_to_UP_SA n q"] 
+            SA_poly_to_Qp_poly_add[of  a n "coord_ring_to_UP_SA n p" "coord_ring_to_UP_SA n q"]
       by presburger
     show "eval_at_point Q\<^sub>p (t # a) (p \<oplus>\<^bsub>Q\<^sub>p [\<X>\<^bsub>Suc n\<^esub>]\<^esub> q) = SA_poly_to_Qp_poly n a (coord_ring_to_UP_SA n (p \<oplus>\<^bsub>Q\<^sub>p [\<X>\<^bsub>Suc n\<^esub>]\<^esub> q)) \<bullet> t"
-      using 0 1  assms SA_poly_to_Qp_poly_closed[of a n] SA_poly_to_Qp_poly_closed A(1) A(2) A(3) A(4) UPQ.to_fun_plus coord_ring_to_UP_SA_closed 
+      using 0 1  assms SA_poly_to_Qp_poly_closed[of a n] SA_poly_to_Qp_poly_closed A(1) A(2) A(3) A(4) UPQ.to_fun_plus coord_ring_to_UP_SA_closed
       by presburger
   qed
   show "\<And>p i. p \<in> carrier (Q\<^sub>p [\<X>\<^bsub>Suc n\<^esub>]) \<Longrightarrow>
            i < Suc n \<Longrightarrow>
            eval_at_point Q\<^sub>p (t # a) p = SA_poly_to_Qp_poly n a (coord_ring_to_UP_SA n p) \<bullet> t \<Longrightarrow>
            eval_at_point Q\<^sub>p (t # a) (p \<otimes>\<^bsub>Q\<^sub>p [\<X>\<^bsub>Suc n\<^esub>]\<^esub> pvar Q\<^sub>p i) = SA_poly_to_Qp_poly n a (coord_ring_to_UP_SA n (p \<otimes>\<^bsub>Q\<^sub>p [\<X>\<^bsub>Suc n\<^esub>]\<^esub> pvar Q\<^sub>p i)) \<bullet> t"
   proof- fix p i assume A: "p \<in> carrier (Q\<^sub>p [\<X>\<^bsub>Suc n\<^esub>])" "i < Suc n"
            "eval_at_point Q\<^sub>p (t # a) p = SA_poly_to_Qp_poly n a (coord_ring_to_UP_SA n p) \<bullet> t"
     show " eval_at_point Q\<^sub>p (t # a) (p \<otimes>\<^bsub>Q\<^sub>p [\<X>\<^bsub>Suc n\<^esub>]\<^esub> pvar Q\<^sub>p i) = SA_poly_to_Qp_poly n a (coord_ring_to_UP_SA n (p \<otimes>\<^bsub>Q\<^sub>p [\<X>\<^bsub>Suc n\<^esub>]\<^esub> pvar Q\<^sub>p i)) \<bullet> t"
     proof-
       have 0: "eval_at_point Q\<^sub>p (t # a) (p \<otimes>\<^bsub>Q\<^sub>p [\<X>\<^bsub>Suc n\<^esub>]\<^esub> pvar Q\<^sub>p i) = eval_at_point Q\<^sub>p (t # a) p \<otimes>  eval_at_point Q\<^sub>p (t # a) (pvar Q\<^sub>p i)"
         using A(1) A(2) eval_at_point_mult local.pvar_closed ta_closed by blast
       have 1: "coord_ring_to_UP_SA n (p \<otimes>\<^bsub>Q\<^sub>p [\<X>\<^bsub>Suc n\<^esub>]\<^esub> pvar Q\<^sub>p i) = coord_ring_to_UP_SA n p \<otimes>\<^bsub>UP (SA n)\<^esub> coord_ring_to_UP_SA n (pvar Q\<^sub>p i)"
         using A(1) A(2) assms(1) coord_ring_to_UP_SA_mult local.pvar_closed by blast
       hence 2: "SA_poly_to_Qp_poly n a (coord_ring_to_UP_SA n (p \<otimes>\<^bsub>Q\<^sub>p [\<X>\<^bsub>Suc n\<^esub>]\<^esub> pvar Q\<^sub>p i)) =
               SA_poly_to_Qp_poly n a (coord_ring_to_UP_SA n p) \<otimes>\<^bsub>UP Q\<^sub>p\<^esub>SA_poly_to_Qp_poly n a (coord_ring_to_UP_SA n (pvar Q\<^sub>p i))"
-        using SA_poly_to_Qp_poly_mult coord_ring_to_UP_SA_closed A(1) A(2) assms local.pvar_closed 
+        using SA_poly_to_Qp_poly_mult coord_ring_to_UP_SA_closed A(1) A(2) assms local.pvar_closed
         by presburger
       show "eval_at_point Q\<^sub>p (t # a) (p \<otimes>\<^bsub>Q\<^sub>p [\<X>\<^bsub>Suc n\<^esub>]\<^esub> pvar Q\<^sub>p i) = SA_poly_to_Qp_poly n a (coord_ring_to_UP_SA n (p \<otimes>\<^bsub>Q\<^sub>p [\<X>\<^bsub>Suc n\<^esub>]\<^esub> pvar Q\<^sub>p i)) \<bullet> t"
       proof(cases  "i = 0")
         case True
         have T0: "SA_poly_to_Qp_poly n a (coord_ring_to_UP_SA n (p \<otimes>\<^bsub>Q\<^sub>p [\<X>\<^bsub>Suc n\<^esub>]\<^esub> pvar Q\<^sub>p i)) =
               SA_poly_to_Qp_poly n a (coord_ring_to_UP_SA n p) \<otimes>\<^bsub>UP Q\<^sub>p\<^esub> up_ring.monom (UP Q\<^sub>p) \<one> 1"
-          using True coord_ring_to_UP_SA_pvar_0 SA_poly_to_Qp_poly_monom  
+          using True coord_ring_to_UP_SA_pvar_0 SA_poly_to_Qp_poly_monom
           by (metis "2" function_one_eval Qp.one_closed SA_car SA_one assms constant_function_in_semialg_functions function_one_as_constant)
         have T1: "eval_at_point Q\<^sub>p (t # a) (p \<otimes>\<^bsub>Q\<^sub>p [\<X>\<^bsub>Suc n\<^esub>]\<^esub> pvar Q\<^sub>p i) = eval_at_point Q\<^sub>p (t # a) p \<otimes>  t"
-          using 0 True ta_closed eval_pvar[of 0 "Suc n" "t#a"] 
+          using 0 True ta_closed eval_pvar[of 0 "Suc n" "t#a"]
           by (metis A(2) nth_Cons_0)
-        then show ?thesis 
+        then show ?thesis
           using T0 A SA_poly_to_Qp_poly_closed[of a n  "coord_ring_to_UP_SA n p"] UPQ.to_fun_X[of t] to_fun_mult
-                coord_ring_to_UP_SA_closed[of  ] UPQ.X_closed[of ] unfolding X_poly_def 
-          using assms UPQ.to_fun_mult by presburger                 
+                coord_ring_to_UP_SA_closed[of  ] UPQ.X_closed[of ] unfolding X_poly_def
+          using assms UPQ.to_fun_mult by presburger
       next
         case False
         have "\<vv>\<^bsub>n, i - 1\<^esub> a = a!(i-1)"
           by (metis A(2) False Qp.varE Suc_diff_1 Suc_less_eq assms  less_Suc_eq_0_disj)
         hence F0: "SA_poly_to_Qp_poly n a (coord_ring_to_UP_SA n (p \<otimes>\<^bsub>Q\<^sub>p [\<X>\<^bsub>Suc n\<^esub>]\<^esub> pvar Q\<^sub>p i)) =
               SA_poly_to_Qp_poly n a (coord_ring_to_UP_SA n p) \<otimes>\<^bsub>UP Q\<^sub>p\<^esub> to_polynomial Q\<^sub>p (a!(i-1))"
-          using False coord_ring_to_UP_SA_pvar_Suc[of i n] SA_poly_to_Qp_poly_extends_apply[of a n "\<vv>\<^bsub>n, i - 1\<^esub>"] 
+          using False coord_ring_to_UP_SA_pvar_Suc[of i n] SA_poly_to_Qp_poly_extends_apply[of a n "\<vv>\<^bsub>n, i - 1\<^esub>"]
           by (metis (no_types, lifting) "2" A(2) Qp_ev_hom_in_SA Qp_ev_hom_pvar Suc_diff_1 Suc_less_eq assms  local.pvar_closed neq0_conv)
         have F1: "eval_at_point Q\<^sub>p (t # a) (p \<otimes>\<^bsub>Q\<^sub>p [\<X>\<^bsub>Suc n\<^esub>]\<^esub> pvar Q\<^sub>p i) = eval_at_point Q\<^sub>p (t # a) p \<otimes> (a!(i-1))"
-          using 0 False ta_closed eval_pvar[of i "Suc n" "t#a"] 
+          using 0 False ta_closed eval_pvar[of i "Suc n" "t#a"]
           by (metis A(2) nth_Cons')
-        then show ?thesis 
+        then show ?thesis
           using F0 A SA_poly_to_Qp_poly_closed[of a n "coord_ring_to_UP_SA n p"] to_fun_mult
-                coord_ring_to_UP_SA_closed[of p n]   False UPQ.to_fun_to_poly UPQ.to_poly_closed assms 
+                coord_ring_to_UP_SA_closed[of p n]   False UPQ.to_fun_to_poly UPQ.to_poly_closed assms
               eval_at_point_closed eval_pvar local.pvar_closed neq0_conv nth_Cons_pos ta_closed
           by (metis (no_types, lifting) UPQ.to_fun_mult)
       qed
     qed
   qed
 qed
 
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsubsection\<open>Gluing Semialgebraic Polynomials\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 
-definition SA_poly_glue where 
+definition SA_poly_glue where
 "SA_poly_glue m S f g = (\<lambda> n. fun_glue m S (f n) (g n))"
 
-lemma SA_poly_glue_closed: 
+lemma SA_poly_glue_closed:
   assumes "f \<in> carrier (UP (SA m))"
   assumes "g \<in> carrier (UP (SA m))"
   assumes "is_semialgebraic m S"
   shows "SA_poly_glue m S f g \<in> carrier (UP (SA m))"
 proof(rule UP_car_memI[of "max (deg (SA m) f) (deg (SA m) g)"])
   fix n assume A: "max (deg (SA m) f) (deg (SA m) g) < n" show "SA_poly_glue m S f g n = \<zero>\<^bsub>SA m\<^esub>"
     unfolding SA_poly_glue_def
-  proof- 
+  proof-
     have 0: "n > (deg (SA m) f)"
       using A by simp
     have 1: "n > (deg (SA m) g)"
       using A by simp
     have 2: "f n = \<zero>\<^bsub>SA m\<^esub>"
       using 0 assms UPSA.deg_leE by blast
     have 3: "g n = \<zero>\<^bsub>SA m\<^esub>"
       using 1 assms UPSA.deg_leE by blast
     show "fun_glue m S (f n) (g n) = \<zero>\<^bsub>SA m\<^esub>"
       unfolding SA_zero function_ring_def ring_record_simps function_zero_def 2 3
-    proof fix x 
+    proof fix x
       show " fun_glue m S (\<lambda>x\<in>carrier (Q\<^sub>p\<^bsup>m\<^esup>). \<zero>) (\<lambda>x\<in>carrier (Q\<^sub>p\<^bsup>m\<^esup>). \<zero>) x = (\<lambda>x\<in>carrier (Q\<^sub>p\<^bsup>m\<^esup>). \<zero>) x"
         apply(cases "x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)")
         unfolding fun_glue_def restrict_def apply presburger
-        by auto  
+        by auto
     qed
   qed
   next
   show " \<And>n. SA_poly_glue m S f g n \<in> carrier (SA m)"
     unfolding SA_poly_glue_def apply(rule fun_glue_closed)
     by(rule cfs_closed, rule assms, rule cfs_closed, rule assms, rule assms)
 qed
 
-lemma SA_poly_glue_deg: 
+lemma SA_poly_glue_deg:
   assumes "f \<in> carrier (UP (SA m))"
   assumes "g \<in> carrier (UP (SA m))"
   assumes "is_semialgebraic m S"
   assumes "deg (SA m) f \<le> d"
   assumes "deg (SA m) g \<le> d"
   shows "deg (SA m) (SA_poly_glue m S f g) \<le> d"
   apply(rule deg_leqI,  rule SA_poly_glue_closed, rule assms, rule assms, rule assms)
 proof- fix n assume A: "d < n"
   show "SA_poly_glue m S f g n = \<zero>\<^bsub>SA m\<^esub>"
     unfolding SA_poly_glue_def
-  proof- 
+  proof-
     have 0: "n > (deg (SA m) f)"
       using A assms by linarith
     have 1: "n > (deg (SA m) g)"
       using A assms by linarith
     have 2: "f n = \<zero>\<^bsub>SA m\<^esub>"
       using 0 assms UPSA.deg_leE by blast
     have 3: "g n = \<zero>\<^bsub>SA m\<^esub>"
       using 1 assms UPSA.deg_leE by blast
     show "fun_glue m S (f n) (g n) = \<zero>\<^bsub>SA m\<^esub>"
       unfolding SA_zero function_ring_def ring_record_simps function_zero_def 2 3
-    proof fix x 
+    proof fix x
       show " fun_glue m S (\<lambda>x\<in>carrier (Q\<^sub>p\<^bsup>m\<^esup>). \<zero>) (\<lambda>x\<in>carrier (Q\<^sub>p\<^bsup>m\<^esup>). \<zero>) x = (\<lambda>x\<in>carrier (Q\<^sub>p\<^bsup>m\<^esup>). \<zero>) x"
         apply(cases "x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)")
         unfolding fun_glue_def restrict_def apply presburger
-        by auto  
+        by auto
     qed
   qed
 qed
 
-lemma UP_SA_cfs_closed: 
+lemma UP_SA_cfs_closed:
   assumes "g \<in> carrier (UP (SA m))"
   shows "g k \<in> carrier (SA m)"
-  using assms UP_ring.cfs_closed[of "SA m" g k] SA_is_ring[of m] unfolding UP_ring_def 
-  by blast 
-
-
-lemma SA_poly_glue_cfs1: 
+  using assms UP_ring.cfs_closed[of "SA m" g k] SA_is_ring[of m] unfolding UP_ring_def
+  by blast
+
+
+lemma SA_poly_glue_cfs1:
   assumes "f \<in> carrier (UP (SA m))"
   assumes "g \<in> carrier (UP (SA m))"
   assumes "is_semialgebraic m S"
   assumes "x \<in> S"
   shows "(SA_poly_glue m S f g) n x = f n x"
-  unfolding SA_poly_glue_def fun_glue_def restrict_def 
-  using assms 
+  unfolding SA_poly_glue_def fun_glue_def restrict_def
+  using assms
   by (metis SA_car local.function_ring_not_car padic_fields.UP_SA_cfs_closed padic_fields_axioms semialg_functions_memE(2))
 
 lemma SA_poly_glue_cfs2:
   assumes "f \<in> carrier (UP (SA m))"
   assumes "g \<in> carrier (UP (SA m))"
   assumes "is_semialgebraic m S"
   assumes "x \<notin> S"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
-  shows "(SA_poly_glue m S f g) n x = g n x" 
-  unfolding SA_poly_glue_def fun_glue_def restrict_def 
+  shows "(SA_poly_glue m S f g) n x = g n x"
+  unfolding SA_poly_glue_def fun_glue_def restrict_def
   using assms by meson
 
 lemma SA_poly_glue_to_Qp_poly1:
   assumes "f \<in> carrier (UP (SA m))"
   assumes "g \<in> carrier (UP (SA m))"
   assumes "is_semialgebraic m S"
   assumes "x \<in> S"
   shows "SA_poly_to_Qp_poly m x (SA_poly_glue m S f g) = SA_poly_to_Qp_poly m x f"
-proof fix n 
+proof fix n
   have x_closed: "x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
     using assms is_semialgebraic_closed by blast
   have 0: "SA_poly_to_Qp_poly m x (SA_poly_glue m S f g) n = (SA_poly_glue m S f g) n x"
     by(rule SA_poly_to_Qp_poly_coeff[of x m "SA_poly_glue m S f g"], rule x_closed, rule SA_poly_glue_closed
           , rule assms, rule assms, rule assms)
   have 1: "(SA_poly_glue m S f g) n x = f n x"
     by(rule SA_poly_glue_cfs1 , rule assms, rule assms, rule assms, rule assms)
   show "SA_poly_to_Qp_poly m x (SA_poly_glue m S f g) n = SA_poly_to_Qp_poly m x f n"
-    unfolding 0 1 using SA_poly_to_Qp_poly_coeff[of x m f n] assms (1) x_closed by blast 
+    unfolding 0 1 using SA_poly_to_Qp_poly_coeff[of x m f n] assms (1) x_closed by blast
 qed
 
 lemma SA_poly_glue_to_Qp_poly2:
   assumes "f \<in> carrier (UP (SA m))"
   assumes "g \<in> carrier (UP (SA m))"
   assumes "is_semialgebraic m S"
   assumes "x \<notin> S"
   assumes "x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
   shows "SA_poly_to_Qp_poly m x (SA_poly_glue m S f g) = SA_poly_to_Qp_poly m x g"
-proof fix n 
+proof fix n
   have x_closed: "x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
     using assms is_semialgebraic_closed by blast
   have 0: "SA_poly_to_Qp_poly m x (SA_poly_glue m S f g) n = (SA_poly_glue m S f g) n x"
     by(rule SA_poly_to_Qp_poly_coeff[of x m "SA_poly_glue m S f g"], rule x_closed, rule SA_poly_glue_closed
           , rule assms, rule assms, rule assms)
   have 1: "(SA_poly_glue m S f g) n x = g n x"
     by(rule SA_poly_glue_cfs2 , rule assms, rule assms, rule assms, rule assms, rule x_closed)
   show "SA_poly_to_Qp_poly m x (SA_poly_glue m S f g) n = SA_poly_to_Qp_poly m x g n"
-    unfolding 0 1 using SA_poly_to_Qp_poly_coeff[of x m g n] assms x_closed by blast 
-qed
- 
+    unfolding 0 1 using SA_poly_to_Qp_poly_coeff[of x m g n] assms x_closed by blast
+qed
+
 
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsubsection\<open>Polynomials over the Valuation Ring\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 
 definition integral_on where
 "integral_on m B = {f \<in> carrier (UP (SA m)). (\<forall>x \<in> B. \<forall>i. SA_poly_to_Qp_poly m x f i \<in> \<O>\<^sub>p)}"
 
-lemma integral_on_memI: 
+lemma integral_on_memI:
   assumes "f \<in> carrier (UP (SA m))"
   assumes "\<And>x i. x \<in> B \<Longrightarrow> SA_poly_to_Qp_poly m x f i \<in> \<O>\<^sub>p"
   shows "f \<in> integral_on m B"
   unfolding integral_on_def mem_Collect_eq using assms by blast
 
-lemma integral_on_memE: 
+lemma integral_on_memE:
   assumes "f \<in> integral_on m B"
   shows  "f \<in> carrier (UP (SA m))"
           "\<And>x. x \<in> B \<Longrightarrow> SA_poly_to_Qp_poly m x f i \<in> \<O>\<^sub>p"
- using assms unfolding integral_on_def mem_Collect_eq apply blast 
- using assms unfolding integral_on_def mem_Collect_eq by blast 
+ using assms unfolding integral_on_def mem_Collect_eq apply blast
+ using assms unfolding integral_on_def mem_Collect_eq by blast
 
 lemma one_integral_on:
   assumes "B \<subseteq> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
   shows "\<one> \<^bsub>UP (SA m)\<^esub> \<in> integral_on m B"
   apply(rule integral_on_memI)
   apply blast
 proof- fix x i assume A: "x \<in> B"
   have 0: "SA_poly_to_Qp_poly m x \<one>\<^bsub>UP (SA m)\<^esub> = \<one>\<^bsub>UP Q\<^sub>p\<^esub>"
     apply(rule ring_hom_one[of _ "UP (SA m)"])
-    using SA_poly_to_Qp_poly_is_hom[of x m] A assms by blast 
+    using SA_poly_to_Qp_poly_is_hom[of x m] A assms by blast
   show "SA_poly_to_Qp_poly m x \<one>\<^bsub>UP (SA m)\<^esub> i \<in> \<O>\<^sub>p"
-    unfolding 0 
+    unfolding 0
     apply(rule val_ring_memI)
      apply(rule UPQ.cfs_closed)
      apply blast
     apply(cases "i = 0")
      apply (metis Qp.add.nat_pow_eone Qp.one_closed UPQ.cfs_one val_of_nat_inc val_one)
     by (metis Qp.int_inc_zero UPQ.cfs_one val_of_int_inc)
 qed
 
 lemma integral_on_plus:
   assumes "B \<subseteq> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
   assumes "f \<in> integral_on m B"
   assumes "g \<in> integral_on m B"
   shows "f \<oplus>\<^bsub>UP (SA m)\<^esub> g \<in> integral_on m B"
 proof(rule integral_on_memI)
   show "f \<oplus>\<^bsub>UP (SA m)\<^esub> g \<in> carrier (UP (SA m))"
     using assms integral_on_memE by blast
   show "\<And>x i. x \<in> B \<Longrightarrow> SA_poly_to_Qp_poly m x (f \<oplus>\<^bsub>UP (SA m)\<^esub> g) i \<in> \<O>\<^sub>p"
   proof- fix x i assume A: "x \<in> B"
     have 0: "SA_poly_to_Qp_poly m x (f \<oplus>\<^bsub>UP (SA m)\<^esub> g) = SA_poly_to_Qp_poly m x f
                                                           \<oplus>\<^bsub>UP Q\<^sub>p\<^esub>  SA_poly_to_Qp_poly m x g"
       apply(rule SA_poly_to_Qp_poly_add)
       using A assms apply blast using assms integral_on_memE apply blast
       using assms integral_on_memE by blast
     have 1: "SA_poly_to_Qp_poly m x (f \<oplus>\<^bsub>UP (SA m)\<^esub> g) i = SA_poly_to_Qp_poly m x f i
                                                           \<oplus>  SA_poly_to_Qp_poly m x g i"
       unfolding 0 apply(rule UPQ.cfs_add)
        apply(rule SA_poly_to_Qp_poly_closed)
       using  A assms apply blast
       using assms integral_on_memE  apply blast
        apply(rule SA_poly_to_Qp_poly_closed)
       using  A assms apply blast
       using assms integral_on_memE  by blast
     show "SA_poly_to_Qp_poly m x (f \<oplus>\<^bsub>UP (SA m)\<^esub> g) i \<in> \<O>\<^sub>p"
-      unfolding 1 using assms integral_on_memE 
+      unfolding 1 using assms integral_on_memE
       using A val_ring_add_closed by presburger
   qed
 qed
 
 lemma integral_on_times:
   assumes "B \<subseteq> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
   assumes "f \<in> integral_on m B"
   assumes "g \<in> integral_on m B"
   shows "f \<otimes>\<^bsub>UP (SA m)\<^esub> g \<in> integral_on m B"
 proof(rule integral_on_memI)
   show "f \<otimes>\<^bsub>UP (SA m)\<^esub> g \<in> carrier (UP (SA m))"
     using assms integral_on_memE by blast
   show "\<And>x i. x \<in> B \<Longrightarrow> SA_poly_to_Qp_poly m x (f \<otimes>\<^bsub>UP (SA m)\<^esub> g) i \<in> \<O>\<^sub>p"
   proof- fix x i assume A: "x \<in> B"
     have 0: "SA_poly_to_Qp_poly m x (f \<otimes>\<^bsub>UP (SA m)\<^esub> g) = SA_poly_to_Qp_poly m x f
                                                           \<otimes>\<^bsub>UP Q\<^sub>p\<^esub>  SA_poly_to_Qp_poly m x g"
       apply(rule SA_poly_to_Qp_poly_mult)
       using A assms apply blast using assms integral_on_memE apply blast
       using assms integral_on_memE by blast
     obtain S where S_def: "S = UP (Q\<^sub>p \<lparr> carrier := \<O>\<^sub>p \<rparr>)"
-      by blast 
+      by blast
     have 1: "cring S"
-      unfolding S_def apply(rule UPQ.UP_ring_subring_is_ring) 
+      unfolding S_def apply(rule UPQ.UP_ring_subring_is_ring)
       using val_ring_subring by blast
     have 2: " carrier S = {h \<in> carrier (UP Q\<^sub>p). \<forall>n. h n \<in> \<O>\<^sub>p}"
       unfolding S_def  using UPQ.UP_ring_subring_car[of \<O>\<^sub>p] val_ring_subring by blast
     have 3: "SA_poly_to_Qp_poly m x f \<in> carrier S"
-      unfolding 2 mem_Collect_eq using assms integral_on_memE SA_poly_to_Qp_poly_closed A 
-      by blast 
+      unfolding 2 mem_Collect_eq using assms integral_on_memE SA_poly_to_Qp_poly_closed A
+      by blast
     have 4: "SA_poly_to_Qp_poly m x g \<in> carrier S"
-      unfolding 2 mem_Collect_eq using assms integral_on_memE SA_poly_to_Qp_poly_closed A 
-      by blast 
+      unfolding 2 mem_Collect_eq using assms integral_on_memE SA_poly_to_Qp_poly_closed A
+      by blast
     have 5: "SA_poly_to_Qp_poly m x (f \<otimes>\<^bsub>UP (SA m)\<^esub> g) \<in> carrier S"
-      unfolding 0 
-      using  cring.cring_simprules(5)[of S]3 4 1 UPQ.UP_ring_subring_mult[of \<O>\<^sub>p "SA_poly_to_Qp_poly m x f" "SA_poly_to_Qp_poly m x g"] 
-      using S_def val_ring_subring by metis 
+      unfolding 0
+      using  cring.cring_simprules(5)[of S]3 4 1 UPQ.UP_ring_subring_mult[of \<O>\<^sub>p "SA_poly_to_Qp_poly m x f" "SA_poly_to_Qp_poly m x g"]
+      using S_def val_ring_subring by metis
     show "SA_poly_to_Qp_poly m x (f \<otimes>\<^bsub>UP (SA m)\<^esub> g) i \<in> \<O>\<^sub>p"
-      using 5 unfolding 2 mem_Collect_eq by blast 
+      using 5 unfolding 2 mem_Collect_eq by blast
   qed
 qed
 
 lemma integral_on_a_minus:
   assumes "B \<subseteq> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
   assumes "f \<in> integral_on m B"
   shows "\<ominus>\<^bsub>UP (SA m)\<^esub> f \<in> integral_on m B"
   apply(rule integral_on_memI)
-  using assms integral_on_memE(1)[of f m B] 
+  using assms integral_on_memE(1)[of f m B]
   apply blast
 proof- fix x i assume A: "x \<in> B"
   have 0: "SA_poly_to_Qp_poly m x (\<ominus>\<^bsub>UP (SA m)\<^esub> f) = \<ominus>\<^bsub>UP Q\<^sub>p\<^esub> SA_poly_to_Qp_poly m x f"
     apply(rule  UP_cring.ring_hom_uminus[of "UP Q\<^sub>p" "UP (SA m)" "SA_poly_to_Qp_poly m x" f ] )
     unfolding UP_cring_def
        apply (simp add: UPQ.M_cring)
-    using UPSA.P.is_ring apply blast
-    apply(rule SA_poly_to_Qp_poly_is_hom)
+      apply (simp add: UPSA.P.ring_axioms)
+     apply(rule SA_poly_to_Qp_poly_is_hom)
     using A assms apply blast
-    using assms integral_on_memE by blast 
+    using assms integral_on_memE by blast
   have 2: "\<And>f. f \<in> carrier (UP Q\<^sub>p) \<Longrightarrow> (\<ominus>\<^bsub>UP Q\<^sub>p\<^esub> f) i = \<ominus> (f i)"
     using UPQ.cfs_a_inv by blast
   have 1: "SA_poly_to_Qp_poly m x (\<ominus>\<^bsub>UP (SA m)\<^esub> f) i = \<ominus> (SA_poly_to_Qp_poly m x f) i "
     unfolding 0 apply(rule 2)
-    using integral_on_memE  assms A SA_poly_to_Qp_poly_closed[of x m f] by blast 
+    using integral_on_memE  assms A SA_poly_to_Qp_poly_closed[of x m f] by blast
   show "SA_poly_to_Qp_poly m x (\<ominus>\<^bsub>UP (SA m)\<^esub> f) i \<in> \<O>\<^sub>p"
     unfolding 1 using A assms integral_on_memE(2)[of f m B x i]
     using val_ring_ainv_closed by blast
 qed
 
 lemma integral_on_subring:
   assumes "B \<subseteq> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
   shows "subring (integral_on m B) (UP (SA m))"
 proof(rule subringI)
   show "integral_on m B \<subseteq> carrier (UP (SA m))"
-    unfolding integral_on_def by blast 
+    unfolding integral_on_def by blast
   show "\<one>\<^bsub>UP (SA m)\<^esub> \<in> integral_on m B"
-    using one_integral_on assms by blast 
+    using one_integral_on assms by blast
   show " \<And>h. h \<in> integral_on m B \<Longrightarrow> \<ominus>\<^bsub>UP (SA m)\<^esub> h \<in> integral_on m B"
-    using integral_on_a_minus assms by blast 
+    using integral_on_a_minus assms by blast
   show "\<And>h1 h2. h1 \<in> integral_on m B \<Longrightarrow> h2 \<in> integral_on m B \<Longrightarrow> h1 \<otimes>\<^bsub>UP (SA m)\<^esub> h2 \<in> integral_on m B"
-    using integral_on_times assms by blast 
+    using integral_on_times assms by blast
   show "\<And>h1 h2. h1 \<in> integral_on m B \<Longrightarrow> h2 \<in> integral_on m B \<Longrightarrow> h1 \<oplus>\<^bsub>UP (SA m)\<^esub> h2 \<in> integral_on m B"
-    using integral_on_plus assms by blast 
+    using integral_on_plus assms by blast
 qed
 
 lemma val_ring_add_pow:
   assumes "a \<in> carrier Q\<^sub>p"
   assumes "val a \<ge> 0"
   shows "val ([(n::nat)]\<cdot>a) \<ge> 0"
 proof-
   have 0: "[(n::nat)]\<cdot>a = ([n]\<cdot>\<one>)\<otimes>a"
     using assms Qp.add_pow_ldistr Qp.cring_simprules(12) Qp.one_closed by presburger
-  show ?thesis unfolding 0 using assms 
+  show ?thesis unfolding 0 using assms
     by (meson Qp.nat_inc_closed val_ring_memE val_of_nat_inc val_ringI val_ring_times_closed)
 qed
 
 lemma val_ring_poly_eval:
   assumes "f \<in> carrier (UP Q\<^sub>p)"
   assumes "\<And> i. f i \<in> \<O>\<^sub>p"
   shows "\<And>x. x \<in> \<O>\<^sub>p \<Longrightarrow> f \<bullet> x \<in> \<O>\<^sub>p"
 proof- fix x assume A: "x \<in> \<O>\<^sub>p"
   obtain S where S_def: "S = (Q\<^sub>p \<lparr> carrier := \<O>\<^sub>p \<rparr>)"
-    by blast 
+    by blast
   have 0: "UP_cring S"
     unfolding S_def apply(rule UPQ.UP_ring_subring(1))
     using val_ring_subring by blast
   have 1: "to_function Q\<^sub>p f x = to_function S f x"
     unfolding S_def  apply(rule UPQ.UP_subring_eval)
     using val_ring_subring apply blast
     apply(rule UPQ.poly_cfs_subring)    using val_ring_subring apply blast
     using assms apply blast
-    using assms apply blast using A by blast 
+    using assms apply blast using A by blast
   have 2: "f \<in> carrier (UP S)"
-    unfolding S_def 
+    unfolding S_def
     using UPQ.UP_ring_subring_car[of \<O>\<^sub>p] assms val_ring_subring by blast
   have 3: "to_function S f x \<in> \<O>\<^sub>p"
       using UPQ.UP_subring_eval_closed[of \<O>\<^sub>p f x]
     using 1 0 UP_cring.to_fun_closed[of S f x]
-    unfolding S_def 
+    unfolding S_def
     by (metis "2" A S_def UPQ.to_fun_def val_ring_subring)
   thus "f \<bullet> x \<in> \<O>\<^sub>p"
     using 1 UPQ.to_fun_def by presburger
 qed
 
 lemma SA_poly_constant_res_class_semialg':
   assumes "f \<in> carrier (UP (SA m))"
   assumes "\<And>i x. x \<in> B \<Longrightarrow> f i x \<in> \<O>\<^sub>p"
   assumes "deg (SA m) f \<le> d"
   assumes "C \<in> poly_res_classes n d"
   assumes "is_semialgebraic m B"
   shows "is_semialgebraic m {x \<in> B. SA_poly_to_Qp_poly m x f \<in> C}"
 proof-
   obtain g where g_def: "g = SA_poly_glue m B f (\<one>\<^bsub>UP (SA m)\<^esub>)"
-    by blast 
+    by blast
   have g_closed: "g \<in> carrier (UP (SA m))"
     unfolding g_def by(rule SA_poly_glue_closed, rule assms, blast, rule assms)
   have g_deg: "deg (SA m) g \<le> d"
     unfolding g_def apply(rule SA_poly_glue_deg, rule assms, blast, rule assms, rule assms)
-    unfolding deg_one by blast 
+    unfolding deg_one by blast
   have 0: "{x \<in> B. SA_poly_to_Qp_poly m x f \<in> C} = B \<inter> {x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). SA_poly_to_Qp_poly m x g \<in> C}"
     apply(rule equalityI', rule IntI, blast)
     unfolding mem_Collect_eq apply(rule conjI)
     using assms is_semialgebraic_closed apply blast
     unfolding g_def using SA_poly_glue_cfs1[of f m "\<one>\<^bsub>UP (SA m)\<^esub>" B] assms
     using SA_poly_glue_to_Qp_poly1 UPSA.P.cring_simprules(6) apply presburger
     unfolding g_def using SA_poly_glue_cfs1[of f m "\<one>\<^bsub>UP (SA m)\<^esub>" B] assms
-    using SA_poly_glue_to_Qp_poly1 UPSA.P.cring_simprules(6) 
+    using SA_poly_glue_to_Qp_poly1 UPSA.P.cring_simprules(6)
     by (metis (no_types, lifting) Int_iff mem_Collect_eq)
   have 1: "\<And>i x. x \<in> B \<Longrightarrow> g i x = f i x"
     unfolding g_def by(rule SA_poly_glue_cfs1, rule assms, blast, rule assms, blast)
   have 2: "\<And>i x. x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<Longrightarrow> g i x \<in> \<O>\<^sub>p"
   proof- fix i x assume A: "x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
     show "g i x \<in> \<O>\<^sub>p"
       unfolding g_def apply(cases "x \<in> B")
       using SA_poly_glue_cfs1[of f m "\<one>\<^bsub>UP (SA m)\<^esub>" B x i]
-            assms(1) assms(2)[of x i] 1 
+            assms(1) assms(2)[of x i] 1
       using UPSA.P.cring_simprules(6) assms(5) apply presburger
-      using SA_poly_glue_cfs2[of g m "\<one>\<^bsub>UP (SA m)\<^esub>" B x i] g_closed assms A 
-      by (metis (mono_tags, opaque_lifting) SA_poly_glue_cfs2 SA_poly_to_Qp_poly_coeff 
+      using SA_poly_glue_cfs2[of g m "\<one>\<^bsub>UP (SA m)\<^esub>" B x i] g_closed assms A
+      by (metis (mono_tags, opaque_lifting) SA_poly_glue_cfs2 SA_poly_to_Qp_poly_coeff
           UPSA.P.cring_simprules(6) carrier_is_semialgebraic g_def integral_on_memE(2) is_semialgebraic_closed one_integral_on )
   qed
   have 3: "\<And>i x. x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<Longrightarrow> x \<notin> B \<Longrightarrow> g i x = \<one>\<^bsub>UP (SA m)\<^esub> i x"
     unfolding g_def apply(rule SA_poly_glue_cfs2[of f m "\<one>\<^bsub>UP (SA m)\<^esub>" B ])
-    by(rule assms, blast, rule assms, blast, blast)  
+    by(rule assms, blast, rule assms, blast, blast)
   have 4: "\<And>i x. x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<Longrightarrow> x \<notin> B \<Longrightarrow> g i x \<in> \<O>\<^sub>p "
     using 3 cfs_one[of m] 2 by blast
   have 5: "\<And>i x. x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<Longrightarrow> g i x \<in> \<O>\<^sub>p"
     using 1 4 assms 2  by blast
   have 6: "is_semialgebraic m {x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>). SA_poly_to_Qp_poly m x g \<in> C}"
     by(rule SA_poly_constant_res_class_semialg[of _ _ d _ n], rule g_closed, rule 5, blast, rule g_deg, rule assms)
-  show ?thesis unfolding 0 
+  show ?thesis unfolding 0
     by(rule intersection_is_semialg, rule assms, rule 6)
 qed
 
 lemma SA_poly_constant_res_class_decomp:
   assumes "f \<in> carrier (UP (SA m))"
   assumes "\<And>i x. x \<in> B \<Longrightarrow> f i x \<in> \<O>\<^sub>p"
   assumes "deg (SA m) f \<le> d"
   assumes "is_semialgebraic m B"
   shows "B = (\<Union> C \<in> poly_res_classes n d. {x \<in> B. SA_poly_to_Qp_poly m x f \<in> C})"
 proof(rule equalityI')fix x assume A: "x \<in> B"
   obtain g where g_def: "g = SA_poly_glue m B f (\<one>\<^bsub>UP (SA m)\<^esub>)"
-    by blast 
+    by blast
   have g_closed: "g \<in> carrier (UP (SA m))"
     unfolding g_def by(rule SA_poly_glue_closed, rule assms, blast, rule assms)
   have g_deg: "deg (SA m) g \<le> d"
     unfolding g_def apply(rule SA_poly_glue_deg, rule assms, blast, rule assms, rule assms)
-    unfolding deg_one by blast 
+    unfolding deg_one by blast
   have 1: "\<And>i x. x \<in> B \<Longrightarrow> g i x = f i x"
     unfolding g_def by(rule SA_poly_glue_cfs1, rule assms, blast, rule assms, blast)
   have 2: "\<And>i x. x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<Longrightarrow> g i x \<in> \<O>\<^sub>p"
   proof- fix i x assume A: "x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
     show "g i x \<in> \<O>\<^sub>p"
       unfolding g_def apply(cases "x \<in> B")
       using SA_poly_glue_cfs1[of f m "\<one>\<^bsub>UP (SA m)\<^esub>" B x i]
-            assms(1) assms(2)[of x i] 1 
+            assms(1) assms(2)[of x i] 1
       using UPSA.P.cring_simprules(6) assms(4) apply presburger
-      using SA_poly_glue_cfs2[of g m "\<one>\<^bsub>UP (SA m)\<^esub>" B x i] g_closed assms A 
-      by (metis (mono_tags, opaque_lifting) SA_poly_glue_cfs2 SA_poly_to_Qp_poly_coeff 
+      using SA_poly_glue_cfs2[of g m "\<one>\<^bsub>UP (SA m)\<^esub>" B x i] g_closed assms A
+      by (metis (mono_tags, opaque_lifting) SA_poly_glue_cfs2 SA_poly_to_Qp_poly_coeff
           UPSA.P.cring_simprules(6) carrier_is_semialgebraic g_def integral_on_memE(2) is_semialgebraic_closed one_integral_on )
   qed
   have 3: "\<And>i x. x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<Longrightarrow> x \<notin> B \<Longrightarrow> g i x = \<one>\<^bsub>UP (SA m)\<^esub> i x"
     unfolding g_def apply(rule SA_poly_glue_cfs2[of f m "\<one>\<^bsub>UP (SA m)\<^esub>" B ])
-    by(rule assms, blast, rule assms, blast, blast)  
+    by(rule assms, blast, rule assms, blast, blast)
   have 4: "\<And>i x. x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<Longrightarrow> x \<notin> B \<Longrightarrow> g i x \<in> \<O>\<^sub>p "
     using 3 cfs_one[of m] 2 by blast
   have 5: "\<And>i x. x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<Longrightarrow> g i x \<in> \<O>\<^sub>p"
     using 1 4 assms 2  by blast
   have 6: "\<And> i x. x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<Longrightarrow> SA_poly_to_Qp_poly m x g i = g i x"
     by(rule SA_poly_to_Qp_poly_coeff, blast, rule g_closed)
   have 7: "\<And> x. x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<Longrightarrow> SA_poly_to_Qp_poly m x g \<in> val_ring_polys_grad d"
     apply(rule val_ring_polys_grad_memI, rule SA_poly_to_Qp_poly_closed, blast, rule g_closed)
     unfolding  6 apply(rule 5, blast)
     using g_closed  SA_poly_to_Qp_poly_deg_bound[of g m] g_deg le_trans by blast
   have 8: "\<And>x. x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<Longrightarrow> SA_poly_to_Qp_poly m x g \<in> poly_res_class n d (SA_poly_to_Qp_poly m x g)"
     by(rule poly_res_class_refl, rule 7)
   have 9: "\<And>x. x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>) \<Longrightarrow> poly_res_class n d (SA_poly_to_Qp_poly m x g) \<in> poly_res_classes n d"
-    unfolding poly_res_classes_def using 7 by blast 
+    unfolding poly_res_classes_def using 7 by blast
   have 10: "\<And>x. x \<in> B \<Longrightarrow> SA_poly_to_Qp_poly m x g = SA_poly_to_Qp_poly m x f"
     unfolding g_def by(rule SA_poly_glue_to_Qp_poly1, rule assms, blast, rule assms, blast)
   have 11: "x \<in> carrier (Q\<^sub>p\<^bsup>m\<^esup>)"
-    using A is_semialgebraic_closed assms by blast 
+    using A is_semialgebraic_closed assms by blast
   have 12: "SA_poly_to_Qp_poly m x g = SA_poly_to_Qp_poly m x f"
     by(rule 10, rule A)
   have 13: "x \<in> {x \<in> B. SA_poly_to_Qp_poly m x f \<in> poly_res_class n d (SA_poly_to_Qp_poly m x g)}"
-    using 11 10[of x] A 9[of x] 8[of x] unfolding 12 mem_Collect_eq  by blast 
+    using 11 10[of x] A 9[of x] 8[of x] unfolding 12 mem_Collect_eq  by blast
   show  "x \<in> (\<Union>C\<in>poly_res_classes n d. {x \<in> B. SA_poly_to_Qp_poly m x f \<in> C})"
-    using 13 9[of x] 11  unfolding 12 mem_simps(8) mem_Collect_eq by auto 
-next 
+    using 13 9[of x] 11  unfolding 12 mem_simps(8) mem_Collect_eq by auto
+next
   show "\<And>x. x \<in> (\<Union>C\<in>poly_res_classes n d. {x \<in> B. SA_poly_to_Qp_poly m x f \<in> C}) \<Longrightarrow> x \<in> B"
-    by blast 
+    by blast
 qed
 
 end
 
 context UP_cring
 begin
 
 lemma pderiv_deg_bound:
   assumes "p \<in> carrier P"
   assumes "deg R p \<le> (Suc d)"
   shows "deg R (pderiv p) \<le> d"
-proof- 
+proof-
   have "deg R p \<le> (Suc d) \<longrightarrow> deg R (pderiv p) \<le> d"
     apply(rule poly_induct[of p])
       apply (simp add: assms(1))
     using deg_zero pderiv_deg_0 apply presburger
-  proof fix p assume A: "(\<And>q. q \<in> carrier P \<Longrightarrow> deg R q < deg R p \<Longrightarrow> deg R q \<le> Suc d \<longrightarrow> deg R (pderiv q) \<le> d)"  
+  proof fix p assume A: "(\<And>q. q \<in> carrier P \<Longrightarrow> deg R q < deg R p \<Longrightarrow> deg R q \<le> Suc d \<longrightarrow> deg R (pderiv q) \<le> d)"
                         "p \<in> carrier P" "0 < deg R p" "deg R p \<le> Suc d"
     obtain q where q_def: "q \<in> carrier P \<and> deg R q < deg R p \<and> p = q \<oplus>\<^bsub>P\<^esub> ltrm p"
-      using A  ltrm_decomp by metis 
+      using A  ltrm_decomp by metis
     have 0: "deg R (pderiv (ltrm p)) \<le> d"
-    proof- 
+    proof-
       have 1: "pderiv (ltrm p) = up_ring.monom P ([deg R p] \<cdot> p (deg R p)) (deg R p - 1)"
-        using pderiv_monom[of "lcf p" "deg R p"] A  P_def UP_car_memE(1) by auto 
-      show ?thesis unfolding 1 
+        using pderiv_monom[of "lcf p" "deg R p"] A  P_def UP_car_memE(1) by auto
+      show ?thesis unfolding 1
         by (metis (no_types, lifting) A(2) A(3) A(4) R.add_pow_closed Suc_diff_1 cfs_closed deg_monom_le le_trans not_less_eq_eq)
     qed
     have "deg R q \<le> Suc d" using A q_def  by linarith
     then have 1: "deg R (pderiv q) \<le> d"
       using A q_def by blast
     hence "max (deg R (pderiv q)) (deg R (pderiv (up_ring.monom P (p (deg R p)) (deg R p)))) \<le>  d"
       using 0 max.bounded_iff by blast
     thus "deg R (pderiv p) \<le> d "
       using q_def pderiv_add pderiv_monom[of "lcf p" "deg R p"] A deg_add[of "pderiv q" "pderiv (ltrm p)"]
       by (metis "0" "1" ltrm_closed bound_deg_sum pderiv_closed)
   qed
-  thus ?thesis 
+  thus ?thesis
     using assms(2) by blast
 qed
 
-lemma(in cring) minus_zero: 
+lemma(in cring) minus_zero:
 "a \<in> carrier R \<Longrightarrow> a \<ominus> \<zero> = a"
-  unfolding a_minus_def 
+  unfolding a_minus_def
   by (metis add.l_cancel_one' cring_simprules(2) cring_simprules(22))
 
-lemma (in UP_cring) taylor_expansion_at_zero: 
+lemma (in UP_cring) taylor_expansion_at_zero:
   assumes "g \<in> carrier (UP R)"
   shows "taylor_expansion R \<zero> g = g"
-proof- 
+proof-
   have 0: "X_plus \<zero> = X_poly R"
-    unfolding X_poly_plus_def 
+    unfolding X_poly_plus_def
     by (metis ctrm_degree lcf_eq P.r_zero P_def R.zero_closed UP_cring.ctrm_is_poly UP_cring.to_poly_inverse UP_zero_closed X_closed deg_nzero_nzero is_UP_cring to_fun_ctrm to_fun_zero)
-  show ?thesis 
-    unfolding taylor_expansion_def 0 
+  show ?thesis
+    unfolding taylor_expansion_def 0
     using assms UP_cring.X_sub is_UP_cring by blast
 qed
 end
 (**************************************************************************************************)
 (**************************************************************************************************)
 subsection\<open>Partitioning Semialgebraic Sets By Zero Sets of Function\<close>
 (**************************************************************************************************)
 (**************************************************************************************************)
 
 context padic_fields
 begin
 
 definition SA_funs_to_SA_decomp where
 "SA_funs_to_SA_decomp n Fs S = atoms_of ((\<inter>) S ` ((SA_zero_set n ` Fs) \<union> (SA_nonzero_set n ` Fs))) "
 
-lemma SA_funs_to_SA_decomp_closed_0: 
+lemma SA_funs_to_SA_decomp_closed_0:
   assumes "Fs \<subseteq> carrier (SA n)"
   assumes "is_semialgebraic n S"
   shows "(\<inter>) S ` ((SA_zero_set n ` Fs) \<union> (SA_nonzero_set n ` Fs)) \<subseteq> semialg_sets n"
 proof(rule subsetI)
   fix x assume A: "x \<in> (\<inter>) S ` (SA_zero_set n ` Fs \<union> SA_nonzero_set n ` Fs)"
   show " x \<in> semialg_sets n"
   proof(cases "x \<in> (\<inter>) S ` SA_zero_set n ` Fs")
     case True
     then obtain f where f_def: "f \<in> Fs \<and> x = S \<inter> SA_zero_set n f"
-      by blast 
-    then show "x \<in> semialg_sets n" using assms SA_zero_set_is_semialgebraic  
+      by blast
+    then show "x \<in> semialg_sets n" using assms SA_zero_set_is_semialgebraic
       by (meson is_semialgebraicE semialg_intersect subset_iff)
   next
     case False
     then have "x \<in> (\<inter>) S ` SA_nonzero_set n ` Fs"
       using A by blast
     then obtain f where f_def: "f \<in> Fs \<and> x = S \<inter> SA_nonzero_set n f"
-      using A by blast 
-    then show "x \<in> semialg_sets n" using assms SA_nonzero_set_is_semialgebraic 
-      by (meson padic_fields.is_semialgebraicE padic_fields.semialg_intersect padic_fields_axioms subset_iff)   
+      using A by blast
+    then show "x \<in> semialg_sets n" using assms SA_nonzero_set_is_semialgebraic
+      by (meson padic_fields.is_semialgebraicE padic_fields.semialg_intersect padic_fields_axioms subset_iff)
   qed
 qed
 
-lemma SA_funs_to_SA_decomp_closed: 
+lemma SA_funs_to_SA_decomp_closed:
   assumes "finite Fs"
   assumes "Fs \<subseteq> carrier (SA n)"
   assumes "is_semialgebraic n S"
   shows "SA_funs_to_SA_decomp n Fs S \<subseteq> semialg_sets n"
   unfolding SA_funs_to_SA_decomp_def semialg_sets_def
   apply(rule atoms_of_gen_boolean_algebra)
-  using SA_funs_to_SA_decomp_closed_0[of Fs n S]  assms  unfolding semialg_sets_def 
-   apply blast 
-  using assms 
+  using SA_funs_to_SA_decomp_closed_0[of Fs n S]  assms  unfolding semialg_sets_def
+   apply blast
+  using assms
   by blast
 
-lemma SA_funs_to_SA_decomp_finite: 
+lemma SA_funs_to_SA_decomp_finite:
   assumes "finite Fs"
   assumes "Fs \<subseteq> carrier (SA n)"
   assumes "is_semialgebraic n S"
   shows "finite (SA_funs_to_SA_decomp n Fs S)"
-  unfolding SA_funs_to_SA_decomp_def 
+  unfolding SA_funs_to_SA_decomp_def
   apply(rule finite_set_imp_finite_atoms)
-  using assms by blast 
-
-lemma SA_funs_to_SA_decomp_disjoint: 
+  using assms by blast
+
+lemma SA_funs_to_SA_decomp_disjoint:
   assumes "finite Fs"
   assumes "Fs \<subseteq> carrier (SA n)"
   assumes "is_semialgebraic n S"
   shows "disjoint (SA_funs_to_SA_decomp n Fs S)"
   apply(rule disjointI) unfolding SA_funs_to_SA_decomp_def
   apply(rule atoms_of_disjoint[of _ " ((\<inter>) S ` (SA_zero_set n ` Fs \<union> SA_nonzero_set n ` Fs))"])
-    apply blast apply blast by blast 
+    apply blast apply blast by blast
 
 lemma pre_SA_funs_to_SA_decomp_in_algebra:
   shows " ((\<inter>) S ` (SA_zero_set n ` Fs \<union> SA_nonzero_set n ` Fs)) \<subseteq> gen_boolean_algebra S (SA_zero_set n ` Fs \<union> SA_nonzero_set n ` Fs)"
 proof(rule subsetI) fix x assume A: " x \<in>  (\<inter>) S ` (SA_zero_set n ` Fs \<union> SA_nonzero_set n ` Fs)"
   then obtain A where A_def: "A \<in> (SA_zero_set n ` Fs \<union> SA_nonzero_set n ` Fs) \<and> x = S \<inter> A"
-    by blast 
+    by blast
   then show "x \<in> gen_boolean_algebra S (SA_zero_set n ` Fs \<union> SA_nonzero_set n ` Fs)"
     using gen_boolean_algebra.generator[of A "SA_zero_set n ` Fs \<union> SA_nonzero_set n ` Fs" S]
-    by (metis inf.commute)  
+    by (metis inf.commute)
 qed
 
 lemma SA_funs_to_SA_decomp_in_algebra:
   assumes "finite Fs"
   shows "SA_funs_to_SA_decomp n Fs S \<subseteq> gen_boolean_algebra S (SA_zero_set n ` Fs \<union> SA_nonzero_set n ` Fs)"
   unfolding SA_funs_to_SA_decomp_def  apply(rule atoms_of_gen_boolean_algebra)
-  using pre_SA_funs_to_SA_decomp_in_algebra[of S n Fs] apply blast 
+  using pre_SA_funs_to_SA_decomp_in_algebra[of S n Fs] apply blast
   using assms by blast
 
-lemma SA_funs_to_SA_decomp_subset: 
+lemma SA_funs_to_SA_decomp_subset:
   assumes "finite Fs"
   assumes "Fs \<subseteq> carrier (SA n)"
   assumes "is_semialgebraic n S"
   assumes "A \<in> SA_funs_to_SA_decomp n Fs S"
   shows "A \<subseteq> S"
 proof-
   have "A \<in> gen_boolean_algebra S (SA_zero_set n ` Fs \<union> SA_nonzero_set n ` Fs)"
-  using assms SA_funs_to_SA_decomp_in_algebra[of Fs n S] 
-        atoms_of_gen_boolean_algebra[of _  S "(SA_zero_set n ` Fs \<union> SA_nonzero_set n ` Fs)"] 
-  unfolding SA_funs_to_SA_decomp_def by blast 
-  then show ?thesis using gen_boolean_algebra_subset by blast  
-qed
-
-lemma SA_funs_to_SA_decomp_memE: 
+  using assms SA_funs_to_SA_decomp_in_algebra[of Fs n S]
+        atoms_of_gen_boolean_algebra[of _  S "(SA_zero_set n ` Fs \<union> SA_nonzero_set n ` Fs)"]
+  unfolding SA_funs_to_SA_decomp_def by blast
+  then show ?thesis using gen_boolean_algebra_subset by blast
+qed
+
+lemma SA_funs_to_SA_decomp_memE:
   assumes "finite Fs"
   assumes "Fs \<subseteq> carrier (SA n)"
   assumes "is_semialgebraic n S"
   assumes "A \<in> (SA_funs_to_SA_decomp n Fs S)"
   assumes "f \<in> Fs"
   shows "A \<subseteq> SA_zero_set n f \<or> A \<subseteq> SA_nonzero_set n f"
 proof(cases "A \<subseteq> S \<inter> SA_zero_set n f")
 case True
-  then show ?thesis 
+  then show ?thesis
     by blast
 next
   case False
   have 0: "S \<inter> SA_zero_set n f \<in> (\<inter>) S ` (SA_zero_set n ` Fs \<union> SA_nonzero_set n ` Fs)"
-    using assms 
+    using assms
     by blast
   then have 1: "A \<inter> (S \<inter> SA_zero_set n f) = {}"
     using False assms atoms_are_minimal[of A "((\<inter>) S ` (SA_zero_set n ` Fs \<union> SA_nonzero_set n ` Fs))" "S \<inter> SA_zero_set n f"]
     unfolding SA_funs_to_SA_decomp_def
     by blast
   have 2: "A \<subseteq> S"
     using assms SA_funs_to_SA_decomp_subset by blast
-  then show ?thesis 
+  then show ?thesis
     using 0 1 False zero_set_nonzero_set_covers_semialg_set[of n S] assms(3) by auto
 qed
 
-lemma SA_funs_to_SA_decomp_covers: 
+lemma SA_funs_to_SA_decomp_covers:
   assumes "finite Fs"
   assumes "Fs \<noteq> {}"
   assumes "Fs \<subseteq> carrier (SA n)"
   assumes "is_semialgebraic n S"
   shows "S = \<Union> (SA_funs_to_SA_decomp n Fs S)"
 proof-
   have 0: "S = \<Union> ((\<inter>) S ` ((SA_zero_set n ` Fs) \<union> (SA_nonzero_set n ` Fs)))"
-  proof 
+  proof
     obtain f where f_def: "f \<in> Fs"
-      using assms by blast 
+      using assms by blast
     have 0: "S \<inter> SA_nonzero_set n f \<in> ((\<inter>) S ` ((SA_zero_set n ` Fs) \<union> (SA_nonzero_set n ` Fs)))"
-      using f_def by blast 
+      using f_def by blast
     have 1: "S \<inter> SA_zero_set n f \<in> ((\<inter>) S ` ((SA_zero_set n ` Fs) \<union> (SA_nonzero_set n ` Fs)))"
-      using f_def by blast 
+      using f_def by blast
     have 2: "S = S \<inter> SA_zero_set n f \<union> S \<inter> SA_nonzero_set n f"
       by (simp add: assms(4) zero_set_nonzero_set_covers_semialg_set)
     then show "S \<subseteq> \<Union> ((\<inter>) S ` (SA_zero_set n ` Fs \<union> SA_nonzero_set n ` Fs))"
-      using 0 1 Sup_upper2 Un_subset_iff subset_refl by blast          
+      using 0 1 Sup_upper2 Un_subset_iff subset_refl by blast
     show "\<Union> ((\<inter>) S ` (SA_zero_set n ` Fs \<union> SA_nonzero_set n ` Fs)) \<subseteq> S"
       by blast
   qed
-  show ?thesis 
-    unfolding SA_funs_to_SA_decomp_def atoms_of_covers' using 0 by blast 
+  show ?thesis
+    unfolding SA_funs_to_SA_decomp_def atoms_of_covers' using 0 by blast
 qed
 
 end
 end