diff --git a/src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy b/src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
--- a/src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
+++ b/src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
@@ -1,2076 +1,2036 @@
 (*  Title:      HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
     Author:     Amine Chaieb
 *)
 
 section \<open>Implementation and verification of multivariate polynomials\<close>
 
 theory Reflected_Multivariate_Polynomial
   imports Complex_Main Rat_Pair Polynomial_List
 begin
 
 subsection \<open>Datatype of polynomial expressions\<close>
 
 datatype poly = C Num | Bound nat | Add poly poly | Sub poly poly
   | Mul poly poly| Neg poly| Pw poly nat| CN poly nat poly
 
 abbreviation poly_0 :: "poly" ("0\<^sub>p") where "0\<^sub>p \<equiv> C (0\<^sub>N)"
 abbreviation poly_p :: "int \<Rightarrow> poly" ("'((_)')\<^sub>p") where "(i)\<^sub>p \<equiv> C (i)\<^sub>N"
 
 
 subsection\<open>Boundedness, substitution and all that\<close>
 
 primrec polysize:: "poly \<Rightarrow> nat"
   where
     "polysize (C c) = 1"
   | "polysize (Bound n) = 1"
   | "polysize (Neg p) = 1 + polysize p"
   | "polysize (Add p q) = 1 + polysize p + polysize q"
   | "polysize (Sub p q) = 1 + polysize p + polysize q"
   | "polysize (Mul p q) = 1 + polysize p + polysize q"
   | "polysize (Pw p n) = 1 + polysize p"
   | "polysize (CN c n p) = 4 + polysize c + polysize p"
 
 primrec polybound0:: "poly \<Rightarrow> bool" \<comment> \<open>a poly is INDEPENDENT of Bound 0\<close>
   where
     "polybound0 (C c) \<longleftrightarrow> True"
   | "polybound0 (Bound n) \<longleftrightarrow> n > 0"
   | "polybound0 (Neg a) \<longleftrightarrow> polybound0 a"
   | "polybound0 (Add a b) \<longleftrightarrow> polybound0 a \<and> polybound0 b"
   | "polybound0 (Sub a b) \<longleftrightarrow> polybound0 a \<and> polybound0 b"
   | "polybound0 (Mul a b) \<longleftrightarrow> polybound0 a \<and> polybound0 b"
   | "polybound0 (Pw p n) \<longleftrightarrow> polybound0 p"
   | "polybound0 (CN c n p) \<longleftrightarrow> n \<noteq> 0 \<and> polybound0 c \<and> polybound0 p"
 
 primrec polysubst0:: "poly \<Rightarrow> poly \<Rightarrow> poly" \<comment> \<open>substitute a poly into a poly for Bound 0\<close>
   where
     "polysubst0 t (C c) = C c"
   | "polysubst0 t (Bound n) = (if n = 0 then t else Bound n)"
   | "polysubst0 t (Neg a) = Neg (polysubst0 t a)"
   | "polysubst0 t (Add a b) = Add (polysubst0 t a) (polysubst0 t b)"
   | "polysubst0 t (Sub a b) = Sub (polysubst0 t a) (polysubst0 t b)"
   | "polysubst0 t (Mul a b) = Mul (polysubst0 t a) (polysubst0 t b)"
   | "polysubst0 t (Pw p n) = Pw (polysubst0 t p) n"
   | "polysubst0 t (CN c n p) =
       (if n = 0 then Add (polysubst0 t c) (Mul t (polysubst0 t p))
        else CN (polysubst0 t c) n (polysubst0 t p))"
 
 fun decrpoly:: "poly \<Rightarrow> poly"
   where
     "decrpoly (Bound n) = Bound (n - 1)"
   | "decrpoly (Neg a) = Neg (decrpoly a)"
   | "decrpoly (Add a b) = Add (decrpoly a) (decrpoly b)"
   | "decrpoly (Sub a b) = Sub (decrpoly a) (decrpoly b)"
   | "decrpoly (Mul a b) = Mul (decrpoly a) (decrpoly b)"
   | "decrpoly (Pw p n) = Pw (decrpoly p) n"
   | "decrpoly (CN c n p) = CN (decrpoly c) (n - 1) (decrpoly p)"
   | "decrpoly a = a"
 
 
 subsection \<open>Degrees and heads and coefficients\<close>
 
 fun degree :: "poly \<Rightarrow> nat"
   where
     "degree (CN c 0 p) = 1 + degree p"
   | "degree p = 0"
 
 fun head :: "poly \<Rightarrow> poly"
   where
     "head (CN c 0 p) = head p"
   | "head p = p"
 
 text \<open>More general notions of degree and head.\<close>
 
 fun degreen :: "poly \<Rightarrow> nat \<Rightarrow> nat"
   where
     "degreen (CN c n p) = (\<lambda>m. if n = m then 1 + degreen p n else 0)"
   | "degreen p = (\<lambda>m. 0)"
 
 fun headn :: "poly \<Rightarrow> nat \<Rightarrow> poly"
   where
     "headn (CN c n p) = (\<lambda>m. if n \<le> m then headn p m else CN c n p)"
   | "headn p = (\<lambda>m. p)"
 
 fun coefficients :: "poly \<Rightarrow> poly list"
   where
     "coefficients (CN c 0 p) = c # coefficients p"
   | "coefficients p = [p]"
 
 fun isconstant :: "poly \<Rightarrow> bool"
   where
     "isconstant (CN c 0 p) = False"
   | "isconstant p = True"
 
 fun behead :: "poly \<Rightarrow> poly"
   where
     "behead (CN c 0 p) = (let p' = behead p in if p' = 0\<^sub>p then c else CN c 0 p')"
   | "behead p = 0\<^sub>p"
 
 fun headconst :: "poly \<Rightarrow> Num"
   where
     "headconst (CN c n p) = headconst p"
   | "headconst (C n) = n"
 
 
 subsection \<open>Operations for normalization\<close>
 
 declare if_cong[fundef_cong del]
 declare let_cong[fundef_cong del]
 
 fun polyadd :: "poly \<Rightarrow> poly \<Rightarrow> poly"  (infixl "+\<^sub>p" 60)
   where
     "polyadd (C c) (C c') = C (c +\<^sub>N c')"
   | "polyadd (C c) (CN c' n' p') = CN (polyadd (C c) c') n' p'"
   | "polyadd (CN c n p) (C c') = CN (polyadd c (C c')) n p"
   | "polyadd (CN c n p) (CN c' n' p') =
       (if n < n' then CN (polyadd c (CN c' n' p')) n p
        else if n' < n then CN (polyadd (CN c n p) c') n' p'
        else
         let
           cc' = polyadd c c';
           pp' = polyadd p p'
         in if pp' = 0\<^sub>p then cc' else CN cc' n pp')"
   | "polyadd a b = Add a b"
 
 fun polyneg :: "poly \<Rightarrow> poly" ("~\<^sub>p")
   where
     "polyneg (C c) = C (~\<^sub>N c)"
   | "polyneg (CN c n p) = CN (polyneg c) n (polyneg p)"
   | "polyneg a = Neg a"
 
 definition polysub :: "poly \<Rightarrow> poly \<Rightarrow> poly"  (infixl "-\<^sub>p" 60)
   where "p -\<^sub>p q = polyadd p (polyneg q)"
 
 fun polymul :: "poly \<Rightarrow> poly \<Rightarrow> poly"  (infixl "*\<^sub>p" 60)
   where
     "polymul (C c) (C c') = C (c *\<^sub>N c')"
   | "polymul (C c) (CN c' n' p') =
       (if c = 0\<^sub>N then 0\<^sub>p else CN (polymul (C c) c') n' (polymul (C c) p'))"
   | "polymul (CN c n p) (C c') =
       (if c' = 0\<^sub>N  then 0\<^sub>p else CN (polymul c (C c')) n (polymul p (C c')))"
   | "polymul (CN c n p) (CN c' n' p') =
       (if n < n' then CN (polymul c (CN c' n' p')) n (polymul p (CN c' n' p'))
        else if n' < n then CN (polymul (CN c n p) c') n' (polymul (CN c n p) p')
        else polyadd (polymul (CN c n p) c') (CN 0\<^sub>p n' (polymul (CN c n p) p')))"
   | "polymul a b = Mul a b"
 
 declare if_cong[fundef_cong]
 declare let_cong[fundef_cong]
 
 fun polypow :: "nat \<Rightarrow> poly \<Rightarrow> poly"
   where
     "polypow 0 = (\<lambda>p. (1)\<^sub>p)"
   | "polypow n =
       (\<lambda>p.
         let
           q = polypow (n div 2) p;
           d = polymul q q
         in if even n then d else polymul p d)"
 
 abbreviation poly_pow :: "poly \<Rightarrow> nat \<Rightarrow> poly"  (infixl "^\<^sub>p" 60)
   where "a ^\<^sub>p k \<equiv> polypow k a"
 
 function polynate :: "poly \<Rightarrow> poly"
   where
     "polynate (Bound n) = CN 0\<^sub>p n (1)\<^sub>p"
   | "polynate (Add p q) = polynate p +\<^sub>p polynate q"
   | "polynate (Sub p q) = polynate p -\<^sub>p polynate q"
   | "polynate (Mul p q) = polynate p *\<^sub>p polynate q"
   | "polynate (Neg p) = ~\<^sub>p (polynate p)"
   | "polynate (Pw p n) = polynate p ^\<^sub>p n"
   | "polynate (CN c n p) = polynate (Add c (Mul (Bound n) p))"
   | "polynate (C c) = C (normNum c)"
   by pat_completeness auto
 termination by (relation "measure polysize") auto
 
 fun poly_cmul :: "Num \<Rightarrow> poly \<Rightarrow> poly"
 where
   "poly_cmul y (C x) = C (y *\<^sub>N x)"
 | "poly_cmul y (CN c n p) = CN (poly_cmul y c) n (poly_cmul y p)"
 | "poly_cmul y p = C y *\<^sub>p p"
 
 definition monic :: "poly \<Rightarrow> poly \<times> bool"
   where "monic p =
     (let h = headconst p
      in if h = 0\<^sub>N then (p, False) else (C (Ninv h) *\<^sub>p p, 0>\<^sub>N h))"
 
 
 subsection \<open>Pseudo-division\<close>
 
 definition shift1 :: "poly \<Rightarrow> poly"
   where "shift1 p = CN 0\<^sub>p 0 p"
 
 abbreviation funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"
   where "funpow \<equiv> compow"
 
 partial_function (tailrec) polydivide_aux :: "poly \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> nat \<times> poly"
 where
   "polydivide_aux a n p k s =
     (if s = 0\<^sub>p then (k, s)
      else
       let
         b = head s;
         m = degree s
       in
         if m < n then (k,s)
         else
           let p' = funpow (m - n) shift1 p
           in
             if a = b then polydivide_aux a n p k (s -\<^sub>p p')
             else polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (b *\<^sub>p p')))"
 
 definition polydivide :: "poly \<Rightarrow> poly \<Rightarrow> nat \<times> poly"
   where "polydivide s p = polydivide_aux (head p) (degree p) p 0 s"
 
 fun poly_deriv_aux :: "nat \<Rightarrow> poly \<Rightarrow> poly"
   where
     "poly_deriv_aux n (CN c 0 p) = CN (poly_cmul ((int n)\<^sub>N) c) 0 (poly_deriv_aux (n + 1) p)"
   | "poly_deriv_aux n p = poly_cmul ((int n)\<^sub>N) p"
 
 fun poly_deriv :: "poly \<Rightarrow> poly"
   where
     "poly_deriv (CN c 0 p) = poly_deriv_aux 1 p"
   | "poly_deriv p = 0\<^sub>p"
 
 
 subsection \<open>Semantics of the polynomial representation\<close>
 
 primrec Ipoly :: "'a list \<Rightarrow> poly \<Rightarrow> 'a::{field_char_0,power}"
   where
     "Ipoly bs (C c) = INum c"
   | "Ipoly bs (Bound n) = bs!n"
   | "Ipoly bs (Neg a) = - Ipoly bs a"
   | "Ipoly bs (Add a b) = Ipoly bs a + Ipoly bs b"
   | "Ipoly bs (Sub a b) = Ipoly bs a - Ipoly bs b"
   | "Ipoly bs (Mul a b) = Ipoly bs a * Ipoly bs b"
   | "Ipoly bs (Pw t n) = Ipoly bs t ^ n"
   | "Ipoly bs (CN c n p) = Ipoly bs c + (bs!n) * Ipoly bs p"
 
 abbreviation Ipoly_syntax :: "poly \<Rightarrow> 'a list \<Rightarrow>'a::{field_char_0,power}"  ("\<lparr>_\<rparr>\<^sub>p\<^bsup>_\<^esup>")
   where "\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<equiv> Ipoly bs p"
 
 lemma Ipoly_CInt: "Ipoly bs (C (i, 1)) = of_int i"
   by (simp add: INum_def)
 
 lemma Ipoly_CRat: "Ipoly bs (C (i, j)) = of_int i / of_int j"
   by (simp  add: INum_def)
 
 lemmas RIpoly_eqs = Ipoly.simps(2-7) Ipoly_CInt Ipoly_CRat
 
 
 subsection \<open>Normal form and normalization\<close>
 
 fun isnpolyh:: "poly \<Rightarrow> nat \<Rightarrow> bool"
   where
     "isnpolyh (C c) = (\<lambda>k. isnormNum c)"
   | "isnpolyh (CN c n p) = (\<lambda>k. n \<ge> k \<and> isnpolyh c (Suc n) \<and> isnpolyh p n \<and> p \<noteq> 0\<^sub>p)"
   | "isnpolyh p = (\<lambda>k. False)"
 
 lemma isnpolyh_mono: "n' \<le> n \<Longrightarrow> isnpolyh p n \<Longrightarrow> isnpolyh p n'"
   by (induct p rule: isnpolyh.induct) auto
 
 definition isnpoly :: "poly \<Rightarrow> bool"
   where "isnpoly p = isnpolyh p 0"
 
 text \<open>polyadd preserves normal forms\<close>
 
 lemma polyadd_normh: "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> isnpolyh (polyadd p q) (min n0 n1)"
 proof (induct p q arbitrary: n0 n1 rule: polyadd.induct)
   case (2 ab c' n' p' n0 n1)
   from 2 have  th1: "isnpolyh (C ab) (Suc n')"
     by simp
   from 2(3) have th2: "isnpolyh c' (Suc n')" and nplen1: "n' \<ge> n1"
     by simp_all
   with isnpolyh_mono have cp: "isnpolyh c' (Suc n')"
     by simp
   with 2(1)[OF th1 th2] have th3:"isnpolyh (C ab +\<^sub>p c') (Suc n')"
     by simp
   from nplen1 have n01len1: "min n0 n1 \<le> n'"
     by simp
   then show ?case using 2 th3
     by simp
 next
   case (3 c' n' p' ab n1 n0)
   from 3 have  th1: "isnpolyh (C ab) (Suc n')"
     by simp
   from 3(2) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1"
     by simp_all
   with isnpolyh_mono have cp: "isnpolyh c' (Suc n')"
     by simp
   with 3(1)[OF th2 th1] have th3:"isnpolyh (c' +\<^sub>p C ab) (Suc n')"
     by simp
   from nplen1 have n01len1: "min n0 n1 \<le> n'"
     by simp
   then show ?case using 3 th3
     by simp
 next
   case (4 c n p c' n' p' n0 n1)
   then have nc: "isnpolyh c (Suc n)" and np: "isnpolyh p n"
     by simp_all
   from 4 have nc': "isnpolyh c' (Suc n')" and np': "isnpolyh p' n'"
     by simp_all
   from 4 have ngen0: "n \<ge> n0"
     by simp
   from 4 have n'gen1: "n' \<ge> n1"
     by simp
   consider (eq) "n = n'" | (lt) "n < n'" | (gt) "n > n'"
     by arith
   then show ?case
   proof cases
     case eq
     with "4.hyps"(3)[OF nc nc']
     have ncc':"isnpolyh (c +\<^sub>p c') (Suc n)"
       by auto
     then have ncc'n01: "isnpolyh (c +\<^sub>p c') (min n0 n1)"
       using isnpolyh_mono[where n'="min n0 n1" and n="Suc n"] ngen0 n'gen1
       by auto
     from eq "4.hyps"(4)[OF np np'] have npp': "isnpolyh (p +\<^sub>p p') n"
       by simp
     have minle: "min n0 n1 \<le> n'"
       using ngen0 n'gen1 eq by simp
     from minle npp' ncc'n01 4 eq ngen0 n'gen1 ncc' show ?thesis
       by (simp add: Let_def)
   next
     case lt
     have "min n0 n1 \<le> n0"
       by simp
     with 4 lt have th1:"min n0 n1 \<le> n"
       by auto
     from 4 have th21: "isnpolyh c (Suc n)"
       by simp
     from 4 have th22: "isnpolyh (CN c' n' p') n'"
       by simp
     from lt have th23: "min (Suc n) n' = Suc n"
       by arith
     from "4.hyps"(1)[OF th21 th22] have "isnpolyh (polyadd c (CN c' n' p')) (Suc n)"
       using th23 by simp
     with 4 lt th1 show ?thesis
       by simp
   next
     case gt
     then have gt': "n' < n \<and> \<not> n < n'"
       by simp
     have "min n0 n1 \<le> n1"
       by simp
     with 4 gt have th1: "min n0 n1 \<le> n'"
       by auto
     from 4 have th21: "isnpolyh c' (Suc n')"
       by simp_all
     from 4 have th22: "isnpolyh (CN c n p) n"
       by simp
     from gt have th23: "min n (Suc n') = Suc n'"
       by arith
     from "4.hyps"(2)[OF th22 th21] have "isnpolyh (polyadd (CN c n p) c') (Suc n')"
       using th23 by simp
     with 4 gt th1 show ?thesis
       by simp
   qed
 qed auto
 
 lemma polyadd[simp]: "Ipoly bs (polyadd p q) = Ipoly bs p + Ipoly bs q"
   by (induct p q rule: polyadd.induct)
      (auto simp add: Let_def field_simps distrib_left[symmetric] simp del: distrib_left_NO_MATCH)
 
 lemma polyadd_norm: "isnpoly p \<Longrightarrow> isnpoly q \<Longrightarrow> isnpoly (polyadd p q)"
   using polyadd_normh[of p 0 q 0] isnpoly_def by simp
 
 text \<open>The degree of addition and other general lemmas needed for the normal form of polymul.\<close>
 
 lemma polyadd_different_degreen:
   assumes "isnpolyh p n0"
     and "isnpolyh q n1"
     and "degreen p m \<noteq> degreen q m"
     and "m \<le> min n0 n1"
   shows "degreen (polyadd p q) m = max (degreen p m) (degreen q m)"
   using assms
 proof (induct p q arbitrary: m n0 n1 rule: polyadd.induct)
   case (4 c n p c' n' p' m n0 n1)
   show ?case
   proof (cases "n = n'")
     case True
     with 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7)
     show ?thesis by (auto simp: Let_def)
   next
     case False
     with 4 show ?thesis by auto
   qed
 qed auto
 
 lemma headnz[simp]: "isnpolyh p n \<Longrightarrow> p \<noteq> 0\<^sub>p \<Longrightarrow> headn p m \<noteq> 0\<^sub>p"
   by (induct p arbitrary: n rule: headn.induct) auto
 
 lemma degree_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> degree p = 0"
   by (induct p arbitrary: n rule: degree.induct) auto
 
 lemma degreen_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> degreen p m = 0"
   by (induct p arbitrary: n rule: degreen.induct) auto
 
 lemma degree_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> degree p = 0"
   by (induct p arbitrary: n rule: degree.induct) auto
 
 lemma degree_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degree c = 0"
   using degree_isnpolyh_Suc by auto
 
 lemma degreen_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degreen c n = 0"
   using degreen_0 by auto
 
 
 lemma degreen_polyadd:
   assumes np: "isnpolyh p n0"
     and nq: "isnpolyh q n1"
     and m: "m \<le> max n0 n1"
   shows "degreen (p +\<^sub>p q) m \<le> max (degreen p m) (degreen q m)"
   using np nq m
 proof (induct p q arbitrary: n0 n1 m rule: polyadd.induct)
   case (2 c c' n' p' n0 n1)
   then show ?case
     by (cases n') simp_all
 next
   case (3 c n p c' n0 n1)
   then show ?case
     by (cases n) auto
 next
   case (4 c n p c' n' p' n0 n1 m)
   show ?case
   proof (cases "n = n'")
     case True
     with 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7)
     show ?thesis by (auto simp: Let_def)
   next
     case False
     then show ?thesis by simp
   qed
 qed auto
 
 lemma polyadd_eq_const_degreen:
   assumes "isnpolyh p n0"
     and "isnpolyh q n1"
     and "polyadd p q = C c"
   shows "degreen p m = degreen q m"
   using assms
 proof (induct p q arbitrary: m n0 n1 c rule: polyadd.induct)
   case (4 c n p c' n' p' m n0 n1 x)
   consider "n = n'" | "n > n' \<or> n < n'" by arith
   then show ?case
   proof cases
     case 1
     with 4 show ?thesis
       by (cases "p +\<^sub>p p' = 0\<^sub>p") (auto simp add: Let_def)
   next
     case 2
     with 4 show ?thesis by auto
   qed
 qed simp_all
 
 lemma polymul_properties:
   assumes "SORT_CONSTRAINT('a::field_char_0)"
     and np: "isnpolyh p n0"
     and nq: "isnpolyh q n1"
     and m: "m \<le> min n0 n1"
   shows "isnpolyh (p *\<^sub>p q) (min n0 n1)"
     and "p *\<^sub>p q = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p \<or> q = 0\<^sub>p"
     and "degreen (p *\<^sub>p q) m = (if p = 0\<^sub>p \<or> q = 0\<^sub>p then 0 else degreen p m + degreen q m)"
   using np nq m
 proof (induct p q arbitrary: n0 n1 m rule: polymul.induct)
   case (2 c c' n' p')
   {
     case (1 n0 n1)
     with "2.hyps"(4-6)[of n' n' n'] and "2.hyps"(1-3)[of "Suc n'" "Suc n'" n']
     show ?case by (auto simp add: min_def)
   next
     case (2 n0 n1)
     then show ?case by auto
   next
     case (3 n0 n1)
     then show ?case using "2.hyps" by auto
   }
 next
   case (3 c n p c')
   {
     case (1 n0 n1)
     with "3.hyps"(4-6)[of n n n] and "3.hyps"(1-3)[of "Suc n" "Suc n" n]
     show ?case by (auto simp add: min_def)
   next
     case (2 n0 n1)
     then show ?case by auto
   next
     case (3 n0 n1)
     then show ?case  using "3.hyps" by auto
   }
 next
   case (4 c n p c' n' p')
   let ?cnp = "CN c n p" let ?cnp' = "CN c' n' p'"
   {
     case (1 n0 n1)
     then have cnp: "isnpolyh ?cnp n"
       and cnp': "isnpolyh ?cnp' n'"
       and np: "isnpolyh p n"
       and nc: "isnpolyh c (Suc n)"
       and np': "isnpolyh p' n'"
       and nc': "isnpolyh c' (Suc n')"
       and nn0: "n \<ge> n0"
       and nn1: "n' \<ge> n1"
       by simp_all
     consider "n < n'" | "n' < n" | "n' = n" by arith
     then show ?case
     proof cases
       case 1
       with "4.hyps"(4-5)[OF np cnp', of n] and "4.hyps"(1)[OF nc cnp', of n] nn0 cnp
       show ?thesis by (simp add: min_def)
     next
       case 2
       with "4.hyps"(16-17)[OF cnp np', of "n'"] and "4.hyps"(13)[OF cnp nc', of "Suc n'"] nn1 cnp'
       show ?thesis by (cases "Suc n' = n") (simp_all add: min_def)
     next
       case 3
       with "4.hyps"(16-17)[OF cnp np', of n] and "4.hyps"(13)[OF cnp nc', of n] cnp cnp' nn1 nn0
       show ?thesis
         by (auto intro!: polyadd_normh) (simp_all add: min_def isnpolyh_mono[OF nn0])
     qed
   next
     fix n0 n1 m
     assume np: "isnpolyh ?cnp n0"
     assume np':"isnpolyh ?cnp' n1"
     assume m: "m \<le> min n0 n1"
     let ?d = "degreen (?cnp *\<^sub>p ?cnp') m"
     let ?d1 = "degreen ?cnp m"
     let ?d2 = "degreen ?cnp' m"
     let ?eq = "?d = (if ?cnp = 0\<^sub>p \<or> ?cnp' = 0\<^sub>p then 0  else ?d1 + ?d2)"
     consider "n' < n \<or> n < n'" | "n' = n" by linarith
     then show ?eq
     proof cases
       case 1
       with "4.hyps"(3,6,18) np np' m show ?thesis by auto
     next
       case 2
       have nn': "n' = n" by fact
       then have nn: "\<not> n' < n \<and> \<not> n < n'" by arith
       from "4.hyps"(16,18)[of n n' n]
         "4.hyps"(13,14)[of n "Suc n'" n]
         np np' nn'
       have norm:
         "isnpolyh ?cnp n"
         "isnpolyh c' (Suc n)"
         "isnpolyh (?cnp *\<^sub>p c') n"
         "isnpolyh p' n"
         "isnpolyh (?cnp *\<^sub>p p') n"
         "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
         "?cnp *\<^sub>p c' = 0\<^sub>p \<longleftrightarrow> c' = 0\<^sub>p"
         "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p"
         by (auto simp add: min_def)
       show ?thesis
       proof (cases "m = n")
         case mn: True
         from "4.hyps"(17,18)[OF norm(1,4), of n]
           "4.hyps"(13,15)[OF norm(1,2), of n] norm nn' mn
         have degs:
           "degreen (?cnp *\<^sub>p c') n = (if c' = 0\<^sub>p then 0 else ?d1 + degreen c' n)"
           "degreen (?cnp *\<^sub>p p') n = ?d1  + degreen p' n"
           by (simp_all add: min_def)
         from degs norm have th1: "degreen (?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
           by simp
         then have neq: "degreen (?cnp *\<^sub>p c') n \<noteq> degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
           by simp
         have nmin: "n \<le> min n n"
           by (simp add: min_def)
         from polyadd_different_degreen[OF norm(3,6) neq nmin] th1
         have deg: "degreen (CN c n p *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n =
             degreen (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
           by simp
         from "4.hyps"(16-18)[OF norm(1,4), of n]
           "4.hyps"(13-15)[OF norm(1,2), of n]
           mn norm m nn' deg
         show ?thesis by simp
       next
         case mn: False
         then have mn': "m < n"
           using m np by auto
         from nn' m np have max1: "m \<le> max n n"
           by simp
         then have min1: "m \<le> min n n"
           by simp
         then have min2: "m \<le> min n (Suc n)"
           by simp
         from "4.hyps"(16-18)[OF norm(1,4) min1]
           "4.hyps"(13-15)[OF norm(1,2) min2]
           degreen_polyadd[OF norm(3,6) max1]
         have "degreen (?cnp *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n (?cnp *\<^sub>p p')) m \<le>
             max (degreen (?cnp *\<^sub>p c') m) (degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) m)"
           using mn nn' np np' by simp
         with "4.hyps"(16-18)[OF norm(1,4) min1]
           "4.hyps"(13-15)[OF norm(1,2) min2]
           degreen_0[OF norm(3) mn']
           nn' mn np np'
         show ?thesis by clarsimp
       qed
     qed
   }
   {
     case (2 n0 n1)
     then have np: "isnpolyh ?cnp n0"
       and np': "isnpolyh ?cnp' n1"
       and m: "m \<le> min n0 n1"
       by simp_all
     then have mn: "m \<le> n" by simp
     let ?c0p = "CN 0\<^sub>p n (?cnp *\<^sub>p p')"
     have False if C: "?cnp *\<^sub>p c' +\<^sub>p ?c0p = 0\<^sub>p" "n' = n"
     proof -
       from C have nn: "\<not> n' < n \<and> \<not> n < n'"
         by simp
       from "4.hyps"(16-18) [of n n n]
         "4.hyps"(13-15)[of n "Suc n" n]
         np np' C(2) mn
       have norm:
         "isnpolyh ?cnp n"
         "isnpolyh c' (Suc n)"
         "isnpolyh (?cnp *\<^sub>p c') n"
         "isnpolyh p' n"
         "isnpolyh (?cnp *\<^sub>p p') n"
         "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
         "?cnp *\<^sub>p c' = 0\<^sub>p \<longleftrightarrow> c' = 0\<^sub>p"
         "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p"
         "degreen (?cnp *\<^sub>p c') n = (if c' = 0\<^sub>p then 0 else degreen ?cnp n + degreen c' n)"
         "degreen (?cnp *\<^sub>p p') n = degreen ?cnp n + degreen p' n"
         by (simp_all add: min_def)
       from norm have cn: "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
         by simp
       have degneq: "degreen (?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
         using norm by simp
       from polyadd_eq_const_degreen[OF norm(3) cn C(1), where m="n"] degneq
       show ?thesis by simp
     qed
     then show ?case using "4.hyps" by clarsimp
   }
 qed auto
 
 lemma polymul[simp]: "Ipoly bs (p *\<^sub>p q) = Ipoly bs p * Ipoly bs q"
   by (induct p q rule: polymul.induct) (auto simp add: field_simps)
 
 lemma polymul_normh:
   assumes "SORT_CONSTRAINT('a::field_char_0)"
   shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> isnpolyh (p *\<^sub>p q) (min n0 n1)"
   using polymul_properties(1) by blast
 
 lemma polymul_eq0_iff:
   assumes "SORT_CONSTRAINT('a::field_char_0)"
   shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> p *\<^sub>p q = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p \<or> q = 0\<^sub>p"
   using polymul_properties(2) by blast
 
 lemma polymul_degreen:
   assumes "SORT_CONSTRAINT('a::field_char_0)"
   shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> m \<le> min n0 n1 \<Longrightarrow>
     degreen (p *\<^sub>p q) m = (if p = 0\<^sub>p \<or> q = 0\<^sub>p then 0 else degreen p m + degreen q m)"
   by (fact polymul_properties(3))
 
 lemma polymul_norm:
   assumes "SORT_CONSTRAINT('a::field_char_0)"
   shows "isnpoly p \<Longrightarrow> isnpoly q \<Longrightarrow> isnpoly (polymul p q)"
   using polymul_normh[of "p" "0" "q" "0"] isnpoly_def by simp
 
 lemma headconst_zero: "isnpolyh p n0 \<Longrightarrow> headconst p = 0\<^sub>N \<longleftrightarrow> p = 0\<^sub>p"
   by (induct p arbitrary: n0 rule: headconst.induct) auto
 
 lemma headconst_isnormNum: "isnpolyh p n0 \<Longrightarrow> isnormNum (headconst p)"
   by (induct p arbitrary: n0) auto
 
 lemma monic_eqI:
   assumes np: "isnpolyh p n0"
   shows "INum (headconst p) * Ipoly bs (fst (monic p)) =
     (Ipoly bs p ::'a::{field_char_0, power})"
   unfolding monic_def Let_def
 proof (cases "headconst p = 0\<^sub>N", simp_all add: headconst_zero[OF np])
   let ?h = "headconst p"
   assume pz: "p \<noteq> 0\<^sub>p"
   {
     assume hz: "INum ?h = (0::'a)"
     from headconst_isnormNum[OF np] have norm: "isnormNum ?h" "isnormNum 0\<^sub>N"
       by simp_all
     from isnormNum_unique[where ?'a = 'a, OF norm] hz have "?h = 0\<^sub>N"
       by simp
     with headconst_zero[OF np] have "p = 0\<^sub>p"
       by blast
     with pz have False
       by blast
   }
   then show "INum (headconst p) = (0::'a) \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0"
     by blast
 qed
 
 
 text \<open>polyneg is a negation and preserves normal forms\<close>
 
 lemma polyneg[simp]: "Ipoly bs (polyneg p) = - Ipoly bs p"
   by (induct p rule: polyneg.induct) auto
 
 lemma polyneg0: "isnpolyh p n \<Longrightarrow> (~\<^sub>p p) = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
   by (induct p arbitrary: n rule: polyneg.induct) (auto simp add: Nneg_def)
 
 lemma polyneg_polyneg: "isnpolyh p n0 \<Longrightarrow> ~\<^sub>p (~\<^sub>p p) = p"
   by (induct p arbitrary: n0 rule: polyneg.induct) auto
 
 lemma polyneg_normh: "isnpolyh p n \<Longrightarrow> isnpolyh (polyneg p) n"
   by (induct p arbitrary: n rule: polyneg.induct) (auto simp add: polyneg0)
 
 lemma polyneg_norm: "isnpoly p \<Longrightarrow> isnpoly (polyneg p)"
   using isnpoly_def polyneg_normh by simp
 
 
 text \<open>polysub is a substraction and preserves normal forms\<close>
 
 lemma polysub[simp]: "Ipoly bs (polysub p q) = Ipoly bs p - Ipoly bs q"
   by (simp add: polysub_def)
 
 lemma polysub_normh: "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> isnpolyh (polysub p q) (min n0 n1)"
   by (simp add: polysub_def polyneg_normh polyadd_normh)
 
 lemma polysub_norm: "isnpoly p \<Longrightarrow> isnpoly q \<Longrightarrow> isnpoly (polysub p q)"
   using polyadd_norm polyneg_norm by (simp add: polysub_def)
 
 lemma polysub_same_0[simp]:
   assumes "SORT_CONSTRAINT('a::field_char_0)"
   shows "isnpolyh p n0 \<Longrightarrow> polysub p p = 0\<^sub>p"
   unfolding polysub_def split_def fst_conv snd_conv
   by (induct p arbitrary: n0) (auto simp add: Let_def Nsub0[simplified Nsub_def])
 
 lemma polysub_0:
   assumes "SORT_CONSTRAINT('a::field_char_0)"
   shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> p -\<^sub>p q = 0\<^sub>p \<longleftrightarrow> p = q"
   unfolding polysub_def split_def fst_conv snd_conv
   by (induct p q arbitrary: n0 n1 rule:polyadd.induct)
     (auto simp: Nsub0[simplified Nsub_def] Let_def)
 
 text \<open>polypow is a power function and preserves normal forms\<close>
 
 lemma polypow[simp]: "Ipoly bs (polypow n p) = (Ipoly bs p :: 'a::field_char_0) ^ n"
 proof (induct n rule: polypow.induct)
   case 1
   then show ?case by simp
 next
   case (2 n)
   let ?q = "polypow ((Suc n) div 2) p"
   let ?d = "polymul ?q ?q"
   consider "odd (Suc n)" | "even (Suc n)" by auto
   then show ?case
   proof cases
     case odd: 1
     have *: "(Suc (Suc (Suc 0) * (Suc n div Suc (Suc 0)))) = Suc n div 2 + Suc n div 2 + 1"
       by arith
     from odd have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs (polymul p ?d)"
       by (simp add: Let_def)
     also have "\<dots> = (Ipoly bs p) * (Ipoly bs p)^(Suc n div 2) * (Ipoly bs p)^(Suc n div 2)"
       using "2.hyps" by simp
     also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2 + 1)"
       by (simp only: power_add power_one_right) simp
     also have "\<dots> = (Ipoly bs p) ^ (Suc (Suc (Suc 0) * (Suc n div Suc (Suc 0))))"
       by (simp only: *)
     finally show ?thesis
       unfolding numeral_2_eq_2 [symmetric]
       using odd_two_times_div_two_nat [OF odd] by simp
   next
     case even: 2
     from even have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs ?d"
       by (simp add: Let_def)
     also have "\<dots> = (Ipoly bs p) ^ (2 * (Suc n div 2))"
       using "2.hyps" by (simp only: mult_2 power_add) simp
     finally show ?thesis
       using even_two_times_div_two [OF even] by simp
   qed
 qed
 
 lemma polypow_normh:
   assumes "SORT_CONSTRAINT('a::field_char_0)"
   shows "isnpolyh p n \<Longrightarrow> isnpolyh (polypow k p) n"
 proof (induct k arbitrary: n rule: polypow.induct)
   case 1
   then show ?case by auto
 next
   case (2 k n)
   let ?q = "polypow (Suc k div 2) p"
   let ?d = "polymul ?q ?q"
   from 2 have *: "isnpolyh ?q n" and **: "isnpolyh p n"
     by blast+
   from polymul_normh[OF * *] have dn: "isnpolyh ?d n"
     by simp
   from polymul_normh[OF ** dn] have on: "isnpolyh (polymul p ?d) n"
     by simp
   from dn on show ?case by (simp, unfold Let_def) auto
 qed
 
 lemma polypow_norm:
   assumes "SORT_CONSTRAINT('a::field_char_0)"
   shows "isnpoly p \<Longrightarrow> isnpoly (polypow k p)"
   by (simp add: polypow_normh isnpoly_def)
 
 text \<open>Finally the whole normalization\<close>
 
 lemma polynate [simp]: "Ipoly bs (polynate p) = (Ipoly bs p :: 'a ::field_char_0)"
   by (induct p rule:polynate.induct) auto
 
 lemma polynate_norm[simp]:
   assumes "SORT_CONSTRAINT('a::field_char_0)"
   shows "isnpoly (polynate p)"
   by (induct p rule: polynate.induct)
      (simp_all add: polyadd_norm polymul_norm polysub_norm polyneg_norm polypow_norm,
       simp_all add: isnpoly_def)
 
 text \<open>shift1\<close>
 
 lemma shift1: "Ipoly bs (shift1 p) = Ipoly bs (Mul (Bound 0) p)"
   by (simp add: shift1_def)
 
 lemma shift1_isnpoly:
   assumes "isnpoly p"
     and "p \<noteq> 0\<^sub>p"
   shows "isnpoly (shift1 p) "
   using assms by (simp add: shift1_def isnpoly_def)
 
 lemma shift1_nz[simp]:"shift1 p \<noteq> 0\<^sub>p"
   by (simp add: shift1_def)
 
 lemma funpow_shift1_isnpoly: "isnpoly p \<Longrightarrow> p \<noteq> 0\<^sub>p \<Longrightarrow> isnpoly (funpow n shift1 p)"
   by (induct n arbitrary: p) (auto simp add: shift1_isnpoly funpow_swap1)
 
 lemma funpow_isnpolyh:
   assumes "\<And>p. isnpolyh p n \<Longrightarrow> isnpolyh (f p) n"
     and "isnpolyh p n"
   shows "isnpolyh (funpow k f p) n"
   using assms by (induct k arbitrary: p) auto
 
 lemma funpow_shift1:
   "(Ipoly bs (funpow n shift1 p) :: 'a :: field_char_0) =
     Ipoly bs (Mul (Pw (Bound 0) n) p)"
   by (induct n arbitrary: p) (simp_all add: shift1_isnpoly shift1)
 
 lemma shift1_isnpolyh: "isnpolyh p n0 \<Longrightarrow> p \<noteq> 0\<^sub>p \<Longrightarrow> isnpolyh (shift1 p) 0"
   using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by (simp add: shift1_def)
 
 lemma funpow_shift1_1:
   "(Ipoly bs (funpow n shift1 p) :: 'a :: field_char_0) =
     Ipoly bs (funpow n shift1 (1)\<^sub>p *\<^sub>p p)"
   by (simp add: funpow_shift1)
 
 lemma poly_cmul[simp]: "Ipoly bs (poly_cmul c p) = Ipoly bs (Mul (C c) p)"
   by (induct p rule: poly_cmul.induct) (auto simp add: field_simps)
 
 lemma behead:
   assumes "isnpolyh p n"
   shows "Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) =
     (Ipoly bs p :: 'a :: field_char_0)"
   using assms
 proof (induct p arbitrary: n rule: behead.induct)
   case (1 c p n)
   then have pn: "isnpolyh p n" by simp
   from 1(1)[OF pn]
   have th:"Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = Ipoly bs p" .
   then show ?case using "1.hyps"
     apply (simp add: Let_def,cases "behead p = 0\<^sub>p")
     apply (simp_all add: th[symmetric] field_simps)
     done
 qed (auto simp add: Let_def)
 
 lemma behead_isnpolyh:
   assumes "isnpolyh p n"
   shows "isnpolyh (behead p) n"
   using assms by (induct p rule: behead.induct) (auto simp add: Let_def isnpolyh_mono)
 
 
 subsection \<open>Miscellaneous lemmas about indexes, decrementation, substitution  etc ...\<close>
 
 lemma isnpolyh_polybound0: "isnpolyh p (Suc n) \<Longrightarrow> polybound0 p"
 proof (induct p arbitrary: n rule: poly.induct, auto, goal_cases)
   case prems: (1 c n p n')
   then have "n = Suc (n - 1)"
     by simp
   then have "isnpolyh p (Suc (n - 1))"
     using \<open>isnpolyh p n\<close> by simp
   with prems(2) show ?case
     by simp
 qed
 
 lemma isconstant_polybound0: "isnpolyh p n0 \<Longrightarrow> isconstant p \<longleftrightarrow> polybound0 p"
   by (induct p arbitrary: n0 rule: isconstant.induct) (auto simp add: isnpolyh_polybound0)
 
 lemma decrpoly_zero[simp]: "decrpoly p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
   by (induct p) auto
 
 lemma decrpoly_normh: "isnpolyh p n0 \<Longrightarrow> polybound0 p \<Longrightarrow> isnpolyh (decrpoly p) (n0 - 1)"
-  apply (induct p arbitrary: n0)
-         apply auto
-  apply atomize
-  apply (rename_tac nat a b, erule_tac x = "Suc nat" in allE)
-  apply auto
-  done
+  by (induct p arbitrary: n0) force+
 
 lemma head_polybound0: "isnpolyh p n0 \<Longrightarrow> polybound0 (head p)"
   by (induct p  arbitrary: n0 rule: head.induct) (auto intro: isnpolyh_polybound0)
 
 lemma polybound0_I:
   assumes "polybound0 a"
   shows "Ipoly (b # bs) a = Ipoly (b' # bs) a"
   using assms by (induct a rule: poly.induct) auto
 
 lemma polysubst0_I: "Ipoly (b # bs) (polysubst0 a t) = Ipoly ((Ipoly (b # bs) a) # bs) t"
   by (induct t) simp_all
 
 lemma polysubst0_I':
   assumes "polybound0 a"
   shows "Ipoly (b # bs) (polysubst0 a t) = Ipoly ((Ipoly (b' # bs) a) # bs) t"
   by (induct t) (simp_all add: polybound0_I[OF assms, where b="b" and b'="b'"])
 
 lemma decrpoly:
   assumes "polybound0 t"
   shows "Ipoly (x # bs) t = Ipoly bs (decrpoly t)"
   using assms by (induct t rule: decrpoly.induct) simp_all
 
 lemma polysubst0_polybound0:
   assumes "polybound0 t"
   shows "polybound0 (polysubst0 t a)"
   using assms by (induct a rule: poly.induct) auto
 
 lemma degree0_polybound0: "isnpolyh p n \<Longrightarrow> degree p = 0 \<Longrightarrow> polybound0 p"
   by (induct p arbitrary: n rule: degree.induct) (auto simp add: isnpolyh_polybound0)
 
 primrec maxindex :: "poly \<Rightarrow> nat"
   where
     "maxindex (Bound n) = n + 1"
   | "maxindex (CN c n p) = max  (n + 1) (max (maxindex c) (maxindex p))"
   | "maxindex (Add p q) = max (maxindex p) (maxindex q)"
   | "maxindex (Sub p q) = max (maxindex p) (maxindex q)"
   | "maxindex (Mul p q) = max (maxindex p) (maxindex q)"
   | "maxindex (Neg p) = maxindex p"
   | "maxindex (Pw p n) = maxindex p"
   | "maxindex (C x) = 0"
 
 definition wf_bs :: "'a list \<Rightarrow> poly \<Rightarrow> bool"
   where "wf_bs bs p \<longleftrightarrow> length bs \<ge> maxindex p"
 
 lemma wf_bs_coefficients: "wf_bs bs p \<Longrightarrow> \<forall>c \<in> set (coefficients p). wf_bs bs c"
 proof (induct p rule: coefficients.induct)
   case (1 c p)
   show ?case
   proof
     fix x
     assume "x \<in> set (coefficients (CN c 0 p))"
     then consider "x = c" | "x \<in> set (coefficients p)"
       by auto
     then show "wf_bs bs x"
     proof cases
       case prems: 1
       then show ?thesis
         using "1.prems" by (simp add: wf_bs_def)
     next
       case prems: 2
       from "1.prems" have "wf_bs bs p"
         by (simp add: wf_bs_def)
       with "1.hyps" prems show ?thesis
         by blast
     qed
   qed
 qed simp_all
 
 lemma maxindex_coefficients: "\<forall>c \<in> set (coefficients p). maxindex c \<le> maxindex p"
   by (induct p rule: coefficients.induct) auto
 
 lemma wf_bs_I: "wf_bs bs p \<Longrightarrow> Ipoly (bs @ bs') p = Ipoly bs p"
   by (induct p) (auto simp add: nth_append wf_bs_def)
 
 lemma take_maxindex_wf:
   assumes wf: "wf_bs bs p"
   shows "Ipoly (take (maxindex p) bs) p = Ipoly bs p"
 proof -
   let ?ip = "maxindex p"
   let ?tbs = "take ?ip bs"
   from wf have "length ?tbs = ?ip"
     unfolding wf_bs_def by simp
   then have wf': "wf_bs ?tbs p"
     unfolding wf_bs_def by  simp
   have eq: "bs = ?tbs @ drop ?ip bs"
     by simp
   from wf_bs_I[OF wf', of "drop ?ip bs"] show ?thesis
     using eq by simp
 qed
 
 lemma decr_maxindex: "polybound0 p \<Longrightarrow> maxindex (decrpoly p) = maxindex p - 1"
   by (induct p) auto
 
 lemma wf_bs_insensitive: "length bs = length bs' \<Longrightarrow> wf_bs bs p = wf_bs bs' p"
   by (simp add: wf_bs_def)
 
 lemma wf_bs_insensitive': "wf_bs (x # bs) p = wf_bs (y # bs) p"
   by (simp add: wf_bs_def)
 
 lemma wf_bs_coefficients': "\<forall>c \<in> set (coefficients p). wf_bs bs c \<Longrightarrow> wf_bs (x # bs) p"
   by (induct p rule: coefficients.induct) (auto simp add: wf_bs_def)
 
 lemma coefficients_Nil[simp]: "coefficients p \<noteq> []"
   by (induct p rule: coefficients.induct) simp_all
 
 lemma coefficients_head: "last (coefficients p) = head p"
   by (induct p rule: coefficients.induct) auto
 
 lemma wf_bs_decrpoly: "wf_bs bs (decrpoly p) \<Longrightarrow> wf_bs (x # bs) p"
   unfolding wf_bs_def by (induct p rule: decrpoly.induct) auto
 
 lemma length_le_list_ex: "length xs \<le> n \<Longrightarrow> \<exists>ys. length (xs @ ys) = n"
   by (rule exI[where x="replicate (n - length xs) z" for z]) simp
 
 lemma isnpolyh_Suc_const: "isnpolyh p (Suc n) \<Longrightarrow> isconstant p"
-  apply (cases p)
-         apply auto
-  apply (rename_tac nat a, case_tac "nat")
-   apply simp_all
-  done
+  by (simp add: isconstant_polybound0 isnpolyh_polybound0)
 
 lemma wf_bs_polyadd: "wf_bs bs p \<and> wf_bs bs q \<longrightarrow> wf_bs bs (p +\<^sub>p q)"
   by (induct p q rule: polyadd.induct) (auto simp add: Let_def wf_bs_def)
 
 lemma wf_bs_polyul: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p *\<^sub>p q)"
-  apply (induct p q arbitrary: bs rule: polymul.induct)
-                      apply (simp_all add: wf_bs_polyadd wf_bs_def)
-  apply clarsimp
-  apply (rule wf_bs_polyadd[unfolded wf_bs_def, rule_format])
-  apply auto
-  done
+proof (induct p q rule: polymul.induct)
+  case (4 c n p c' n' p')
+  then show ?case
+    apply (simp add: wf_bs_def)
+    by (metis Suc_eq_plus1 max.bounded_iff max_0L maxindex.simps(2) maxindex.simps(8) wf_bs_def wf_bs_polyadd)
+qed (simp_all add: wf_bs_def)
 
 lemma wf_bs_polyneg: "wf_bs bs p \<Longrightarrow> wf_bs bs (~\<^sub>p p)"
   by (induct p rule: polyneg.induct) (auto simp: wf_bs_def)
 
 lemma wf_bs_polysub: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p -\<^sub>p q)"
   unfolding polysub_def split_def fst_conv snd_conv
   using wf_bs_polyadd wf_bs_polyneg by blast
 
 
 subsection \<open>Canonicity of polynomial representation, see lemma isnpolyh_unique\<close>
 
 definition "polypoly bs p = map (Ipoly bs) (coefficients p)"
 definition "polypoly' bs p = map (Ipoly bs \<circ> decrpoly) (coefficients p)"
 definition "poly_nate bs p = map (Ipoly bs \<circ> decrpoly) (coefficients (polynate p))"
 
 lemma coefficients_normh: "isnpolyh p n0 \<Longrightarrow> \<forall>q \<in> set (coefficients p). isnpolyh q n0"
 proof (induct p arbitrary: n0 rule: coefficients.induct)
   case (1 c p n0)
   have cp: "isnpolyh (CN c 0 p) n0"
     by fact
   then have norm: "isnpolyh c 0" "isnpolyh p 0" "p \<noteq> 0\<^sub>p" "n0 = 0"
     by (auto simp add: isnpolyh_mono[where n'=0])
   from "1.hyps"[OF norm(2)] norm(1) norm(4) show ?case
     by simp
 qed auto
 
 lemma coefficients_isconst: "isnpolyh p n \<Longrightarrow> \<forall>q \<in> set (coefficients p). isconstant q"
   by (induct p arbitrary: n rule: coefficients.induct) (auto simp add: isnpolyh_Suc_const)
 
 lemma polypoly_polypoly':
   assumes np: "isnpolyh p n0"
   shows "polypoly (x # bs) p = polypoly' bs p"
 proof -
   let ?cf = "set (coefficients p)"
   from coefficients_normh[OF np] have cn_norm: "\<forall> q\<in> ?cf. isnpolyh q n0" .
   have "polybound0 q" if "q \<in> ?cf" for q
   proof -
     from that cn_norm have *: "isnpolyh q n0"
       by blast
     from coefficients_isconst[OF np] that have "isconstant q"
       by blast
     with isconstant_polybound0[OF *] show ?thesis
       by blast
   qed
   then have "\<forall>q \<in> ?cf. polybound0 q" ..
   then have "\<forall>q \<in> ?cf. Ipoly (x # bs) q = Ipoly bs (decrpoly q)"
     using polybound0_I[where b=x and bs=bs and b'=y] decrpoly[where x=x and bs=bs]
     by auto
   then show ?thesis
     unfolding polypoly_def polypoly'_def by simp
 qed
 
 lemma polypoly_poly:
   assumes "isnpolyh p n0"
   shows "Ipoly (x # bs) p = poly (polypoly (x # bs) p) x"
   using assms
   by (induct p arbitrary: n0 bs rule: coefficients.induct) (auto simp add: polypoly_def)
 
 lemma polypoly'_poly:
   assumes "isnpolyh p n0"
   shows "\<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup> = poly (polypoly' bs p) x"
   using polypoly_poly[OF assms, simplified polypoly_polypoly'[OF assms]] .
 
 
 lemma polypoly_poly_polybound0:
   assumes "isnpolyh p n0"
     and "polybound0 p"
   shows "polypoly bs p = [Ipoly bs p]"
   using assms
   unfolding polypoly_def
-  apply (cases p)
-         apply auto
-  apply (rename_tac nat a, case_tac nat)
-   apply auto
-  done
+  by (cases p; use gr0_conv_Suc in force)
 
 lemma head_isnpolyh: "isnpolyh p n0 \<Longrightarrow> isnpolyh (head p) n0"
   by (induct p rule: head.induct) auto
 
 lemma headn_nz[simp]: "isnpolyh p n0 \<Longrightarrow> headn p m = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
   by (cases p) auto
 
 lemma head_eq_headn0: "head p = headn p 0"
   by (induct p rule: head.induct) simp_all
 
 lemma head_nz[simp]: "isnpolyh p n0 \<Longrightarrow> head p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
   by (simp add: head_eq_headn0)
 
 lemma isnpolyh_zero_iff:
   assumes nq: "isnpolyh p n0"
     and eq :"\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a::{field_char_0, power})"
   shows "p = 0\<^sub>p"
   using nq eq
 proof (induct "maxindex p" arbitrary: p n0 rule: less_induct)
   case less
   note np = \<open>isnpolyh p n0\<close> and zp = \<open>\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)\<close>
   show "p = 0\<^sub>p"
   proof (cases "maxindex p = 0")
     case True
     with np obtain c where "p = C c" by (cases p) auto
     with zp np show ?thesis by (simp add: wf_bs_def)
   next
     case nz: False
     let ?h = "head p"
     let ?hd = "decrpoly ?h"
     let ?ihd = "maxindex ?hd"
     from head_isnpolyh[OF np] head_polybound0[OF np]
     have h: "isnpolyh ?h n0" "polybound0 ?h"
       by simp_all
     then have nhd: "isnpolyh ?hd (n0 - 1)"
       using decrpoly_normh by blast
 
     from maxindex_coefficients[of p] coefficients_head[of p, symmetric]
     have mihn: "maxindex ?h \<le> maxindex p"
       by auto
     with decr_maxindex[OF h(2)] nz have ihd_lt_n: "?ihd < maxindex p"
       by auto
 
     have "\<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" if bs: "wf_bs bs ?hd" for bs :: "'a list"
     proof -
       let ?ts = "take ?ihd bs"
       let ?rs = "drop ?ihd bs"
       from bs have ts: "wf_bs ?ts ?hd"
         by (simp add: wf_bs_def)
       have bs_ts_eq: "?ts @ ?rs = bs"
         by simp
       from wf_bs_decrpoly[OF ts] have tsh: " \<forall>x. wf_bs (x # ?ts) ?h"
         by simp
       from ihd_lt_n have "\<forall>x. length (x # ?ts) \<le> maxindex p"
         by simp
       with length_le_list_ex obtain xs where xs: "length ((x # ?ts) @ xs) = maxindex p"
         by blast
       then have "\<forall>x. wf_bs ((x # ?ts) @ xs) p"
         by (simp add: wf_bs_def)
       with zp have "\<forall>x. Ipoly ((x # ?ts) @ xs) p = 0"
         by blast
       then have "\<forall>x. Ipoly (x # (?ts @ xs)) p = 0"
         by simp
       with polypoly_poly[OF np, where ?'a = 'a] polypoly_polypoly'[OF np, where ?'a = 'a]
       have "\<forall>x. poly (polypoly' (?ts @ xs) p) x = poly [] x"
         by simp
       then have "poly (polypoly' (?ts @ xs) p) = poly []"
         by auto
       then have "\<forall>c \<in> set (coefficients p). Ipoly (?ts @ xs) (decrpoly c) = 0"
         using poly_zero[where ?'a='a] by (simp add: polypoly'_def)
       with coefficients_head[of p, symmetric]
       have *: "Ipoly (?ts @ xs) ?hd = 0"
         by simp
       from bs have wf'': "wf_bs ?ts ?hd"
         by (simp add: wf_bs_def)
       with * wf_bs_I[of ?ts ?hd xs] have "Ipoly ?ts ?hd = 0"
         by simp
       with wf'' wf_bs_I[of ?ts ?hd ?rs] bs_ts_eq show ?thesis
         by simp
     qed
     then have hdz: "\<forall>bs. wf_bs bs ?hd \<longrightarrow> \<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)"
       by blast
     from less(1)[OF ihd_lt_n nhd] hdz have "?hd = 0\<^sub>p"
       by blast
     then have "?h = 0\<^sub>p" by simp
     with head_nz[OF np] show ?thesis by simp
   qed
 qed
 
 lemma isnpolyh_unique:
   assumes np: "isnpolyh p n0"
     and nq: "isnpolyh q n1"
   shows "(\<forall>bs. \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (\<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup> :: 'a::{field_char_0,power})) \<longleftrightarrow> p = q"
 proof auto
   assume "\<forall>bs. (\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> ::'a) = \<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
   then have "\<forall>bs.\<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup>= (0::'a)"
     by simp
   then have "\<forall>bs. wf_bs bs (p -\<^sub>p q) \<longrightarrow> \<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)"
     using wf_bs_polysub[where p=p and q=q] by auto
   with isnpolyh_zero_iff[OF polysub_normh[OF np nq]] polysub_0[OF np nq] show "p = q"
     by blast
 qed
 
 
 text \<open>Consequences of unicity on the algorithms for polynomial normalization.\<close>
 
 lemma polyadd_commute:
   assumes "SORT_CONSTRAINT('a::field_char_0)"
     and np: "isnpolyh p n0"
     and nq: "isnpolyh q n1"
   shows "p +\<^sub>p q = q +\<^sub>p p"
   using isnpolyh_unique[OF polyadd_normh[OF np nq] polyadd_normh[OF nq np]]
   by simp
 
 lemma zero_normh: "isnpolyh 0\<^sub>p n"
   by simp
 
 lemma one_normh: "isnpolyh (1)\<^sub>p n"
   by simp
 
 lemma polyadd_0[simp]:
   assumes "SORT_CONSTRAINT('a::field_char_0)"
     and np: "isnpolyh p n0"
   shows "p +\<^sub>p 0\<^sub>p = p"
     and "0\<^sub>p +\<^sub>p p = p"
   using isnpolyh_unique[OF polyadd_normh[OF np zero_normh] np]
     isnpolyh_unique[OF polyadd_normh[OF zero_normh np] np] by simp_all
 
 lemma polymul_1[simp]:
   assumes "SORT_CONSTRAINT('a::field_char_0)"
     and np: "isnpolyh p n0"
   shows "p *\<^sub>p (1)\<^sub>p = p"
     and "(1)\<^sub>p *\<^sub>p p = p"
   using isnpolyh_unique[OF polymul_normh[OF np one_normh] np]
     isnpolyh_unique[OF polymul_normh[OF one_normh np] np] by simp_all
 
 lemma polymul_0[simp]:
   assumes "SORT_CONSTRAINT('a::field_char_0)"
     and np: "isnpolyh p n0"
   shows "p *\<^sub>p 0\<^sub>p = 0\<^sub>p"
     and "0\<^sub>p *\<^sub>p p = 0\<^sub>p"
   using isnpolyh_unique[OF polymul_normh[OF np zero_normh] zero_normh]
     isnpolyh_unique[OF polymul_normh[OF zero_normh np] zero_normh] by simp_all
 
 lemma polymul_commute:
   assumes "SORT_CONSTRAINT('a::field_char_0)"
     and np: "isnpolyh p n0"
     and nq: "isnpolyh q n1"
   shows "p *\<^sub>p q = q *\<^sub>p p"
   using isnpolyh_unique[OF polymul_normh[OF np nq] polymul_normh[OF nq np],
     where ?'a = "'a::{field_char_0, power}"]
   by simp
 
 declare polyneg_polyneg [simp]
 
 lemma isnpolyh_polynate_id [simp]:
   assumes "SORT_CONSTRAINT('a::field_char_0)"
     and np: "isnpolyh p n0"
   shows "polynate p = p"
   using isnpolyh_unique[where ?'a= "'a::field_char_0",
       OF polynate_norm[of p, unfolded isnpoly_def] np]
     polynate[where ?'a = "'a::field_char_0"]
   by simp
 
 lemma polynate_idempotent[simp]:
   assumes "SORT_CONSTRAINT('a::field_char_0)"
   shows "polynate (polynate p) = polynate p"
   using isnpolyh_polynate_id[OF polynate_norm[of p, unfolded isnpoly_def]] .
 
 lemma poly_nate_polypoly': "poly_nate bs p = polypoly' bs (polynate p)"
   unfolding poly_nate_def polypoly'_def ..
 
 lemma poly_nate_poly:
   "poly (poly_nate bs p) = (\<lambda>x:: 'a ::field_char_0. \<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup>)"
   using polypoly'_poly[OF polynate_norm[unfolded isnpoly_def], symmetric, of bs p]
   unfolding poly_nate_polypoly' by auto
 
 
 subsection \<open>Heads, degrees and all that\<close>
 
 lemma degree_eq_degreen0: "degree p = degreen p 0"
   by (induct p rule: degree.induct) simp_all
 
 lemma degree_polyneg:
   assumes "isnpolyh p n"
   shows "degree (polyneg p) = degree p"
-  apply (induct p rule: polyneg.induct)
-  using assms
-         apply simp_all
-  apply (case_tac na)
-   apply auto
-  done
+proof (induct p rule: polyneg.induct)
+  case (2 c n p)
+  then show ?case
+  by simp (smt (verit) degree.elims degree.simps(1) poly.inject(8))
+qed auto
 
 lemma degree_polyadd:
   assumes np: "isnpolyh p n0"
     and nq: "isnpolyh q n1"
   shows "degree (p +\<^sub>p q) \<le> max (degree p) (degree q)"
   using degreen_polyadd[OF np nq, where m= "0"] degree_eq_degreen0 by simp
 
 
 lemma degree_polysub:
   assumes np: "isnpolyh p n0"
     and nq: "isnpolyh q n1"
   shows "degree (p -\<^sub>p q) \<le> max (degree p) (degree q)"
 proof-
   from nq have nq': "isnpolyh (~\<^sub>p q) n1"
     using polyneg_normh by simp
   from degree_polyadd[OF np nq'] show ?thesis
     by (simp add: polysub_def degree_polyneg[OF nq])
 qed
 
 lemma degree_polysub_samehead:
   assumes "SORT_CONSTRAINT('a::field_char_0)"
     and np: "isnpolyh p n0"
     and nq: "isnpolyh q n1"
     and h: "head p = head q"
     and d: "degree p = degree q"
   shows "degree (p -\<^sub>p q) < degree p \<or> (p -\<^sub>p q = 0\<^sub>p)"
   unfolding polysub_def split_def fst_conv snd_conv
   using np nq h d
 proof (induct p q rule: polyadd.induct)
   case (1 c c')
   then show ?case
     by (simp add: Nsub_def Nsub0[simplified Nsub_def])
 next
   case (2 c c' n' p')
   from 2 have "degree (C c) = degree (CN c' n' p')"
     by simp
   then have nz: "n' > 0"
     by (cases n') auto
   then have "head (CN c' n' p') = CN c' n' p'"
     by (cases n') auto
   with 2 show ?case
     by simp
 next
   case (3 c n p c')
   then have "degree (C c') = degree (CN c n p)"
     by simp
   then have nz: "n > 0"
     by (cases n) auto
   then have "head (CN c n p) = CN c n p"
     by (cases n) auto
   with 3 show ?case by simp
 next
   case (4 c n p c' n' p')
   then have H:
     "isnpolyh (CN c n p) n0"
     "isnpolyh (CN c' n' p') n1"
     "head (CN c n p) = head (CN c' n' p')"
     "degree (CN c n p) = degree (CN c' n' p')"
     by simp_all
   then have degc: "degree c = 0" and degc': "degree c' = 0"
     by simp_all
   then have degnc: "degree (~\<^sub>p c) = 0" and degnc': "degree (~\<^sub>p c') = 0"
     using H(1-2) degree_polyneg by auto
   from H have cnh: "isnpolyh c (Suc n)" and c'nh: "isnpolyh c' (Suc n')"
     by simp_all
   from degree_polysub[OF cnh c'nh, simplified polysub_def] degc degc'
   have degcmc': "degree (c +\<^sub>p  ~\<^sub>pc') = 0"
     by simp
   from H have pnh: "isnpolyh p n" and p'nh: "isnpolyh p' n'"
     by auto
   consider "n = n'" | "n < n'" | "n > n'"
     by arith
   then show ?case
   proof cases
     case nn': 1
     consider "n = 0" | "n > 0" by arith
     then show ?thesis
     proof cases
       case 1
       with 4 nn' show ?thesis
         by (auto simp add: Let_def degcmc')
     next
       case 2
       with nn' H(3) have "c = c'" and "p = p'"
         by (cases n; auto)+
       with nn' 4 show ?thesis
         using polysub_same_0[OF p'nh, simplified polysub_def split_def fst_conv snd_conv]
         using polysub_same_0[OF c'nh, simplified polysub_def]
         by (simp add: Let_def)
     qed
   next
     case nn': 2
     then have n'p: "n' > 0"
       by simp
     then have headcnp':"head (CN c' n' p') = CN c' n' p'"
       by (cases n') simp_all
     with 4 nn' have degcnp': "degree (CN c' n' p') = 0"
       and degcnpeq: "degree (CN c n p) = degree (CN c' n' p')"
       by (cases n', simp_all)
     then have "n > 0"
       by (cases n) simp_all
     then have headcnp: "head (CN c n p) = CN c n p"
       by (cases n) auto
     from H(3) headcnp headcnp' nn' show ?thesis
       by auto
   next
     case nn': 3
     then have np: "n > 0" by simp
     then have headcnp:"head (CN c n p) = CN c n p"
       by (cases n) simp_all
     from 4 have degcnpeq: "degree (CN c' n' p') = degree (CN c n p)"
       by simp
     from np have degcnp: "degree (CN c n p) = 0"
       by (cases n) simp_all
     with degcnpeq have "n' > 0"
       by (cases n') simp_all
     then have headcnp': "head (CN c' n' p') = CN c' n' p'"
       by (cases n') auto
     from H(3) headcnp headcnp' nn' show ?thesis by auto
   qed
 qed auto
 
 lemma shift1_head : "isnpolyh p n0 \<Longrightarrow> head (shift1 p) = head p"
   by (induct p arbitrary: n0 rule: head.induct) (simp_all add: shift1_def)
 
 lemma funpow_shift1_head: "isnpolyh p n0 \<Longrightarrow> p \<noteq> 0\<^sub>p \<Longrightarrow> head (funpow k shift1 p) = head p"
 proof (induct k arbitrary: n0 p)
   case 0
   then show ?case
     by auto
 next
   case (Suc k n0 p)
   then have "isnpolyh (shift1 p) 0"
     by (simp add: shift1_isnpolyh)
   with Suc have "head (funpow k shift1 (shift1 p)) = head (shift1 p)"
     and "head (shift1 p) = head p"
     by (simp_all add: shift1_head)
   then show ?case
     by (simp add: funpow_swap1)
 qed
 
 lemma shift1_degree: "degree (shift1 p) = 1 + degree p"
   by (simp add: shift1_def)
 
 lemma funpow_shift1_degree: "degree (funpow k shift1 p) = k + degree p "
   by (induct k arbitrary: p) (auto simp add: shift1_degree)
 
 lemma funpow_shift1_nz: "p \<noteq> 0\<^sub>p \<Longrightarrow> funpow n shift1 p \<noteq> 0\<^sub>p"
   by (induct n arbitrary: p) simp_all
 
 lemma head_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> head p = p"
   by (induct p arbitrary: n rule: degree.induct) auto
 lemma headn_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> headn p m = p"
   by (induct p arbitrary: n rule: degreen.induct) auto
 lemma head_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> head p = p"
   by (induct p arbitrary: n rule: degree.induct) auto
 lemma head_head[simp]: "isnpolyh p n0 \<Longrightarrow> head (head p) = head p"
   by (induct p rule: head.induct) auto
 
 lemma polyadd_eq_const_degree:
   "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> polyadd p q = C c \<Longrightarrow> degree p = degree q"
-  using polyadd_eq_const_degreen degree_eq_degreen0 by simp
+  by (metis degree_eq_degreen0 polyadd_eq_const_degreen)
+
+lemma polyadd_head_aux:
+  assumes "isnpolyh p n0" "isnpolyh q n1"
+  shows "head (p +\<^sub>p q)
+           = (if degree p < degree q then head q 
+              else if degree p > degree q then head p else head (p +\<^sub>p q))"
+  using assms
+proof (induct p q rule: polyadd.induct)
+  case (4 c n p c' n' p')
+  then show ?case
+    by (auto simp add: degree_eq_degreen0 Let_def; metis head_nz)
+qed (auto simp: degree_eq_degreen0)
 
 lemma polyadd_head:
-  assumes np: "isnpolyh p n0"
-    and nq: "isnpolyh q n1"
-    and deg: "degree p \<noteq> degree q"
-  shows "head (p +\<^sub>p q) = (if degree p < degree q then head q else head p)"
-  using np nq deg
-  apply (induct p q arbitrary: n0 n1 rule: polyadd.induct)
-                      apply simp_all
-    apply (case_tac n', simp, simp)
-   apply (case_tac n, simp, simp)
-  apply (case_tac n, case_tac n', simp add: Let_def)
-    apply (auto simp add: polyadd_eq_const_degree)[2]
-    apply (metis head_nz)
-   apply (metis head_nz)
-  apply (metis degree.simps(9) gr0_conv_Suc head.simps(1) less_Suc0 not_less_eq)
-  done
+  assumes "isnpolyh p n0" and "isnpolyh q n1" and "degree p \<noteq> degree q"
+  shows "head (p +\<^sub>p q) = (if degree p < degree q then head q  else head p)"
+  by (meson assms nat_neq_iff polyadd_head_aux)
 
 lemma polymul_head_polyeq:
   assumes "SORT_CONSTRAINT('a::field_char_0)"
   shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> p \<noteq> 0\<^sub>p \<Longrightarrow> q \<noteq> 0\<^sub>p \<Longrightarrow> head (p *\<^sub>p q) = head p *\<^sub>p head q"
 proof (induct p q arbitrary: n0 n1 rule: polymul.induct)
   case (2 c c' n' p' n0 n1)
   then have "isnpolyh (head (CN c' n' p')) n1" "isnormNum c"
     by (simp_all add: head_isnpolyh)
   then show ?case
     using 2 by (cases n') auto
 next
   case (3 c n p c' n0 n1)
   then have "isnpolyh (head (CN c n p)) n0" "isnormNum c'"
     by (simp_all add: head_isnpolyh)
   then show ?case
     using 3 by (cases n) auto
 next
   case (4 c n p c' n' p' n0 n1)
   then have norm: "isnpolyh p n" "isnpolyh c (Suc n)" "isnpolyh p' n'" "isnpolyh c' (Suc n')"
     "isnpolyh (CN c n p) n" "isnpolyh (CN c' n' p') n'"
     by simp_all
   consider "n < n'" | "n' < n" | "n' = n" by arith
   then show ?case
   proof cases
     case nn': 1
     then show ?thesis
       using norm "4.hyps"(2)[OF norm(1,6)] "4.hyps"(1)[OF norm(2,6)]
-      apply simp
-      apply (cases n)
-       apply simp
-      apply (cases n')
-       apply simp_all
-      done
+      by (simp add: head_eq_headn0)
   next
     case nn': 2
     then show ?thesis
       using norm "4.hyps"(6) [OF norm(5,3)] "4.hyps"(5)[OF norm(5,4)]
-      apply simp
-      apply (cases n')
-       apply simp
-      apply (cases n)
-       apply auto
-      done
+      by (simp add: head_eq_headn0)
   next
     case nn': 3
     from nn' polymul_normh[OF norm(5,4)]
     have ncnpc': "isnpolyh (CN c n p *\<^sub>p c') n" by (simp add: min_def)
     from nn' polymul_normh[OF norm(5,3)] norm
     have ncnpp': "isnpolyh (CN c n p *\<^sub>p p') n" by simp
     from nn' ncnpp' polymul_eq0_iff[OF norm(5,3)] norm(6)
     have ncnpp0': "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
     from polyadd_normh[OF ncnpc' ncnpp0']
     have nth: "isnpolyh ((CN c n p *\<^sub>p c') +\<^sub>p (CN 0\<^sub>p n (CN c n p *\<^sub>p p'))) n"
       by (simp add: min_def)
     consider "n > 0" | "n = 0" by auto
     then show ?thesis
     proof cases
       case np: 1
       with nn' head_isnpolyh_Suc'[OF np nth]
         head_isnpolyh_Suc'[OF np norm(5)] head_isnpolyh_Suc'[OF np norm(6)[simplified nn']]
       show ?thesis by simp
     next
       case nz: 2
       from polymul_degreen[OF norm(5,4), where m="0"]
         polymul_degreen[OF norm(5,3), where m="0"] nn' nz degree_eq_degreen0
         norm(5,6) degree_npolyhCN[OF norm(6)]
       have dth: "degree (CN c 0 p *\<^sub>p c') < degree (CN 0\<^sub>p 0 (CN c 0 p *\<^sub>p p'))"
         by simp
       then have dth': "degree (CN c 0 p *\<^sub>p c') \<noteq> degree (CN 0\<^sub>p 0 (CN c 0 p *\<^sub>p p'))"
         by simp
       from polyadd_head[OF ncnpc'[simplified nz] ncnpp0'[simplified nz] dth'] dth
       show ?thesis
         using norm "4.hyps"(6)[OF norm(5,3)] "4.hyps"(5)[OF norm(5,4)] nn' nz
         by simp
     qed
   qed
 qed simp_all
 
 lemma degree_coefficients: "degree p = length (coefficients p) - 1"
   by (induct p rule: degree.induct) auto
 
 lemma degree_head[simp]: "degree (head p) = 0"
   by (induct p rule: head.induct) auto
 
 lemma degree_CN: "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<le> 1 + degree p"
   by (cases n) simp_all
 
 lemma degree_CN': "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<ge>  degree p"
   by (cases n) simp_all
 
 lemma polyadd_different_degree:
   "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> degree p \<noteq> degree q \<Longrightarrow>
     degree (polyadd p q) = max (degree p) (degree q)"
   using polyadd_different_degreen degree_eq_degreen0 by simp
 
 lemma degreen_polyneg: "isnpolyh p n0 \<Longrightarrow> degreen (~\<^sub>p p) m = degreen p m"
   by (induct p arbitrary: n0 rule: polyneg.induct) auto
 
 lemma degree_polymul:
   assumes "SORT_CONSTRAINT('a::field_char_0)"
     and np: "isnpolyh p n0"
     and nq: "isnpolyh q n1"
   shows "degree (p *\<^sub>p q) \<le> degree p + degree q"
   using polymul_degreen[OF np nq, where m="0"]  degree_eq_degreen0 by simp
 
 lemma polyneg_degree: "isnpolyh p n \<Longrightarrow> degree (polyneg p) = degree p"
   by (induct p arbitrary: n rule: degree.induct) auto
 
 lemma polyneg_head: "isnpolyh p n \<Longrightarrow> head (polyneg p) = polyneg (head p)"
   by (induct p arbitrary: n rule: degree.induct) auto
 
 
 subsection \<open>Correctness of polynomial pseudo division\<close>
 
 lemma polydivide_aux_properties:
   assumes "SORT_CONSTRAINT('a::field_char_0)"
     and np: "isnpolyh p n0"
     and ns: "isnpolyh s n1"
     and ap: "head p = a"
     and ndp: "degree p = n"
     and pnz: "p \<noteq> 0\<^sub>p"
   shows "polydivide_aux a n p k s = (k', r) \<longrightarrow> k' \<ge> k \<and> (degree r = 0 \<or> degree r < degree p) \<and>
     (\<exists>nr. isnpolyh r nr) \<and> (\<exists>q n1. isnpolyh q n1 \<and> (polypow (k' - k) a) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"
   using ns
 proof (induct "degree s" arbitrary: s k k' r n1 rule: less_induct)
   case less
   let ?qths = "\<exists>q n1. isnpolyh q n1 \<and> (a ^\<^sub>p (k' - k) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"
   let ?ths = "polydivide_aux a n p k s = (k', r) \<longrightarrow>  k \<le> k' \<and>
     (degree r = 0 \<or> degree r < degree p) \<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths"
   let ?b = "head s"
   let ?p' = "funpow (degree s - n) shift1 p"
   let ?xdn = "funpow (degree s - n) shift1 (1)\<^sub>p"
   let ?akk' = "a ^\<^sub>p (k' - k)"
   note ns = \<open>isnpolyh s n1\<close>
   from np have np0: "isnpolyh p 0"
     using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by simp
   have np': "isnpolyh ?p' 0"
     using funpow_shift1_isnpoly[OF np0[simplified isnpoly_def[symmetric]] pnz, where n="degree s - n"] isnpoly_def
     by simp
   have headp': "head ?p' = head p"
     using funpow_shift1_head[OF np pnz] by simp
   from funpow_shift1_isnpoly[where p="(1)\<^sub>p"] have nxdn: "isnpolyh ?xdn 0"
     by (simp add: isnpoly_def)
   from polypow_normh [OF head_isnpolyh[OF np0], where k="k' - k"] ap
   have nakk':"isnpolyh ?akk' 0" by blast
   show ?ths
   proof (cases "s = 0\<^sub>p")
     case True
     with np show ?thesis
       apply (clarsimp simp: polydivide_aux.simps)
-      apply (rule exI[where x="0\<^sub>p"])
-      apply simp
-      done
+      by (metis polyadd_0(1) polymul_0(1) zero_normh)
   next
     case sz: False
     show ?thesis
     proof (cases "degree s < n")
       case True
       then show ?thesis
         using ns ndp np polydivide_aux.simps
-        apply auto
-        apply (rule exI[where x="0\<^sub>p"])
-        apply simp
-        done
+        apply clarsimp
+        by (metis polyadd_0(2) polymul_0(1) polymul_1(2) zero_normh)
     next
       case dn': False
       then have dn: "degree s \<ge> n"
         by arith
       have degsp': "degree s = degree ?p'"
         using dn ndp funpow_shift1_degree[where k = "degree s - n" and p="p"]
         by simp
       show ?thesis
       proof (cases "?b = a")
         case ba: True
         then have headsp': "head s = head ?p'"
           using ap headp' by simp
         have nr: "isnpolyh (s -\<^sub>p ?p') 0"
           using polysub_normh[OF ns np'] by simp
         from degree_polysub_samehead[OF ns np' headsp' degsp']
         consider "degree (s -\<^sub>p ?p') < degree s" | "s -\<^sub>p ?p' = 0\<^sub>p" by auto
         then show ?thesis
         proof cases
           case deglt: 1
           from polydivide_aux.simps sz dn' ba
           have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
             by (simp add: Let_def)
           have "k \<le> k' \<and> (degree r = 0 \<or> degree r < degree p) \<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths"
             if h1: "polydivide_aux a n p k s = (k', r)"
           proof -
             from less(1)[OF deglt nr, of k k' r] trans[OF eq[symmetric] h1]
             have kk': "k \<le> k'"
               and nr: "\<exists>nr. isnpolyh r nr"
               and dr: "degree r = 0 \<or> degree r < degree p"
               and q1: "\<exists>q nq. isnpolyh q nq \<and> a ^\<^sub>p k' - k *\<^sub>p (s -\<^sub>p ?p') = p *\<^sub>p q +\<^sub>p r"
               by auto
             from q1 obtain q n1 where nq: "isnpolyh q n1"
               and asp: "a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p') = p *\<^sub>p q +\<^sub>p r"
               by blast
             from nr obtain nr where nr': "isnpolyh r nr"
               by blast
             from polymul_normh[OF nakk' ns] have nakks': "isnpolyh (a ^\<^sub>p (k' - k) *\<^sub>p s) 0"
               by simp
             from polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]
             have nq': "isnpolyh (?akk' *\<^sub>p ?xdn +\<^sub>p q) 0" by simp
             from polyadd_normh[OF polymul_normh[OF np
               polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]] nr']
             have nqr': "isnpolyh (p *\<^sub>p (?akk' *\<^sub>p ?xdn +\<^sub>p q) +\<^sub>p r) 0"
               by simp
             from asp have "\<forall>bs :: 'a::field_char_0 list.
               Ipoly bs (a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p')) = Ipoly bs (p *\<^sub>p q +\<^sub>p r)"
               by simp
             then have "\<forall>bs :: 'a::field_char_0 list.
               Ipoly bs (a^\<^sub>p (k' - k)*\<^sub>p s) =
               Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs ?p' + Ipoly bs p * Ipoly bs q + Ipoly bs r"
               by (simp add: field_simps)
             then have "\<forall>bs :: 'a::field_char_0 list.
               Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
               Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 (1)\<^sub>p *\<^sub>p p) +
               Ipoly bs p * Ipoly bs q + Ipoly bs r"
               by (auto simp only: funpow_shift1_1)
             then have "\<forall>bs:: 'a::field_char_0 list.
               Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
               Ipoly bs p * (Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 (1)\<^sub>p) +
               Ipoly bs q) + Ipoly bs r"
               by (simp add: field_simps)
             then have "\<forall>bs:: 'a::field_char_0 list.
               Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
               Ipoly bs (p *\<^sub>p ((a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 (1)\<^sub>p) +\<^sub>p q) +\<^sub>p r)"
               by simp
             with isnpolyh_unique[OF nakks' nqr']
             have "a ^\<^sub>p (k' - k) *\<^sub>p s =
               p *\<^sub>p ((a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 (1)\<^sub>p) +\<^sub>p q) +\<^sub>p r"
               by blast
             with nq' have ?qths
               apply (rule_tac x="(a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 (1)\<^sub>p) +\<^sub>p q" in exI)
               apply (rule_tac x="0" in exI)
               apply simp
               done
             with kk' nr dr show ?thesis
               by blast
           qed
           then show ?thesis by blast
         next
           case spz: 2
           from spz isnpolyh_unique[OF polysub_normh[OF ns np'], where q="0\<^sub>p", symmetric, where ?'a = "'a::field_char_0"]
           have "\<forall>bs:: 'a::field_char_0 list. Ipoly bs s = Ipoly bs ?p'"
             by simp
           with np nxdn have "\<forall>bs:: 'a::field_char_0 list. Ipoly bs s = Ipoly bs (?xdn *\<^sub>p p)"
             by (simp only: funpow_shift1_1) simp
           then have sp': "s = ?xdn *\<^sub>p p"
             using isnpolyh_unique[OF ns polymul_normh[OF nxdn np]]
             by blast
           have ?thesis if h1: "polydivide_aux a n p k s = (k', r)"
           proof -
             from sz dn' ba
             have "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
               by (simp add: Let_def polydivide_aux.simps)
             also have "\<dots> = (k,0\<^sub>p)"
               using spz by (simp add: polydivide_aux.simps)
             finally have "(k', r) = (k, 0\<^sub>p)"
               by (simp add: h1)
             with sp'[symmetric] ns np nxdn polyadd_0(1)[OF polymul_normh[OF np nxdn]]
               polyadd_0(2)[OF polymul_normh[OF np nxdn]] show ?thesis
               apply auto
               apply (rule exI[where x="?xdn"])
               apply (auto simp add: polymul_commute[of p])
               done
           qed
           then show ?thesis by blast
         qed
       next
         case ba: False
         from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns]
           polymul_normh[OF head_isnpolyh[OF ns] np']]
         have nth: "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0"
           by (simp add: min_def)
         have nzths: "a *\<^sub>p s \<noteq> 0\<^sub>p" "?b *\<^sub>p ?p' \<noteq> 0\<^sub>p"
           using polymul_eq0_iff[OF head_isnpolyh[OF np0, simplified ap] ns]
             polymul_eq0_iff[OF head_isnpolyh[OF ns] np']head_nz[OF np0] ap pnz sz head_nz[OF ns]
             funpow_shift1_nz[OF pnz]
           by simp_all
         from polymul_head_polyeq[OF head_isnpolyh[OF np] ns] head_nz[OF np] sz ap head_head[OF np] pnz
           polymul_head_polyeq[OF head_isnpolyh[OF ns] np'] head_nz [OF ns] sz
           funpow_shift1_nz[OF pnz, where n="degree s - n"]
         have hdth: "head (a *\<^sub>p s) = head (?b *\<^sub>p ?p')"
           using head_head[OF ns] funpow_shift1_head[OF np pnz]
             polymul_commute[OF head_isnpolyh[OF np] head_isnpolyh[OF ns]]
           by (simp add: ap)
         from polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]
           head_nz[OF np] pnz sz ap[symmetric]
           funpow_shift1_nz[OF pnz, where n="degree s - n"]
           polymul_degreen[OF head_isnpolyh[OF ns] np', where m="0"] head_nz[OF ns]
           ndp dn
         have degth: "degree (a *\<^sub>p s) = degree (?b *\<^sub>p ?p')"
           by (simp add: degree_eq_degreen0[symmetric] funpow_shift1_degree)
 
         consider "degree ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) < degree s" | "a *\<^sub>p s -\<^sub>p (?b *\<^sub>p ?p') = 0\<^sub>p"
           using degree_polysub_samehead[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns]
             polymul_normh[OF head_isnpolyh[OF ns] np'] hdth degth]
             polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]
             head_nz[OF np] pnz sz ap[symmetric]
           by (auto simp add: degree_eq_degreen0[symmetric])
         then show ?thesis
         proof cases
           case dth: 1
           from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np] ns]
             polymul_normh[OF head_isnpolyh[OF ns]np']] ap
           have nasbp': "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0"
             by simp
           have ?thesis if h1: "polydivide_aux a n p k s = (k', r)"
           proof -
             from h1 polydivide_aux.simps sz dn' ba
             have eq:"polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) = (k',r)"
               by (simp add: Let_def)
             with less(1)[OF dth nasbp', of "Suc k" k' r]
             obtain q nq nr where kk': "Suc k \<le> k'"
               and nr: "isnpolyh r nr"
               and nq: "isnpolyh q nq"
               and dr: "degree r = 0 \<or> degree r < degree p"
               and qr: "a ^\<^sub>p (k' - Suc k) *\<^sub>p ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) = p *\<^sub>p q +\<^sub>p r"
               by auto
             from kk' have kk'': "Suc (k' - Suc k) = k' - k"
               by arith
             have "Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
                 Ipoly bs p * (Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn) + Ipoly bs r"
               for bs :: "'a::field_char_0 list"
             proof -
               from qr isnpolyh_unique[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - Suc k", simplified ap] nasbp', symmetric]
               have "Ipoly bs (a ^\<^sub>p (k' - Suc k) *\<^sub>p ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p'))) = Ipoly bs (p *\<^sub>p q +\<^sub>p r)"
                 by simp
               then have "Ipoly bs a ^ (Suc (k' - Suc k)) * Ipoly bs s =
                 Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?p' + Ipoly bs r"
                 by (simp add: field_simps)
               then have "Ipoly bs a ^ (k' - k)  * Ipoly bs s =
                 Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn * Ipoly bs p + Ipoly bs r"
                 by (simp add: kk'' funpow_shift1_1[where n="degree s - n" and p="p"])
               then show ?thesis
                 by (simp add: field_simps)
             qed
             then have ieq: "\<forall>bs :: 'a::field_char_0 list.
                 Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
                 Ipoly bs (p *\<^sub>p (q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)) +\<^sub>p r)"
               by auto
             let ?q = "q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)"
             from polyadd_normh[OF nq polymul_normh[OF polymul_normh[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - Suc k"] head_isnpolyh[OF ns], simplified ap] nxdn]]
             have nqw: "isnpolyh ?q 0"
               by simp
             from ieq isnpolyh_unique[OF polymul_normh[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - k"] ns, simplified ap] polyadd_normh[OF polymul_normh[OF np nqw] nr]]
             have asth: "(a ^\<^sub>p (k' - k) *\<^sub>p s) = p *\<^sub>p (q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)) +\<^sub>p r"
               by blast
             from dr kk' nr h1 asth nqw show ?thesis
               apply simp
               apply (rule conjI)
               apply (rule exI[where x="nr"], simp)
               apply (rule exI[where x="(q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn))"], simp)
               apply (rule exI[where x="0"], simp)
               done
           qed
           then show ?thesis by blast
         next
           case spz: 2
           have hth: "\<forall>bs :: 'a::field_char_0 list.
             Ipoly bs (a *\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))"
           proof
             fix bs :: "'a::field_char_0 list"
             from isnpolyh_unique[OF nth, where ?'a="'a" and q="0\<^sub>p",simplified,symmetric] spz
             have "Ipoly bs (a*\<^sub>p s) = Ipoly bs ?b * Ipoly bs ?p'"
               by simp
             then have "Ipoly bs (a*\<^sub>p s) = Ipoly bs (?b *\<^sub>p ?xdn) * Ipoly bs p"
               by (simp add: funpow_shift1_1[where n="degree s - n" and p="p"])
             then show "Ipoly bs (a*\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))"
               by simp
           qed
           from hth have asq: "a *\<^sub>p s = p *\<^sub>p (?b *\<^sub>p ?xdn)"
             using isnpolyh_unique[where ?'a = "'a::field_char_0", OF polymul_normh[OF head_isnpolyh[OF np] ns]
                     polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]],
               simplified ap]
             by simp
           have ?ths if h1: "polydivide_aux a n p k s = (k', r)"
           proof -
             from h1 sz ba dn' spz polydivide_aux.simps polydivide_aux.simps
             have "(k', r) = (Suc k, 0\<^sub>p)"
               by (simp add: Let_def)
             with h1 np head_isnpolyh[OF np, simplified ap] ns polymul_normh[OF head_isnpolyh[OF ns] nxdn]
               polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]] asq
             show ?thesis
               apply (clarsimp simp add: Let_def)
               apply (rule exI[where x="?b *\<^sub>p ?xdn"])
               apply simp
               apply (rule exI[where x="0"], simp)
               done
           qed
           then show ?thesis by blast
         qed
       qed
     qed
   qed
 qed
 
 lemma polydivide_properties:
   assumes "SORT_CONSTRAINT('a::field_char_0)"
     and np: "isnpolyh p n0"
     and ns: "isnpolyh s n1"
     and pnz: "p \<noteq> 0\<^sub>p"
   shows "\<exists>k r. polydivide s p = (k, r) \<and>
     (\<exists>nr. isnpolyh r nr) \<and> (degree r = 0 \<or> degree r < degree p) \<and>
     (\<exists>q n1. isnpolyh q n1 \<and> polypow k (head p) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"
 proof -
   have trv: "head p = head p" "degree p = degree p"
     by simp_all
   from polydivide_def[where s="s" and p="p"] have ex: "\<exists> k r. polydivide s p = (k,r)"
     by auto
   then obtain k r where kr: "polydivide s p = (k,r)"
     by blast
   from trans[OF polydivide_def[where s="s"and p="p", symmetric] kr]
     polydivide_aux_properties[OF np ns trv pnz, where k="0" and k'="k" and r="r"]
   have "(degree r = 0 \<or> degree r < degree p) \<and>
     (\<exists>nr. isnpolyh r nr) \<and> (\<exists>q n1. isnpolyh q n1 \<and> head p ^\<^sub>p k - 0 *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"
     by blast
   with kr show ?thesis
-    apply -
-    apply (rule exI[where x="k"])
-    apply (rule exI[where x="r"])
-    apply simp
-    done
+    by auto
 qed
 
 
 subsection \<open>More about polypoly and pnormal etc\<close>
 
 definition "isnonconstant p \<longleftrightarrow> \<not> isconstant p"
 
 lemma isnonconstant_pnormal_iff:
   assumes "isnonconstant p"
   shows "pnormal (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
 proof
   let ?p = "polypoly bs p"
   assume *: "pnormal ?p"
   have "coefficients p \<noteq> []"
     using assms by (cases p) auto
   from coefficients_head[of p] last_map[OF this, of "Ipoly bs"] pnormal_last_nonzero[OF *]
   show "Ipoly bs (head p) \<noteq> 0"
     by (simp add: polypoly_def)
 next
   assume *: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
   let ?p = "polypoly bs p"
   have csz: "coefficients p \<noteq> []"
     using assms by (cases p) auto
   then have pz: "?p \<noteq> []"
     by (simp add: polypoly_def)
   then have lg: "length ?p > 0"
     by simp
   from * coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]
   have lz: "last ?p \<noteq> 0"
     by (simp add: polypoly_def)
   from pnormal_last_length[OF lg lz] show "pnormal ?p" .
 qed
 
 lemma isnonconstant_coefficients_length: "isnonconstant p \<Longrightarrow> length (coefficients p) > 1"
   unfolding isnonconstant_def
-  apply (cases p)
-  apply simp_all
-  apply (rename_tac nat a, case_tac nat)
-  apply auto
-  done
+  using isconstant.elims(3) by fastforce
 
 lemma isnonconstant_nonconstant:
   assumes "isnonconstant p"
   shows "nonconstant (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
 proof
   let ?p = "polypoly bs p"
   assume "nonconstant ?p"
   with isnonconstant_pnormal_iff[OF assms, of bs] show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
     unfolding nonconstant_def by blast
 next
   let ?p = "polypoly bs p"
   assume "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
   with isnonconstant_pnormal_iff[OF assms, of bs] have pn: "pnormal ?p"
     by blast
   have False if H: "?p = [x]" for x
   proof -
     from H have "length (coefficients p) = 1"
       by (auto simp: polypoly_def)
     with isnonconstant_coefficients_length[OF assms]
     show ?thesis by arith
   qed
   then show "nonconstant ?p"
     using pn unfolding nonconstant_def by blast
 qed
 
 lemma pnormal_length: "p \<noteq> [] \<Longrightarrow> pnormal p \<longleftrightarrow> length (pnormalize p) = length p"
-  apply (induct p)
-   apply (simp_all add: pnormal_def)
-  apply (case_tac "p = []")
-   apply simp_all
-  done
+  by (induct p) (auto simp: pnormal_id)
 
 lemma degree_degree:
   assumes "isnonconstant p"
   shows "degree p = Polynomial_List.degree (polypoly bs p) \<longleftrightarrow> \<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
     (is "?lhs \<longleftrightarrow> ?rhs")
 proof
   let ?p = "polypoly bs p"
   {
     assume ?lhs
     from isnonconstant_coefficients_length[OF assms] have "?p \<noteq> []"
       by (auto simp: polypoly_def)
     from \<open>?lhs\<close> degree_coefficients[of p] isnonconstant_coefficients_length[OF assms]
     have "length (pnormalize ?p) = length ?p"
       by (simp add: Polynomial_List.degree_def polypoly_def)
     with pnormal_length[OF \<open>?p \<noteq> []\<close>] have "pnormal ?p"
       by blast
     with isnonconstant_pnormal_iff[OF assms] show ?rhs
       by blast
   next
     assume ?rhs
     with isnonconstant_pnormal_iff[OF assms] have "pnormal ?p"
       by blast
     with degree_coefficients[of p] isnonconstant_coefficients_length[OF assms] show ?lhs
       by (auto simp: polypoly_def pnormal_def Polynomial_List.degree_def)
   }
 qed
 
 
 section \<open>Swaps -- division by a certain variable\<close>
 
 primrec swap :: "nat \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> poly"
   where
     "swap n m (C x) = C x"
   | "swap n m (Bound k) = Bound (if k = n then m else if k = m then n else k)"
   | "swap n m (Neg t) = Neg (swap n m t)"
   | "swap n m (Add s t) = Add (swap n m s) (swap n m t)"
   | "swap n m (Sub s t) = Sub (swap n m s) (swap n m t)"
   | "swap n m (Mul s t) = Mul (swap n m s) (swap n m t)"
   | "swap n m (Pw t k) = Pw (swap n m t) k"
   | "swap n m (CN c k p) = CN (swap n m c) (if k = n then m else if k=m then n else k) (swap n m p)"
 
 lemma swap:
   assumes "n < length bs"
     and "m < length bs"
   shows "Ipoly bs (swap n m t) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
 proof (induct t)
   case (Bound k)
   then show ?case
     using assms by simp
 next
   case (CN c k p)
   then show ?case
     using assms by simp
 qed simp_all
 
 lemma swap_swap_id [simp]: "swap n m (swap m n t) = t"
   by (induct t) simp_all
 
 lemma swap_commute: "swap n m p = swap m n p"
   by (induct p) simp_all
 
 lemma swap_same_id[simp]: "swap n n t = t"
   by (induct t) simp_all
 
 definition "swapnorm n m t = polynate (swap n m t)"
 
 lemma swapnorm:
   assumes nbs: "n < length bs"
     and mbs: "m < length bs"
   shows "((Ipoly bs (swapnorm n m t) :: 'a::field_char_0)) =
     Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
   using swap[OF assms] swapnorm_def by simp
 
 lemma swapnorm_isnpoly [simp]:
   assumes "SORT_CONSTRAINT('a::field_char_0)"
   shows "isnpoly (swapnorm n m p)"
   unfolding swapnorm_def by simp
 
 definition "polydivideby n s p =
   (let
     ss = swapnorm 0 n s;
     sp = swapnorm 0 n p;
     h = head sp;
     (k, r) = polydivide ss sp
    in (k, swapnorm 0 n h, swapnorm 0 n r))"
 
 lemma swap_nz [simp]: "swap n m p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
   by (induct p) simp_all
 
 fun isweaknpoly :: "poly \<Rightarrow> bool"
   where
     "isweaknpoly (C c) = True"
   | "isweaknpoly (CN c n p) \<longleftrightarrow> isweaknpoly c \<and> isweaknpoly p"
   | "isweaknpoly p = False"
 
 lemma isnpolyh_isweaknpoly: "isnpolyh p n0 \<Longrightarrow> isweaknpoly p"
   by (induct p arbitrary: n0) auto
 
 lemma swap_isweanpoly: "isweaknpoly p \<Longrightarrow> isweaknpoly (swap n m p)"
   by (induct p) auto
 
 end