diff --git a/src/HOL/Computational_Algebra/Formal_Power_Series.thy b/src/HOL/Computational_Algebra/Formal_Power_Series.thy
--- a/src/HOL/Computational_Algebra/Formal_Power_Series.thy
+++ b/src/HOL/Computational_Algebra/Formal_Power_Series.thy
@@ -1,6465 +1,6244 @@
 (*  Title:      HOL/Computational_Algebra/Formal_Power_Series.thy
     Author:     Amine Chaieb, University of Cambridge
     Author:     Jeremy Sylvestre, University of Alberta (Augustana Campus)
     Author:     Manuel Eberl, TU München
 *)
 
 section \<open>A formalization of formal power series\<close>
 
 theory Formal_Power_Series
 imports
   Complex_Main
   Euclidean_Algorithm
 begin
 
 
 subsection \<open>The type of formal power series\<close>
 
 typedef 'a fps = "{f :: nat \<Rightarrow> 'a. True}"
   morphisms fps_nth Abs_fps
   by simp
 
 notation fps_nth (infixl "$" 75)
 
 lemma expand_fps_eq: "p = q \<longleftrightarrow> (\<forall>n. p $ n = q $ n)"
   by (simp add: fps_nth_inject [symmetric] fun_eq_iff)
 
 lemmas fps_eq_iff = expand_fps_eq
 
 lemma fps_ext: "(\<And>n. p $ n = q $ n) \<Longrightarrow> p = q"
   by (simp add: expand_fps_eq)
 
 lemma fps_nth_Abs_fps [simp]: "Abs_fps f $ n = f n"
   by (simp add: Abs_fps_inverse)
 
 text \<open>Definition of the basic elements 0 and 1 and the basic operations of addition,
   negation and multiplication.\<close>
 
 instantiation fps :: (zero) zero
 begin
   definition fps_zero_def: "0 = Abs_fps (\<lambda>n. 0)"
   instance ..
 end
 
 lemma fps_zero_nth [simp]: "0 $ n = 0"
   unfolding fps_zero_def by simp
 
 lemma fps_nonzero_nth: "f \<noteq> 0 \<longleftrightarrow> (\<exists> n. f $n \<noteq> 0)"
   by (simp add: expand_fps_eq)
 
 lemma fps_nonzero_nth_minimal: "f \<noteq> 0 \<longleftrightarrow> (\<exists>n. f $ n \<noteq> 0 \<and> (\<forall>m < n. f $ m = 0))"
   (is "?lhs \<longleftrightarrow> ?rhs")
 proof
   let ?n = "LEAST n. f $ n \<noteq> 0"
   show ?rhs if ?lhs
   proof -
     from that have "\<exists>n. f $ n \<noteq> 0"
       by (simp add: fps_nonzero_nth)
     then have "f $ ?n \<noteq> 0"
       by (rule LeastI_ex)
     moreover have "\<forall>m<?n. f $ m = 0"
       by (auto dest: not_less_Least)
     ultimately have "f $ ?n \<noteq> 0 \<and> (\<forall>m<?n. f $ m = 0)" ..
     then show ?thesis ..
   qed
   show ?lhs if ?rhs
     using that by (auto simp add: expand_fps_eq)
 qed
 
 lemma fps_nonzeroI: "f$n \<noteq> 0 \<Longrightarrow> f \<noteq> 0"
   by auto
 
 instantiation fps :: ("{one, zero}") one
 begin
   definition fps_one_def: "1 = Abs_fps (\<lambda>n. if n = 0 then 1 else 0)"
   instance ..
 end
 
 lemma fps_one_nth [simp]: "1 $ n = (if n = 0 then 1 else 0)"
   unfolding fps_one_def by simp
 
 instantiation fps :: (plus) plus
 begin
   definition fps_plus_def: "(+) = (\<lambda>f g. Abs_fps (\<lambda>n. f $ n + g $ n))"
   instance ..
 end
 
 lemma fps_add_nth [simp]: "(f + g) $ n = f $ n + g $ n"
   unfolding fps_plus_def by simp
 
 instantiation fps :: (minus) minus
 begin
   definition fps_minus_def: "(-) = (\<lambda>f g. Abs_fps (\<lambda>n. f $ n - g $ n))"
   instance ..
 end
 
 lemma fps_sub_nth [simp]: "(f - g) $ n = f $ n - g $ n"
   unfolding fps_minus_def by simp
 
 instantiation fps :: (uminus) uminus
 begin
   definition fps_uminus_def: "uminus = (\<lambda>f. Abs_fps (\<lambda>n. - (f $ n)))"
   instance ..
 end
 
 lemma fps_neg_nth [simp]: "(- f) $ n = - (f $ n)"
   unfolding fps_uminus_def by simp
 
 lemma fps_neg_0 [simp]: "-(0::'a::group_add fps) = 0"
   by (rule iffD2, rule fps_eq_iff, auto)
 
 instantiation fps :: ("{comm_monoid_add, times}") times
 begin
   definition fps_times_def: "(*) = (\<lambda>f g. Abs_fps (\<lambda>n. \<Sum>i=0..n. f $ i * g $ (n - i)))"
   instance ..
 end
 
 lemma fps_mult_nth: "(f * g) $ n = (\<Sum>i=0..n. f$i * g$(n - i))"
   unfolding fps_times_def by simp
 
 lemma fps_mult_nth_0 [simp]: "(f * g) $ 0 = f $ 0 * g $ 0"
   unfolding fps_times_def by simp
 
-lemma fps_mult_nth_1 [simp]: "(f * g) $ 1 = f$0 * g$1 + f$1 * g$0"
+lemma fps_mult_nth_1: "(f * g) $ 1 = f$0 * g$1 + f$1 * g$0"
+  by (simp add: fps_mult_nth)
+
+lemma fps_mult_nth_1' [simp]: "(f * g) $ Suc 0 = f$0 * g$Suc 0 + f$Suc 0 * g$0"
   by (simp add: fps_mult_nth)
 
 lemmas mult_nth_0 = fps_mult_nth_0
 lemmas mult_nth_1 = fps_mult_nth_1
 
 instance fps :: ("{comm_monoid_add, mult_zero}") mult_zero
 proof
   fix a :: "'a fps"
   show "0 * a = 0" by (simp add: fps_ext fps_mult_nth)
   show "a * 0 = 0" by (simp add: fps_ext fps_mult_nth)
 qed
 
 declare atLeastAtMost_iff [presburger]
 declare Bex_def [presburger]
 declare Ball_def [presburger]
 
 lemma mult_delta_left:
   fixes x y :: "'a::mult_zero"
   shows "(if b then x else 0) * y = (if b then x * y else 0)"
   by simp
 
 lemma mult_delta_right:
   fixes x y :: "'a::mult_zero"
   shows "x * (if b then y else 0) = (if b then x * y else 0)"
   by simp
 
 lemma fps_one_mult:
   fixes f :: "'a::{comm_monoid_add, mult_zero, monoid_mult} fps"
   shows "1 * f = f"
   and   "f * 1 = f"
   by    (simp_all add: fps_ext fps_mult_nth mult_delta_left mult_delta_right)
 
 
 subsection \<open>Subdegrees\<close>
 
 definition subdegree :: "('a::zero) fps \<Rightarrow> nat" where
   "subdegree f = (if f = 0 then 0 else LEAST n. f$n \<noteq> 0)"
 
 lemma subdegreeI:
   assumes "f $ d \<noteq> 0" and "\<And>i. i < d \<Longrightarrow> f $ i = 0"
   shows   "subdegree f = d"
 proof-
   from assms(1) have "f \<noteq> 0" by auto
   moreover from assms(1) have "(LEAST i. f $ i \<noteq> 0) = d"
   proof (rule Least_equality)
     fix e assume "f $ e \<noteq> 0"
     with assms(2) have "\<not>(e < d)" by blast
     thus "e \<ge> d" by simp
   qed
   ultimately show ?thesis unfolding subdegree_def by simp
 qed
 
 lemma nth_subdegree_nonzero [simp,intro]: "f \<noteq> 0 \<Longrightarrow> f $ subdegree f \<noteq> 0"
 proof-
   assume "f \<noteq> 0"
   hence "subdegree f = (LEAST n. f $ n \<noteq> 0)" by (simp add: subdegree_def)
   also from \<open>f \<noteq> 0\<close> have "\<exists>n. f$n \<noteq> 0" using fps_nonzero_nth by blast
   from LeastI_ex[OF this] have "f $ (LEAST n. f $ n \<noteq> 0) \<noteq> 0" .
   finally show ?thesis .
 qed
 
 lemma nth_less_subdegree_zero [dest]: "n < subdegree f \<Longrightarrow> f $ n = 0"
 proof (cases "f = 0")
   assume "f \<noteq> 0" and less: "n < subdegree f"
   note less
   also from \<open>f \<noteq> 0\<close> have "subdegree f = (LEAST n. f $ n \<noteq> 0)" by (simp add: subdegree_def)
   finally show "f $ n = 0" using not_less_Least by blast
 qed simp_all
 
 lemma subdegree_geI:
   assumes "f \<noteq> 0" "\<And>i. i < n \<Longrightarrow> f$i = 0"
   shows   "subdegree f \<ge> n"
 proof (rule ccontr)
   assume "\<not>(subdegree f \<ge> n)"
   with assms(2) have "f $ subdegree f = 0" by simp
   moreover from assms(1) have "f $ subdegree f \<noteq> 0" by simp
   ultimately show False by contradiction
 qed
 
 lemma subdegree_greaterI:
   assumes "f \<noteq> 0" "\<And>i. i \<le> n \<Longrightarrow> f$i = 0"
   shows   "subdegree f > n"
 proof (rule ccontr)
   assume "\<not>(subdegree f > n)"
   with assms(2) have "f $ subdegree f = 0" by simp
   moreover from assms(1) have "f $ subdegree f \<noteq> 0" by simp
   ultimately show False by contradiction
 qed
 
 lemma subdegree_leI:
   "f $ n \<noteq> 0 \<Longrightarrow> subdegree f \<le> n"
   by (rule leI) auto
 
 lemma subdegree_0 [simp]: "subdegree 0 = 0"
   by (simp add: subdegree_def)
 
 lemma subdegree_1 [simp]: "subdegree 1 = 0"
   by  (cases "(1::'a) = 0")
       (auto intro: subdegreeI fps_ext simp: subdegree_def)
 
 lemma subdegree_eq_0_iff: "subdegree f = 0 \<longleftrightarrow> f = 0 \<or> f $ 0 \<noteq> 0"
 proof (cases "f = 0")
   assume "f \<noteq> 0"
   thus ?thesis
     using nth_subdegree_nonzero[OF \<open>f \<noteq> 0\<close>] by (fastforce intro!: subdegreeI)
 qed simp_all
 
 lemma subdegree_eq_0 [simp]: "f $ 0 \<noteq> 0 \<Longrightarrow> subdegree f = 0"
   by (simp add: subdegree_eq_0_iff)
 
 lemma nth_subdegree_zero_iff [simp]: "f $ subdegree f = 0 \<longleftrightarrow> f = 0"
   by (cases "f = 0") auto
 
 lemma fps_nonzero_subdegree_nonzeroI: "subdegree f > 0 \<Longrightarrow> f \<noteq> 0"
  by auto
 
 lemma subdegree_uminus [simp]:
   "subdegree (-(f::('a::group_add) fps)) = subdegree f"
 proof (cases "f=0")
   case False thus ?thesis by (force intro: subdegreeI)
 qed simp
 
 lemma subdegree_minus_commute [simp]:
   "subdegree (f-(g::('a::group_add) fps)) = subdegree (g - f)"
 proof (-, cases "g-f=0")
   case True
   have "\<And>n. (f - g) $ n = -((g - f) $ n)" by simp
   with True have "f - g = 0" by (intro fps_ext) simp
   with True show ?thesis by simp
 next
   case False show ?thesis
     using nth_subdegree_nonzero[OF False] by (fastforce intro: subdegreeI)
 qed
 
 lemma subdegree_add_ge':
   fixes   f g :: "'a::monoid_add fps"
   assumes "f + g \<noteq> 0"
   shows   "subdegree (f + g) \<ge> min (subdegree f) (subdegree g)"
   using   assms
   by      (force intro: subdegree_geI)
 
 lemma subdegree_add_ge:
   assumes "f \<noteq> -(g :: ('a :: group_add) fps)"
   shows   "subdegree (f + g) \<ge> min (subdegree f) (subdegree g)"
 proof (rule subdegree_add_ge')
   have "f + g = 0 \<Longrightarrow> False"
   proof-
     assume fg: "f + g = 0"
     have "\<And>n. f $ n = - g $ n"
     proof-
       fix n
       from fg have "(f + g) $ n = 0" by simp
       hence "f $ n + g $ n - g $ n = - g $ n" by simp
       thus "f $ n = - g $ n" by simp      
     qed
     with assms show False by (auto intro: fps_ext)
   qed
   thus "f + g \<noteq> 0" by fast
 qed
 
 lemma subdegree_add_eq1:
   assumes "f \<noteq> 0"
   and     "subdegree f < subdegree (g :: 'a::monoid_add fps)"
   shows   "subdegree (f + g) = subdegree f"
   using   assms
   by      (auto intro: subdegreeI simp: nth_less_subdegree_zero)
 
 lemma subdegree_add_eq2:
   assumes "g \<noteq> 0"
   and     "subdegree g < subdegree (f :: 'a :: monoid_add fps)"
   shows   "subdegree (f + g) = subdegree g"
   using   assms
   by      (auto intro: subdegreeI simp: nth_less_subdegree_zero)
 
 lemma subdegree_diff_eq1:
   assumes "f \<noteq> 0"
   and     "subdegree f < subdegree (g :: 'a :: group_add fps)"
   shows   "subdegree (f - g) = subdegree f"
   using   assms
   by      (auto intro: subdegreeI simp: nth_less_subdegree_zero)
 
 lemma subdegree_diff_eq1_cancel:
   assumes "f \<noteq> 0"
   and     "subdegree f < subdegree (g :: 'a :: cancel_comm_monoid_add fps)"
   shows   "subdegree (f - g) = subdegree f"
   using   assms
   by      (auto intro: subdegreeI simp: nth_less_subdegree_zero)
 
 lemma subdegree_diff_eq2:
   assumes "g \<noteq> 0"
   and     "subdegree g < subdegree (f :: 'a :: group_add fps)"
   shows   "subdegree (f - g) = subdegree g"
   using   assms
   by      (auto intro: subdegreeI simp: nth_less_subdegree_zero)
 
 lemma subdegree_diff_ge [simp]:
   assumes "f \<noteq> (g :: 'a :: group_add fps)"
   shows   "subdegree (f - g) \<ge> min (subdegree f) (subdegree g)"
 proof-
   from assms have "f = - (- g) \<Longrightarrow> False" using expand_fps_eq by fastforce
   hence "f \<noteq> - (- g)" by fast
   moreover have "f + - g = f - g" by (simp add: fps_ext)
   ultimately show ?thesis
     using subdegree_add_ge[of f "-g"] by simp
 qed
 
 lemma subdegree_diff_ge':
   fixes   f g :: "'a :: comm_monoid_diff fps"
   assumes "f - g \<noteq> 0"
   shows   "subdegree (f - g) \<ge> subdegree f"
   using   assms
   by      (auto intro: subdegree_geI simp: nth_less_subdegree_zero)
 
 lemma nth_subdegree_mult_left [simp]:
   fixes f g :: "('a :: {mult_zero,comm_monoid_add}) fps"
   shows "(f * g) $ (subdegree f) = f $ subdegree f * g $ 0"
   by    (cases "subdegree f") (simp_all add: fps_mult_nth nth_less_subdegree_zero)
 
 lemma nth_subdegree_mult_right [simp]:
   fixes f g :: "('a :: {mult_zero,comm_monoid_add}) fps"
   shows "(f * g) $ (subdegree g) = f $ 0 * g $ subdegree g"
   by    (cases "subdegree g") (simp_all add: fps_mult_nth nth_less_subdegree_zero sum.atLeast_Suc_atMost)
 
 lemma nth_subdegree_mult [simp]:
   fixes f g :: "('a :: {mult_zero,comm_monoid_add}) fps"
   shows "(f * g) $ (subdegree f + subdegree g) = f $ subdegree f * g $ subdegree g"
 proof-
   let ?n = "subdegree f + subdegree g"
   have "(f * g) $ ?n = (\<Sum>i=0..?n. f$i * g$(?n-i))"
     by (simp add: fps_mult_nth)
   also have "... = (\<Sum>i=0..?n. if i = subdegree f then f$i * g$(?n-i) else 0)"
   proof (intro sum.cong)
     fix x assume x: "x \<in> {0..?n}"
     hence "x = subdegree f \<or> x < subdegree f \<or> ?n - x < subdegree g" by auto
     thus "f $ x * g $ (?n - x) = (if x = subdegree f then f $ x * g $ (?n - x) else 0)"
       by (elim disjE conjE) auto
   qed auto
   also have "... = f $ subdegree f * g $ subdegree g" by simp
   finally show ?thesis .
 qed
 
 lemma fps_mult_nth_eq0:
   fixes f g :: "'a::{comm_monoid_add,mult_zero} fps"
   assumes "n < subdegree f + subdegree g"
   shows   "(f*g) $ n = 0"
 proof-
   have "\<And>i. i\<in>{0..n} \<Longrightarrow> f$i * g$(n - i) = 0"
   proof-
     fix i assume i: "i\<in>{0..n}"
     show "f$i * g$(n - i) = 0"
     proof (cases "i < subdegree f \<or> n - i < subdegree g")
       case False with assms i show ?thesis by auto
     qed (auto simp: nth_less_subdegree_zero)
   qed
   thus "(f * g) $ n = 0" by (simp add: fps_mult_nth)
 qed
 
 lemma fps_mult_subdegree_ge:
   fixes   f g :: "'a::{comm_monoid_add,mult_zero} fps"
   assumes "f*g \<noteq> 0"
   shows   "subdegree (f*g) \<ge> subdegree f + subdegree g"
   using   assms fps_mult_nth_eq0
   by      (intro subdegree_geI) simp
 
 lemma subdegree_mult':
   fixes   f g :: "'a::{comm_monoid_add,mult_zero} fps"
   assumes "f $ subdegree f * g $ subdegree g \<noteq> 0"
   shows   "subdegree (f*g) = subdegree f + subdegree g"
 proof-
   from assms have "(f * g) $ (subdegree f + subdegree g) \<noteq> 0" by simp
   hence "f*g \<noteq> 0" by fastforce
   hence "subdegree (f*g) \<ge> subdegree f + subdegree g" using fps_mult_subdegree_ge by fast
   moreover from assms have "subdegree (f*g) \<le> subdegree f + subdegree g"
     by (intro subdegree_leI) simp
   ultimately show ?thesis by simp
 qed
 
 lemma subdegree_mult [simp]:
   fixes   f g :: "'a :: {semiring_no_zero_divisors} fps"
   assumes "f \<noteq> 0" "g \<noteq> 0"
   shows   "subdegree (f * g) = subdegree f + subdegree g"
   using   assms
   by      (intro subdegree_mult') simp
 
 lemma fps_mult_nth_conv_upto_subdegree_left:
   fixes f g :: "('a :: {mult_zero,comm_monoid_add}) fps"
   shows "(f * g) $ n = (\<Sum>i=subdegree f..n. f $ i * g $ (n - i))"
 proof (cases "subdegree f \<le> n")
   case True
   hence "{0..n} = {0..<subdegree f} \<union> {subdegree f..n}" by auto
   moreover have "{0..<subdegree f} \<inter> {subdegree f..n} = {}" by auto
   ultimately show ?thesis
     using nth_less_subdegree_zero[of _ f]
     by    (simp add: fps_mult_nth sum.union_disjoint)
 qed (simp add: fps_mult_nth nth_less_subdegree_zero)
 
 lemma fps_mult_nth_conv_upto_subdegree_right:
   fixes f g :: "('a :: {mult_zero,comm_monoid_add}) fps"
   shows "(f * g) $ n = (\<Sum>i=0..n - subdegree g. f $ i * g $ (n - i))"
 proof-
   have "{0..n} = {0..n - subdegree g} \<union> {n - subdegree g<..n}" by auto
   moreover have "{0..n - subdegree g} \<inter> {n - subdegree g<..n} = {}" by auto
   moreover have "\<forall>i\<in>{n - subdegree g<..n}. g $ (n - i) = 0"
     using nth_less_subdegree_zero[of _ g] by auto
   ultimately show ?thesis by (simp add: fps_mult_nth sum.union_disjoint)
 qed
 
 lemma fps_mult_nth_conv_inside_subdegrees:
   fixes f g :: "('a :: {mult_zero,comm_monoid_add}) fps"
   shows "(f * g) $ n = (\<Sum>i=subdegree f..n - subdegree g. f $ i * g $ (n - i))"
 proof (cases "subdegree f \<le> n - subdegree g")
   case True
   hence "{subdegree f..n} = {subdegree f..n - subdegree g} \<union> {n - subdegree g<..n}"
     by auto
   moreover have "{subdegree f..n - subdegree g} \<inter> {n - subdegree g<..n} = {}" by auto
   moreover have "\<forall>i\<in>{n - subdegree g<..n}. f $ i * g $ (n - i) = 0"
     using nth_less_subdegree_zero[of _ g] by auto
   ultimately show ?thesis
     using fps_mult_nth_conv_upto_subdegree_left[of f g n]
     by    (simp add: sum.union_disjoint)
 next
   case False
   hence 1: "subdegree f > n - subdegree g" by simp
   show ?thesis
   proof (cases "f*g = 0")
     case False
     with 1 have "n < subdegree (f*g)" using fps_mult_subdegree_ge[of f g] by simp
     with 1 show ?thesis by auto
   qed (simp add: 1)
 qed
 
 lemma fps_mult_nth_outside_subdegrees:
   fixes f g :: "('a :: {mult_zero,comm_monoid_add}) fps"
   shows "n < subdegree f \<Longrightarrow> (f * g) $ n = 0"
   and   "n < subdegree g \<Longrightarrow> (f * g) $ n = 0"
   by    (auto simp: fps_mult_nth_conv_inside_subdegrees)
 
 
 subsection \<open>Formal power series form a commutative ring with unity, if the range of sequences
   they represent is a commutative ring with unity\<close>
 
 instance fps :: (semigroup_add) semigroup_add
 proof
   fix a b c :: "'a fps"
   show "a + b + c = a + (b + c)"
     by (simp add: fps_ext add.assoc)
 qed
 
 instance fps :: (ab_semigroup_add) ab_semigroup_add
 proof
   fix a b :: "'a fps"
   show "a + b = b + a"
     by (simp add: fps_ext add.commute)
 qed
 
 instance fps :: (monoid_add) monoid_add
 proof
   fix a :: "'a fps"
   show "0 + a = a" by (simp add: fps_ext)
   show "a + 0 = a" by (simp add: fps_ext)
 qed
 
 instance fps :: (comm_monoid_add) comm_monoid_add
 proof
   fix a :: "'a fps"
   show "0 + a = a" by (simp add: fps_ext)
 qed
 
 instance fps :: (cancel_semigroup_add) cancel_semigroup_add
 proof
   fix a b c :: "'a fps"
   show "b = c" if "a + b = a + c"
     using that by (simp add: expand_fps_eq)
   show "b = c" if "b + a = c + a"
     using that by (simp add: expand_fps_eq)
 qed
 
 instance fps :: (cancel_ab_semigroup_add) cancel_ab_semigroup_add
 proof
   fix a b c :: "'a fps"
   show "a + b - a = b"
     by (simp add: expand_fps_eq)
   show "a - b - c = a - (b + c)"
     by (simp add: expand_fps_eq diff_diff_eq)
 qed
 
 instance fps :: (cancel_comm_monoid_add) cancel_comm_monoid_add ..
 
 instance fps :: (group_add) group_add
 proof
   fix a b :: "'a fps"
   show "- a + a = 0" by (simp add: fps_ext)
   show "a + - b = a - b" by (simp add: fps_ext)
 qed
 
 instance fps :: (ab_group_add) ab_group_add
 proof
   fix a b :: "'a fps"
   show "- a + a = 0" by (simp add: fps_ext)
   show "a - b = a + - b" by (simp add: fps_ext)
 qed
 
 instance fps :: (zero_neq_one) zero_neq_one
   by standard (simp add: expand_fps_eq)
 
 lemma fps_mult_assoc_lemma:
   fixes k :: nat
     and f :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a::comm_monoid_add"
   shows "(\<Sum>j=0..k. \<Sum>i=0..j. f i (j - i) (n - j)) =
          (\<Sum>j=0..k. \<Sum>i=0..k - j. f j i (n - j - i))"
   by (induct k) (simp_all add: Suc_diff_le sum.distrib add.assoc)
 
 instance fps :: (semiring_0) semiring_0
 proof
   fix a b c :: "'a fps"
   show "(a + b) * c = a * c + b * c"
     by (simp add: expand_fps_eq fps_mult_nth distrib_right sum.distrib)
   show "a * (b + c) = a * b + a * c"
     by (simp add: expand_fps_eq fps_mult_nth distrib_left sum.distrib)
   show "(a * b) * c = a * (b * c)"
   proof (rule fps_ext)
     fix n :: nat
     have "(\<Sum>j=0..n. \<Sum>i=0..j. a$i * b$(j - i) * c$(n - j)) =
           (\<Sum>j=0..n. \<Sum>i=0..n - j. a$j * b$i * c$(n - j - i))"
       by (rule fps_mult_assoc_lemma)
     then show "((a * b) * c) $ n = (a * (b * c)) $ n"
       by (simp add: fps_mult_nth sum_distrib_left sum_distrib_right mult.assoc)
   qed
 qed
 
 instance fps :: (semiring_0_cancel) semiring_0_cancel ..
 
 lemma fps_mult_commute_lemma:
   fixes n :: nat
     and f :: "nat \<Rightarrow> nat \<Rightarrow> 'a::comm_monoid_add"
   shows "(\<Sum>i=0..n. f i (n - i)) = (\<Sum>i=0..n. f (n - i) i)"
   by (rule sum.reindex_bij_witness[where i="(-) n" and j="(-) n"]) auto
 
 instance fps :: (comm_semiring_0) comm_semiring_0
 proof
   fix a b c :: "'a fps"
   show "a * b = b * a"
   proof (rule fps_ext)
     fix n :: nat
     have "(\<Sum>i=0..n. a$i * b$(n - i)) = (\<Sum>i=0..n. a$(n - i) * b$i)"
       by (rule fps_mult_commute_lemma)
     then show "(a * b) $ n = (b * a) $ n"
       by (simp add: fps_mult_nth mult.commute)
   qed 
 qed (simp add: distrib_right)
 
 instance fps :: (comm_semiring_0_cancel) comm_semiring_0_cancel ..
 
 instance fps :: (semiring_1) semiring_1
 proof
   fix a :: "'a fps"
   show "1 * a = a" "a * 1 = a" by (simp_all add: fps_one_mult)
 qed
 
 instance fps :: (comm_semiring_1) comm_semiring_1
   by standard simp
 
 instance fps :: (semiring_1_cancel) semiring_1_cancel ..
 
 
 subsection \<open>Selection of the nth power of the implicit variable in the infinite sum\<close>
 
 lemma fps_square_nth: "(f^2) $ n = (\<Sum>k\<le>n. f $ k * f $ (n - k))"
   by (simp add: power2_eq_square fps_mult_nth atLeast0AtMost)
 
 lemma fps_sum_nth: "sum f S $ n = sum (\<lambda>k. (f k) $ n) S"
 proof (cases "finite S")
   case True
   then show ?thesis by (induct set: finite) auto
 next
   case False
   then show ?thesis by simp
 qed
 
 
 subsection \<open>Injection of the basic ring elements and multiplication by scalars\<close>
 
 definition "fps_const c = Abs_fps (\<lambda>n. if n = 0 then c else 0)"
 
 lemma fps_nth_fps_const [simp]: "fps_const c $ n = (if n = 0 then c else 0)"
   unfolding fps_const_def by simp
 
 lemma fps_const_0_eq_0 [simp]: "fps_const 0 = 0"
   by (simp add: fps_ext)
 
 lemma fps_const_nonzero_eq_nonzero: "c \<noteq> 0 \<Longrightarrow> fps_const c \<noteq> 0"
   using fps_nonzeroI[of "fps_const c" 0] by simp
 
 lemma fps_const_1_eq_1 [simp]: "fps_const 1 = 1"
   by (simp add: fps_ext)
 
 lemma subdegree_fps_const [simp]: "subdegree (fps_const c) = 0"
   by (cases "c = 0") (auto intro!: subdegreeI)
 
 lemma fps_const_neg [simp]: "- (fps_const (c::'a::group_add)) = fps_const (- c)"
   by (simp add: fps_ext)
 
 lemma fps_const_add [simp]: "fps_const (c::'a::monoid_add) + fps_const d = fps_const (c + d)"
   by (simp add: fps_ext)
 
 lemma fps_const_add_left: "fps_const (c::'a::monoid_add) + f =
     Abs_fps (\<lambda>n. if n = 0 then c + f$0 else f$n)"
   by (simp add: fps_ext)
 
 lemma fps_const_add_right: "f + fps_const (c::'a::monoid_add) =
     Abs_fps (\<lambda>n. if n = 0 then f$0 + c else f$n)"
   by (simp add: fps_ext)
 
 lemma fps_const_sub [simp]: "fps_const (c::'a::group_add) - fps_const d = fps_const (c - d)"
   by (simp add: fps_ext)
 
 lemmas fps_const_minus = fps_const_sub
 
 lemma fps_const_mult[simp]:
   fixes c d :: "'a::{comm_monoid_add,mult_zero}"
   shows "fps_const c * fps_const d = fps_const (c * d)"
   by    (simp add: fps_eq_iff fps_mult_nth sum.neutral)
 
 lemma fps_const_mult_left:
   "fps_const (c::'a::{comm_monoid_add,mult_zero}) * f = Abs_fps (\<lambda>n. c * f$n)"
   unfolding fps_eq_iff fps_mult_nth
   by (simp add: fps_const_def mult_delta_left)
 
 lemma fps_const_mult_right:
   "f * fps_const (c::'a::{comm_monoid_add,mult_zero}) = Abs_fps (\<lambda>n. f$n * c)"
   unfolding fps_eq_iff fps_mult_nth
   by (simp add: fps_const_def mult_delta_right)
 
 lemma fps_mult_left_const_nth [simp]:
   "(fps_const (c::'a::{comm_monoid_add,mult_zero}) * f)$n = c* f$n"
   by (simp add: fps_mult_nth mult_delta_left)
 
 lemma fps_mult_right_const_nth [simp]:
   "(f * fps_const (c::'a::{comm_monoid_add,mult_zero}))$n = f$n * c"
   by (simp add: fps_mult_nth mult_delta_right)
 
 lemma fps_const_power [simp]: "fps_const c ^ n = fps_const (c^n)"
   by (induct n) auto
 
 
 subsection \<open>Formal power series form an integral domain\<close>
 
 instance fps :: (ring) ring ..
 
 instance fps :: (comm_ring) comm_ring ..
 
 instance fps :: (ring_1) ring_1 ..
 
 instance fps :: (comm_ring_1) comm_ring_1 ..
 
 instance fps :: (semiring_no_zero_divisors) semiring_no_zero_divisors
 proof
   fix a b :: "'a fps"
   assume "a \<noteq> 0" and "b \<noteq> 0"
   hence "(a * b) $ (subdegree a + subdegree b) \<noteq> 0" by simp
   thus "a * b \<noteq> 0" using fps_nonzero_nth by fast
 qed
 
 instance fps :: (semiring_1_no_zero_divisors) semiring_1_no_zero_divisors ..
 
 instance fps :: ("{cancel_semigroup_add,semiring_no_zero_divisors_cancel}")
   semiring_no_zero_divisors_cancel
 proof
   fix a b c :: "'a fps"
   show "(a * c = b * c) = (c = 0 \<or> a = b)"
   proof
     assume ab: "a * c = b * c"
     have "c \<noteq> 0 \<Longrightarrow> a = b"
     proof (rule fps_ext)
       fix n
       assume c: "c \<noteq> 0"
       show "a $ n = b $ n"
       proof (induct n rule: nat_less_induct)
         case (1 n)
         with ab c show ?case
           using fps_mult_nth_conv_upto_subdegree_right[of a c "subdegree c + n"]
                 fps_mult_nth_conv_upto_subdegree_right[of b c "subdegree c + n"]
           by    (cases n) auto
       qed
     qed
     thus "c = 0 \<or> a = b" by fast
   qed auto
   show "(c * a = c * b) = (c = 0 \<or> a = b)"
   proof
     assume ab: "c * a = c * b"
     have "c \<noteq> 0 \<Longrightarrow> a = b"
     proof (rule fps_ext)
       fix n
       assume c: "c \<noteq> 0"
       show "a $ n = b $ n"
       proof (induct n rule: nat_less_induct)
         case (1 n)
         moreover have "\<forall>i\<in>{Suc (subdegree c)..subdegree c + n}. subdegree c + n - i < n" by auto
         ultimately show ?case
           using ab c fps_mult_nth_conv_upto_subdegree_left[of c a "subdegree c + n"]
                 fps_mult_nth_conv_upto_subdegree_left[of c b "subdegree c + n"]
           by    (simp add: sum.atLeast_Suc_atMost)
       qed
     qed    
     thus "c = 0 \<or> a = b" by fast
   qed auto
 qed
 
 instance fps :: (ring_no_zero_divisors) ring_no_zero_divisors ..
 
 instance fps :: (ring_1_no_zero_divisors) ring_1_no_zero_divisors ..
 
 instance fps :: (idom) idom ..
 
 lemma fps_numeral_fps_const: "numeral k = fps_const (numeral k)"
   by (induct k) (simp_all only: numeral.simps fps_const_1_eq_1 fps_const_add [symmetric])
 
 lemmas numeral_fps_const = fps_numeral_fps_const
 
 lemma neg_numeral_fps_const:
   "(- numeral k :: 'a :: ring_1 fps) = fps_const (- numeral k)"
   by (simp add: numeral_fps_const)
 
 lemma fps_numeral_nth: "numeral n $ i = (if i = 0 then numeral n else 0)"
   by (simp add: numeral_fps_const)
 
 lemma fps_numeral_nth_0 [simp]: "numeral n $ 0 = numeral n"
   by (simp add: numeral_fps_const)
 
 lemma subdegree_numeral [simp]: "subdegree (numeral n) = 0"
   by (simp add: numeral_fps_const)
 
 lemma fps_of_nat: "fps_const (of_nat c) = of_nat c"
   by (induction c) (simp_all add: fps_const_add [symmetric] del: fps_const_add)
 
 lemma fps_of_int: "fps_const (of_int c) = of_int c"
   by (induction c) (simp_all add: fps_const_minus [symmetric] fps_of_nat fps_const_neg [symmetric] 
                              del: fps_const_minus fps_const_neg)
 
 lemma fps_nth_of_nat [simp]:
   "(of_nat c) $ n = (if n=0 then of_nat c else 0)"
   by (simp add: fps_of_nat[symmetric])
 
 lemma fps_nth_of_int [simp]:
   "(of_int c) $ n = (if n=0 then of_int c else 0)"
   by (simp add: fps_of_int[symmetric])
 
 lemma fps_mult_of_nat_nth [simp]:
   shows "(of_nat k * f) $ n = of_nat k * f$n"
   and   "(f * of_nat k ) $ n = f$n * of_nat k"
   by    (simp_all add: fps_of_nat[symmetric])
 
 lemma fps_mult_of_int_nth [simp]:
   shows "(of_int k * f) $ n = of_int k * f$n"
   and   "(f * of_int k ) $ n = f$n * of_int k"
   by    (simp_all add: fps_of_int[symmetric])
 
 lemma numeral_neq_fps_zero [simp]: "(numeral f :: 'a :: field_char_0 fps) \<noteq> 0"
 proof
   assume "numeral f = (0 :: 'a fps)"
   from arg_cong[of _ _ "\<lambda>F. F $ 0", OF this] show False by simp
 qed 
 
 lemma subdegree_power_ge:
   "f^n \<noteq> 0 \<Longrightarrow> subdegree (f^n) \<ge> n * subdegree f"
 proof (induct n)
   case (Suc n) thus ?case using fps_mult_subdegree_ge by fastforce
 qed simp
 
 lemma fps_pow_nth_below_subdegree:
   "k < n * subdegree f \<Longrightarrow> (f^n) $ k = 0"
 proof (cases "f^n = 0")
   case False
   assume "k < n * subdegree f"
   with False have "k < subdegree (f^n)" using subdegree_power_ge[of f n] by simp
   thus "(f^n) $ k = 0" by auto
 qed simp
 
 lemma fps_pow_base [simp]:
   "(f ^ n) $ (n * subdegree f) = (f $ subdegree f) ^ n"
 proof (induct n)
   case (Suc n)
   show ?case
   proof (cases "Suc n * subdegree f < subdegree f + subdegree (f^n)")
     case True with Suc show ?thesis
       by (auto simp: fps_mult_nth_eq0 distrib_right)
   next
     case False
     hence "\<forall>i\<in>{Suc (subdegree f)..Suc n * subdegree f - subdegree (f ^ n)}.
             f ^ n $ (Suc n * subdegree f - i) = 0"
      by (auto simp: fps_pow_nth_below_subdegree)
    with False Suc show ?thesis
       using fps_mult_nth_conv_inside_subdegrees[of f "f^n" "Suc n * subdegree f"]
             sum.atLeast_Suc_atMost[of
               "subdegree f"
               "Suc n * subdegree f - subdegree (f ^ n)"
               "\<lambda>i. f $ i * f ^ n $ (Suc n * subdegree f - i)"
             ]
       by    simp
   qed
 qed simp
 
 lemma subdegree_power_eqI:
   fixes f :: "'a::semiring_1 fps"
   shows "(f $ subdegree f) ^ n \<noteq> 0 \<Longrightarrow> subdegree (f ^ n) = n * subdegree f"
 proof (induct n)
   case (Suc n)
   from Suc have 1: "subdegree (f ^ n) = n * subdegree f" by fastforce
   with Suc(2) have "f $ subdegree f * f ^ n $ subdegree (f ^ n) \<noteq> 0" by simp
   with 1 show ?case using subdegree_mult'[of f "f^n"] by simp
 qed simp
 
 lemma subdegree_power [simp]:
   "subdegree ((f :: ('a :: semiring_1_no_zero_divisors) fps) ^ n) = n * subdegree f"
   by (cases "f = 0"; induction n) simp_all
 
 
 subsection \<open>The efps_Xtractor series fps_X\<close>
 
 lemma minus_one_power_iff: "(- (1::'a::ring_1)) ^ n = (if even n then 1 else - 1)"
   by (induct n) auto
 
 definition "fps_X = Abs_fps (\<lambda>n. if n = 1 then 1 else 0)"
 
 lemma subdegree_fps_X [simp]: "subdegree (fps_X :: ('a :: zero_neq_one) fps) = 1"
   by (auto intro!: subdegreeI simp: fps_X_def)
 
 lemma fps_X_mult_nth [simp]:
   fixes f :: "'a::{comm_monoid_add,mult_zero,monoid_mult} fps"
   shows "(fps_X * f) $ n = (if n = 0 then 0 else f $ (n - 1))"
 proof (cases n)
   case (Suc m)
   moreover have "(fps_X * f) $ Suc m = f $ (Suc m - 1)"
   proof (cases m)
     case 0 thus ?thesis using fps_mult_nth_1[of "fps_X" f] by (simp add: fps_X_def)
   next
     case (Suc k) thus ?thesis by (simp add: fps_mult_nth fps_X_def sum.atLeast_Suc_atMost)
   qed
   ultimately show ?thesis by simp
 qed (simp add: fps_X_def)
 
 lemma fps_X_mult_right_nth [simp]:
   fixes a :: "'a::{comm_monoid_add,mult_zero,monoid_mult} fps"
   shows "(a * fps_X) $ n = (if n = 0 then 0 else a $ (n - 1))"
 proof (cases n)
   case (Suc m)
   moreover have "(a * fps_X) $ Suc m = a $ (Suc m - 1)"
   proof (cases m)
     case 0 thus ?thesis using fps_mult_nth_1[of a "fps_X"] by (simp add: fps_X_def)
   next
     case (Suc k)
     hence "(a * fps_X) $ Suc m = (\<Sum>i=0..k. a$i * fps_X$(Suc m - i)) + a$(Suc k)"
       by (simp add: fps_mult_nth fps_X_def)
     moreover have "\<forall>i\<in>{0..k}. a$i * fps_X$(Suc m - i) = 0" by (auto simp: Suc fps_X_def)
     ultimately show ?thesis by (simp add: Suc)
   qed
   ultimately show ?thesis by simp
 qed (simp add: fps_X_def)
 
 lemma fps_mult_fps_X_commute:
   fixes a :: "'a::{comm_monoid_add,mult_zero,monoid_mult} fps"
   shows "fps_X * a = a * fps_X" 
   by (simp add: fps_eq_iff)
 
 lemma fps_mult_fps_X_power_commute: "fps_X ^ k * a = a * fps_X ^ k"
 proof (induct k)
   case (Suc k)
   hence "fps_X ^ Suc k * a = a * fps_X * fps_X ^ k"
     by (simp add: mult.assoc fps_mult_fps_X_commute[symmetric])
   thus ?case by (simp add: mult.assoc)
 qed simp
 
 lemma fps_subdegree_mult_fps_X:
   fixes   f :: "'a::{comm_monoid_add,mult_zero,monoid_mult} fps"
   assumes "f \<noteq> 0"
   shows   "subdegree (fps_X * f) = subdegree f + 1"
   and     "subdegree (f * fps_X) = subdegree f + 1"
 proof-
   show "subdegree (fps_X * f) = subdegree f + 1"
   proof (intro subdegreeI)
     fix i :: nat assume i: "i < subdegree f + 1"
     show "(fps_X * f) $ i = 0"
     proof (cases "i=0")
       case False with i show ?thesis by (simp add: nth_less_subdegree_zero)
     next
       case True thus ?thesis using fps_X_mult_nth[of f i] by simp
     qed
   qed (simp add: assms)
   thus "subdegree (f * fps_X) = subdegree f + 1"
     by (simp add: fps_mult_fps_X_commute)
 qed
 
 lemma fps_mult_fps_X_nonzero:
   fixes   f :: "'a::{comm_monoid_add,mult_zero,monoid_mult} fps"
   assumes "f \<noteq> 0"
   shows   "fps_X * f \<noteq> 0"
   and     "f * fps_X \<noteq> 0"
   using   assms fps_subdegree_mult_fps_X[of f]
           fps_nonzero_subdegree_nonzeroI[of "fps_X * f"]
           fps_nonzero_subdegree_nonzeroI[of "f * fps_X"]
   by      auto
 
 lemma fps_mult_fps_X_power_nonzero:
   assumes "f \<noteq> 0"
   shows   "fps_X ^ n * f \<noteq> 0"
   and     "f * fps_X ^ n \<noteq> 0"
 proof -
   show "fps_X ^ n * f \<noteq> 0"
     by (induct n) (simp_all add: assms mult.assoc fps_mult_fps_X_nonzero(1))
   thus "f * fps_X ^ n \<noteq> 0"
     by (simp add: fps_mult_fps_X_power_commute)
 qed
 
 lemma fps_X_power_iff: "fps_X ^ n = Abs_fps (\<lambda>m. if m = n then 1 else 0)"
   by (induction n) (auto simp: fps_eq_iff)
 
 lemma fps_X_nth[simp]: "fps_X$n = (if n = 1 then 1 else 0)"
   by (simp add: fps_X_def)
 
 lemma fps_X_power_nth[simp]: "(fps_X^k) $n = (if n = k then 1 else 0)"
   by (simp add: fps_X_power_iff)
 
 lemma fps_X_power_subdegree: "subdegree (fps_X^n) = n"
   by (auto intro: subdegreeI)
 
 lemma fps_X_power_mult_nth:
   "(fps_X^k * f) $ n = (if n < k then 0 else f $ (n - k))"
   by  (cases "n<k")
       (simp_all add: fps_mult_nth_conv_upto_subdegree_left fps_X_power_subdegree sum.atLeast_Suc_atMost)
 
 lemma fps_X_power_mult_right_nth:
   "(f * fps_X^k) $ n = (if n < k then 0 else f $ (n - k))"
   using fps_mult_fps_X_power_commute[of k f] fps_X_power_mult_nth[of k f] by simp
 
 lemma fps_subdegree_mult_fps_X_power:
   assumes "f \<noteq> 0"
   shows   "subdegree (fps_X ^ n * f) = subdegree f + n"
   and     "subdegree (f * fps_X ^ n) = subdegree f + n"
 proof -
   from assms show "subdegree (fps_X ^ n * f) = subdegree f + n"
     by (induct n)
        (simp_all add: algebra_simps fps_subdegree_mult_fps_X(1) fps_mult_fps_X_power_nonzero(1))
   thus "subdegree (f * fps_X ^ n) = subdegree f + n"
     by (simp add: fps_mult_fps_X_power_commute)
 qed
 
 lemma fps_mult_fps_X_plus_1_nth:
   "((1+fps_X)*a) $n = (if n = 0 then (a$n :: 'a::semiring_1) else a$n + a$(n - 1))"
 proof (cases n)
   case 0
   then show ?thesis
     by (simp add: fps_mult_nth)
 next
   case (Suc m)
   have "((1 + fps_X)*a) $ n = sum (\<lambda>i. (1 + fps_X) $ i * a $ (n - i)) {0..n}"
     by (simp add: fps_mult_nth)
   also have "\<dots> = sum (\<lambda>i. (1+fps_X)$i * a$(n-i)) {0.. 1}"
     unfolding Suc by (rule sum.mono_neutral_right) auto
   also have "\<dots> = (if n = 0 then a$n else a$n + a$(n - 1))"
     by (simp add: Suc)
   finally show ?thesis .
 qed
 
 lemma fps_mult_right_fps_X_plus_1_nth:
   fixes a :: "'a :: semiring_1 fps"
   shows "(a*(1+fps_X)) $ n = (if n = 0 then a$n else a$n + a$(n - 1))"
   using fps_mult_fps_X_plus_1_nth
   by    (simp add: distrib_left fps_mult_fps_X_commute distrib_right)
 
 lemma fps_X_neq_fps_const [simp]: "(fps_X :: 'a :: zero_neq_one fps) \<noteq> fps_const c"
 proof
   assume "(fps_X::'a fps) = fps_const (c::'a)"
   hence "fps_X$1 = (fps_const (c::'a))$1" by (simp only:)
   thus False by auto
 qed
 
 lemma fps_X_neq_zero [simp]: "(fps_X :: 'a :: zero_neq_one fps) \<noteq> 0"
   by (simp only: fps_const_0_eq_0[symmetric] fps_X_neq_fps_const) simp
 
 lemma fps_X_neq_one [simp]: "(fps_X :: 'a :: zero_neq_one fps) \<noteq> 1"
   by (simp only: fps_const_1_eq_1[symmetric] fps_X_neq_fps_const) simp
 
 lemma fps_X_neq_numeral [simp]: "fps_X \<noteq> numeral c"
   by (simp only: numeral_fps_const fps_X_neq_fps_const) simp
 
 lemma fps_X_pow_eq_fps_X_pow_iff [simp]: "fps_X ^ m = fps_X ^ n \<longleftrightarrow> m = n"
 proof
   assume "(fps_X :: 'a fps) ^ m = fps_X ^ n"
   hence "(fps_X :: 'a fps) ^ m $ m = fps_X ^ n $ m" by (simp only:)
   thus "m = n" by (simp split: if_split_asm)
 qed simp_all
 
 
 subsection \<open>Shifting and slicing\<close>
 
 definition fps_shift :: "nat \<Rightarrow> 'a fps \<Rightarrow> 'a fps" where
   "fps_shift n f = Abs_fps (\<lambda>i. f $ (i + n))"
 
 lemma fps_shift_nth [simp]: "fps_shift n f $ i = f $ (i + n)"
   by (simp add: fps_shift_def)
 
 lemma fps_shift_0 [simp]: "fps_shift 0 f = f"
   by (intro fps_ext) (simp add: fps_shift_def)
 
 lemma fps_shift_zero [simp]: "fps_shift n 0 = 0"
   by (intro fps_ext) (simp add: fps_shift_def)
 
 lemma fps_shift_one: "fps_shift n 1 = (if n = 0 then 1 else 0)"
   by (intro fps_ext) (simp add: fps_shift_def)
 
 lemma fps_shift_fps_const: "fps_shift n (fps_const c) = (if n = 0 then fps_const c else 0)"
   by (intro fps_ext) (simp add: fps_shift_def)
 
 lemma fps_shift_numeral: "fps_shift n (numeral c) = (if n = 0 then numeral c else 0)"
   by (simp add: numeral_fps_const fps_shift_fps_const)
 
 lemma fps_shift_fps_X [simp]:
   "n \<ge> 1 \<Longrightarrow> fps_shift n fps_X = (if n = 1 then 1 else 0)"
   by (intro fps_ext) (auto simp: fps_X_def)
 
 lemma fps_shift_fps_X_power [simp]:
   "n \<le> m \<Longrightarrow> fps_shift n (fps_X ^ m) = fps_X ^ (m - n)"
  by (intro fps_ext) auto
 
 lemma fps_shift_subdegree [simp]:
   "n \<le> subdegree f \<Longrightarrow> subdegree (fps_shift n f) = subdegree f - n"
   by (cases "f=0") (auto intro: subdegreeI simp: nth_less_subdegree_zero)
 
 lemma fps_shift_fps_shift:
   "fps_shift (m + n) f = fps_shift m (fps_shift n f)"
   by (rule fps_ext) (simp add: add_ac)
 
 lemma fps_shift_fps_shift_reorder:
   "fps_shift m (fps_shift n f) = fps_shift n (fps_shift m f)"
   using fps_shift_fps_shift[of m n f] fps_shift_fps_shift[of n m f] by (simp add: add.commute)
 
 lemma fps_shift_rev_shift:
   "m \<le> n \<Longrightarrow> fps_shift n (Abs_fps (\<lambda>k. if k<m then 0 else f $ (k-m))) = fps_shift (n-m) f"
   "m > n \<Longrightarrow> fps_shift n (Abs_fps (\<lambda>k. if k<m then 0 else f $ (k-m))) =
     Abs_fps (\<lambda>k. if k<m-n then 0 else f $ (k-(m-n)))"
 proof -
   assume "m \<le> n"
   thus "fps_shift n (Abs_fps (\<lambda>k. if k<m then 0 else f $ (k-m))) = fps_shift (n-m) f"
     by (intro fps_ext) auto
 next
   assume mn: "m > n"
   hence "\<And>k. k \<ge> m-n \<Longrightarrow> k+n-m = k - (m-n)" by auto
   thus
     "fps_shift n (Abs_fps (\<lambda>k. if k<m then 0 else f $ (k-m))) =
       Abs_fps (\<lambda>k. if k<m-n then 0 else f $ (k-(m-n)))"
     by (intro fps_ext) auto
 qed
 
 lemma fps_shift_add:
   "fps_shift n (f + g) = fps_shift n f + fps_shift n g"
   by (simp add: fps_eq_iff)
 
 lemma fps_shift_diff:
   "fps_shift n (f - g) = fps_shift n f - fps_shift n g"
   by (auto intro: fps_ext)
 
 lemma fps_shift_uminus:
   "fps_shift n (-f) = - fps_shift n f"
   by (auto intro: fps_ext)
 
 lemma fps_shift_mult:
   assumes "n \<le> subdegree (g :: 'b :: {comm_monoid_add, mult_zero} fps)"
   shows "fps_shift n (h*g) = h * fps_shift n g"
 proof-
   have case1: "\<And>a b::'b fps. 1 \<le> subdegree b \<Longrightarrow> fps_shift 1 (a*b) = a * fps_shift 1 b"
   proof (rule fps_ext)
     fix a b :: "'b fps"
     and n :: nat
     assume b: "1 \<le> subdegree b"
     have "\<And>i. i \<le> n \<Longrightarrow> n + 1 - i = (n-i) + 1"
       by (simp add: algebra_simps)
     with b show "fps_shift 1 (a*b) $ n = (a * fps_shift 1 b) $ n"
       by (simp add: fps_mult_nth nth_less_subdegree_zero)
   qed
   have "n \<le> subdegree g \<Longrightarrow> fps_shift n (h*g) = h * fps_shift n g"
   proof (induct n)
     case (Suc n)
     have "fps_shift (Suc n) (h*g) = fps_shift 1 (fps_shift n (h*g))"
       by (simp add: fps_shift_fps_shift[symmetric])
     also have "\<dots> = h * (fps_shift 1 (fps_shift n g))"
       using Suc case1 by force
     finally show ?case by (simp add: fps_shift_fps_shift[symmetric])
   qed simp
   with assms show ?thesis by fast
 qed
 
 lemma fps_shift_mult_right_noncomm:
   assumes "n \<le> subdegree (g :: 'b :: {comm_monoid_add, mult_zero} fps)"
   shows "fps_shift n (g*h) = fps_shift n g * h"
 proof-
   have case1: "\<And>a b::'b fps. 1 \<le> subdegree a \<Longrightarrow> fps_shift 1 (a*b) = fps_shift 1 a * b"
   proof (rule fps_ext)
     fix a b :: "'b fps"
     and n :: nat
     assume "1 \<le> subdegree a"
     hence "fps_shift 1 (a*b) $ n = (\<Sum>i=Suc 0..Suc n. a$i * b$(n+1-i))"
       using sum.atLeast_Suc_atMost[of 0 "n+1" "\<lambda>i. a$i * b$(n+1-i)"]
       by    (simp add: fps_mult_nth nth_less_subdegree_zero)
     thus "fps_shift 1 (a*b) $ n = (fps_shift 1 a * b) $ n"
       using sum.shift_bounds_cl_Suc_ivl[of "\<lambda>i. a$i * b$(n+1-i)" 0 n]
       by    (simp add: fps_mult_nth)
   qed
   have "n \<le> subdegree g \<Longrightarrow> fps_shift n (g*h) = fps_shift n g * h"
   proof (induct n)
     case (Suc n)
     have "fps_shift (Suc n) (g*h) = fps_shift 1 (fps_shift n (g*h))"
       by (simp add: fps_shift_fps_shift[symmetric])
     also have "\<dots> = (fps_shift 1 (fps_shift n g)) * h"
       using Suc case1 by force
     finally show ?case by (simp add: fps_shift_fps_shift[symmetric])
   qed simp
   with assms show ?thesis by fast
 qed
 
 lemma fps_shift_mult_right:
   assumes "n \<le> subdegree (g :: 'b :: comm_semiring_0 fps)"
   shows   "fps_shift n (g*h) = h * fps_shift n g"
   by      (simp add: assms fps_shift_mult_right_noncomm mult.commute)
 
 lemma fps_shift_mult_both:
   fixes   f g :: "'a::{comm_monoid_add, mult_zero} fps"
   assumes "m \<le> subdegree f" "n \<le> subdegree g"
   shows   "fps_shift m f * fps_shift n g = fps_shift (m+n) (f*g)"
   using   assms
   by      (simp add: fps_shift_mult fps_shift_mult_right_noncomm fps_shift_fps_shift)
 
 lemma fps_shift_subdegree_zero_iff [simp]:
   "fps_shift (subdegree f) f = 0 \<longleftrightarrow> f = 0"
   by (subst (1) nth_subdegree_zero_iff[symmetric], cases "f = 0")
      (simp_all del: nth_subdegree_zero_iff)
 
 lemma fps_shift_times_fps_X:
   fixes f g :: "'a::{comm_monoid_add,mult_zero,monoid_mult} fps"
   shows "1 \<le> subdegree f \<Longrightarrow> fps_shift 1 f * fps_X = f"
   by (intro fps_ext) (simp add: nth_less_subdegree_zero)
 
 lemma fps_shift_times_fps_X' [simp]:
   fixes f :: "'a::{comm_monoid_add,mult_zero,monoid_mult} fps"
   shows "fps_shift 1 (f * fps_X) = f"
   by (intro fps_ext) (simp add: nth_less_subdegree_zero)
 
 lemma fps_shift_times_fps_X'':
   fixes f :: "'a::{comm_monoid_add,mult_zero,monoid_mult} fps"
   shows "1 \<le> n \<Longrightarrow> fps_shift n (f * fps_X) = fps_shift (n - 1) f"
   by (intro fps_ext) (simp add: nth_less_subdegree_zero)
 
 lemma fps_shift_times_fps_X_power:
   "n \<le> subdegree f \<Longrightarrow> fps_shift n f * fps_X ^ n = f"
   by (intro fps_ext) (simp add: fps_X_power_mult_right_nth nth_less_subdegree_zero)
 
 lemma fps_shift_times_fps_X_power' [simp]:
   "fps_shift n (f * fps_X^n) = f"
   by (intro fps_ext) (simp add: fps_X_power_mult_right_nth nth_less_subdegree_zero)
 
 lemma fps_shift_times_fps_X_power'':
   "m \<le> n \<Longrightarrow> fps_shift n (f * fps_X^m) = fps_shift (n - m) f"
   by (intro fps_ext) (simp add: fps_X_power_mult_right_nth nth_less_subdegree_zero)
 
 lemma fps_shift_times_fps_X_power''':
   "m > n \<Longrightarrow> fps_shift n (f * fps_X^m) = f * fps_X^(m - n)"
 proof (cases "f=0")
   case False
   assume m: "m>n"
   hence "m = n + (m-n)" by auto
   with False m show ?thesis
     using power_add[of "fps_X::'a fps" n "m-n"]
           fps_shift_mult_right_noncomm[of n "f * fps_X^n" "fps_X^(m-n)"]
     by    (simp add: mult.assoc fps_subdegree_mult_fps_X_power(2))
 qed simp
 
 lemma subdegree_decompose:
   "f = fps_shift (subdegree f) f * fps_X ^ subdegree f"
   by (rule fps_ext) (auto simp: fps_X_power_mult_right_nth)
 
 lemma subdegree_decompose':
   "n \<le> subdegree f \<Longrightarrow> f = fps_shift n f * fps_X^n"
   by (rule fps_ext) (auto simp: fps_X_power_mult_right_nth intro!: nth_less_subdegree_zero)
 
 instantiation fps :: (zero) unit_factor
 begin
 definition fps_unit_factor_def [simp]:
   "unit_factor f = fps_shift (subdegree f) f"
 instance ..
 end
 
 lemma fps_unit_factor_zero_iff: "unit_factor (f::'a::zero fps) = 0 \<longleftrightarrow> f = 0"
   by simp
 
 lemma fps_unit_factor_nth_0: "f \<noteq> 0 \<Longrightarrow> unit_factor f $ 0 \<noteq> 0"
   by simp
 
 lemma fps_X_unit_factor: "unit_factor (fps_X :: 'a :: zero_neq_one fps) = 1"
  by (intro fps_ext) auto
 
 lemma fps_X_power_unit_factor: "unit_factor (fps_X ^ n) = 1"
 proof-
   define X :: "'a fps" where "X \<equiv> fps_X"
   hence "unit_factor (X^n) = fps_shift n (X^n)"
     by (simp add: fps_X_power_subdegree)
   moreover have "fps_shift n (X^n) = 1"
     by (auto intro: fps_ext simp: fps_X_power_iff X_def)
   ultimately show ?thesis by (simp add: X_def)
 qed
 
 lemma fps_unit_factor_decompose:
   "f = unit_factor f * fps_X ^ subdegree f"
   by (simp add: subdegree_decompose)
 
 lemma fps_unit_factor_decompose':
   "f = fps_X ^ subdegree f * unit_factor f"
   using fps_unit_factor_decompose by (simp add: fps_mult_fps_X_power_commute)
 
 lemma fps_unit_factor_uminus:
   "unit_factor (-f) = - unit_factor (f::'a::group_add fps)"
   by    (simp add: fps_shift_uminus)
 
 lemma fps_unit_factor_shift:
   assumes "n \<le> subdegree f"
   shows   "unit_factor (fps_shift n f) = unit_factor f"
   by      (simp add: assms fps_shift_fps_shift[symmetric])
 
 lemma fps_unit_factor_mult_fps_X:
   fixes f :: "'a::{comm_monoid_add,monoid_mult,mult_zero} fps"
   shows "unit_factor (fps_X * f) = unit_factor f"
   and   "unit_factor (f * fps_X) = unit_factor f"
 proof -
   show "unit_factor (fps_X * f) = unit_factor f"
     by (cases "f=0") (auto intro: fps_ext simp: fps_subdegree_mult_fps_X(1))
   thus "unit_factor (f * fps_X) = unit_factor f" by (simp add: fps_mult_fps_X_commute)
 qed
 
 lemma fps_unit_factor_mult_fps_X_power:
   shows "unit_factor (fps_X ^ n * f) = unit_factor f"
   and   "unit_factor (f * fps_X ^ n) = unit_factor f"
 proof -
   show "unit_factor (fps_X ^ n * f) = unit_factor f"
   proof (induct n)
     case (Suc m) thus ?case
       using fps_unit_factor_mult_fps_X(1)[of "fps_X ^ m * f"] by (simp add: mult.assoc)
   qed simp
   thus "unit_factor (f * fps_X ^ n) = unit_factor f"
     by (simp add: fps_mult_fps_X_power_commute)
 qed
 
 lemma fps_unit_factor_mult_unit_factor:
   fixes f g :: "'a::{comm_monoid_add,mult_zero} fps"
   shows "unit_factor (f * unit_factor g) = unit_factor (f * g)"
   and   "unit_factor (unit_factor f * g) = unit_factor (f * g)"
 proof -
   show "unit_factor (f * unit_factor g) = unit_factor (f * g)"
   proof (cases "f*g = 0")
     case False thus ?thesis
       using fps_mult_subdegree_ge[of f g] fps_unit_factor_shift[of "subdegree g" "f*g"]
       by    (simp add: fps_shift_mult)
   next
     case True
     moreover have "f * unit_factor g = fps_shift (subdegree g) (f*g)"
       by (simp add: fps_shift_mult)
     ultimately show ?thesis by simp
   qed
   show "unit_factor (unit_factor f * g) = unit_factor (f * g)"
   proof (cases "f*g = 0")
     case False thus ?thesis
       using fps_mult_subdegree_ge[of f g] fps_unit_factor_shift[of "subdegree f" "f*g"]
       by    (simp add: fps_shift_mult_right_noncomm)
   next
     case True
     moreover have "unit_factor f * g = fps_shift (subdegree f) (f*g)"
       by (simp add: fps_shift_mult_right_noncomm)
     ultimately show ?thesis by simp
   qed
 qed
 
 lemma fps_unit_factor_mult_both_unit_factor:
   fixes f g :: "'a::{comm_monoid_add,mult_zero} fps"
   shows "unit_factor (unit_factor f * unit_factor g) = unit_factor (f * g)"
   using fps_unit_factor_mult_unit_factor(1)[of "unit_factor f" g]
         fps_unit_factor_mult_unit_factor(2)[of f g]
   by    simp
 
 lemma fps_unit_factor_mult':
   fixes   f g :: "'a::{comm_monoid_add,mult_zero} fps"
   assumes "f $ subdegree f * g $ subdegree g \<noteq> 0"
   shows   "unit_factor (f * g) = unit_factor f * unit_factor g"
   using   assms
   by      (simp add: subdegree_mult' fps_shift_mult_both)
 
 lemma fps_unit_factor_mult:
   fixes f g :: "'a::semiring_no_zero_divisors fps"
   shows "unit_factor (f * g) = unit_factor f * unit_factor g"
   using fps_unit_factor_mult'[of f g]
   by    (cases "f=0 \<or> g=0") auto
 
 definition "fps_cutoff n f = Abs_fps (\<lambda>i. if i < n then f$i else 0)"
 
 lemma fps_cutoff_nth [simp]: "fps_cutoff n f $ i = (if i < n then f$i else 0)"
   unfolding fps_cutoff_def by simp
 
 lemma fps_cutoff_zero_iff: "fps_cutoff n f = 0 \<longleftrightarrow> (f = 0 \<or> n \<le> subdegree f)"
 proof
   assume A: "fps_cutoff n f = 0"
   thus "f = 0 \<or> n \<le> subdegree f"
   proof (cases "f = 0")
     assume "f \<noteq> 0"
     with A have "n \<le> subdegree f"
       by (intro subdegree_geI) (simp_all add: fps_eq_iff split: if_split_asm)
     thus ?thesis ..
   qed simp
 qed (auto simp: fps_eq_iff intro: nth_less_subdegree_zero)
 
 lemma fps_cutoff_0 [simp]: "fps_cutoff 0 f = 0"
   by (simp add: fps_eq_iff)
 
 lemma fps_cutoff_zero [simp]: "fps_cutoff n 0 = 0"
   by (simp add: fps_eq_iff)
 
 lemma fps_cutoff_one: "fps_cutoff n 1 = (if n = 0 then 0 else 1)"
   by (simp add: fps_eq_iff)
 
 lemma fps_cutoff_fps_const: "fps_cutoff n (fps_const c) = (if n = 0 then 0 else fps_const c)"
   by (simp add: fps_eq_iff)
 
 lemma fps_cutoff_numeral: "fps_cutoff n (numeral c) = (if n = 0 then 0 else numeral c)"
   by (simp add: numeral_fps_const fps_cutoff_fps_const)
 
 lemma fps_shift_cutoff:
   "fps_shift n f * fps_X^n + fps_cutoff n f = f"
   by (simp add: fps_eq_iff fps_X_power_mult_right_nth)
 
 lemma fps_shift_cutoff':
   "fps_X^n * fps_shift n f + fps_cutoff n f = f"
   by (simp add: fps_eq_iff fps_X_power_mult_nth)
 
 lemma fps_cutoff_left_mult_nth:
   "k < n \<Longrightarrow> (fps_cutoff n f * g) $ k = (f * g) $ k"
   by (simp add: fps_mult_nth)
 
 lemma fps_cutoff_right_mult_nth:
   assumes "k < n"
   shows   "(f * fps_cutoff n g) $ k = (f * g) $ k"
 proof-
   from assms have "\<forall>i\<in>{0..k}. fps_cutoff n g $ (k - i) = g $ (k - i)" by auto
   thus ?thesis by (simp add: fps_mult_nth)
 qed
 
 subsection \<open>Formal Power series form a metric space\<close>
 
 instantiation fps :: ("{minus,zero}") dist
 begin
 
 definition
   dist_fps_def: "dist (a :: 'a fps) b = (if a = b then 0 else inverse (2 ^ subdegree (a - b)))"
 
 lemma dist_fps_ge0: "dist (a :: 'a fps) b \<ge> 0"
   by (simp add: dist_fps_def)
 
 instance ..
 
 end
 
 instantiation fps :: (group_add) metric_space
 begin
 
 definition uniformity_fps_def [code del]:
   "(uniformity :: ('a fps \<times> 'a fps) filter) = (INF e\<in>{0 <..}. principal {(x, y). dist x y < e})"
 
 definition open_fps_def' [code del]:
   "open (U :: 'a fps set) \<longleftrightarrow> (\<forall>x\<in>U. eventually (\<lambda>(x', y). x' = x \<longrightarrow> y \<in> U) uniformity)"
 
 lemma dist_fps_sym: "dist (a :: 'a fps) b = dist b a"
   by (simp add: dist_fps_def)
 
 instance
 proof
   show th: "dist a b = 0 \<longleftrightarrow> a = b" for a b :: "'a fps"
     by (simp add: dist_fps_def split: if_split_asm)
   then have th'[simp]: "dist a a = 0" for a :: "'a fps" by simp
 
   fix a b c :: "'a fps"
   consider "a = b" | "c = a \<or> c = b" | "a \<noteq> b" "a \<noteq> c" "b \<noteq> c" by blast
   then show "dist a b \<le> dist a c + dist b c"
   proof cases
     case 1
     then show ?thesis by (simp add: dist_fps_def)
   next
     case 2
     then show ?thesis
       by (cases "c = a") (simp_all add: th dist_fps_sym)
   next
     case neq: 3
     have False if "dist a b > dist a c + dist b c"
     proof -
       let ?n = "subdegree (a - b)"
       from neq have "dist a b > 0" "dist b c > 0" and "dist a c > 0" by (simp_all add: dist_fps_def)
       with that have "dist a b > dist a c" and "dist a b > dist b c" by simp_all
       with neq have "?n < subdegree (a - c)" and "?n < subdegree (b - c)"
         by (simp_all add: dist_fps_def field_simps)
       hence "(a - c) $ ?n = 0" and "(b - c) $ ?n = 0"
         by (simp_all only: nth_less_subdegree_zero)
       hence "(a - b) $ ?n = 0" by simp
       moreover from neq have "(a - b) $ ?n \<noteq> 0" by (intro nth_subdegree_nonzero) simp_all
       ultimately show False by contradiction
     qed
     thus ?thesis by (auto simp add: not_le[symmetric])
   qed
 qed (rule open_fps_def' uniformity_fps_def)+
 
 end
 
 declare uniformity_Abort[where 'a="'a :: group_add fps", code]
 
 lemma open_fps_def: "open (S :: 'a::group_add fps set) = (\<forall>a \<in> S. \<exists>r. r >0 \<and> {y. dist y a < r} \<subseteq> S)"
   unfolding open_dist subset_eq by simp
 
 text \<open>The infinite sums and justification of the notation in textbooks.\<close>
 
 lemma reals_power_lt_ex:
   fixes x y :: real
   assumes xp: "x > 0"
     and y1: "y > 1"
   shows "\<exists>k>0. (1/y)^k < x"
 proof -
   have yp: "y > 0"
     using y1 by simp
   from reals_Archimedean2[of "max 0 (- log y x) + 1"]
   obtain k :: nat where k: "real k > max 0 (- log y x) + 1"
     by blast
   from k have kp: "k > 0"
     by simp
   from k have "real k > - log y x"
     by simp
   then have "ln y * real k > - ln x"
     unfolding log_def
     using ln_gt_zero_iff[OF yp] y1
     by (simp add: minus_divide_left field_simps del: minus_divide_left[symmetric])
   then have "ln y * real k + ln x > 0"
     by simp
   then have "exp (real k * ln y + ln x) > exp 0"
     by (simp add: ac_simps)
   then have "y ^ k * x > 1"
     unfolding exp_zero exp_add exp_of_nat_mult exp_ln [OF xp] exp_ln [OF yp]
     by simp
   then have "x > (1 / y)^k" using yp
     by (simp add: field_simps)
   then show ?thesis
     using kp by blast
 qed
 
 lemma fps_sum_rep_nth: "(sum (\<lambda>i. fps_const(a$i)*fps_X^i) {0..m})$n = (if n \<le> m then a$n else 0)"
   by (simp add: fps_sum_nth if_distrib cong del: if_weak_cong)
 
 lemma fps_notation: "(\<lambda>n. sum (\<lambda>i. fps_const(a$i) * fps_X^i) {0..n}) \<longlonglongrightarrow> a"
   (is "?s \<longlonglongrightarrow> a")
 proof -
   have "\<exists>n0. \<forall>n \<ge> n0. dist (?s n) a < r" if "r > 0" for r
   proof -
     obtain n0 where n0: "(1/2)^n0 < r" "n0 > 0"
       using reals_power_lt_ex[OF \<open>r > 0\<close>, of 2] by auto
     show ?thesis
     proof -
       have "dist (?s n) a < r" if nn0: "n \<ge> n0" for n
       proof -
         from that have thnn0: "(1/2)^n \<le> (1/2 :: real)^n0"
           by (simp add: field_split_simps)
         show ?thesis
         proof (cases "?s n = a")
           case True
           then show ?thesis
             unfolding dist_eq_0_iff[of "?s n" a, symmetric]
             using \<open>r > 0\<close> by (simp del: dist_eq_0_iff)
         next
           case False
           from False have dth: "dist (?s n) a = (1/2)^subdegree (?s n - a)"
             by (simp add: dist_fps_def field_simps)
           from False have kn: "subdegree (?s n - a) > n"
             by (intro subdegree_greaterI) (simp_all add: fps_sum_rep_nth)
           then have "dist (?s n) a < (1/2)^n"
             by (simp add: field_simps dist_fps_def)
           also have "\<dots> \<le> (1/2)^n0"
             using nn0 by (simp add: field_split_simps)
           also have "\<dots> < r"
             using n0 by simp
           finally show ?thesis .
         qed
       qed
       then show ?thesis by blast
     qed
   qed
   then show ?thesis
     unfolding lim_sequentially by blast
 qed
 
 
 subsection \<open>Inverses and division of formal power series\<close>
 
 declare sum.cong[fundef_cong]
 
 fun fps_left_inverse_constructor ::
   "'a::{comm_monoid_add,times,uminus} fps \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"
 where
   "fps_left_inverse_constructor f a 0 = a"
 | "fps_left_inverse_constructor f a (Suc n) =
     - sum (\<lambda>i. fps_left_inverse_constructor f a i * f$(Suc n - i)) {0..n} * a"
 
 \<comment> \<open>This will construct a left inverse for f in case that x * f$0 = 1\<close>
 abbreviation "fps_left_inverse \<equiv> (\<lambda>f x. Abs_fps (fps_left_inverse_constructor f x))"
 
 fun fps_right_inverse_constructor ::
   "'a::{comm_monoid_add,times,uminus} fps \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"
 where
   "fps_right_inverse_constructor f a 0 = a"
 | "fps_right_inverse_constructor f a n =
     - a * sum (\<lambda>i. f$i * fps_right_inverse_constructor f a (n - i)) {1..n}"
 
 \<comment> \<open>This will construct a right inverse for f in case that f$0 * y = 1\<close>
 abbreviation "fps_right_inverse \<equiv> (\<lambda>f y. Abs_fps (fps_right_inverse_constructor f y))"
 
 instantiation fps :: ("{comm_monoid_add,inverse,times,uminus}") inverse
 begin
 
 \<comment> \<open>For backwards compatibility.\<close>
 abbreviation natfun_inverse:: "'a fps \<Rightarrow> nat \<Rightarrow> 'a"
   where "natfun_inverse f \<equiv> fps_right_inverse_constructor f (inverse (f$0))"
 
 definition fps_inverse_def: "inverse f = Abs_fps (natfun_inverse f)"
 \<comment> \<open>
   With scalars from a (possibly non-commutative) ring, this defines a right inverse.
   Furthermore, if scalars are of class @{class mult_zero} and satisfy
   condition @{term "inverse 0 = 0"}, then this will evaluate to zero when
   the zeroth term is zero.
 \<close>
 
 definition fps_divide_def: "f div g = fps_shift (subdegree g) (f * inverse (unit_factor g))"
 \<comment> \<open>
   If scalars are of class @{class mult_zero} and satisfy condition
   @{term "inverse 0 = 0"}, then div by zero will equal zero.
 \<close>
 
 instance ..
 
 end
 
 lemma fps_lr_inverse_0_iff:
   "(fps_left_inverse f x) $ 0 = 0 \<longleftrightarrow> x = 0"
   "(fps_right_inverse f x) $ 0 = 0 \<longleftrightarrow> x = 0"
   by auto
 
 lemma fps_inverse_0_iff': "(inverse f) $ 0 = 0 \<longleftrightarrow> inverse (f $ 0) = 0"
   by (simp add: fps_inverse_def fps_lr_inverse_0_iff(2))
 
 lemma fps_inverse_0_iff[simp]: "(inverse f) $ 0 = (0::'a::division_ring) \<longleftrightarrow> f $ 0 = 0"
   by (simp add: fps_inverse_0_iff')
 
 lemma fps_lr_inverse_nth_0:
   "(fps_left_inverse f x) $ 0 = x" "(fps_right_inverse f x) $ 0 = x"
   by auto
 
 lemma fps_inverse_nth_0 [simp]: "(inverse f) $ 0 = inverse (f $ 0)"
   by (simp add: fps_inverse_def)
 
 lemma fps_lr_inverse_starting0:
   fixes f :: "'a::{comm_monoid_add,mult_zero,uminus} fps"
   and   g :: "'b::{ab_group_add,mult_zero} fps"
   shows "fps_left_inverse f 0 = 0"
   and   "fps_right_inverse g 0 = 0"
 proof-
   show "fps_left_inverse f 0 = 0"
   proof (rule fps_ext)
     fix n show "fps_left_inverse f 0 $ n = 0 $ n"
       by (cases n) (simp_all add: fps_inverse_def)
   qed
   show "fps_right_inverse g 0 = 0"
   proof (rule fps_ext)
     fix n show "fps_right_inverse g 0 $ n = 0 $ n"
       by (cases n) (simp_all add: fps_inverse_def)
   qed
 qed
 
 lemma fps_lr_inverse_eq0_imp_starting0:
   "fps_left_inverse f x = 0 \<Longrightarrow> x = 0"
   "fps_right_inverse f x = 0 \<Longrightarrow> x = 0"
 proof-
   assume A: "fps_left_inverse f x = 0"
   have "0 = fps_left_inverse f x $ 0" by (subst A) simp
   thus "x = 0" by simp
 next
   assume A: "fps_right_inverse f x = 0"
   have "0 = fps_right_inverse f x $ 0" by (subst A) simp
   thus "x = 0" by simp
 qed
 
 lemma fps_lr_inverse_eq_0_iff:
   fixes x :: "'a::{comm_monoid_add,mult_zero,uminus}"
   and   y :: "'b::{ab_group_add,mult_zero}"
   shows "fps_left_inverse f x = 0 \<longleftrightarrow> x = 0"
   and   "fps_right_inverse g y = 0 \<longleftrightarrow> y = 0"
   using fps_lr_inverse_starting0 fps_lr_inverse_eq0_imp_starting0
   by    auto
 
 lemma fps_inverse_eq_0_iff':
   fixes f :: "'a::{ab_group_add,inverse,mult_zero} fps"
   shows "inverse f = 0 \<longleftrightarrow> inverse (f $ 0) = 0"
   by    (simp add: fps_inverse_def fps_lr_inverse_eq_0_iff(2))
 
 lemma fps_inverse_eq_0_iff[simp]: "inverse f = (0:: ('a::division_ring) fps) \<longleftrightarrow> f $ 0 = 0"
   using fps_inverse_eq_0_iff'[of f] by simp
 
 lemmas fps_inverse_eq_0' = iffD2[OF fps_inverse_eq_0_iff']
 lemmas fps_inverse_eq_0  = iffD2[OF fps_inverse_eq_0_iff]
 
 lemma fps_const_lr_inverse:
   fixes a :: "'a::{ab_group_add,mult_zero}"
   and   b :: "'b::{comm_monoid_add,mult_zero,uminus}"
   shows "fps_left_inverse (fps_const a) x = fps_const x"
   and   "fps_right_inverse (fps_const b) y = fps_const y"
 proof-
   show "fps_left_inverse (fps_const a) x = fps_const x"
   proof (rule fps_ext)
     fix n show "fps_left_inverse (fps_const a) x $ n = fps_const x $ n"
       by (cases n) auto
   qed
   show "fps_right_inverse (fps_const b) y = fps_const y"
   proof (rule fps_ext)
     fix n show "fps_right_inverse (fps_const b) y $ n = fps_const y $ n"
       by (cases n) auto
   qed
 qed
 
 lemma fps_const_inverse:
   fixes     a :: "'a::{comm_monoid_add,inverse,mult_zero,uminus}"
   shows     "inverse (fps_const a) = fps_const (inverse a)"
   unfolding fps_inverse_def
   by        (simp add: fps_const_lr_inverse(2))
 
 lemma fps_lr_inverse_zero:
   fixes x :: "'a::{ab_group_add,mult_zero}"
   and   y :: "'b::{comm_monoid_add,mult_zero,uminus}"
   shows "fps_left_inverse 0 x = fps_const x"
   and   "fps_right_inverse 0 y = fps_const y"
   using fps_const_lr_inverse[of 0]
   by    simp_all
 
 lemma fps_inverse_zero_conv_fps_const:
   "inverse (0::'a::{comm_monoid_add,mult_zero,uminus,inverse} fps) = fps_const (inverse 0)"
   using fps_lr_inverse_zero(2)[of "inverse (0::'a)"] by (simp add: fps_inverse_def)
 
 lemma fps_inverse_zero':
   assumes "inverse (0::'a::{comm_monoid_add,inverse,mult_zero,uminus}) = 0"
   shows   "inverse (0::'a fps) = 0"
   by      (simp add: assms fps_inverse_zero_conv_fps_const)
 
 lemma fps_inverse_zero [simp]:
   "inverse (0::'a::division_ring fps) = 0"
   by (rule fps_inverse_zero'[OF inverse_zero])
 
 lemma fps_lr_inverse_one:
   fixes x :: "'a::{ab_group_add,mult_zero,one}"
   and   y :: "'b::{comm_monoid_add,mult_zero,uminus,one}"
   shows "fps_left_inverse 1 x = fps_const x"
   and   "fps_right_inverse 1 y = fps_const y"
   using fps_const_lr_inverse[of 1]
   by    simp_all
 
 lemma fps_lr_inverse_one_one:
   "fps_left_inverse 1 1 = (1::'a::{ab_group_add,mult_zero,one} fps)"
   "fps_right_inverse 1 1 = (1::'b::{comm_monoid_add,mult_zero,uminus,one} fps)"
   by (simp_all add: fps_lr_inverse_one)
 
 lemma fps_inverse_one':
   assumes "inverse (1::'a::{comm_monoid_add,inverse,mult_zero,uminus,one}) = 1"
   shows   "inverse (1 :: 'a fps) = 1"
   using   assms fps_lr_inverse_one_one(2)
   by      (simp add: fps_inverse_def)
 
 lemma fps_inverse_one [simp]: "inverse (1 :: 'a :: division_ring fps) = 1"
   by (rule fps_inverse_one'[OF inverse_1])
 
 lemma fps_lr_inverse_minus:
   fixes f :: "'a::ring_1 fps"
   shows "fps_left_inverse (-f) (-x) = - fps_left_inverse f x"
   and   "fps_right_inverse (-f) (-x) = - fps_right_inverse f x"
 proof-
 
   show "fps_left_inverse (-f) (-x) = - fps_left_inverse f x"
   proof (intro fps_ext)
     fix n show "fps_left_inverse (-f) (-x) $ n = - fps_left_inverse f x $ n"
     proof (induct n rule: nat_less_induct)
       case (1 n) thus ?case by (cases n) (simp_all add: sum_negf algebra_simps)
     qed
   qed
 
   show "fps_right_inverse (-f) (-x) = - fps_right_inverse f x"
   proof (intro fps_ext)
     fix n show "fps_right_inverse (-f) (-x) $ n = - fps_right_inverse f x $ n"
     proof (induct n rule: nat_less_induct)
       case (1 n) show ?case
       proof (cases n)
         case (Suc m)
         with 1 have
           "\<forall>i\<in>{1..Suc m}. fps_right_inverse (-f) (-x) $ (Suc m - i) =
             - fps_right_inverse f x $ (Suc m - i)"
           by auto
         with Suc show ?thesis by (simp add: sum_negf algebra_simps)
       qed simp
     qed
   qed
 
 qed
 
 lemma fps_inverse_minus [simp]: "inverse (-f) = -inverse (f :: 'a :: division_ring fps)"
   by (simp add: fps_inverse_def fps_lr_inverse_minus(2))
 
 lemma fps_left_inverse:
   fixes   f :: "'a::ring_1 fps"
   assumes f0: "x * f$0 = 1"
   shows   "fps_left_inverse f x * f = 1"
 proof (rule fps_ext)
   fix n show "(fps_left_inverse f x * f) $ n = 1 $ n"
     by (cases n) (simp_all add: f0 fps_mult_nth mult.assoc)
 qed
 
 lemma fps_right_inverse:
   fixes   f :: "'a::ring_1 fps"
   assumes f0: "f$0 * y = 1"
   shows   "f * fps_right_inverse f y = 1"
 proof (rule fps_ext)
   fix n
   show "(f * fps_right_inverse f y) $ n = 1 $ n"
   proof (cases n)
     case (Suc k)
     moreover from Suc have "fps_right_inverse f y $ n =
             - y * sum (\<lambda>i. f$i * fps_right_inverse_constructor f y (n - i)) {1..n}"
       by simp
     hence
       "(f * fps_right_inverse f y) $ n =
         - 1 * sum (\<lambda>i. f$i * fps_right_inverse_constructor f y (n - i)) {1..n} +
         sum (\<lambda>i. f$i * (fps_right_inverse_constructor f y (n - i))) {1..n}"
       by (simp add: fps_mult_nth sum.atLeast_Suc_atMost mult.assoc f0[symmetric])
     thus "(f * fps_right_inverse f y) $ n = 1 $ n" by (simp add: Suc)
   qed (simp add: f0 fps_inverse_def)
 qed
 
 \<comment> \<open>
   It is possible in a ring for an element to have a left inverse but not a right inverse, or
   vice versa. But when an element has both, they must be the same.
 \<close>
 lemma fps_left_inverse_eq_fps_right_inverse:
   fixes   f :: "'a::ring_1 fps"
   assumes f0: "x * f$0 = 1" "f $ 0 * y = 1"
   \<comment> \<open>These assumptions imply x equals y, but no need to assume that.\<close>
   shows   "fps_left_inverse f x = fps_right_inverse f y"
 proof-
   from f0(2) have "f * fps_right_inverse f y = 1"
       by (simp add: fps_right_inverse)
   hence "fps_left_inverse f x * f * fps_right_inverse f y = fps_left_inverse f x"
     by (simp add: mult.assoc)
   moreover from f0(1) have
     "fps_left_inverse f x * f * fps_right_inverse f y = fps_right_inverse f y"
     by (simp add: fps_left_inverse)
   ultimately show ?thesis by simp
 qed
 
 lemma fps_left_inverse_eq_fps_right_inverse_comm:
   fixes   f :: "'a::comm_ring_1 fps"
   assumes f0: "x * f$0 = 1"
   shows   "fps_left_inverse f x = fps_right_inverse f x"
   using   assms fps_left_inverse_eq_fps_right_inverse[of x f x]
   by      (simp add: mult.commute)
 
 lemma fps_left_inverse':
   fixes   f :: "'a::ring_1 fps"
   assumes "x * f$0 = 1" "f$0 * y = 1"
   \<comment> \<open>These assumptions imply x equals y, but no need to assume that.\<close>
   shows   "fps_right_inverse f y * f = 1"
   using   assms fps_left_inverse_eq_fps_right_inverse[of x f y] fps_left_inverse[of x f]
   by      simp
 
 lemma fps_right_inverse':
   fixes   f :: "'a::ring_1 fps"
   assumes "x * f$0 = 1" "f$0 * y = 1"
   \<comment> \<open>These assumptions imply x equals y, but no need to assume that.\<close>
   shows   "f * fps_left_inverse f x = 1"
   using   assms fps_left_inverse_eq_fps_right_inverse[of x f y] fps_right_inverse[of f y]
   by      simp
 
 lemma inverse_mult_eq_1 [intro]:
   assumes "f$0 \<noteq> (0::'a::division_ring)"
   shows   "inverse f * f = 1"
   using   fps_left_inverse'[of "inverse (f$0)"]
   by      (simp add: assms fps_inverse_def)
 
 lemma inverse_mult_eq_1':
   assumes "f$0 \<noteq> (0::'a::division_ring)"
   shows   "f * inverse f = 1"
   using   assms fps_right_inverse
   by      (force simp: fps_inverse_def)
 
 lemma fps_mult_left_inverse_unit_factor:
   fixes   f :: "'a::ring_1 fps"
   assumes "x * f $ subdegree f = 1"
   shows   "fps_left_inverse (unit_factor f) x * f = fps_X ^ subdegree f"
 proof-
   have
     "fps_left_inverse (unit_factor f) x * f =
       fps_left_inverse (unit_factor f) x * unit_factor f * fps_X ^ subdegree f"
     using fps_unit_factor_decompose[of f] by (simp add: mult.assoc)
   with assms show ?thesis by (simp add: fps_left_inverse)
 qed
 
 lemma fps_mult_right_inverse_unit_factor:
   fixes   f :: "'a::ring_1 fps"
   assumes "f $ subdegree f * y = 1"
   shows   "f * fps_right_inverse (unit_factor f) y = fps_X ^ subdegree f"
 proof-
   have
     "f * fps_right_inverse (unit_factor f) y =
       fps_X ^ subdegree f * (unit_factor f * fps_right_inverse (unit_factor f) y)"
     using fps_unit_factor_decompose'[of f] by (simp add: mult.assoc[symmetric])
   with assms show ?thesis by (simp add: fps_right_inverse)
 qed
 
 lemma fps_mult_right_inverse_unit_factor_divring:
   "(f :: 'a::division_ring fps) \<noteq> 0 \<Longrightarrow> f * inverse (unit_factor f) = fps_X ^ subdegree f"
   using   fps_mult_right_inverse_unit_factor[of f]
   by      (simp add: fps_inverse_def)
 
 lemma fps_left_inverse_idempotent_ring1:
   fixes   f :: "'a::ring_1 fps"
   assumes "x * f$0 = 1" "y * x = 1"
   \<comment> \<open>These assumptions imply y equals f$0, but no need to assume that.\<close>
   shows   "fps_left_inverse (fps_left_inverse f x) y = f"
 proof-
   from assms(1) have
     "fps_left_inverse (fps_left_inverse f x) y * fps_left_inverse f x * f =
       fps_left_inverse (fps_left_inverse f x) y"
     by (simp add: fps_left_inverse mult.assoc)
   moreover from assms(2) have
     "fps_left_inverse (fps_left_inverse f x) y * fps_left_inverse f x = 1"
     by (simp add: fps_left_inverse)
   ultimately show ?thesis by simp
 qed
 
 lemma fps_left_inverse_idempotent_comm_ring1:
   fixes   f :: "'a::comm_ring_1 fps"
   assumes "x * f$0 = 1"
   shows   "fps_left_inverse (fps_left_inverse f x) (f$0) = f"
   using   assms fps_left_inverse_idempotent_ring1[of x f "f$0"]
   by      (simp add: mult.commute)
 
 lemma fps_right_inverse_idempotent_ring1:
   fixes   f :: "'a::ring_1 fps"
   assumes "f$0 * x = 1" "x * y = 1"
   \<comment> \<open>These assumptions imply y equals f$0, but no need to assume that.\<close>
   shows   "fps_right_inverse (fps_right_inverse f x) y = f"
 proof-
   from assms(1) have "f * (fps_right_inverse f x * fps_right_inverse (fps_right_inverse f x) y) =
       fps_right_inverse (fps_right_inverse f x) y"
     by (simp add: fps_right_inverse mult.assoc[symmetric])
   moreover from assms(2) have
     "fps_right_inverse f x * fps_right_inverse (fps_right_inverse f x) y = 1"
     by (simp add: fps_right_inverse)
   ultimately show ?thesis by simp
 qed
 
 lemma fps_right_inverse_idempotent_comm_ring1:
   fixes   f :: "'a::comm_ring_1 fps"
   assumes "f$0 * x = 1"
   shows   "fps_right_inverse (fps_right_inverse f x) (f$0) = f"
   using   assms fps_right_inverse_idempotent_ring1[of f x "f$0"]
   by      (simp add: mult.commute)
 
 lemma fps_inverse_idempotent[intro, simp]:
   "f$0 \<noteq> (0::'a::division_ring) \<Longrightarrow> inverse (inverse f) = f"
   using fps_right_inverse_idempotent_ring1[of f]
   by    (simp add: fps_inverse_def)
 
 lemma fps_lr_inverse_unique_ring1:
   fixes   f g :: "'a :: ring_1 fps"
   assumes fg: "f * g = 1" "g$0 * f$0 = 1"
   shows   "fps_left_inverse g (f$0) = f"
   and     "fps_right_inverse f (g$0) = g"
 proof-
 
   show "fps_left_inverse g (f$0) = f"
   proof (intro fps_ext)
     fix n show "fps_left_inverse g (f$0) $ n = f $ n"
     proof (induct n rule: nat_less_induct)
       case (1 n) show ?case
       proof (cases n)
         case (Suc k)
         hence "\<forall>i\<in>{0..k}. fps_left_inverse g (f$0) $ i = f $ i" using 1 by simp
         hence "fps_left_inverse g (f$0) $ Suc k = f $ Suc k - 1 $ Suc k * f$0"
           by (simp add: fps_mult_nth fg(1)[symmetric] distrib_right mult.assoc fg(2))
         with Suc show ?thesis by simp
       qed simp
     qed
   qed
 
   show "fps_right_inverse f (g$0) = g"
   proof (intro fps_ext)
     fix n show "fps_right_inverse f (g$0) $ n = g $ n"
     proof (induct n rule: nat_less_induct)
       case (1 n) show ?case
       proof (cases n)
         case (Suc k)
         hence "\<forall>i\<in>{1..Suc k}. fps_right_inverse f (g$0) $ (Suc k - i) = g $ (Suc k - i)"
           using 1 by auto
         hence
           "fps_right_inverse f (g$0) $ Suc k = 1 * g $ Suc k - g$0 * 1 $ Suc k"
           by (simp add: fps_mult_nth fg(1)[symmetric] algebra_simps fg(2)[symmetric] sum.atLeast_Suc_atMost)
         with Suc show ?thesis by simp
       qed simp
     qed
   qed
 
 qed
 
 lemma fps_lr_inverse_unique_divring:
   fixes   f g :: "'a ::division_ring fps"
   assumes fg: "f * g = 1"
   shows   "fps_left_inverse g (f$0) = f"
   and     "fps_right_inverse f (g$0) = g"
 proof-
   from fg have "f$0 * g$0 = 1" using fps_mult_nth_0[of f g] by simp
   hence "g$0 * f$0 = 1" using inverse_unique[of "f$0"] left_inverse[of "f$0"] by force
   thus "fps_left_inverse g (f$0) = f" "fps_right_inverse f (g$0) = g"
     using fg fps_lr_inverse_unique_ring1 by auto
 qed
 
 lemma fps_inverse_unique:
   fixes   f g :: "'a :: division_ring fps"
   assumes fg: "f * g = 1"
   shows   "inverse f = g"
 proof -
   from fg have if0: "inverse (f$0) = g$0" "f$0 \<noteq> 0"
     using inverse_unique[of "f$0"] fps_mult_nth_0[of f g] by auto
   with fg have "fps_right_inverse f (g$0) = g"
     using left_inverse[of "f$0"] by (intro fps_lr_inverse_unique_ring1(2)) simp_all
   with if0(1) show ?thesis by (simp add: fps_inverse_def)
 qed
 
 lemma inverse_fps_numeral:
   "inverse (numeral n :: ('a :: field_char_0) fps) = fps_const (inverse (numeral n))"
   by (intro fps_inverse_unique fps_ext) (simp_all add: fps_numeral_nth)
 
 lemma inverse_fps_of_nat:
   "inverse (of_nat n :: 'a :: {semiring_1,times,uminus,inverse} fps) =
     fps_const (inverse (of_nat n))"
   by (simp add: fps_of_nat fps_const_inverse[symmetric])
 
-lemma sum_zero_lemma:
-  fixes n::nat
-  assumes "0 < n"
-  shows "(\<Sum>i = 0..n. if n = i then 1 else if n - i = 1 then - 1 else 0) = (0::'a::field)"
-proof -
-  let ?f = "\<lambda>i. if n = i then 1 else if n - i = 1 then - 1 else 0"
-  let ?g = "\<lambda>i. if i = n then 1 else if i = n - 1 then - 1 else 0"
-  let ?h = "\<lambda>i. if i=n - 1 then - 1 else 0"
-  have th1: "sum ?f {0..n} = sum ?g {0..n}"
-    by (rule sum.cong) auto
-  have th2: "sum ?g {0..n - 1} = sum ?h {0..n - 1}"
-    apply (rule sum.cong)
-    using assms
-    apply auto
-    done
-  have eq: "{0 .. n} = {0.. n - 1} \<union> {n}"
-    by auto
-  from assms have d: "{0.. n - 1} \<inter> {n} = {}"
-    by auto
-  have f: "finite {0.. n - 1}" "finite {n}"
-    by auto
-  show ?thesis
-    unfolding th1
-    apply (simp add: sum.union_disjoint[OF f d, unfolded eq[symmetric]] del: One_nat_def)
-    unfolding th2
-    apply simp
-    done
-qed
-
 lemma fps_lr_inverse_mult_ring1:
   fixes   f g :: "'a::ring_1 fps"
   assumes x: "x * f$0 = 1" "f$0 * x = 1"
   and     y: "y * g$0 = 1" "g$0 * y = 1"
   shows   "fps_left_inverse (f * g) (y*x) = fps_left_inverse g y * fps_left_inverse f x"
   and     "fps_right_inverse (f * g) (y*x) = fps_right_inverse g y * fps_right_inverse f x"
 proof -
   define h where "h \<equiv> fps_left_inverse g y * fps_left_inverse f x"
   hence h0: "h$0 = y*x" by simp
   have "fps_left_inverse (f*g) (h$0) = h"
   proof (intro fps_lr_inverse_unique_ring1(1))
     from h_def
       have  "h * (f * g) = fps_left_inverse g y * (fps_left_inverse f x * f) * g"
       by    (simp add: mult.assoc)
     thus "h * (f * g) = 1"
       using fps_left_inverse[OF x(1)] fps_left_inverse[OF y(1)] by simp
     from h_def have "(f*g)$0 * h$0 = f$0 * 1 * x"
       by (simp add: mult.assoc y(2)[symmetric])
     with x(2) show "(f * g) $ 0 * h $ 0 = 1" by simp
   qed
   with h_def
     show  "fps_left_inverse (f * g) (y*x) = fps_left_inverse g y * fps_left_inverse f x"
     by    simp
 next
   define h where "h \<equiv> fps_right_inverse g y * fps_right_inverse f x"
   hence h0: "h$0 = y*x" by simp
   have "fps_right_inverse (f*g) (h$0) = h"
   proof (intro fps_lr_inverse_unique_ring1(2))
     from h_def
       have  "f * g * h = f * (g * fps_right_inverse g y) * fps_right_inverse f x"
       by    (simp add: mult.assoc)
     thus "f * g * h = 1"
       using fps_right_inverse[OF x(2)] fps_right_inverse[OF y(2)] by simp
     from h_def have "h$0 * (f*g)$0 = y * 1 * g$0"
       by (simp add: mult.assoc x(1)[symmetric])
     with y(1) show "h$0 * (f*g)$0  = 1" by simp
   qed
   with h_def
     show  "fps_right_inverse (f * g) (y*x) = fps_right_inverse g y * fps_right_inverse f x"
     by    simp
 qed
 
 lemma fps_lr_inverse_mult_divring:
   fixes f g :: "'a::division_ring fps"
   shows "fps_left_inverse (f * g) (inverse ((f*g)$0)) =
           fps_left_inverse g (inverse (g$0)) * fps_left_inverse f (inverse (f$0))"
   and   "fps_right_inverse (f * g) (inverse ((f*g)$0)) =
           fps_right_inverse g (inverse (g$0)) * fps_right_inverse f (inverse (f$0))"
 proof-
   show "fps_left_inverse (f * g) (inverse ((f*g)$0)) =
           fps_left_inverse g (inverse (g$0)) * fps_left_inverse f (inverse (f$0))"
   proof (cases "f$0 = 0 \<or> g$0 = 0")
     case True
     hence "fps_left_inverse (f * g) (inverse ((f*g)$0)) = 0"
       by (simp add: fps_lr_inverse_eq_0_iff(1))
     moreover from True have
       "fps_left_inverse g (inverse (g$0)) * fps_left_inverse f (inverse (f$0)) = 0"
       by (auto simp: fps_lr_inverse_eq_0_iff(1))
     ultimately show ?thesis by simp
   next
     case False
     hence "fps_left_inverse (f * g) (inverse (g$0) * inverse (f$0)) =
             fps_left_inverse g (inverse (g$0)) * fps_left_inverse f (inverse (f$0))"
       by  (intro fps_lr_inverse_mult_ring1(1)) simp_all
     with False show ?thesis by (simp add: nonzero_inverse_mult_distrib)
   qed
   show "fps_right_inverse (f * g) (inverse ((f*g)$0)) =
           fps_right_inverse g (inverse (g$0)) * fps_right_inverse f (inverse (f$0))"
   proof (cases "f$0 = 0 \<or> g$0 = 0")
     case True
     from True have "fps_right_inverse (f * g) (inverse ((f*g)$0)) = 0"
       by (simp add: fps_lr_inverse_eq_0_iff(2))
     moreover from True have
       "fps_right_inverse g (inverse (g$0)) * fps_right_inverse f (inverse (f$0)) = 0"
       by (auto simp: fps_lr_inverse_eq_0_iff(2))
     ultimately show ?thesis by simp
   next
     case False
     hence "fps_right_inverse (f * g) (inverse (g$0) * inverse (f$0)) =
             fps_right_inverse g (inverse (g$0)) * fps_right_inverse f (inverse (f$0))"
       by  (intro fps_lr_inverse_mult_ring1(2)) simp_all
     with False show ?thesis by (simp add: nonzero_inverse_mult_distrib)
   qed
 qed
 
 lemma fps_inverse_mult_divring:
   "inverse (f * g) = inverse g * inverse (f :: 'a::division_ring fps)"
   using fps_lr_inverse_mult_divring(2) by (simp add: fps_inverse_def)
 
 lemma fps_inverse_mult: "inverse (f * g :: 'a::field fps) = inverse f * inverse g"
   by (simp add: fps_inverse_mult_divring)
 
 lemma fps_lr_inverse_gp_ring1:
   fixes   ones ones_inv :: "'a :: ring_1 fps"
   defines "ones \<equiv> Abs_fps (\<lambda>n. 1)"
   and     "ones_inv \<equiv> Abs_fps (\<lambda>n. if n=0 then 1 else if n=1 then - 1 else 0)"
   shows   "fps_left_inverse ones 1 = ones_inv"
   and     "fps_right_inverse ones 1 = ones_inv"
 proof-
   show "fps_left_inverse ones 1 = ones_inv"
   proof (rule fps_ext)
     fix n
     show "fps_left_inverse ones 1 $ n = ones_inv $ n"
     proof (induct n rule: nat_less_induct)
       case (1 n) show ?case
       proof (cases n)
         case (Suc m)
         have m: "n = Suc m" by fact
         moreover have "fps_left_inverse ones 1 $ Suc m = ones_inv $ Suc m"
         proof (cases m)
           case (Suc k) thus ?thesis
             using Suc m 1 by (simp add: ones_def ones_inv_def sum.atLeast_Suc_atMost)
         qed (simp add: ones_def ones_inv_def)
         ultimately show ?thesis by simp
       qed (simp add: ones_inv_def)
     qed
   qed
   moreover have "fps_right_inverse ones 1 = fps_left_inverse ones 1"
     by (auto intro: fps_left_inverse_eq_fps_right_inverse[symmetric] simp: ones_def)
   ultimately show "fps_right_inverse ones 1 = ones_inv" by simp
 qed
 
 lemma fps_lr_inverse_gp_ring1':
   fixes   ones :: "'a :: ring_1 fps"
   defines "ones \<equiv> Abs_fps (\<lambda>n. 1)"
   shows   "fps_left_inverse ones 1 = 1 - fps_X"
   and     "fps_right_inverse ones 1 = 1 - fps_X"
 proof-
   define ones_inv :: "'a :: ring_1 fps" 
     where "ones_inv \<equiv> Abs_fps (\<lambda>n. if n=0 then 1 else if n=1 then - 1 else 0)"
   hence "fps_left_inverse ones 1 = ones_inv"
   and   "fps_right_inverse ones 1 = ones_inv"
     using ones_def fps_lr_inverse_gp_ring1 by auto
   thus "fps_left_inverse ones 1 = 1 - fps_X"
   and   "fps_right_inverse ones 1 = 1 - fps_X"
     by (auto intro: fps_ext simp: ones_inv_def)
 qed
 
 lemma fps_inverse_gp:
   "inverse (Abs_fps(\<lambda>n. (1::'a::division_ring))) =
     Abs_fps (\<lambda>n. if n= 0 then 1 else if n=1 then - 1 else 0)"
   using fps_lr_inverse_gp_ring1(2) by (simp add: fps_inverse_def)
 
 lemma fps_inverse_gp': "inverse (Abs_fps (\<lambda>n. 1::'a::division_ring)) = 1 - fps_X"
   by (simp add: fps_inverse_def fps_lr_inverse_gp_ring1'(2))
 
 lemma fps_lr_inverse_one_minus_fps_X:
   fixes   ones :: "'a :: ring_1 fps"
   defines "ones \<equiv> Abs_fps (\<lambda>n. 1)"
   shows "fps_left_inverse (1 - fps_X) 1 = ones"
   and   "fps_right_inverse (1 - fps_X) 1 = ones"
 proof-
   have "fps_left_inverse ones 1 = 1 - fps_X"
     using fps_lr_inverse_gp_ring1'(1) by (simp add: ones_def)
   thus "fps_left_inverse (1 - fps_X) 1 = ones"
     using fps_left_inverse_idempotent_ring1[of 1 ones 1] by (simp add: ones_def)
   have "fps_right_inverse ones 1 = 1 - fps_X"
     using fps_lr_inverse_gp_ring1'(2) by (simp add: ones_def)
   thus "fps_right_inverse (1 - fps_X) 1 = ones"
     using fps_right_inverse_idempotent_ring1[of ones 1 1] by (simp add: ones_def)
 qed
 
 lemma fps_inverse_one_minus_fps_X:
   fixes   ones :: "'a :: division_ring fps"
   defines "ones \<equiv> Abs_fps (\<lambda>n. 1)"
   shows   "inverse (1 - fps_X) = ones"
   by      (simp add: fps_inverse_def assms fps_lr_inverse_one_minus_fps_X(2))
 
 lemma fps_lr_one_over_one_minus_fps_X_squared:
   shows   "fps_left_inverse ((1 - fps_X)^2) (1::'a::ring_1) = Abs_fps (\<lambda>n. of_nat (n+1))"
           "fps_right_inverse ((1 - fps_X)^2) (1::'a) = Abs_fps (\<lambda>n. of_nat (n+1))"
 proof-
   define  f invf2 :: "'a fps"
     where "f \<equiv> (1 - fps_X)"
     and   "invf2 \<equiv> Abs_fps (\<lambda>n. of_nat (n+1))"
 
   have f2_nth_simps:
     "f^2 $ 1 = - of_nat 2" "f^2 $ 2 = 1" "\<And>n. n>2 \<Longrightarrow> f^2 $ n = 0"
       by (simp_all add: power2_eq_square f_def fps_mult_nth sum.atLeast_Suc_atMost)
 
   show "fps_left_inverse (f^2) 1 = invf2"
   proof (intro fps_ext)
     fix n show "fps_left_inverse (f^2) 1 $ n = invf2 $ n"
     proof (induct n rule: nat_less_induct)
       case (1 t)
       hence induct_assm:
         "\<And>m. m < t \<Longrightarrow> fps_left_inverse (f\<^sup>2) 1 $ m = invf2 $ m"
         by fast
       show ?case
       proof (cases t)
         case (Suc m)
         have m: "t = Suc m" by fact
         moreover have "fps_left_inverse (f^2) 1 $ Suc m = invf2 $ Suc m"
         proof (cases m)
           case 0 thus ?thesis using f2_nth_simps(1) by (simp add: invf2_def)
         next
           case (Suc l)
           have l: "m = Suc l" by fact
           moreover have "fps_left_inverse (f^2) 1 $ Suc (Suc l) = invf2 $ Suc (Suc l)"
           proof (cases l)
             case 0 thus ?thesis using f2_nth_simps(1,2) by (simp add: Suc_1[symmetric] invf2_def)
           next
             case (Suc k)
             from Suc l m
               have A: "fps_left_inverse (f\<^sup>2) 1 $ Suc (Suc k) = invf2 $ Suc (Suc k)"
               and  B: "fps_left_inverse (f\<^sup>2) 1 $ Suc k = invf2 $ Suc k"
               using induct_assm[of "Suc k"] induct_assm[of "Suc (Suc k)"]
               by    auto
             have times2: "\<And>a::nat. 2*a = a + a" by simp
             have "\<forall>i\<in>{0..k}. (f^2)$(Suc (Suc (Suc k)) - i) = 0"
               using f2_nth_simps(3) by auto
             hence
               "fps_left_inverse (f^2) 1 $ Suc (Suc (Suc k)) =
                 fps_left_inverse (f\<^sup>2) 1 $ Suc (Suc k) * of_nat 2 -
                 fps_left_inverse (f\<^sup>2) 1 $ Suc k"
               using sum.ub_add_nat f2_nth_simps(1,2) by simp
             also have "\<dots> = of_nat (2 * Suc (Suc (Suc k))) - of_nat (Suc (Suc k))"
               by (subst A, subst B) (simp add: invf2_def mult.commute)
             also have "\<dots> = of_nat (Suc (Suc (Suc k)) + 1)"
               by (subst times2[of "Suc (Suc (Suc k))"]) simp
             finally have
               "fps_left_inverse (f^2) 1 $ Suc (Suc (Suc k)) = invf2 $ Suc (Suc (Suc k))"
                by (simp add: invf2_def)
             with Suc show ?thesis by simp
           qed
           ultimately show ?thesis by simp
         qed
         ultimately show ?thesis by simp
       qed (simp add: invf2_def)
     qed
   qed
 
   moreover have "fps_right_inverse (f^2) 1 = fps_left_inverse (f^2) 1"
     by  (auto
           intro: fps_left_inverse_eq_fps_right_inverse[symmetric]
           simp: f_def power2_eq_square
         )
   ultimately show "fps_right_inverse (f^2) 1 = invf2"
     by simp
 
 qed
 
 lemma fps_one_over_one_minus_fps_X_squared':
   assumes "inverse (1::'a::{ring_1,inverse}) = 1"
   shows   "inverse ((1 - fps_X)^2 :: 'a  fps) = Abs_fps (\<lambda>n. of_nat (n+1))"
   using   assms fps_lr_one_over_one_minus_fps_X_squared(2)
   by      (simp add: fps_inverse_def power2_eq_square)
 
 lemma fps_one_over_one_minus_fps_X_squared:
   "inverse ((1 - fps_X)^2 :: 'a :: division_ring fps) = Abs_fps (\<lambda>n. of_nat (n+1))"
   by (rule fps_one_over_one_minus_fps_X_squared'[OF inverse_1])
 
 lemma fps_lr_inverse_fps_X_plus1:
   "fps_left_inverse (1 + fps_X) (1::'a::ring_1) = Abs_fps (\<lambda>n. (-1)^n)"
   "fps_right_inverse (1 + fps_X) (1::'a) = Abs_fps (\<lambda>n. (-1)^n)"
 proof-
 
   show "fps_left_inverse (1 + fps_X) (1::'a) = Abs_fps (\<lambda>n. (-1)^n)"
   proof (rule fps_ext)
     fix n show "fps_left_inverse (1 + fps_X) (1::'a) $ n = Abs_fps (\<lambda>n. (-1)^n) $ n"
     proof (induct n rule: nat_less_induct)
       case (1 n) show ?case
       proof (cases n)
         case (Suc m)
         have m: "n = Suc m" by fact
         from Suc 1 have
           A:  "fps_left_inverse (1 + fps_X) (1::'a) $ n =
                 - (\<Sum>i=0..m. (- 1)^i * (1 + fps_X) $ (Suc m - i))"
           by simp
         show ?thesis
         proof (cases m)
           case (Suc l)
           have "\<forall>i\<in>{0..l}. ((1::'a fps) + fps_X) $ (Suc (Suc l) - i) = 0" by auto
           with Suc A m show ?thesis by simp
         qed (simp add: m A)
       qed simp
     qed
   qed
 
   moreover have
     "fps_right_inverse (1 + fps_X) (1::'a) = fps_left_inverse (1 + fps_X) 1"
     by (intro fps_left_inverse_eq_fps_right_inverse[symmetric]) simp_all
   ultimately show "fps_right_inverse (1 + fps_X) (1::'a) = Abs_fps (\<lambda>n. (-1)^n)" by simp
 
 qed
 
 lemma fps_inverse_fps_X_plus1':
   assumes "inverse (1::'a::{ring_1,inverse}) = 1"
   shows   "inverse (1 + fps_X) = Abs_fps (\<lambda>n. (- (1::'a)) ^ n)"
   using   assms fps_lr_inverse_fps_X_plus1(2)
   by      (simp add: fps_inverse_def)
 
 lemma fps_inverse_fps_X_plus1:
   "inverse (1 + fps_X) = Abs_fps (\<lambda>n. (- (1::'a::division_ring)) ^ n)"
   by (rule fps_inverse_fps_X_plus1'[OF inverse_1])
 
 lemma subdegree_lr_inverse:
   fixes x :: "'a::{comm_monoid_add,mult_zero,uminus}"
   and   y :: "'b::{ab_group_add,mult_zero}"
   shows "subdegree (fps_left_inverse f x) = 0"
   and   "subdegree (fps_right_inverse g y) = 0"
 proof-
   show "subdegree (fps_left_inverse f x) = 0"
     using fps_lr_inverse_eq_0_iff(1) subdegree_eq_0_iff by fastforce
   show "subdegree (fps_right_inverse g y) = 0"
     using fps_lr_inverse_eq_0_iff(2) subdegree_eq_0_iff by fastforce
 qed
 
 lemma subdegree_inverse [simp]:
   fixes f :: "'a::{ab_group_add,inverse,mult_zero} fps"
   shows "subdegree (inverse f) = 0"
   using subdegree_lr_inverse(2)
   by    (simp add: fps_inverse_def)
 
 lemma fps_div_zero [simp]:
   "0 div (g :: 'a :: {comm_monoid_add,inverse,mult_zero,uminus} fps) = 0"
   by (simp add: fps_divide_def)
 
 lemma fps_div_by_zero':
   fixes   g :: "'a::{comm_monoid_add,inverse,mult_zero,uminus} fps"
   assumes "inverse (0::'a) = 0"
   shows   "g div 0 = 0"
   by      (simp add: fps_divide_def assms fps_inverse_zero')
 
 lemma fps_div_by_zero [simp]: "(g::'a::division_ring fps) div 0 = 0"
   by    (rule fps_div_by_zero'[OF inverse_zero])
 
 lemma fps_divide_unit': "subdegree g = 0 \<Longrightarrow> f div g = f * inverse g"
   by (simp add: fps_divide_def)
 
 lemma fps_divide_unit: "g$0 \<noteq> 0 \<Longrightarrow> f div g = f * inverse g"
   by (intro fps_divide_unit') (simp add: subdegree_eq_0_iff)
 
 lemma fps_divide_nth_0':
   "subdegree (g::'a::division_ring fps) = 0 \<Longrightarrow> (f div g) $ 0 = f $ 0 / (g $ 0)"
   by (simp add: fps_divide_unit' divide_inverse)
 
 lemma fps_divide_nth_0 [simp]:
   "g $ 0 \<noteq> 0 \<Longrightarrow> (f div g) $ 0 = f $ 0 / (g $ 0 :: _ :: division_ring)"
   by (simp add: fps_divide_nth_0')
 
 lemma fps_divide_nth_below:
   fixes f g :: "'a::{comm_monoid_add,uminus,mult_zero,inverse} fps"
   shows "n < subdegree f - subdegree g \<Longrightarrow> (f div g) $ n = 0"
   by    (simp add: fps_divide_def fps_mult_nth_eq0)
 
 lemma fps_divide_nth_base:
   fixes   f g :: "'a::division_ring fps"
   assumes "subdegree g \<le> subdegree f"
   shows   "(f div g) $ (subdegree f - subdegree g) = f $ subdegree f * inverse (g $ subdegree g)"
   by      (simp add: assms fps_divide_def fps_divide_unit')
 
 lemma fps_divide_subdegree_ge:
   fixes   f g :: "'a::{comm_monoid_add,uminus,mult_zero,inverse} fps"
   assumes "f / g \<noteq> 0"
   shows   "subdegree (f / g) \<ge> subdegree f - subdegree g"
   by      (intro subdegree_geI) (simp_all add: assms fps_divide_nth_below)
 
 lemma fps_divide_subdegree:
   fixes   f g :: "'a::division_ring fps"
   assumes "f \<noteq> 0" "g \<noteq> 0" "subdegree g \<le> subdegree f"
   shows   "subdegree (f / g) = subdegree f - subdegree g"
 proof (intro antisym)
   from assms have 1: "(f div g) $ (subdegree f - subdegree g) \<noteq> 0"
     using fps_divide_nth_base[of g f] by simp
   thus "subdegree (f / g) \<le> subdegree f - subdegree g" by (intro subdegree_leI) simp
   from 1 have "f / g \<noteq> 0" by (auto intro: fps_nonzeroI)
   thus "subdegree f - subdegree g \<le> subdegree (f / g)" by (rule fps_divide_subdegree_ge)
 qed
 
 lemma fps_divide_shift_numer:
   fixes   f g :: "'a::{inverse,comm_monoid_add,uminus,mult_zero} fps"
   assumes "n \<le> subdegree f"
   shows   "fps_shift n f / g = fps_shift n (f/g)"
   using   assms fps_shift_mult_right_noncomm[of n f "inverse (unit_factor g)"]
           fps_shift_fps_shift_reorder[of "subdegree g" n "f * inverse (unit_factor g)"]
   by      (simp add: fps_divide_def)
 
 lemma fps_divide_shift_denom:
   fixes   f g :: "'a::{inverse,comm_monoid_add,uminus,mult_zero} fps"
   assumes "n \<le> subdegree g" "subdegree g \<le> subdegree f"
   shows   "f / fps_shift n g = Abs_fps (\<lambda>k. if k<n then 0 else (f/g) $ (k-n))"
 proof (intro fps_ext)
   fix k
   from assms(1) have LHS:
     "(f / fps_shift n g) $ k = (f * inverse (unit_factor g)) $ (k + (subdegree g - n))"
     using fps_unit_factor_shift[of n g]
     by    (simp add: fps_divide_def)
   show "(f / fps_shift n g) $ k = Abs_fps (\<lambda>k. if k<n then 0 else (f/g) $ (k-n)) $ k"
   proof (cases "k<n")
     case True with assms LHS show ?thesis using fps_mult_nth_eq0[of _ f] by simp
   next
     case False
     hence "(f/g) $ (k-n) = (f * inverse (unit_factor g)) $ ((k-n) + subdegree g)"
       by (simp add: fps_divide_def)
     with False LHS assms(1) show ?thesis by auto
   qed
 qed
 
 lemma fps_divide_unit_factor_numer:
   fixes   f g :: "'a::{inverse,comm_monoid_add,uminus,mult_zero} fps"
   shows   "unit_factor f / g = fps_shift (subdegree f) (f/g)"
   by      (simp add: fps_divide_shift_numer)
 
 lemma fps_divide_unit_factor_denom:
   fixes   f g :: "'a::{inverse,comm_monoid_add,uminus,mult_zero} fps"
   assumes "subdegree g \<le> subdegree f"
   shows
     "f / unit_factor g = Abs_fps (\<lambda>k. if k<subdegree g then 0 else (f/g) $ (k-subdegree g))"
   by      (simp add: assms fps_divide_shift_denom)
 
 lemma fps_divide_unit_factor_both':
   fixes   f g :: "'a::{inverse,comm_monoid_add,uminus,mult_zero} fps"
   assumes "subdegree g \<le> subdegree f"
   shows   "unit_factor f / unit_factor g = fps_shift (subdegree f - subdegree g) (f / g)"
   using   assms fps_divide_unit_factor_numer[of f "unit_factor g"]
           fps_divide_unit_factor_denom[of g f]
           fps_shift_rev_shift(1)[of "subdegree g" "subdegree f" "f/g"]
   by      simp
 
 lemma fps_divide_unit_factor_both:
   fixes   f g :: "'a::division_ring fps"
   assumes "subdegree g \<le> subdegree f"
   shows   "unit_factor f / unit_factor g = unit_factor (f / g)"
   using   assms fps_divide_unit_factor_both'[of g f] fps_divide_subdegree[of f g]
   by      (cases "f=0 \<or> g=0") auto
 
 lemma fps_divide_self:
   "(f::'a::division_ring fps) \<noteq> 0 \<Longrightarrow> f / f = 1"
   using   fps_mult_right_inverse_unit_factor_divring[of f]
   by      (simp add: fps_divide_def)
 
 lemma fps_divide_add:
   fixes f g h :: "'a::{semiring_0,inverse,uminus} fps"
   shows "(f + g) / h = f / h + g / h"
   by    (simp add: fps_divide_def algebra_simps fps_shift_add)
 
 lemma fps_divide_diff:
   fixes f g h :: "'a::{ring,inverse} fps"
   shows "(f - g) / h = f / h - g / h"
   by    (simp add: fps_divide_def algebra_simps fps_shift_diff)
 
 lemma fps_divide_uminus:
   fixes f g h :: "'a::{ring,inverse} fps"
   shows "(- f) / g = - (f / g)"
   by    (simp add: fps_divide_def algebra_simps fps_shift_uminus)
 
 lemma fps_divide_uminus':
   fixes f g h :: "'a::division_ring fps"
   shows "f / (- g) = - (f / g)"
   by (simp add: fps_divide_def fps_unit_factor_uminus fps_shift_uminus)
 
 lemma fps_divide_times:
   fixes   f g h :: "'a::{semiring_0,inverse,uminus} fps"
   assumes "subdegree h \<le> subdegree g"
   shows   "(f * g) / h = f * (g / h)"
   using   assms fps_mult_subdegree_ge[of g "inverse (unit_factor h)"]
           fps_shift_mult[of "subdegree h" "g * inverse (unit_factor h)" f]
   by      (fastforce simp add: fps_divide_def mult.assoc)
 
 lemma fps_divide_times2:
   fixes   f g h :: "'a::{comm_semiring_0,inverse,uminus} fps"
   assumes "subdegree h \<le> subdegree f"
   shows   "(f * g) / h = (f / h) * g"
   using   assms fps_divide_times[of h f g]
   by      (simp add: mult.commute)
 
 lemma fps_times_divide_eq:
   fixes   f g :: "'a::field fps"
   assumes "g \<noteq> 0" and "subdegree f \<ge> subdegree g"
   shows   "f div g * g = f"
   using   assms fps_divide_times2[of g f g]
   by      (simp add: fps_divide_times fps_divide_self)
 
 lemma fps_divide_times_eq:
   "(g :: 'a::division_ring fps) \<noteq> 0 \<Longrightarrow> (f * g) div g = f"
   by (simp add: fps_divide_times fps_divide_self)
 
 lemma fps_divide_by_mult':
   fixes   f g h :: "'a :: division_ring fps"
   assumes "subdegree h \<le> subdegree f"
   shows   "f / (g * h) = f / h / g"
 proof (cases "f=0 \<or> g=0 \<or> h=0")
   case False with assms show ?thesis
     using fps_unit_factor_mult[of g h]
     by    (auto simp:
             fps_divide_def fps_shift_fps_shift fps_inverse_mult_divring mult.assoc
             fps_shift_mult_right_noncomm
           )
 qed auto
 
 lemma fps_divide_by_mult:
   fixes   f g h :: "'a :: field fps"
   assumes "subdegree g \<le> subdegree f"
   shows   "f / (g * h) = f / g / h"
 proof-
   have "f / (g * h) = f / (h * g)" by (simp add: mult.commute)
   also have "\<dots> = f / g / h" using fps_divide_by_mult'[OF assms] by simp
   finally show ?thesis by simp
 qed
 
 lemma fps_divide_cancel:
   fixes   f g h :: "'a :: division_ring fps"
   shows "h \<noteq> 0 \<Longrightarrow> (f * h) div (g * h) = f div g"
   by    (cases "f=0")
         (auto simp: fps_divide_by_mult' fps_divide_times_eq)
 
 lemma fps_divide_1':
   fixes   a :: "'a::{comm_monoid_add,inverse,mult_zero,uminus,zero_neq_one,monoid_mult} fps"
   assumes "inverse (1::'a) = 1"
   shows   "a / 1 = a"
   using   assms fps_inverse_one' fps_one_mult(2)[of a]
   by      (force simp: fps_divide_def)
 
 lemma fps_divide_1 [simp]: "(a :: 'a::division_ring fps) / 1 = a"
   by (rule fps_divide_1'[OF inverse_1])
 
 lemma fps_divide_X':
   fixes   f :: "'a::{comm_monoid_add,inverse,mult_zero,uminus,zero_neq_one,monoid_mult} fps"
   assumes "inverse (1::'a) = 1"
   shows   "f / fps_X = fps_shift 1 f"
   using   assms fps_one_mult(2)[of f]
   by      (simp add: fps_divide_def fps_X_unit_factor fps_inverse_one')
 
 lemma fps_divide_X [simp]: "a / fps_X = fps_shift 1 (a::'a::division_ring fps)"
   by (rule fps_divide_X'[OF inverse_1])
 
 lemma fps_divide_X_power':
   fixes   f :: "'a::{semiring_1,inverse,uminus} fps"
   assumes "inverse (1::'a) = 1"
   shows   "f / (fps_X ^ n) = fps_shift n f"
   using   fps_inverse_one'[OF assms] fps_one_mult(2)[of f]
   by      (simp add: fps_divide_def fps_X_power_subdegree)
 
 lemma fps_divide_X_power [simp]: "a / (fps_X ^ n) = fps_shift n (a::'a::division_ring fps)"
   by (rule fps_divide_X_power'[OF inverse_1])
 
 lemma fps_divide_shift_denom_conv_times_fps_X_power:
   fixes   f g :: "'a::{semiring_1,inverse,uminus} fps"
   assumes "n \<le> subdegree g" "subdegree g \<le> subdegree f"
   shows   "f / fps_shift n g = f / g * fps_X ^ n"
   using   assms
   by      (intro fps_ext) (simp_all add: fps_divide_shift_denom fps_X_power_mult_right_nth)
 
 lemma fps_divide_unit_factor_denom_conv_times_fps_X_power:
   fixes   f g :: "'a::{semiring_1,inverse,uminus} fps"
   assumes "subdegree g \<le> subdegree f"
   shows   "f / unit_factor g = f / g * fps_X ^ subdegree g"
   by      (simp add: assms fps_divide_shift_denom_conv_times_fps_X_power)
 
 lemma fps_shift_altdef':
   fixes   f :: "'a::{semiring_1,inverse,uminus} fps"
   assumes "inverse (1::'a) = 1"
   shows   "fps_shift n f = f div fps_X^n"
   using   assms 
   by      (simp add:
             fps_divide_def fps_X_power_subdegree fps_X_power_unit_factor fps_inverse_one'
           )
 
 lemma fps_shift_altdef:
   "fps_shift n f = (f :: 'a :: division_ring fps) div fps_X^n"
   by (rule fps_shift_altdef'[OF inverse_1])
 
 lemma fps_div_fps_X_power_nth':
   fixes   f :: "'a::{semiring_1,inverse,uminus} fps"
   assumes "inverse (1::'a) = 1"
   shows   "(f div fps_X^n) $ k = f $ (k + n)"
   using   assms
   by      (simp add: fps_shift_altdef' [symmetric])
 
 lemma fps_div_fps_X_power_nth: "((f :: 'a :: division_ring fps) div fps_X^n) $ k = f $ (k + n)"
   by (rule fps_div_fps_X_power_nth'[OF inverse_1])
 
 lemma fps_div_fps_X_nth':
   fixes   f :: "'a::{semiring_1,inverse,uminus} fps"
   assumes "inverse (1::'a) = 1"
   shows   "(f div fps_X) $ k = f $ Suc k"
   using   assms fps_div_fps_X_power_nth'[of f 1]
   by      simp
 
 lemma fps_div_fps_X_nth: "((f :: 'a :: division_ring fps) div fps_X) $ k = f $ Suc k"
   by (rule fps_div_fps_X_nth'[OF inverse_1])
 
 lemma divide_fps_const':
   fixes c :: "'a :: {inverse,comm_monoid_add,uminus,mult_zero}"
   shows   "f / fps_const c = f * fps_const (inverse c)"
   by      (simp add: fps_divide_def fps_const_inverse)
 
 lemma divide_fps_const [simp]:
   fixes c :: "'a :: {comm_semiring_0,inverse,uminus}"
   shows "f / fps_const c = fps_const (inverse c) * f"
   by    (simp add: divide_fps_const' mult.commute)
 
 lemma fps_const_divide: "fps_const (x :: _ :: division_ring) / fps_const y = fps_const (x / y)"
   by (simp add: fps_divide_def fps_const_inverse divide_inverse)
 
 lemma fps_numeral_divide_divide:
   "x / numeral b / numeral c = (x / numeral (b * c) :: 'a :: field fps)"
   by (simp add: fps_divide_by_mult[symmetric])
 
 lemma fps_numeral_mult_divide:
   "numeral b * x / numeral c = (numeral b / numeral c * x :: 'a :: field fps)"
   by (simp add: fps_divide_times2)
 
 lemmas fps_numeral_simps = 
   fps_numeral_divide_divide fps_numeral_mult_divide inverse_fps_numeral neg_numeral_fps_const
 
 lemma fps_is_left_unit_iff_zeroth_is_left_unit:
   fixes f :: "'a :: ring_1 fps"
   shows "(\<exists>g. 1 = f * g) \<longleftrightarrow> (\<exists>k. 1 = f$0 * k)"
 proof
   assume "\<exists>g. 1 = f * g"
   then obtain g where "1 = f * g" by fast
   hence "1 = f$0 * g$0" using fps_mult_nth_0[of f g] by simp
   thus "\<exists>k. 1 = f$0 * k" by auto
 next
   assume "\<exists>k. 1 = f$0 * k"
   then obtain k where "1 = f$0 * k" by fast
   hence "1 = f * fps_right_inverse f k"
     using fps_right_inverse by simp
   thus "\<exists>g. 1 = f * g" by fast
 qed
 
 lemma fps_is_right_unit_iff_zeroth_is_right_unit:
   fixes f :: "'a :: ring_1 fps"
   shows "(\<exists>g. 1 = g * f) \<longleftrightarrow> (\<exists>k. 1 = k * f$0)"
 proof
   assume "\<exists>g. 1 = g * f"
   then obtain g where "1 = g * f" by fast
   hence "1 = g$0 * f$0" using fps_mult_nth_0[of g f] by simp
   thus "\<exists>k. 1 = k * f$0" by auto
 next
   assume "\<exists>k. 1 = k * f$0"
   then obtain k where "1 = k * f$0" by fast
   hence "1 = fps_left_inverse f k * f"
     using fps_left_inverse by simp
   thus "\<exists>g. 1 = g * f" by fast
 qed
 
 lemma fps_is_unit_iff [simp]: "(f :: 'a :: field fps) dvd 1 \<longleftrightarrow> f $ 0 \<noteq> 0"
 proof
   assume "f dvd 1"
   then obtain g where "1 = f * g" by (elim dvdE)
   from this[symmetric] have "(f*g) $ 0 = 1" by simp
   thus "f $ 0 \<noteq> 0" by auto
 next
   assume A: "f $ 0 \<noteq> 0"
   thus "f dvd 1" by (simp add: inverse_mult_eq_1[OF A, symmetric])
 qed
 
 lemma subdegree_eq_0_left:
   fixes   f :: "'a::{comm_monoid_add,zero_neq_one,mult_zero} fps"
   assumes "\<exists>g. 1 = f * g"
   shows   "subdegree f = 0"
 proof (intro subdegree_eq_0)
   from assms obtain g where "1 = f * g" by fast
   hence "f$0 * g$0 = 1" using fps_mult_nth_0[of f g] by simp
   thus "f$0 \<noteq> 0" by auto
 qed
 
 lemma subdegree_eq_0_right:
   fixes   f :: "'a::{comm_monoid_add,zero_neq_one,mult_zero} fps"
   assumes "\<exists>g. 1 = g * f"
   shows   "subdegree f = 0"
 proof (intro subdegree_eq_0)
   from assms obtain g where "1 = g * f" by fast
   hence "g$0 * f$0 = 1" using fps_mult_nth_0[of g f] by simp
   thus "f$0 \<noteq> 0" by auto
 qed
 
 lemma subdegree_eq_0' [simp]: "(f :: 'a :: field fps) dvd 1 \<Longrightarrow> subdegree f = 0"
   by simp
 
 lemma fps_dvd1_left_trivial_unit_factor:
   fixes   f :: "'a::{comm_monoid_add, zero_neq_one, mult_zero} fps"
   assumes "\<exists>g. 1 = f * g"
   shows   "unit_factor f = f"
   using   assms subdegree_eq_0_left
   by      fastforce
 
 lemma fps_dvd1_right_trivial_unit_factor:
   fixes   f :: "'a::{comm_monoid_add, zero_neq_one, mult_zero} fps"
   assumes "\<exists>g. 1 = g * f"
   shows   "unit_factor f = f"
   using   assms subdegree_eq_0_right
   by      fastforce
 
 lemma fps_dvd1_trivial_unit_factor:
   "(f :: 'a::comm_semiring_1 fps) dvd 1 \<Longrightarrow> unit_factor f = f"
   unfolding dvd_def by (rule fps_dvd1_left_trivial_unit_factor) simp
 
 lemma fps_unit_dvd_left:
   fixes   f :: "'a :: division_ring fps"
   assumes "f $ 0 \<noteq> 0"
   shows   "\<exists>g. 1 = f * g"
   using   assms fps_is_left_unit_iff_zeroth_is_left_unit right_inverse
   by      fastforce
 
 lemma fps_unit_dvd_right:
   fixes   f :: "'a :: division_ring fps"
   assumes "f $ 0 \<noteq> 0"
   shows   "\<exists>g. 1 = g * f"
   using   assms fps_is_right_unit_iff_zeroth_is_right_unit left_inverse
   by      fastforce
 
 lemma fps_unit_dvd [simp]: "(f $ 0 :: 'a :: field) \<noteq> 0 \<Longrightarrow> f dvd g"
   using fps_unit_dvd_left dvd_trans[of f 1] by simp
 
 lemma dvd_left_imp_subdegree_le:
   fixes   f g :: "'a::{comm_monoid_add,mult_zero} fps"
   assumes "\<exists>k. g = f * k" "g \<noteq> 0"
   shows   "subdegree f \<le> subdegree g"
   using   assms fps_mult_subdegree_ge
   by      fastforce
 
 lemma dvd_right_imp_subdegree_le:
   fixes   f g :: "'a::{comm_monoid_add,mult_zero} fps"
   assumes "\<exists>k. g = k * f" "g \<noteq> 0"
   shows   "subdegree f \<le> subdegree g"
   using   assms fps_mult_subdegree_ge
   by      fastforce
 
 lemma dvd_imp_subdegree_le:
   "f dvd g \<Longrightarrow> g \<noteq> 0 \<Longrightarrow> subdegree f \<le> subdegree g"
   using dvd_left_imp_subdegree_le by fast
 
 lemma subdegree_le_imp_dvd_left_ring1:
   fixes   f g :: "'a :: ring_1 fps"
   assumes "\<exists>y. f $ subdegree f * y = 1" "subdegree f \<le> subdegree g"
   shows   "\<exists>k. g = f * k"
 proof-
   define h :: "'a fps" where "h \<equiv> fps_X ^ (subdegree g - subdegree f)"
   from assms(1) obtain y where "f $ subdegree f * y = 1" by fast
   hence "unit_factor f $ 0 * y = 1" by simp
   from this obtain k where "1 = unit_factor f * k"
     using fps_is_left_unit_iff_zeroth_is_left_unit[of "unit_factor f"] by auto
   hence "fps_X ^ subdegree f = fps_X ^ subdegree f * unit_factor f * k"
     by (simp add: mult.assoc)
   moreover have "fps_X ^ subdegree f * unit_factor f = f"
     by (rule fps_unit_factor_decompose'[symmetric])
   ultimately have
     "fps_X ^ (subdegree f + (subdegree g - subdegree f)) = f * k * h"
     by (simp add: power_add h_def)
   hence "g = f * (k * h * unit_factor g)"
     using fps_unit_factor_decompose'[of g]
     by    (simp add: assms(2) mult.assoc)
   thus ?thesis by fast
 qed
 
 lemma subdegree_le_imp_dvd_left_divring:
   fixes   f g :: "'a :: division_ring fps"
   assumes "f \<noteq> 0" "subdegree f \<le> subdegree g"
   shows   "\<exists>k. g = f * k"
 proof (intro subdegree_le_imp_dvd_left_ring1)
   from assms(1) have "f $ subdegree f \<noteq> 0" by simp
   thus "\<exists>y. f $ subdegree f * y = 1" using right_inverse by blast
 qed (rule assms(2))
 
 lemma subdegree_le_imp_dvd_right_ring1:
   fixes   f g :: "'a :: ring_1 fps"
   assumes "\<exists>x. x * f $ subdegree f = 1" "subdegree f \<le> subdegree g"
   shows   "\<exists>k. g = k * f"
 proof-
   define h :: "'a fps" where "h \<equiv> fps_X ^ (subdegree g - subdegree f)"
   from assms(1) obtain x where "x * f $ subdegree f = 1" by fast
   hence "x * unit_factor f $ 0 = 1" by simp
   from this obtain k where "1 = k * unit_factor f"
     using fps_is_right_unit_iff_zeroth_is_right_unit[of "unit_factor f"] by auto
   hence "fps_X ^ subdegree f = k * (unit_factor f * fps_X ^ subdegree f)"
     by (simp add: mult.assoc[symmetric])
   moreover have "unit_factor f * fps_X ^ subdegree f = f"
     by (rule fps_unit_factor_decompose[symmetric])
   ultimately have "fps_X ^ (subdegree g - subdegree f + subdegree f) = h * k * f"
     by (simp add: power_add h_def mult.assoc)
   hence "g = unit_factor g * h * k * f"
     using fps_unit_factor_decompose[of g]
     by    (simp add: assms(2) mult.assoc)
   thus ?thesis by fast
 qed
 
 lemma subdegree_le_imp_dvd_right_divring:
   fixes   f g :: "'a :: division_ring fps"
   assumes "f \<noteq> 0" "subdegree f \<le> subdegree g"
   shows   "\<exists>k. g = k * f"
 proof (intro subdegree_le_imp_dvd_right_ring1)
   from assms(1) have "f $ subdegree f \<noteq> 0" by simp
   thus "\<exists>x. x * f $ subdegree f = 1" using left_inverse by blast
 qed (rule assms(2))
 
 lemma fps_dvd_iff:
   assumes "(f :: 'a :: field fps) \<noteq> 0" "g \<noteq> 0"
   shows   "f dvd g \<longleftrightarrow> subdegree f \<le> subdegree g"
 proof
   assume "subdegree f \<le> subdegree g"
   with assms show "f dvd g"
     using subdegree_le_imp_dvd_left_divring
     by    (auto intro: dvdI)
 qed (simp add: assms dvd_imp_subdegree_le)
 
 lemma subdegree_div':
   fixes   p q :: "'a::division_ring fps"
   assumes "\<exists>k. p = k * q"
   shows   "subdegree (p div q) = subdegree p - subdegree q"
 proof (cases "p = 0")
   case False
   from assms(1) obtain k where k: "p = k * q" by blast
   with False have "subdegree (p div q) = subdegree k" by (simp add: fps_divide_times_eq)
   moreover have "k $ subdegree k * q $ subdegree q \<noteq> 0"
   proof
     assume "k $ subdegree k * q $ subdegree q = 0"
     hence "k $ subdegree k * q $ subdegree q * inverse (q $ subdegree q) = 0" by simp
     with False k show False by (simp add: mult.assoc)
   qed
   ultimately show ?thesis by (simp add: k subdegree_mult')
 qed simp
 
 lemma subdegree_div:
   fixes     p q :: "'a :: field fps"
   assumes   "q dvd p"
   shows     "subdegree (p div q) = subdegree p - subdegree q"
   using     assms
   unfolding dvd_def
   by        (auto intro: subdegree_div')
 
 lemma subdegree_div_unit':
   fixes   p q :: "'a :: {ab_group_add,mult_zero,inverse} fps"
   assumes "q $ 0 \<noteq> 0" "p $ subdegree p * inverse (q $ 0) \<noteq> 0"
   shows   "subdegree (p div q) = subdegree p"
   using   assms subdegree_mult'[of p "inverse q"]
   by      (auto simp add: fps_divide_unit)
 
 lemma subdegree_div_unit'':
   fixes   p q :: "'a :: {ring_no_zero_divisors,inverse} fps"
   assumes "q $ 0 \<noteq> 0" "inverse (q $ 0) \<noteq> 0"
   shows   "subdegree (p div q) = subdegree p"
   by      (cases "p = 0") (auto intro: subdegree_div_unit' simp: assms)
 
 lemma subdegree_div_unit:
   fixes   p q :: "'a :: division_ring fps"
   assumes "q $ 0 \<noteq> 0"
   shows   "subdegree (p div q) = subdegree p"
   by      (intro subdegree_div_unit'') (simp_all add: assms)
 
 instantiation fps :: ("{comm_semiring_1,inverse,uminus}") modulo
 begin
 
 definition fps_mod_def:
   "f mod g = (if g = 0 then f else
      let h = unit_factor g in  fps_cutoff (subdegree g) (f * inverse h) * h)"
 
 instance ..
 
 end
 
 lemma fps_mod_zero [simp]:
   "(f::'a::{comm_semiring_1,inverse,uminus} fps) mod 0 = f"
   by (simp add: fps_mod_def)
 
 lemma fps_mod_eq_zero:
   assumes "g \<noteq> 0" and "subdegree f \<ge> subdegree g"
   shows   "f mod g = 0"
 proof (cases "f * inverse (unit_factor g) = 0")
   case False
   have "fps_cutoff (subdegree g) (f * inverse (unit_factor g)) = 0"
     using False assms(2) fps_mult_subdegree_ge fps_cutoff_zero_iff by force
   with assms(1) show ?thesis by (simp add: fps_mod_def Let_def)
 qed (simp add: assms fps_mod_def)
 
 lemma fps_mod_unit [simp]: "g$0 \<noteq> 0 \<Longrightarrow> f mod g = 0"
   by (intro fps_mod_eq_zero) auto
 
 lemma subdegree_mod:
   assumes "subdegree (f::'a::field fps) < subdegree g"
   shows   "subdegree (f mod g) = subdegree f"
 proof (cases "f = 0")
   case False
   with assms show ?thesis
     by  (intro subdegreeI)
         (auto simp: inverse_mult_eq_1 fps_mod_def Let_def fps_cutoff_left_mult_nth mult.assoc)
 qed (simp add: fps_mod_def)
 
 instance fps :: (field) idom_modulo
 proof
 
   fix f g :: "'a fps"
 
   define n where "n = subdegree g"
   define h where "h = f * inverse (unit_factor g)"
 
   show "f div g * g + f mod g = f"
   proof (cases "g = 0")
     case False
     with n_def h_def have
       "f div g * g + f mod g = (fps_shift n h * fps_X ^ n + fps_cutoff n h) * unit_factor g"
       by (simp add: fps_divide_def fps_mod_def Let_def subdegree_decompose algebra_simps)
     with False show ?thesis
       by (simp add: fps_shift_cutoff h_def inverse_mult_eq_1)
   qed auto
 
 qed (rule fps_divide_times_eq, simp_all add: fps_divide_def)
 
 instantiation fps :: (field) normalization_semidom_multiplicative
 begin
 
 definition fps_normalize_def [simp]:
   "normalize f = (if f = 0 then 0 else fps_X ^ subdegree f)"
 
 instance proof
   fix f g :: "'a fps"
   assume "is_unit f"
   thus "unit_factor (f * g) = f * unit_factor g"
     using fps_unit_factor_mult[of f g] by simp
 next
   fix f g :: "'a fps"
   show "unit_factor f * normalize f = f"
     by (simp add: fps_shift_times_fps_X_power)
 next
   fix f g :: "'a fps"
   show "unit_factor (f * g) = unit_factor f * unit_factor g"
     using fps_unit_factor_mult[of f g] by simp
 qed (simp_all add: fps_divide_def Let_def)
 
 end
 
 
 subsection \<open>Formal power series form a Euclidean ring\<close>
 
 instantiation fps :: (field) euclidean_ring_cancel
 begin
 
 definition fps_euclidean_size_def:
   "euclidean_size f = (if f = 0 then 0 else 2 ^ subdegree f)"
 
 instance proof
   fix f g :: "'a fps" assume [simp]: "g \<noteq> 0"
   show "euclidean_size f \<le> euclidean_size (f * g)"
     by (cases "f = 0") (simp_all add: fps_euclidean_size_def)
   show "euclidean_size (f mod g) < euclidean_size g"
-    apply (cases "f = 0", simp add: fps_euclidean_size_def)
-    apply (rule disjE[OF le_less_linear[of "subdegree g" "subdegree f"]])
-    apply (simp_all add: fps_mod_eq_zero fps_euclidean_size_def subdegree_mod)
-    done
+  proof (cases "f = 0")
+    case True
+    then show ?thesis
+      by (simp add: fps_euclidean_size_def)
+  next
+    case False
+    then show ?thesis 
+      using le_less_linear[of "subdegree g" "subdegree f"]
+      by (force simp add: fps_mod_eq_zero fps_euclidean_size_def subdegree_mod)
+  qed
 next
   fix f g h :: "'a fps" assume [simp]: "h \<noteq> 0"
   show "(h * f) div (h * g) = f div g"
     by (simp add: fps_divide_cancel mult.commute)
   show "(f + g * h) div h = g + f div h"
     by (simp add: fps_divide_add fps_divide_times_eq)
 qed (simp add: fps_euclidean_size_def)
 
 end
 
 instance fps :: (field) normalization_euclidean_semiring ..
 
 instantiation fps :: (field) euclidean_ring_gcd
 begin
 definition fps_gcd_def: "(gcd :: 'a fps \<Rightarrow> _) = Euclidean_Algorithm.gcd"
 definition fps_lcm_def: "(lcm :: 'a fps \<Rightarrow> _) = Euclidean_Algorithm.lcm"
 definition fps_Gcd_def: "(Gcd :: 'a fps set \<Rightarrow> _) = Euclidean_Algorithm.Gcd"
 definition fps_Lcm_def: "(Lcm :: 'a fps set \<Rightarrow> _) = Euclidean_Algorithm.Lcm"
 instance by standard (simp_all add: fps_gcd_def fps_lcm_def fps_Gcd_def fps_Lcm_def)
 end
 
 lemma fps_gcd:
   assumes [simp]: "f \<noteq> 0" "g \<noteq> 0"
   shows   "gcd f g = fps_X ^ min (subdegree f) (subdegree g)"
 proof -
   let ?m = "min (subdegree f) (subdegree g)"
   show "gcd f g = fps_X ^ ?m"
   proof (rule sym, rule gcdI)
     fix d assume "d dvd f" "d dvd g"
     thus "d dvd fps_X ^ ?m" by (cases "d = 0") (simp_all add: fps_dvd_iff)
   qed (simp_all add: fps_dvd_iff)
 qed
 
 lemma fps_gcd_altdef: "gcd f g =
   (if f = 0 \<and> g = 0 then 0 else
    if f = 0 then fps_X ^ subdegree g else
    if g = 0 then fps_X ^ subdegree f else
      fps_X ^ min (subdegree f) (subdegree g))"
   by (simp add: fps_gcd)
 
 lemma fps_lcm:
   assumes [simp]: "f \<noteq> 0" "g \<noteq> 0"
   shows   "lcm f g = fps_X ^ max (subdegree f) (subdegree g)"
 proof -
   let ?m = "max (subdegree f) (subdegree g)"
   show "lcm f g = fps_X ^ ?m"
   proof (rule sym, rule lcmI)
     fix d assume "f dvd d" "g dvd d"
     thus "fps_X ^ ?m dvd d" by (cases "d = 0") (simp_all add: fps_dvd_iff)
   qed (simp_all add: fps_dvd_iff)
 qed
 
 lemma fps_lcm_altdef: "lcm f g =
   (if f = 0 \<or> g = 0 then 0 else fps_X ^ max (subdegree f) (subdegree g))"
   by (simp add: fps_lcm)
 
 lemma fps_Gcd:
   assumes "A - {0} \<noteq> {}"
   shows   "Gcd A = fps_X ^ (INF f\<in>A-{0}. subdegree f)"
 proof (rule sym, rule GcdI)
   fix f assume "f \<in> A"
   thus "fps_X ^ (INF f\<in>A - {0}. subdegree f) dvd f"
     by (cases "f = 0") (auto simp: fps_dvd_iff intro!: cINF_lower)
 next
   fix d assume d: "\<And>f. f \<in> A \<Longrightarrow> d dvd f"
   from assms obtain f where "f \<in> A - {0}" by auto
   with d[of f] have [simp]: "d \<noteq> 0" by auto
   from d assms have "subdegree d \<le> (INF f\<in>A-{0}. subdegree f)"
     by (intro cINF_greatest) (simp_all add: fps_dvd_iff[symmetric])
   with d assms show "d dvd fps_X ^ (INF f\<in>A-{0}. subdegree f)" by (simp add: fps_dvd_iff)
 qed simp_all
 
 lemma fps_Gcd_altdef: "Gcd A =
   (if A \<subseteq> {0} then 0 else fps_X ^ (INF f\<in>A-{0}. subdegree f))"
   using fps_Gcd by auto
 
 lemma fps_Lcm:
   assumes "A \<noteq> {}" "0 \<notin> A" "bdd_above (subdegree`A)"
   shows   "Lcm A = fps_X ^ (SUP f\<in>A. subdegree f)"
 proof (rule sym, rule LcmI)
   fix f assume "f \<in> A"
   moreover from assms(3) have "bdd_above (subdegree ` A)" by auto
   ultimately show "f dvd fps_X ^ (SUP f\<in>A. subdegree f)" using assms(2)
     by (cases "f = 0") (auto simp: fps_dvd_iff intro!: cSUP_upper)
 next
   fix d assume d: "\<And>f. f \<in> A \<Longrightarrow> f dvd d"
   from assms obtain f where f: "f \<in> A" "f \<noteq> 0" by auto
   show "fps_X ^ (SUP f\<in>A. subdegree f) dvd d"
   proof (cases "d = 0")
     assume "d \<noteq> 0"
     moreover from d have "\<And>f. f \<in> A \<Longrightarrow> f \<noteq> 0 \<Longrightarrow> f dvd d" by blast
     ultimately have "subdegree d \<ge> (SUP f\<in>A. subdegree f)" using assms
       by (intro cSUP_least) (auto simp: fps_dvd_iff)
     with \<open>d \<noteq> 0\<close> show ?thesis by (simp add: fps_dvd_iff)
   qed simp_all
 qed simp_all
 
 lemma fps_Lcm_altdef:
   "Lcm A =
      (if 0 \<in> A \<or> \<not>bdd_above (subdegree`A) then 0 else
       if A = {} then 1 else fps_X ^ (SUP f\<in>A. subdegree f))"
 proof (cases "bdd_above (subdegree`A)")
   assume unbounded: "\<not>bdd_above (subdegree`A)"
   have "Lcm A = 0"
   proof (rule ccontr)
     assume "Lcm A \<noteq> 0"
     from unbounded obtain f where f: "f \<in> A" "subdegree (Lcm A) < subdegree f"
       unfolding bdd_above_def by (auto simp: not_le)
     moreover from f and \<open>Lcm A \<noteq> 0\<close> have "subdegree f \<le> subdegree (Lcm A)"
       by (intro dvd_imp_subdegree_le dvd_Lcm) simp_all
     ultimately show False by simp
   qed
   with unbounded show ?thesis by simp
 qed (simp_all add: fps_Lcm Lcm_eq_0_I)
 
 
 
 subsection \<open>Formal Derivatives, and the MacLaurin theorem around 0\<close>
 
 definition "fps_deriv f = Abs_fps (\<lambda>n. of_nat (n + 1) * f $ (n + 1))"
 
 lemma fps_deriv_nth[simp]: "fps_deriv f $ n = of_nat (n + 1) * f $ (n + 1)"
   by (simp add: fps_deriv_def)
 
 lemma fps_0th_higher_deriv: 
   "(fps_deriv ^^ n) f $ 0 = fact n * f $ n"
   by  (induction n arbitrary: f)
       (simp_all add: funpow_Suc_right mult_of_nat_commute algebra_simps del: funpow.simps)
 
 lemma fps_deriv_mult[simp]:
   "fps_deriv (f * g) = f * fps_deriv g + fps_deriv f * g"
 proof (intro fps_ext)
   fix n
   have LHS: "fps_deriv (f * g) $ n = (\<Sum>i=0..Suc n. of_nat (n+1) * f$i * g$(Suc n - i))"
     by (simp add: fps_mult_nth sum_distrib_left algebra_simps)
 
   have "\<forall>i\<in>{1..n}. n - (i - 1) = n - i + 1" by auto
   moreover have
     "(\<Sum>i=0..n. of_nat (i+1) * f$(i+1) * g$(n - i)) =
       (\<Sum>i=1..Suc n. of_nat i * f$i * g$(n - (i - 1)))"
     by (intro sum.reindex_bij_witness[where i="\<lambda>x. x-1" and j="\<lambda>x. x+1"]) auto
   ultimately have
     "(f * fps_deriv g + fps_deriv f * g) $ n =
       of_nat (Suc n) * f$0 * g$(Suc n) +
       (\<Sum>i=1..n. (of_nat (n - i + 1) + of_nat i) * f $ i * g $ (n - i + 1)) +
           of_nat (Suc n) * f$(Suc n) * g$0"
     by (simp add: fps_mult_nth algebra_simps mult_of_nat_commute sum.atLeast_Suc_atMost sum.distrib)
   moreover have
     "\<forall>i\<in>{1..n}.
       (of_nat (n - i + 1) + of_nat i) * f $ i * g $ (n - i + 1) =
       of_nat (n + 1) * f $ i * g $ (Suc n - i)"
   proof
     fix i assume i: "i \<in> {1..n}"
     from i have "of_nat (n - i + 1) + (of_nat i :: 'a) = of_nat (n + 1)"
       using of_nat_add[of "n-i+1" i,symmetric] by simp
     moreover from i have "Suc n - i = n - i + 1" by auto
     ultimately show "(of_nat (n - i + 1) + of_nat i) * f $ i * g $ (n - i + 1) =
             of_nat (n + 1) * f $ i * g $ (Suc n - i)"
       by simp
   qed
   ultimately have
     "(f * fps_deriv g + fps_deriv f * g) $ n =
       (\<Sum>i=0..Suc n. of_nat (Suc n) * f $ i * g $ (Suc n - i))"
     by (simp add: sum.atLeast_Suc_atMost)
   with LHS show "fps_deriv (f * g) $ n = (f * fps_deriv g + fps_deriv f * g) $ n"
     by simp
 qed
 
 lemma fps_deriv_fps_X[simp]: "fps_deriv fps_X = 1"
   by (simp add: fps_deriv_def fps_X_def fps_eq_iff)
 
 lemma fps_deriv_neg[simp]:
   "fps_deriv (- (f:: 'a::ring_1 fps)) = - (fps_deriv f)"
   by (simp add: fps_eq_iff fps_deriv_def)
 
 lemma fps_deriv_add[simp]: "fps_deriv (f + g) = fps_deriv f + fps_deriv g"
   by (auto intro: fps_ext simp: algebra_simps)
 
 lemma fps_deriv_sub[simp]:
   "fps_deriv ((f:: 'a::ring_1 fps) - g) = fps_deriv f - fps_deriv g"
   using fps_deriv_add [of f "- g"] by simp
 
 lemma fps_deriv_const[simp]: "fps_deriv (fps_const c) = 0"
   by (simp add: fps_ext fps_deriv_def fps_const_def)
 
 lemma fps_deriv_of_nat [simp]: "fps_deriv (of_nat n) = 0"
   by (simp add: fps_of_nat [symmetric])
 
 lemma fps_deriv_of_int [simp]: "fps_deriv (of_int n) = 0"
   by (simp add: fps_of_int [symmetric])
 
 lemma fps_deriv_numeral [simp]: "fps_deriv (numeral n) = 0"
   by (simp add: numeral_fps_const)    
 
 lemma fps_deriv_mult_const_left[simp]:
   "fps_deriv (fps_const c * f) = fps_const c * fps_deriv f"
   by simp
 
 lemma fps_deriv_linear[simp]:
   "fps_deriv (fps_const a * f + fps_const b * g) =
     fps_const a * fps_deriv f + fps_const b * fps_deriv g"
   by simp
 
 lemma fps_deriv_0[simp]: "fps_deriv 0 = 0"
   by (simp add: fps_deriv_def fps_eq_iff)
 
 lemma fps_deriv_1[simp]: "fps_deriv 1 = 0"
   by (simp add: fps_deriv_def fps_eq_iff)
 
 lemma fps_deriv_mult_const_right[simp]:
   "fps_deriv (f * fps_const c) = fps_deriv f * fps_const c"
   by simp
 
 lemma fps_deriv_sum:
   "fps_deriv (sum f S) = sum (\<lambda>i. fps_deriv (f i)) S"
 proof (cases "finite S")
   case False
   then show ?thesis by simp
 next
   case True
   show ?thesis by (induct rule: finite_induct [OF True]) simp_all
 qed
 
 lemma fps_deriv_eq_0_iff [simp]:
   "fps_deriv f = 0 \<longleftrightarrow> f = fps_const (f$0 :: 'a::{semiring_no_zero_divisors,semiring_char_0})"
 proof
   assume f: "fps_deriv f = 0"
   show "f = fps_const (f$0)"
   proof (intro fps_ext)
     fix n show "f $ n = fps_const (f$0) $ n"
     proof (cases n)
       case (Suc m)
       have "(of_nat (Suc m) :: 'a) \<noteq> 0" by (rule of_nat_neq_0)
       with f Suc show ?thesis using fps_deriv_nth[of f] by auto
     qed simp
   qed
 next
   show "f = fps_const (f$0) \<Longrightarrow> fps_deriv f = 0" using fps_deriv_const[of "f$0"] by simp
 qed
 
 lemma fps_deriv_eq_iff:
   fixes f g :: "'a::{ring_1_no_zero_divisors,semiring_char_0} fps"
   shows "fps_deriv f = fps_deriv g \<longleftrightarrow> (f = fps_const(f$0 - g$0) + g)"
 proof -
   have "fps_deriv f = fps_deriv g \<longleftrightarrow> fps_deriv (f - g) = 0"
     using fps_deriv_sub[of f g]
     by simp
   also have "\<dots> \<longleftrightarrow> f - g = fps_const ((f - g) $ 0)"
     unfolding fps_deriv_eq_0_iff ..
   finally show ?thesis
     by (simp add: field_simps)
 qed
 
 lemma fps_deriv_eq_iff_ex:
   fixes f g :: "'a::{ring_1_no_zero_divisors,semiring_char_0} fps"
   shows "(fps_deriv f = fps_deriv g) \<longleftrightarrow> (\<exists>c. f = fps_const c + g)"
   by    (auto simp: fps_deriv_eq_iff)
 
 
 fun fps_nth_deriv :: "nat \<Rightarrow> 'a::semiring_1 fps \<Rightarrow> 'a fps"
 where
   "fps_nth_deriv 0 f = f"
 | "fps_nth_deriv (Suc n) f = fps_nth_deriv n (fps_deriv f)"
 
 lemma fps_nth_deriv_commute: "fps_nth_deriv (Suc n) f = fps_deriv (fps_nth_deriv n f)"
   by (induct n arbitrary: f) auto
 
 lemma fps_nth_deriv_linear[simp]:
   "fps_nth_deriv n (fps_const a * f + fps_const b * g) =
     fps_const a * fps_nth_deriv n f + fps_const b * fps_nth_deriv n g"
   by (induct n arbitrary: f g) auto
 
 lemma fps_nth_deriv_neg[simp]:
   "fps_nth_deriv n (- (f :: 'a::ring_1 fps)) = - (fps_nth_deriv n f)"
   by (induct n arbitrary: f) simp_all
 
 lemma fps_nth_deriv_add[simp]:
   "fps_nth_deriv n ((f :: 'a::ring_1 fps) + g) = fps_nth_deriv n f + fps_nth_deriv n g"
   using fps_nth_deriv_linear[of n 1 f 1 g] by simp
 
 lemma fps_nth_deriv_sub[simp]:
   "fps_nth_deriv n ((f :: 'a::ring_1 fps) - g) = fps_nth_deriv n f - fps_nth_deriv n g"
   using fps_nth_deriv_add [of n f "- g"] by simp
 
 lemma fps_nth_deriv_0[simp]: "fps_nth_deriv n 0 = 0"
   by (induct n) simp_all
 
 lemma fps_nth_deriv_1[simp]: "fps_nth_deriv n 1 = (if n = 0 then 1 else 0)"
   by (induct n) simp_all
 
 lemma fps_nth_deriv_const[simp]:
   "fps_nth_deriv n (fps_const c) = (if n = 0 then fps_const c else 0)"
   by (cases n) simp_all
 
 lemma fps_nth_deriv_mult_const_left[simp]:
   "fps_nth_deriv n (fps_const c * f) = fps_const c * fps_nth_deriv n f"
   using fps_nth_deriv_linear[of n "c" f 0 0 ] by simp
 
 lemma fps_nth_deriv_mult_const_right[simp]:
   "fps_nth_deriv n (f * fps_const c) = fps_nth_deriv n f * fps_const c"
   by (induct n arbitrary: f) auto
 
 lemma fps_nth_deriv_sum:
   "fps_nth_deriv n (sum f S) = sum (\<lambda>i. fps_nth_deriv n (f i :: 'a::ring_1 fps)) S"
 proof (cases "finite S")
   case True
   show ?thesis by (induct rule: finite_induct [OF True]) simp_all
 next
   case False
   then show ?thesis by simp
 qed
 
 lemma fps_deriv_maclauren_0:
   "(fps_nth_deriv k (f :: 'a::comm_semiring_1 fps)) $ 0 = of_nat (fact k) * f $ k"
   by (induct k arbitrary: f) (simp_all add: field_simps)
 
 lemma fps_deriv_lr_inverse:
   fixes   x y :: "'a::ring_1"
   assumes "x * f$0 = 1" "f$0 * y = 1"
   \<comment> \<open>These assumptions imply x equals y, but no need to assume that.\<close>
   shows   "fps_deriv (fps_left_inverse f x) =
             - fps_left_inverse f x * fps_deriv f * fps_left_inverse f x"
   and     "fps_deriv (fps_right_inverse f y) =
             - fps_right_inverse f y * fps_deriv f * fps_right_inverse f y"
 proof-
 
   define L where "L \<equiv> fps_left_inverse f x"
   hence "fps_deriv (L * f) = 0" using fps_left_inverse[OF assms(1)] by simp
   with assms show "fps_deriv L = - L * fps_deriv f * L"
     using fps_right_inverse'[OF assms]
     by    (simp add: minus_unique mult.assoc L_def)
 
   define R where "R \<equiv> fps_right_inverse f y"
   hence "fps_deriv (f * R) = 0" using fps_right_inverse[OF assms(2)] by simp
   hence 1: "f * fps_deriv R + fps_deriv f * R = 0" by simp
   have "R * f * fps_deriv R = - R * fps_deriv f * R"
     using iffD2[OF eq_neg_iff_add_eq_0, OF 1] by (simp add: mult.assoc)
   thus "fps_deriv R = - R * fps_deriv f * R"
     using fps_left_inverse'[OF assms] by (simp add: R_def)
 
 qed
 
 lemma fps_deriv_lr_inverse_comm:
   fixes   x :: "'a::comm_ring_1"
   assumes "x * f$0 = 1"
   shows   "fps_deriv (fps_left_inverse f x) = - fps_deriv f * (fps_left_inverse f x)\<^sup>2"
   and     "fps_deriv (fps_right_inverse f x) = - fps_deriv f * (fps_right_inverse f x)\<^sup>2"
   using   assms fps_deriv_lr_inverse[of x f x]
   by      (simp_all add: mult.commute power2_eq_square)
 
 lemma fps_inverse_deriv_divring:
   fixes   a :: "'a::division_ring fps"
   assumes "a$0 \<noteq> 0"
   shows   "fps_deriv (inverse a) = - inverse a * fps_deriv a * inverse a"
   using   assms fps_deriv_lr_inverse(2)[of "inverse (a$0)" a "inverse (a$0)"]
   by      (simp add: fps_inverse_def)
 
 lemma fps_inverse_deriv:
   fixes   a :: "'a::field fps"
   assumes "a$0 \<noteq> 0"
   shows   "fps_deriv (inverse a) = - fps_deriv a * (inverse a)\<^sup>2"
   using   assms fps_deriv_lr_inverse_comm(2)[of "inverse (a$0)" a]
   by      (simp add: fps_inverse_def)
 
 lemma fps_inverse_deriv':
   fixes a :: "'a::field fps"
   assumes a0: "a $ 0 \<noteq> 0"
   shows "fps_deriv (inverse a) = - fps_deriv a / a\<^sup>2"
   using fps_inverse_deriv[OF a0] a0
   by (simp add: fps_divide_unit power2_eq_square fps_inverse_mult)
 
 (* FIXME: The last part of this proof should go through by simp once we have a proper
    theorem collection for simplifying division on rings *)
 lemma fps_divide_deriv:
   assumes "b dvd (a :: 'a :: field fps)"
   shows   "fps_deriv (a / b) = (fps_deriv a * b - a * fps_deriv b) / b^2"
 proof -
   have eq_divide_imp: "c \<noteq> 0 \<Longrightarrow> a * c = b \<Longrightarrow> a = b div c" for a b c :: "'a :: field fps"
     by (drule sym) (simp add: mult.assoc)
   from assms have "a = a / b * b" by simp
   also have "fps_deriv (a / b * b) = fps_deriv (a / b) * b + a / b * fps_deriv b" by simp
   finally have "fps_deriv (a / b) * b^2 = fps_deriv a * b - a * fps_deriv b" using assms
     by (simp add: power2_eq_square algebra_simps)
   thus ?thesis by (cases "b = 0") (simp_all add: eq_divide_imp)
 qed
 
 lemma fps_nth_deriv_fps_X[simp]: "fps_nth_deriv n fps_X = (if n = 0 then fps_X else if n=1 then 1 else 0)"
   by (cases n) simp_all
 
 
 subsection \<open>Powers\<close>
 
 lemma fps_power_zeroth: "(a^n) $ 0 = (a$0)^n"
   by (induct n) auto
 
 lemma fps_power_zeroth_eq_one: "a$0 = 1 \<Longrightarrow> a^n $ 0 = 1"
   by (simp add: fps_power_zeroth)
 
 lemma fps_power_first:
   fixes a :: "'a::comm_semiring_1 fps"
   shows "(a^n) $ 1 = of_nat n * (a$0)^(n-1) * a$1"
 proof (cases n)
   case (Suc m)
   have "(a ^ Suc m) $ 1 = of_nat (Suc m) * (a$0)^(Suc m - 1) * a$1"
   proof (induct m)
     case (Suc k)
     hence "(a ^ Suc (Suc k)) $ 1 =
             a$0 * of_nat (Suc k) * (a $ 0)^k * a$1 + a$1 * ((a$0)^(Suc k))"
       using fps_mult_nth_1[of a] by (simp add: fps_power_zeroth[symmetric] mult.assoc)
     thus ?case by (simp add: algebra_simps)
   qed simp
   with Suc show ?thesis by simp
 qed simp
 
 lemma fps_power_first_eq: "a $ 0 = 1 \<Longrightarrow> a^n $ 1 = of_nat n * a$1"
 proof (induct n)
   case (Suc n)
   show ?case unfolding power_Suc fps_mult_nth
     using Suc.hyps[OF \<open>a$0 = 1\<close>] \<open>a$0 = 1\<close> fps_power_zeroth_eq_one[OF \<open>a$0=1\<close>]
     by (simp add: algebra_simps)
 qed simp
 
 lemma fps_power_first_eq':
   assumes "a $ 1 = 1"
   shows   "a^n $ 1 = of_nat n * (a$0)^(n-1)"
 proof (cases n)
   case (Suc m)
   from assms have "(a ^ Suc m) $ 1 = of_nat (Suc m) * (a$0)^(Suc m - 1)"
     using fps_mult_nth_1[of a]
     by    (induct m)
           (simp_all add: algebra_simps mult_of_nat_commute fps_power_zeroth)
   with Suc show ?thesis by simp
 qed simp
 
 lemmas startsby_one_power = fps_power_zeroth_eq_one
 
 lemma startsby_zero_power: "a $ 0 = 0 \<Longrightarrow> n > 0 \<Longrightarrow> a^n $0 = 0"
   by (simp add: fps_power_zeroth zero_power)
 
 lemma startsby_power: "a $0 = v \<Longrightarrow> a^n $0 = v^n"
   by (simp add: fps_power_zeroth)
 
 lemma startsby_nonzero_power:
   fixes a :: "'a::semiring_1_no_zero_divisors fps"
   shows "a $ 0 \<noteq> 0 \<Longrightarrow> a^n $ 0 \<noteq> 0"
   by    (simp add: startsby_power)
 
 lemma startsby_zero_power_iff[simp]:
   "a^n $0 = (0::'a::semiring_1_no_zero_divisors) \<longleftrightarrow> n \<noteq> 0 \<and> a$0 = 0"
 proof
   show "a ^ n $ 0 = 0 \<Longrightarrow> n \<noteq> 0 \<and> a $ 0 = 0"
   proof
     assume a: "a^n $ 0 = 0"
     thus "a $ 0 = 0" using startsby_nonzero_power by auto
     have "n = 0 \<Longrightarrow> a^n $ 0 = 1" by simp
     with a show "n \<noteq> 0" by fastforce
   qed
   show "n \<noteq> 0 \<and> a $ 0 = 0 \<Longrightarrow> a ^ n $ 0 = 0"
     by (cases n) auto
 qed
 
 lemma startsby_zero_power_prefix:
   assumes a0: "a $ 0 = 0"
   shows "\<forall>n < k. a ^ k $ n = 0"
 proof (induct k rule: nat_less_induct, clarify)
   case (1 k)
   fix j :: nat assume j: "j < k"
   show "a ^ k $ j = 0"
   proof (cases k)
     case 0 with j show ?thesis by simp
   next
     case (Suc i)
     with 1 j have "\<forall>m\<in>{0<..j}. a ^ i $ (j - m) = 0" by auto
     with Suc a0 show ?thesis by (simp add: fps_mult_nth sum.atLeast_Suc_atMost)
   qed
 qed
 
 lemma startsby_zero_sum_depends:
   assumes a0: "a $0 = 0"
     and kn: "n \<ge> k"
   shows "sum (\<lambda>i. (a ^ i)$k) {0 .. n} = sum (\<lambda>i. (a ^ i)$k) {0 .. k}"
-  apply (rule sum.mono_neutral_right)
-  using kn
-  apply auto
-  apply (rule startsby_zero_power_prefix[rule_format, OF a0])
-  apply arith
-  done
+proof (intro strip sum.mono_neutral_right)
+  show "\<And>i. i \<in> {0..n} - {0..k} \<Longrightarrow> a ^ i $ k = 0"
+    by (simp add: a0 startsby_zero_power_prefix)
+qed (use kn in auto)
 
 lemma startsby_zero_power_nth_same:
   assumes a0: "a$0 = 0"
   shows   "a^n $ n = (a$1) ^ n"
 proof (induct n)
   case (Suc n)
   have "\<forall>i\<in>{Suc 1..Suc n}. a ^ n $ (Suc n - i) = 0"
     using a0 startsby_zero_power_prefix[of a n] by auto
   thus ?case
     using a0 Suc sum.atLeast_Suc_atMost[of 0 "Suc n" "\<lambda>i. a $ i * a ^ n $ (Suc n - i)"]
           sum.atLeast_Suc_atMost[of 1 "Suc n" "\<lambda>i. a $ i * a ^ n $ (Suc n - i)"]
     by    (simp add: fps_mult_nth)
 qed simp
 
 lemma fps_lr_inverse_power:
   fixes a :: "'a::ring_1 fps"
   assumes "x * a$0 = 1" "a$0 * x = 1"
   shows "fps_left_inverse (a^n) (x^n) = fps_left_inverse a x ^ n"
   and   "fps_right_inverse (a^n) (x^n) = fps_right_inverse a x ^ n"
 proof-
 
   from assms have xn: "\<And>n. x^n * (a^n $ 0) = 1" "\<And>n. (a^n $ 0) * x^n = 1" 
     by (simp_all add: left_right_inverse_power fps_power_zeroth)
 
   show "fps_left_inverse (a^n) (x^n) = fps_left_inverse a x ^ n"
   proof (induct n)
     case 0
     then show ?case by (simp add: fps_lr_inverse_one_one(1))
   next
     case (Suc n)
     with assms show ?case
       using xn fps_lr_inverse_mult_ring1(1)[of x a "x^n" "a^n"]
       by    (simp add: power_Suc2[symmetric])
   qed
 
   moreover have "fps_right_inverse (a^n) (x^n) = fps_left_inverse (a^n) (x^n)"
     using xn by (intro fps_left_inverse_eq_fps_right_inverse[symmetric])
   moreover have "fps_right_inverse a x = fps_left_inverse a x"
     using assms by (intro fps_left_inverse_eq_fps_right_inverse[symmetric])
   ultimately show "fps_right_inverse (a^n) (x^n) = fps_right_inverse a x ^ n"
     by simp
 
 qed
 
 lemma fps_inverse_power:
   fixes a :: "'a::division_ring fps"
   shows "inverse (a^n) = inverse a ^ n"
 proof (cases "n=0" "a$0 = 0" rule: case_split[case_product case_split])
   case False_True
   hence LHS: "inverse (a^n) = 0" and RHS: "inverse a ^ n = 0"
     by (simp_all add: startsby_zero_power)
   show ?thesis using trans_sym[OF LHS RHS] by fast
 next
   case False_False
   from False_False(2) show ?thesis
     by  (simp add:
           fps_inverse_def fps_power_zeroth power_inverse fps_lr_inverse_power(2)[symmetric]
         )
 qed auto
 
 lemma fps_deriv_power':
   fixes a :: "'a::comm_semiring_1 fps"
   shows "fps_deriv (a ^ n) = (of_nat n) * fps_deriv a * a ^ (n - 1)"
 proof (cases n)
   case (Suc m)
   moreover have "fps_deriv (a^Suc m) = of_nat (Suc m) * fps_deriv a * a^m"
     by (induct m) (simp_all add: algebra_simps)
   ultimately show ?thesis by simp
 qed simp
 
 lemma fps_deriv_power:
   fixes a :: "'a::comm_semiring_1 fps"
   shows "fps_deriv (a ^ n) = fps_const (of_nat n) * fps_deriv a * a ^ (n - 1)"
   by (simp add: fps_deriv_power' fps_of_nat)
 
 
 subsection \<open>Integration\<close>
 
 definition fps_integral :: "'a::{semiring_1,inverse} fps \<Rightarrow> 'a \<Rightarrow> 'a fps"
   where "fps_integral a a0 =
           Abs_fps (\<lambda>n. if n=0 then a0 else inverse (of_nat n) * a$(n - 1))"
 
 abbreviation "fps_integral0 a \<equiv> fps_integral a 0"
 
 lemma fps_integral_nth_0_Suc [simp]:
   fixes a :: "'a::{semiring_1,inverse} fps"
   shows "fps_integral a a0 $ 0 = a0"
   and   "fps_integral a a0 $ Suc n = inverse (of_nat (Suc n)) * a $ n"
   by    (auto simp: fps_integral_def)
 
 lemma fps_integral_conv_plus_const:
   "fps_integral a a0 = fps_integral a 0 + fps_const a0"
   unfolding fps_integral_def by (intro fps_ext) simp
 
 lemma fps_deriv_fps_integral:
   fixes a :: "'a::{division_ring,ring_char_0} fps"
   shows "fps_deriv (fps_integral a a0) = a"
 proof (intro fps_ext)
   fix n
   have "(of_nat (Suc n) :: 'a) \<noteq> 0" by (rule of_nat_neq_0)
   hence "of_nat (Suc n) * inverse (of_nat (Suc n) :: 'a) = 1" by simp
   moreover have
     "fps_deriv (fps_integral a a0) $ n = of_nat (Suc n) * inverse (of_nat (Suc n)) * a $ n"
     by (simp add: mult.assoc)
   ultimately show "fps_deriv (fps_integral a a0) $ n = a $ n" by simp
 qed
 
 lemma fps_integral0_deriv:
   fixes a :: "'a::{division_ring,ring_char_0} fps"
   shows "fps_integral0 (fps_deriv a) = a - fps_const (a$0)"
 proof (intro fps_ext)
   fix n
   show "fps_integral0 (fps_deriv a) $ n = (a - fps_const (a$0)) $ n"
   proof (cases n)
     case (Suc m)
     have "(of_nat (Suc m) :: 'a) \<noteq> 0" by (rule of_nat_neq_0)
     hence "inverse (of_nat (Suc m) :: 'a) * of_nat (Suc m) = 1" by simp
     moreover have
       "fps_integral0 (fps_deriv a) $ Suc m =
         inverse (of_nat (Suc m)) * of_nat (Suc m) * a $ (Suc m)"
       by (simp add: mult.assoc)
     ultimately show ?thesis using Suc by simp
   qed simp
 qed
 
 lemma fps_integral_deriv:
   fixes a :: "'a::{division_ring,ring_char_0} fps"
   shows "fps_integral (fps_deriv a) (a$0) = a"
   using fps_integral_conv_plus_const[of "fps_deriv a" "a$0"]
   by    (simp add: fps_integral0_deriv)
 
 lemma fps_integral0_zero:
   "fps_integral0 (0::'a::{semiring_1,inverse} fps) = 0"
   by (intro fps_ext) (simp add: fps_integral_def)
 
 lemma fps_integral0_fps_const':
   fixes   c :: "'a::{semiring_1,inverse}"
   assumes "inverse (1::'a) = 1"
   shows   "fps_integral0 (fps_const c) = fps_const c * fps_X"
 proof (intro fps_ext)
   fix n
   show "fps_integral0 (fps_const c) $ n = (fps_const c * fps_X) $ n"
     by (cases n) (simp_all add: assms mult_delta_right)
 qed
 
 lemma fps_integral0_fps_const:
   fixes c :: "'a::division_ring"
   shows "fps_integral0 (fps_const c) = fps_const c * fps_X"
   by    (rule fps_integral0_fps_const'[OF inverse_1])
 
 lemma fps_integral0_one':
   assumes "inverse (1::'a::{semiring_1,inverse}) = 1"
   shows   "fps_integral0 (1::'a fps) = fps_X"
   using   assms fps_integral0_fps_const'[of "1::'a"]
   by      simp
 
 lemma fps_integral0_one:
   "fps_integral0 (1::'a::division_ring fps) = fps_X"
   by (rule fps_integral0_one'[OF inverse_1])
 
 lemma fps_integral0_fps_const_mult_left:
   fixes a :: "'a::division_ring fps"
   shows "fps_integral0 (fps_const c * a) = fps_const c * fps_integral0 a"
 proof (intro fps_ext)
   fix n
   show "fps_integral0 (fps_const c * a) $ n = (fps_const c * fps_integral0 a) $ n"
     using mult_inverse_of_nat_commute[of n c, symmetric]
           mult.assoc[of "inverse (of_nat n)" c "a$(n-1)"]
           mult.assoc[of c "inverse (of_nat n)" "a$(n-1)"]
     by    (simp add: fps_integral_def)
 qed
 
 lemma fps_integral0_fps_const_mult_right:
   fixes a :: "'a::{semiring_1,inverse} fps"
   shows "fps_integral0 (a * fps_const c) = fps_integral0 a * fps_const c"
   by    (intro fps_ext) (simp add: fps_integral_def algebra_simps)
 
 lemma fps_integral0_neg:
   fixes a :: "'a::{ring_1,inverse} fps"
   shows "fps_integral0 (-a) = - fps_integral0 a"
   using fps_integral0_fps_const_mult_right[of a "-1"]
   by    (simp add: fps_const_neg[symmetric])
 
 lemma fps_integral0_add:
   "fps_integral0 (a+b) = fps_integral0 a + fps_integral0 b"
   by (intro fps_ext) (simp add: fps_integral_def algebra_simps)
 
 lemma fps_integral0_linear:
   fixes a b :: "'a::division_ring"
   shows "fps_integral0 (fps_const a * f + fps_const b * g) =
           fps_const a * fps_integral0 f + fps_const b * fps_integral0 g"
   by    (simp add: fps_integral0_add fps_integral0_fps_const_mult_left)
   
 lemma fps_integral0_linear2:
   "fps_integral0 (f * fps_const a + g * fps_const b) =
     fps_integral0 f * fps_const a + fps_integral0 g * fps_const b"
   by (simp add: fps_integral0_add fps_integral0_fps_const_mult_right)
 
 lemma fps_integral_linear:
   fixes a b a0 b0 :: "'a::division_ring"
   shows
   "fps_integral (fps_const a * f + fps_const b * g) (a*a0 + b*b0) =
     fps_const a * fps_integral f a0 + fps_const b * fps_integral g b0"
   using fps_integral_conv_plus_const[of
           "fps_const a * f + fps_const b * g"
           "a*a0 + b*b0"
         ]
         fps_integral_conv_plus_const[of f a0] fps_integral_conv_plus_const[of g b0]
   by    (simp add: fps_integral0_linear algebra_simps)
 
 lemma fps_integral0_sub:
   fixes a b :: "'a::{ring_1,inverse} fps"
   shows "fps_integral0 (a-b) = fps_integral0 a - fps_integral0 b"
   using fps_integral0_linear2[of a 1 b "-1"]
   by    (simp add: fps_const_neg[symmetric])
 
 lemma fps_integral0_of_nat:
   "fps_integral0 (of_nat n :: 'a::division_ring fps) = of_nat n * fps_X"
   using fps_integral0_fps_const[of "of_nat n :: 'a"] by (simp add: fps_of_nat)
 
 lemma fps_integral0_sum:
   "fps_integral0 (sum f S) = sum (\<lambda>i. fps_integral0 (f i)) S"
 proof (cases "finite S")
   case True show ?thesis
     by  (induct rule: finite_induct [OF True])
         (simp_all add: fps_integral0_zero fps_integral0_add)
 qed (simp add: fps_integral0_zero)
 
 lemma fps_integral0_by_parts:
   fixes a b :: "'a::{division_ring,ring_char_0} fps"
   shows
     "fps_integral0 (a * b) =
       a * fps_integral0 b - fps_integral0 (fps_deriv a * fps_integral0 b)"
 proof-
   have "fps_integral0 (fps_deriv (a * fps_integral0 b)) = a * fps_integral0 b"
     using fps_integral0_deriv[of "(a * fps_integral0 b)"] by simp
   moreover have
     "fps_integral0 (a * b) =
       fps_integral0 (fps_deriv (a * fps_integral0 b)) -
       fps_integral0 (fps_deriv a * fps_integral0 b)"
     by (auto simp: fps_deriv_fps_integral fps_integral0_sub[symmetric])
   ultimately show ?thesis by simp
 qed
 
 lemma fps_integral0_fps_X:
   "fps_integral0 (fps_X::'a::{semiring_1,inverse} fps) =
     fps_const (inverse (of_nat 2)) * fps_X\<^sup>2"
   by (intro fps_ext) (auto simp: fps_integral_def)
 
 lemma fps_integral0_fps_X_power:
   "fps_integral0 ((fps_X::'a::{semiring_1,inverse} fps) ^ n) =
             fps_const (inverse (of_nat (Suc n))) * fps_X ^ Suc n"
 proof (intro fps_ext)
   fix k show
     "fps_integral0 ((fps_X::'a fps) ^ n) $ k =
       (fps_const (inverse (of_nat (Suc n))) * fps_X ^ Suc n) $ k"
     by (cases k) simp_all
 qed
 
 
 subsection \<open>Composition of FPSs\<close>
 
 definition fps_compose :: "'a::semiring_1 fps \<Rightarrow> 'a fps \<Rightarrow> 'a fps"  (infixl "oo" 55)
   where "a oo b = Abs_fps (\<lambda>n. sum (\<lambda>i. a$i * (b^i$n)) {0..n})"
 
 lemma fps_compose_nth: "(a oo b)$n = sum (\<lambda>i. a$i * (b^i$n)) {0..n}"
   by (simp add: fps_compose_def)
 
 lemma fps_compose_nth_0 [simp]: "(f oo g) $ 0 = f $ 0"
   by (simp add: fps_compose_nth)
 
 lemma fps_compose_fps_X[simp]: "a oo fps_X = (a :: 'a::comm_ring_1 fps)"
   by (simp add: fps_ext fps_compose_def mult_delta_right)
 
 lemma fps_const_compose[simp]: "fps_const (a::'a::comm_ring_1) oo b = fps_const a"
   by (simp add: fps_eq_iff fps_compose_nth mult_delta_left)
 
 lemma numeral_compose[simp]: "(numeral k :: 'a::comm_ring_1 fps) oo b = numeral k"
   unfolding numeral_fps_const by simp
 
 lemma neg_numeral_compose[simp]: "(- numeral k :: 'a::comm_ring_1 fps) oo b = - numeral k"
   unfolding neg_numeral_fps_const by simp
 
 lemma fps_X_fps_compose_startby0[simp]: "a$0 = 0 \<Longrightarrow> fps_X oo a = (a :: 'a::comm_ring_1 fps)"
   by (simp add: fps_eq_iff fps_compose_def mult_delta_left not_le)
 
 
 subsection \<open>Rules from Herbert Wilf's Generatingfunctionology\<close>
 
 subsubsection \<open>Rule 1\<close>
   (* {a_{n+k}}_0^infty Corresponds to (f - sum (\<lambda>i. a_i * x^i))/x^h, for h>0*)
 
 lemma fps_power_mult_eq_shift:
   "fps_X^Suc k * Abs_fps (\<lambda>n. a (n + Suc k)) =
     Abs_fps a - sum (\<lambda>i. fps_const (a i :: 'a::comm_ring_1) * fps_X^i) {0 .. k}"
   (is "?lhs = ?rhs")
 proof -
   have "?lhs $ n = ?rhs $ n" for n :: nat
   proof -
     have "?lhs $ n = (if n < Suc k then 0 else a n)"
       unfolding fps_X_power_mult_nth by auto
     also have "\<dots> = ?rhs $ n"
     proof (induct k)
       case 0
       then show ?case
         by (simp add: fps_sum_nth)
     next
       case (Suc k)
       have "(Abs_fps a - sum (\<lambda>i. fps_const (a i :: 'a) * fps_X^i) {0 .. Suc k})$n =
         (Abs_fps a - sum (\<lambda>i. fps_const (a i :: 'a) * fps_X^i) {0 .. k} -
           fps_const (a (Suc k)) * fps_X^ Suc k) $ n"
         by (simp add: field_simps)
       also have "\<dots> = (if n < Suc k then 0 else a n) - (fps_const (a (Suc k)) * fps_X^ Suc k)$n"
         using Suc.hyps[symmetric] unfolding fps_sub_nth by simp
       also have "\<dots> = (if n < Suc (Suc k) then 0 else a n)"
         unfolding fps_X_power_mult_right_nth
-        apply (auto simp add: not_less fps_const_def)
-        apply (rule cong[of a a, OF refl])
-        apply arith
-        done
+        by (simp add: not_less le_less_Suc_eq)
       finally show ?case
         by simp
     qed
     finally show ?thesis .
   qed
   then show ?thesis
     by (simp add: fps_eq_iff)
 qed
 
 
 subsubsection \<open>Rule 2\<close>
 
   (* We can not reach the form of Wilf, but still near to it using rewrite rules*)
   (* If f reprents {a_n} and P is a polynomial, then
         P(xD) f represents {P(n) a_n}*)
 
 definition "fps_XD = (*) fps_X \<circ> fps_deriv"
 
 lemma fps_XD_add[simp]:"fps_XD (a + b) = fps_XD a + fps_XD (b :: 'a::comm_ring_1 fps)"
   by (simp add: fps_XD_def field_simps)
 
 lemma fps_XD_mult_const[simp]:"fps_XD (fps_const (c::'a::comm_ring_1) * a) = fps_const c * fps_XD a"
   by (simp add: fps_XD_def field_simps)
 
 lemma fps_XD_linear[simp]: "fps_XD (fps_const c * a + fps_const d * b) =
     fps_const c * fps_XD a + fps_const d * fps_XD (b :: 'a::comm_ring_1 fps)"
   by simp
 
 lemma fps_XDN_linear:
   "(fps_XD ^^ n) (fps_const c * a + fps_const d * b) =
     fps_const c * (fps_XD ^^ n) a + fps_const d * (fps_XD ^^ n) (b :: 'a::comm_ring_1 fps)"
   by (induct n) simp_all
 
 lemma fps_mult_fps_X_deriv_shift: "fps_X* fps_deriv a = Abs_fps (\<lambda>n. of_nat n* a$n)"
   by (simp add: fps_eq_iff)
 
 lemma fps_mult_fps_XD_shift:
   "(fps_XD ^^ k) (a :: 'a::comm_ring_1 fps) = Abs_fps (\<lambda>n. (of_nat n ^ k) * a$n)"
   by (induct k arbitrary: a) (simp_all add: fps_XD_def fps_eq_iff field_simps del: One_nat_def)
 
 
 subsubsection \<open>Rule 3\<close>
 
 text \<open>Rule 3 is trivial and is given by \<open>fps_times_def\<close>.\<close>
 
 
 subsubsection \<open>Rule 5 --- summation and "division" by (1 - fps_X)\<close>
 
 lemma fps_divide_fps_X_minus1_sum_lemma:
   "a = ((1::'a::ring_1 fps) - fps_X) * Abs_fps (\<lambda>n. sum (\<lambda>i. a $ i) {0..n})"
 proof (rule fps_ext)
   define f g :: "'a fps"
     where "f \<equiv> 1 - fps_X"
     and   "g \<equiv> Abs_fps (\<lambda>n. sum (\<lambda>i. a $ i) {0..n})"
   fix n show "a $ n= (f * g) $ n"
   proof (cases n)
     case (Suc m)
     hence "(f * g) $ n = g $ Suc m - g $ m"
       using fps_mult_nth[of f g "Suc m"]
             sum.atLeast_Suc_atMost[of 0 "Suc m" "\<lambda>i. f $ i * g $ (Suc m - i)"]
             sum.atLeast_Suc_atMost[of 1 "Suc m" "\<lambda>i. f $ i * g $ (Suc m - i)"]
       by    (simp add: f_def)
     with Suc show ?thesis by (simp add: g_def)
   qed (simp add: f_def g_def)
 qed
 
 lemma fps_divide_fps_X_minus1_sum_ring1:
   assumes "inverse 1 = (1::'a::{ring_1,inverse})"
   shows   "a /((1::'a fps) - fps_X) = Abs_fps (\<lambda>n. sum (\<lambda>i. a $ i) {0..n})"
 proof-
   from assms have "a /((1::'a fps) - fps_X) = a * Abs_fps (\<lambda>n. 1)"
     by (simp add: fps_divide_def fps_inverse_def fps_lr_inverse_one_minus_fps_X(2))
   thus ?thesis by (auto intro: fps_ext simp: fps_mult_nth)
 qed
 
 lemma fps_divide_fps_X_minus1_sum:
   "a /((1::'a::division_ring fps) - fps_X) = Abs_fps (\<lambda>n. sum (\<lambda>i. a $ i) {0..n})"
   using fps_divide_fps_X_minus1_sum_ring1[of a] by simp
 
 
 subsubsection \<open>Rule 4 in its more general form: generalizes Rule 3 for an arbitrary
   finite product of FPS, also the relvant instance of powers of a FPS\<close>
 
 definition "natpermute n k = {l :: nat list. length l = k \<and> sum_list l = n}"
 
 lemma natlist_trivial_1: "natpermute n 1 = {[n]}"
-  apply (auto simp add: natpermute_def)
-  apply (case_tac x)
-  apply auto
-  done
+proof -
+  have "\<lbrakk>length xs = 1; n = sum_list xs\<rbrakk> \<Longrightarrow> xs = [sum_list xs]" for xs
+    by (cases xs) auto
+  then show ?thesis
+    by (auto simp add: natpermute_def)
+qed
+
+lemma natlist_trivial_Suc0 [simp]: "natpermute n (Suc 0) = {[n]}"
+  using natlist_trivial_1 by force
 
 lemma append_natpermute_less_eq:
   assumes "xs @ ys \<in> natpermute n k"
   shows "sum_list xs \<le> n"
     and "sum_list ys \<le> n"
 proof -
   from assms have "sum_list (xs @ ys) = n"
     by (simp add: natpermute_def)
   then have "sum_list xs + sum_list ys = n"
     by simp
   then show "sum_list xs \<le> n" and "sum_list ys \<le> n"
     by simp_all
 qed
 
 lemma natpermute_split:
   assumes "h \<le> k"
   shows "natpermute n k =
     (\<Union>m \<in>{0..n}. {l1 @ l2 |l1 l2. l1 \<in> natpermute m h \<and> l2 \<in> natpermute (n - m) (k - h)})"
   (is "?L = ?R" is "_ = (\<Union>m \<in>{0..n}. ?S m)")
 proof
   show "?R \<subseteq> ?L"
   proof
     fix l
     assume l: "l \<in> ?R"
     from l obtain m xs ys where h: "m \<in> {0..n}"
       and xs: "xs \<in> natpermute m h"
       and ys: "ys \<in> natpermute (n - m) (k - h)"
       and leq: "l = xs@ys" by blast
     from xs have xs': "sum_list xs = m"
       by (simp add: natpermute_def)
     from ys have ys': "sum_list ys = n - m"
       by (simp add: natpermute_def)
     show "l \<in> ?L" using leq xs ys h
-      apply (clarsimp simp add: natpermute_def)
-      unfolding xs' ys'
-      using assms xs ys
-      unfolding natpermute_def
-      apply simp
-      done
+      using assms by (force simp add: natpermute_def)
   qed
   show "?L \<subseteq> ?R"
   proof
     fix l
     assume l: "l \<in> natpermute n k"
     let ?xs = "take h l"
     let ?ys = "drop h l"
     let ?m = "sum_list ?xs"
     from l have ls: "sum_list (?xs @ ?ys) = n"
       by (simp add: natpermute_def)
     have xs: "?xs \<in> natpermute ?m h" using l assms
       by (simp add: natpermute_def)
     have l_take_drop: "sum_list l = sum_list (take h l @ drop h l)"
       by simp
     then have ys: "?ys \<in> natpermute (n - ?m) (k - h)"
       using l assms ls by (auto simp add: natpermute_def simp del: append_take_drop_id)
     from ls have m: "?m \<in> {0..n}"
       by (simp add: l_take_drop del: append_take_drop_id)
-    from xs ys ls show "l \<in> ?R"
-      apply auto
-      apply (rule bexI [where x = "?m"])
-      apply (rule exI [where x = "?xs"])
-      apply (rule exI [where x = "?ys"])
-      using ls l
-      apply (auto simp add: natpermute_def l_take_drop simp del: append_take_drop_id)
-      apply simp
-      done
+    have "sum_list (take h l) \<le> sum_list l"
+      using l_take_drop ls m by presburger
+    with xs ys ls l show "l \<in> ?R"
+      by simp (metis append_take_drop_id m)
   qed
 qed
 
 lemma natpermute_0: "natpermute n 0 = (if n = 0 then {[]} else {})"
   by (auto simp add: natpermute_def)
 
 lemma natpermute_0'[simp]: "natpermute 0 k = (if k = 0 then {[]} else {replicate k 0})"
-  apply (auto simp add: set_replicate_conv_if natpermute_def)
-  apply (rule nth_equalityI)
-  apply simp_all
-  done
+  by (auto simp add: set_replicate_conv_if natpermute_def replicate_length_same)
 
 lemma natpermute_finite: "finite (natpermute n k)"
 proof (induct k arbitrary: n)
   case 0
   then show ?case
-    apply (subst natpermute_split[of 0 0, simplified])
-    apply (simp add: natpermute_0)
-    done
+    by (simp add: natpermute_0)
 next
   case (Suc k)
-  then show ?case unfolding natpermute_split [of k "Suc k", simplified]
-    apply -
-    apply (rule finite_UN_I)
-    apply simp
-    unfolding One_nat_def[symmetric] natlist_trivial_1
-    apply simp
-    done
+  then show ?case
+    using natpermute_split [of k "Suc k"] finite_UN_I by simp
 qed
 
 lemma natpermute_contain_maximal:
   "{xs \<in> natpermute n (k + 1). n \<in> set xs} = (\<Union>i\<in>{0 .. k}. {(replicate (k + 1) 0) [i:=n]})"
   (is "?A = ?B")
 proof
   show "?A \<subseteq> ?B"
   proof
     fix xs
     assume "xs \<in> ?A"
     then have H: "xs \<in> natpermute n (k + 1)" and n: "n \<in> set xs"
       by blast+
     then obtain i where i: "i \<in> {0.. k}" "xs!i = n"
       unfolding in_set_conv_nth by (auto simp add: less_Suc_eq_le natpermute_def)
     have eqs: "({0..k} - {i}) \<union> {i} = {0..k}"
       using i by auto
     have f: "finite({0..k} - {i})" "finite {i}"
       by auto
     have d: "({0..k} - {i}) \<inter> {i} = {}"
       using i by auto
     from H have "n = sum (nth xs) {0..k}"
-      apply (simp add: natpermute_def)
-      apply (auto simp add: atLeastLessThanSuc_atLeastAtMost sum_list_sum_nth)
-      done
+      by (auto simp add: natpermute_def atLeastLessThanSuc_atLeastAtMost sum_list_sum_nth)
     also have "\<dots> = n + sum (nth xs) ({0..k} - {i})"
       unfolding sum.union_disjoint[OF f d, unfolded eqs] using i by simp
     finally have zxs: "\<forall> j\<in> {0..k} - {i}. xs!j = 0"
       by auto
     from H have xsl: "length xs = k+1"
       by (simp add: natpermute_def)
     from i have i': "i < length (replicate (k+1) 0)"   "i < k+1"
       unfolding length_replicate by presburger+
     have "xs = (replicate (k+1) 0) [i := n]"
     proof (rule nth_equalityI)
       show "length xs = length ((replicate (k + 1) 0)[i := n])"
         by (metis length_list_update length_replicate xsl)
       show "xs ! j = (replicate (k + 1) 0)[i := n] ! j" if "j < length xs" for j
       proof (cases "j = i")
         case True
         then show ?thesis
           by (metis i'(1) i(2) nth_list_update)
       next
         case False
         with that show ?thesis
           by (simp add: xsl zxs del: replicate.simps split: nat.split)
       qed
     qed
     then show "xs \<in> ?B" using i by blast
   qed
   show "?B \<subseteq> ?A"
   proof
     fix xs
     assume "xs \<in> ?B"
     then obtain i where i: "i \<in> {0..k}" and xs: "xs = (replicate (k + 1) 0) [i:=n]"
       by auto
     have nxs: "n \<in> set xs"
-      unfolding xs
-      apply (rule set_update_memI)
-      using i apply simp
-      done
+      unfolding xs using set_update_memI i 
+      by (metis Suc_eq_plus1 atLeast0AtMost atMost_iff le_simps(2) length_replicate)
     have xsl: "length xs = k + 1"
       by (simp only: xs length_replicate length_list_update)
     have "sum_list xs = sum (nth xs) {0..<k+1}"
       unfolding sum_list_sum_nth xsl ..
     also have "\<dots> = sum (\<lambda>j. if j = i then n else 0) {0..< k+1}"
       by (rule sum.cong) (simp_all add: xs del: replicate.simps)
     also have "\<dots> = n" using i by simp
     finally have "xs \<in> natpermute n (k + 1)"
       using xsl unfolding natpermute_def mem_Collect_eq by blast
     then show "xs \<in> ?A"
       using nxs by blast
   qed
 qed
 
 text \<open>The general form.\<close>
 lemma fps_prod_nth:
   fixes m :: nat
     and a :: "nat \<Rightarrow> 'a::comm_ring_1 fps"
   shows "(prod a {0 .. m}) $ n =
     sum (\<lambda>v. prod (\<lambda>j. (a j) $ (v!j)) {0..m}) (natpermute n (m+1))"
   (is "?P m n")
 proof (induct m arbitrary: n rule: nat_less_induct)
   fix m n assume H: "\<forall>m' < m. \<forall>n. ?P m' n"
   show "?P m n"
   proof (cases m)
     case 0
     then show ?thesis
-      apply simp
-      unfolding natlist_trivial_1[where n = n, unfolded One_nat_def]
-      apply simp
-      done
+      by simp
   next
     case (Suc k)
     then have km: "k < m" by arith
     have u0: "{0 .. k} \<union> {m} = {0..m}"
       using Suc by (simp add: set_eq_iff) presburger
     have f0: "finite {0 .. k}" "finite {m}" by auto
     have d0: "{0 .. k} \<inter> {m} = {}" using Suc by auto
     have "(prod a {0 .. m}) $ n = (prod a {0 .. k} * a m) $ n"
       unfolding prod.union_disjoint[OF f0 d0, unfolded u0] by simp
-    also have "\<dots> = (\<Sum>i = 0..n. (\<Sum>v\<in>natpermute i (k + 1). \<Prod>j\<in>{0..k}. a j $ v ! j) * a m $ (n - i))"
-      unfolding fps_mult_nth H[rule_format, OF km] ..
+    also have "\<dots> = (\<Sum>i = 0..n. (\<Sum>v\<in>natpermute i (k + 1).
+            (\<Prod>j = 0..k. a j $ v ! j) * a m $ (n - i)))"
+      unfolding fps_mult_nth H[rule_format, OF km] sum_distrib_right ..
+    also have "... = (\<Sum>i = 0..n.
+                       \<Sum>v\<in>(\<lambda>l1. l1 @ [n - i]) ` natpermute i (Suc k).
+                        (\<Prod>j = 0..k. a j $ v ! j) * a (Suc k) $ v ! Suc k)"
+      by (intro sum.cong [OF refl sym] sum.reindex_cong) (auto simp: inj_on_def natpermute_def nth_append Suc)
+    also have "... = (\<Sum>v\<in>(\<Union>x\<in>{0..n}. {l1 @ [n - x] |l1. l1 \<in> natpermute x (Suc k)}).
+                      (\<Prod>j = 0..k. a j $ v ! j) * a (Suc k) $ v ! Suc k)"
+      by (subst sum.UNION_disjoint) (auto simp add: natpermute_finite setcompr_eq_image)
     also have "\<dots> = (\<Sum>v\<in>natpermute n (m + 1). \<Prod>j\<in>{0..m}. a j $ v ! j)"
-      apply (simp add: Suc)
-      unfolding natpermute_split[of m "m + 1", simplified, of n,
-        unfolded natlist_trivial_1[unfolded One_nat_def] Suc]
-      apply (subst sum.UNION_disjoint)
-      apply simp
-      apply simp
-      unfolding image_Collect[symmetric]
-      apply clarsimp
-      apply (rule finite_imageI)
-      apply (rule natpermute_finite)
-      apply (clarsimp simp add: set_eq_iff)
-      apply auto
-      apply (rule sum.cong)
-      apply (rule refl)
-      unfolding sum_distrib_right
-      apply (rule sym)
-      apply (rule_tac l = "\<lambda>xs. xs @ [n - x]" in sum.reindex_cong)
-      apply (simp add: inj_on_def)
-      apply auto
-      unfolding prod.union_disjoint[OF f0 d0, unfolded u0, unfolded Suc]
-      apply (clarsimp simp add: natpermute_def nth_append)
-      done
+      using natpermute_split[of m "m + 1"] by (simp add: Suc)
     finally show ?thesis .
   qed
 qed
 
 text \<open>The special form for powers.\<close>
 lemma fps_power_nth_Suc:
   fixes m :: nat
     and a :: "'a::comm_ring_1 fps"
   shows "(a ^ Suc m)$n = sum (\<lambda>v. prod (\<lambda>j. a $ (v!j)) {0..m}) (natpermute n (m+1))"
 proof -
   have th0: "a^Suc m = prod (\<lambda>i. a) {0..m}"
     by (simp add: prod_constant)
   show ?thesis unfolding th0 fps_prod_nth ..
 qed
 
 lemma fps_power_nth:
   fixes m :: nat
     and a :: "'a::comm_ring_1 fps"
   shows "(a ^m)$n =
     (if m=0 then 1$n else sum (\<lambda>v. prod (\<lambda>j. a $ (v!j)) {0..m - 1}) (natpermute n m))"
   by (cases m) (simp_all add: fps_power_nth_Suc del: power_Suc)
 
 lemmas fps_nth_power_0 = fps_power_zeroth
 
 lemma natpermute_max_card:
   assumes n0: "n \<noteq> 0"
   shows "card {xs \<in> natpermute n (k + 1). n \<in> set xs} = k + 1"
   unfolding natpermute_contain_maximal
 proof -
   let ?A = "\<lambda>i. {(replicate (k + 1) 0)[i := n]}"
   let ?K = "{0 ..k}"
   have fK: "finite ?K"
     by simp
   have fAK: "\<forall>i\<in>?K. finite (?A i)"
     by auto
   have d: "\<forall>i\<in> ?K. \<forall>j\<in> ?K. i \<noteq> j \<longrightarrow>
     {(replicate (k + 1) 0)[i := n]} \<inter> {(replicate (k + 1) 0)[j := n]} = {}"
   proof clarify
     fix i j
     assume i: "i \<in> ?K" and j: "j \<in> ?K" and ij: "i \<noteq> j"
     have False if eq: "(replicate (k+1) 0)[i:=n] = (replicate (k+1) 0)[j:= n]"
     proof -
       have "(replicate (k+1) 0) [i:=n] ! i = n"
         using i by (simp del: replicate.simps)
       moreover
       have "(replicate (k+1) 0) [j:=n] ! i = 0"
         using i ij by (simp del: replicate.simps)
       ultimately show ?thesis
         using eq n0 by (simp del: replicate.simps)
     qed
     then show "{(replicate (k + 1) 0)[i := n]} \<inter> {(replicate (k + 1) 0)[j := n]} = {}"
       by auto
   qed
   from card_UN_disjoint[OF fK fAK d]
   show "card (\<Union>i\<in>{0..k}. {(replicate (k + 1) 0)[i := n]}) = k + 1"
     by simp
 qed
 
 lemma fps_power_Suc_nth:
   fixes f :: "'a :: comm_ring_1 fps"
   assumes k: "k > 0"
   shows "(f ^ Suc m) $ k = 
            of_nat (Suc m) * (f $ k * (f $ 0) ^ m) +
            (\<Sum>v\<in>{v\<in>natpermute k (m+1). k \<notin> set v}. \<Prod>j = 0..m. f $ v ! j)"
 proof -
   define A B 
     where "A = {v\<in>natpermute k (m+1). k \<in> set v}" 
       and  "B = {v\<in>natpermute k (m+1). k \<notin> set v}"
   have [simp]: "finite A" "finite B" "A \<inter> B = {}" by (auto simp: A_def B_def natpermute_finite)
 
   from natpermute_max_card[of k m] k have card_A: "card A = m + 1" by (simp add: A_def)
   {
     fix v assume v: "v \<in> A"
     from v have [simp]: "length v = Suc m" by (simp add: A_def natpermute_def)
     from v have "\<exists>j. j \<le> m \<and> v ! j = k" 
       by (auto simp: set_conv_nth A_def natpermute_def less_Suc_eq_le)
     then guess j by (elim exE conjE) note j = this
     
     from v have "k = sum_list v" by (simp add: A_def natpermute_def)
     also have "\<dots> = (\<Sum>i=0..m. v ! i)"
       by (simp add: sum_list_sum_nth atLeastLessThanSuc_atLeastAtMost del: sum.op_ivl_Suc)
     also from j have "{0..m} = insert j ({0..m}-{j})" by auto
     also from j have "(\<Sum>i\<in>\<dots>. v ! i) = k + (\<Sum>i\<in>{0..m}-{j}. v ! i)"
       by (subst sum.insert) simp_all
     finally have "(\<Sum>i\<in>{0..m}-{j}. v ! i) = 0" by simp
     hence zero: "v ! i = 0" if "i \<in> {0..m}-{j}" for i using that
       by (subst (asm) sum_eq_0_iff) auto
       
     from j have "{0..m} = insert j ({0..m} - {j})" by auto
     also from j have "(\<Prod>i\<in>\<dots>. f $ (v ! i)) = f $ k * (\<Prod>i\<in>{0..m} - {j}. f $ (v ! i))"
       by (subst prod.insert) auto
     also have "(\<Prod>i\<in>{0..m} - {j}. f $ (v ! i)) = (\<Prod>i\<in>{0..m} - {j}. f $ 0)"
       by (intro prod.cong) (simp_all add: zero)
     also from j have "\<dots> = (f $ 0) ^ m" by (subst prod_constant) simp_all
     finally have "(\<Prod>j = 0..m. f $ (v ! j)) = f $ k * (f $ 0) ^ m" .
   } note A = this
   
   have "(f ^ Suc m) $ k = (\<Sum>v\<in>natpermute k (m + 1). \<Prod>j = 0..m. f $ v ! j)"
     by (rule fps_power_nth_Suc)
   also have "natpermute k (m+1) = A \<union> B" unfolding A_def B_def by blast
   also have "(\<Sum>v\<in>\<dots>. \<Prod>j = 0..m. f $ (v ! j)) = 
                (\<Sum>v\<in>A. \<Prod>j = 0..m. f $ (v ! j)) + (\<Sum>v\<in>B. \<Prod>j = 0..m. f $ (v ! j))"
     by (intro sum.union_disjoint) simp_all   
   also have "(\<Sum>v\<in>A. \<Prod>j = 0..m. f $ (v ! j)) = of_nat (Suc m) * (f $ k * (f $ 0) ^ m)"
     by (simp add: A card_A)
   finally show ?thesis by (simp add: B_def)
 qed 
   
 lemma fps_power_Suc_eqD:
   fixes f g :: "'a :: {idom,semiring_char_0} fps"
   assumes "f ^ Suc m = g ^ Suc m" "f $ 0 = g $ 0" "f $ 0 \<noteq> 0"
   shows   "f = g"
 proof (rule fps_ext)
   fix k :: nat
   show "f $ k = g $ k"
   proof (induction k rule: less_induct)
     case (less k)
     show ?case
     proof (cases "k = 0")
       case False
       let ?h = "\<lambda>f. (\<Sum>v | v \<in> natpermute k (m + 1) \<and> k \<notin> set v. \<Prod>j = 0..m. f $ v ! j)"
       from False fps_power_Suc_nth[of k f m] fps_power_Suc_nth[of k g m]
         have "f $ k * (of_nat (Suc m) * (f $ 0) ^ m) + ?h f =
                 g $ k * (of_nat (Suc m) * (f $ 0) ^ m) + ?h g" using assms 
         by (simp add: mult_ac del: power_Suc of_nat_Suc)
       also have "v ! i < k" if "v \<in> {v\<in>natpermute k (m+1). k \<notin> set v}" "i \<le> m" for v i
         using that elem_le_sum_list[of i v] unfolding natpermute_def
         by (auto simp: set_conv_nth dest!: spec[of _ i])
       hence "?h f = ?h g"
         by (intro sum.cong refl prod.cong less lessI) (simp add: natpermute_def)
       finally have "f $ k * (of_nat (Suc m) * (f $ 0) ^ m) = g $ k * (of_nat (Suc m) * (f $ 0) ^ m)"
         by simp
       with assms show "f $ k = g $ k" 
         by (subst (asm) mult_right_cancel) (auto simp del: of_nat_Suc)
     qed (simp_all add: assms)
   qed
 qed
 
 lemma fps_power_Suc_eqD':
   fixes f g :: "'a :: {idom,semiring_char_0} fps"
   assumes "f ^ Suc m = g ^ Suc m" "f $ subdegree f = g $ subdegree g"
   shows   "f = g"
 proof (cases "f = 0")
   case False
   have "Suc m * subdegree f = subdegree (f ^ Suc m)"
     by (rule subdegree_power [symmetric])
   also have "f ^ Suc m = g ^ Suc m" by fact
   also have "subdegree \<dots> = Suc m * subdegree g" by (rule subdegree_power)
   finally have [simp]: "subdegree f = subdegree g"
     by (subst (asm) Suc_mult_cancel1)
   have "fps_shift (subdegree f) f * fps_X ^ subdegree f = f"
     by (rule subdegree_decompose [symmetric])
   also have "\<dots> ^ Suc m = g ^ Suc m" by fact
   also have "g = fps_shift (subdegree g) g * fps_X ^ subdegree g"
     by (rule subdegree_decompose)
   also have "subdegree f = subdegree g" by fact
   finally have "fps_shift (subdegree g) f ^ Suc m = fps_shift (subdegree g) g ^ Suc m"
     by (simp add: algebra_simps power_mult_distrib del: power_Suc)
   hence "fps_shift (subdegree g) f = fps_shift (subdegree g) g"
     by (rule fps_power_Suc_eqD) (insert assms False, auto)
   with subdegree_decompose[of f] subdegree_decompose[of g] show ?thesis by simp
 qed (insert assms, simp_all)
 
 lemma fps_power_eqD':
   fixes f g :: "'a :: {idom,semiring_char_0} fps"
   assumes "f ^ m = g ^ m" "f $ subdegree f = g $ subdegree g" "m > 0"
   shows   "f = g"
   using fps_power_Suc_eqD'[of f "m-1" g] assms by simp
 
 lemma fps_power_eqD:
   fixes f g :: "'a :: {idom,semiring_char_0} fps"
   assumes "f ^ m = g ^ m" "f $ 0 = g $ 0" "f $ 0 \<noteq> 0" "m > 0"
   shows   "f = g"
   by (rule fps_power_eqD'[of f m g]) (insert assms, simp_all)
 
 lemma fps_compose_inj_right:
   assumes a0: "a$0 = (0::'a::idom)"
     and a1: "a$1 \<noteq> 0"
   shows "(b oo a = c oo a) \<longleftrightarrow> b = c"
   (is "?lhs \<longleftrightarrow>?rhs")
 proof
   show ?lhs if ?rhs using that by simp
   show ?rhs if ?lhs
   proof -
     have "b$n = c$n" for n
     proof (induct n rule: nat_less_induct)
       fix n
       assume H: "\<forall>m<n. b$m = c$m"
       show "b$n = c$n"
       proof (cases n)
         case 0
         from \<open>?lhs\<close> have "(b oo a)$n = (c oo a)$n"
           by simp
         then show ?thesis
           using 0 by (simp add: fps_compose_nth)
       next
         case (Suc n1)
         have f: "finite {0 .. n1}" "finite {n}" by simp_all
         have eq: "{0 .. n1} \<union> {n} = {0 .. n}" using Suc by auto
         have d: "{0 .. n1} \<inter> {n} = {}" using Suc by auto
         have seq: "(\<Sum>i = 0..n1. b $ i * a ^ i $ n) = (\<Sum>i = 0..n1. c $ i * a ^ i $ n)"
-          apply (rule sum.cong)
-          using H Suc
-          apply auto
-          done
+          using H Suc by auto
         have th0: "(b oo a) $n = (\<Sum>i = 0..n1. c $ i * a ^ i $ n) + b$n * (a$1)^n"
           unfolding fps_compose_nth sum.union_disjoint[OF f d, unfolded eq] seq
           using startsby_zero_power_nth_same[OF a0]
           by simp
         have th1: "(c oo a) $n = (\<Sum>i = 0..n1. c $ i * a ^ i $ n) + c$n * (a$1)^n"
           unfolding fps_compose_nth sum.union_disjoint[OF f d, unfolded eq]
           using startsby_zero_power_nth_same[OF a0]
           by simp
         from \<open>?lhs\<close>[unfolded fps_eq_iff, rule_format, of n] th0 th1 a1
         show ?thesis by auto
       qed
     qed
     then show ?rhs by (simp add: fps_eq_iff)
   qed
 qed
 
 
 subsection \<open>Radicals\<close>
 
 declare prod.cong [fundef_cong]
 
 function radical :: "(nat \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a::field fps \<Rightarrow> nat \<Rightarrow> 'a"
 where
   "radical r 0 a 0 = 1"
 | "radical r 0 a (Suc n) = 0"
 | "radical r (Suc k) a 0 = r (Suc k) (a$0)"
 | "radical r (Suc k) a (Suc n) =
     (a$ Suc n - sum (\<lambda>xs. prod (\<lambda>j. radical r (Suc k) a (xs ! j)) {0..k})
       {xs. xs \<in> natpermute (Suc n) (Suc k) \<and> Suc n \<notin> set xs}) /
     (of_nat (Suc k) * (radical r (Suc k) a 0)^k)"
   by pat_completeness auto
 
 termination radical
 proof
   let ?R = "measure (\<lambda>(r, k, a, n). n)"
   {
     show "wf ?R" by auto
   next
     fix r :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"
     and a :: "'a fps"
     and k n xs i
     assume xs: "xs \<in> {xs \<in> natpermute (Suc n) (Suc k). Suc n \<notin> set xs}" and i: "i \<in> {0..k}"
     have False if c: "Suc n \<le> xs ! i"
     proof -
       from xs i have "xs !i \<noteq> Suc n"
         by (simp add: in_set_conv_nth natpermute_def)
       with c have c': "Suc n < xs!i" by arith
       have fths: "finite {0 ..< i}" "finite {i}" "finite {i+1..<Suc k}"
         by simp_all
       have d: "{0 ..< i} \<inter> ({i} \<union> {i+1 ..< Suc k}) = {}" "{i} \<inter> {i+1..< Suc k} = {}"
         by auto
       have eqs: "{0..<Suc k} = {0 ..< i} \<union> ({i} \<union> {i+1 ..< Suc k})"
         using i by auto
       from xs have "Suc n = sum_list xs"
         by (simp add: natpermute_def)
       also have "\<dots> = sum (nth xs) {0..<Suc k}" using xs
         by (simp add: natpermute_def sum_list_sum_nth)
       also have "\<dots> = xs!i + sum (nth xs) {0..<i} + sum (nth xs) {i+1..<Suc k}"
         unfolding eqs  sum.union_disjoint[OF fths(1) finite_UnI[OF fths(2,3)] d(1)]
         unfolding sum.union_disjoint[OF fths(2) fths(3) d(2)]
         by simp
       finally show ?thesis using c' by simp
     qed
     then show "((r, Suc k, a, xs!i), r, Suc k, a, Suc n) \<in> ?R"
-      apply auto
-      apply (metis not_less)
-      done
+      using not_less by auto
   next
     fix r :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"
     and a :: "'a fps"
     and k n
     show "((r, Suc k, a, 0), r, Suc k, a, Suc n) \<in> ?R" by simp
   }
 qed
 
 definition "fps_radical r n a = Abs_fps (radical r n a)"
 
+lemma radical_0 [simp]: "\<And>n. 0 < n \<Longrightarrow> radical r 0 a n = 0"
+  using radical.elims by blast
+
 lemma fps_radical0[simp]: "fps_radical r 0 a = 1"
-  apply (auto simp add: fps_eq_iff fps_radical_def)
-  apply (case_tac n)
-  apply auto
-  done
+  by (auto simp add: fps_eq_iff fps_radical_def)
 
 lemma fps_radical_nth_0[simp]: "fps_radical r n a $ 0 = (if n = 0 then 1 else r n (a$0))"
   by (cases n) (simp_all add: fps_radical_def)
 
 lemma fps_radical_power_nth[simp]:
   assumes r: "(r k (a$0)) ^ k = a$0"
   shows "fps_radical r k a ^ k $ 0 = (if k = 0 then 1 else a$0)"
 proof (cases k)
   case 0
   then show ?thesis by simp
 next
   case (Suc h)
   have eq1: "fps_radical r k a ^ k $ 0 = (\<Prod>j\<in>{0..h}. fps_radical r k a $ (replicate k 0) ! j)"
     unfolding fps_power_nth Suc by simp
   also have "\<dots> = (\<Prod>j\<in>{0..h}. r k (a$0))"
-    apply (rule prod.cong)
-    apply simp
-    using Suc
-    apply (subgoal_tac "replicate k 0 ! x = 0")
-    apply (auto intro: nth_replicate simp del: replicate.simps)
-    done
+  proof (rule prod.cong [OF refl])
+    show "fps_radical r k a $ replicate k 0 ! j = r k (a $ 0)" if "j \<in> {0..h}" for j
+    proof -
+      have "j < Suc h"
+        using that by presburger
+      then show ?thesis
+        by (metis Suc fps_radical_nth_0 nth_replicate old.nat.distinct(2))
+    qed
+  qed
   also have "\<dots> = a$0"
-    using r Suc by (simp add: prod_constant)
+    using r Suc by simp
   finally show ?thesis
     using Suc by simp
 qed
 
 lemma power_radical:
   fixes a:: "'a::field_char_0 fps"
   assumes a0: "a$0 \<noteq> 0"
   shows "(r (Suc k) (a$0)) ^ Suc k = a$0 \<longleftrightarrow> (fps_radical r (Suc k) a) ^ (Suc k) = a"
     (is "?lhs \<longleftrightarrow> ?rhs")
 proof
   let ?r = "fps_radical r (Suc k) a"
   show ?rhs if r0: ?lhs
   proof -
     from a0 r0 have r00: "r (Suc k) (a$0) \<noteq> 0" by auto
     have "?r ^ Suc k $ z = a$z" for z
     proof (induct z rule: nat_less_induct)
       fix n
       assume H: "\<forall>m<n. ?r ^ Suc k $ m = a$m"
       show "?r ^ Suc k $ n = a $n"
       proof (cases n)
         case 0
         then show ?thesis
           using fps_radical_power_nth[of r "Suc k" a, OF r0] by simp
       next
         case (Suc n1)
         then have "n \<noteq> 0" by simp
         let ?Pnk = "natpermute n (k + 1)"
         let ?Pnkn = "{xs \<in> ?Pnk. n \<in> set xs}"
         let ?Pnknn = "{xs \<in> ?Pnk. n \<notin> set xs}"
         have eq: "?Pnkn \<union> ?Pnknn = ?Pnk" by blast
         have d: "?Pnkn \<inter> ?Pnknn = {}" by blast
         have f: "finite ?Pnkn" "finite ?Pnknn"
           using finite_Un[of ?Pnkn ?Pnknn, unfolded eq]
           by (metis natpermute_finite)+
         let ?f = "\<lambda>v. \<Prod>j\<in>{0..k}. ?r $ v ! j"
         have "sum ?f ?Pnkn = sum (\<lambda>v. ?r $ n * r (Suc k) (a $ 0) ^ k) ?Pnkn"
         proof (rule sum.cong)
           fix v assume v: "v \<in> {xs \<in> natpermute n (k + 1). n \<in> set xs}"
           let ?ths = "(\<Prod>j\<in>{0..k}. fps_radical r (Suc k) a $ v ! j) =
             fps_radical r (Suc k) a $ n * r (Suc k) (a $ 0) ^ k"
           from v obtain i where i: "i \<in> {0..k}" "v = (replicate (k+1) 0) [i:= n]"
             unfolding natpermute_contain_maximal by auto
           have "(\<Prod>j\<in>{0..k}. fps_radical r (Suc k) a $ v ! j) =
-              (\<Prod>j\<in>{0..k}. if j = i then fps_radical r (Suc k) a $ n else r (Suc k) (a$0))"
-            apply (rule prod.cong, simp)
-            using i r0
-            apply (simp del: replicate.simps)
-            done
+                (\<Prod>j\<in>{0..k}. if j = i then fps_radical r (Suc k) a $ n else r (Suc k) (a$0))"
+              using i r0 by (auto simp del: replicate.simps intro: prod.cong)
           also have "\<dots> = (fps_radical r (Suc k) a $ n) * r (Suc k) (a$0) ^ k"
             using i r0 by (simp add: prod_gen_delta)
           finally show ?ths .
         qed rule
         then have "sum ?f ?Pnkn = of_nat (k+1) * ?r $ n * r (Suc k) (a $ 0) ^ k"
           by (simp add: natpermute_max_card[OF \<open>n \<noteq> 0\<close>, simplified])
         also have "\<dots> = a$n - sum ?f ?Pnknn"
           unfolding Suc using r00 a0 by (simp add: field_simps fps_radical_def del: of_nat_Suc)
         finally have fn: "sum ?f ?Pnkn = a$n - sum ?f ?Pnknn" .
         have "(?r ^ Suc k)$n = sum ?f ?Pnkn + sum ?f ?Pnknn"
           unfolding fps_power_nth_Suc sum.union_disjoint[OF f d, unfolded eq] ..
         also have "\<dots> = a$n" unfolding fn by simp
         finally show ?thesis .
       qed
     qed
     then show ?thesis using r0 by (simp add: fps_eq_iff)
   qed
   show ?lhs if ?rhs
   proof -
     from that have "((fps_radical r (Suc k) a) ^ (Suc k))$0 = a$0"
       by simp
     then show ?thesis
       unfolding fps_power_nth_Suc
       by (simp add: prod_constant del: replicate.simps)
   qed
 qed
 
-(*
-lemma power_radical:
-  fixes a:: "'a::field_char_0 fps"
-  assumes r0: "(r (Suc k) (a$0)) ^ Suc k = a$0" and a0: "a$0 \<noteq> 0"
-  shows "(fps_radical r (Suc k) a) ^ (Suc k) = a"
-proof-
-  let ?r = "fps_radical r (Suc k) a"
-  from a0 r0 have r00: "r (Suc k) (a$0) \<noteq> 0" by auto
-  {fix z have "?r ^ Suc k $ z = a$z"
-    proof(induct z rule: nat_less_induct)
-      fix n assume H: "\<forall>m<n. ?r ^ Suc k $ m = a$m"
-      {assume "n = 0" then have "?r ^ Suc k $ n = a $n"
-          using fps_radical_power_nth[of r "Suc k" a, OF r0] by simp}
-      moreover
-      {fix n1 assume n1: "n = Suc n1"
-        have fK: "finite {0..k}" by simp
-        have nz: "n \<noteq> 0" using n1 by arith
-        let ?Pnk = "natpermute n (k + 1)"
-        let ?Pnkn = "{xs \<in> ?Pnk. n \<in> set xs}"
-        let ?Pnknn = "{xs \<in> ?Pnk. n \<notin> set xs}"
-        have eq: "?Pnkn \<union> ?Pnknn = ?Pnk" by blast
-        have d: "?Pnkn \<inter> ?Pnknn = {}" by blast
-        have f: "finite ?Pnkn" "finite ?Pnknn"
-          using finite_Un[of ?Pnkn ?Pnknn, unfolded eq]
-          by (metis natpermute_finite)+
-        let ?f = "\<lambda>v. \<Prod>j\<in>{0..k}. ?r $ v ! j"
-        have "sum ?f ?Pnkn = sum (\<lambda>v. ?r $ n * r (Suc k) (a $ 0) ^ k) ?Pnkn"
-        proof(rule sum.cong2)
-          fix v assume v: "v \<in> {xs \<in> natpermute n (k + 1). n \<in> set xs}"
-          let ?ths = "(\<Prod>j\<in>{0..k}. fps_radical r (Suc k) a $ v ! j) = fps_radical r (Suc k) a $ n * r (Suc k) (a $ 0) ^ k"
-          from v obtain i where i: "i \<in> {0..k}" "v = replicate (k+1) 0 [i:= n]"
-            unfolding natpermute_contain_maximal by auto
-          have "(\<Prod>j\<in>{0..k}. fps_radical r (Suc k) a $ v ! j) = (\<Prod>j\<in>{0..k}. if j = i then fps_radical r (Suc k) a $ n else r (Suc k) (a$0))"
-            apply (rule prod.cong, simp)
-            using i r0 by (simp del: replicate.simps)
-          also have "\<dots> = (fps_radical r (Suc k) a $ n) * r (Suc k) (a$0) ^ k"
-            unfolding prod_gen_delta[OF fK] using i r0 by simp
-          finally show ?ths .
-        qed
-        then have "sum ?f ?Pnkn = of_nat (k+1) * ?r $ n * r (Suc k) (a $ 0) ^ k"
-          by (simp add: natpermute_max_card[OF nz, simplified])
-        also have "\<dots> = a$n - sum ?f ?Pnknn"
-          unfolding n1 using r00 a0 by (simp add: field_simps fps_radical_def del: of_nat_Suc )
-        finally have fn: "sum ?f ?Pnkn = a$n - sum ?f ?Pnknn" .
-        have "(?r ^ Suc k)$n = sum ?f ?Pnkn + sum ?f ?Pnknn"
-          unfolding fps_power_nth_Suc sum.union_disjoint[OF f d, unfolded eq] ..
-        also have "\<dots> = a$n" unfolding fn by simp
-        finally have "?r ^ Suc k $ n = a $n" .}
-      ultimately  show "?r ^ Suc k $ n = a $n" by (cases n, auto)
-  qed }
-  then show ?thesis by (simp add: fps_eq_iff)
-qed
-
-*)
-lemma eq_divide_imp':
-  fixes c :: "'a::field"
-  shows "c \<noteq> 0 \<Longrightarrow> a * c = b \<Longrightarrow> a = b / c"
-  by (simp add: field_simps)
-
 lemma radical_unique:
   assumes r0: "(r (Suc k) (b$0)) ^ Suc k = b$0"
     and a0: "r (Suc k) (b$0 ::'a::field_char_0) = a$0"
     and b0: "b$0 \<noteq> 0"
   shows "a^(Suc k) = b \<longleftrightarrow> a = fps_radical r (Suc k) b"
     (is "?lhs \<longleftrightarrow> ?rhs" is "_ \<longleftrightarrow> a = ?r")
 proof
   show ?lhs if ?rhs
     using that using power_radical[OF b0, of r k, unfolded r0] by simp
   show ?rhs if ?lhs
   proof -
     have r00: "r (Suc k) (b$0) \<noteq> 0" using b0 r0 by auto
     have ceq: "card {0..k} = Suc k" by simp
     from a0 have a0r0: "a$0 = ?r$0" by simp
     have "a $ n = ?r $ n" for n
     proof (induct n rule: nat_less_induct)
       fix n
       assume h: "\<forall>m<n. a$m = ?r $m"
       show "a$n = ?r $ n"
       proof (cases n)
         case 0
         then show ?thesis using a0 by simp
       next
         case (Suc n1)
         have fK: "finite {0..k}" by simp
         have nz: "n \<noteq> 0" using Suc by simp
         let ?Pnk = "natpermute n (Suc k)"
         let ?Pnkn = "{xs \<in> ?Pnk. n \<in> set xs}"
         let ?Pnknn = "{xs \<in> ?Pnk. n \<notin> set xs}"
         have eq: "?Pnkn \<union> ?Pnknn = ?Pnk" by blast
         have d: "?Pnkn \<inter> ?Pnknn = {}" by blast
         have f: "finite ?Pnkn" "finite ?Pnknn"
           using finite_Un[of ?Pnkn ?Pnknn, unfolded eq]
           by (metis natpermute_finite)+
         let ?f = "\<lambda>v. \<Prod>j\<in>{0..k}. ?r $ v ! j"
         let ?g = "\<lambda>v. \<Prod>j\<in>{0..k}. a $ v ! j"
         have "sum ?g ?Pnkn = sum (\<lambda>v. a $ n * (?r$0)^k) ?Pnkn"
         proof (rule sum.cong)
           fix v
           assume v: "v \<in> {xs \<in> natpermute n (Suc k). n \<in> set xs}"
           let ?ths = "(\<Prod>j\<in>{0..k}. a $ v ! j) = a $ n * (?r$0)^k"
           from v obtain i where i: "i \<in> {0..k}" "v = (replicate (k+1) 0) [i:= n]"
             unfolding Suc_eq_plus1 natpermute_contain_maximal
             by (auto simp del: replicate.simps)
           have "(\<Prod>j\<in>{0..k}. a $ v ! j) = (\<Prod>j\<in>{0..k}. if j = i then a $ n else r (Suc k) (b$0))"
-            apply (rule prod.cong, simp)
-            using i a0
-            apply (simp del: replicate.simps)
-            done
+            using i a0 by (auto simp del: replicate.simps intro: prod.cong)
           also have "\<dots> = a $ n * (?r $ 0)^k"
             using i by (simp add: prod_gen_delta)
           finally show ?ths .
         qed rule
         then have th0: "sum ?g ?Pnkn = of_nat (k+1) * a $ n * (?r $ 0)^k"
           by (simp add: natpermute_max_card[OF nz, simplified])
         have th1: "sum ?g ?Pnknn = sum ?f ?Pnknn"
         proof (rule sum.cong, rule refl, rule prod.cong, simp)
           fix xs i
           assume xs: "xs \<in> ?Pnknn" and i: "i \<in> {0..k}"
           have False if c: "n \<le> xs ! i"
           proof -
             from xs i have "xs ! i \<noteq> n"
               by (simp add: in_set_conv_nth natpermute_def)
             with c have c': "n < xs!i" by arith
             have fths: "finite {0 ..< i}" "finite {i}" "finite {i+1..<Suc k}"
               by simp_all
             have d: "{0 ..< i} \<inter> ({i} \<union> {i+1 ..< Suc k}) = {}" "{i} \<inter> {i+1..< Suc k} = {}"
               by auto
             have eqs: "{0..<Suc k} = {0 ..< i} \<union> ({i} \<union> {i+1 ..< Suc k})"
               using i by auto
             from xs have "n = sum_list xs"
               by (simp add: natpermute_def)
             also have "\<dots> = sum (nth xs) {0..<Suc k}"
               using xs by (simp add: natpermute_def sum_list_sum_nth)
             also have "\<dots> = xs!i + sum (nth xs) {0..<i} + sum (nth xs) {i+1..<Suc k}"
               unfolding eqs  sum.union_disjoint[OF fths(1) finite_UnI[OF fths(2,3)] d(1)]
               unfolding sum.union_disjoint[OF fths(2) fths(3) d(2)]
               by simp
             finally show ?thesis using c' by simp
           qed
           then have thn: "xs!i < n" by presburger
           from h[rule_format, OF thn] show "a$(xs !i) = ?r$(xs!i)" .
         qed
         have th00: "\<And>x::'a. of_nat (Suc k) * (x * inverse (of_nat (Suc k))) = x"
           by (simp add: field_simps del: of_nat_Suc)
         from \<open>?lhs\<close> have "b$n = a^Suc k $ n"
           by (simp add: fps_eq_iff)
         also have "a ^ Suc k$n = sum ?g ?Pnkn + sum ?g ?Pnknn"
           unfolding fps_power_nth_Suc
           using sum.union_disjoint[OF f d, unfolded Suc_eq_plus1[symmetric],
             unfolded eq, of ?g] by simp
         also have "\<dots> = of_nat (k+1) * a $ n * (?r $ 0)^k + sum ?f ?Pnknn"
           unfolding th0 th1 ..
-        finally have "of_nat (k+1) * a $ n * (?r $ 0)^k = b$n - sum ?f ?Pnknn"
+        finally have \<section>: "of_nat (k+1) * a $ n * (?r $ 0)^k = b$n - sum ?f ?Pnknn"
           by simp
-        then have "a$n = (b$n - sum ?f ?Pnknn) / (of_nat (k+1) * (?r $ 0)^k)"
-          apply -
-          apply (rule eq_divide_imp')
-          using r00
-          apply (simp del: of_nat_Suc)
-          apply (simp add: ac_simps)
-          done
+        have "a$n = (b$n - sum ?f ?Pnknn) / (of_nat (k+1) * (?r $ 0)^k)"
+          apply (rule eq_divide_imp)
+          using r00 \<section> by (simp_all add: ac_simps del: of_nat_Suc)
         then show ?thesis
-          apply (simp del: of_nat_Suc)
           unfolding fps_radical_def Suc
-          apply (simp add: field_simps Suc th00 del: of_nat_Suc)
-          done
+          by (simp del: of_nat_Suc)
       qed
     qed
     then show ?rhs by (simp add: fps_eq_iff)
   qed
 qed
 
 
 lemma radical_power:
   assumes r0: "r (Suc k) ((a$0) ^ Suc k) = a$0"
     and a0: "(a$0 :: 'a::field_char_0) \<noteq> 0"
   shows "(fps_radical r (Suc k) (a ^ Suc k)) = a"
 proof -
   let ?ak = "a^ Suc k"
   have ak0: "?ak $ 0 = (a$0) ^ Suc k"
     by (simp add: fps_nth_power_0 del: power_Suc)
   from r0 have th0: "r (Suc k) (a ^ Suc k $ 0) ^ Suc k = a ^ Suc k $ 0"
     using ak0 by auto
   from r0 ak0 have th1: "r (Suc k) (a ^ Suc k $ 0) = a $ 0"
     by auto
   from ak0 a0 have ak00: "?ak $ 0 \<noteq>0 "
     by auto
   from radical_unique[of r k ?ak a, OF th0 th1 ak00] show ?thesis
     by metis
 qed
 
 lemma fps_deriv_radical':
   fixes a :: "'a::field_char_0 fps"
   assumes r0: "(r (Suc k) (a$0)) ^ Suc k = a$0"
     and a0: "a$0 \<noteq> 0"
   shows "fps_deriv (fps_radical r (Suc k) a) =
     fps_deriv a / ((of_nat (Suc k)) * (fps_radical r (Suc k) a) ^ k)"
 proof -
   let ?r = "fps_radical r (Suc k) a"
   let ?w = "(of_nat (Suc k)) * ?r ^ k"
   from a0 r0 have r0': "r (Suc k) (a$0) \<noteq> 0"
     by auto
   from r0' have w0: "?w $ 0 \<noteq> 0"
     by (simp del: of_nat_Suc)
   note th0 = inverse_mult_eq_1[OF w0]
   let ?iw = "inverse ?w"
   from iffD1[OF power_radical[of a r], OF a0 r0]
   have "fps_deriv (?r ^ Suc k) = fps_deriv a"
     by simp
   then have "fps_deriv ?r * ?w = fps_deriv a"
     by (simp add: fps_deriv_power' ac_simps del: power_Suc)
   then have "?iw * fps_deriv ?r * ?w = ?iw * fps_deriv a"
     by simp
   with a0 r0 have "fps_deriv ?r * (?iw * ?w) = fps_deriv a / ?w"
     by (subst fps_divide_unit) (auto simp del: of_nat_Suc)
   then show ?thesis unfolding th0 by simp
 qed
 
 lemma fps_deriv_radical:
   fixes a :: "'a::field_char_0 fps"
   assumes r0: "(r (Suc k) (a$0)) ^ Suc k = a$0"
     and a0: "a$0 \<noteq> 0"
   shows "fps_deriv (fps_radical r (Suc k) a) =
     fps_deriv a / (fps_const (of_nat (Suc k)) * (fps_radical r (Suc k) a) ^ k)"
   using fps_deriv_radical'[of r k a, OF r0 a0]
   by (simp add: fps_of_nat[symmetric])
 
 lemma radical_mult_distrib:
   fixes a :: "'a::field_char_0 fps"
   assumes k: "k > 0"
     and ra0: "r k (a $ 0) ^ k = a $ 0"
     and rb0: "r k (b $ 0) ^ k = b $ 0"
     and a0: "a $ 0 \<noteq> 0"
     and b0: "b $ 0 \<noteq> 0"
   shows "r k ((a * b) $ 0) = r k (a $ 0) * r k (b $ 0) \<longleftrightarrow>
     fps_radical r k (a * b) = fps_radical r k a * fps_radical r k b"
     (is "?lhs \<longleftrightarrow> ?rhs")
 proof
   show ?rhs if r0': ?lhs
   proof -
     from r0' have r0: "(r k ((a * b) $ 0)) ^ k = (a * b) $ 0"
       by (simp add: fps_mult_nth ra0 rb0 power_mult_distrib)
     show ?thesis
     proof (cases k)
       case 0
       then show ?thesis using r0' by simp
     next
       case (Suc h)
       let ?ra = "fps_radical r (Suc h) a"
       let ?rb = "fps_radical r (Suc h) b"
       have th0: "r (Suc h) ((a * b) $ 0) = (fps_radical r (Suc h) a * fps_radical r (Suc h) b) $ 0"
         using r0' Suc by (simp add: fps_mult_nth)
       have ab0: "(a*b) $ 0 \<noteq> 0"
         using a0 b0 by (simp add: fps_mult_nth)
       from radical_unique[of r h "a*b" "fps_radical r (Suc h) a * fps_radical r (Suc h) b", OF r0[unfolded Suc] th0 ab0, symmetric]
         iffD1[OF power_radical[of _ r], OF a0 ra0[unfolded Suc]] iffD1[OF power_radical[of _ r], OF b0 rb0[unfolded Suc]] Suc r0'
       show ?thesis
         by (auto simp add: power_mult_distrib simp del: power_Suc)
     qed
   qed
   show ?lhs if ?rhs
   proof -
     from that have "(fps_radical r k (a * b)) $ 0 = (fps_radical r k a * fps_radical r k b) $ 0"
       by simp
     then show ?thesis
       using k by (simp add: fps_mult_nth)
   qed
 qed
 
 (*
 lemma radical_mult_distrib:
   fixes a:: "'a::field_char_0 fps"
   assumes
   ra0: "r k (a $ 0) ^ k = a $ 0"
   and rb0: "r k (b $ 0) ^ k = b $ 0"
   and r0': "r k ((a * b) $ 0) = r k (a $ 0) * r k (b $ 0)"
   and a0: "a$0 \<noteq> 0"
   and b0: "b$0 \<noteq> 0"
   shows "fps_radical r (k) (a*b) = fps_radical r (k) a * fps_radical r (k) (b)"
 proof-
   from r0' have r0: "(r (k) ((a*b)$0)) ^ k = (a*b)$0"
     by (simp add: fps_mult_nth ra0 rb0 power_mult_distrib)
   {assume "k=0" then have ?thesis by simp}
   moreover
   {fix h assume k: "k = Suc h"
   let ?ra = "fps_radical r (Suc h) a"
   let ?rb = "fps_radical r (Suc h) b"
   have th0: "r (Suc h) ((a * b) $ 0) = (fps_radical r (Suc h) a * fps_radical r (Suc h) b) $ 0"
     using r0' k by (simp add: fps_mult_nth)
   have ab0: "(a*b) $ 0 \<noteq> 0" using a0 b0 by (simp add: fps_mult_nth)
   from radical_unique[of r h "a*b" "fps_radical r (Suc h) a * fps_radical r (Suc h) b", OF r0[unfolded k] th0 ab0, symmetric]
     power_radical[of r, OF ra0[unfolded k] a0] power_radical[of r, OF rb0[unfolded k] b0] k
   have ?thesis by (auto simp add: power_mult_distrib simp del: power_Suc)}
 ultimately show ?thesis by (cases k, auto)
 qed
 *)
 
 lemma radical_divide:
   fixes a :: "'a::field_char_0 fps"
   assumes kp: "k > 0"
     and ra0: "(r k (a $ 0)) ^ k = a $ 0"
     and rb0: "(r k (b $ 0)) ^ k = b $ 0"
     and a0: "a$0 \<noteq> 0"
     and b0: "b$0 \<noteq> 0"
   shows "r k ((a $ 0) / (b$0)) = r k (a$0) / r k (b $ 0) \<longleftrightarrow>
     fps_radical r k (a/b) = fps_radical r k a / fps_radical r k b"
   (is "?lhs = ?rhs")
 proof
   let ?r = "fps_radical r k"
   from kp obtain h where k: "k = Suc h"
     by (cases k) auto
   have ra0': "r k (a$0) \<noteq> 0" using a0 ra0 k by auto
   have rb0': "r k (b$0) \<noteq> 0" using b0 rb0 k by auto
 
   show ?lhs if ?rhs
   proof -
     from that have "?r (a/b) $ 0 = (?r a / ?r b)$0"
       by simp
     then show ?thesis
       using k a0 b0 rb0' by (simp add: fps_divide_unit fps_mult_nth fps_inverse_def divide_inverse)
   qed
   show ?rhs if ?lhs
   proof -
     from a0 b0 have ab0[simp]: "(a/b)$0 = a$0 / b$0"
       by (simp add: fps_divide_def fps_mult_nth divide_inverse fps_inverse_def)
     have th0: "r k ((a/b)$0) ^ k = (a/b)$0"
       by (simp add: \<open>?lhs\<close> power_divide ra0 rb0)
     from a0 b0 ra0' rb0' kp \<open>?lhs\<close>
     have th1: "r k ((a / b) $ 0) = (fps_radical r k a / fps_radical r k b) $ 0"
       by (simp add: fps_divide_unit fps_mult_nth fps_inverse_def divide_inverse)
     from a0 b0 ra0' rb0' kp have ab0': "(a / b) $ 0 \<noteq> 0"
       by (simp add: fps_divide_unit fps_mult_nth fps_inverse_def nonzero_imp_inverse_nonzero)
     note tha[simp] = iffD1[OF power_radical[where r=r and k=h], OF a0 ra0[unfolded k], unfolded k[symmetric]]
     note thb[simp] = iffD1[OF power_radical[where r=r and k=h], OF b0 rb0[unfolded k], unfolded k[symmetric]]
     from b0 rb0' have th2: "(?r a / ?r b)^k = a/b"
       by (simp add: fps_divide_unit power_mult_distrib fps_inverse_power[symmetric])
 
     from iffD1[OF radical_unique[where r=r and a="?r a / ?r b" and b="a/b" and k=h], symmetric, unfolded k[symmetric], OF th0 th1 ab0' th2]
     show ?thesis .
   qed
 qed
 
 lemma radical_inverse:
   fixes a :: "'a::field_char_0 fps"
   assumes k: "k > 0"
     and ra0: "r k (a $ 0) ^ k = a $ 0"
     and r1: "(r k 1)^k = 1"
     and a0: "a$0 \<noteq> 0"
   shows "r k (inverse (a $ 0)) = r k 1 / (r k (a $ 0)) \<longleftrightarrow>
     fps_radical r k (inverse a) = fps_radical r k 1 / fps_radical r k a"
   using radical_divide[where k=k and r=r and a=1 and b=a, OF k ] ra0 r1 a0
   by (simp add: divide_inverse fps_divide_def)
 
 
 subsection \<open>Derivative of composition\<close>
 
 lemma fps_compose_deriv:
   fixes a :: "'a::idom fps"
   assumes b0: "b$0 = 0"
   shows "fps_deriv (a oo b) = ((fps_deriv a) oo b) * fps_deriv b"
 proof -
   have "(fps_deriv (a oo b))$n = (((fps_deriv a) oo b) * (fps_deriv b)) $n" for n
   proof -
     have "(fps_deriv (a oo b))$n = sum (\<lambda>i. a $ i * (fps_deriv (b^i))$n) {0.. Suc n}"
       by (simp add: fps_compose_def field_simps sum_distrib_left del: of_nat_Suc)
     also have "\<dots> = sum (\<lambda>i. a$i * ((fps_const (of_nat i)) * (fps_deriv b * (b^(i - 1))))$n) {0.. Suc n}"
       by (simp add: field_simps fps_deriv_power del: fps_mult_left_const_nth of_nat_Suc)
     also have "\<dots> = sum (\<lambda>i. of_nat i * a$i * (((b^(i - 1)) * fps_deriv b))$n) {0.. Suc n}"
       unfolding fps_mult_left_const_nth  by (simp add: field_simps)
     also have "\<dots> = sum (\<lambda>i. of_nat i * a$i * (sum (\<lambda>j. (b^ (i - 1))$j * (fps_deriv b)$(n - j)) {0..n})) {0.. Suc n}"
       unfolding fps_mult_nth ..
     also have "\<dots> = sum (\<lambda>i. of_nat i * a$i * (sum (\<lambda>j. (b^ (i - 1))$j * (fps_deriv b)$(n - j)) {0..n})) {1.. Suc n}"
-      apply (rule sum.mono_neutral_right)
-      apply (auto simp add: mult_delta_left not_le)
-      done
+      by (intro sum.mono_neutral_right) (auto simp add: mult_delta_left not_le)
     also have "\<dots> = sum (\<lambda>i. of_nat (i + 1) * a$(i+1) * (sum (\<lambda>j. (b^ i)$j * of_nat (n - j + 1) * b$(n - j + 1)) {0..n})) {0.. n}"
       unfolding fps_deriv_nth
       by (rule sum.reindex_cong [of Suc]) (simp_all add: mult.assoc)
     finally have th0: "(fps_deriv (a oo b))$n =
       sum (\<lambda>i. of_nat (i + 1) * a$(i+1) * (sum (\<lambda>j. (b^ i)$j * of_nat (n - j + 1) * b$(n - j + 1)) {0..n})) {0.. n}" .
 
     have "(((fps_deriv a) oo b) * (fps_deriv b))$n = sum (\<lambda>i. (fps_deriv b)$ (n - i) * ((fps_deriv a) oo b)$i) {0..n}"
       unfolding fps_mult_nth by (simp add: ac_simps)
     also have "\<dots> = sum (\<lambda>i. sum (\<lambda>j. of_nat (n - i +1) * b$(n - i + 1) * of_nat (j + 1) * a$(j+1) * (b^j)$i) {0..n}) {0..n}"
       unfolding fps_deriv_nth fps_compose_nth sum_distrib_left mult.assoc
-      apply (rule sum.cong)
-      apply (rule refl)
-      apply (rule sum.mono_neutral_left)
-      apply (simp_all add: subset_eq)
-      apply clarify
-      apply (subgoal_tac "b^i$x = 0")
-      apply simp
-      apply (rule startsby_zero_power_prefix[OF b0, rule_format])
-      apply simp
-      done
+      by (auto simp: subset_eq b0 startsby_zero_power_prefix sum.mono_neutral_left intro: sum.cong)
     also have "\<dots> = sum (\<lambda>i. of_nat (i + 1) * a$(i+1) * (sum (\<lambda>j. (b^ i)$j * of_nat (n - j + 1) * b$(n - j + 1)) {0..n})) {0.. n}"
       unfolding sum_distrib_left
-      apply (subst sum.swap)
-      apply (rule sum.cong, rule refl)+
-      apply simp
-      done
+      by (subst sum.swap) (force intro: sum.cong)
     finally show ?thesis
       unfolding th0 by simp
   qed
   then show ?thesis by (simp add: fps_eq_iff)
 qed
 
 
 subsection \<open>Finite FPS (i.e. polynomials) and fps_X\<close>
 
 lemma fps_poly_sum_fps_X:
   assumes "\<forall>i > n. a$i = 0"
   shows "a = sum (\<lambda>i. fps_const (a$i) * fps_X^i) {0..n}" (is "a = ?r")
 proof -
   have "a$i = ?r$i" for i
     unfolding fps_sum_nth fps_mult_left_const_nth fps_X_power_nth
     by (simp add: mult_delta_right assms)
   then show ?thesis
     unfolding fps_eq_iff by blast
 qed
 
 
 subsection \<open>Compositional inverses\<close>
 
 fun compinv :: "'a fps \<Rightarrow> nat \<Rightarrow> 'a::field"
 where
   "compinv a 0 = fps_X$0"
 | "compinv a (Suc n) =
     (fps_X$ Suc n - sum (\<lambda>i. (compinv a i) * (a^i)$Suc n) {0 .. n}) / (a$1) ^ Suc n"
 
 definition "fps_inv a = Abs_fps (compinv a)"
 
 lemma fps_inv:
   assumes a0: "a$0 = 0"
     and a1: "a$1 \<noteq> 0"
   shows "fps_inv a oo a = fps_X"
 proof -
   let ?i = "fps_inv a oo a"
   have "?i $n = fps_X$n" for n
   proof (induct n rule: nat_less_induct)
     fix n
     assume h: "\<forall>m<n. ?i$m = fps_X$m"
     show "?i $ n = fps_X$n"
     proof (cases n)
       case 0
       then show ?thesis using a0
         by (simp add: fps_compose_nth fps_inv_def)
     next
       case (Suc n1)
       have "?i $ n = sum (\<lambda>i. (fps_inv a $ i) * (a^i)$n) {0 .. n1} + fps_inv a $ Suc n1 * (a $ 1)^ Suc n1"
         by (simp only: fps_compose_nth) (simp add: Suc startsby_zero_power_nth_same [OF a0] del: power_Suc)
       also have "\<dots> = sum (\<lambda>i. (fps_inv a $ i) * (a^i)$n) {0 .. n1} +
         (fps_X$ Suc n1 - sum (\<lambda>i. (fps_inv a $ i) * (a^i)$n) {0 .. n1})"
         using a0 a1 Suc by (simp add: fps_inv_def)
       also have "\<dots> = fps_X$n" using Suc by simp
       finally show ?thesis .
     qed
   qed
   then show ?thesis
     by (simp add: fps_eq_iff)
 qed
 
 
 fun gcompinv :: "'a fps \<Rightarrow> 'a fps \<Rightarrow> nat \<Rightarrow> 'a::field"
 where
   "gcompinv b a 0 = b$0"
 | "gcompinv b a (Suc n) =
     (b$ Suc n - sum (\<lambda>i. (gcompinv b a i) * (a^i)$Suc n) {0 .. n}) / (a$1) ^ Suc n"
 
 definition "fps_ginv b a = Abs_fps (gcompinv b a)"
 
 lemma fps_ginv:
   assumes a0: "a$0 = 0"
     and a1: "a$1 \<noteq> 0"
   shows "fps_ginv b a oo a = b"
 proof -
   let ?i = "fps_ginv b a oo a"
   have "?i $n = b$n" for n
   proof (induct n rule: nat_less_induct)
     fix n
     assume h: "\<forall>m<n. ?i$m = b$m"
     show "?i $ n = b$n"
     proof (cases n)
       case 0
       then show ?thesis using a0
         by (simp add: fps_compose_nth fps_ginv_def)
     next
       case (Suc n1)
       have "?i $ n = sum (\<lambda>i. (fps_ginv b a $ i) * (a^i)$n) {0 .. n1} + fps_ginv b a $ Suc n1 * (a $ 1)^ Suc n1"
         by (simp only: fps_compose_nth) (simp add: Suc startsby_zero_power_nth_same [OF a0] del: power_Suc)
       also have "\<dots> = sum (\<lambda>i. (fps_ginv b a $ i) * (a^i)$n) {0 .. n1} +
         (b$ Suc n1 - sum (\<lambda>i. (fps_ginv b a $ i) * (a^i)$n) {0 .. n1})"
         using a0 a1 Suc by (simp add: fps_ginv_def)
       also have "\<dots> = b$n" using Suc by simp
       finally show ?thesis .
     qed
   qed
   then show ?thesis
     by (simp add: fps_eq_iff)
 qed
 
 lemma fps_inv_ginv: "fps_inv = fps_ginv fps_X"
-  apply (auto simp add: fun_eq_iff fps_eq_iff fps_inv_def fps_ginv_def)
-  apply (induct_tac n rule: nat_less_induct)
-  apply auto
-  apply (case_tac na)
-  apply simp
-  apply simp
-  done
+proof -
+  have "compinv x n = gcompinv fps_X x n" for n and x :: "'a fps"
+  proof (induction n rule: nat_less_induct)
+    case (1 n)
+    then show ?case
+      by (cases n) auto
+  qed
+  then show ?thesis
+    by (auto simp add: fun_eq_iff fps_eq_iff fps_inv_def fps_ginv_def)
+qed
 
 lemma fps_compose_1[simp]: "1 oo a = 1"
   by (simp add: fps_eq_iff fps_compose_nth mult_delta_left)
 
 lemma fps_compose_0[simp]: "0 oo a = 0"
   by (simp add: fps_eq_iff fps_compose_nth)
 
 lemma fps_compose_0_right[simp]: "a oo 0 = fps_const (a $ 0)"
   by (simp add: fps_eq_iff fps_compose_nth power_0_left sum.neutral)
 
 lemma fps_compose_add_distrib: "(a + b) oo c = (a oo c) + (b oo c)"
   by (simp add: fps_eq_iff fps_compose_nth field_simps sum.distrib)
 
 lemma fps_compose_sum_distrib: "(sum f S) oo a = sum (\<lambda>i. f i oo a) S"
 proof (cases "finite S")
   case True
   show ?thesis
   proof (rule finite_induct[OF True])
     show "sum f {} oo a = (\<Sum>i\<in>{}. f i oo a)"
       by simp
   next
     fix x F
     assume fF: "finite F"
       and xF: "x \<notin> F"
       and h: "sum f F oo a = sum (\<lambda>i. f i oo a) F"
     show "sum f (insert x F) oo a  = sum (\<lambda>i. f i oo a) (insert x F)"
       using fF xF h by (simp add: fps_compose_add_distrib)
   qed
 next
   case False
   then show ?thesis by simp
 qed
 
 lemma convolution_eq:
   "sum (\<lambda>i. a (i :: nat) * b (n - i)) {0 .. n} =
     sum (\<lambda>(i,j). a i * b j) {(i,j). i \<le> n \<and> j \<le> n \<and> i + j = n}"
   by (rule sum.reindex_bij_witness[where i=fst and j="\<lambda>i. (i, n - i)"]) auto
 
 lemma product_composition_lemma:
   assumes c0: "c$0 = (0::'a::idom)"
     and d0: "d$0 = 0"
   shows "((a oo c) * (b oo d))$n =
     sum (\<lambda>(k,m). a$k * b$m * (c^k * d^m) $ n) {(k,m). k + m \<le> n}"  (is "?l = ?r")
 proof -
   let ?S = "{(k::nat, m::nat). k + m \<le> n}"
   have s: "?S \<subseteq> {0..n} \<times> {0..n}" by (simp add: subset_eq)
   have f: "finite {(k::nat, m::nat). k + m \<le> n}"
-    apply (rule finite_subset[OF s])
-    apply auto
-    done
-  have "?r =  sum (\<lambda>i. sum (\<lambda>(k,m). a$k * (c^k)$i * b$m * (d^m) $ (n - i)) {(k,m). k + m \<le> n}) {0..n}"
-    apply (simp add: fps_mult_nth sum_distrib_left)
-    apply (subst sum.swap)
-    apply (rule sum.cong)
-    apply (auto simp add: field_simps)
-    done
-  also have "\<dots> = ?l"
-    apply (simp add: fps_mult_nth fps_compose_nth sum_product)
-    apply (rule sum.cong)
-    apply (rule refl)
+    by (auto intro: finite_subset[OF s])
+  have "?r = (\<Sum>(k, m) \<in> {(k, m). k + m \<le> n}. \<Sum>j = 0..n. a $ k * b $ m * (c ^ k $ j * d ^ m $ (n - j)))"
+    by (simp add: fps_mult_nth sum_distrib_left)
+  also have "\<dots> = (\<Sum>i = 0..n. \<Sum>(k,m)\<in>{(k,m). k+m \<le> n}. a $ k * c ^ k $ i * b $ m * d ^ m $ (n-i))"
+    unfolding sum.swap [where A = "{0..n}"] by (auto simp add: field_simps intro: sum.cong)
+  also have "... = (\<Sum>i = 0..n.
+                    \<Sum>q = 0..i. \<Sum>j = 0..n - i. a $ q * c ^ q $ i * (b $ j * d ^ j $ (n - i)))"
+    apply (rule sum.cong [OF refl])
     apply (simp add: sum.cartesian_product mult.assoc)
-    apply (rule sum.mono_neutral_right[OF f])
-    apply (simp add: subset_eq)
-    apply presburger
-    apply clarsimp
-    apply (rule ccontr)
-    apply (clarsimp simp add: not_le)
-    apply (case_tac "x < aa")
-    apply simp
-    apply (frule_tac startsby_zero_power_prefix[rule_format, OF c0])
-    apply blast
-    apply simp
-    apply (frule_tac startsby_zero_power_prefix[rule_format, OF d0])
-    apply blast
-    done
+    apply (rule sum.mono_neutral_right[OF f], force)
+    by clarsimp (meson c0 d0 leI startsby_zero_power_prefix)
+  also have "\<dots> = ?l"
+    by (simp add: fps_mult_nth fps_compose_nth sum_product)
   finally show ?thesis by simp
 qed
 
-lemma product_composition_lemma':
-  assumes c0: "c$0 = (0::'a::idom)"
-    and d0: "d$0 = 0"
-  shows "((a oo c) * (b oo d))$n =
-    sum (\<lambda>k. sum (\<lambda>m. a$k * b$m * (c^k * d^m) $ n) {0..n}) {0..n}"  (is "?l = ?r")
-  unfolding product_composition_lemma[OF c0 d0]
-  unfolding sum.cartesian_product
-  apply (rule sum.mono_neutral_left)
-  apply simp
-  apply (clarsimp simp add: subset_eq)
-  apply clarsimp
-  apply (rule ccontr)
-  apply (subgoal_tac "(c^aa * d^ba) $ n = 0")
-  apply simp
-  unfolding fps_mult_nth
-  apply (rule sum.neutral)
-  apply (clarsimp simp add: not_le)
-  apply (case_tac "x < aa")
-  apply (rule startsby_zero_power_prefix[OF c0, rule_format])
-  apply simp
-  apply (subgoal_tac "n - x < ba")
-  apply (frule_tac k = "ba" in startsby_zero_power_prefix[OF d0, rule_format])
-  apply simp
-  apply arith
-  done
-
-
 lemma sum_pair_less_iff:
   "sum (\<lambda>((k::nat),m). a k * b m * c (k + m)) {(k,m). k + m \<le> n} =
     sum (\<lambda>s. sum (\<lambda>i. a i * b (s - i) * c s) {0..s}) {0..n}"
   (is "?l = ?r")
 proof -
-  let ?KM = "{(k,m). k + m \<le> n}"
-  let ?f = "\<lambda>s. \<Union>i\<in>{0..s}. {(i, s - i)}"
-  have th0: "?KM = \<Union> (?f ` {0..n})"
+  have th0: "{(k, m). k + m \<le> n} = (\<Union>s\<in>{0..n}. \<Union>i\<in>{0..s}. {(i, s - i)})"
     by auto
-  show "?l = ?r "
+  show "?l = ?r"
     unfolding th0
-    apply (subst sum.UNION_disjoint)
-    apply auto
-    apply (subst sum.UNION_disjoint)
-    apply auto
-    done
+    by (simp add: sum.UNION_disjoint eq_diff_iff disjoint_iff)
 qed
 
 lemma fps_compose_mult_distrib_lemma:
   assumes c0: "c$0 = (0::'a::idom)"
   shows "((a oo c) * (b oo c))$n = sum (\<lambda>s. sum (\<lambda>i. a$i * b$(s - i) * (c^s) $ n) {0..s}) {0..n}"
   unfolding product_composition_lemma[OF c0 c0] power_add[symmetric]
   unfolding sum_pair_less_iff[where a = "\<lambda>k. a$k" and b="\<lambda>m. b$m" and c="\<lambda>s. (c ^ s)$n" and n = n] ..
 
 lemma fps_compose_mult_distrib:
   assumes c0: "c $ 0 = (0::'a::idom)"
   shows "(a * b) oo c = (a oo c) * (b oo c)"
-  apply (simp add: fps_eq_iff fps_compose_mult_distrib_lemma [OF c0])
-  apply (simp add: fps_compose_nth fps_mult_nth sum_distrib_right)
-  done
+proof (clarsimp simp add: fps_eq_iff fps_compose_mult_distrib_lemma [OF c0])
+  show "(a * b oo c) $ n = (\<Sum>s = 0..n. \<Sum>i = 0..s. a $ i * b $ (s - i) * c ^ s $ n)" for n
+    by (simp add: fps_compose_nth fps_mult_nth sum_distrib_right)
+qed
+
 
 lemma fps_compose_prod_distrib:
   assumes c0: "c$0 = (0::'a::idom)"
   shows "prod a S oo c = prod (\<lambda>k. a k oo c) S"
-  apply (cases "finite S")
-  apply simp_all
-  apply (induct S rule: finite_induct)
-  apply simp
-  apply (simp add: fps_compose_mult_distrib[OF c0])
-  done
+proof (induct S rule: infinite_finite_induct)
+next
+  case (insert)
+  then show ?case   
+    by (simp add: fps_compose_mult_distrib[OF c0])
+qed auto
 
 lemma fps_compose_divide:
   assumes [simp]: "g dvd f" "h $ 0 = 0"
   shows   "fps_compose f h = fps_compose (f / g :: 'a :: field fps) h * fps_compose g h"
 proof -
   have "f = (f / g) * g" by simp
   also have "fps_compose \<dots> h = fps_compose (f / g) h * fps_compose g h"
     by (subst fps_compose_mult_distrib) simp_all
   finally show ?thesis .
 qed
 
 lemma fps_compose_divide_distrib:
   assumes "g dvd f" "h $ 0 = 0" "fps_compose g h \<noteq> 0"
   shows   "fps_compose (f / g :: 'a :: field fps) h = fps_compose f h / fps_compose g h"
   using fps_compose_divide[OF assms(1,2)] assms(3) by simp
 
 lemma fps_compose_power:
   assumes c0: "c$0 = (0::'a::idom)"
   shows "(a oo c)^n = a^n oo c"
 proof (cases n)
   case 0
   then show ?thesis by simp
 next
   case (Suc m)
   have "(\<Prod>n = 0..m. a) oo c = (\<Prod>n = 0..m. a oo c)"
     using c0 fps_compose_prod_distrib by blast
   moreover have th0: "a^n = prod (\<lambda>k. a) {0..m}" "(a oo c) ^ n = prod (\<lambda>k. a oo c) {0..m}"
     by (simp_all add: prod_constant Suc)
   ultimately show ?thesis
     by presburger
 qed
 
 lemma fps_compose_uminus: "- (a::'a::ring_1 fps) oo c = - (a oo c)"
   by (simp add: fps_eq_iff fps_compose_nth field_simps sum_negf[symmetric])
     
 lemma fps_compose_sub_distrib: "(a - b) oo (c::'a::ring_1 fps) = (a oo c) - (b oo c)"
   using fps_compose_add_distrib [of a "- b" c] by (simp add: fps_compose_uminus)
 
 lemma fps_X_fps_compose: "fps_X oo a = Abs_fps (\<lambda>n. if n = 0 then (0::'a::comm_ring_1) else a$n)"
   by (simp add: fps_eq_iff fps_compose_nth mult_delta_left)
 
 lemma fps_inverse_compose:
   assumes b0: "(b$0 :: 'a::field) = 0"
     and a0: "a$0 \<noteq> 0"
   shows "inverse a oo b = inverse (a oo b)"
 proof -
   let ?ia = "inverse a"
   let ?ab = "a oo b"
   let ?iab = "inverse ?ab"
 
   from a0 have ia0: "?ia $ 0 \<noteq> 0" by simp
   from a0 have ab0: "?ab $ 0 \<noteq> 0" by (simp add: fps_compose_def)
   have "(?ia oo b) *  (a oo b) = 1"
     unfolding fps_compose_mult_distrib[OF b0, symmetric]
     unfolding inverse_mult_eq_1[OF a0]
     fps_compose_1 ..
 
   then have "(?ia oo b) *  (a oo b) * ?iab  = 1 * ?iab" by simp
   then have "(?ia oo b) *  (?iab * (a oo b))  = ?iab" by simp
   then show ?thesis unfolding inverse_mult_eq_1[OF ab0] by simp
 qed
 
 lemma fps_divide_compose:
   assumes c0: "(c$0 :: 'a::field) = 0"
     and b0: "b$0 \<noteq> 0"
   shows "(a/b) oo c = (a oo c) / (b oo c)"
     using b0 c0 by (simp add: fps_divide_unit fps_inverse_compose fps_compose_mult_distrib)
 
 lemma gp:
   assumes a0: "a$0 = (0::'a::field)"
   shows "(Abs_fps (\<lambda>n. 1)) oo a = 1/(1 - a)"
     (is "?one oo a = _")
 proof -
   have o0: "?one $ 0 \<noteq> 0" by simp
   have th0: "(1 - fps_X) $ 0 \<noteq> (0::'a)" by simp
   from fps_inverse_gp[where ?'a = 'a]
   have "inverse ?one = 1 - fps_X" by (simp add: fps_eq_iff)
   then have "inverse (inverse ?one) = inverse (1 - fps_X)" by simp
   then have th: "?one = 1/(1 - fps_X)" unfolding fps_inverse_idempotent[OF o0]
     by (simp add: fps_divide_def)
   show ?thesis
     unfolding th
     unfolding fps_divide_compose[OF a0 th0]
     fps_compose_1 fps_compose_sub_distrib fps_X_fps_compose_startby0[OF a0] ..
 qed
 
 lemma fps_compose_radical:
   assumes b0: "b$0 = (0::'a::field_char_0)"
     and ra0: "r (Suc k) (a$0) ^ Suc k = a$0"
     and a0: "a$0 \<noteq> 0"
   shows "fps_radical r (Suc k)  a oo b = fps_radical r (Suc k) (a oo b)"
 proof -
   let ?r = "fps_radical r (Suc k)"
   let ?ab = "a oo b"
   have ab0: "?ab $ 0 = a$0"
     by (simp add: fps_compose_def)
   from ab0 a0 ra0 have rab0: "?ab $ 0 \<noteq> 0" "r (Suc k) (?ab $ 0) ^ Suc k = ?ab $ 0"
     by simp_all
   have th00: "r (Suc k) ((a oo b) $ 0) = (fps_radical r (Suc k) a oo b) $ 0"
     by (simp add: ab0 fps_compose_def)
   have th0: "(?r a oo b) ^ (Suc k) = a  oo b"
     unfolding fps_compose_power[OF b0]
     unfolding iffD1[OF power_radical[of a r k], OF a0 ra0]  ..
   from iffD1[OF radical_unique[where r=r and k=k and b= ?ab and a = "?r a oo b", OF rab0(2) th00 rab0(1)], OF th0]
   show ?thesis  .
 qed
 
 lemma fps_const_mult_apply_left: "fps_const c * (a oo b) = (fps_const c * a) oo b"
   by (simp add: fps_eq_iff fps_compose_nth sum_distrib_left mult.assoc)
 
 lemma fps_const_mult_apply_right:
   "(a oo b) * fps_const (c::'a::comm_semiring_1) = (fps_const c * a) oo b"
   by (simp add: fps_const_mult_apply_left mult.commute)
 
 lemma fps_compose_assoc:
   assumes c0: "c$0 = (0::'a::idom)"
     and b0: "b$0 = 0"
   shows "a oo (b oo c) = a oo b oo c" (is "?l = ?r")
 proof -
   have "?l$n = ?r$n" for n
   proof -
     have "?l$n = (sum (\<lambda>i. (fps_const (a$i) * b^i) oo c) {0..n})$n"
       by (simp add: fps_compose_nth fps_compose_power[OF c0] fps_const_mult_apply_left
         sum_distrib_left mult.assoc fps_sum_nth)
     also have "\<dots> = ((sum (\<lambda>i. fps_const (a$i) * b^i) {0..n}) oo c)$n"
       by (simp add: fps_compose_sum_distrib)
+    also have "... = (\<Sum>i = 0..n. \<Sum>j = 0..n. a $ j * (b ^ j $ i * c ^ i $ n))"
+      by (simp add: fps_compose_nth fps_sum_nth sum_distrib_right mult.assoc)
+    also have "... = (\<Sum>i = 0..n. \<Sum>j = 0..i. a $ j * (b ^ j $ i * c ^ i $ n))"
+      by (intro sum.cong [OF refl] sum.mono_neutral_right; simp add: b0 startsby_zero_power_prefix)
     also have "\<dots> = ?r$n"
-      apply (simp add: fps_compose_nth fps_sum_nth sum_distrib_right mult.assoc)
-      apply (rule sum.cong)
-      apply (rule refl)
-      apply (rule sum.mono_neutral_right)
-      apply (auto simp add: not_le)
-      apply (erule startsby_zero_power_prefix[OF b0, rule_format])
-      done
+      by (simp add: fps_compose_nth  sum_distrib_right mult.assoc)
     finally show ?thesis .
   qed
   then show ?thesis
     by (simp add: fps_eq_iff)
 qed
 
 
 lemma fps_X_power_compose:
   assumes a0: "a$0=0"
   shows "fps_X^k oo a = (a::'a::idom fps)^k"
   (is "?l = ?r")
 proof (cases k)
   case 0
   then show ?thesis by simp
 next
   case (Suc h)
   have "?l $ n = ?r $n" for n
   proof -
     consider "k > n" | "k \<le> n" by arith
     then show ?thesis
     proof cases
       case 1
       then show ?thesis
         using a0 startsby_zero_power_prefix[OF a0] Suc
         by (simp add: fps_compose_nth del: power_Suc)
     next
       case 2
       then show ?thesis
         by (simp add: fps_compose_nth mult_delta_left)
     qed
   qed
   then show ?thesis
     unfolding fps_eq_iff by blast
 qed
 
 lemma fps_inv_right:
   assumes a0: "a$0 = 0"
     and a1: "a$1 \<noteq> 0"
   shows "a oo fps_inv a = fps_X"
 proof -
   let ?ia = "fps_inv a"
   let ?iaa = "a oo fps_inv a"
   have th0: "?ia $ 0 = 0"
     by (simp add: fps_inv_def)
   have th1: "?iaa $ 0 = 0"
     using a0 a1 by (simp add: fps_inv_def fps_compose_nth)
   have th2: "fps_X$0 = 0"
     by simp
   from fps_inv[OF a0 a1] have "a oo (fps_inv a oo a) = a oo fps_X"
     by simp
   then have "(a oo fps_inv a) oo a = fps_X oo a"
     by (simp add: fps_compose_assoc[OF a0 th0] fps_X_fps_compose_startby0[OF a0])
   with fps_compose_inj_right[OF a0 a1] show ?thesis
     by simp
 qed
 
 lemma fps_inv_deriv:
   assumes a0: "a$0 = (0::'a::field)"
     and a1: "a$1 \<noteq> 0"
   shows "fps_deriv (fps_inv a) = inverse (fps_deriv a oo fps_inv a)"
 proof -
   let ?ia = "fps_inv a"
   let ?d = "fps_deriv a oo ?ia"
   let ?dia = "fps_deriv ?ia"
   have ia0: "?ia$0 = 0"
     by (simp add: fps_inv_def)
   have th0: "?d$0 \<noteq> 0"
     using a1 by (simp add: fps_compose_nth)
   from fps_inv_right[OF a0 a1] have "?d * ?dia = 1"
     by (simp add: fps_compose_deriv[OF ia0, of a, symmetric] )
   then have "inverse ?d * ?d * ?dia = inverse ?d * 1"
     by simp
   with inverse_mult_eq_1 [OF th0] show "?dia = inverse ?d"
     by simp
 qed
 
 lemma fps_inv_idempotent:
   assumes a0: "a$0 = 0"
     and a1: "a$1 \<noteq> 0"
   shows "fps_inv (fps_inv a) = a"
 proof -
   let ?r = "fps_inv"
   have ra0: "?r a $ 0 = 0"
     by (simp add: fps_inv_def)
   from a1 have ra1: "?r a $ 1 \<noteq> 0"
     by (simp add: fps_inv_def field_simps)
   have fps_X0: "fps_X$0 = 0"
     by simp
   from fps_inv[OF ra0 ra1] have "?r (?r a) oo ?r a = fps_X" .
   then have "?r (?r a) oo ?r a oo a = fps_X oo a"
     by simp
   then have "?r (?r a) oo (?r a oo a) = a"
     unfolding fps_X_fps_compose_startby0[OF a0]
     unfolding fps_compose_assoc[OF a0 ra0, symmetric] .
   then show ?thesis
     unfolding fps_inv[OF a0 a1] by simp
 qed
 
 lemma fps_ginv_ginv:
   assumes a0: "a$0 = 0"
     and a1: "a$1 \<noteq> 0"
     and c0: "c$0 = 0"
     and  c1: "c$1 \<noteq> 0"
   shows "fps_ginv b (fps_ginv c a) = b oo a oo fps_inv c"
 proof -
   let ?r = "fps_ginv"
   from c0 have rca0: "?r c a $0 = 0"
     by (simp add: fps_ginv_def)
   from a1 c1 have rca1: "?r c a $ 1 \<noteq> 0"
     by (simp add: fps_ginv_def field_simps)
   from fps_ginv[OF rca0 rca1]
   have "?r b (?r c a) oo ?r c a = b" .
   then have "?r b (?r c a) oo ?r c a oo a = b oo a"
     by simp
   then have "?r b (?r c a) oo (?r c a oo a) = b oo a"
-    apply (subst fps_compose_assoc)
-    using a0 c0
-    apply (simp_all add: fps_ginv_def)
-    done
+    by (simp add: a0 fps_compose_assoc rca0)
   then have "?r b (?r c a) oo c = b oo a"
     unfolding fps_ginv[OF a0 a1] .
   then have "?r b (?r c a) oo c oo fps_inv c= b oo a oo fps_inv c"
     by simp
   then have "?r b (?r c a) oo (c oo fps_inv c) = b oo a oo fps_inv c"
-    apply (subst fps_compose_assoc)
-    using a0 c0
-    apply (simp_all add: fps_inv_def)
-    done
+    by (metis c0 c1 fps_compose_assoc fps_compose_nth_0 fps_inv fps_inv_right)
   then show ?thesis
     unfolding fps_inv_right[OF c0 c1] by simp
 qed
 
 lemma fps_ginv_deriv:
   assumes a0:"a$0 = (0::'a::field)"
     and a1: "a$1 \<noteq> 0"
   shows "fps_deriv (fps_ginv b a) = (fps_deriv b / fps_deriv a) oo fps_ginv fps_X a"
 proof -
   let ?ia = "fps_ginv b a"
   let ?ifps_Xa = "fps_ginv fps_X a"
   let ?d = "fps_deriv"
   let ?dia = "?d ?ia"
   have ifps_Xa0: "?ifps_Xa $ 0 = 0"
     by (simp add: fps_ginv_def)
   have da0: "?d a $ 0 \<noteq> 0"
     using a1 by simp
   from fps_ginv[OF a0 a1, of b] have "?d (?ia oo a) = fps_deriv b"
     by simp
   then have "(?d ?ia oo a) * ?d a = ?d b"
     unfolding fps_compose_deriv[OF a0] .
   then have "(?d ?ia oo a) * ?d a * inverse (?d a) = ?d b * inverse (?d a)"
     by simp
   with a1 have "(?d ?ia oo a) * (inverse (?d a) * ?d a) = ?d b / ?d a"
     by (simp add: fps_divide_unit)
   then have "(?d ?ia oo a) oo ?ifps_Xa =  (?d b / ?d a) oo ?ifps_Xa"
     unfolding inverse_mult_eq_1[OF da0] by simp
   then have "?d ?ia oo (a oo ?ifps_Xa) =  (?d b / ?d a) oo ?ifps_Xa"
     unfolding fps_compose_assoc[OF ifps_Xa0 a0] .
   then show ?thesis unfolding fps_inv_ginv[symmetric]
     unfolding fps_inv_right[OF a0 a1] by simp
 qed
 
 lemma fps_compose_linear:
   "fps_compose (f :: 'a :: comm_ring_1 fps) (fps_const c * fps_X) = Abs_fps (\<lambda>n. c^n * f $ n)"
   by (simp add: fps_eq_iff fps_compose_def power_mult_distrib
                 if_distrib cong: if_cong)
               
 lemma fps_compose_uminus': 
   "fps_compose f (-fps_X :: 'a :: comm_ring_1 fps) = Abs_fps (\<lambda>n. (-1)^n * f $ n)"
   using fps_compose_linear[of f "-1"] 
   by (simp only: fps_const_neg [symmetric] fps_const_1_eq_1) simp
 
 subsection \<open>Elementary series\<close>
 
 subsubsection \<open>Exponential series\<close>
 
 definition "fps_exp x = Abs_fps (\<lambda>n. x^n / of_nat (fact n))"
 
 lemma fps_exp_deriv[simp]: "fps_deriv (fps_exp a) = fps_const (a::'a::field_char_0) * fps_exp a" 
   (is "?l = ?r")
 proof -
   have "?l$n = ?r $ n" for n
-    apply (auto simp add: fps_exp_def field_simps power_Suc[symmetric]
-      simp del: fact_Suc of_nat_Suc power_Suc)
-    apply (simp add: field_simps)
-    done
+    using of_nat_neq_0 by (auto simp add: fps_exp_def divide_simps)
   then show ?thesis
     by (simp add: fps_eq_iff)
 qed
 
 lemma fps_exp_unique_ODE:
   "fps_deriv a = fps_const c * a \<longleftrightarrow> a = fps_const (a$0) * fps_exp (c::'a::field_char_0)"
   (is "?lhs \<longleftrightarrow> ?rhs")
 proof
   show ?rhs if ?lhs
   proof -
     from that have th: "\<And>n. a $ Suc n = c * a$n / of_nat (Suc n)"
       by (simp add: fps_deriv_def fps_eq_iff field_simps del: of_nat_Suc)
     have th': "a$n = a$0 * c ^ n/ (fact n)" for n
     proof (induct n)
       case 0
       then show ?case by simp
     next
       case Suc
       then show ?case
-        unfolding th
-        using fact_gt_zero
-        apply (simp add: field_simps del: of_nat_Suc fact_Suc)
-        apply simp
-        done
+        by (simp add: th divide_simps)
     qed
     show ?thesis
       by (auto simp add: fps_eq_iff fps_const_mult_left fps_exp_def intro: th')
   qed
   show ?lhs if ?rhs
     using that by (metis fps_exp_deriv fps_deriv_mult_const_left mult.left_commute)
 qed
 
 lemma fps_exp_add_mult: "fps_exp (a + b) = fps_exp (a::'a::field_char_0) * fps_exp b" (is "?l = ?r")
 proof -
   have "fps_deriv ?r = fps_const (a + b) * ?r"
     by (simp add: fps_const_add[symmetric] field_simps del: fps_const_add)
   then have "?r = ?l"
     by (simp only: fps_exp_unique_ODE) (simp add: fps_mult_nth fps_exp_def)
   then show ?thesis ..
 qed
 
 lemma fps_exp_nth[simp]: "fps_exp a $ n = a^n / of_nat (fact n)"
   by (simp add: fps_exp_def)
 
 lemma fps_exp_0[simp]: "fps_exp (0::'a::field) = 1"
   by (simp add: fps_eq_iff power_0_left)
 
 lemma fps_exp_neg: "fps_exp (- a) = inverse (fps_exp (a::'a::field_char_0))"
 proof -
   from fps_exp_add_mult[of a "- a"] have th0: "fps_exp a * fps_exp (- a) = 1" by simp
   from fps_inverse_unique[OF th0] show ?thesis by simp
 qed
 
 lemma fps_exp_nth_deriv[simp]: 
   "fps_nth_deriv n (fps_exp (a::'a::field_char_0)) = (fps_const a)^n * (fps_exp a)"
   by (induct n) auto
 
 lemma fps_X_compose_fps_exp[simp]: "fps_X oo fps_exp (a::'a::field) = fps_exp a - 1"
   by (simp add: fps_eq_iff fps_X_fps_compose)
 
 lemma fps_inv_fps_exp_compose:
   assumes a: "a \<noteq> 0"
   shows "fps_inv (fps_exp a - 1) oo (fps_exp a - 1) = fps_X"
     and "(fps_exp a - 1) oo fps_inv (fps_exp a - 1) = fps_X"
 proof -
   let ?b = "fps_exp a - 1"
   have b0: "?b $ 0 = 0"
     by simp
   have b1: "?b $ 1 \<noteq> 0"
     by (simp add: a)
   from fps_inv[OF b0 b1] show "fps_inv (fps_exp a - 1) oo (fps_exp a - 1) = fps_X" .
   from fps_inv_right[OF b0 b1] show "(fps_exp a - 1) oo fps_inv (fps_exp a - 1) = fps_X" .
 qed
 
 lemma fps_exp_power_mult: "(fps_exp (c::'a::field_char_0))^n = fps_exp (of_nat n * c)"
   by (induct n) (simp_all add: field_simps fps_exp_add_mult)
 
 lemma radical_fps_exp:
   assumes r: "r (Suc k) 1 = 1"
   shows "fps_radical r (Suc k) (fps_exp (c::'a::field_char_0)) = fps_exp (c / of_nat (Suc k))"
 proof -
   let ?ck = "(c / of_nat (Suc k))"
   let ?r = "fps_radical r (Suc k)"
   have eq0[simp]: "?ck * of_nat (Suc k) = c" "of_nat (Suc k) * ?ck = c"
     by (simp_all del: of_nat_Suc)
   have th0: "fps_exp ?ck ^ (Suc k) = fps_exp c" unfolding fps_exp_power_mult eq0 ..
   have th: "r (Suc k) (fps_exp c $0) ^ Suc k = fps_exp c $ 0"
     "r (Suc k) (fps_exp c $ 0) = fps_exp ?ck $ 0" "fps_exp c $ 0 \<noteq> 0" using r by simp_all
   from th0 radical_unique[where r=r and k=k, OF th] show ?thesis
     by auto
 qed
 
 lemma fps_exp_compose_linear [simp]: 
   "fps_exp (d::'a::field_char_0) oo (fps_const c * fps_X) = fps_exp (c * d)"
   by (simp add: fps_compose_linear fps_exp_def fps_eq_iff power_mult_distrib)
   
 lemma fps_fps_exp_compose_minus [simp]: 
   "fps_compose (fps_exp c) (-fps_X) = fps_exp (-c :: 'a :: field_char_0)"
   using fps_exp_compose_linear[of c "-1 :: 'a"] 
   unfolding fps_const_neg [symmetric] fps_const_1_eq_1 by simp
 
 lemma fps_exp_eq_iff [simp]: "fps_exp c = fps_exp d \<longleftrightarrow> c = (d :: 'a :: field_char_0)"
 proof
   assume "fps_exp c = fps_exp d"
   from arg_cong[of _ _ "\<lambda>F. F $ 1", OF this] show "c = d" by simp
 qed simp_all
 
 lemma fps_exp_eq_fps_const_iff [simp]: 
   "fps_exp (c :: 'a :: field_char_0) = fps_const c' \<longleftrightarrow> c = 0 \<and> c' = 1"
 proof
   assume "c = 0 \<and> c' = 1"
   thus "fps_exp c = fps_const c'" by (simp add: fps_eq_iff)
 next
   assume "fps_exp c = fps_const c'"
   from arg_cong[of _ _ "\<lambda>F. F $ 1", OF this] arg_cong[of _ _ "\<lambda>F. F $ 0", OF this] 
     show "c = 0 \<and> c' = 1" by simp_all
 qed
 
 lemma fps_exp_neq_0 [simp]: "\<not>fps_exp (c :: 'a :: field_char_0) = 0"
   unfolding fps_const_0_eq_0 [symmetric] fps_exp_eq_fps_const_iff by simp  
 
 lemma fps_exp_eq_1_iff [simp]: "fps_exp (c :: 'a :: field_char_0) = 1 \<longleftrightarrow> c = 0"
   unfolding fps_const_1_eq_1 [symmetric] fps_exp_eq_fps_const_iff by simp
     
 lemma fps_exp_neq_numeral_iff [simp]: 
   "fps_exp (c :: 'a :: field_char_0) = numeral n \<longleftrightarrow> c = 0 \<and> n = Num.One"
   unfolding numeral_fps_const fps_exp_eq_fps_const_iff by simp
 
 
 subsubsection \<open>Logarithmic series\<close>
 
 lemma Abs_fps_if_0:
   "Abs_fps (\<lambda>n. if n = 0 then (v::'a::ring_1) else f n) =
     fps_const v + fps_X * Abs_fps (\<lambda>n. f (Suc n))"
   by (simp add: fps_eq_iff)
 
 definition fps_ln :: "'a::field_char_0 \<Rightarrow> 'a fps"
   where "fps_ln c = fps_const (1/c) * Abs_fps (\<lambda>n. if n = 0 then 0 else (- 1) ^ (n - 1) / of_nat n)"
 
 lemma fps_ln_deriv: "fps_deriv (fps_ln c) = fps_const (1/c) * inverse (1 + fps_X)"
   unfolding fps_inverse_fps_X_plus1
   by (simp add: fps_ln_def fps_eq_iff del: of_nat_Suc)
 
 lemma fps_ln_nth: "fps_ln c $ n = (if n = 0 then 0 else 1/c * ((- 1) ^ (n - 1) / of_nat n))"
   by (simp add: fps_ln_def field_simps)
 
 lemma fps_ln_0 [simp]: "fps_ln c $ 0 = 0" by (simp add: fps_ln_def)
 
 lemma fps_ln_fps_exp_inv:
   fixes a :: "'a::field_char_0"
   assumes a: "a \<noteq> 0"
   shows "fps_ln a = fps_inv (fps_exp a - 1)"  (is "?l = ?r")
 proof -
   let ?b = "fps_exp a - 1"
   have b0: "?b $ 0 = 0" by simp
   have b1: "?b $ 1 \<noteq> 0" by (simp add: a)
   have "fps_deriv (fps_exp a - 1) oo fps_inv (fps_exp a - 1) =
     (fps_const a * (fps_exp a - 1) + fps_const a) oo fps_inv (fps_exp a - 1)"
     by (simp add: field_simps)
   also have "\<dots> = fps_const a * (fps_X + 1)"
-    apply (simp add: fps_compose_add_distrib fps_const_mult_apply_left[symmetric] fps_inv_right[OF b0 b1])
-    apply (simp add: field_simps)
-    done
+    by (simp add: fps_compose_add_distrib fps_inv_right[OF b0 b1] distrib_left flip: fps_const_mult_apply_left)
   finally have eq: "fps_deriv (fps_exp a - 1) oo fps_inv (fps_exp a - 1) = fps_const a * (fps_X + 1)" .
   from fps_inv_deriv[OF b0 b1, unfolded eq]
   have "fps_deriv (fps_inv ?b) = fps_const (inverse a) / (fps_X + 1)"
     using a by (simp add: fps_const_inverse eq fps_divide_def fps_inverse_mult)
   then have "fps_deriv ?l = fps_deriv ?r"
     by (simp add: fps_ln_deriv add.commute fps_divide_def divide_inverse)
   then show ?thesis unfolding fps_deriv_eq_iff
     by (simp add: fps_ln_nth fps_inv_def)
 qed
 
 lemma fps_ln_mult_add:
   assumes c0: "c\<noteq>0"
     and d0: "d\<noteq>0"
   shows "fps_ln c + fps_ln d = fps_const (c+d) * fps_ln (c*d)"
   (is "?r = ?l")
 proof-
   from c0 d0 have eq: "1/c + 1/d = (c+d)/(c*d)" by (simp add: field_simps)
   have "fps_deriv ?r = fps_const (1/c + 1/d) * inverse (1 + fps_X)"
     by (simp add: fps_ln_deriv fps_const_add[symmetric] algebra_simps del: fps_const_add)
   also have "\<dots> = fps_deriv ?l"
-    apply (simp add: fps_ln_deriv)
-    apply (simp add: fps_eq_iff eq)
-    done
+    by (simp add: eq fps_ln_deriv)
   finally show ?thesis
     unfolding fps_deriv_eq_iff by simp
 qed
 
 lemma fps_X_dvd_fps_ln [simp]: "fps_X dvd fps_ln c"
 proof -
   have "fps_ln c = fps_X * Abs_fps (\<lambda>n. (-1) ^ n / (of_nat (Suc n) * c))"
     by (intro fps_ext) (simp add: fps_ln_def of_nat_diff)
   thus ?thesis by simp
 qed
 
 
 subsubsection \<open>Binomial series\<close>
 
 definition "fps_binomial a = Abs_fps (\<lambda>n. a gchoose n)"
 
 lemma fps_binomial_nth[simp]: "fps_binomial a $ n = a gchoose n"
   by (simp add: fps_binomial_def)
 
 lemma fps_binomial_ODE_unique:
   fixes c :: "'a::field_char_0"
   shows "fps_deriv a = (fps_const c * a) / (1 + fps_X) \<longleftrightarrow> a = fps_const (a$0) * fps_binomial c"
   (is "?lhs \<longleftrightarrow> ?rhs")
 proof
   let ?da = "fps_deriv a"
   let ?x1 = "(1 + fps_X):: 'a fps"
   let ?l = "?x1 * ?da"
   let ?r = "fps_const c * a"
 
   have eq: "?l = ?r \<longleftrightarrow> ?lhs"
   proof -
     have x10: "?x1 $ 0 \<noteq> 0" by simp
     have "?l = ?r \<longleftrightarrow> inverse ?x1 * ?l = inverse ?x1 * ?r" by simp
     also have "\<dots> \<longleftrightarrow> ?da = (fps_const c * a) / ?x1"
-      apply (simp only: fps_divide_def  mult.assoc[symmetric] inverse_mult_eq_1[OF x10])
-      apply (simp add: field_simps)
-      done
+      unfolding fps_divide_def  mult.assoc[symmetric] inverse_mult_eq_1[OF x10]
+      by (simp add: field_simps)
     finally show ?thesis .
   qed
 
   show ?rhs if ?lhs
   proof -
     from eq that have h: "?l = ?r" ..
     have th0: "a$ Suc n = ((c - of_nat n) / of_nat (Suc n)) * a $n" for n
     proof -
       from h have "?l $ n = ?r $ n" by simp
       then show ?thesis
-        apply (simp add: field_simps del: of_nat_Suc)
-        apply (cases n)
-        apply (simp_all add: field_simps del: of_nat_Suc)
-        done
+        by (simp add: field_simps del: of_nat_Suc split: if_split_asm)
     qed
     have th1: "a $ n = (c gchoose n) * a $ 0" for n
     proof (induct n)
       case 0
       then show ?case by simp
     next
       case (Suc m)
-      then show ?case
+      have "(c - of_nat m) * (c gchoose m) = (c gchoose Suc m) * of_nat (Suc m)"
+        by (metis gbinomial_absorb_comp gbinomial_absorption mult.commute)
+      with Suc show ?case
         unfolding th0
-        apply (simp add: field_simps del: of_nat_Suc)
-        unfolding mult.assoc[symmetric] gbinomial_mult_1
-        apply (simp add: field_simps)
-        done
+        by (simp add: divide_simps del: of_nat_Suc)
     qed
     show ?thesis
-      apply (simp add: fps_eq_iff)
-      apply (subst th1)
-      apply (simp add: field_simps)
-      done
+      by (metis expand_fps_eq fps_binomial_nth fps_mult_right_const_nth mult.commute th1)
   qed
 
   show ?lhs if ?rhs
   proof -
     have th00: "x * (a $ 0 * y) = a $ 0 * (x * y)" for x y
       by (simp add: mult.commute)
-    have "?l = ?r"
-      apply (subst \<open>?rhs\<close>)
-      apply (subst (2) \<open>?rhs\<close>)
-      apply (clarsimp simp add: fps_eq_iff field_simps)
-      unfolding mult.assoc[symmetric] th00 gbinomial_mult_1
-      apply (simp add: field_simps gbinomial_mult_1)
-      done
+    have "?l = (1 + fps_X) * fps_deriv (fps_const (a $ 0) * fps_binomial c)"
+      using that by auto
+    also have "... = fps_const c * (fps_const (a $ 0) * fps_binomial c)"
+    proof (clarsimp simp add: fps_eq_iff algebra_simps)
+      show "a $ 0 * (c gchoose Suc n) + (of_nat n * ((c gchoose n) * a $ 0) + of_nat n * (a $ 0 * (c gchoose Suc n))) 
+         = c * ((c gchoose n) * a $ 0)" for n
+      unfolding mult.assoc[symmetric]  
+      by (simp add: field_simps gbinomial_mult_1)
+  qed
+    also have "... = ?r"
+      using that by auto
+    finally have "?l = ?r" .
     with eq show ?thesis ..
   qed
 qed
 
 lemma fps_binomial_ODE_unique':
   "(fps_deriv a = fps_const c * a / (1 + fps_X) \<and> a $ 0 = 1) \<longleftrightarrow> (a = fps_binomial c)"
   by (subst fps_binomial_ODE_unique) auto
 
 lemma fps_binomial_deriv: "fps_deriv (fps_binomial c) = fps_const c * fps_binomial c / (1 + fps_X)"
 proof -
   let ?a = "fps_binomial c"
   have th0: "?a = fps_const (?a$0) * ?a" by (simp)
   from iffD2[OF fps_binomial_ODE_unique, OF th0] show ?thesis .
 qed
 
 lemma fps_binomial_add_mult: "fps_binomial (c+d) = fps_binomial c * fps_binomial d" (is "?l = ?r")
 proof -
   let ?P = "?r - ?l"
   let ?b = "fps_binomial"
   let ?db = "\<lambda>x. fps_deriv (?b x)"
   have "fps_deriv ?P = ?db c * ?b d + ?b c * ?db d - ?db (c + d)"  by simp
   also have "\<dots> = inverse (1 + fps_X) *
       (fps_const c * ?b c * ?b d + fps_const d * ?b c * ?b d - fps_const (c+d) * ?b (c + d))"
     unfolding fps_binomial_deriv
     by (simp add: fps_divide_def field_simps)
   also have "\<dots> = (fps_const (c + d)/ (1 + fps_X)) * ?P"
     by (simp add: field_simps fps_divide_unit fps_const_add[symmetric] del: fps_const_add)
   finally have th0: "fps_deriv ?P = fps_const (c+d) * ?P / (1 + fps_X)"
     by (simp add: fps_divide_def)
   have "?P = fps_const (?P$0) * ?b (c + d)"
     unfolding fps_binomial_ODE_unique[symmetric]
     using th0 by simp
   then have "?P = 0" by (simp add: fps_mult_nth)
   then show ?thesis by simp
 qed
 
 lemma fps_binomial_minus_one: "fps_binomial (- 1) = inverse (1 + fps_X)"
   (is "?l = inverse ?r")
 proof-
   have th: "?r$0 \<noteq> 0" by simp
   have th': "fps_deriv (inverse ?r) = fps_const (- 1) * inverse ?r / (1 + fps_X)"
     by (simp add: fps_inverse_deriv[OF th] fps_divide_def
       power2_eq_square mult.commute fps_const_neg[symmetric] del: fps_const_neg)
   have eq: "inverse ?r $ 0 = 1"
     by (simp add: fps_inverse_def)
   from iffD1[OF fps_binomial_ODE_unique[of "inverse (1 + fps_X)" "- 1"] th'] eq
   show ?thesis by (simp add: fps_inverse_def)
 qed
 
 lemma fps_binomial_of_nat: "fps_binomial (of_nat n) = (1 + fps_X :: 'a :: field_char_0 fps) ^ n"
 proof (cases "n = 0")
   case [simp]: True
   have "fps_deriv ((1 + fps_X) ^ n :: 'a fps) = 0" by simp
   also have "\<dots> = fps_const (of_nat n) * (1 + fps_X) ^ n / (1 + fps_X)" by (simp add: fps_binomial_def)
   finally show ?thesis by (subst sym, subst fps_binomial_ODE_unique' [symmetric]) simp_all
 next
   case False
   have "fps_deriv ((1 + fps_X) ^ n :: 'a fps) = fps_const (of_nat n) * (1 + fps_X) ^ (n - 1)"
     by (simp add: fps_deriv_power)
   also have "(1 + fps_X :: 'a fps) $ 0 \<noteq> 0" by simp
   hence "(1 + fps_X :: 'a fps) \<noteq> 0" by (intro notI) (simp only: , simp)
   with False have "(1 + fps_X :: 'a fps) ^ (n - 1) = (1 + fps_X) ^ n / (1 + fps_X)"
     by (cases n) (simp_all )
   also have "fps_const (of_nat n :: 'a) * ((1 + fps_X) ^ n / (1 + fps_X)) =
                fps_const (of_nat n) * (1 + fps_X) ^ n / (1 + fps_X)"
     by (simp add: unit_div_mult_swap)
   finally show ?thesis
     by (subst sym, subst fps_binomial_ODE_unique' [symmetric]) (simp_all add: fps_power_nth)
 qed
 
 lemma fps_binomial_0 [simp]: "fps_binomial 0 = 1"
   using fps_binomial_of_nat[of 0] by simp
   
 lemma fps_binomial_power: "fps_binomial a ^ n = fps_binomial (of_nat n * a)"
   by (induction n) (simp_all add: fps_binomial_add_mult ring_distribs)
 
 lemma fps_binomial_1: "fps_binomial 1 = 1 + fps_X"
   using fps_binomial_of_nat[of 1] by simp
 
 lemma fps_binomial_minus_of_nat:
   "fps_binomial (- of_nat n) = inverse ((1 + fps_X :: 'a :: field_char_0 fps) ^ n)"
   by (rule sym, rule fps_inverse_unique)
      (simp add: fps_binomial_of_nat [symmetric] fps_binomial_add_mult [symmetric])
 
 lemma one_minus_const_fps_X_power:
   "c \<noteq> 0 \<Longrightarrow> (1 - fps_const c * fps_X) ^ n =
      fps_compose (fps_binomial (of_nat n)) (-fps_const c * fps_X)"
   by (subst fps_binomial_of_nat)
      (simp add: fps_compose_power [symmetric] fps_compose_add_distrib fps_const_neg [symmetric] 
            del: fps_const_neg)
 
 lemma one_minus_fps_X_const_neg_power:
   "inverse ((1 - fps_const c * fps_X) ^ n) = 
        fps_compose (fps_binomial (-of_nat n)) (-fps_const c * fps_X)"
 proof (cases "c = 0")
   case False
   thus ?thesis
   by (subst fps_binomial_minus_of_nat)
      (simp add: fps_compose_power [symmetric] fps_inverse_compose fps_compose_add_distrib
                 fps_const_neg [symmetric] del: fps_const_neg)
 qed simp
 
 lemma fps_X_plus_const_power:
   "c \<noteq> 0 \<Longrightarrow> (fps_X + fps_const c) ^ n =
      fps_const (c^n) * fps_compose (fps_binomial (of_nat n)) (fps_const (inverse c) * fps_X)"
   by (subst fps_binomial_of_nat)
      (simp add: fps_compose_power [symmetric] fps_binomial_of_nat fps_compose_add_distrib
                 fps_const_power [symmetric] power_mult_distrib [symmetric] 
                 algebra_simps inverse_mult_eq_1' del: fps_const_power)
 
 lemma fps_X_plus_const_neg_power:
   "c \<noteq> 0 \<Longrightarrow> inverse ((fps_X + fps_const c) ^ n) =
      fps_const (inverse c^n) * fps_compose (fps_binomial (-of_nat n)) (fps_const (inverse c) * fps_X)"
   by (subst fps_binomial_minus_of_nat)
      (simp add: fps_compose_power [symmetric] fps_binomial_of_nat fps_compose_add_distrib
                 fps_const_power [symmetric] power_mult_distrib [symmetric] fps_inverse_compose 
                 algebra_simps fps_const_inverse [symmetric] fps_inverse_mult [symmetric]
                 fps_inverse_power [symmetric] inverse_mult_eq_1'
            del: fps_const_power)
 
 
 lemma one_minus_const_fps_X_neg_power':
-  "n > 0 \<Longrightarrow> inverse ((1 - fps_const (c :: 'a :: field_char_0) * fps_X) ^ n) =
-       Abs_fps (\<lambda>k. of_nat ((n + k - 1) choose k) * c^k)"
-  apply (rule fps_ext)
-  apply (subst one_minus_fps_X_const_neg_power, subst fps_const_neg, subst fps_compose_linear)
-  apply (simp add: power_mult_distrib [symmetric] mult.assoc [symmetric] 
-                   gbinomial_minus binomial_gbinomial of_nat_diff)
-  done
+  fixes c :: "'a :: field_char_0"
+  assumes "n > 0"
+  shows "inverse ((1 - fps_const c * fps_X) ^ n) = Abs_fps (\<lambda>k. of_nat ((n + k - 1) choose k) * c^k)"
+proof -
+  have \<section>: "\<And>j. Abs_fps (\<lambda>na. (- c) ^ na * fps_binomial (- of_nat n) $ na) $ j =
+          Abs_fps (\<lambda>k. of_nat (n + k - 1 choose k) * c ^ k) $ j"
+    using assms
+    by (simp add: gbinomial_minus binomial_gbinomial of_nat_diff flip: power_mult_distrib mult.assoc)
+  show ?thesis
+    apply (rule fps_ext)
+    using \<section> 
+    by (metis (no_types, lifting) one_minus_fps_X_const_neg_power fps_const_neg fps_compose_linear fps_nth_Abs_fps)
+qed
 
 text \<open>Vandermonde's Identity as a consequence.\<close>
 lemma gbinomial_Vandermonde:
   "sum (\<lambda>k. (a gchoose k) * (b gchoose (n - k))) {0..n} = (a + b) gchoose n"
 proof -
   let ?ba = "fps_binomial a"
   let ?bb = "fps_binomial b"
   let ?bab = "fps_binomial (a + b)"
   from fps_binomial_add_mult[of a b] have "?bab $ n = (?ba * ?bb)$n" by simp
   then show ?thesis by (simp add: fps_mult_nth)
 qed
 
 lemma binomial_Vandermonde:
   "sum (\<lambda>k. (a choose k) * (b choose (n - k))) {0..n} = (a + b) choose n"
   using gbinomial_Vandermonde[of "(of_nat a)" "of_nat b" n]
   by (simp only: binomial_gbinomial[symmetric] of_nat_mult[symmetric]
                  of_nat_sum[symmetric] of_nat_add[symmetric] of_nat_eq_iff)
 
 lemma binomial_Vandermonde_same: "sum (\<lambda>k. (n choose k)\<^sup>2) {0..n} = (2 * n) choose n"
   using binomial_Vandermonde[of n n n, symmetric]
   unfolding mult_2
-  apply (simp add: power2_eq_square)
-  apply (rule sum.cong)
-  apply (auto intro:  binomial_symmetric)
-  done
+  by (metis atMost_atLeast0 choose_square_sum mult_2)
 
 lemma Vandermonde_pochhammer_lemma:
   fixes a :: "'a::field_char_0"
-  assumes b: "\<forall>j\<in>{0 ..<n}. b \<noteq> of_nat j"
+  assumes b: "\<And>j. j<n \<Longrightarrow> b \<noteq> of_nat j"
   shows "sum (\<lambda>k. (pochhammer (- a) k * pochhammer (- (of_nat n)) k) /
       (of_nat (fact k) * pochhammer (b - of_nat n + 1) k)) {0..n} =
     pochhammer (- (a + b)) n / pochhammer (- b) n"
   (is "?l = ?r")
 proof -
   let ?m1 = "\<lambda>m. (- 1 :: 'a) ^ m"
   let ?f = "\<lambda>m. of_nat (fact m)"
   let ?p = "\<lambda>(x::'a). pochhammer (- x)"
   from b have bn0: "?p b n \<noteq> 0"
     unfolding pochhammer_eq_0_iff by simp
   have th00:
     "b gchoose (n - k) =
         (?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k)"
       (is ?gchoose)
     "pochhammer (1 + b - of_nat n) k \<noteq> 0"
       (is ?pochhammer)
     if kn: "k \<in> {0..n}" for k
   proof -
     from kn have "k \<le> n" by simp
     have nz: "pochhammer (1 + b - of_nat n) n \<noteq> 0"
     proof
       assume "pochhammer (1 + b - of_nat n) n = 0"
       then have c: "pochhammer (b - of_nat n + 1) n = 0"
         by (simp add: algebra_simps)
       then obtain j where j: "j < n" "b - of_nat n + 1 = - of_nat j"
         unfolding pochhammer_eq_0_iff by blast
       from j have "b = of_nat n - of_nat j - of_nat 1"
         by (simp add: algebra_simps)
       then have "b = of_nat (n - j - 1)"
         using j kn by (simp add: of_nat_diff)
-      with b show False using j by auto
+      then show False 
+        using \<open>j < n\<close> j b by auto
     qed
 
     from nz kn [simplified] have nz': "pochhammer (1 + b - of_nat n) k \<noteq> 0"
       by (rule pochhammer_neq_0_mono)
 
     consider "k = 0 \<or> n = 0" | "k \<noteq> 0" "n \<noteq> 0"
       by blast
     then have "b gchoose (n - k) =
       (?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k)"
     proof cases
       case 1
       then show ?thesis
         using kn by (cases "k = 0") (simp_all add: gbinomial_pochhammer)
     next
       case neq: 2
       then obtain m where m: "n = Suc m"
         by (cases n) auto
       from neq(1) obtain h where h: "k = Suc h"
         by (cases k) auto
       show ?thesis
       proof (cases "k = n")
         case True
-        then show ?thesis
-          using pochhammer_minus'[where k=k and b=b]
-          apply (simp add: pochhammer_same)
-          using bn0
-          apply (simp add: field_simps power_add[symmetric])
-          done
+        with pochhammer_minus'[where k=k and b=b] bn0 show ?thesis       
+          by (simp add: pochhammer_same)
       next
         case False
         with kn have kn': "k < n"
           by simp
+        have "h \<le> m"
+          using \<open>k \<le> n\<close> h m by blast
         have m1nk: "?m1 n = prod (\<lambda>i. - 1) {..m}" "?m1 k = prod (\<lambda>i. - 1) {0..h}"
-          by (simp_all add: prod_constant m h)
+          by (simp_all add: m h)
         have bnz0: "pochhammer (b - of_nat n + 1) k \<noteq> 0"
           using bn0 kn
           unfolding pochhammer_eq_0_iff
-          apply auto
-          apply (erule_tac x= "n - ka - 1" in allE)
-          apply (auto simp add: algebra_simps of_nat_diff)
-          done
+          by (metis add.commute add_diff_eq nz' pochhammer_eq_0_iff)
         have eq1: "prod (\<lambda>k. (1::'a) + of_nat m - of_nat k) {..h} =
           prod of_nat {Suc (m - h) .. Suc m}"
           using kn' h m
           by (intro prod.reindex_bij_witness[where i="\<lambda>k. Suc m - k" and j="\<lambda>k. Suc m - k"])
              (auto simp: of_nat_diff)
-        have th1: "(?m1 k * ?p (of_nat n) k) / ?f n = 1 / of_nat(fact (n - k))"
-          apply (simp add: pochhammer_minus field_simps)
-          using \<open>k \<le> n\<close> apply (simp add: fact_split [of k n])
-          apply (simp add: pochhammer_prod)
+        have "(\<Prod>i = 0..<k. 1 + of_nat n - of_nat k + of_nat i) = (\<Prod>x = n - k..<n. (1::'a) + of_nat x)"
+          using \<open>k \<le> n\<close> 
           using prod.atLeastLessThan_shift_bounds [where ?'a = 'a, of "\<lambda>i. 1 + of_nat i" 0 "n - k" k]
-          apply (auto simp add: of_nat_diff field_simps)
-          done
-        have th20: "?m1 n * ?p b n = prod (\<lambda>i. b - of_nat i) {0..m}"
-          apply (simp add: pochhammer_minus field_simps m)
-          apply (auto simp add: pochhammer_prod_rev of_nat_diff prod.atLeast_Suc_atMost_Suc_shift simp del: prod.cl_ivl_Suc)
-          done
-        have th21:"pochhammer (b - of_nat n + 1) k = prod (\<lambda>i. b - of_nat i) {n - k .. n - 1}"
-          using kn apply (simp add: pochhammer_prod_rev m h prod.atLeast_Suc_atMost_Suc_shift del: prod.op_ivl_Suc del: prod.cl_ivl_Suc)
-          using prod.atLeastAtMost_shift_0 [of "m - h" m, where ?'a = 'a]
-          apply (auto simp add: of_nat_diff field_simps)
-          done
+          by (auto simp add: of_nat_diff field_simps)
+        then have "fact (n - k) * pochhammer ((1::'a) + of_nat n - of_nat k) k = fact n"
+          using \<open>k \<le> n\<close> 
+          by (auto simp add: fact_split [of k n] pochhammer_prod field_simps)
+        then have th1: "(?m1 k * ?p (of_nat n) k) / ?f n = 1 / of_nat(fact (n - k))"
+          by (simp add: pochhammer_minus field_simps)
+        have "?m1 n * ?p b n = pochhammer (b - of_nat m) (Suc m)"
+          by (simp add: pochhammer_minus field_simps m)
+        also have "... = (\<Prod>i = 0..m. b - of_nat i)"
+          by (auto simp add: pochhammer_prod_rev of_nat_diff prod.atLeast_Suc_atMost_Suc_shift simp del: prod.cl_ivl_Suc)
+        finally have th20: "?m1 n * ?p b n = prod (\<lambda>i. b - of_nat i) {0..m}" .
+        have "(\<Prod>x = 0..h. b - of_nat m + of_nat (h - x)) = (\<Prod>i = m - h..m. b - of_nat i)"
+          using \<open>h \<le> m\<close> prod.atLeastAtMost_shift_0 [of "m - h" m, where ?'a = 'a]
+          by (auto simp add: of_nat_diff field_simps)
+        then have th21:"pochhammer (b - of_nat n + 1) k = prod (\<lambda>i. b - of_nat i) {n - k .. n - 1}"
+          using kn by (simp add: pochhammer_prod_rev m h prod.atLeast_Suc_atMost_Suc_shift del: prod.op_ivl_Suc del: prod.cl_ivl_Suc)
         have "?m1 n * ?p b n =
           prod (\<lambda>i. b - of_nat i) {0.. n - k - 1} * pochhammer (b - of_nat n + 1) k"
-          using kn' m h unfolding th20 th21 apply simp
-          apply (subst prod.union_disjoint [symmetric])
-          apply auto
-          apply (rule prod.cong)
-          apply auto
-          done
+          using kn' m h unfolding th20 th21 
+          by (auto simp flip: prod.union_disjoint intro: prod.cong)
         then have th2: "(?m1 n * ?p b n)/pochhammer (b - of_nat n + 1) k =
           prod (\<lambda>i. b - of_nat i) {0.. n - k - 1}"
           using nz' by (simp add: field_simps)
         have "(?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k) =
           ((?m1 k * ?p (of_nat n) k) / ?f n) * ((?m1 n * ?p b n)/pochhammer (b - of_nat n + 1) k)"
           using bnz0
           by (simp add: field_simps)
         also have "\<dots> = b gchoose (n - k)"
           unfolding th1 th2
           using kn' m h
-          apply (simp add: field_simps gbinomial_mult_fact)
-          apply (rule prod.cong)
-          apply auto
-          done
+          by (auto simp: field_simps gbinomial_mult_fact intro: prod.cong)
         finally show ?thesis by simp
       qed
     qed
     then show ?gchoose and ?pochhammer
-      apply (cases "n = 0")
-      using nz'
-      apply auto
-      done
+      using nz' by force+
   qed
   have "?r = ((a + b) gchoose n) * (of_nat (fact n) / (?m1 n * pochhammer (- b) n))"
     unfolding gbinomial_pochhammer
     using bn0 by (auto simp add: field_simps)
   also have "\<dots> = ?l"
+    using bn0
     unfolding gbinomial_Vandermonde[symmetric]
     apply (simp add: th00)
-    unfolding gbinomial_pochhammer
-    using bn0
-    apply (simp add: sum_distrib_right sum_distrib_left field_simps)
-    done
+    by (simp add: gbinomial_pochhammer sum_distrib_right sum_distrib_left field_simps)
   finally show ?thesis by simp
 qed
 
 lemma Vandermonde_pochhammer:
   fixes a :: "'a::field_char_0"
   assumes c: "\<forall>i \<in> {0..< n}. c \<noteq> - of_nat i"
   shows "sum (\<lambda>k. (pochhammer a k * pochhammer (- (of_nat n)) k) /
     (of_nat (fact k) * pochhammer c k)) {0..n} = pochhammer (c - a) n / pochhammer c n"
 proof -
   let ?a = "- a"
   let ?b = "c + of_nat n - 1"
-  have h: "\<forall> j \<in>{0..< n}. ?b \<noteq> of_nat j"
-    using c
-    apply (auto simp add: algebra_simps of_nat_diff)
-    apply (erule_tac x = "n - j - 1" in ballE)
-    apply (auto simp add: of_nat_diff algebra_simps)
-    done
+  have h: "?b \<noteq> of_nat j" if "j < n" for j
+  proof -
+    have "c \<noteq> - of_nat (n - j - 1)"
+      using c that by auto
+    with that show ?thesis
+      by (auto simp add: algebra_simps of_nat_diff)
+  qed
   have th0: "pochhammer (- (?a + ?b)) n = (- 1)^n * pochhammer (c - a) n"
     unfolding pochhammer_minus
     by (simp add: algebra_simps)
   have th1: "pochhammer (- ?b) n = (- 1)^n * pochhammer c n"
     unfolding pochhammer_minus
     by simp
   have nz: "pochhammer c n \<noteq> 0" using c
     by (simp add: pochhammer_eq_0_iff)
   from Vandermonde_pochhammer_lemma[where a = "?a" and b="?b" and n=n, OF h, unfolded th0 th1]
   show ?thesis
     using nz by (simp add: field_simps sum_distrib_left)
 qed
 
 
 subsubsection \<open>Formal trigonometric functions\<close>
 
 definition "fps_sin (c::'a::field_char_0) =
   Abs_fps (\<lambda>n. if even n then 0 else (- 1) ^((n - 1) div 2) * c^n /(of_nat (fact n)))"
 
 definition "fps_cos (c::'a::field_char_0) =
   Abs_fps (\<lambda>n. if even n then (- 1) ^ (n div 2) * c^n / (of_nat (fact n)) else 0)"
 
 lemma fps_sin_0 [simp]: "fps_sin 0 = 0"
   by (intro fps_ext) (auto simp: fps_sin_def elim!: oddE)
 
 lemma fps_cos_0 [simp]: "fps_cos 0 = 1"
   by (intro fps_ext) (simp add: fps_cos_def)
 
 lemma fps_sin_deriv:
   "fps_deriv (fps_sin c) = fps_const c * fps_cos c"
   (is "?lhs = ?rhs")
 proof (rule fps_ext)
   fix n :: nat
   show "?lhs $ n = ?rhs $ n"
   proof (cases "even n")
     case True
     have "?lhs$n = of_nat (n+1) * (fps_sin c $ (n+1))" by simp
     also have "\<dots> = of_nat (n+1) * ((- 1)^(n div 2) * c^Suc n / of_nat (fact (Suc n)))"
       using True by (simp add: fps_sin_def)
     also have "\<dots> = (- 1)^(n div 2) * c^Suc n * (of_nat (n+1) / (of_nat (Suc n) * of_nat (fact n)))"
       unfolding fact_Suc of_nat_mult
       by (simp add: field_simps del: of_nat_add of_nat_Suc)
     also have "\<dots> = (- 1)^(n div 2) * c^Suc n / of_nat (fact n)"
       by (simp add: field_simps del: of_nat_add of_nat_Suc)
     finally show ?thesis
       using True by (simp add: fps_cos_def field_simps)
   next
     case False
     then show ?thesis
       by (simp_all add: fps_deriv_def fps_sin_def fps_cos_def)
   qed
 qed
 
 lemma fps_cos_deriv: "fps_deriv (fps_cos c) = fps_const (- c)* (fps_sin c)"
   (is "?lhs = ?rhs")
 proof (rule fps_ext)
   have th0: "- ((- 1::'a) ^ n) = (- 1)^Suc n" for n
     by simp
   show "?lhs $ n = ?rhs $ n" for n
   proof (cases "even n")
     case False
     then have n0: "n \<noteq> 0" by presburger
     from False have th1: "Suc ((n - 1) div 2) = Suc n div 2"
       by (cases n) simp_all
     have "?lhs$n = of_nat (n+1) * (fps_cos c $ (n+1))" by simp
     also have "\<dots> = of_nat (n+1) * ((- 1)^((n + 1) div 2) * c^Suc n / of_nat (fact (Suc n)))"
       using False by (simp add: fps_cos_def)
     also have "\<dots> = (- 1)^((n + 1) div 2) * c^Suc n * (of_nat (n+1) / (of_nat (Suc n) * of_nat (fact n)))"
       unfolding fact_Suc of_nat_mult
       by (simp add: field_simps del: of_nat_add of_nat_Suc)
     also have "\<dots> = (- 1)^((n + 1) div 2) * c^Suc n / of_nat (fact n)"
       by (simp add: field_simps del: of_nat_add of_nat_Suc)
     also have "\<dots> = (- ((- 1)^((n - 1) div 2))) * c^Suc n / of_nat (fact n)"
       unfolding th0 unfolding th1 by simp
     finally show ?thesis
       using False by (simp add: fps_sin_def field_simps)
   next
     case True
     then show ?thesis
       by (simp_all add: fps_deriv_def fps_sin_def fps_cos_def)
   qed
 qed
 
 lemma fps_sin_cos_sum_of_squares: "(fps_cos c)\<^sup>2 + (fps_sin c)\<^sup>2 = 1"
   (is "?lhs = _")
 proof -
   have "fps_deriv ?lhs = 0"
-    apply (simp add:  fps_deriv_power fps_sin_deriv fps_cos_deriv)
-    apply (simp add: field_simps fps_const_neg[symmetric] del: fps_const_neg)
-    done
+    by (simp add:  fps_deriv_power fps_sin_deriv fps_cos_deriv field_simps flip: fps_const_neg)
   then have "?lhs = fps_const (?lhs $ 0)"
     unfolding fps_deriv_eq_0_iff .
   also have "\<dots> = 1"
     by (simp add: fps_eq_iff numeral_2_eq_2 fps_mult_nth fps_cos_def fps_sin_def)
   finally show ?thesis .
 qed
 
 lemma fps_sin_nth_0 [simp]: "fps_sin c $ 0 = 0"
   unfolding fps_sin_def by simp
 
-lemma fps_sin_nth_1 [simp]: "fps_sin c $ 1 = c"
+lemma fps_sin_nth_1 [simp]: "fps_sin c $ Suc 0 = c"
   unfolding fps_sin_def by simp
 
 lemma fps_sin_nth_add_2:
     "fps_sin c $ (n + 2) = - (c * c * fps_sin c $ n / (of_nat (n + 1) * of_nat (n + 2)))"
-  unfolding fps_sin_def
-  apply (cases n)
-  apply simp
-  apply (simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq del: of_nat_Suc fact_Suc)
-  apply simp
-  done
+proof (cases n)
+  case (Suc n')
+  then show ?thesis
+    unfolding fps_sin_def by (simp add: field_simps)
+qed (auto simp: fps_sin_def)
+
 
 lemma fps_cos_nth_0 [simp]: "fps_cos c $ 0 = 1"
   unfolding fps_cos_def by simp
 
-lemma fps_cos_nth_1 [simp]: "fps_cos c $ 1 = 0"
+lemma fps_cos_nth_1 [simp]: "fps_cos c $ Suc 0 = 0"
   unfolding fps_cos_def by simp
 
 lemma fps_cos_nth_add_2:
   "fps_cos c $ (n + 2) = - (c * c * fps_cos c $ n / (of_nat (n + 1) * of_nat (n + 2)))"
-  unfolding fps_cos_def
-  apply (simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq del: of_nat_Suc fact_Suc)
-  apply simp
-  done
+proof (cases n)
+  case (Suc n')
+  then show ?thesis
+    unfolding fps_cos_def by (simp add: field_simps)
+qed (auto simp: fps_cos_def)
 
 lemma nat_add_1_add_1: "(n::nat) + 1 + 1 = n + 2"
   by simp
 
 lemma eq_fps_sin:
-  assumes 0: "a $ 0 = 0"
-    and 1: "a $ 1 = c"
-    and 2: "fps_deriv (fps_deriv a) = - (fps_const c * fps_const c * a)"
-  shows "a = fps_sin c"
-  apply (rule fps_ext)
-  apply (induct_tac n rule: nat_induct2)
-  apply (simp add: 0)
-  apply (simp add: 1 del: One_nat_def)
-  apply (rename_tac m, cut_tac f="\<lambda>a. a $ m" in arg_cong [OF 2])
-  apply (simp add: nat_add_1_add_1 fps_sin_nth_add_2
-              del: One_nat_def of_nat_Suc of_nat_add add_2_eq_Suc')
-  apply (subst minus_divide_left)
-  apply (subst nonzero_eq_divide_eq)
-  apply (simp del: of_nat_add of_nat_Suc)
-  apply (simp only: ac_simps)
-  done
+  assumes a0: "a $ 0 = 0"
+    and a1: "a $ 1 = c"
+    and a2: "fps_deriv (fps_deriv a) = - (fps_const c * fps_const c * a)"
+  shows "fps_sin c = a"
+proof (rule fps_ext)
+  fix n
+  show "fps_sin c $ n = a $ n"
+  proof (induction n rule: nat_induct2)
+    case (step n)
+    then have "of_nat (n + 1) * (of_nat (n + 2) * a $ (n + 2)) =
+     - (c * c * fps_sin c $ n)"
+      using a2
+      by (metis fps_const_mult fps_deriv_nth fps_mult_left_const_nth fps_neg_nth nat_add_1_add_1)
+    with step show ?case
+      by (metis (no_types, lifting) a0 add.commute add.inverse_inverse fps_sin_nth_0 fps_sin_nth_add_2 mult_divide_mult_cancel_left_if mult_minus_right nonzero_mult_div_cancel_left not_less_zero of_nat_eq_0_iff plus_1_eq_Suc zero_less_Suc)
+  qed (use assms in auto)
+qed
 
 lemma eq_fps_cos:
-  assumes 0: "a $ 0 = 1"
-    and 1: "a $ 1 = 0"
-    and 2: "fps_deriv (fps_deriv a) = - (fps_const c * fps_const c * a)"
-  shows "a = fps_cos c"
-  apply (rule fps_ext)
-  apply (induct_tac n rule: nat_induct2)
-  apply (simp add: 0)
-  apply (simp add: 1 del: One_nat_def)
-  apply (rename_tac m, cut_tac f="\<lambda>a. a $ m" in arg_cong [OF 2])
-  apply (simp add: nat_add_1_add_1 fps_cos_nth_add_2
-              del: One_nat_def of_nat_Suc of_nat_add add_2_eq_Suc')
-  apply (subst minus_divide_left)
-  apply (subst nonzero_eq_divide_eq)
-  apply (simp del: of_nat_add of_nat_Suc)
-  apply (simp only: ac_simps)
-  done
+  assumes a0: "a $ 0 = 1"
+    and a1: "a $ 1 = 0"
+    and a2: "fps_deriv (fps_deriv a) = - (fps_const c * fps_const c * a)"
+  shows "fps_cos c = a"
+proof (rule fps_ext)
+  fix n
+  show "fps_cos c $ n = a $ n"
+  proof (induction n rule: nat_induct2)
+    case (step n)
+    then have "of_nat (n + 1) * (of_nat (n + 2) * a $ (n + 2)) =
+     - (c * c * fps_cos c $ n)"
+      using a2
+      by (metis fps_const_mult fps_deriv_nth fps_mult_left_const_nth fps_neg_nth nat_add_1_add_1)
+    with step show ?case
+      by (metis (no_types, lifting) a0 add.commute add.inverse_inverse fps_cos_nth_0 fps_cos_nth_add_2 mult_divide_mult_cancel_left_if mult_minus_right nonzero_mult_div_cancel_left not_less_zero of_nat_eq_0_iff plus_1_eq_Suc zero_less_Suc)
+  qed (use assms in auto)
+qed
 
 lemma fps_sin_add: "fps_sin (a + b) = fps_sin a * fps_cos b + fps_cos a * fps_sin b"
-  apply (rule eq_fps_sin [symmetric], simp, simp del: One_nat_def)
-  apply (simp del: fps_const_neg fps_const_add fps_const_mult
-              add: fps_const_add [symmetric] fps_const_neg [symmetric]
-                   fps_sin_deriv fps_cos_deriv algebra_simps)
-  done
+proof -
+  have "fps_deriv (fps_deriv (fps_sin a * fps_cos b + fps_cos a * fps_sin b)) =
+         - (fps_const (a + b) * fps_const (a + b) * (fps_sin a * fps_cos b + fps_cos a * fps_sin b))"
+    by (simp flip: fps_const_neg fps_const_add fps_const_mult
+        add: fps_sin_deriv fps_cos_deriv algebra_simps)
+  then show ?thesis
+    by (auto intro: eq_fps_sin)
+qed
+
 
 lemma fps_cos_add: "fps_cos (a + b) = fps_cos a * fps_cos b - fps_sin a * fps_sin b"
-  apply (rule eq_fps_cos [symmetric], simp, simp del: One_nat_def)
-  apply (simp del: fps_const_neg fps_const_add fps_const_mult
-              add: fps_const_add [symmetric] fps_const_neg [symmetric]
-                   fps_sin_deriv fps_cos_deriv algebra_simps)
-  done
+proof -
+  have "fps_deriv
+     (fps_deriv (fps_cos a * fps_cos b - fps_sin a * fps_sin b)) =
+    - (fps_const (a + b) * fps_const (a + b) *
+       (fps_cos a * fps_cos b - fps_sin a * fps_sin b))"
+    by (simp flip: fps_const_neg fps_const_add fps_const_mult
+        add: fps_sin_deriv fps_cos_deriv algebra_simps)
+  then show ?thesis
+    by (auto intro: eq_fps_cos)
+qed
 
 lemma fps_sin_even: "fps_sin (- c) = - fps_sin c"
   by (simp add: fps_eq_iff fps_sin_def)
 
 lemma fps_cos_odd: "fps_cos (- c) = fps_cos c"
   by (simp add: fps_eq_iff fps_cos_def)
 
 definition "fps_tan c = fps_sin c / fps_cos c"
 
 lemma fps_tan_0 [simp]: "fps_tan 0 = 0"
   by (simp add: fps_tan_def)
 
 lemma fps_tan_deriv: "fps_deriv (fps_tan c) = fps_const c / (fps_cos c)\<^sup>2"
 proof -
   have th0: "fps_cos c $ 0 \<noteq> 0" by (simp add: fps_cos_def)
   from this have "fps_cos c \<noteq> 0" by (intro notI) simp
   hence "fps_deriv (fps_tan c) =
            fps_const c * (fps_cos c^2 + fps_sin c^2) / (fps_cos c^2)"
     by (simp add: fps_tan_def fps_divide_deriv power2_eq_square algebra_simps
                   fps_sin_deriv fps_cos_deriv fps_const_neg[symmetric] div_mult_swap
              del: fps_const_neg)
   also note fps_sin_cos_sum_of_squares
   finally show ?thesis by simp
 qed
 
 text \<open>Connection to @{const "fps_exp"} over the complex numbers --- Euler and de Moivre.\<close>
 
 lemma fps_exp_ii_sin_cos: "fps_exp (\<i> * c) = fps_cos c + fps_const \<i> * fps_sin c"
   (is "?l = ?r")
 proof -
   have "?l $ n = ?r $ n" for n
   proof (cases "even n")
     case True
     then obtain m where m: "n = 2 * m" ..
     show ?thesis
       by (simp add: m fps_sin_def fps_cos_def power_mult_distrib power_mult power_minus [of "c ^ 2"])
   next
     case False
     then obtain m where m: "n = 2 * m + 1" ..
     show ?thesis
       by (simp add: m fps_sin_def fps_cos_def power_mult_distrib
         power_mult power_minus [of "c ^ 2"])
   qed
   then show ?thesis
     by (simp add: fps_eq_iff)
 qed
 
 lemma fps_exp_minus_ii_sin_cos: "fps_exp (- (\<i> * c)) = fps_cos c - fps_const \<i> * fps_sin c"
   unfolding minus_mult_right fps_exp_ii_sin_cos by (simp add: fps_sin_even fps_cos_odd)
 
 lemma fps_cos_fps_exp_ii: "fps_cos c = (fps_exp (\<i> * c) + fps_exp (- \<i> * c)) / fps_const 2"
 proof -
   have th: "fps_cos c + fps_cos c = fps_cos c * fps_const 2"
     by (simp add: numeral_fps_const)
   show ?thesis
     unfolding fps_exp_ii_sin_cos minus_mult_commute
     by (simp add: fps_sin_even fps_cos_odd numeral_fps_const fps_divide_unit fps_const_inverse th)
 qed
 
 lemma fps_sin_fps_exp_ii: "fps_sin c = (fps_exp (\<i> * c) - fps_exp (- \<i> * c)) / fps_const (2*\<i>)"
 proof -
   have th: "fps_const \<i> * fps_sin c + fps_const \<i> * fps_sin c = fps_sin c * fps_const (2 * \<i>)"
     by (simp add: fps_eq_iff numeral_fps_const)
   show ?thesis
     unfolding fps_exp_ii_sin_cos minus_mult_commute
     by (simp add: fps_sin_even fps_cos_odd fps_divide_unit fps_const_inverse th)
 qed
 
 lemma fps_tan_fps_exp_ii:
   "fps_tan c = (fps_exp (\<i> * c) - fps_exp (- \<i> * c)) / 
       (fps_const \<i> * (fps_exp (\<i> * c) + fps_exp (- \<i> * c)))"
-  unfolding fps_tan_def fps_sin_fps_exp_ii fps_cos_fps_exp_ii mult_minus_left fps_exp_neg
-  apply (simp add: fps_divide_unit fps_inverse_mult fps_const_mult[symmetric] fps_const_inverse del: fps_const_mult)
-  apply simp
-  done
+  unfolding fps_tan_def fps_sin_fps_exp_ii fps_cos_fps_exp_ii
+  by (simp add: fps_divide_unit fps_inverse_mult fps_const_inverse)
 
 lemma fps_demoivre:
   "(fps_cos a + fps_const \<i> * fps_sin a)^n =
     fps_cos (of_nat n * a) + fps_const \<i> * fps_sin (of_nat n * a)"
   unfolding fps_exp_ii_sin_cos[symmetric] fps_exp_power_mult
   by (simp add: ac_simps)
 
 
 subsection \<open>Hypergeometric series\<close>
 
 definition "fps_hypergeo as bs (c::'a::field_char_0) =
   Abs_fps (\<lambda>n. (foldl (\<lambda>r a. r* pochhammer a n) 1 as * c^n) /
     (foldl (\<lambda>r b. r * pochhammer b n) 1 bs * of_nat (fact n)))"
 
 lemma fps_hypergeo_nth[simp]: "fps_hypergeo as bs c $ n =
   (foldl (\<lambda>r a. r* pochhammer a n) 1 as * c^n) /
     (foldl (\<lambda>r b. r * pochhammer b n) 1 bs * of_nat (fact n))"
   by (simp add: fps_hypergeo_def)
 
 lemma foldl_mult_start:
   fixes v :: "'a::comm_ring_1"
   shows "foldl (\<lambda>r x. r * f x) v as * x = foldl (\<lambda>r x. r * f x) (v * x) as "
   by (induct as arbitrary: x v) (auto simp add: algebra_simps)
 
 lemma foldr_mult_foldl:
   fixes v :: "'a::comm_ring_1"
   shows "foldr (\<lambda>x r. r * f x) as v = foldl (\<lambda>r x. r * f x) v as"
   by (induct as arbitrary: v) (simp_all add: foldl_mult_start)
 
 lemma fps_hypergeo_nth_alt:
   "fps_hypergeo as bs c $ n = foldr (\<lambda>a r. r * pochhammer a n) as (c ^ n) /
     foldr (\<lambda>b r. r * pochhammer b n) bs (of_nat (fact n))"
   by (simp add: foldl_mult_start foldr_mult_foldl)
 
 lemma fps_hypergeo_fps_exp[simp]: "fps_hypergeo [] [] c = fps_exp c"
   by (simp add: fps_eq_iff)
 
 lemma fps_hypergeo_1_0[simp]: "fps_hypergeo [1] [] c = 1/(1 - fps_const c * fps_X)"
 proof -
   let ?a = "(Abs_fps (\<lambda>n. 1)) oo (fps_const c * fps_X)"
   have th0: "(fps_const c * fps_X) $ 0 = 0" by simp
   show ?thesis unfolding gp[OF th0, symmetric]
     by (simp add: fps_eq_iff pochhammer_fact[symmetric]
       fps_compose_nth power_mult_distrib if_distrib cong del: if_weak_cong)
 qed
 
 lemma fps_hypergeo_B[simp]: "fps_hypergeo [-a] [] (- 1) = fps_binomial a"
   by (simp add: fps_eq_iff gbinomial_pochhammer algebra_simps)
 
 lemma fps_hypergeo_0[simp]: "fps_hypergeo as bs c $ 0 = 1"
-  apply simp
-  apply (subgoal_tac "\<forall>as. foldl (\<lambda>(r::'a) (a::'a). r) 1 as = 1")
-  apply auto
-  apply (induct_tac as)
-  apply auto
-  done
+proof -
+  have "foldl (\<lambda>(r::'a) (a::'a). r) 1 as = 1" for as
+    by (induction as) auto
+  then show ?thesis
+    by auto
+qed
 
 lemma foldl_prod_prod:
   "foldl (\<lambda>(r::'b::comm_ring_1) (x::'a::comm_ring_1). r * f x) v as * foldl (\<lambda>r x. r * g x) w as =
     foldl (\<lambda>r x. r * f x * g x) (v * w) as"
   by (induct as arbitrary: v w) (simp_all add: algebra_simps)
 
 
 lemma fps_hypergeo_rec:
   "fps_hypergeo as bs c $ Suc n = ((foldl (\<lambda>r a. r* (a + of_nat n)) c as) /
     (foldl (\<lambda>r b. r * (b + of_nat n)) (of_nat (Suc n)) bs )) * fps_hypergeo as bs c $ n"
-  apply (simp del: of_nat_Suc of_nat_add fact_Suc)
-  apply (simp add: foldl_mult_start del: fact_Suc of_nat_Suc)
+  apply (simp add: foldl_mult_start del: of_nat_Suc of_nat_add fact_Suc)
   unfolding foldl_prod_prod[unfolded foldl_mult_start] pochhammer_Suc
-  apply (simp add: algebra_simps)
-  done
+  by (simp add: algebra_simps)
 
 lemma fps_XD_nth[simp]: "fps_XD a $ n = of_nat n * a$n"
   by (simp add: fps_XD_def)
 
 lemma fps_XD_0th[simp]: "fps_XD a $ 0 = 0"
   by simp
 lemma fps_XD_Suc[simp]:" fps_XD a $ Suc n = of_nat (Suc n) * a $ Suc n"
   by simp
 
 definition "fps_XDp c a = fps_XD a + fps_const c * a"
 
 lemma fps_XDp_nth[simp]: "fps_XDp c a $ n = (c + of_nat n) * a$n"
   by (simp add: fps_XDp_def algebra_simps)
 
 lemma fps_XDp_commute: "fps_XDp b \<circ> fps_XDp (c::'a::comm_ring_1) = fps_XDp c \<circ> fps_XDp b"
   by (simp add: fps_XDp_def fun_eq_iff fps_eq_iff algebra_simps)
 
 lemma fps_XDp0 [simp]: "fps_XDp 0 = fps_XD"
   by (simp add: fun_eq_iff fps_eq_iff)
 
 lemma fps_XDp_fps_integral [simp]:
   fixes  a :: "'a::{division_ring,ring_char_0} fps"
   shows  "fps_XDp 0 (fps_integral a c) = fps_X * a"
   using  fps_deriv_fps_integral[of a c]
   by     (simp add: fps_XD_def)
 
 lemma fps_hypergeo_minus_nat:
   "fps_hypergeo [- of_nat n] [- of_nat (n + m)] (c::'a::field_char_0) $ k =
     (if k \<le> n then
       pochhammer (- of_nat n) k * c ^ k / (pochhammer (- of_nat (n + m)) k * of_nat (fact k))
      else 0)"
   "fps_hypergeo [- of_nat m] [- of_nat (m + n)] (c::'a::field_char_0) $ k =
     (if k \<le> m then
       pochhammer (- of_nat m) k * c ^ k / (pochhammer (- of_nat (m + n)) k * of_nat (fact k))
      else 0)"
   by (simp_all add: pochhammer_eq_0_iff)
 
-lemma sum_eq_if: "sum f {(n::nat) .. m} = (if m < n then 0 else f n + sum f {n+1 .. m})"
-  apply simp
-  apply (subst sum.insert[symmetric])
-  apply (auto simp add: not_less sum.atLeast_Suc_atMost)
-  done
-
 lemma pochhammer_rec_if: "pochhammer a n = (if n = 0 then 1 else a * pochhammer (a + 1) (n - 1))"
   by (cases n) (simp_all add: pochhammer_rec)
 
 lemma fps_XDp_foldr_nth [simp]: "foldr (\<lambda>c r. fps_XDp c \<circ> r) cs (\<lambda>c. fps_XDp c a) c0 $ n =
     foldr (\<lambda>c r. (c + of_nat n) * r) cs (c0 + of_nat n) * a$n"
   by (induct cs arbitrary: c0) (simp_all add: algebra_simps)
 
 lemma genric_fps_XDp_foldr_nth:
   assumes f: "\<forall>n c a. f c a $ n = (of_nat n + k c) * a$n"
   shows "foldr (\<lambda>c r. f c \<circ> r) cs (\<lambda>c. g c a) c0 $ n =
     foldr (\<lambda>c r. (k c + of_nat n) * r) cs (g c0 a $ n)"
   by (induct cs arbitrary: c0) (simp_all add: algebra_simps f)
 
 lemma dist_less_imp_nth_equal:
   assumes "dist f g < inverse (2 ^ i)"
     and"j \<le> i"
   shows "f $ j = g $ j"
 proof (rule ccontr)
   assume "f $ j \<noteq> g $ j"
   hence "f \<noteq> g" by auto
   with assms have "i < subdegree (f - g)"
     by (simp add: if_split_asm dist_fps_def)
   also have "\<dots> \<le> j"
     using \<open>f $ j \<noteq> g $ j\<close> by (intro subdegree_leI) simp_all
   finally show False using \<open>j \<le> i\<close> by simp
 qed
 
 lemma nth_equal_imp_dist_less:
   assumes "\<And>j. j \<le> i \<Longrightarrow> f $ j = g $ j"
   shows "dist f g < inverse (2 ^ i)"
 proof (cases "f = g")
   case True
   then show ?thesis by simp
 next
   case False
   with assms have "dist f g = inverse (2 ^ subdegree (f - g))"
     by (simp add: if_split_asm dist_fps_def)
   moreover
   from assms and False have "i < subdegree (f - g)"
     by (intro subdegree_greaterI) simp_all
   ultimately show ?thesis by simp
 qed
 
 lemma dist_less_eq_nth_equal: "dist f g < inverse (2 ^ i) \<longleftrightarrow> (\<forall>j \<le> i. f $ j = g $ j)"
   using dist_less_imp_nth_equal nth_equal_imp_dist_less by blast
 
 instance fps :: (comm_ring_1) complete_space
 proof
   fix fps_X :: "nat \<Rightarrow> 'a fps"
   assume "Cauchy fps_X"
   obtain M where M: "\<forall>i. \<forall>m \<ge> M i. \<forall>j \<le> i. fps_X (M i) $ j = fps_X m $ j"
   proof -
     have "\<exists>M. \<forall>m \<ge> M. \<forall>j\<le>i. fps_X M $ j = fps_X m $ j" for i
     proof -
       have "0 < inverse ((2::real)^i)" by simp
       from metric_CauchyD[OF \<open>Cauchy fps_X\<close> this] dist_less_imp_nth_equal
       show ?thesis by blast
     qed
     then show ?thesis using that by metis
   qed
 
   show "convergent fps_X"
   proof (rule convergentI)
     show "fps_X \<longlonglongrightarrow> Abs_fps (\<lambda>i. fps_X (M i) $ i)"
       unfolding tendsto_iff
     proof safe
       fix e::real assume e: "0 < e"
       have "(\<lambda>n. inverse (2 ^ n) :: real) \<longlonglongrightarrow> 0" by (rule LIMSEQ_inverse_realpow_zero) simp_all
       from this and e have "eventually (\<lambda>i. inverse (2 ^ i) < e) sequentially"
         by (rule order_tendstoD)
       then obtain i where "inverse (2 ^ i) < e"
         by (auto simp: eventually_sequentially)
       have "eventually (\<lambda>x. M i \<le> x) sequentially"
         by (auto simp: eventually_sequentially)
       then show "eventually (\<lambda>x. dist (fps_X x) (Abs_fps (\<lambda>i. fps_X (M i) $ i)) < e) sequentially"
       proof eventually_elim
         fix x
         assume x: "M i \<le> x"
         have "fps_X (M i) $ j = fps_X (M j) $ j" if "j \<le> i" for j
           using M that by (metis nat_le_linear)
         with x have "dist (fps_X x) (Abs_fps (\<lambda>j. fps_X (M j) $ j)) < inverse (2 ^ i)"
           using M by (force simp: dist_less_eq_nth_equal)
         also note \<open>inverse (2 ^ i) < e\<close>
         finally show "dist (fps_X x) (Abs_fps (\<lambda>j. fps_X (M j) $ j)) < e" .
       qed
     qed
   qed
 qed
 
 (* TODO: Figure out better notation for this thing *)
 no_notation fps_nth (infixl "$" 75)
 
 bundle fps_notation
 begin
 notation fps_nth (infixl "$" 75)
 end
 
 end
diff --git a/src/HOL/Set_Interval.thy b/src/HOL/Set_Interval.thy
--- a/src/HOL/Set_Interval.thy
+++ b/src/HOL/Set_Interval.thy
@@ -1,2517 +1,2522 @@
 (*  Title:      HOL/Set_Interval.thy
     Author:     Tobias Nipkow, Clemens Ballarin, Jeremy Avigad
 
 lessThan, greaterThan, atLeast, atMost and two-sided intervals
 
 Modern convention: Ixy stands for an interval where x and y
 describe the lower and upper bound and x,y : {c,o,i}
 where c = closed, o = open, i = infinite.
 Examples: Ico = {_ ..< _} and Ici = {_ ..}
 *)
 
 section \<open>Set intervals\<close>
 
 theory Set_Interval
 imports Divides
 begin
 
 (* Belongs in Finite_Set but 2 is not available there *)
 lemma card_2_iff: "card S = 2 \<longleftrightarrow> (\<exists>x y. S = {x,y} \<and> x \<noteq> y)"
   by (auto simp: card_Suc_eq numeral_eq_Suc)
 
 lemma card_2_iff': "card S = 2 \<longleftrightarrow> (\<exists>x\<in>S. \<exists>y\<in>S. x \<noteq> y \<and> (\<forall>z\<in>S. z = x \<or> z = y))"
   by (auto simp: card_Suc_eq numeral_eq_Suc)
 
 context ord
 begin
 
 definition
   lessThan    :: "'a => 'a set" ("(1{..<_})") where
   "{..<u} == {x. x < u}"
 
 definition
   atMost      :: "'a => 'a set" ("(1{.._})") where
   "{..u} == {x. x \<le> u}"
 
 definition
   greaterThan :: "'a => 'a set" ("(1{_<..})") where
   "{l<..} == {x. l<x}"
 
 definition
   atLeast     :: "'a => 'a set" ("(1{_..})") where
   "{l..} == {x. l\<le>x}"
 
 definition
   greaterThanLessThan :: "'a => 'a => 'a set"  ("(1{_<..<_})") where
   "{l<..<u} == {l<..} Int {..<u}"
 
 definition
   atLeastLessThan :: "'a => 'a => 'a set"      ("(1{_..<_})") where
   "{l..<u} == {l..} Int {..<u}"
 
 definition
   greaterThanAtMost :: "'a => 'a => 'a set"    ("(1{_<.._})") where
   "{l<..u} == {l<..} Int {..u}"
 
 definition
   atLeastAtMost :: "'a => 'a => 'a set"        ("(1{_.._})") where
   "{l..u} == {l..} Int {..u}"
 
 end
 
 
 text\<open>A note of warning when using \<^term>\<open>{..<n}\<close> on type \<^typ>\<open>nat\<close>: it is equivalent to \<^term>\<open>{0::nat..<n}\<close> but some lemmas involving
 \<^term>\<open>{m..<n}\<close> may not exist in \<^term>\<open>{..<n}\<close>-form as well.\<close>
 
 syntax (ASCII)
   "_UNION_le"   :: "'a => 'a => 'b set => 'b set"       ("(3UN _<=_./ _)" [0, 0, 10] 10)
   "_UNION_less" :: "'a => 'a => 'b set => 'b set"       ("(3UN _<_./ _)" [0, 0, 10] 10)
   "_INTER_le"   :: "'a => 'a => 'b set => 'b set"       ("(3INT _<=_./ _)" [0, 0, 10] 10)
   "_INTER_less" :: "'a => 'a => 'b set => 'b set"       ("(3INT _<_./ _)" [0, 0, 10] 10)
 
 syntax (latex output)
   "_UNION_le"   :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Union>(\<open>unbreakable\<close>_ \<le> _)/ _)" [0, 0, 10] 10)
   "_UNION_less" :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Union>(\<open>unbreakable\<close>_ < _)/ _)" [0, 0, 10] 10)
   "_INTER_le"   :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Inter>(\<open>unbreakable\<close>_ \<le> _)/ _)" [0, 0, 10] 10)
   "_INTER_less" :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Inter>(\<open>unbreakable\<close>_ < _)/ _)" [0, 0, 10] 10)
 
 syntax
   "_UNION_le"   :: "'a => 'a => 'b set => 'b set"       ("(3\<Union>_\<le>_./ _)" [0, 0, 10] 10)
   "_UNION_less" :: "'a => 'a => 'b set => 'b set"       ("(3\<Union>_<_./ _)" [0, 0, 10] 10)
   "_INTER_le"   :: "'a => 'a => 'b set => 'b set"       ("(3\<Inter>_\<le>_./ _)" [0, 0, 10] 10)
   "_INTER_less" :: "'a => 'a => 'b set => 'b set"       ("(3\<Inter>_<_./ _)" [0, 0, 10] 10)
 
 translations
   "\<Union>i\<le>n. A" \<rightleftharpoons> "\<Union>i\<in>{..n}. A"
   "\<Union>i<n. A" \<rightleftharpoons> "\<Union>i\<in>{..<n}. A"
   "\<Inter>i\<le>n. A" \<rightleftharpoons> "\<Inter>i\<in>{..n}. A"
   "\<Inter>i<n. A" \<rightleftharpoons> "\<Inter>i\<in>{..<n}. A"
 
 
 subsection \<open>Various equivalences\<close>
 
 lemma (in ord) lessThan_iff [iff]: "(i \<in> lessThan k) = (i<k)"
 by (simp add: lessThan_def)
 
 lemma Compl_lessThan [simp]:
     "!!k:: 'a::linorder. -lessThan k = atLeast k"
   by (auto simp add: lessThan_def atLeast_def)
 
 lemma single_Diff_lessThan [simp]: "!!k:: 'a::preorder. {k} - lessThan k = {k}"
   by auto
 
 lemma (in ord) greaterThan_iff [iff]: "(i \<in> greaterThan k) = (k<i)"
   by (simp add: greaterThan_def)
 
 lemma Compl_greaterThan [simp]:
     "!!k:: 'a::linorder. -greaterThan k = atMost k"
   by (auto simp add: greaterThan_def atMost_def)
 
 lemma Compl_atMost [simp]: "!!k:: 'a::linorder. -atMost k = greaterThan k"
   apply (subst Compl_greaterThan [symmetric])
   apply (rule double_complement)
   done
 
 lemma (in ord) atLeast_iff [iff]: "(i \<in> atLeast k) = (k<=i)"
 by (simp add: atLeast_def)
 
 lemma Compl_atLeast [simp]: "!!k:: 'a::linorder. -atLeast k = lessThan k"
   by (auto simp add: lessThan_def atLeast_def)
 
 lemma (in ord) atMost_iff [iff]: "(i \<in> atMost k) = (i<=k)"
 by (simp add: atMost_def)
 
 lemma atMost_Int_atLeast: "!!n:: 'a::order. atMost n Int atLeast n = {n}"
 by (blast intro: order_antisym)
 
 lemma (in linorder) lessThan_Int_lessThan: "{ a <..} \<inter> { b <..} = { max a b <..}"
   by auto
 
 lemma (in linorder) greaterThan_Int_greaterThan: "{..< a} \<inter> {..< b} = {..< min a b}"
   by auto
 
 subsection \<open>Logical Equivalences for Set Inclusion and Equality\<close>
 
 lemma atLeast_empty_triv [simp]: "{{}..} = UNIV"
   by auto
 
 lemma atMost_UNIV_triv [simp]: "{..UNIV} = UNIV"
   by auto
 
 lemma atLeast_subset_iff [iff]:
      "(atLeast x \<subseteq> atLeast y) = (y \<le> (x::'a::preorder))"
 by (blast intro: order_trans)
 
 lemma atLeast_eq_iff [iff]:
      "(atLeast x = atLeast y) = (x = (y::'a::order))"
 by (blast intro: order_antisym order_trans)
 
 lemma greaterThan_subset_iff [iff]:
      "(greaterThan x \<subseteq> greaterThan y) = (y \<le> (x::'a::linorder))"
   unfolding greaterThan_def by (auto simp: linorder_not_less [symmetric])
 
 lemma greaterThan_eq_iff [iff]:
      "(greaterThan x = greaterThan y) = (x = (y::'a::linorder))"
   by (auto simp: elim!: equalityE)
 
 lemma atMost_subset_iff [iff]: "(atMost x \<subseteq> atMost y) = (x \<le> (y::'a::preorder))"
   by (blast intro: order_trans)
 
 lemma atMost_eq_iff [iff]: "(atMost x = atMost y) = (x = (y::'a::order))"
   by (blast intro: order_antisym order_trans)
 
 lemma lessThan_subset_iff [iff]:
      "(lessThan x \<subseteq> lessThan y) = (x \<le> (y::'a::linorder))"
   unfolding lessThan_def by (auto simp: linorder_not_less [symmetric])
 
 lemma lessThan_eq_iff [iff]:
      "(lessThan x = lessThan y) = (x = (y::'a::linorder))"
   by (auto simp: elim!: equalityE)
 
 lemma lessThan_strict_subset_iff:
   fixes m n :: "'a::linorder"
   shows "{..<m} < {..<n} \<longleftrightarrow> m < n"
   by (metis leD lessThan_subset_iff linorder_linear not_less_iff_gr_or_eq psubset_eq)
 
 lemma (in linorder) Ici_subset_Ioi_iff: "{a ..} \<subseteq> {b <..} \<longleftrightarrow> b < a"
   by auto
 
 lemma (in linorder) Iic_subset_Iio_iff: "{.. a} \<subseteq> {..< b} \<longleftrightarrow> a < b"
   by auto
 
 lemma (in preorder) Ioi_le_Ico: "{a <..} \<subseteq> {a ..}"
   by (auto intro: less_imp_le)
 
 subsection \<open>Two-sided intervals\<close>
 
 context ord
 begin
 
 lemma greaterThanLessThan_iff [simp]: "(i \<in> {l<..<u}) = (l < i \<and> i < u)"
   by (simp add: greaterThanLessThan_def)
 
 lemma atLeastLessThan_iff [simp]: "(i \<in> {l..<u}) = (l \<le> i \<and> i < u)"
   by (simp add: atLeastLessThan_def)
 
 lemma greaterThanAtMost_iff [simp]: "(i \<in> {l<..u}) = (l < i \<and> i \<le> u)"
   by (simp add: greaterThanAtMost_def)
 
 lemma atLeastAtMost_iff [simp]: "(i \<in> {l..u}) = (l \<le> i \<and> i \<le> u)"
   by (simp add: atLeastAtMost_def)
 
 text \<open>The above four lemmas could be declared as iffs. Unfortunately this
 breaks many proofs. Since it only helps blast, it is better to leave them
 alone.\<close>
 
 lemma greaterThanLessThan_eq: "{ a <..< b} = { a <..} \<inter> {..< b }"
   by auto
 
 lemma (in order) atLeastLessThan_eq_atLeastAtMost_diff:
   "{a..<b} = {a..b} - {b}"
   by (auto simp add: atLeastLessThan_def atLeastAtMost_def)
 
 lemma (in order) greaterThanAtMost_eq_atLeastAtMost_diff:
   "{a<..b} = {a..b} - {a}"
   by (auto simp add: greaterThanAtMost_def atLeastAtMost_def)
 
 end
 
 subsubsection\<open>Emptyness, singletons, subset\<close>
 
 context preorder
 begin
 
 lemma atLeastatMost_empty_iff[simp]:
   "{a..b} = {} \<longleftrightarrow> (\<not> a \<le> b)"
   by auto (blast intro: order_trans)
 
 lemma atLeastatMost_empty_iff2[simp]:
   "{} = {a..b} \<longleftrightarrow> (\<not> a \<le> b)"
   by auto (blast intro: order_trans)
 
 lemma atLeastLessThan_empty_iff[simp]:
   "{a..<b} = {} \<longleftrightarrow> (\<not> a < b)"
   by auto (blast intro: le_less_trans)
 
 lemma atLeastLessThan_empty_iff2[simp]:
   "{} = {a..<b} \<longleftrightarrow> (\<not> a < b)"
   by auto (blast intro: le_less_trans)
 
 lemma greaterThanAtMost_empty_iff[simp]: "{k<..l} = {} \<longleftrightarrow> \<not> k < l"
   by auto (blast intro: less_le_trans)
 
 lemma greaterThanAtMost_empty_iff2[simp]: "{} = {k<..l} \<longleftrightarrow> \<not> k < l"
   by auto (blast intro: less_le_trans)
 
 lemma atLeastatMost_subset_iff[simp]:
   "{a..b} \<le> {c..d} \<longleftrightarrow> (\<not> a \<le> b) \<or> c \<le> a \<and> b \<le> d"
   unfolding atLeastAtMost_def atLeast_def atMost_def
   by (blast intro: order_trans)
 
 lemma atLeastatMost_psubset_iff:
   "{a..b} < {c..d} \<longleftrightarrow>
    ((\<not> a \<le> b) \<or> c \<le> a \<and> b \<le> d \<and> (c < a \<or> b < d)) \<and> c \<le> d"
   by(simp add: psubset_eq set_eq_iff less_le_not_le)(blast intro: order_trans)
 
 lemma atLeastAtMost_subseteq_atLeastLessThan_iff:
   "{a..b} \<subseteq> {c ..< d} \<longleftrightarrow> (a \<le> b \<longrightarrow> c \<le> a \<and> b < d)" 
   by auto (blast intro: local.order_trans local.le_less_trans elim: )+
 
 lemma Icc_subset_Ici_iff[simp]:
   "{l..h} \<subseteq> {l'..} = (\<not> l\<le>h \<or> l\<ge>l')"
   by(auto simp: subset_eq intro: order_trans)
 
 lemma Icc_subset_Iic_iff[simp]:
   "{l..h} \<subseteq> {..h'} = (\<not> l\<le>h \<or> h\<le>h')"
   by(auto simp: subset_eq intro: order_trans)
 
 lemma not_Ici_eq_empty[simp]: "{l..} \<noteq> {}"
   by(auto simp: set_eq_iff)
 
 lemma not_Iic_eq_empty[simp]: "{..h} \<noteq> {}"
   by(auto simp: set_eq_iff)
 
 lemmas not_empty_eq_Ici_eq_empty[simp] = not_Ici_eq_empty[symmetric]
 lemmas not_empty_eq_Iic_eq_empty[simp] = not_Iic_eq_empty[symmetric]
 
 end
 
 context order
 begin
 
 lemma atLeastatMost_empty[simp]:
   "b < a \<Longrightarrow> {a..b} = {}"
   by(auto simp: atLeastAtMost_def atLeast_def atMost_def)
 
 lemma atLeastLessThan_empty[simp]:
   "b \<le> a \<Longrightarrow> {a..<b} = {}"
   by(auto simp: atLeastLessThan_def)
 
 lemma greaterThanAtMost_empty[simp]: "l \<le> k ==> {k<..l} = {}"
   by(auto simp:greaterThanAtMost_def greaterThan_def atMost_def)
 
 lemma greaterThanLessThan_empty[simp]:"l \<le> k ==> {k<..<l} = {}"
   by(auto simp:greaterThanLessThan_def greaterThan_def lessThan_def)
 
 lemma atLeastAtMost_singleton [simp]: "{a..a} = {a}"
   by (auto simp add: atLeastAtMost_def atMost_def atLeast_def)
 
 lemma atLeastAtMost_singleton': "a = b \<Longrightarrow> {a .. b} = {a}" by simp
 
 lemma Icc_eq_Icc[simp]:
   "{l..h} = {l'..h'} = (l=l' \<and> h=h' \<or> \<not> l\<le>h \<and> \<not> l'\<le>h')"
   by(simp add: order_class.eq_iff)(auto intro: order_trans)
 
 lemma atLeastAtMost_singleton_iff[simp]:
   "{a .. b} = {c} \<longleftrightarrow> a = b \<and> b = c"
 proof
   assume "{a..b} = {c}"
   hence *: "\<not> (\<not> a \<le> b)" unfolding atLeastatMost_empty_iff[symmetric] by simp
   with \<open>{a..b} = {c}\<close> have "c \<le> a \<and> b \<le> c" by auto
   with * show "a = b \<and> b = c" by auto
 qed simp
 
 end
 
 context no_top
 begin
 
 (* also holds for no_bot but no_top should suffice *)
 lemma not_UNIV_le_Icc[simp]: "\<not> UNIV \<subseteq> {l..h}"
 using gt_ex[of h] by(auto simp: subset_eq less_le_not_le)
 
 lemma not_UNIV_le_Iic[simp]: "\<not> UNIV \<subseteq> {..h}"
 using gt_ex[of h] by(auto simp: subset_eq less_le_not_le)
 
 lemma not_Ici_le_Icc[simp]: "\<not> {l..} \<subseteq> {l'..h'}"
 using gt_ex[of h']
 by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)
 
 lemma not_Ici_le_Iic[simp]: "\<not> {l..} \<subseteq> {..h'}"
 using gt_ex[of h']
 by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)
 
 end
 
 context no_bot
 begin
 
 lemma not_UNIV_le_Ici[simp]: "\<not> UNIV \<subseteq> {l..}"
 using lt_ex[of l] by(auto simp: subset_eq less_le_not_le)
 
 lemma not_Iic_le_Icc[simp]: "\<not> {..h} \<subseteq> {l'..h'}"
 using lt_ex[of l']
 by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)
 
 lemma not_Iic_le_Ici[simp]: "\<not> {..h} \<subseteq> {l'..}"
 using lt_ex[of l']
 by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)
 
 end
 
 
 context no_top
 begin
 
 (* also holds for no_bot but no_top should suffice *)
 lemma not_UNIV_eq_Icc[simp]: "\<not> UNIV = {l'..h'}"
 using gt_ex[of h'] by(auto simp: set_eq_iff  less_le_not_le)
 
 lemmas not_Icc_eq_UNIV[simp] = not_UNIV_eq_Icc[symmetric]
 
 lemma not_UNIV_eq_Iic[simp]: "\<not> UNIV = {..h'}"
 using gt_ex[of h'] by(auto simp: set_eq_iff  less_le_not_le)
 
 lemmas not_Iic_eq_UNIV[simp] = not_UNIV_eq_Iic[symmetric]
 
 lemma not_Icc_eq_Ici[simp]: "\<not> {l..h} = {l'..}"
 unfolding atLeastAtMost_def using not_Ici_le_Iic[of l'] by blast
 
 lemmas not_Ici_eq_Icc[simp] = not_Icc_eq_Ici[symmetric]
 
 (* also holds for no_bot but no_top should suffice *)
 lemma not_Iic_eq_Ici[simp]: "\<not> {..h} = {l'..}"
 using not_Ici_le_Iic[of l' h] by blast
 
 lemmas not_Ici_eq_Iic[simp] = not_Iic_eq_Ici[symmetric]
 
 end
 
 context no_bot
 begin
 
 lemma not_UNIV_eq_Ici[simp]: "\<not> UNIV = {l'..}"
 using lt_ex[of l'] by(auto simp: set_eq_iff  less_le_not_le)
 
 lemmas not_Ici_eq_UNIV[simp] = not_UNIV_eq_Ici[symmetric]
 
 lemma not_Icc_eq_Iic[simp]: "\<not> {l..h} = {..h'}"
 unfolding atLeastAtMost_def using not_Iic_le_Ici[of h'] by blast
 
 lemmas not_Iic_eq_Icc[simp] = not_Icc_eq_Iic[symmetric]
 
 end
 
 
 context dense_linorder
 begin
 
 lemma greaterThanLessThan_empty_iff[simp]:
   "{ a <..< b } = {} \<longleftrightarrow> b \<le> a"
   using dense[of a b] by (cases "a < b") auto
 
 lemma greaterThanLessThan_empty_iff2[simp]:
   "{} = { a <..< b } \<longleftrightarrow> b \<le> a"
   using dense[of a b] by (cases "a < b") auto
 
 lemma atLeastLessThan_subseteq_atLeastAtMost_iff:
   "{a ..< b} \<subseteq> { c .. d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
   using dense[of "max a d" "b"]
   by (force simp: subset_eq Ball_def not_less[symmetric])
 
 lemma greaterThanAtMost_subseteq_atLeastAtMost_iff:
   "{a <.. b} \<subseteq> { c .. d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
   using dense[of "a" "min c b"]
   by (force simp: subset_eq Ball_def not_less[symmetric])
 
 lemma greaterThanLessThan_subseteq_atLeastAtMost_iff:
   "{a <..< b} \<subseteq> { c .. d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
   using dense[of "a" "min c b"] dense[of "max a d" "b"]
   by (force simp: subset_eq Ball_def not_less[symmetric])
 
 lemma greaterThanLessThan_subseteq_greaterThanLessThan:
   "{a <..< b} \<subseteq> {c <..< d} \<longleftrightarrow> (a < b \<longrightarrow> a \<ge> c \<and> b \<le> d)"
   using dense[of "a" "min c b"] dense[of "max a d" "b"]
   by (force simp: subset_eq Ball_def not_less[symmetric])
 
 lemma greaterThanAtMost_subseteq_atLeastLessThan_iff:
   "{a <.. b} \<subseteq> { c ..< d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b < d)"
   using dense[of "a" "min c b"]
   by (force simp: subset_eq Ball_def not_less[symmetric])
 
 lemma greaterThanLessThan_subseteq_atLeastLessThan_iff:
   "{a <..< b} \<subseteq> { c ..< d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
   using dense[of "a" "min c b"] dense[of "max a d" "b"]
   by (force simp: subset_eq Ball_def not_less[symmetric])
 
 lemma greaterThanLessThan_subseteq_greaterThanAtMost_iff:
   "{a <..< b} \<subseteq> { c <.. d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
   using dense[of "a" "min c b"] dense[of "max a d" "b"]
   by (force simp: subset_eq Ball_def not_less[symmetric])
 
 end
 
 context no_top
 begin
 
 lemma greaterThan_non_empty[simp]: "{x <..} \<noteq> {}"
   using gt_ex[of x] by auto
 
 end
 
 context no_bot
 begin
 
 lemma lessThan_non_empty[simp]: "{..< x} \<noteq> {}"
   using lt_ex[of x] by auto
 
 end
 
 lemma (in linorder) atLeastLessThan_subset_iff:
   "{a..<b} \<subseteq> {c..<d} \<Longrightarrow> b \<le> a \<or> c\<le>a \<and> b\<le>d"
   apply (auto simp:subset_eq Ball_def not_le)
   apply(frule_tac x=a in spec)
   apply(erule_tac x=d in allE)
   apply (auto simp: )
   done
 
 lemma atLeastLessThan_inj:
   fixes a b c d :: "'a::linorder"
   assumes eq: "{a ..< b} = {c ..< d}" and "a < b" "c < d"
   shows "a = c" "b = d"
 using assms by (metis atLeastLessThan_subset_iff eq less_le_not_le antisym_conv2 subset_refl)+
 
 lemma atLeastLessThan_eq_iff:
   fixes a b c d :: "'a::linorder"
   assumes "a < b" "c < d"
   shows "{a ..< b} = {c ..< d} \<longleftrightarrow> a = c \<and> b = d"
   using atLeastLessThan_inj assms by auto
 
 lemma (in linorder) Ioc_inj: "{a <.. b} = {c <.. d} \<longleftrightarrow> (b \<le> a \<and> d \<le> c) \<or> a = c \<and> b = d"
   by (metis eq_iff greaterThanAtMost_empty_iff2 greaterThanAtMost_iff le_cases not_le)
 
 lemma (in order) Iio_Int_singleton: "{..<k} \<inter> {x} = (if x < k then {x} else {})"
   by auto
 
 lemma (in linorder) Ioc_subset_iff: "{a<..b} \<subseteq> {c<..d} \<longleftrightarrow> (b \<le> a \<or> c \<le> a \<and> b \<le> d)"
   by (auto simp: subset_eq Ball_def) (metis less_le not_less)
 
 lemma (in order_bot) atLeast_eq_UNIV_iff: "{x..} = UNIV \<longleftrightarrow> x = bot"
 by (auto simp: set_eq_iff intro: le_bot)
 
 lemma (in order_top) atMost_eq_UNIV_iff: "{..x} = UNIV \<longleftrightarrow> x = top"
 by (auto simp: set_eq_iff intro: top_le)
 
 lemma (in bounded_lattice) atLeastAtMost_eq_UNIV_iff:
   "{x..y} = UNIV \<longleftrightarrow> (x = bot \<and> y = top)"
 by (auto simp: set_eq_iff intro: top_le le_bot)
 
 lemma Iio_eq_empty_iff: "{..< n::'a::{linorder, order_bot}} = {} \<longleftrightarrow> n = bot"
   by (auto simp: set_eq_iff not_less le_bot)
 
 lemma lessThan_empty_iff: "{..< n::nat} = {} \<longleftrightarrow> n = 0"
   by (simp add: Iio_eq_empty_iff bot_nat_def)
 
 lemma mono_image_least:
   assumes f_mono: "mono f" and f_img: "f ` {m ..< n} = {m' ..< n'}" "m < n"
   shows "f m = m'"
 proof -
   from f_img have "{m' ..< n'} \<noteq> {}"
     by (metis atLeastLessThan_empty_iff image_is_empty)
   with f_img have "m' \<in> f ` {m ..< n}" by auto
   then obtain k where "f k = m'" "m \<le> k" by auto
   moreover have "m' \<le> f m" using f_img by auto
   ultimately show "f m = m'"
     using f_mono by (auto elim: monoE[where x=m and y=k])
 qed
 
 
 subsection \<open>Infinite intervals\<close>
 
 context dense_linorder
 begin
 
 lemma infinite_Ioo:
   assumes "a < b"
   shows "\<not> finite {a<..<b}"
 proof
   assume fin: "finite {a<..<b}"
   moreover have ne: "{a<..<b} \<noteq> {}"
     using \<open>a < b\<close> by auto
   ultimately have "a < Max {a <..< b}" "Max {a <..< b} < b"
     using Max_in[of "{a <..< b}"] by auto
   then obtain x where "Max {a <..< b} < x" "x < b"
     using dense[of "Max {a<..<b}" b] by auto
   then have "x \<in> {a <..< b}"
     using \<open>a < Max {a <..< b}\<close> by auto
   then have "x \<le> Max {a <..< b}"
     using fin by auto
   with \<open>Max {a <..< b} < x\<close> show False by auto
 qed
 
 lemma infinite_Icc: "a < b \<Longrightarrow> \<not> finite {a .. b}"
   using greaterThanLessThan_subseteq_atLeastAtMost_iff[of a b a b] infinite_Ioo[of a b]
   by (auto dest: finite_subset)
 
 lemma infinite_Ico: "a < b \<Longrightarrow> \<not> finite {a ..< b}"
   using greaterThanLessThan_subseteq_atLeastLessThan_iff[of a b a b] infinite_Ioo[of a b]
   by (auto dest: finite_subset)
 
 lemma infinite_Ioc: "a < b \<Longrightarrow> \<not> finite {a <.. b}"
   using greaterThanLessThan_subseteq_greaterThanAtMost_iff[of a b a b] infinite_Ioo[of a b]
   by (auto dest: finite_subset)
 
 lemma infinite_Ioo_iff [simp]: "infinite {a<..<b} \<longleftrightarrow> a < b"
   using not_less_iff_gr_or_eq by (fastforce simp: infinite_Ioo)
 
 lemma infinite_Icc_iff [simp]: "infinite {a .. b} \<longleftrightarrow> a < b"
   using not_less_iff_gr_or_eq by (fastforce simp: infinite_Icc)
 
 lemma infinite_Ico_iff [simp]: "infinite {a..<b} \<longleftrightarrow> a < b"
   using not_less_iff_gr_or_eq by (fastforce simp: infinite_Ico)
 
 lemma infinite_Ioc_iff [simp]: "infinite {a<..b} \<longleftrightarrow> a < b"
   using not_less_iff_gr_or_eq by (fastforce simp: infinite_Ioc)
 
 end
 
 lemma infinite_Iio: "\<not> finite {..< a :: 'a :: {no_bot, linorder}}"
 proof
   assume "finite {..< a}"
   then have *: "\<And>x. x < a \<Longrightarrow> Min {..< a} \<le> x"
     by auto
   obtain x where "x < a"
     using lt_ex by auto
 
   obtain y where "y < Min {..< a}"
     using lt_ex by auto
   also have "Min {..< a} \<le> x"
     using \<open>x < a\<close> by fact
   also note \<open>x < a\<close>
   finally have "Min {..< a} \<le> y"
     by fact
   with \<open>y < Min {..< a}\<close> show False by auto
 qed
 
 lemma infinite_Iic: "\<not> finite {.. a :: 'a :: {no_bot, linorder}}"
   using infinite_Iio[of a] finite_subset[of "{..< a}" "{.. a}"]
   by (auto simp: subset_eq less_imp_le)
 
 lemma infinite_Ioi: "\<not> finite {a :: 'a :: {no_top, linorder} <..}"
 proof
   assume "finite {a <..}"
   then have *: "\<And>x. a < x \<Longrightarrow> x \<le> Max {a <..}"
     by auto
 
   obtain y where "Max {a <..} < y"
     using gt_ex by auto
 
   obtain x where x: "a < x"
     using gt_ex by auto
   also from x have "x \<le> Max {a <..}"
     by fact
   also note \<open>Max {a <..} < y\<close>
   finally have "y \<le> Max { a <..}"
     by fact
   with \<open>Max {a <..} < y\<close> show False by auto
 qed
 
 lemma infinite_Ici: "\<not> finite {a :: 'a :: {no_top, linorder} ..}"
   using infinite_Ioi[of a] finite_subset[of "{a <..}" "{a ..}"]
   by (auto simp: subset_eq less_imp_le)
 
 subsubsection \<open>Intersection\<close>
 
 context linorder
 begin
 
 lemma Int_atLeastAtMost[simp]: "{a..b} Int {c..d} = {max a c .. min b d}"
 by auto
 
 lemma Int_atLeastAtMostR1[simp]: "{..b} Int {c..d} = {c .. min b d}"
 by auto
 
 lemma Int_atLeastAtMostR2[simp]: "{a..} Int {c..d} = {max a c .. d}"
 by auto
 
 lemma Int_atLeastAtMostL1[simp]: "{a..b} Int {..d} = {a .. min b d}"
 by auto
 
 lemma Int_atLeastAtMostL2[simp]: "{a..b} Int {c..} = {max a c .. b}"
 by auto
 
 lemma Int_atLeastLessThan[simp]: "{a..<b} Int {c..<d} = {max a c ..< min b d}"
 by auto
 
 lemma Int_greaterThanAtMost[simp]: "{a<..b} Int {c<..d} = {max a c <.. min b d}"
 by auto
 
 lemma Int_greaterThanLessThan[simp]: "{a<..<b} Int {c<..<d} = {max a c <..< min b d}"
 by auto
 
 lemma Int_atMost[simp]: "{..a} \<inter> {..b} = {.. min a b}"
   by (auto simp: min_def)
 
 lemma Ioc_disjoint: "{a<..b} \<inter> {c<..d} = {} \<longleftrightarrow> b \<le> a \<or> d \<le> c \<or> b \<le> c \<or> d \<le> a"
   by auto
 
 end
 
 context complete_lattice
 begin
 
 lemma
   shows Sup_atLeast[simp]: "Sup {x ..} = top"
     and Sup_greaterThanAtLeast[simp]: "x < top \<Longrightarrow> Sup {x <..} = top"
     and Sup_atMost[simp]: "Sup {.. y} = y"
     and Sup_atLeastAtMost[simp]: "x \<le> y \<Longrightarrow> Sup { x .. y} = y"
     and Sup_greaterThanAtMost[simp]: "x < y \<Longrightarrow> Sup { x <.. y} = y"
   by (auto intro!: Sup_eqI)
 
 lemma
   shows Inf_atMost[simp]: "Inf {.. x} = bot"
     and Inf_atMostLessThan[simp]: "top < x \<Longrightarrow> Inf {..< x} = bot"
     and Inf_atLeast[simp]: "Inf {x ..} = x"
     and Inf_atLeastAtMost[simp]: "x \<le> y \<Longrightarrow> Inf { x .. y} = x"
     and Inf_atLeastLessThan[simp]: "x < y \<Longrightarrow> Inf { x ..< y} = x"
   by (auto intro!: Inf_eqI)
 
 end
 
 lemma
   fixes x y :: "'a :: {complete_lattice, dense_linorder}"
   shows Sup_lessThan[simp]: "Sup {..< y} = y"
     and Sup_atLeastLessThan[simp]: "x < y \<Longrightarrow> Sup { x ..< y} = y"
     and Sup_greaterThanLessThan[simp]: "x < y \<Longrightarrow> Sup { x <..< y} = y"
     and Inf_greaterThan[simp]: "Inf {x <..} = x"
     and Inf_greaterThanAtMost[simp]: "x < y \<Longrightarrow> Inf { x <.. y} = x"
     and Inf_greaterThanLessThan[simp]: "x < y \<Longrightarrow> Inf { x <..< y} = x"
   by (auto intro!: Inf_eqI Sup_eqI intro: dense_le dense_le_bounded dense_ge dense_ge_bounded)
 
 subsection \<open>Intervals of natural numbers\<close>
 
 subsubsection \<open>The Constant \<^term>\<open>lessThan\<close>\<close>
 
 lemma lessThan_0 [simp]: "lessThan (0::nat) = {}"
 by (simp add: lessThan_def)
 
 lemma lessThan_Suc: "lessThan (Suc k) = insert k (lessThan k)"
 by (simp add: lessThan_def less_Suc_eq, blast)
 
 text \<open>The following proof is convenient in induction proofs where
 new elements get indices at the beginning. So it is used to transform
 \<^term>\<open>{..<Suc n}\<close> to \<^term>\<open>0::nat\<close> and \<^term>\<open>{..< n}\<close>.\<close>
 
 lemma zero_notin_Suc_image [simp]: "0 \<notin> Suc ` A"
   by auto
 
 lemma lessThan_Suc_eq_insert_0: "{..<Suc n} = insert 0 (Suc ` {..<n})"
   by (auto simp: image_iff less_Suc_eq_0_disj)
 
 lemma lessThan_Suc_atMost: "lessThan (Suc k) = atMost k"
 by (simp add: lessThan_def atMost_def less_Suc_eq_le)
 
 lemma atMost_Suc_eq_insert_0: "{.. Suc n} = insert 0 (Suc ` {.. n})"
   unfolding lessThan_Suc_atMost[symmetric] lessThan_Suc_eq_insert_0[of "Suc n"] ..
 
 lemma UN_lessThan_UNIV: "(\<Union>m::nat. lessThan m) = UNIV"
 by blast
 
 subsubsection \<open>The Constant \<^term>\<open>greaterThan\<close>\<close>
 
 lemma greaterThan_0: "greaterThan 0 = range Suc"
   unfolding greaterThan_def
   by (blast dest: gr0_conv_Suc [THEN iffD1])
 
 lemma greaterThan_Suc: "greaterThan (Suc k) = greaterThan k - {Suc k}"
   unfolding greaterThan_def
   by (auto elim: linorder_neqE)
 
 lemma INT_greaterThan_UNIV: "(\<Inter>m::nat. greaterThan m) = {}"
   by blast
 
 subsubsection \<open>The Constant \<^term>\<open>atLeast\<close>\<close>
 
 lemma atLeast_0 [simp]: "atLeast (0::nat) = UNIV"
 by (unfold atLeast_def UNIV_def, simp)
 
 lemma atLeast_Suc: "atLeast (Suc k) = atLeast k - {k}"
   unfolding atLeast_def by (auto simp: order_le_less Suc_le_eq)
 
 lemma atLeast_Suc_greaterThan: "atLeast (Suc k) = greaterThan k"
   by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)
 
 lemma UN_atLeast_UNIV: "(\<Union>m::nat. atLeast m) = UNIV"
   by blast
 
 subsubsection \<open>The Constant \<^term>\<open>atMost\<close>\<close>
 
 lemma atMost_0 [simp]: "atMost (0::nat) = {0}"
   by (simp add: atMost_def)
 
 lemma atMost_Suc: "atMost (Suc k) = insert (Suc k) (atMost k)"
   unfolding atMost_def by (auto simp add: less_Suc_eq order_le_less)
 
 lemma UN_atMost_UNIV: "(\<Union>m::nat. atMost m) = UNIV"
   by blast
 
 subsubsection \<open>The Constant \<^term>\<open>atLeastLessThan\<close>\<close>
 
 text\<open>The orientation of the following 2 rules is tricky. The lhs is
 defined in terms of the rhs.  Hence the chosen orientation makes sense
 in this theory --- the reverse orientation complicates proofs (eg
 nontermination). But outside, when the definition of the lhs is rarely
 used, the opposite orientation seems preferable because it reduces a
 specific concept to a more general one.\<close>
 
 lemma atLeast0LessThan [code_abbrev]: "{0::nat..<n} = {..<n}"
   by(simp add:lessThan_def atLeastLessThan_def)
 
 lemma atLeast0AtMost [code_abbrev]: "{0..n::nat} = {..n}"
   by(simp add:atMost_def atLeastAtMost_def)
 
 lemma lessThan_atLeast0: "{..<n} = {0::nat..<n}"
   by (simp add: atLeast0LessThan)
 
 lemma atMost_atLeast0: "{..n} = {0::nat..n}"
   by (simp add: atLeast0AtMost)
 
 lemma atLeastLessThan0: "{m..<0::nat} = {}"
   by (simp add: atLeastLessThan_def)
 
 lemma atLeast0_lessThan_Suc: "{0..<Suc n} = insert n {0..<n}"
   by (simp add: atLeast0LessThan lessThan_Suc)
 
 lemma atLeast0_lessThan_Suc_eq_insert_0: "{0..<Suc n} = insert 0 (Suc ` {0..<n})"
   by (simp add: atLeast0LessThan lessThan_Suc_eq_insert_0)
 
 
 subsubsection \<open>The Constant \<^term>\<open>atLeastAtMost\<close>\<close>
 
 lemma Icc_eq_insert_lb_nat: "m \<le> n \<Longrightarrow> {m..n} = insert m {Suc m..n}"
 by auto
 
 lemma atLeast0_atMost_Suc:
   "{0..Suc n} = insert (Suc n) {0..n}"
   by (simp add: atLeast0AtMost atMost_Suc)
 
 lemma atLeast0_atMost_Suc_eq_insert_0:
   "{0..Suc n} = insert 0 (Suc ` {0..n})"
   by (simp add: atLeast0AtMost atMost_Suc_eq_insert_0)
 
 
 subsubsection \<open>Intervals of nats with \<^term>\<open>Suc\<close>\<close>
 
 text\<open>Not a simprule because the RHS is too messy.\<close>
 lemma atLeastLessThanSuc:
     "{m..<Suc n} = (if m \<le> n then insert n {m..<n} else {})"
 by (auto simp add: atLeastLessThan_def)
 
 lemma atLeastLessThan_singleton [simp]: "{m..<Suc m} = {m}"
 by (auto simp add: atLeastLessThan_def)
 
 lemma atLeastLessThanSuc_atLeastAtMost: "{l..<Suc u} = {l..u}"
   by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)
 
 lemma atLeastSucAtMost_greaterThanAtMost: "{Suc l..u} = {l<..u}"
   by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def
       greaterThanAtMost_def)
 
 lemma atLeastSucLessThan_greaterThanLessThan: "{Suc l..<u} = {l<..<u}"
   by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def
     greaterThanLessThan_def)
 
 lemma atLeastAtMostSuc_conv: "m \<le> Suc n \<Longrightarrow> {m..Suc n} = insert (Suc n) {m..n}"
   by auto
 
 lemma atLeastAtMost_insertL: "m \<le> n \<Longrightarrow> insert m {Suc m..n} = {m ..n}"
   by auto
 
 text \<open>The analogous result is useful on \<^typ>\<open>int\<close>:\<close>
 (* here, because we don't have an own int section *)
 lemma atLeastAtMostPlus1_int_conv:
   "m \<le> 1+n \<Longrightarrow> {m..1+n} = insert (1+n) {m..n::int}"
   by (auto intro: set_eqI)
 
 lemma atLeastLessThan_add_Un: "i \<le> j \<Longrightarrow> {i..<j+k} = {i..<j} \<union> {j..<j+k::nat}"
   by (induct k) (simp_all add: atLeastLessThanSuc)
 
 
 subsubsection \<open>Intervals and numerals\<close>
 
 lemma lessThan_nat_numeral:  \<comment> \<open>Evaluation for specific numerals\<close>
   "lessThan (numeral k :: nat) = insert (pred_numeral k) (lessThan (pred_numeral k))"
   by (simp add: numeral_eq_Suc lessThan_Suc)
 
 lemma atMost_nat_numeral:  \<comment> \<open>Evaluation for specific numerals\<close>
   "atMost (numeral k :: nat) = insert (numeral k) (atMost (pred_numeral k))"
   by (simp add: numeral_eq_Suc atMost_Suc)
 
 lemma atLeastLessThan_nat_numeral:  \<comment> \<open>Evaluation for specific numerals\<close>
   "atLeastLessThan m (numeral k :: nat) =
      (if m \<le> (pred_numeral k) then insert (pred_numeral k) (atLeastLessThan m (pred_numeral k))
                  else {})"
   by (simp add: numeral_eq_Suc atLeastLessThanSuc)
 
 
 subsubsection \<open>Image\<close>
 
 context linordered_semidom
 begin
 
 lemma image_add_atLeast[simp]: "plus k ` {i..} = {k + i..}"
 proof -
   have "n = k + (n - k)" if "i + k \<le> n" for n
   proof -
     have "n = (n - (k + i)) + (k + i)" using that
       by (metis add_commute le_add_diff_inverse)
     then show "n = k + (n - k)"
       by (metis local.add_diff_cancel_left' add_assoc add_commute)
   qed
   then show ?thesis
     by (fastforce simp: add_le_imp_le_diff add.commute)
 qed
 
 lemma image_add_atLeastAtMost [simp]:
   "plus k ` {i..j} = {i + k..j + k}" (is "?A = ?B")
 proof
   show "?A \<subseteq> ?B"
     by (auto simp add: ac_simps)
 next
   show "?B \<subseteq> ?A"
   proof
     fix n
     assume "n \<in> ?B"
     then have "i \<le> n - k"
       by (simp add: add_le_imp_le_diff)
     have "n = n - k + k"
     proof -
       from \<open>n \<in> ?B\<close> have "n = n - (i + k) + (i + k)"
         by simp
       also have "\<dots> = n - k - i + i + k"
         by (simp add: algebra_simps)
       also have "\<dots> = n - k + k"
         using \<open>i \<le> n - k\<close> by simp
       finally show ?thesis .
     qed
     moreover have "n - k \<in> {i..j}"
       using \<open>n \<in> ?B\<close>
       by (auto simp: add_le_imp_le_diff add_le_add_imp_diff_le)
     ultimately show "n \<in> ?A"
       by (simp add: ac_simps) 
   qed
 qed
 
 lemma image_add_atLeastAtMost' [simp]:
   "(\<lambda>n. n + k) ` {i..j} = {i + k..j + k}"
   by (simp add: add.commute [of _ k])
 
 lemma image_add_atLeastLessThan [simp]:
   "plus k ` {i..<j} = {i + k..<j + k}"
   by (simp add: image_set_diff atLeastLessThan_eq_atLeastAtMost_diff ac_simps)
 
 lemma image_add_atLeastLessThan' [simp]:
   "(\<lambda>n. n + k) ` {i..<j} = {i + k..<j + k}"
   by (simp add: add.commute [of _ k])
 
 lemma image_add_greaterThanAtMost[simp]: "(+) c ` {a<..b} = {c + a<..c + b}"
   by (simp add: image_set_diff greaterThanAtMost_eq_atLeastAtMost_diff ac_simps)
 
 end
 
 context ordered_ab_group_add
 begin
 
 lemma
   fixes x :: 'a
   shows image_uminus_greaterThan[simp]: "uminus ` {x<..} = {..<-x}"
   and image_uminus_atLeast[simp]: "uminus ` {x..} = {..-x}"
 proof safe
   fix y assume "y < -x"
   hence *:  "x < -y" using neg_less_iff_less[of "-y" x] by simp
   have "- (-y) \<in> uminus ` {x<..}"
     by (rule imageI) (simp add: *)
   thus "y \<in> uminus ` {x<..}" by simp
 next
   fix y assume "y \<le> -x"
   have "- (-y) \<in> uminus ` {x..}"
     by (rule imageI) (insert \<open>y \<le> -x\<close>[THEN le_imp_neg_le], simp)
   thus "y \<in> uminus ` {x..}" by simp
 qed simp_all
 
 lemma
   fixes x :: 'a
   shows image_uminus_lessThan[simp]: "uminus ` {..<x} = {-x<..}"
   and image_uminus_atMost[simp]: "uminus ` {..x} = {-x..}"
 proof -
   have "uminus ` {..<x} = uminus ` uminus ` {-x<..}"
     and "uminus ` {..x} = uminus ` uminus ` {-x..}" by simp_all
   thus "uminus ` {..<x} = {-x<..}" and "uminus ` {..x} = {-x..}"
     by (simp_all add: image_image
         del: image_uminus_greaterThan image_uminus_atLeast)
 qed
 
 lemma
   fixes x :: 'a
   shows image_uminus_atLeastAtMost[simp]: "uminus ` {x..y} = {-y..-x}"
   and image_uminus_greaterThanAtMost[simp]: "uminus ` {x<..y} = {-y..<-x}"
   and image_uminus_atLeastLessThan[simp]: "uminus ` {x..<y} = {-y<..-x}"
   and image_uminus_greaterThanLessThan[simp]: "uminus ` {x<..<y} = {-y<..<-x}"
   by (simp_all add: atLeastAtMost_def greaterThanAtMost_def atLeastLessThan_def
       greaterThanLessThan_def image_Int[OF inj_uminus] Int_commute)
 
 lemma image_add_atMost[simp]: "(+) c ` {..a} = {..c + a}"
   by (auto intro!: image_eqI[where x="x - c" for x] simp: algebra_simps)
 
 end
 
 lemma image_Suc_atLeastAtMost [simp]:
   "Suc ` {i..j} = {Suc i..Suc j}"
   using image_add_atLeastAtMost [of 1 i j]
     by (simp only: plus_1_eq_Suc) simp
 
 lemma image_Suc_atLeastLessThan [simp]:
   "Suc ` {i..<j} = {Suc i..<Suc j}"
   using image_add_atLeastLessThan [of 1 i j]
     by (simp only: plus_1_eq_Suc) simp
 
 corollary image_Suc_atMost:
   "Suc ` {..n} = {1..Suc n}"
   by (simp add: atMost_atLeast0 atLeastLessThanSuc_atLeastAtMost)
 
 corollary image_Suc_lessThan:
   "Suc ` {..<n} = {1..n}"
   by (simp add: lessThan_atLeast0 atLeastLessThanSuc_atLeastAtMost)
 
 lemma image_diff_atLeastAtMost [simp]:
   fixes d::"'a::linordered_idom" shows "((-) d ` {a..b}) = {d-b..d-a}"
   apply auto
   apply (rule_tac x="d-x" in rev_image_eqI, auto)
   done
 
 lemma image_diff_atLeastLessThan [simp]:
   fixes a b c::"'a::linordered_idom"
   shows "(-) c ` {a..<b} = {c - b<..c - a}"
 proof -
   have "(-) c ` {a..<b} = (+) c ` uminus ` {a ..<b}"
     unfolding image_image by simp
   also have "\<dots> = {c - b<..c - a}" by simp
   finally show ?thesis by simp
 qed
 
 lemma image_minus_const_greaterThanAtMost[simp]:
   fixes a b c::"'a::linordered_idom"
   shows "(-) c ` {a<..b} = {c - b..<c - a}"
 proof -
   have "(-) c ` {a<..b} = (+) c ` uminus ` {a<..b}"
     unfolding image_image by simp
   also have "\<dots> = {c - b..<c - a}" by simp
   finally show ?thesis by simp
 qed
 
 lemma image_minus_const_atLeast[simp]:
   fixes a c::"'a::linordered_idom"
   shows "(-) c ` {a..} = {..c - a}"
 proof -
   have "(-) c ` {a..} = (+) c ` uminus ` {a ..}"
     unfolding image_image by simp
   also have "\<dots> = {..c - a}" by simp
   finally show ?thesis by simp
 qed
 
 lemma image_minus_const_AtMost[simp]:
   fixes b c::"'a::linordered_idom"
   shows "(-) c ` {..b} = {c - b..}"
 proof -
   have "(-) c ` {..b} = (+) c ` uminus ` {..b}"
     unfolding image_image by simp
   also have "\<dots> = {c - b..}" by simp
   finally show ?thesis by simp
 qed
 
 lemma image_minus_const_atLeastAtMost' [simp]:
   "(\<lambda>t. t-d)`{a..b} = {a-d..b-d}" for d::"'a::linordered_idom"
   by (metis (no_types, lifting) diff_conv_add_uminus image_add_atLeastAtMost' image_cong)
 
 context linordered_field
 begin
 
 lemma image_mult_atLeastAtMost [simp]:
   "((*) d ` {a..b}) = {d*a..d*b}" if "d>0"
   using that
   by (auto simp: field_simps mult_le_cancel_right intro: rev_image_eqI [where x="x/d" for x])
 
 lemma image_divide_atLeastAtMost [simp]:
   "((\<lambda>c. c / d) ` {a..b}) = {a/d..b/d}" if "d>0"
 proof -
   from that have "inverse d > 0"
     by simp
   with image_mult_atLeastAtMost [of "inverse d" a b]
   have "(*) (inverse d) ` {a..b} = {inverse d * a..inverse d * b}"
     by blast
   moreover have "(*) (inverse d) = (\<lambda>c. c / d)"
     by (simp add: fun_eq_iff field_simps)
   ultimately show ?thesis
     by simp
 qed
 
 lemma image_mult_atLeastAtMost_if:
   "(*) c ` {x .. y} =
     (if c > 0 then {c * x .. c * y} else if x \<le> y then {c * y .. c * x} else {})"
 proof (cases "c = 0 \<or> x > y")
   case True
   then show ?thesis
     by auto
 next
   case False
   then have "x \<le> y"
     by auto
   from False consider "c < 0"| "c > 0"
     by (auto simp add: neq_iff)
   then show ?thesis
   proof cases
     case 1
     have "(*) c ` {x..y} = {c * y..c * x}"
     proof (rule set_eqI)
       fix d
       from 1 have "inj (\<lambda>z. z / c)"
         by (auto intro: injI)
       then have "d \<in> (*) c ` {x..y} \<longleftrightarrow> d / c \<in> (\<lambda>z. z div c) ` (*) c ` {x..y}"
         by (subst inj_image_mem_iff) simp_all
       also have "\<dots> \<longleftrightarrow> d / c \<in> {x..y}"
         using 1 by (simp add: image_image)
       also have "\<dots> \<longleftrightarrow> d \<in> {c * y..c * x}"
         by (auto simp add: field_simps 1)
       finally show "d \<in> (*) c ` {x..y} \<longleftrightarrow> d \<in> {c * y..c * x}" .
     qed
     with \<open>x \<le> y\<close> show ?thesis
       by auto
   qed (simp add: mult_left_mono_neg)
 qed
 
 lemma image_mult_atLeastAtMost_if':
   "(\<lambda>x. x * c) ` {x..y} =
     (if x \<le> y then if c > 0 then {x * c .. y * c} else {y * c .. x * c} else {})"
   using image_mult_atLeastAtMost_if [of c x y] by (auto simp add: ac_simps)
 
 lemma image_affinity_atLeastAtMost:
   "((\<lambda>x. m * x + c) ` {a..b}) = (if {a..b} = {} then {}
             else if 0 \<le> m then {m * a + c .. m * b + c}
             else {m * b + c .. m * a + c})"
 proof -
   have *: "(\<lambda>x. m * x + c) = ((\<lambda>x. x + c) \<circ> (*) m)"
     by (simp add: fun_eq_iff)
   show ?thesis by (simp only: * image_comp [symmetric] image_mult_atLeastAtMost_if)
     (auto simp add: mult_le_cancel_left)
 qed
 
 lemma image_affinity_atLeastAtMost_diff:
   "((\<lambda>x. m*x - c) ` {a..b}) = (if {a..b}={} then {}
             else if 0 \<le> m then {m*a - c .. m*b - c}
             else {m*b - c .. m*a - c})"
   using image_affinity_atLeastAtMost [of m "-c" a b]
   by simp
 
 lemma image_affinity_atLeastAtMost_div:
   "((\<lambda>x. x/m + c) ` {a..b}) = (if {a..b}={} then {}
             else if 0 \<le> m then {a/m + c .. b/m + c}
             else {b/m + c .. a/m + c})"
   using image_affinity_atLeastAtMost [of "inverse m" c a b]
   by (simp add: field_class.field_divide_inverse algebra_simps inverse_eq_divide)
 
 lemma image_affinity_atLeastAtMost_div_diff:
   "((\<lambda>x. x/m - c) ` {a..b}) = (if {a..b}={} then {}
             else if 0 \<le> m then {a/m - c .. b/m - c}
             else {b/m - c .. a/m - c})"
   using image_affinity_atLeastAtMost_diff [of "inverse m" c a b]
   by (simp add: field_class.field_divide_inverse algebra_simps inverse_eq_divide)
 
 end
 
 lemma atLeast1_lessThan_eq_remove0:
   "{Suc 0..<n} = {..<n} - {0}"
   by auto
 
 lemma atLeast1_atMost_eq_remove0:
   "{Suc 0..n} = {..n} - {0}"
   by auto
 
 lemma image_add_int_atLeastLessThan:
     "(\<lambda>x. x + (l::int)) ` {0..<u-l} = {l..<u}"
   apply (auto simp add: image_def)
   apply (rule_tac x = "x - l" in bexI)
   apply auto
   done
 
 lemma image_minus_const_atLeastLessThan_nat:
   fixes c :: nat
   shows "(\<lambda>i. i - c) ` {x ..< y} =
       (if c < y then {x - c ..< y - c} else if x < y then {0} else {})"
     (is "_ = ?right")
 proof safe
   fix a assume a: "a \<in> ?right"
   show "a \<in> (\<lambda>i. i - c) ` {x ..< y}"
   proof cases
     assume "c < y" with a show ?thesis
       by (auto intro!: image_eqI[of _ _ "a + c"])
   next
     assume "\<not> c < y" with a show ?thesis
       by (auto intro!: image_eqI[of _ _ x] split: if_split_asm)
   qed
 qed auto
 
 lemma image_int_atLeastLessThan:
   "int ` {a..<b} = {int a..<int b}"
   by (auto intro!: image_eqI [where x = "nat x" for x])
 
 lemma image_int_atLeastAtMost:
   "int ` {a..b} = {int a..int b}"
   by (auto intro!: image_eqI [where x = "nat x" for x])
 
 
 subsubsection \<open>Finiteness\<close>
 
 lemma finite_lessThan [iff]: fixes k :: nat shows "finite {..<k}"
   by (induct k) (simp_all add: lessThan_Suc)
 
 lemma finite_atMost [iff]: fixes k :: nat shows "finite {..k}"
   by (induct k) (simp_all add: atMost_Suc)
 
 lemma finite_greaterThanLessThan [iff]:
   fixes l :: nat shows "finite {l<..<u}"
   by (simp add: greaterThanLessThan_def)
 
 lemma finite_atLeastLessThan [iff]:
   fixes l :: nat shows "finite {l..<u}"
   by (simp add: atLeastLessThan_def)
 
 lemma finite_greaterThanAtMost [iff]:
   fixes l :: nat shows "finite {l<..u}"
   by (simp add: greaterThanAtMost_def)
 
 lemma finite_atLeastAtMost [iff]:
   fixes l :: nat shows "finite {l..u}"
   by (simp add: atLeastAtMost_def)
 
 text \<open>A bounded set of natural numbers is finite.\<close>
 lemma bounded_nat_set_is_finite: "(\<forall>i\<in>N. i < (n::nat)) \<Longrightarrow> finite N"
   by (rule finite_subset [OF _ finite_lessThan]) auto
 
 text \<open>A set of natural numbers is finite iff it is bounded.\<close>
 lemma finite_nat_set_iff_bounded:
   "finite(N::nat set) = (\<exists>m. \<forall>n\<in>N. n<m)" (is "?F = ?B")
 proof
   assume f:?F  show ?B
     using Max_ge[OF \<open>?F\<close>, simplified less_Suc_eq_le[symmetric]] by blast
 next
   assume ?B show ?F using \<open>?B\<close> by(blast intro:bounded_nat_set_is_finite)
 qed
 
 lemma finite_nat_set_iff_bounded_le: "finite(N::nat set) = (\<exists>m. \<forall>n\<in>N. n\<le>m)"
   unfolding finite_nat_set_iff_bounded
   by (blast dest:less_imp_le_nat le_imp_less_Suc)
 
 lemma finite_less_ub:
      "!!f::nat=>nat. (!!n. n \<le> f n) ==> finite {n. f n \<le> u}"
 by (rule_tac B="{..u}" in finite_subset, auto intro: order_trans)
 
 lemma bounded_Max_nat:
   fixes P :: "nat \<Rightarrow> bool"
   assumes x: "P x" and M: "\<And>x. P x \<Longrightarrow> x \<le> M"
   obtains m where "P m" "\<And>x. P x \<Longrightarrow> x \<le> m"
 proof -
   have "finite {x. P x}"
     using M finite_nat_set_iff_bounded_le by auto
   then have "Max {x. P x} \<in> {x. P x}"
     using Max_in x by auto
   then show ?thesis
     by (simp add: \<open>finite {x. P x}\<close> that)
 qed
 
 
 text\<open>Any subset of an interval of natural numbers the size of the
 subset is exactly that interval.\<close>
 
 lemma subset_card_intvl_is_intvl:
   assumes "A \<subseteq> {k..<k + card A}"
   shows "A = {k..<k + card A}"
 proof (cases "finite A")
   case True
   from this and assms show ?thesis
   proof (induct A rule: finite_linorder_max_induct)
     case empty thus ?case by auto
   next
     case (insert b A)
     hence *: "b \<notin> A" by auto
     with insert have "A \<le> {k..<k + card A}" and "b = k + card A"
       by fastforce+
     with insert * show ?case by auto
   qed
 next
   case False
   with assms show ?thesis by simp
 qed
 
 
 subsubsection \<open>Proving Inclusions and Equalities between Unions\<close>
 
 lemma UN_le_eq_Un0:
   "(\<Union>i\<le>n::nat. M i) = (\<Union>i\<in>{1..n}. M i) \<union> M 0" (is "?A = ?B")
 proof
   show "?A \<subseteq> ?B"
   proof
     fix x assume "x \<in> ?A"
     then obtain i where i: "i\<le>n" "x \<in> M i" by auto
     show "x \<in> ?B"
     proof(cases i)
       case 0 with i show ?thesis by simp
     next
       case (Suc j) with i show ?thesis by auto
     qed
   qed
 next
   show "?B \<subseteq> ?A" by fastforce
 qed
 
 lemma UN_le_add_shift:
   "(\<Union>i\<le>n::nat. M(i+k)) = (\<Union>i\<in>{k..n+k}. M i)" (is "?A = ?B")
 proof
   show "?A \<subseteq> ?B" by fastforce
 next
   show "?B \<subseteq> ?A"
   proof
     fix x assume "x \<in> ?B"
     then obtain i where i: "i \<in> {k..n+k}" "x \<in> M(i)" by auto
     hence "i-k\<le>n \<and> x \<in> M((i-k)+k)" by auto
     thus "x \<in> ?A" by blast
   qed
 qed
 
 lemma UN_le_add_shift_strict:
   "(\<Union>i<n::nat. M(i+k)) = (\<Union>i\<in>{k..<n+k}. M i)" (is "?A = ?B")
 proof
   show "?B \<subseteq> ?A"
   proof
     fix x assume "x \<in> ?B"
     then obtain i where i: "i \<in> {k..<n+k}" "x \<in> M(i)" by auto
     then have "i - k < n \<and> x \<in> M((i-k) + k)" by auto
     then show "x \<in> ?A" using UN_le_add_shift by blast
   qed
 qed (fastforce)
 
 lemma UN_UN_finite_eq: "(\<Union>n::nat. \<Union>i\<in>{0..<n}. A i) = (\<Union>n. A n)"
   by (auto simp add: atLeast0LessThan)
 
 lemma UN_finite_subset:
   "(\<And>n::nat. (\<Union>i\<in>{0..<n}. A i) \<subseteq> C) \<Longrightarrow> (\<Union>n. A n) \<subseteq> C"
   by (subst UN_UN_finite_eq [symmetric]) blast
 
 lemma UN_finite2_subset:
   assumes "\<And>n::nat. (\<Union>i\<in>{0..<n}. A i) \<subseteq> (\<Union>i\<in>{0..<n + k}. B i)"
   shows "(\<Union>n. A n) \<subseteq> (\<Union>n. B n)"
 proof (rule UN_finite_subset, rule)
   fix n and a
   from assms have "(\<Union>i\<in>{0..<n}. A i) \<subseteq> (\<Union>i\<in>{0..<n + k}. B i)" .
   moreover assume "a \<in> (\<Union>i\<in>{0..<n}. A i)"
   ultimately have "a \<in> (\<Union>i\<in>{0..<n + k}. B i)" by blast
   then show "a \<in> (\<Union>i. B i)" by (auto simp add: UN_UN_finite_eq)
 qed
 
 lemma UN_finite2_eq:
   "(\<And>n::nat. (\<Union>i\<in>{0..<n}. A i) = (\<Union>i\<in>{0..<n + k}. B i)) \<Longrightarrow>
     (\<Union>n. A n) = (\<Union>n. B n)"
   apply (rule subset_antisym [OF UN_finite_subset UN_finite2_subset])
    apply auto
   apply (force simp add: atLeastLessThan_add_Un [of 0])+
   done
 
 
 subsubsection \<open>Cardinality\<close>
 
 lemma card_lessThan [simp]: "card {..<u} = u"
   by (induct u, simp_all add: lessThan_Suc)
 
 lemma card_atMost [simp]: "card {..u} = Suc u"
   by (simp add: lessThan_Suc_atMost [THEN sym])
 
 lemma card_atLeastLessThan [simp]: "card {l..<u} = u - l"
 proof -
   have "{l..<u} = (\<lambda>x. x + l) ` {..<u-l}"
     apply (auto simp add: image_def atLeastLessThan_def lessThan_def)
     apply (rule_tac x = "x - l" in exI)
     apply arith
     done
   then have "card {l..<u} = card {..<u-l}"
     by (simp add: card_image inj_on_def)
   then show ?thesis
     by simp
 qed
 
 lemma card_atLeastAtMost [simp]: "card {l..u} = Suc u - l"
   by (subst atLeastLessThanSuc_atLeastAtMost [THEN sym], simp)
 
 lemma card_greaterThanAtMost [simp]: "card {l<..u} = u - l"
   by (subst atLeastSucAtMost_greaterThanAtMost [THEN sym], simp)
 
 lemma card_greaterThanLessThan [simp]: "card {l<..<u} = u - Suc l"
   by (subst atLeastSucLessThan_greaterThanLessThan [THEN sym], simp)
 
 lemma subset_eq_atLeast0_lessThan_finite:
   fixes n :: nat
   assumes "N \<subseteq> {0..<n}"
   shows "finite N"
   using assms finite_atLeastLessThan by (rule finite_subset)
 
 lemma subset_eq_atLeast0_atMost_finite:
   fixes n :: nat
   assumes "N \<subseteq> {0..n}"
   shows "finite N"
   using assms finite_atLeastAtMost by (rule finite_subset)
 
 lemma ex_bij_betw_nat_finite:
   "finite M \<Longrightarrow> \<exists>h. bij_betw h {0..<card M} M"
   apply(drule finite_imp_nat_seg_image_inj_on)
   apply(auto simp:atLeast0LessThan[symmetric] lessThan_def[symmetric] card_image bij_betw_def)
   done
 
 lemma ex_bij_betw_finite_nat:
   "finite M \<Longrightarrow> \<exists>h. bij_betw h M {0..<card M}"
   by (blast dest: ex_bij_betw_nat_finite bij_betw_inv)
 
 lemma finite_same_card_bij:
   "finite A \<Longrightarrow> finite B \<Longrightarrow> card A = card B \<Longrightarrow> \<exists>h. bij_betw h A B"
   apply(drule ex_bij_betw_finite_nat)
   apply(drule ex_bij_betw_nat_finite)
   apply(auto intro!:bij_betw_trans)
   done
 
 lemma ex_bij_betw_nat_finite_1:
   "finite M \<Longrightarrow> \<exists>h. bij_betw h {1 .. card M} M"
   by (rule finite_same_card_bij) auto
 
 lemma bij_betw_iff_card:
   assumes "finite A" "finite B"
   shows "(\<exists>f. bij_betw f A B) \<longleftrightarrow> (card A = card B)"
 proof
   assume "card A = card B"
   moreover obtain f where "bij_betw f A {0 ..< card A}"
     using assms ex_bij_betw_finite_nat by blast
   moreover obtain g where "bij_betw g {0 ..< card B} B"
     using assms ex_bij_betw_nat_finite by blast
   ultimately have "bij_betw (g \<circ> f) A B"
     by (auto simp: bij_betw_trans)
   thus "(\<exists>f. bij_betw f A B)" by blast
 qed (auto simp: bij_betw_same_card)
 
 lemma subset_eq_atLeast0_lessThan_card:
   fixes n :: nat
   assumes "N \<subseteq> {0..<n}"
   shows "card N \<le> n"
 proof -
   from assms finite_lessThan have "card N \<le> card {0..<n}"
     using card_mono by blast
   then show ?thesis by simp
 qed
 
 text \<open>Relational version of @{thm [source] card_inj_on_le}:\<close>
 lemma card_le_if_inj_on_rel:
 assumes "finite B"
   "\<And>a. a \<in> A \<Longrightarrow> \<exists>b. b\<in>B \<and> r a b"
   "\<And>a1 a2 b. \<lbrakk> a1 \<in> A;  a2 \<in> A;  b \<in> B;  r a1 b;  r a2 b \<rbrakk> \<Longrightarrow> a1 = a2"
 shows "card A \<le> card B"
 proof -
   let ?P = "\<lambda>a b. b \<in> B \<and> r a b"
   let ?f = "\<lambda>a. SOME b. ?P a b"
   have 1: "?f ` A \<subseteq> B"  by (auto intro: someI2_ex[OF assms(2)])
   have "inj_on ?f A"
   proof (auto simp: inj_on_def)
     fix a1 a2 assume asms: "a1 \<in> A" "a2 \<in> A" "?f a1 = ?f a2"
     have 0: "?f a1 \<in> B" using "1" \<open>a1 \<in> A\<close> by blast
     have 1: "r a1 (?f a1)" using someI_ex[OF assms(2)[OF \<open>a1 \<in> A\<close>]] by blast
     have 2: "r a2 (?f a1)" using someI_ex[OF assms(2)[OF \<open>a2 \<in> A\<close>]] asms(3) by auto
     show "a1 = a2" using assms(3)[OF asms(1,2) 0 1 2] .
   qed
   with 1 show ?thesis using card_inj_on_le[of ?f A B] assms(1) by simp
 qed
 
 
 subsection \<open>Intervals of integers\<close>
 
 lemma atLeastLessThanPlusOne_atLeastAtMost_int: "{l..<u+1} = {l..(u::int)}"
   by (auto simp add: atLeastAtMost_def atLeastLessThan_def)
 
 lemma atLeastPlusOneAtMost_greaterThanAtMost_int: "{l+1..u} = {l<..(u::int)}"
   by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)
 
 lemma atLeastPlusOneLessThan_greaterThanLessThan_int:
     "{l+1..<u} = {l<..<u::int}"
   by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)
 
 subsubsection \<open>Finiteness\<close>
 
 lemma image_atLeastZeroLessThan_int: "0 \<le> u ==>
     {(0::int)..<u} = int ` {..<nat u}"
   unfolding image_def lessThan_def
   apply auto
   apply (rule_tac x = "nat x" in exI)
   apply (auto simp add: zless_nat_eq_int_zless [THEN sym])
   done
 
 lemma finite_atLeastZeroLessThan_int: "finite {(0::int)..<u}"
 proof (cases "0 \<le> u")
   case True
   then show ?thesis
     by (auto simp: image_atLeastZeroLessThan_int)
 qed auto
 
 lemma finite_atLeastLessThan_int [iff]: "finite {l..<u::int}"
   by (simp only: image_add_int_atLeastLessThan [symmetric, of l] finite_imageI finite_atLeastZeroLessThan_int)
 
 lemma finite_atLeastAtMost_int [iff]: "finite {l..(u::int)}"
   by (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym], simp)
 
 lemma finite_greaterThanAtMost_int [iff]: "finite {l<..(u::int)}"
   by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
 
 lemma finite_greaterThanLessThan_int [iff]: "finite {l<..<u::int}"
   by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
 
 
 subsubsection \<open>Cardinality\<close>
 
 lemma card_atLeastZeroLessThan_int: "card {(0::int)..<u} = nat u"
 proof (cases "0 \<le> u")
   case True
   then show ?thesis
     by (auto simp: image_atLeastZeroLessThan_int card_image inj_on_def)    
 qed auto
 
 lemma card_atLeastLessThan_int [simp]: "card {l..<u} = nat (u - l)"
 proof -
   have "card {l..<u} = card {0..<u-l}"
     apply (subst image_add_int_atLeastLessThan [symmetric])
     apply (rule card_image)
     apply (simp add: inj_on_def)
     done
   then show ?thesis
     by (simp add: card_atLeastZeroLessThan_int)
 qed
 
 lemma card_atLeastAtMost_int [simp]: "card {l..u} = nat (u - l + 1)"
   apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])
   apply (auto simp add: algebra_simps)
   done
 
 lemma card_greaterThanAtMost_int [simp]: "card {l<..u} = nat (u - l)"
   by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
 
 lemma card_greaterThanLessThan_int [simp]: "card {l<..<u} = nat (u - (l + 1))"
   by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
 
 lemma finite_M_bounded_by_nat: "finite {k. P k \<and> k < (i::nat)}"
 proof -
   have "{k. P k \<and> k < i} \<subseteq> {..<i}" by auto
   with finite_lessThan[of "i"] show ?thesis by (simp add: finite_subset)
 qed
 
 lemma card_less:
   assumes zero_in_M: "0 \<in> M"
   shows "card {k \<in> M. k < Suc i} \<noteq> 0"
 proof -
   from zero_in_M have "{k \<in> M. k < Suc i} \<noteq> {}" by auto
   with finite_M_bounded_by_nat show ?thesis by (auto simp add: card_eq_0_iff)
 qed
 
 lemma card_less_Suc2: 
   assumes "0 \<notin> M" shows "card {k. Suc k \<in> M \<and> k < i} = card {k \<in> M. k < Suc i}"
 proof -
   have *: "\<lbrakk>j \<in> M; j < Suc i\<rbrakk> \<Longrightarrow> j - Suc 0 < i \<and> Suc (j - Suc 0) \<in> M \<and> Suc 0 \<le> j" for j
     by (cases j) (use assms in auto)
   show ?thesis
   proof (rule card_bij_eq)
     show "inj_on Suc {k. Suc k \<in> M \<and> k < i}"
       by force
     show "inj_on (\<lambda>x. x - Suc 0) {k \<in> M. k < Suc i}"
       by (rule inj_on_diff_nat) (use * in blast)
   qed (use * in auto)
 qed
 
 lemma card_less_Suc:
   assumes "0 \<in> M"
     shows "Suc (card {k. Suc k \<in> M \<and> k < i}) = card {k \<in> M. k < Suc i}"
 proof -
   have "Suc (card {k. Suc k \<in> M \<and> k < i}) = Suc (card {k. Suc k \<in> M - {0} \<and> k < i})"
     by simp
   also have "\<dots> = Suc (card {k \<in> M - {0}. k < Suc i})"
     apply (subst card_less_Suc2)
     using assms by auto
   also have "\<dots> = Suc (card ({k \<in> M. k < Suc i} - {0}))"
     by (force intro: arg_cong [where f=card])
   also have "\<dots> = card (insert 0 ({k \<in> M. k < Suc i} - {0}))"
     by (simp add: card.insert_remove)
   also have "... = card {k \<in> M. k < Suc i}"
     using assms
     by (force simp add: intro: arg_cong [where f=card])
   finally show ?thesis.
 qed
 
 
 subsection \<open>Lemmas useful with the summation operator sum\<close>
 
 text \<open>For examples, see Algebra/poly/UnivPoly2.thy\<close>
 
 subsubsection \<open>Disjoint Unions\<close>
 
 text \<open>Singletons and open intervals\<close>
 
 lemma ivl_disj_un_singleton:
   "{l::'a::linorder} Un {l<..} = {l..}"
   "{..<u} Un {u::'a::linorder} = {..u}"
   "(l::'a::linorder) < u ==> {l} Un {l<..<u} = {l..<u}"
   "(l::'a::linorder) < u ==> {l<..<u} Un {u} = {l<..u}"
   "(l::'a::linorder) \<le> u ==> {l} Un {l<..u} = {l..u}"
   "(l::'a::linorder) \<le> u ==> {l..<u} Un {u} = {l..u}"
 by auto
 
 text \<open>One- and two-sided intervals\<close>
 
 lemma ivl_disj_un_one:
   "(l::'a::linorder) < u ==> {..l} Un {l<..<u} = {..<u}"
   "(l::'a::linorder) \<le> u ==> {..<l} Un {l..<u} = {..<u}"
   "(l::'a::linorder) \<le> u ==> {..l} Un {l<..u} = {..u}"
   "(l::'a::linorder) \<le> u ==> {..<l} Un {l..u} = {..u}"
   "(l::'a::linorder) \<le> u ==> {l<..u} Un {u<..} = {l<..}"
   "(l::'a::linorder) < u ==> {l<..<u} Un {u..} = {l<..}"
   "(l::'a::linorder) \<le> u ==> {l..u} Un {u<..} = {l..}"
   "(l::'a::linorder) \<le> u ==> {l..<u} Un {u..} = {l..}"
 by auto
 
 text \<open>Two- and two-sided intervals\<close>
 
 lemma ivl_disj_un_two:
   "[| (l::'a::linorder) < m; m \<le> u |] ==> {l<..<m} Un {m..<u} = {l<..<u}"
   "[| (l::'a::linorder) \<le> m; m < u |] ==> {l<..m} Un {m<..<u} = {l<..<u}"
   "[| (l::'a::linorder) \<le> m; m \<le> u |] ==> {l..<m} Un {m..<u} = {l..<u}"
   "[| (l::'a::linorder) \<le> m; m < u |] ==> {l..m} Un {m<..<u} = {l..<u}"
   "[| (l::'a::linorder) < m; m \<le> u |] ==> {l<..<m} Un {m..u} = {l<..u}"
   "[| (l::'a::linorder) \<le> m; m \<le> u |] ==> {l<..m} Un {m<..u} = {l<..u}"
   "[| (l::'a::linorder) \<le> m; m \<le> u |] ==> {l..<m} Un {m..u} = {l..u}"
   "[| (l::'a::linorder) \<le> m; m \<le> u |] ==> {l..m} Un {m<..u} = {l..u}"
 by auto
 
 lemma ivl_disj_un_two_touch:
   "[| (l::'a::linorder) < m; m < u |] ==> {l<..m} Un {m..<u} = {l<..<u}"
   "[| (l::'a::linorder) \<le> m; m < u |] ==> {l..m} Un {m..<u} = {l..<u}"
   "[| (l::'a::linorder) < m; m \<le> u |] ==> {l<..m} Un {m..u} = {l<..u}"
   "[| (l::'a::linorder) \<le> m; m \<le> u |] ==> {l..m} Un {m..u} = {l..u}"
 by auto
 
 lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two ivl_disj_un_two_touch
 
 subsubsection \<open>Disjoint Intersections\<close>
 
 text \<open>One- and two-sided intervals\<close>
 
 lemma ivl_disj_int_one:
   "{..l::'a::order} Int {l<..<u} = {}"
   "{..<l} Int {l..<u} = {}"
   "{..l} Int {l<..u} = {}"
   "{..<l} Int {l..u} = {}"
   "{l<..u} Int {u<..} = {}"
   "{l<..<u} Int {u..} = {}"
   "{l..u} Int {u<..} = {}"
   "{l..<u} Int {u..} = {}"
   by auto
 
 text \<open>Two- and two-sided intervals\<close>
 
 lemma ivl_disj_int_two:
   "{l::'a::order<..<m} Int {m..<u} = {}"
   "{l<..m} Int {m<..<u} = {}"
   "{l..<m} Int {m..<u} = {}"
   "{l..m} Int {m<..<u} = {}"
   "{l<..<m} Int {m..u} = {}"
   "{l<..m} Int {m<..u} = {}"
   "{l..<m} Int {m..u} = {}"
   "{l..m} Int {m<..u} = {}"
   by auto
 
 lemmas ivl_disj_int = ivl_disj_int_one ivl_disj_int_two
 
 subsubsection \<open>Some Differences\<close>
 
 lemma ivl_diff[simp]:
  "i \<le> n \<Longrightarrow> {i..<m} - {i..<n} = {n..<(m::'a::linorder)}"
 by(auto)
 
 lemma (in linorder) lessThan_minus_lessThan [simp]:
   "{..< n} - {..< m} = {m ..< n}"
   by auto
 
 lemma (in linorder) atLeastAtMost_diff_ends:
   "{a..b} - {a, b} = {a<..<b}"
   by auto
 
 
 subsubsection \<open>Some Subset Conditions\<close>
 
 lemma ivl_subset [simp]: "({i..<j} \<subseteq> {m..<n}) = (j \<le> i \<or> m \<le> i \<and> j \<le> (n::'a::linorder))"
   using linorder_class.le_less_linear[of i n]
   apply (auto simp: linorder_not_le)
    apply (force intro: leI)+
   done
 
 lemma obtain_subset_with_card_n:
   assumes "n \<le> card S"
   obtains T where "T \<subseteq> S" "card T = n" "finite T"
 proof -
   obtain n' where "card S = n + n'" 
     by (metis assms le_add_diff_inverse)
   with that show thesis
   proof (induct n' arbitrary: S)
     case 0 
     then show ?case
       by (cases "finite S") auto
   next
     case Suc 
     then show ?case 
       by (simp add: card_Suc_eq) (metis subset_insertI2)
   qed
 qed
 
 subsection \<open>Generic big monoid operation over intervals\<close>
 
 context semiring_char_0
 begin
 
 lemma inj_on_of_nat [simp]:
   "inj_on of_nat N"
   by rule simp
 
 lemma bij_betw_of_nat [simp]:
   "bij_betw of_nat N A \<longleftrightarrow> of_nat ` N = A"
   by (simp add: bij_betw_def)
 
 end
 
 context comm_monoid_set
 begin
 
 lemma atLeastLessThan_reindex:
   "F g {h m..<h n} = F (g \<circ> h) {m..<n}"
   if "bij_betw h {m..<n} {h m..<h n}" for m n ::nat
 proof -
   from that have "inj_on h {m..<n}" and "h ` {m..<n} = {h m..<h n}"
     by (simp_all add: bij_betw_def)
   then show ?thesis
     using reindex [of h "{m..<n}" g] by simp
 qed
 
 lemma atLeastAtMost_reindex:
   "F g {h m..h n} = F (g \<circ> h) {m..n}"
   if "bij_betw h {m..n} {h m..h n}" for m n ::nat
 proof -
   from that have "inj_on h {m..n}" and "h ` {m..n} = {h m..h n}"
     by (simp_all add: bij_betw_def)
   then show ?thesis
     using reindex [of h "{m..n}" g] by simp
 qed
 
 lemma atLeastLessThan_shift_bounds:
   "F g {m + k..<n + k} = F (g \<circ> plus k) {m..<n}"
   for m n k :: nat
   using atLeastLessThan_reindex [of "plus k" m n g]
   by (simp add: ac_simps)
 
 lemma atLeastAtMost_shift_bounds:
   "F g {m + k..n + k} = F (g \<circ> plus k) {m..n}"
   for m n k :: nat
   using atLeastAtMost_reindex [of "plus k" m n g]
   by (simp add: ac_simps)
 
 lemma atLeast_Suc_lessThan_Suc_shift:
   "F g {Suc m..<Suc n} = F (g \<circ> Suc) {m..<n}"
   using atLeastLessThan_shift_bounds [of _ _ 1]
   by (simp add: plus_1_eq_Suc)
 
 lemma atLeast_Suc_atMost_Suc_shift:
   "F g {Suc m..Suc n} = F (g \<circ> Suc) {m..n}"
   using atLeastAtMost_shift_bounds [of _ _ 1]
   by (simp add: plus_1_eq_Suc)
 
 lemma atLeast_int_lessThan_int_shift:
   "F g {int m..<int n} = F (g \<circ> int) {m..<n}"
   by (rule atLeastLessThan_reindex)
     (simp add: image_int_atLeastLessThan)
 
 lemma atLeast_int_atMost_int_shift:
   "F g {int m..int n} = F (g \<circ> int) {m..n}"
   by (rule atLeastAtMost_reindex)
     (simp add: image_int_atLeastAtMost)
 
 lemma atLeast0_lessThan_Suc:
   "F g {0..<Suc n} = F g {0..<n} \<^bold>* g n"
   by (simp add: atLeast0_lessThan_Suc ac_simps)
 
 lemma atLeast0_atMost_Suc:
   "F g {0..Suc n} = F g {0..n} \<^bold>* g (Suc n)"
   by (simp add: atLeast0_atMost_Suc ac_simps)
 
 lemma atLeast0_lessThan_Suc_shift:
   "F g {0..<Suc n} = g 0 \<^bold>* F (g \<circ> Suc) {0..<n}"
   by (simp add: atLeast0_lessThan_Suc_eq_insert_0 atLeast_Suc_lessThan_Suc_shift)
 
 lemma atLeast0_atMost_Suc_shift:
   "F g {0..Suc n} = g 0 \<^bold>* F (g \<circ> Suc) {0..n}"
   by (simp add: atLeast0_atMost_Suc_eq_insert_0 atLeast_Suc_atMost_Suc_shift)
 
 lemma atLeast_Suc_lessThan:
   "F g {m..<n} = g m \<^bold>* F g {Suc m..<n}" if "m < n"
 proof -
   from that have "{m..<n} = insert m {Suc m..<n}"
     by auto
   then show ?thesis by simp
 qed
 
 lemma atLeast_Suc_atMost:
   "F g {m..n} = g m \<^bold>* F g {Suc m..n}" if "m \<le> n"
 proof -
   from that have "{m..n} = insert m {Suc m..n}"
     by auto
   then show ?thesis by simp
 qed
 
 lemma ivl_cong:
   "a = c \<Longrightarrow> b = d \<Longrightarrow> (\<And>x. c \<le> x \<Longrightarrow> x < d \<Longrightarrow> g x = h x)
     \<Longrightarrow> F g {a..<b} = F h {c..<d}"
   by (rule cong) simp_all
 
 lemma atLeastLessThan_shift_0:
   fixes m n p :: nat
   shows "F g {m..<n} = F (g \<circ> plus m) {0..<n - m}"
   using atLeastLessThan_shift_bounds [of g 0 m "n - m"]
   by (cases "m \<le> n") simp_all
 
 lemma atLeastAtMost_shift_0:
   fixes m n p :: nat
   assumes "m \<le> n"
   shows "F g {m..n} = F (g \<circ> plus m) {0..n - m}"
   using assms atLeastAtMost_shift_bounds [of g 0 m "n - m"] by simp
 
 lemma atLeastLessThan_concat:
   fixes m n p :: nat
   shows "m \<le> n \<Longrightarrow> n \<le> p \<Longrightarrow> F g {m..<n} \<^bold>* F g {n..<p} = F g {m..<p}"
   by (simp add: union_disjoint [symmetric] ivl_disj_un)
 
 lemma atLeastLessThan_rev:
   "F g {n..<m} = F (\<lambda>i. g (m + n - Suc i)) {n..<m}"
   by (rule reindex_bij_witness [where i="\<lambda>i. m + n - Suc i" and j="\<lambda>i. m + n - Suc i"], auto)
 
 lemma atLeastAtMost_rev:
   fixes n m :: nat
   shows "F g {n..m} = F (\<lambda>i. g (m + n - i)) {n..m}"
   by (rule reindex_bij_witness [where i="\<lambda>i. m + n - i" and j="\<lambda>i. m + n - i"]) auto
 
 lemma atLeastLessThan_rev_at_least_Suc_atMost:
   "F g {n..<m} = F (\<lambda>i. g (m + n - i)) {Suc n..m}"
   unfolding atLeastLessThan_rev [of g n m]
   by (cases m) (simp_all add: atLeast_Suc_atMost_Suc_shift atLeastLessThanSuc_atLeastAtMost)
 
 end
 
 
 subsection \<open>Summation indexed over intervals\<close>
 
 syntax (ASCII)
   "_from_to_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(SUM _ = _.._./ _)" [0,0,0,10] 10)
   "_from_upto_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(SUM _ = _..<_./ _)" [0,0,0,10] 10)
   "_upt_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(SUM _<_./ _)" [0,0,10] 10)
   "_upto_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(SUM _<=_./ _)" [0,0,10] 10)
 
 syntax (latex_sum output)
   "_from_to_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
  ("(3\<^latex>\<open>$\\sum_{\<close>_ = _\<^latex>\<open>}^{\<close>_\<^latex>\<open>}$\<close> _)" [0,0,0,10] 10)
   "_from_upto_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
  ("(3\<^latex>\<open>$\\sum_{\<close>_ = _\<^latex>\<open>}^{<\<close>_\<^latex>\<open>}$\<close> _)" [0,0,0,10] 10)
   "_upt_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
  ("(3\<^latex>\<open>$\\sum_{\<close>_ < _\<^latex>\<open>}$\<close> _)" [0,0,10] 10)
   "_upto_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
  ("(3\<^latex>\<open>$\\sum_{\<close>_ \<le> _\<^latex>\<open>}$\<close> _)" [0,0,10] 10)
 
 syntax
   "_from_to_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
   "_from_upto_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
   "_upt_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sum>_<_./ _)" [0,0,10] 10)
   "_upto_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
 
 translations
   "\<Sum>x=a..b. t" == "CONST sum (\<lambda>x. t) {a..b}"
   "\<Sum>x=a..<b. t" == "CONST sum (\<lambda>x. t) {a..<b}"
   "\<Sum>i\<le>n. t" == "CONST sum (\<lambda>i. t) {..n}"
   "\<Sum>i<n. t" == "CONST sum (\<lambda>i. t) {..<n}"
 
 text\<open>The above introduces some pretty alternative syntaxes for
 summation over intervals:
 \begin{center}
 \begin{tabular}{lll}
 Old & New & \LaTeX\\
 @{term[source]"\<Sum>x\<in>{a..b}. e"} & \<^term>\<open>\<Sum>x=a..b. e\<close> & @{term[mode=latex_sum]"\<Sum>x=a..b. e"}\\
 @{term[source]"\<Sum>x\<in>{a..<b}. e"} & \<^term>\<open>\<Sum>x=a..<b. e\<close> & @{term[mode=latex_sum]"\<Sum>x=a..<b. e"}\\
 @{term[source]"\<Sum>x\<in>{..b}. e"} & \<^term>\<open>\<Sum>x\<le>b. e\<close> & @{term[mode=latex_sum]"\<Sum>x\<le>b. e"}\\
 @{term[source]"\<Sum>x\<in>{..<b}. e"} & \<^term>\<open>\<Sum>x<b. e\<close> & @{term[mode=latex_sum]"\<Sum>x<b. e"}
 \end{tabular}
 \end{center}
 The left column shows the term before introduction of the new syntax,
 the middle column shows the new (default) syntax, and the right column
 shows a special syntax. The latter is only meaningful for latex output
 and has to be activated explicitly by setting the print mode to
 \<open>latex_sum\<close> (e.g.\ via \<open>mode = latex_sum\<close> in
 antiquotations). It is not the default \LaTeX\ output because it only
 works well with italic-style formulae, not tt-style.
 
 Note that for uniformity on \<^typ>\<open>nat\<close> it is better to use
 \<^term>\<open>\<Sum>x::nat=0..<n. e\<close> rather than \<open>\<Sum>x<n. e\<close>: \<open>sum\<close> may
 not provide all lemmas available for \<^term>\<open>{m..<n}\<close> also in the
 special form for \<^term>\<open>{..<n}\<close>.\<close>
 
 text\<open>This congruence rule should be used for sums over intervals as
 the standard theorem @{text[source]sum.cong} does not work well
 with the simplifier who adds the unsimplified premise \<^term>\<open>x\<in>B\<close> to
 the context.\<close>
 
 context comm_monoid_set
 begin
 
 lemma zero_middle:
   assumes "1 \<le> p" "k \<le> p"
   shows "F (\<lambda>j. if j < k then g j else if j = k then \<^bold>1 else h (j - Suc 0)) {..p}
        = F (\<lambda>j. if j < k then g j else h j) {..p - Suc 0}"  (is "?lhs = ?rhs")
 proof -
   have [simp]: "{..p - Suc 0} \<inter> {j. j < k} = {..<k}" "{..p - Suc 0} \<inter> - {j. j < k} = {k..p - Suc 0}"
     using assms by auto
   have "?lhs = F g {..<k} \<^bold>* F (\<lambda>j. if j = k then \<^bold>1 else h (j - Suc 0)) {k..p}"
     using union_disjoint [of "{..<k}" "{k..p}"] assms
     by (simp add: ivl_disj_int_one ivl_disj_un_one)
   also have "\<dots> = F g {..<k} \<^bold>* F (\<lambda>j.  h (j - Suc 0)) {Suc k..p}"
     by (simp add: atLeast_Suc_atMost [of k p] assms)
   also have "\<dots> = F g {..<k} \<^bold>* F h {k .. p - Suc 0}"
     using reindex [of Suc "{k..p - Suc 0}"] assms by simp
   also have "\<dots> = ?rhs"
     by (simp add: If_cases)
   finally show ?thesis .
 qed
 
 lemma atMost_Suc [simp]:
   "F g {..Suc n} = F g {..n} \<^bold>* g (Suc n)"
   by (simp add: atMost_Suc ac_simps)
 
 lemma lessThan_Suc [simp]:
   "F g {..<Suc n} = F g {..<n} \<^bold>* g n"
   by (simp add: lessThan_Suc ac_simps)
 
 lemma cl_ivl_Suc [simp]:
   "F g {m..Suc n} = (if Suc n < m then \<^bold>1 else F g {m..n} \<^bold>* g(Suc n))"
   by (auto simp: ac_simps atLeastAtMostSuc_conv)
 
 lemma op_ivl_Suc [simp]:
   "F g {m..<Suc n} = (if n < m then \<^bold>1 else F g {m..<n} \<^bold>* g(n))"
   by (auto simp: ac_simps atLeastLessThanSuc)
 
 lemma head:
   fixes n :: nat
   assumes mn: "m \<le> n"
   shows "F g {m..n} = g m \<^bold>* F g {m<..n}" (is "?lhs = ?rhs")
 proof -
   from mn
   have "{m..n} = {m} \<union> {m<..n}"
     by (auto intro: ivl_disj_un_singleton)
   hence "?lhs = F g ({m} \<union> {m<..n})"
     by (simp add: atLeast0LessThan)
   also have "\<dots> = ?rhs" by simp
   finally show ?thesis .
 qed
 
+lemma last_plus: 
+  fixes n::nat  shows "m \<le> n \<Longrightarrow> F g {m..n} = g n \<^bold>* F g {m..<n}"
+  by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost commute)
+
+lemma head_if:
+  fixes n :: nat
+  shows "F g {m..n} = (if n < m then \<^bold>1 else  F g {m..<n} \<^bold>* g(n))"
+  by (simp add: commute last_plus)
+
 lemma ub_add_nat: 
   assumes "(m::nat) \<le> n + 1"
   shows "F g {m..n + p} = F g {m..n} \<^bold>* F g {n + 1..n + p}"
 proof-
   have "{m .. n+p} = {m..n} \<union> {n+1..n+p}" using \<open>m \<le> n+1\<close> by auto
   thus ?thesis by (auto simp: ivl_disj_int union_disjoint atLeastSucAtMost_greaterThanAtMost)
 qed
 
 lemma nat_group: 
   fixes k::nat shows "F (\<lambda>m. F g {m * k ..< m*k + k}) {..<n} = F g {..< n * k}"
 proof (cases k)
   case (Suc l)
   then have "k > 0"
     by auto
   then show ?thesis
     by (induct n) (simp_all add: atLeastLessThan_concat add.commute atLeast0LessThan[symmetric])
 qed auto   
 
 lemma triangle_reindex:
   fixes n :: nat
   shows "F (\<lambda>(i,j). g i j) {(i,j). i+j < n} = F (\<lambda>k. F (\<lambda>i. g i (k - i)) {..k}) {..<n}"
   apply (simp add: Sigma)
   apply (rule reindex_bij_witness[where j="\<lambda>(i, j). (i+j, i)" and i="\<lambda>(k, i). (i, k - i)"])
   apply auto
   done
 
 lemma triangle_reindex_eq:
   fixes n :: nat
   shows "F (\<lambda>(i,j). g i j) {(i,j). i+j \<le> n} = F (\<lambda>k. F (\<lambda>i. g i (k - i)) {..k}) {..n}"
 using triangle_reindex [of g "Suc n"]
 by (simp only: Nat.less_Suc_eq_le lessThan_Suc_atMost)
 
 lemma nat_diff_reindex: "F (\<lambda>i. g (n - Suc i)) {..<n} = F g {..<n}"
   by (rule reindex_bij_witness[where i="\<lambda>i. n - Suc i" and j="\<lambda>i. n - Suc i"]) auto
 
 lemma shift_bounds_nat_ivl:
   "F g {m+k..<n+k} = F (\<lambda>i. g(i + k)){m..<n::nat}"
 by (induct "n", auto simp: atLeastLessThanSuc)
 
 lemma shift_bounds_cl_nat_ivl:
   "F g {m+k..n+k} = F (\<lambda>i. g(i + k)){m..n::nat}"
   by (rule reindex_bij_witness[where i="\<lambda>i. i + k" and j="\<lambda>i. i - k"]) auto
 
 corollary shift_bounds_cl_Suc_ivl:
   "F g {Suc m..Suc n} = F (\<lambda>i. g(Suc i)){m..n}"
 by (simp add: shift_bounds_cl_nat_ivl[where k="Suc 0", simplified])
 
 corollary Suc_reindex_ivl: "m \<le> n \<Longrightarrow> F g {m..n} \<^bold>* g (Suc n) = g m \<^bold>* F (\<lambda>i. g (Suc i)) {m..n}"
   by (simp add: assoc atLeast_Suc_atMost flip: shift_bounds_cl_Suc_ivl)
 
 corollary shift_bounds_Suc_ivl:
   "F g {Suc m..<Suc n} = F (\<lambda>i. g(Suc i)){m..<n}"
 by (simp add: shift_bounds_nat_ivl[where k="Suc 0", simplified])
 
 lemma atMost_Suc_shift:
   shows "F g {..Suc n} = g 0 \<^bold>* F (\<lambda>i. g (Suc i)) {..n}"
 proof (induct n)
   case 0 show ?case by simp
 next
   case (Suc n) note IH = this
   have "F g {..Suc (Suc n)} = F g {..Suc n} \<^bold>* g (Suc (Suc n))"
     by (rule atMost_Suc)
   also have "F g {..Suc n}  = g 0 \<^bold>* F (\<lambda>i. g (Suc i)) {..n}"
     by (rule IH)
   also have "g 0 \<^bold>* F (\<lambda>i. g (Suc i)) {..n} \<^bold>* g (Suc (Suc n)) =
              g 0 \<^bold>* (F (\<lambda>i. g (Suc i)) {..n} \<^bold>* g (Suc (Suc n)))"
     by (rule assoc)
   also have "F (\<lambda>i. g (Suc i)) {..n} \<^bold>* g (Suc (Suc n)) = F (\<lambda>i. g (Suc i)) {..Suc n}"
     by (rule atMost_Suc [symmetric])
   finally show ?case .
 qed
 
 lemma lessThan_Suc_shift:
   "F g {..<Suc n} = g 0 \<^bold>* F (\<lambda>i. g (Suc i)) {..<n}"
   by (induction n) (simp_all add: ac_simps)
 
 lemma atMost_shift:
   "F g {..n} = g 0 \<^bold>* F (\<lambda>i. g (Suc i)) {..<n}"
   by (metis atLeast0AtMost atLeast0LessThan atLeastLessThanSuc_atLeastAtMost 
        atLeastSucAtMost_greaterThanAtMost le0 head shift_bounds_Suc_ivl)
 
-lemma last_plus: 
-  fixes n::nat  shows "m \<le> n \<Longrightarrow> F g {m..n} = g n \<^bold>* F g {m..<n}"
-  by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost commute)
-
 lemma nested_swap:
      "F (\<lambda>i. F (\<lambda>j. a i j) {0..<i}) {0..n} = F (\<lambda>j. F (\<lambda>i. a i j) {Suc j..n}) {0..<n}"
   by (induction n) (auto simp: distrib)
 
 lemma nested_swap':
      "F (\<lambda>i. F (\<lambda>j. a i j) {..<i}) {..n} = F (\<lambda>j. F (\<lambda>i. a i j) {Suc j..n}) {..<n}"
   by (induction n) (auto simp: distrib)
 
 lemma atLeast1_atMost_eq:
   "F g {Suc 0..n} = F (\<lambda>k. g (Suc k)) {..<n}"
 proof -
   have "F g {Suc 0..n} = F g (Suc ` {..<n})"
     by (simp add: image_Suc_lessThan)
   also have "\<dots> = F (\<lambda>k. g (Suc k)) {..<n}"
     by (simp add: reindex)
   finally show ?thesis .
 qed
 
 lemma atLeastLessThan_Suc: "a \<le> b \<Longrightarrow> F g {a..<Suc b} = F g {a..<b} \<^bold>* g b"
   by (simp add: atLeastLessThanSuc commute)
 
 lemma nat_ivl_Suc':
   assumes "m \<le> Suc n"
   shows   "F g {m..Suc n} = g (Suc n) \<^bold>* F g {m..n}"
 proof -
   from assms have "{m..Suc n} = insert (Suc n) {m..n}" by auto
   also have "F g \<dots> = g (Suc n) \<^bold>* F g {m..n}" by simp
   finally show ?thesis .
 qed
 
 lemma in_pairs: "F g {2*m..Suc(2*n)} = F (\<lambda>i. g(2*i) \<^bold>* g(Suc(2*i))) {m..n}"
 proof (induction n)
   case 0
   show ?case
     by (cases "m=0") auto
 next
   case (Suc n)
   then show ?case
     by (auto simp: assoc split: if_split_asm)
 qed
 
 lemma in_pairs_0: "F g {..Suc(2*n)} = F (\<lambda>i. g(2*i) \<^bold>* g(Suc(2*i))) {..n}"
   using in_pairs [of _ 0 n] by (simp add: atLeast0AtMost)
 
 end
 
 lemma sum_natinterval_diff:
   fixes f:: "nat \<Rightarrow> ('a::ab_group_add)"
   shows  "sum (\<lambda>k. f k - f(k + 1)) {(m::nat) .. n} =
           (if m \<le> n then f m - f(n + 1) else 0)"
 by (induct n, auto simp add: algebra_simps not_le le_Suc_eq)
 
 lemma sum_diff_nat_ivl:
   fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
   shows "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow> sum f {m..<p} - sum f {m..<n} = sum f {n..<p}"
   using sum.atLeastLessThan_concat [of m n p f,symmetric]
   by (simp add: ac_simps)
 
 lemma sum_diff_distrib: "\<forall>x. Q x \<le> P x  \<Longrightarrow> (\<Sum>x<n. P x) - (\<Sum>x<n. Q x) = (\<Sum>x<n. P x - Q x :: nat)"
   by (subst sum_subtractf_nat) auto
 
 
 context unique_euclidean_semiring_with_bit_shifts
 begin
 
 lemma take_bit_sum:
   "take_bit n a = (\<Sum>k = 0..<n. push_bit k (of_bool (bit a k)))"
   for n :: nat
 proof (induction n arbitrary: a)
   case 0
   then show ?case
     by simp
 next
   case (Suc n)
   have "(\<Sum>k = 0..<Suc n. push_bit k (of_bool (bit a k))) = 
     of_bool (odd a) + (\<Sum>k = Suc 0..<Suc n. push_bit k (of_bool (bit a k)))"
     by (simp add: sum.atLeast_Suc_lessThan ac_simps)
   also have "(\<Sum>k = Suc 0..<Suc n. push_bit k (of_bool (bit a k)))
     = (\<Sum>k = 0..<n. push_bit k (of_bool (bit (a div 2) k))) * 2"
     by (simp only: sum.atLeast_Suc_lessThan_Suc_shift) (simp add: sum_distrib_right push_bit_double drop_bit_Suc bit_Suc)
   finally show ?case
     using Suc [of "a div 2"] by (simp add: ac_simps take_bit_Suc mod_2_eq_odd)
 qed
 
 end
 
 
 subsubsection \<open>Shifting bounds\<close>
 
 context comm_monoid_add
 begin
 
 context
   fixes f :: "nat \<Rightarrow> 'a"
   assumes "f 0 = 0"
 begin
 
 lemma sum_shift_lb_Suc0_0_upt:
   "sum f {Suc 0..<k} = sum f {0..<k}"
 proof (cases k)
   case 0
   then show ?thesis
     by simp
 next
   case (Suc k)
   moreover have "{0..<Suc k} = insert 0 {Suc 0..<Suc k}"
     by auto
   ultimately show ?thesis
     using \<open>f 0 = 0\<close> by simp
 qed
 
 lemma sum_shift_lb_Suc0_0: "sum f {Suc 0..k} = sum f {0..k}"
 proof (cases k)
   case 0
   with \<open>f 0 = 0\<close> show ?thesis
     by simp
 next
   case (Suc k)
   moreover have "{0..Suc k} = insert 0 {Suc 0..Suc k}"
     by auto
   ultimately show ?thesis
     using \<open>f 0 = 0\<close> by simp
 qed
 
 end
 
 end
 
 lemma sum_Suc_diff:
   fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
   assumes "m \<le> Suc n"
   shows "(\<Sum>i = m..n. f(Suc i) - f i) = f (Suc n) - f m"
 using assms by (induct n) (auto simp: le_Suc_eq)
 
 lemma sum_Suc_diff':
   fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
   assumes "m \<le> n"
   shows "(\<Sum>i = m..<n. f (Suc i) - f i) = f n - f m"
 using assms by (induct n) (auto simp: le_Suc_eq)
 
 
 subsubsection \<open>Telescoping\<close>
 
 lemma sum_telescope:
   fixes f::"nat \<Rightarrow> 'a::ab_group_add"
   shows "sum (\<lambda>i. f i - f (Suc i)) {.. i} = f 0 - f (Suc i)"
   by (induct i) simp_all
 
 lemma sum_telescope'':
   assumes "m \<le> n"
   shows   "(\<Sum>k\<in>{Suc m..n}. f k - f (k - 1)) = f n - (f m :: 'a :: ab_group_add)"
   by (rule dec_induct[OF assms]) (simp_all add: algebra_simps)
 
 lemma sum_lessThan_telescope:
   "(\<Sum>n<m. f (Suc n) - f n :: 'a :: ab_group_add) = f m - f 0"
   by (induction m) (simp_all add: algebra_simps)
 
 lemma sum_lessThan_telescope':
   "(\<Sum>n<m. f n - f (Suc n) :: 'a :: ab_group_add) = f 0 - f m"
   by (induction m) (simp_all add: algebra_simps)
 
 
 subsubsection \<open>The formula for geometric sums\<close>
 
 lemma sum_power2: "(\<Sum>i=0..<k. (2::nat)^i) = 2^k-1"
   by (induction k) (auto simp: mult_2)
 
 lemma geometric_sum:
   assumes "x \<noteq> 1"
   shows "(\<Sum>i<n. x ^ i) = (x ^ n - 1) / (x - 1::'a::field)"
 proof -
   from assms obtain y where "y = x - 1" and "y \<noteq> 0" by simp_all
   moreover have "(\<Sum>i<n. (y + 1) ^ i) = ((y + 1) ^ n - 1) / y"
     by (induct n) (simp_all add: field_simps \<open>y \<noteq> 0\<close>)
   ultimately show ?thesis by simp
 qed
 
 lemma diff_power_eq_sum:
   fixes y :: "'a::{comm_ring,monoid_mult}"
   shows
     "x ^ (Suc n) - y ^ (Suc n) =
       (x - y) * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))"
 proof (induct n)
   case (Suc n)
   have "x ^ Suc (Suc n) - y ^ Suc (Suc n) = x * (x * x^n) - y * (y * y ^ n)"
     by simp
   also have "... = y * (x ^ (Suc n) - y ^ (Suc n)) + (x - y) * (x * x^n)"
     by (simp add: algebra_simps)
   also have "... = y * ((x - y) * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))) + (x - y) * (x * x^n)"
     by (simp only: Suc)
   also have "... = (x - y) * (y * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))) + (x - y) * (x * x^n)"
     by (simp only: mult.left_commute)
   also have "... = (x - y) * (\<Sum>p<Suc (Suc n). x ^ p * y ^ (Suc n - p))"
     by (simp add: field_simps Suc_diff_le sum_distrib_right sum_distrib_left)
   finally show ?case .
 qed simp
 
 corollary power_diff_sumr2: \<comment> \<open>\<open>COMPLEX_POLYFUN\<close> in HOL Light\<close>
   fixes x :: "'a::{comm_ring,monoid_mult}"
   shows "x^n - y^n = (x - y) * (\<Sum>i<n. y^(n - Suc i) * x^i)"
 using diff_power_eq_sum[of x "n - 1" y]
 by (cases "n = 0") (simp_all add: field_simps)
 
 lemma power_diff_1_eq:
   fixes x :: "'a::{comm_ring,monoid_mult}"
   shows "x^n - 1 = (x - 1) * (\<Sum>i<n. (x^i))"
 using diff_power_eq_sum [of x _ 1]
   by (cases n) auto
 
 lemma one_diff_power_eq':
   fixes x :: "'a::{comm_ring,monoid_mult}"
   shows "1 - x^n = (1 - x) * (\<Sum>i<n. x^(n - Suc i))"
 using diff_power_eq_sum [of 1 _ x]
   by (cases n) auto
 
 lemma one_diff_power_eq:
   fixes x :: "'a::{comm_ring,monoid_mult}"
   shows "1 - x^n = (1 - x) * (\<Sum>i<n. x^i)"
 by (metis one_diff_power_eq' sum.nat_diff_reindex)
 
 lemma sum_gp_basic:
   fixes x :: "'a::{comm_ring,monoid_mult}"
   shows "(1 - x) * (\<Sum>i\<le>n. x^i) = 1 - x^Suc n"
   by (simp only: one_diff_power_eq lessThan_Suc_atMost)
 
 lemma sum_power_shift:
   fixes x :: "'a::{comm_ring,monoid_mult}"
   assumes "m \<le> n"
   shows "(\<Sum>i=m..n. x^i) = x^m * (\<Sum>i\<le>n-m. x^i)"
 proof -
   have "(\<Sum>i=m..n. x^i) = x^m * (\<Sum>i=m..n. x^(i-m))"
     by (simp add: sum_distrib_left power_add [symmetric])
   also have "(\<Sum>i=m..n. x^(i-m)) = (\<Sum>i\<le>n-m. x^i)"
     using \<open>m \<le> n\<close> by (intro sum.reindex_bij_witness[where j="\<lambda>i. i - m" and i="\<lambda>i. i + m"]) auto
   finally show ?thesis .
 qed
 
 lemma sum_gp_multiplied:
   fixes x :: "'a::{comm_ring,monoid_mult}"
   assumes "m \<le> n"
   shows "(1 - x) * (\<Sum>i=m..n. x^i) = x^m - x^Suc n"
 proof -
   have  "(1 - x) * (\<Sum>i=m..n. x^i) = x^m * (1 - x) * (\<Sum>i\<le>n-m. x^i)"
     by (metis mult.assoc mult.commute assms sum_power_shift)
   also have "... =x^m * (1 - x^Suc(n-m))"
     by (metis mult.assoc sum_gp_basic)
   also have "... = x^m - x^Suc n"
     using assms
     by (simp add: algebra_simps) (metis le_add_diff_inverse power_add)
   finally show ?thesis .
 qed
 
 lemma sum_gp:
   fixes x :: "'a::{comm_ring,division_ring}"
   shows   "(\<Sum>i=m..n. x^i) =
                (if n < m then 0
                 else if x = 1 then of_nat((n + 1) - m)
                 else (x^m - x^Suc n) / (1 - x))"
 using sum_gp_multiplied [of m n x] apply auto
 by (metis eq_iff_diff_eq_0 mult.commute nonzero_divide_eq_eq)
 
 
 subsubsection\<open>Geometric progressions\<close>
 
 lemma sum_gp0:
   fixes x :: "'a::{comm_ring,division_ring}"
   shows "(\<Sum>i\<le>n. x^i) = (if x = 1 then of_nat(n + 1) else (1 - x^Suc n) / (1 - x))"
   using sum_gp_basic[of x n]
   by (simp add: mult.commute field_split_simps)
 
 lemma sum_power_add:
   fixes x :: "'a::{comm_ring,monoid_mult}"
   shows "(\<Sum>i\<in>I. x^(m+i)) = x^m * (\<Sum>i\<in>I. x^i)"
   by (simp add: sum_distrib_left power_add)
 
 lemma sum_gp_offset:
   fixes x :: "'a::{comm_ring,division_ring}"
   shows   "(\<Sum>i=m..m+n. x^i) =
        (if x = 1 then of_nat n + 1 else x^m * (1 - x^Suc n) / (1 - x))"
   using sum_gp [of x m "m+n"]
   by (auto simp: power_add algebra_simps)
 
 lemma sum_gp_strict:
   fixes x :: "'a::{comm_ring,division_ring}"
   shows "(\<Sum>i<n. x^i) = (if x = 1 then of_nat n else (1 - x^n) / (1 - x))"
   by (induct n) (auto simp: algebra_simps field_split_simps)
 
 
 subsubsection \<open>The formulae for arithmetic sums\<close>
 
 context comm_semiring_1
 begin
 
 lemma double_gauss_sum:
   "2 * (\<Sum>i = 0..n. of_nat i) = of_nat n * (of_nat n + 1)"
   by (induct n) (simp_all add: sum.atLeast0_atMost_Suc algebra_simps left_add_twice)
 
 lemma double_gauss_sum_from_Suc_0:
   "2 * (\<Sum>i = Suc 0..n. of_nat i) = of_nat n * (of_nat n + 1)"
 proof -
   have "sum of_nat {Suc 0..n} = sum of_nat (insert 0 {Suc 0..n})"
     by simp
   also have "\<dots> = sum of_nat {0..n}"
     by (cases n) (simp_all add: atLeast0_atMost_Suc_eq_insert_0)
   finally show ?thesis
     by (simp add: double_gauss_sum)
 qed
 
 lemma double_arith_series:
   "2 * (\<Sum>i = 0..n. a + of_nat i * d) = (of_nat n + 1) * (2 * a + of_nat n * d)"
 proof -
   have "(\<Sum>i = 0..n. a + of_nat i * d) = ((\<Sum>i = 0..n. a) + (\<Sum>i = 0..n. of_nat i * d))"
     by (rule sum.distrib)
   also have "\<dots> = (of_nat (Suc n) * a + d * (\<Sum>i = 0..n. of_nat i))"
     by (simp add: sum_distrib_left algebra_simps)
   finally show ?thesis
     by (simp add: algebra_simps double_gauss_sum left_add_twice)
 qed
 
 end
 
 context unique_euclidean_semiring_with_nat
 begin
 
 lemma gauss_sum:
   "(\<Sum>i = 0..n. of_nat i) = of_nat n * (of_nat n + 1) div 2"
   using double_gauss_sum [of n, symmetric] by simp
 
 lemma gauss_sum_from_Suc_0:
   "(\<Sum>i = Suc 0..n. of_nat i) = of_nat n * (of_nat n + 1) div 2"
   using double_gauss_sum_from_Suc_0 [of n, symmetric] by simp
 
 lemma arith_series:
   "(\<Sum>i = 0..n. a + of_nat i * d) = (of_nat n + 1) * (2 * a + of_nat n * d) div 2"
   using double_arith_series [of a d n, symmetric] by simp
 
 end
 
 lemma gauss_sum_nat:
   "\<Sum>{0..n} = (n * Suc n) div 2"
   using gauss_sum [of n, where ?'a = nat] by simp
 
 lemma arith_series_nat:
   "(\<Sum>i = 0..n. a + i * d) = Suc n * (2 * a + n * d) div 2"
   using arith_series [of a d n] by simp
 
 lemma Sum_Icc_int:
   "\<Sum>{m..n} = (n * (n + 1) - m * (m - 1)) div 2"
   if "m \<le> n" for m n :: int
 using that proof (induct i \<equiv> "nat (n - m)" arbitrary: m n)
   case 0
   then have "m = n"
     by arith
   then show ?case
     by (simp add: algebra_simps mult_2 [symmetric])
 next
   case (Suc i)
   have 0: "i = nat((n-1) - m)" "m \<le> n-1" using Suc(2,3) by arith+
   have "\<Sum> {m..n} = \<Sum> {m..1+(n-1)}" by simp
   also have "\<dots> = \<Sum> {m..n-1} + n" using \<open>m \<le> n\<close>
     by(subst atLeastAtMostPlus1_int_conv) simp_all
   also have "\<dots> = ((n-1)*(n-1+1) - m*(m-1)) div 2 + n"
     by(simp add: Suc(1)[OF 0])
   also have "\<dots> = ((n-1)*(n-1+1) - m*(m-1) + 2*n) div 2" by simp
   also have "\<dots> = (n*(n+1) - m*(m-1)) div 2"
     by (simp add: algebra_simps mult_2_right)
   finally show ?case .
 qed
 
 lemma Sum_Icc_nat:
   "\<Sum>{m..n} = (n * (n + 1) - m * (m - 1)) div 2" for m n :: nat
 proof (cases "m \<le> n")
   case True
   then have *: "m * (m - 1) \<le> n * (n + 1)"
     by (meson diff_le_self order_trans le_add1 mult_le_mono)
   have "int (\<Sum>{m..n}) = (\<Sum>{int m..int n})"
     by (simp add: sum.atLeast_int_atMost_int_shift)
   also have "\<dots> = (int n * (int n + 1) - int m * (int m - 1)) div 2"
     using \<open>m \<le> n\<close> by (simp add: Sum_Icc_int)
   also have "\<dots> = int ((n * (n + 1) - m * (m - 1)) div 2)"
     using le_square * by (simp add: algebra_simps of_nat_div of_nat_diff)
   finally show ?thesis
     by (simp only: of_nat_eq_iff)
 next
   case False
   then show ?thesis
     by (auto dest: less_imp_Suc_add simp add: not_le algebra_simps)
 qed
 
 lemma Sum_Ico_nat: 
   "\<Sum>{m..<n} = (n * (n - 1) - m * (m - 1)) div 2" for m n :: nat
   by (cases n) (simp_all add: atLeastLessThanSuc_atLeastAtMost Sum_Icc_nat)
 
 
 subsubsection \<open>Division remainder\<close>
 
 lemma range_mod:
   fixes n :: nat
   assumes "n > 0"
   shows "range (\<lambda>m. m mod n) = {0..<n}" (is "?A = ?B")
 proof (rule set_eqI)
   fix m
   show "m \<in> ?A \<longleftrightarrow> m \<in> ?B"
   proof
     assume "m \<in> ?A"
     with assms show "m \<in> ?B"
       by auto
   next
     assume "m \<in> ?B"
     moreover have "m mod n \<in> ?A"
       by (rule rangeI)
     ultimately show "m \<in> ?A"
       by simp
   qed
 qed
 
 
 subsection \<open>Products indexed over intervals\<close>
 
 syntax (ASCII)
   "_from_to_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(PROD _ = _.._./ _)" [0,0,0,10] 10)
   "_from_upto_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(PROD _ = _..<_./ _)" [0,0,0,10] 10)
   "_upt_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(PROD _<_./ _)" [0,0,10] 10)
   "_upto_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(PROD _<=_./ _)" [0,0,10] 10)
 
 syntax (latex_prod output)
   "_from_to_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
  ("(3\<^latex>\<open>$\\prod_{\<close>_ = _\<^latex>\<open>}^{\<close>_\<^latex>\<open>}$\<close> _)" [0,0,0,10] 10)
   "_from_upto_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
  ("(3\<^latex>\<open>$\\prod_{\<close>_ = _\<^latex>\<open>}^{<\<close>_\<^latex>\<open>}$\<close> _)" [0,0,0,10] 10)
   "_upt_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
  ("(3\<^latex>\<open>$\\prod_{\<close>_ < _\<^latex>\<open>}$\<close> _)" [0,0,10] 10)
   "_upto_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
  ("(3\<^latex>\<open>$\\prod_{\<close>_ \<le> _\<^latex>\<open>}$\<close> _)" [0,0,10] 10)
 
 syntax
   "_from_to_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)
   "_from_upto_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)
   "_upt_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Prod>_<_./ _)" [0,0,10] 10)
   "_upto_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)
 
 translations
   "\<Prod>x=a..b. t" \<rightleftharpoons> "CONST prod (\<lambda>x. t) {a..b}"
   "\<Prod>x=a..<b. t" \<rightleftharpoons> "CONST prod (\<lambda>x. t) {a..<b}"
   "\<Prod>i\<le>n. t" \<rightleftharpoons> "CONST prod (\<lambda>i. t) {..n}"
   "\<Prod>i<n. t" \<rightleftharpoons> "CONST prod (\<lambda>i. t) {..<n}"
 
 lemma prod_int_plus_eq: "prod int {i..i+j} =  \<Prod>{int i..int (i+j)}"
   by (induct j) (auto simp add: atLeastAtMostSuc_conv atLeastAtMostPlus1_int_conv)
 
 lemma prod_int_eq: "prod int {i..j} =  \<Prod>{int i..int j}"
 proof (cases "i \<le> j")
   case True
   then show ?thesis
     by (metis le_iff_add prod_int_plus_eq)
 next
   case False
   then show ?thesis
     by auto
 qed
 
 subsection \<open>Efficient folding over intervals\<close>
 
 function fold_atLeastAtMost_nat where
   [simp del]: "fold_atLeastAtMost_nat f a (b::nat) acc =
                  (if a > b then acc else fold_atLeastAtMost_nat f (a+1) b (f a acc))"
 by pat_completeness auto
 termination by (relation "measure (\<lambda>(_,a,b,_). Suc b - a)") auto
 
 lemma fold_atLeastAtMost_nat:
   assumes "comp_fun_commute f"
   shows   "fold_atLeastAtMost_nat f a b acc = Finite_Set.fold f acc {a..b}"
 using assms
 proof (induction f a b acc rule: fold_atLeastAtMost_nat.induct, goal_cases)
   case (1 f a b acc)
   interpret comp_fun_commute f by fact
   show ?case
   proof (cases "a > b")
     case True
     thus ?thesis by (subst fold_atLeastAtMost_nat.simps) auto
   next
     case False
     with 1 show ?thesis
       by (subst fold_atLeastAtMost_nat.simps)
          (auto simp: atLeastAtMost_insertL[symmetric] fold_fun_left_comm)
   qed
 qed
 
 lemma sum_atLeastAtMost_code:
   "sum f {a..b} = fold_atLeastAtMost_nat (\<lambda>a acc. f a + acc) a b 0"
 proof -
   have "comp_fun_commute (\<lambda>a. (+) (f a))"
     by unfold_locales (auto simp: o_def add_ac)
   thus ?thesis
     by (simp add: sum.eq_fold fold_atLeastAtMost_nat o_def)
 qed
 
 lemma prod_atLeastAtMost_code:
   "prod f {a..b} = fold_atLeastAtMost_nat (\<lambda>a acc. f a * acc) a b 1"
 proof -
   have "comp_fun_commute (\<lambda>a. (*) (f a))"
     by unfold_locales (auto simp: o_def mult_ac)
   thus ?thesis
     by (simp add: prod.eq_fold fold_atLeastAtMost_nat o_def)
 qed
 
 (* TODO: Add support for folding over more kinds of intervals here *)
 
 end