Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
authorManuel Eberl <eberlm@in.tum.de>
Mon, 04 Feb 2019 17:19:04 +0100
changeset 69791 195aeee8b30a
parent 69790 154cf64e403e
child 69792 d21789843f01
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
CONTRIBUTORS
NEWS
src/HOL/Computational_Algebra/Computational_Algebra.thy
src/HOL/Computational_Algebra/Formal_Laurent_Series.thy
src/HOL/Computational_Algebra/Formal_Power_Series.thy
src/HOL/Fields.thy
src/HOL/Int.thy
src/HOL/Orderings.thy
src/HOL/Power.thy
--- a/CONTRIBUTORS	Mon Feb 04 15:39:37 2019 +0100
+++ b/CONTRIBUTORS	Mon Feb 04 17:19:04 2019 +0100
@@ -6,6 +6,9 @@
 Contributions to this Isabelle version
 --------------------------------------
 
+* February 2019: Jeremy Sylvestre
+  Formal Laurent Series and overhaul of Formal power series.
+
 * February 2019: Manuel Eberl
   Exponentiation by squaring, used to implement "power" in monoid_mult and
   fast modular exponentiation.
--- a/NEWS	Mon Feb 04 15:39:37 2019 +0100
+++ b/NEWS	Mon Feb 04 17:19:04 2019 +0100
@@ -87,6 +87,9 @@
 
 *** HOL ***
 
+* Formal Laurent series and overhaul of Formal power series 
+in HOL-Computational_Algebra
+
 * exponentiation by squaring in HOL-Library; used for computing powers in monoid_mult and modular exponentiation in HOL-Number_Theory
 
 * more material on residue rings in HOL-Number_Theory:
--- a/src/HOL/Computational_Algebra/Computational_Algebra.thy	Mon Feb 04 15:39:37 2019 +0100
+++ b/src/HOL/Computational_Algebra/Computational_Algebra.thy	Mon Feb 04 17:19:04 2019 +0100
@@ -5,7 +5,7 @@
 imports
   Euclidean_Algorithm
   Factorial_Ring
-  Formal_Power_Series
+  Formal_Laurent_Series
   Fraction_Field
   Fundamental_Theorem_Algebra
   Group_Closure
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Computational_Algebra/Formal_Laurent_Series.thy	Mon Feb 04 17:19:04 2019 +0100
@@ -0,0 +1,4178 @@
+(*
+  Title:      HOL/Computational_Algebra/Formal_Laurent_Series.thy
+  Author:     Jeremy Sylvestre, University of Alberta (Augustana Campus)
+*)
+
+
+section \<open>A formalization of formal Laurent series\<close>
+
+theory Formal_Laurent_Series
+imports
+  Polynomial_FPS
+begin
+
+
+subsection \<open>The type of formal Laurent series\<close>
+
+subsubsection \<open>Type definition\<close>
+
+typedef (overloaded) 'a fls = "{f::int \<Rightarrow> 'a::zero. \<forall>\<^sub>\<infinity> n::nat. f (- int n) = 0}"
+  morphisms fls_nth Abs_fls
+proof
+  show "(\<lambda>x. 0) \<in> {f::int \<Rightarrow> 'a::zero. \<forall>\<^sub>\<infinity> n::nat. f (- int n) = 0}"
+    by simp
+qed
+
+setup_lifting type_definition_fls
+
+unbundle fps_notation
+notation fls_nth (infixl "$$" 75)
+
+lemmas fls_eqI = iffD1[OF fls_nth_inject, OF iffD2, OF fun_eq_iff, OF allI]
+
+lemma expand_fls_eq: "f = g \<longleftrightarrow> (\<forall>n. f $$ n = g $$ n)"
+  by (simp add: fls_nth_inject[symmetric] fun_eq_iff)
+
+lemma nth_Abs_fls [simp]: "\<forall>\<^sub>\<infinity>n. f (- int n) = 0 \<Longrightarrow> Abs_fls f $$ n = f n"
+ by (simp add: Abs_fls_inverse[OF CollectI])
+
+lemmas nth_Abs_fls_finite_nonzero_neg_nth = nth_Abs_fls[OF iffD2, OF eventually_cofinite]
+lemmas nth_Abs_fls_ex_nat_lower_bound = nth_Abs_fls[OF iffD2, OF MOST_nat]
+lemmas nth_Abs_fls_nat_lower_bound = nth_Abs_fls_ex_nat_lower_bound[OF exI]
+
+lemma nth_Abs_fls_ex_lower_bound:
+  assumes "\<exists>N. \<forall>n<N. f n = 0"
+  shows   "Abs_fls f $$ n = f n"
+proof (intro nth_Abs_fls_ex_nat_lower_bound)
+  from assms obtain N::int where "\<forall>n<N. f n = 0" by fast
+  hence "\<forall>n > (if N < 0 then nat (-N) else 0). f (-int n) = 0" by auto
+  thus "\<exists>M. \<forall>n>M. f (- int n) = 0" by fast
+qed
+
+lemmas nth_Abs_fls_lower_bound = nth_Abs_fls_ex_lower_bound[OF exI]
+
+lemmas MOST_fls_neg_nth_eq_0 [simp] = CollectD[OF fls_nth]
+lemmas fls_finite_nonzero_neg_nth = iffD1[OF eventually_cofinite MOST_fls_neg_nth_eq_0]
+
+lemma fls_nth_vanishes_below_natE:
+  fixes   f :: "'a::zero fls"
+  obtains N :: nat
+  where   "\<forall>n>N. f$$(-int n) = 0"
+  using   iffD1[OF MOST_nat MOST_fls_neg_nth_eq_0]
+  by      blast
+
+lemma fls_nth_vanishes_belowE:
+  fixes   f :: "'a::zero fls"
+  obtains N :: int
+  where   "\<forall>n<N. f$$n = 0"
+proof-
+  obtain K :: nat where K: "\<forall>n>K. f$$(-int n) = 0" by (elim fls_nth_vanishes_below_natE)
+  have "\<forall>n < -int K. f$$n = 0"
+  proof clarify
+    fix n assume n: "n < -int K"
+    define m where "m \<equiv> nat (-n)"
+    with n have "m > K" by simp
+    moreover from n m_def have "f$$n = f $$ (-int m)" by simp
+    ultimately show "f $$ n = 0" using K by simp
+  qed
+  thus "(\<And>N. \<forall>n<N. f $$ n = 0 \<Longrightarrow> thesis) \<Longrightarrow> thesis" by fast
+qed
+
+
+subsubsection \<open>Definition of basic zero, one, constant, X, and inverse X elements\<close>
+
+instantiation fls :: (zero) zero
+begin
+  lift_definition zero_fls :: "'a fls" is "\<lambda>_. 0" by simp
+  instance ..
+end
+
+lemma fls_zero_nth [simp]: "0 $$ n = 0"
+ by (simp add: zero_fls_def)
+
+lemma fls_zero_eqI: "(\<And>n. f$$n = 0) \<Longrightarrow> f = 0"
+  by (fastforce intro: fls_eqI)
+
+lemma fls_nonzeroI: "f$$n \<noteq> 0 \<Longrightarrow> f \<noteq> 0"
+  by auto
+
+lemma fls_nonzero_nth: "f \<noteq> 0 \<longleftrightarrow> (\<exists> n. f $$ n \<noteq> 0)"
+  using fls_zero_eqI by fastforce
+
+lemma fls_trivial_delta_eq_zero [simp]: "b = 0 \<Longrightarrow> Abs_fls (\<lambda>n. if n=a then b else 0) = 0"
+  by (intro fls_zero_eqI) simp
+
+lemma fls_delta_nth [simp]:
+  "Abs_fls (\<lambda>n. if n=a then b else 0) $$ n = (if n=a then b else 0)"
+  using nth_Abs_fls_lower_bound[of a "\<lambda>n. if n=a then b else 0"] by simp
+
+instantiation fls :: ("{zero,one}") one
+begin
+  lift_definition one_fls :: "'a fls" is "\<lambda>k. if k = 0 then 1 else 0"
+    by (simp add: eventually_cofinite)
+  instance ..
+end
+
+lemma fls_one_nth [simp]:
+  "1 $$ n = (if n = 0 then 1 else 0)"
+  by (simp add: one_fls_def eventually_cofinite)
+
+instance fls :: (zero_neq_one) zero_neq_one
+proof (standard, standard)
+  assume "(0::'a fls) = (1::'a fls)"
+  hence "(0::'a fls) $$ 0 = (1::'a fls) $$ 0" by simp
+  thus False by simp
+qed
+
+definition fls_const :: "'a::zero \<Rightarrow> 'a fls"
+  where "fls_const c \<equiv> Abs_fls (\<lambda>n. if n = 0 then c else 0)"
+
+lemma fls_const_nth [simp]: "fls_const c $$ n = (if n = 0 then c else 0)"
+  by (simp add: fls_const_def eventually_cofinite)
+
+lemma fls_const_0 [simp]: "fls_const 0 = 0"
+  unfolding fls_const_def using fls_trivial_delta_eq_zero by fast
+
+lemma fls_const_nonzero: "c \<noteq> 0 \<Longrightarrow> fls_const c \<noteq> 0"
+  using fls_nonzeroI[of "fls_const c" 0] by simp
+
+lemma fls_const_1 [simp]: "fls_const 1 = 1"
+  unfolding fls_const_def one_fls_def ..
+
+lift_definition fls_X :: "'a::{zero,one} fls"
+  is "\<lambda>n. if n = 1 then 1 else 0"
+  by simp
+
+lemma fls_X_nth [simp]:
+  "fls_X $$ n = (if n = 1 then 1 else 0)"
+  by (simp add: fls_X_def)
+
+lemma fls_X_nonzero [simp]: "(fls_X :: 'a :: zero_neq_one fls) \<noteq> 0"
+  by (intro fls_nonzeroI) simp
+
+lift_definition fls_X_inv :: "'a::{zero,one} fls"
+  is "\<lambda>n. if n = -1 then 1 else 0"
+  by (simp add: eventually_cofinite)
+
+lemma fls_X_inv_nth [simp]:
+  "fls_X_inv $$ n = (if n = -1 then 1 else 0)"
+  by (simp add: fls_X_inv_def eventually_cofinite)
+
+lemma fls_X_inv_nonzero [simp]: "(fls_X_inv :: 'a :: zero_neq_one fls) \<noteq> 0"
+  by (intro fls_nonzeroI) simp
+
+
+subsection \<open>Subdegrees\<close>
+
+lemma unique_fls_subdegree:
+  assumes "f \<noteq> 0"
+  shows   "\<exists>!n. f$$n \<noteq> 0 \<and> (\<forall>m. f$$m \<noteq> 0 \<longrightarrow> n \<le> m)"
+proof-
+  obtain N::nat where N: "\<forall>n>N. f$$(-int n) = 0" by (elim fls_nth_vanishes_below_natE)
+  define M where "M \<equiv> -int N"
+  have M: "\<And>m. f$$m \<noteq> 0 \<Longrightarrow> M \<le> m"
+  proof-
+    fix m assume m: "f$$m \<noteq> 0"
+    show "M \<le> m"
+    proof (cases "m<0")
+      case True with m N M_def show ?thesis
+        using allE[OF N, of "nat (-m)" False] by force
+    qed (simp add: M_def)
+  qed
+  have "\<not> (\<forall>k::nat. f$$(M + int k) = 0)"
+  proof
+    assume above0: "\<forall>k::nat. f$$(M + int k) = 0"
+    have "f=0"
+    proof (rule fls_zero_eqI)
+      fix n show "f$$n = 0"
+      proof (cases "M \<le> n")
+        case True
+        define k where "k = nat (n - M)"
+        from True have "n = M + int k" by (simp add: k_def)
+        with above0 show ?thesis by simp
+      next
+        case False with M show ?thesis by auto
+      qed
+    qed
+    with assms show False by fast
+  qed
+  hence ex_k: "\<exists>k::nat. f$$(M + int k) \<noteq> 0" by fast
+  define k where "k \<equiv> (LEAST k::nat. f$$(M + int k) \<noteq> 0)"
+  define n where "n \<equiv> M + int k"
+  from k_def n_def have fn: "f$$n \<noteq> 0" using LeastI_ex[OF ex_k] by simp
+  moreover have "\<forall>m. f$$m \<noteq> 0 \<longrightarrow> n \<le> m"
+  proof (clarify)
+    fix m assume m: "f$$m \<noteq> 0"
+    with M have "M \<le> m" by fast
+    define l where "l = nat (m - M)"
+    from \<open>M \<le> m\<close> have l: "m = M + int l" by (simp add: l_def)
+    with n_def m k_def l show "n \<le> m"
+      using Least_le[of "\<lambda>k. f$$(M + int k) \<noteq> 0" l] by auto
+  qed
+  moreover have "\<And>n'. f$$n' \<noteq> 0 \<Longrightarrow> (\<forall>m. f$$m \<noteq> 0 \<longrightarrow> n' \<le> m) \<Longrightarrow> n' = n"
+  proof-
+    fix n' :: int
+    assume n': "f$$n' \<noteq> 0" "\<forall>m. f$$m \<noteq> 0 \<longrightarrow> n' \<le> m"
+    from n'(1) M have "M \<le> n'" by fast
+    define l where "l = nat (n' - M)"
+    from \<open>M \<le> n'\<close> have l: "n' = M + int l" by (simp add: l_def)
+    with n_def k_def n' fn show "n' = n"
+      using Least_le[of "\<lambda>k. f$$(M + int k) \<noteq> 0" l] by force
+  qed
+  ultimately show ?thesis
+    using ex1I[of "\<lambda>n. f$$n \<noteq> 0 \<and> (\<forall>m. f$$m \<noteq> 0 \<longrightarrow> n \<le> m)" n] by blast
+qed
+
+definition fls_subdegree :: "('a::zero) fls \<Rightarrow> int"
+  where "fls_subdegree f \<equiv> (if f = 0 then 0 else LEAST n::int. f$$n \<noteq> 0)"
+
+lemma fls_zero_subdegree [simp]: "fls_subdegree 0 = 0"
+  by (simp add: fls_subdegree_def)
+
+lemma nth_fls_subdegree_nonzero [simp]: "f \<noteq> 0 \<Longrightarrow> f $$ fls_subdegree f \<noteq> 0"
+  using Least1I[OF unique_fls_subdegree] by (simp add: fls_subdegree_def)
+
+lemma nth_fls_subdegree_zero_iff: "(f $$ fls_subdegree f = 0) \<longleftrightarrow> (f = 0)"
+  using nth_fls_subdegree_nonzero by auto
+
+lemma fls_subdegree_leI: "f $$ n \<noteq> 0 \<Longrightarrow> fls_subdegree f \<le> n"
+  using Least1_le[OF unique_fls_subdegree]
+  by    (auto simp: fls_subdegree_def)
+
+lemma fls_subdegree_leI': "f $$ n \<noteq> 0 \<Longrightarrow> n \<le> m \<Longrightarrow> fls_subdegree f \<le> m"
+  using fls_subdegree_leI by fastforce
+
+lemma fls_eq0_below_subdegree [simp]: "n < fls_subdegree f \<Longrightarrow> f $$ n = 0"
+  using fls_subdegree_leI by fastforce
+
+lemma fls_subdegree_geI: "f \<noteq> 0 \<Longrightarrow> (\<And>k. k < n \<Longrightarrow> f $$ k = 0) \<Longrightarrow> n \<le> fls_subdegree f"
+  using nth_fls_subdegree_nonzero by force
+
+lemma fls_subdegree_ge0I: "(\<And>k. k < 0 \<Longrightarrow> f $$ k = 0) \<Longrightarrow> 0 \<le> fls_subdegree f"
+  using fls_subdegree_geI[of f 0] by (cases "f=0") auto
+
+lemma fls_subdegree_greaterI:
+  assumes "f \<noteq> 0" "\<And>k. k \<le> n \<Longrightarrow> f $$ k = 0"
+  shows   "n < fls_subdegree f"
+  using   assms(1) assms(2)[of "fls_subdegree f"] nth_fls_subdegree_nonzero[of f]
+  by      force
+
+lemma fls_subdegree_eqI: "f $$ n \<noteq> 0 \<Longrightarrow> (\<And>k. k < n \<Longrightarrow> f $$ k = 0) \<Longrightarrow> fls_subdegree f = n"
+  using fls_subdegree_leI fls_subdegree_geI[of f]
+  by    fastforce
+
+lemma fls_delta_subdegree [simp]:
+  "b \<noteq> 0 \<Longrightarrow> fls_subdegree (Abs_fls (\<lambda>n. if n=a then b else 0)) = a"
+  by (intro fls_subdegree_eqI) simp_all
+
+lemma fls_delta0_subdegree: "fls_subdegree (Abs_fls (\<lambda>n. if n=0 then a else 0)) = 0"
+  by (cases "a=0") simp_all
+
+lemma fls_one_subdegree [simp]: "fls_subdegree 1 = 0"
+  by (auto intro: fls_delta0_subdegree simp: one_fls_def)
+
+lemma fls_const_subdegree [simp]: "fls_subdegree (fls_const c) = 0"
+  by (cases "c=0") (auto intro: fls_subdegree_eqI)
+
+lemma fls_X_subdegree [simp]: "fls_subdegree (fls_X::'a::{zero_neq_one} fls) = 1"
+  by (intro fls_subdegree_eqI) simp_all
+
+lemma fls_X_inv_subdegree [simp]: "fls_subdegree (fls_X_inv::'a::{zero_neq_one} fls) = -1"
+  by (intro fls_subdegree_eqI) simp_all
+
+lemma fls_eq_above_subdegreeI:
+  assumes "N \<le> fls_subdegree f" "N \<le> fls_subdegree g" "\<forall>k\<ge>N. f $$ k = g $$ k"
+  shows   "f = g"
+proof (rule fls_eqI)
+  fix n from assms show "f $$ n = g $$ n" by (cases "n < N") auto
+qed
+
+
+subsection \<open>Shifting\<close>
+
+subsubsection \<open>Shift definition\<close>
+
+definition fls_shift :: "int \<Rightarrow> ('a::zero) fls \<Rightarrow> 'a fls"
+  where "fls_shift n f \<equiv> Abs_fls (\<lambda>k. f $$ (k+n))"
+\<comment> \<open>Since the index set is unbounded in both directions, we can shift in either direction.\<close>
+
+lemma fls_shift_nth [simp]: "fls_shift m f $$ n = f $$ (n+m)"
+  unfolding fls_shift_def
+proof (rule nth_Abs_fls_ex_lower_bound)
+  obtain K::int where K: "\<forall>n<K. f$$n = 0" by (elim fls_nth_vanishes_belowE)
+  hence "\<forall>n<K-m. f$$(n+m) = 0" by auto
+  thus "\<exists>N. \<forall>n<N. f $$ (n + m) = 0" by fast
+qed
+
+lemma fls_shift_eq_iff: "(fls_shift m f = fls_shift m g) \<longleftrightarrow> (f = g)"
+proof (rule iffI, rule fls_eqI)
+  fix k
+  assume 1: "fls_shift m f = fls_shift m g"
+  have "f $$ k = fls_shift m g $$ (k - m)" by (simp add: 1[symmetric])  
+  thus "f $$ k = g $$ k" by simp
+qed (intro fls_eqI, simp)
+
+lemma fls_shift_0 [simp]: "fls_shift 0 f = f"
+  by (intro fls_eqI) simp
+
+lemma fls_shift_subdegree [simp]:
+  "f \<noteq> 0 \<Longrightarrow> fls_subdegree (fls_shift n f) = fls_subdegree f - n"
+  by (intro fls_subdegree_eqI) simp_all
+
+lemma fls_shift_fls_shift [simp]: "fls_shift m (fls_shift k f) = fls_shift (k+m) f"
+  by (intro fls_eqI) (simp add: algebra_simps)
+
+lemma fls_shift_fls_shift_reorder:
+  "fls_shift m (fls_shift k f) = fls_shift k (fls_shift m f)"
+  using fls_shift_fls_shift[of m k f] fls_shift_fls_shift[of k m f] by (simp add: add.commute)
+
+lemma fls_shift_zero [simp]: "fls_shift m 0 = 0"
+  by (intro fls_zero_eqI) simp
+
+lemma fls_shift_eq0_iff: "fls_shift m f = 0 \<longleftrightarrow> f = 0"
+  using fls_shift_eq_iff[of m f 0] by simp
+
+lemma fls_shift_nonneg_subdegree: "m \<le> fls_subdegree f \<Longrightarrow> fls_subdegree (fls_shift m f) \<ge> 0"
+  by (cases "f=0") (auto intro: fls_subdegree_geI)
+
+lemma fls_shift_delta:
+  "fls_shift m (Abs_fls (\<lambda>n. if n=a then b else 0)) = Abs_fls (\<lambda>n. if n=a-m then b else 0)"
+  by (intro fls_eqI) simp
+
+lemma fls_shift_const:
+  "fls_shift m (fls_const c) = Abs_fls (\<lambda>n. if n=-m then c else 0)"
+  by (intro fls_eqI) simp
+
+lemma fls_shift_const_nth:
+  "fls_shift m (fls_const c) $$ n = (if n=-m then c else 0)"
+  by (simp add: fls_shift_const)
+
+lemma fls_X_conv_shift_1: "fls_X = fls_shift (-1) 1"
+  by (intro fls_eqI) simp
+
+lemma fls_X_shift_to_one [simp]: "fls_shift 1 fls_X = 1"
+  using fls_shift_fls_shift[of "-1" 1 1] by (simp add: fls_X_conv_shift_1)
+
+lemma fls_X_inv_conv_shift_1: "fls_X_inv = fls_shift 1 1"
+  by (intro fls_eqI) simp
+
+lemma fls_X_inv_shift_to_one [simp]: "fls_shift (-1) fls_X_inv = 1"
+  using fls_shift_fls_shift[of 1 "-1" 1] by (simp add: fls_X_inv_conv_shift_1)
+
+lemma fls_X_fls_X_inv_conv:
+  "fls_X = fls_shift (-2) fls_X_inv" "fls_X_inv = fls_shift 2 fls_X"
+  by (simp_all add: fls_X_conv_shift_1 fls_X_inv_conv_shift_1)
+
+
+subsubsection \<open>Base factor\<close>
+
+text \<open>
+  Similarly to the @{const unit_factor} for formal power series, we can decompose a formal Laurent
+  series as a power of the implied variable times a series of subdegree 0.
+  (See lemma @{text "fls_base_factor_X_power_decompose"}.)
+  But we will call this something other @{const unit_factor}
+  because it will not satisfy assumption @{text "is_unit_unit_factor"} of
+  @{class semidom_divide_unit_factor}.
+\<close>
+
+definition fls_base_factor :: "('a::zero) fls \<Rightarrow> 'a fls"
+  where fls_base_factor_def[simp]: "fls_base_factor f = fls_shift (fls_subdegree f) f"
+
+lemma fls_base_factor_nth: "fls_base_factor f $$ n = f $$ (n + fls_subdegree f)"
+  by simp
+
+lemma fls_base_factor_nonzero [simp]: "f \<noteq> 0 \<Longrightarrow> fls_base_factor f \<noteq> 0"
+  using fls_nonzeroI[of "fls_base_factor f" 0] by simp
+
+lemma fls_base_factor_subdegree [simp]: "fls_subdegree (fls_base_factor f) = 0"
+ by (cases "f=0") auto
+
+lemma fls_base_factor_base [simp]:
+  "fls_base_factor f $$ fls_subdegree (fls_base_factor f) = f $$ fls_subdegree f"
+  using fls_base_factor_subdegree[of f] by simp
+
+lemma fls_conv_base_factor_shift_subdegree:
+  "f = fls_shift (-fls_subdegree f) (fls_base_factor f)"
+  by simp
+
+lemma fls_base_factor_idem:
+  "fls_base_factor (fls_base_factor (f::'a::zero fls)) = fls_base_factor f"
+  using fls_base_factor_subdegree[of f] by simp
+
+lemma fls_base_factor_zero: "fls_base_factor (0::'a::zero fls) = 0"
+  by simp
+
+lemma fls_base_factor_zero_iff: "fls_base_factor (f::'a::zero fls) = 0 \<longleftrightarrow> f = 0"
+proof
+  have "fls_shift (-fls_subdegree f) (fls_shift (fls_subdegree f) f) = f" by simp
+  thus "fls_base_factor f = 0 \<Longrightarrow> f=0" by simp
+qed simp
+
+lemma fls_base_factor_nth_0: "f \<noteq> 0 \<Longrightarrow> fls_base_factor f $$ 0 \<noteq> 0"
+  by simp
+
+lemma fls_base_factor_one: "fls_base_factor (1::'a::{zero,one} fls) = 1"
+  by simp
+
+lemma fls_base_factor_const: "fls_base_factor (fls_const c) = fls_const c"
+  by simp
+
+lemma fls_base_factor_delta:
+  "fls_base_factor (Abs_fls (\<lambda>n. if n=a then c else 0)) = fls_const c"
+  by  (cases "c=0") (auto intro: fls_eqI)
+
+lemma fls_base_factor_X: "fls_base_factor (fls_X::'a::{zero_neq_one} fls) = 1"
+   by simp
+
+lemma fls_base_factor_X_inv: "fls_base_factor (fls_X_inv::'a::{zero_neq_one} fls) = 1"
+   by simp
+
+lemma fls_base_factor_shift [simp]: "fls_base_factor (fls_shift n f) = fls_base_factor f"
+  by (cases "f=0") simp_all
+
+
+subsection \<open>Conversion between formal power and Laurent series\<close>
+
+subsubsection \<open>Converting Laurent to power series\<close>
+
+text \<open>
+  We can truncate a Laurent series at index 0 to create a power series, called the regular part.
+\<close>
+
+lift_definition fls_regpart :: "('a::zero) fls \<Rightarrow> 'a fps"
+  is "\<lambda>f. Abs_fps (\<lambda>n. f (int n))"
+  .
+
+lemma fls_regpart_nth [simp]: "fls_regpart f $ n = f $$ (int n)"
+  by (simp add: fls_regpart_def)
+
+lemma fls_regpart_zero [simp]: "fls_regpart 0 = 0"
+  by (intro fps_ext) simp
+
+lemma fls_regpart_one [simp]: "fls_regpart 1 = 1"
+  by (intro fps_ext) simp
+
+lemma fls_regpart_Abs_fls:
+  "\<forall>\<^sub>\<infinity>n. F (- int n) = 0 \<Longrightarrow> fls_regpart (Abs_fls F) = Abs_fps (\<lambda>n. F (int n))"
+  by (intro fps_ext) auto
+
+lemma fls_regpart_delta:
+  "fls_regpart (Abs_fls (\<lambda>n. if n=a then b else 0)) =
+    (if a < 0 then 0 else Abs_fps (\<lambda>n. if n=nat a then b else 0))"
+  by (rule fps_ext, auto)
+
+lemma fls_regpart_const [simp]: "fls_regpart (fls_const c) = fps_const c"
+  by (intro fps_ext) simp
+
+lemma fls_regpart_fls_X [simp]: "fls_regpart fls_X = fps_X"
+  by (intro fps_ext) simp
+
+lemma fls_regpart_fls_X_inv [simp]: "fls_regpart fls_X_inv = 0"
+  by (intro fps_ext) simp
+
+lemma fls_regpart_eq0_imp_nonpos_subdegree:
+  assumes "fls_regpart f = 0"
+  shows   "fls_subdegree f \<le> 0"
+proof (cases "f=0")
+  case False
+  have "fls_subdegree f \<ge> 0 \<Longrightarrow> f $$ fls_subdegree f = 0"
+  proof-
+    assume "fls_subdegree f \<ge> 0"
+    hence "f $$ (fls_subdegree f) = (fls_regpart f) $ (nat (fls_subdegree f))" by simp
+    with assms show "f $$ (fls_subdegree f) = 0" by simp
+  qed
+  with False show ?thesis by fastforce
+qed simp
+
+lemma fls_subdegree_lt_fls_regpart_subdegree:
+  "fls_subdegree f \<le> int (subdegree (fls_regpart f))"
+  using fls_subdegree_leI nth_subdegree_nonzero[of "fls_regpart f"]
+  by    (cases "(fls_regpart f) = 0")
+        (simp_all add: fls_regpart_eq0_imp_nonpos_subdegree)
+
+lemma fls_regpart_subdegree_conv:
+  assumes "fls_subdegree f \<ge> 0"
+  shows   "subdegree (fls_regpart f) = nat (fls_subdegree f)"
+\<comment>\<open>
+  This is the best we can do since if the subdegree is negative, we might still have the bad luck
+  that the term at index 0 is equal to 0.
+\<close>
+proof (cases "f=0")
+  case False with assms show ?thesis by (intro subdegreeI) simp_all
+qed simp
+
+lemma fls_eq_conv_fps_eqI:
+  assumes "0 \<le> fls_subdegree f" "0 \<le> fls_subdegree g" "fls_regpart f = fls_regpart g"
+  shows   "f = g"
+proof (rule fls_eq_above_subdegreeI, rule assms(1), rule assms(2), clarify)
+  fix k::int assume "0 \<le> k"
+  with assms(3) show "f $$ k = g $$ k"
+    using fls_regpart_nth[of f "nat k"] fls_regpart_nth[of g] by simp
+qed
+
+lemma fls_regpart_shift_conv_fps_shift:
+  "m \<ge> 0 \<Longrightarrow> fls_regpart (fls_shift m f) = fps_shift (nat m) (fls_regpart f)"
+  by (intro fps_ext) simp_all
+
+lemma fps_shift_fls_regpart_conv_fls_shift:
+  "fps_shift m (fls_regpart f) = fls_regpart (fls_shift m f)"
+  by (intro fps_ext) simp_all
+
+lemma fps_unit_factor_fls_regpart:
+  "fls_subdegree f \<ge> 0 \<Longrightarrow> unit_factor (fls_regpart f) = fls_regpart (fls_base_factor f)"
+  by (auto intro: fps_ext simp: fls_regpart_subdegree_conv)
+
+text \<open>
+  The terms below the zeroth form a polynomial in the inverse of the implied variable,
+  called the principle part.
+\<close>
+
+lift_definition fls_prpart :: "('a::zero) fls \<Rightarrow> 'a poly"
+  is "\<lambda>f. Abs_poly (\<lambda>n. if n = 0 then 0 else f (- int n))"
+  .
+
+lemma fls_prpart_coeff [simp]: "coeff (fls_prpart f) n = (if n = 0 then 0 else f $$ (- int n))"
+proof-
+  have "{x. (if x = 0 then 0 else f $$ - int x) \<noteq> 0} \<subseteq> {x. f $$ - int x \<noteq> 0}"
+    by auto
+  hence "finite {x. (if x = 0 then 0 else f $$ - int x) \<noteq> 0}"
+    using fls_finite_nonzero_neg_nth[of f] by (simp add: rev_finite_subset)
+  hence "coeff (fls_prpart f) = (\<lambda>n. if n = 0 then 0 else f $$ (- int n))"
+    using Abs_poly_inverse[OF CollectI, OF iffD2, OF eventually_cofinite]
+    by (simp add: fls_prpart_def)
+  thus ?thesis by simp
+qed
+
+lemma fls_prpart_eq0_iff: "(fls_prpart f = 0) \<longleftrightarrow> (fls_subdegree f \<ge> 0)"
+proof
+  assume 1: "fls_prpart f = 0"
+  show "fls_subdegree f \<ge> 0"
+  proof (intro fls_subdegree_ge0I)
+    fix k::int assume "k < 0"
+    with 1 show "f $$ k = 0" using fls_prpart_coeff[of f "nat (-k)"] by simp
+  qed
+qed (intro poly_eqI, simp)
+
+lemma fls_prpart0 [simp]: "fls_prpart 0 = 0"
+  by (simp add: fls_prpart_eq0_iff)
+
+lemma fls_prpart_one [simp]: "fls_prpart 1 = 0"
+  by (simp add: fls_prpart_eq0_iff)
+
+lemma fls_prpart_delta:
+  "fls_prpart (Abs_fls (\<lambda>n. if n=a then b else 0)) =
+    (if a<0 then Poly (replicate (nat (-a)) 0 @ [b]) else 0)"
+  by (intro poly_eqI) (auto simp: nth_default_def nth_append)
+
+lemma fls_prpart_const [simp]: "fls_prpart (fls_const c) = 0"
+  by (simp add: fls_prpart_eq0_iff)
+
+lemma fls_prpart_X [simp]: "fls_prpart fls_X = 0"
+  by (intro poly_eqI) simp
+
+lemma fls_prpart_X_inv: "fls_prpart fls_X_inv = [:0,1:]"
+proof (intro poly_eqI)
+  fix n show "coeff (fls_prpart fls_X_inv) n = coeff [:0,1:] n"
+  proof (cases n)
+    case (Suc i) thus ?thesis by (cases i) simp_all
+  qed simp
+qed
+
+lemma degree_fls_prpart [simp]:
+  "degree (fls_prpart f) = nat (-fls_subdegree f)"
+proof (cases "f=0")
+  case False show ?thesis unfolding degree_def
+  proof (intro Least_equality)
+    fix N assume N: "\<forall>i>N. coeff (fls_prpart f) i = 0"
+    have "\<forall>i < -int N. f $$ i = 0"
+    proof clarify
+      fix i assume i: "i < -int N"
+      hence "nat (-i) > N" by simp
+      with N i show "f $$ i = 0" using fls_prpart_coeff[of f "nat (-i)"] by auto
+    qed
+    with False have "fls_subdegree f \<ge> -int N" using fls_subdegree_geI by auto
+    thus "nat (- fls_subdegree f) \<le> N" by simp
+  qed auto
+qed simp
+
+lemma fls_prpart_shift:
+  assumes "m \<le> 0"
+  shows   "fls_prpart (fls_shift m f) = pCons 0 (poly_shift (Suc (nat (-m))) (fls_prpart f))"
+proof (intro poly_eqI)
+  fix n
+  define LHS RHS
+    where "LHS \<equiv> fls_prpart (fls_shift m f)"
+    and   "RHS \<equiv> pCons 0 (poly_shift (Suc (nat (-m))) (fls_prpart f))"
+  show "coeff LHS n = coeff RHS n"
+  proof (cases n)
+    case (Suc k)
+    from assms have 1: "-int (Suc k + nat (-m)) = -int (Suc k) + m" by simp
+    have "coeff RHS n = f $$ (-int (Suc k) + m)"
+      using arg_cong[OF 1, of "($$) f"] by (simp add: Suc RHS_def coeff_poly_shift)
+    with Suc show ?thesis by (simp add: LHS_def)
+  qed (simp add: LHS_def RHS_def)
+qed
+
+lemma fls_prpart_base_factor: "fls_prpart (fls_base_factor f) = 0"
+  using fls_base_factor_subdegree[of f] by (simp add: fls_prpart_eq0_iff)
+
+text \<open>The essential data of a formal Laurant series resides from the subdegree up.\<close>
+
+abbreviation fls_base_factor_to_fps :: "('a::zero) fls \<Rightarrow> 'a fps"
+  where "fls_base_factor_to_fps f \<equiv> fls_regpart (fls_base_factor f)"
+
+lemma fls_base_factor_to_fps_conv_fps_shift:
+  assumes "fls_subdegree f \<ge> 0"
+  shows   "fls_base_factor_to_fps f = fps_shift (nat (fls_subdegree f)) (fls_regpart f)"
+  by (simp add: assms fls_regpart_shift_conv_fps_shift)
+
+lemma fls_base_factor_to_fps_nth:
+  "fls_base_factor_to_fps f $ n = f $$ (fls_subdegree f + int n)"
+  by (simp add: algebra_simps)
+
+lemma fls_base_factor_to_fps_base: "f \<noteq> 0 \<Longrightarrow> fls_base_factor_to_fps f $ 0 \<noteq> 0"
+  by simp
+
+lemma fls_base_factor_to_fps_nonzero: "f \<noteq> 0 \<Longrightarrow> fls_base_factor_to_fps f \<noteq> 0"
+  using fps_nonzeroI[of "fls_base_factor_to_fps f" 0] fls_base_factor_to_fps_base by simp
+
+lemma fls_base_factor_to_fps_subdegree [simp]: "subdegree (fls_base_factor_to_fps f) = 0"
+  by (cases "f=0") auto
+
+lemma fls_base_factor_to_fps_trivial:
+  "fls_subdegree f = 0 \<Longrightarrow> fls_base_factor_to_fps f = fls_regpart f"
+  by simp
+
+lemma fls_base_factor_to_fps_zero: "fls_base_factor_to_fps 0 = 0"
+  by simp
+
+lemma fls_base_factor_to_fps_one: "fls_base_factor_to_fps 1 = 1"
+  by simp
+
+lemma fls_base_factor_to_fps_delta:
+  "fls_base_factor_to_fps (Abs_fls (\<lambda>n. if n=a then c else 0)) = fps_const c"
+  using fls_base_factor_delta[of a c] by simp
+
+lemma fls_base_factor_to_fps_const:
+  "fls_base_factor_to_fps (fls_const c) = fps_const c"
+  by simp
+
+lemma fls_base_factor_to_fps_X:
+  "fls_base_factor_to_fps (fls_X::'a::{zero_neq_one} fls) = 1"
+  by simp
+
+lemma fls_base_factor_to_fps_X_inv:
+  "fls_base_factor_to_fps (fls_X_inv::'a::{zero_neq_one} fls) = 1"
+  by simp
+
+lemma fls_base_factor_to_fps_shift:
+  "fls_base_factor_to_fps (fls_shift m f) = fls_base_factor_to_fps f"
+  using fls_base_factor_shift[of m f] by simp
+
+lemma fls_base_factor_to_fps_base_factor:
+  "fls_base_factor_to_fps (fls_base_factor f) = fls_base_factor_to_fps f"
+  using fls_base_factor_to_fps_shift by simp
+
+lemma fps_unit_factor_fls_base_factor:
+  "unit_factor (fls_base_factor_to_fps f) = fls_base_factor_to_fps f"
+  using fls_base_factor_to_fps_subdegree[of f] by simp
+
+subsubsection \<open>Converting power to Laurent series\<close>
+
+text \<open>We can extend a power series by 0s below to create a Laurent series.\<close>
+
+definition fps_to_fls :: "('a::zero) fps \<Rightarrow> 'a fls"
+  where "fps_to_fls f \<equiv> Abs_fls (\<lambda>k::int. if k<0 then 0 else f $ (nat k))"
+
+lemma fps_to_fls_nth [simp]:
+  "(fps_to_fls f) $$ n = (if n < 0 then 0 else f$(nat n))"
+  using     nth_Abs_fls_lower_bound[of 0 "(\<lambda>k::int. if k<0 then 0 else f $ (nat k))"]
+  unfolding fps_to_fls_def
+  by        simp
+
+lemma fps_to_fls_eq_imp_fps_eq:
+  assumes "fps_to_fls f = fps_to_fls g"
+  shows   "f = g"
+proof (intro fps_ext)
+  fix n
+  have "f $ n = fps_to_fls g $$ int n" by (simp add: assms[symmetric])
+  thus "f $ n = g $ n" by simp
+qed
+
+lemma fps_zero_to_fls [simp]: "fps_to_fls 0 = 0"
+  by (intro fls_zero_eqI) simp
+
+lemma fps_to_fls_nonzeroI: "f \<noteq> 0 \<Longrightarrow> fps_to_fls f \<noteq> 0"
+  using fps_to_fls_eq_imp_fps_eq[of f 0] by auto
+
+lemma fps_one_to_fls [simp]: "fps_to_fls 1 = 1"
+  by (intro fls_eqI) simp
+
+lemma fps_to_fls_Abs_fps:
+  "fps_to_fls (Abs_fps F) = Abs_fls (\<lambda>n. if n<0 then 0 else F (nat n))"
+  using nth_Abs_fls_lower_bound[of 0 "(\<lambda>n::int. if n<0 then 0 else F (nat n))"]
+  by    (intro fls_eqI) simp
+
+lemma fps_delta_to_fls:
+  "fps_to_fls (Abs_fps (\<lambda>n. if n=a then b else 0)) = Abs_fls (\<lambda>n. if n=int a then b else 0)"
+  using fls_eqI[of _ "Abs_fls (\<lambda>n. if n=int a then b else 0)"] by force
+
+lemma fps_const_to_fls [simp]: "fps_to_fls (fps_const c) = fls_const c"
+  by (intro fls_eqI) simp
+
+lemma fps_X_to_fls [simp]: "fps_to_fls fps_X = fls_X"
+  by (fastforce intro: fls_eqI)
+
+lemma fps_to_fls_eq_zero_iff: "(fps_to_fls f = 0) \<longleftrightarrow> (f=0)"
+  using fps_to_fls_nonzeroI by auto
+
+lemma fls_subdegree_fls_to_fps_gt0: "fls_subdegree (fps_to_fls f) \<ge> 0"
+proof (cases "f=0")
+  case False show ?thesis
+  proof (rule fls_subdegree_geI, rule fls_nonzeroI)
+    from False show "fps_to_fls f $$ int (subdegree f) \<noteq> 0"
+      by simp
+  qed simp
+qed simp
+
+lemma fls_subdegree_fls_to_fps: "fls_subdegree (fps_to_fls f) = int (subdegree f)"
+proof (cases "f=0")
+  case False
+  have "subdegree f = nat (fls_subdegree (fps_to_fls f))"
+  proof (rule subdegreeI)
+    from False show "f $ (nat (fls_subdegree (fps_to_fls f))) \<noteq> 0"
+      using fls_subdegree_fls_to_fps_gt0[of f] nth_fls_subdegree_nonzero[of "fps_to_fls f"]
+            fps_to_fls_nonzeroI[of f]
+      by    simp
+  next
+    fix k assume k: "k < nat (fls_subdegree (fps_to_fls f))"
+    thus "f $ k = 0"
+      using fls_eq0_below_subdegree[of "int k" "fps_to_fls f"] by simp
+  qed
+  thus ?thesis by (simp add: fls_subdegree_fls_to_fps_gt0)
+qed simp
+
+lemma fps_shift_to_fls [simp]:
+  "n \<le> subdegree f \<Longrightarrow> fps_to_fls (fps_shift n f) = fls_shift (int n) (fps_to_fls f)"
+  by (auto intro: fls_eqI simp: nat_add_distrib nth_less_subdegree_zero)
+
+lemma fls_base_factor_fps_to_fls: "fls_base_factor (fps_to_fls f) = fps_to_fls (unit_factor f)"
+  using nth_less_subdegree_zero[of _ f]
+  by    (auto intro: fls_eqI simp: fls_subdegree_fls_to_fps nat_add_distrib)
+
+lemma fls_regpart_to_fls_trivial [simp]:
+  "fls_subdegree f \<ge> 0 \<Longrightarrow> fps_to_fls (fls_regpart f) = f"
+  by (intro fls_eqI) simp
+
+lemma fls_regpart_fps_trivial [simp]: "fls_regpart (fps_to_fls f) = f"
+  by (intro fps_ext) simp
+
+lemma fps_to_fls_base_factor_to_fps:
+  "fps_to_fls (fls_base_factor_to_fps f) = fls_base_factor f"
+  by (intro fls_eqI) simp
+
+lemma fls_conv_base_factor_to_fps_shift_subdegree:
+  "f = fls_shift (-fls_subdegree f) (fps_to_fls (fls_base_factor_to_fps f))"
+  using fps_to_fls_base_factor_to_fps[of f] fps_to_fls_base_factor_to_fps[of f] by simp
+
+lemma fls_base_factor_to_fps_to_fls:
+  "fls_base_factor_to_fps (fps_to_fls f) = unit_factor f"
+  using fls_base_factor_fps_to_fls[of f] fls_regpart_fps_trivial[of "unit_factor f"]
+  by    simp
+
+abbreviation
+  "fls_regpart_as_fls f \<equiv> fps_to_fls (fls_regpart f)"
+abbreviation
+  "fls_prpart_as_fls f \<equiv>
+    fls_shift (-fls_subdegree f) (fps_to_fls (fps_of_poly (reflect_poly (fls_prpart f))))"
+
+lemma fls_regpart_as_fls_nth:
+  "fls_regpart_as_fls f $$ n = (if n < 0 then 0 else f $$ n)"
+  by simp
+
+lemma fls_regpart_idem:
+  "fls_regpart (fls_regpart_as_fls f) = fls_regpart f"
+  by simp
+
+lemma fls_prpart_as_fls_nth:
+  "fls_prpart_as_fls f $$ n = (if n < 0 then f $$ n else 0)"
+proof (cases "n < fls_subdegree f" "n < 0" rule: case_split[case_product case_split])
+  case False_True
+    hence "nat (-fls_subdegree f) - nat (n - fls_subdegree f) = nat (-n)" by auto
+    with False_True show ?thesis
+      using coeff_reflect_poly[of "fls_prpart f" "nat (n - fls_subdegree f)"] by auto
+  next
+    case False_False thus ?thesis
+      using coeff_reflect_poly[of "fls_prpart f" "nat (n - fls_subdegree f)"] by auto
+qed simp_all
+
+lemma fls_prpart_idem [simp]: "fls_prpart (fls_prpart_as_fls f) = fls_prpart f"
+  using fls_prpart_as_fls_nth[of f] by (intro poly_eqI) simp
+
+lemma fls_regpart_prpart: "fls_regpart (fls_prpart_as_fls f) = 0"
+  using fls_prpart_as_fls_nth[of f] by (intro fps_ext) simp
+
+lemma fls_prpart_regpart: "fls_prpart (fls_regpart_as_fls f) = 0"
+  by (intro poly_eqI) simp
+
+
+subsection \<open>Algebraic structures\<close>
+
+subsubsection \<open>Addition\<close>
+
+instantiation fls :: (monoid_add) plus
+begin
+  lift_definition plus_fls :: "'a fls \<Rightarrow> 'a fls \<Rightarrow> 'a fls" is "\<lambda>f g n. f n + g n"
+  proof-
+    fix f f' :: "int \<Rightarrow> 'a"
+    assume "\<forall>\<^sub>\<infinity>n. f (- int n) = 0" "\<forall>\<^sub>\<infinity>n. f' (- int n) = 0"
+    from this obtain N N' where "\<forall>n>N. f (-int n) = 0" "\<forall>n>N'. f' (-int n) = 0"
+      by (auto simp: MOST_nat)
+    hence "\<forall>n > max N N'. f (-int n) + f' (-int n) = 0" by auto
+    hence "\<exists>K. \<forall>n>K. f (-int n) + f' (-int n) = 0" by fast
+    thus "\<forall>\<^sub>\<infinity>n. f (- int n) + f' (-int n) = 0" by (simp add: MOST_nat)
+  qed
+  instance ..
+end
+
+lemma fls_plus_nth [simp]: "(f + g) $$ n = f $$ n + g $$ n"
+  by transfer simp
+
+lemma fls_plus_const: "fls_const x + fls_const y = fls_const (x+y)"
+  by (intro fls_eqI) simp
+
+lemma fls_plus_subdegree:
+  "f + g \<noteq> 0 \<Longrightarrow> fls_subdegree (f + g) \<ge> min (fls_subdegree f) (fls_subdegree g)"
+  by (auto intro: fls_subdegree_geI)
+
+lemma fls_shift_plus [simp]:
+  "fls_shift m (f + g) = (fls_shift m f) + (fls_shift m g)"
+  by (intro fls_eqI) simp
+
+lemma fls_regpart_plus [simp]: "fls_regpart (f + g) = fls_regpart f + fls_regpart g"
+  by (intro fps_ext) simp
+
+lemma fls_prpart_plus [simp] : "fls_prpart (f + g) = fls_prpart f + fls_prpart g"
+  by (intro poly_eqI) simp
+
+lemma fls_decompose_reg_pr_parts:
+  fixes   f :: "'a :: monoid_add fls"
+  defines "R  \<equiv> fls_regpart_as_fls f"
+  and     "P  \<equiv> fls_prpart_as_fls f"
+  shows   "f = P + R"
+  and     "f = R + P"
+  using   fls_prpart_as_fls_nth[of f]
+  by      (auto intro: fls_eqI simp add: assms)
+
+lemma fps_to_fls_plus [simp]: "fps_to_fls (f + g) = fps_to_fls f + fps_to_fls g"
+  by (intro fls_eqI) simp
+
+instance fls :: (monoid_add) monoid_add
+proof
+  fix a b c :: "'a fls"
+  show "a + b + c = a + (b + c)" by transfer (simp add: add.assoc)
+  show "0 + a = a" by transfer simp
+  show "a + 0 = a" by transfer simp
+qed
+
+instance fls :: (comm_monoid_add) comm_monoid_add
+  by (standard, transfer, auto simp: add.commute)
+
+
+subsubsection \<open>Subtraction and negatives\<close>
+
+instantiation fls :: (group_add) minus
+begin
+  lift_definition minus_fls :: "'a fls \<Rightarrow> 'a fls \<Rightarrow> 'a fls" is "\<lambda>f g n. f n - g n"
+  proof-
+    fix f f' :: "int \<Rightarrow> 'a"
+    assume "\<forall>\<^sub>\<infinity>n. f (- int n) = 0" "\<forall>\<^sub>\<infinity>n. f' (- int n) = 0"
+    from this obtain N N' where "\<forall>n>N. f (-int n) = 0" "\<forall>n>N'. f' (-int n) = 0"
+      by (auto simp: MOST_nat)
+    hence "\<forall>n > max N N'. f (-int n) - f' (-int n) = 0" by auto
+    hence "\<exists>K. \<forall>n>K. f (-int n) - f' (-int n) = 0" by fast
+    thus "\<forall>\<^sub>\<infinity>n. f (- int n) - f' (-int n) = 0" by (simp add: MOST_nat)
+  qed
+  instance ..
+end
+
+lemma fls_minus_nth [simp]: "(f - g) $$ n = f $$ n - g $$ n"
+  by transfer simp
+
+lemma fls_minus_const: "fls_const x - fls_const y = fls_const (x-y)"
+  by (intro fls_eqI) simp
+
+lemma fls_subdegree_minus:
+  "f - g \<noteq> 0 \<Longrightarrow> fls_subdegree (f - g) \<ge> min (fls_subdegree f) (fls_subdegree g)"
+  by (intro fls_subdegree_geI) simp_all
+
+lemma fls_shift_minus [simp]: "fls_shift m (f - g) = (fls_shift m f) - (fls_shift m g)"
+  by (auto intro: fls_eqI)
+
+lemma fls_regpart_minus [simp]: "fls_regpart (f - g) = fls_regpart f - fls_regpart g"
+  by (intro fps_ext) simp
+
+lemma fls_prpart_minus [simp] : "fls_prpart (f - g) = fls_prpart f - fls_prpart g"
+  by (intro poly_eqI) simp
+
+lemma fps_to_fls_minus [simp]: "fps_to_fls (f - g) = fps_to_fls f - fps_to_fls g"
+  by (intro fls_eqI) simp
+
+instantiation fls :: (group_add) uminus
+begin
+  lift_definition uminus_fls :: "'a fls \<Rightarrow> 'a fls" is "\<lambda>f n. - f n"
+  proof-
+    fix f :: "int \<Rightarrow> 'a" assume "\<forall>\<^sub>\<infinity>n. f (- int n) = 0"
+    from this obtain N where "\<forall>n>N. f (-int n) = 0"
+      by (auto simp: MOST_nat)
+    hence "\<forall>n>N. - f (-int n) = 0" by auto
+    hence "\<exists>K. \<forall>n>K. - f (-int n) = 0" by fast
+    thus "\<forall>\<^sub>\<infinity>n. - f (- int n) = 0" by (simp add: MOST_nat)
+  qed
+  instance ..
+end
+
+lemma fls_uminus_nth [simp]: "(-f) $$ n = - (f $$ n)"
+  by transfer simp
+
+lemma fls_const_uminus[simp]: "fls_const (-x) = -fls_const x"
+  by (intro fls_eqI) simp
+
+lemma fls_shift_uminus [simp]: "fls_shift m (- f) = - (fls_shift m f)"
+  by (auto intro: fls_eqI)
+
+lemma fls_regpart_uminus [simp]: "fls_regpart (- f) = - fls_regpart f"
+  by (intro fps_ext) simp
+
+lemma fls_prpart_uminus [simp] : "fls_prpart (- f) = - fls_prpart f"
+  by (intro poly_eqI) simp
+
+lemma fps_to_fls_uminus [simp]: "fps_to_fls (- f) = - fps_to_fls f"
+  by (intro fls_eqI) simp
+
+instance fls :: (group_add) group_add
+proof
+  fix a b :: "'a fls"
+  show "- a + a = 0" by transfer simp
+  show "a + - b = a - b" by transfer simp
+qed
+
+instance fls :: (ab_group_add) ab_group_add
+proof
+  fix a b :: "'a fls"
+  show "- a + a = 0" by transfer simp
+  show "a - b = a + - b" by transfer simp
+qed
+
+lemma fls_uminus_subdegree [simp]: "fls_subdegree (-f) = fls_subdegree f"
+  by (cases "f=0") (auto intro: fls_subdegree_eqI)
+
+lemma fls_subdegree_minus_sym: "fls_subdegree (g - f) = fls_subdegree (f - g)"
+  using fls_uminus_subdegree[of "g-f"] by (simp add: algebra_simps)
+
+lemma fls_regpart_sub_prpart: "fls_regpart (f - fls_prpart_as_fls f) = fls_regpart f"
+  using fls_decompose_reg_pr_parts(2)[of f]
+        add_diff_cancel[of "fls_regpart_as_fls f" "fls_prpart_as_fls f"]
+  by    simp
+
+lemma fls_prpart_sub_regpart: "fls_prpart (f - fls_regpart_as_fls f) = fls_prpart f"
+  using fls_decompose_reg_pr_parts(1)[of f]
+        add_diff_cancel[of "fls_prpart_as_fls f" "fls_regpart_as_fls f"]
+  by    simp
+
+
+subsubsection \<open>Multiplication\<close>
+
+instantiation fls :: ("{comm_monoid_add, times}") times
+begin
+  definition fls_times_def:
+    "(*) = (\<lambda>f g.
+      fls_shift
+        (- (fls_subdegree f + fls_subdegree g))
+        (fps_to_fls (fls_base_factor_to_fps f * fls_base_factor_to_fps g))
+    )"
+  instance ..
+end
+
+lemma fls_times_nth_eq0: "n < fls_subdegree f + fls_subdegree g \<Longrightarrow> (f * g) $$ n = 0"
+  by (simp add: fls_times_def)
+
+lemma fls_times_nth:
+  fixes   f df g dg
+  defines "df \<equiv> fls_subdegree f" and "dg \<equiv> fls_subdegree g"
+  shows   "(f * g) $$ n = (\<Sum>i=df + dg..n. f $$ (i - dg) * g $$ (dg + n - i))"
+  and     "(f * g) $$ n = (\<Sum>i=df..n - dg. f $$ i * g $$ (n - i))"
+  and     "(f * g) $$ n = (\<Sum>i=dg..n - df. f $$ (df + i - dg) * g $$ (dg + n - df - i))"
+  and     "(f * g) $$ n = (\<Sum>i=0..n - (df + dg). f $$ (df + i) * g $$ (n - df - i))"
+proof-
+
+  define dfg where "dfg \<equiv> df + dg"
+
+  show 4: "(f * g) $$ n = (\<Sum>i=0..n - dfg. f $$ (df + i) * g $$ (n - df - i))"
+  proof (cases "n < dfg")
+    case False
+    from False assms have
+      "(f * g) $$ n =
+        (\<Sum>i = 0..nat (n - dfg). f $$ (df + int i) * g $$ (dg + int (nat (n - dfg) - i)))"
+      using fps_mult_nth[of "fls_base_factor_to_fps f" "fls_base_factor_to_fps g"]
+            fls_base_factor_to_fps_nth[of f]
+            fls_base_factor_to_fps_nth[of g]
+      by    (simp add: dfg_def fls_times_def algebra_simps)
+    moreover from False have index:
+      "\<And>i. i \<in> {0..nat (n - dfg)} \<Longrightarrow> dg + int (nat (n - dfg) - i) = n - df - int i"
+      by (auto simp: dfg_def)
+    ultimately have
+      "(f * g) $$ n = (\<Sum>i=0..nat (n - dfg). f $$ (df + int i) * g $$ (n - df - int i))"
+      by simp
+    moreover have
+      "(\<Sum>i=0..nat (n - dfg). f $$ (df + int i) *  g $$ (n - df - int i)) =
+        (\<Sum>i=0..n - dfg. f $$ (df + i) *  g $$ (n - df - i))"
+    proof (intro sum.reindex_cong)
+      show "inj_on nat {0..n - dfg}" by standard auto
+      show "{0..nat (n - dfg)} = nat ` {0..n - dfg}"
+      proof
+        show "{0..nat (n - dfg)} \<subseteq> nat ` {0..n - dfg}"
+        proof
+          fix i assume "i \<in> {0..nat (n - dfg)}"
+          hence i: "i \<ge> 0" "i \<le> nat (n - dfg)" by auto
+          with False have "int i \<ge> 0" "int i \<le> n - dfg" by auto
+          hence "int i \<in> {0..n - dfg}" by simp
+          moreover from i(1) have "i = nat (int i)" by simp
+          ultimately show "i \<in> nat ` {0..n - dfg}" by fast
+        qed
+      qed (auto simp: False)
+    qed (simp add: False)
+    ultimately show "(f * g) $$ n = (\<Sum>i=0..n - dfg. f $$ (df + i) *  g $$ (n - df - i))"
+      by simp
+  qed (simp add: fls_times_nth_eq0 assms dfg_def)
+
+  have
+    "(\<Sum>i=dfg..n. f $$ (i - dg) *  g $$ (dg + n - i)) =
+      (\<Sum>i=0..n - dfg. f $$ (df + i) *  g $$ (n - df - i))"
+  proof (intro sum.reindex_cong)
+    define T where "T \<equiv> \<lambda>i. i + dfg"
+    show "inj_on T {0..n - dfg}" by standard (simp add: T_def)
+  qed (simp_all add: dfg_def algebra_simps)
+  with 4 show 1: "(f * g) $$ n = (\<Sum>i=dfg..n. f $$ (i - dg) *  g $$ (dg + n - i))"
+    by simp
+
+  have
+    "(\<Sum>i=dfg..n. f $$ (i - dg) *  g $$ (dg + n - i)) = (\<Sum>i=df..n - dg. f $$ i *  g $$ (n - i))"
+  proof (intro sum.reindex_cong)
+    define T where "T \<equiv> \<lambda>i. i + dg"
+    show "inj_on T {df..n - dg}" by standard (simp add: T_def)
+  qed (auto simp: dfg_def)
+  with 1 show "(f * g) $$ n = (\<Sum>i=df..n - dg. f $$ i *  g $$ (n - i))"
+    by simp
+
+  have
+    "(\<Sum>i=dfg..n. f $$ (i - dg) *  g $$ (dg + n - i)) =
+      (\<Sum>i=dg..n - df. f $$ (df + i - dg) *  g $$ (dg + n - df - i))"
+  proof (intro sum.reindex_cong)
+    define T where "T \<equiv> \<lambda>i. i + df"
+    show "inj_on T {dg..n - df}" by standard (simp add: T_def)
+  qed (simp_all add: dfg_def algebra_simps)
+  with 1 show "(f * g) $$ n = (\<Sum>i=dg..n - df. f $$ (df + i - dg) *  g $$ (dg + n - df - i))"
+    by simp
+
+qed
+
+lemma fls_times_base [simp]:
+  "(f * g) $$ (fls_subdegree f + fls_subdegree g) =
+    (f $$ fls_subdegree f) * (g $$ fls_subdegree g)"
+  by (simp add: fls_times_nth(1))
+
+instance fls :: ("{comm_monoid_add, mult_zero}") mult_zero
+proof
+  fix a :: "'a fls"
+  have
+    "(0::'a fls) * a =
+      fls_shift (fls_subdegree a) (fps_to_fls ( (0::'a fps)*(fls_base_factor_to_fps a) ))"
+    by (simp add: fls_times_def)
+  moreover have
+    "a * (0::'a fls) =
+      fls_shift (fls_subdegree a) (fps_to_fls ( (fls_base_factor_to_fps a)*(0::'a fps) ))"
+    by (simp add: fls_times_def)
+  ultimately show "0 * a = (0::'a fls)" "a * 0 = (0::'a fls)"
+    by auto
+qed
+
+lemma fls_mult_one:
+  fixes f :: "'a::{comm_monoid_add, mult_zero, monoid_mult} fls"
+  shows "1 * f = f"
+  and   "f * 1 = f"
+  using fls_conv_base_factor_to_fps_shift_subdegree[of f]
+  by    (simp_all add: fls_times_def fps_one_mult)
+
+lemma fls_mult_const_nth [simp]:
+  fixes f :: "'a::{comm_monoid_add, mult_zero} fls"
+  shows "(fls_const x * f) $$ n = x * f$$n"
+  and   "(f * fls_const x ) $$ n = f$$n * x"
+proof-
+  show "(fls_const x * f) $$ n = x * f$$n"
+  proof (cases "n<fls_subdegree f")
+    case False
+    hence "{fls_subdegree f..n} = insert (fls_subdegree f) {fls_subdegree f+1..n}" by auto
+    thus ?thesis by (simp add: fls_times_nth(1))
+  qed (simp add: fls_times_nth_eq0)
+  show "(f * fls_const x ) $$ n = f$$n * x"
+  proof (cases "n<fls_subdegree f")
+    case False
+    hence "{fls_subdegree f..n} = insert n {fls_subdegree f..n-1}" by auto
+    thus ?thesis by (simp add: fls_times_nth(1))
+  qed (simp add: fls_times_nth_eq0)
+qed
+
+lemma fls_const_mult_const[simp]:
+  fixes x y :: "'a::{comm_monoid_add, mult_zero}"
+  shows "fls_const x * fls_const y = fls_const (x*y)"
+  by    (intro fls_eqI) simp
+
+lemma fls_mult_subdegree_ge:
+  fixes   f g :: "'a::{comm_monoid_add,mult_zero} fls"
+  assumes "f*g \<noteq> 0"
+  shows   "fls_subdegree (f*g) \<ge> fls_subdegree f + fls_subdegree g"
+  by      (auto intro: fls_subdegree_geI simp: assms fls_times_nth_eq0)
+
+lemma fls_mult_subdegree_ge_0:
+  fixes   f g :: "'a::{comm_monoid_add,mult_zero} fls"
+  assumes "fls_subdegree f \<ge> 0" "fls_subdegree g \<ge> 0"
+  shows   "fls_subdegree (f*g) \<ge> 0"
+  using   assms fls_mult_subdegree_ge[of f g]
+  by      fastforce
+
+lemma fls_mult_nonzero_base_subdegree_eq:
+  fixes   f g :: "'a::{comm_monoid_add,mult_zero} fls"
+  assumes "f $$ (fls_subdegree f) * g $$ (fls_subdegree g) \<noteq> 0"
+  shows   "fls_subdegree (f*g) = fls_subdegree f + fls_subdegree g"
+proof-
+  from assms have "fls_subdegree (f*g) \<ge> fls_subdegree f + fls_subdegree g"
+    using fls_nonzeroI[of "f*g" "fls_subdegree f + fls_subdegree g"]
+          fls_mult_subdegree_ge[of f g]
+    by    simp
+  moreover from assms have "fls_subdegree (f*g) \<le> fls_subdegree f + fls_subdegree g"
+    by (intro fls_subdegree_leI) simp
+  ultimately show ?thesis by simp
+qed
+
+lemma fls_subdegree_mult [simp]:
+  fixes   f g :: "'a::semiring_no_zero_divisors fls"
+  assumes "f \<noteq> 0" "g \<noteq> 0"
+  shows   "fls_subdegree (f * g) = fls_subdegree f + fls_subdegree g"
+  using   assms
+  by      (auto intro: fls_subdegree_eqI simp: fls_times_nth_eq0)
+
+lemma fls_shifted_times_simps:
+  fixes f g :: "'a::{comm_monoid_add, mult_zero} fls"
+  shows "f * (fls_shift n g) = fls_shift n (f*g)" "(fls_shift n f) * g = fls_shift n (f*g)"
+proof-
+
+  show "f * (fls_shift n g) = fls_shift n (f*g)"
+  proof (cases "g=0")
+    case False
+    hence
+      "f * (fls_shift n g) =
+        fls_shift (- (fls_subdegree f + (fls_subdegree g - n)))
+          (fps_to_fls (fls_base_factor_to_fps f * fls_base_factor_to_fps g))"
+      unfolding fls_times_def by (simp add: fls_base_factor_to_fps_shift)
+    thus "f * (fls_shift n g) = fls_shift n (f*g)"
+      by (simp add: algebra_simps fls_times_def)
+  qed auto
+
+  show "(fls_shift n f)*g = fls_shift n (f*g)"
+  proof (cases "f=0")
+    case False
+    hence
+      "(fls_shift n f)*g =
+        fls_shift (- ((fls_subdegree f - n) + fls_subdegree g))
+          (fps_to_fls (fls_base_factor_to_fps f * fls_base_factor_to_fps g))"
+      unfolding fls_times_def by (simp add: fls_base_factor_to_fps_shift)
+    thus "(fls_shift n f) * g = fls_shift n (f*g)"
+      by (simp add: algebra_simps fls_times_def)
+  qed auto
+
+qed
+
+lemma fls_shifted_times_transfer:
+  fixes f g :: "'a::{comm_monoid_add, mult_zero} fls"
+  shows "fls_shift n f * g = f * fls_shift n g"
+  using fls_shifted_times_simps(1)[of f n g] fls_shifted_times_simps(2)[of n f g]
+  by    simp
+
+lemma fls_times_both_shifted_simp:
+  fixes f g :: "'a::{comm_monoid_add, mult_zero} fls"
+  shows "(fls_shift m f) * (fls_shift n g) = fls_shift (m+n) (f*g)"
+  by    (simp add: fls_shifted_times_simps)
+
+lemma fls_base_factor_mult_base_factor:
+  fixes f g :: "'a::{comm_monoid_add, mult_zero} fls"
+  shows "fls_base_factor (f * fls_base_factor g) = fls_base_factor (f * g)"
+  and   "fls_base_factor (fls_base_factor f * g) = fls_base_factor (f * g)"
+  using fls_base_factor_shift[of "fls_subdegree g" "f*g"]
+        fls_base_factor_shift[of "fls_subdegree f" "f*g"]
+  by    (simp_all add: fls_shifted_times_simps)
+
+lemma fls_base_factor_mult_both_base_factor:
+  fixes f g :: "'a::{comm_monoid_add,mult_zero} fls"
+  shows "fls_base_factor (fls_base_factor f * fls_base_factor g) = fls_base_factor (f * g)"
+  using fls_base_factor_mult_base_factor(1)[of "fls_base_factor f" g]
+        fls_base_factor_mult_base_factor(2)[of f g]
+  by    simp
+
+lemma fls_base_factor_mult:
+  fixes f g :: "'a::semiring_no_zero_divisors fls"
+  shows "fls_base_factor (f * g) = fls_base_factor f * fls_base_factor g"
+  by    (cases "f\<noteq>0 \<and> g\<noteq>0")
+        (auto simp: fls_times_both_shifted_simp)
+
+lemma fls_times_conv_base_factor_times:
+  fixes f g :: "'a::{comm_monoid_add, mult_zero} fls"
+  shows
+    "f * g =
+      fls_shift (-(fls_subdegree f + fls_subdegree g)) (fls_base_factor f * fls_base_factor g)"
+  by (simp add: fls_times_both_shifted_simp)
+
+lemma fls_times_base_factor_conv_shifted_times:
+\<comment> \<open>Convenience form of lemma @{text "fls_times_both_shifted_simp"}.\<close>
+  fixes f g :: "'a::{comm_monoid_add, mult_zero} fls"
+  shows
+    "fls_base_factor f * fls_base_factor g = fls_shift (fls_subdegree f + fls_subdegree g) (f * g)"
+  by (simp add: fls_times_both_shifted_simp)
+
+lemma fls_times_conv_regpart:
+  fixes   f g :: "'a::{comm_monoid_add,mult_zero} fls"
+  assumes "fls_subdegree f \<ge> 0" "fls_subdegree g \<ge> 0"
+  shows "fls_regpart (f * g) = fls_regpart f * fls_regpart g"
+proof-
+  from assms have 1:
+    "f * g =
+      fls_shift (- (fls_subdegree f + fls_subdegree g)) (
+        fps_to_fls (
+          fps_shift (nat (fls_subdegree f) + nat (fls_subdegree g)) (
+            fls_regpart f * fls_regpart g
+          )
+        )
+      )"
+    by (simp add:
+      fls_times_def fls_base_factor_to_fps_conv_fps_shift[symmetric]
+      fls_regpart_subdegree_conv fps_shift_mult_both[symmetric]
+    )
+  show ?thesis
+  proof (cases "fls_regpart f * fls_regpart g = 0")
+    case False
+    with assms have
+      "subdegree (fls_regpart f * fls_regpart g) \<ge>
+        nat (fls_subdegree f) + nat (fls_subdegree g)"
+      by (simp add: fps_mult_subdegree_ge fls_regpart_subdegree_conv[symmetric])
+    with 1 assms show ?thesis by simp
+  qed (simp add: 1)
+qed
+
+lemma fls_base_factor_to_fps_mult_conv_unit_factor:
+  fixes f g :: "'a::{comm_monoid_add,mult_zero} fls"
+  shows
+    "fls_base_factor_to_fps (f * g) =
+      unit_factor (fls_base_factor_to_fps f * fls_base_factor_to_fps g)"
+  using fls_base_factor_mult_both_base_factor[of f g]
+        fps_unit_factor_fls_regpart[of "fls_base_factor f * fls_base_factor g"]
+        fls_base_factor_subdegree[of f] fls_base_factor_subdegree[of g]
+        fls_mult_subdegree_ge_0[of "fls_base_factor f" "fls_base_factor g"]
+        fls_times_conv_regpart[of "fls_base_factor f" "fls_base_factor g"]
+  by    simp
+
+lemma fls_base_factor_to_fps_mult':
+  fixes   f g :: "'a::{comm_monoid_add,mult_zero} fls"
+  assumes "(f $$ fls_subdegree f) * (g $$ fls_subdegree g) \<noteq> 0"
+  shows   "fls_base_factor_to_fps (f * g) = fls_base_factor_to_fps f * fls_base_factor_to_fps g"
+  using   assms fls_mult_nonzero_base_subdegree_eq[of f g]
+          fls_times_base_factor_conv_shifted_times[of f g]
+          fls_times_conv_regpart[of "fls_base_factor f" "fls_base_factor g"]
+          fls_base_factor_subdegree[of f] fls_base_factor_subdegree[of g]
+  by      fastforce
+
+lemma fls_base_factor_to_fps_mult:
+  fixes f g :: "'a::semiring_no_zero_divisors fls"
+  shows "fls_base_factor_to_fps (f * g) = fls_base_factor_to_fps f * fls_base_factor_to_fps g"
+  using fls_base_factor_to_fps_mult'[of f g]
+  by    (cases "f=0 \<or> g=0") auto
+
+lemma fls_times_conv_fps_times:
+  fixes   f g :: "'a::{comm_monoid_add,mult_zero} fls"
+  assumes "fls_subdegree f \<ge> 0" "fls_subdegree g \<ge> 0"
+  shows   "f * g = fps_to_fls (fls_regpart f * fls_regpart g)"
+  using   assms fls_mult_subdegree_ge[of f g]
+  by      (cases "f * g = 0") (simp_all add: fls_times_conv_regpart[symmetric])
+
+lemma fps_times_conv_fls_times:
+  fixes   f g :: "'a::{comm_monoid_add,mult_zero} fps"
+  shows   "f * g = fls_regpart (fps_to_fls f * fps_to_fls g)"
+  using   fls_subdegree_fls_to_fps_gt0 fls_times_conv_regpart[symmetric]
+  by      fastforce
+
+lemma fls_times_fps_to_fls:
+  fixes f g :: "'a::{comm_monoid_add,mult_zero} fps"
+  shows "fps_to_fls (f * g) = fps_to_fls f * fps_to_fls g"
+proof (intro fls_eq_conv_fps_eqI, rule fls_subdegree_fls_to_fps_gt0)
+  show "fls_subdegree (fps_to_fls f * fps_to_fls g) \<ge> 0"
+  proof (cases "fps_to_fls f * fps_to_fls g = 0")
+    case False thus ?thesis
+      using fls_mult_subdegree_ge fls_subdegree_fls_to_fps_gt0[of f]
+            fls_subdegree_fls_to_fps_gt0[of g]
+      by    fastforce
+  qed simp
+qed (simp add: fps_times_conv_fls_times)
+
+lemma fls_X_times_conv_shift:
+  fixes f :: "'a::{comm_monoid_add,mult_zero,monoid_mult} fls"
+  shows "fls_X * f = fls_shift (-1) f" "f * fls_X = fls_shift (-1) f"
+  by    (simp_all add: fls_X_conv_shift_1 fls_mult_one fls_shifted_times_simps)
+
+lemmas fls_X_times_comm = trans_sym[OF fls_X_times_conv_shift]   
+
+lemma fls_subdegree_mult_fls_X:
+  fixes   f :: "'a::{comm_monoid_add,mult_zero,monoid_mult} fls"
+  assumes "f \<noteq> 0"
+  shows   "fls_subdegree (fls_X * f) = fls_subdegree f + 1"
+  and     "fls_subdegree (f * fls_X) = fls_subdegree f + 1"
+  by      (auto simp: fls_X_times_conv_shift assms)
+
+lemma fls_mult_fls_X_nonzero:
+  fixes   f :: "'a::{comm_monoid_add,mult_zero,monoid_mult} fls"
+  assumes "f \<noteq> 0"
+  shows   "fls_X * f \<noteq> 0"
+  and     "f * fls_X \<noteq> 0"
+  by      (auto simp: fls_X_times_conv_shift fls_shift_eq0_iff assms)
+
+lemma fls_base_factor_mult_fls_X:
+  fixes f :: "'a::{comm_monoid_add,monoid_mult,mult_zero} fls"
+  shows "fls_base_factor (fls_X * f) = fls_base_factor f"
+  and   "fls_base_factor (f * fls_X) = fls_base_factor f"
+  using fls_base_factor_shift[of "-1" f]
+  by    (auto simp: fls_X_times_conv_shift)
+
+lemma fls_X_inv_times_conv_shift:
+  fixes f :: "'a::{comm_monoid_add,mult_zero,monoid_mult} fls"
+  shows "fls_X_inv * f = fls_shift 1 f" "f * fls_X_inv = fls_shift 1 f"
+  by    (simp_all add: fls_X_inv_conv_shift_1 fls_mult_one fls_shifted_times_simps)
+
+lemmas fls_X_inv_times_comm = trans_sym[OF fls_X_inv_times_conv_shift]
+
+lemma fls_subdegree_mult_fls_X_inv:
+  fixes   f :: "'a::{comm_monoid_add,mult_zero,monoid_mult} fls"
+  assumes "f \<noteq> 0"
+  shows   "fls_subdegree (fls_X_inv * f) = fls_subdegree f - 1"
+  and     "fls_subdegree (f * fls_X_inv) = fls_subdegree f - 1"
+  by      (auto simp: fls_X_inv_times_conv_shift assms)
+
+lemma fls_mult_fls_X_inv_nonzero:
+  fixes   f :: "'a::{comm_monoid_add,mult_zero,monoid_mult} fls"
+  assumes "f \<noteq> 0"
+  shows   "fls_X_inv * f \<noteq> 0"
+  and     "f * fls_X_inv \<noteq> 0"
+  by      (auto simp: fls_X_inv_times_conv_shift fls_shift_eq0_iff assms)
+
+lemma fls_base_factor_mult_fls_X_inv:
+  fixes f :: "'a::{comm_monoid_add,monoid_mult,mult_zero} fls"
+  shows "fls_base_factor (fls_X_inv * f) = fls_base_factor f"
+  and   "fls_base_factor (f * fls_X_inv) = fls_base_factor f"
+  using fls_base_factor_shift[of 1 f]
+  by    (auto simp: fls_X_inv_times_conv_shift)
+
+lemma fls_mult_assoc_subdegree_ge_0:
+  fixes   f g h :: "'a::semiring_0 fls"
+  assumes "fls_subdegree f \<ge> 0" "fls_subdegree g \<ge> 0" "fls_subdegree h \<ge> 0"
+  shows   "f * g * h = f * (g * h)"
+  using   assms
+  by      (simp add: fls_times_conv_fps_times fls_subdegree_fls_to_fps_gt0 mult.assoc)
+
+lemma fls_mult_assoc_base_factor:
+  fixes a b c :: "'a::semiring_0 fls"
+  shows
+    "fls_base_factor a * fls_base_factor b * fls_base_factor c =
+      fls_base_factor a * (fls_base_factor b * fls_base_factor c)"
+  by    (simp add: fls_mult_assoc_subdegree_ge_0 del: fls_base_factor_def)
+
+lemma fls_mult_distrib_subdegree_ge_0:
+  fixes   f g h :: "'a::semiring_0 fls"
+  assumes "fls_subdegree f \<ge> 0" "fls_subdegree g \<ge> 0" "fls_subdegree h \<ge> 0"
+  shows   "(f + g) * h = f * h + g * h"
+  and     "h * (f + g) = h * f + h * g"
+proof-
+  have "fls_subdegree (f+g) \<ge> 0"
+  proof (cases "f+g = 0")
+    case False
+    with assms(1,2) show ?thesis
+      using fls_plus_subdegree by fastforce
+  qed simp
+  with assms show "(f + g) * h = f * h + g * h" "h * (f + g) = h * f + h * g"
+    using distrib_right[of "fls_regpart f"] distrib_left[of "fls_regpart h"]
+    by    (simp_all add: fls_times_conv_fps_times)
+qed
+
+lemma fls_mult_distrib_base_factor:
+  fixes a b c :: "'a::semiring_0 fls"
+  shows
+    "fls_base_factor a * (fls_base_factor b + fls_base_factor c) =
+      fls_base_factor a * fls_base_factor b + fls_base_factor a * fls_base_factor c"
+  by    (simp add: fls_mult_distrib_subdegree_ge_0 del: fls_base_factor_def)
+
+instance fls :: (semiring_0) semiring_0
+proof
+
+  fix a b c :: "'a fls"
+  have
+    "a * b * c =
+      fls_shift (- (fls_subdegree a + fls_subdegree b + fls_subdegree c))
+        (fls_base_factor a * fls_base_factor b * fls_base_factor c)"
+    by (simp add: fls_times_both_shifted_simp)
+  moreover have
+    "a * (b * c) =
+      fls_shift (- (fls_subdegree a + fls_subdegree b + fls_subdegree c))
+        (fls_base_factor a * fls_base_factor b * fls_base_factor c)"
+    using fls_mult_assoc_base_factor[of a b c] by (simp add: fls_times_both_shifted_simp)
+  ultimately show "a * b * c = a * (b * c)" by simp
+
+  have ab:
+    "fls_subdegree (fls_shift (min (fls_subdegree a) (fls_subdegree b)) a) \<ge> 0"
+    "fls_subdegree (fls_shift (min (fls_subdegree a) (fls_subdegree b)) b) \<ge> 0"
+    by (simp_all add: fls_shift_nonneg_subdegree)
+  have
+    "(a + b) * c =
+      fls_shift (- (min (fls_subdegree a) (fls_subdegree b) + fls_subdegree c)) (
+        (
+          fls_shift (min (fls_subdegree a) (fls_subdegree b)) a +
+          fls_shift (min (fls_subdegree a) (fls_subdegree b)) b
+        ) * fls_base_factor c)"
+    using fls_times_both_shifted_simp[of
+            "-min (fls_subdegree a) (fls_subdegree b)"
+            "fls_shift (min (fls_subdegree a) (fls_subdegree b)) a +
+            fls_shift (min (fls_subdegree a) (fls_subdegree b)) b"
+            "-fls_subdegree c" "fls_base_factor c"
+          ]
+    by    simp
+  also have
+    "\<dots> =
+      fls_shift (-(min (fls_subdegree a) (fls_subdegree b) + fls_subdegree c))
+        (fls_shift (min (fls_subdegree a) (fls_subdegree b)) a * fls_base_factor c)
+      +
+      fls_shift (-(min (fls_subdegree a) (fls_subdegree b) + fls_subdegree c))
+        (fls_shift (min (fls_subdegree a) (fls_subdegree b)) b * fls_base_factor c)"
+    using ab
+    by    (simp add: fls_mult_distrib_subdegree_ge_0(1) del: fls_base_factor_def)
+  finally show "(a + b) * c = a * c + b * c" by (simp add: fls_times_both_shifted_simp)
+
+  have bc:
+    "fls_subdegree (fls_shift (min (fls_subdegree b) (fls_subdegree c)) b) \<ge> 0"
+    "fls_subdegree (fls_shift (min (fls_subdegree b) (fls_subdegree c)) c) \<ge> 0"
+    by (simp_all add: fls_shift_nonneg_subdegree)
+  have
+    "a * (b + c) = 
+      fls_shift (- (fls_subdegree a + min (fls_subdegree b) (fls_subdegree c))) (
+        fls_base_factor a * (
+          fls_shift (min (fls_subdegree b) (fls_subdegree c)) b +
+          fls_shift (min (fls_subdegree b) (fls_subdegree c)) c
+        )
+      )
+    "
+    using fls_times_both_shifted_simp[of
+            "-fls_subdegree a" "fls_base_factor a"
+            "-min (fls_subdegree b) (fls_subdegree c)"
+            "fls_shift (min (fls_subdegree b) (fls_subdegree c)) b +
+            fls_shift (min (fls_subdegree b) (fls_subdegree c)) c"
+          ]
+    by    simp
+  also have
+    "\<dots> =
+      fls_shift (-(fls_subdegree a + min (fls_subdegree b) (fls_subdegree c)))
+        (fls_base_factor a * fls_shift (min (fls_subdegree b) (fls_subdegree c)) b)
+      +
+      fls_shift (-(fls_subdegree a + min (fls_subdegree b) (fls_subdegree c)))
+        (fls_base_factor a * fls_shift (min (fls_subdegree b) (fls_subdegree c)) c)
+    "
+    using bc
+    by    (simp add: fls_mult_distrib_subdegree_ge_0(2) del: fls_base_factor_def)
+  finally show "a * (b + c)  = a * b + a * c" by (simp add: fls_times_both_shifted_simp)
+
+qed
+
+lemma fls_mult_commute_subdegree_ge_0:
+  fixes   f g :: "'a::comm_semiring_0 fls"
+  assumes "fls_subdegree f \<ge> 0" "fls_subdegree g \<ge> 0"
+  shows   "f * g = g * f"
+  using   assms
+  by      (simp add: fls_times_conv_fps_times mult.commute)
+
+lemma fls_mult_commute_base_factor:
+  fixes a b c :: "'a::comm_semiring_0 fls"
+  shows "fls_base_factor a * fls_base_factor b = fls_base_factor b * fls_base_factor a"
+  by    (simp add: fls_mult_commute_subdegree_ge_0 del: fls_base_factor_def)
+
+instance fls :: (comm_semiring_0) comm_semiring_0
+proof
+  fix a b c :: "'a fls"
+  show "a * b = b * a"
+    using fls_times_conv_base_factor_times[of a b] fls_times_conv_base_factor_times[of b a]
+          fls_mult_commute_base_factor[of a b]
+    by    (simp add: add.commute)
+qed (simp add: distrib_right)
+
+instance fls :: (semiring_1) semiring_1
+  by (standard, simp_all add: fls_mult_one)
+
+lemma fls_of_nat: "(of_nat n :: 'a::semiring_1 fls) = fls_const (of_nat n)"
+  by (induct n) (auto intro: fls_eqI)
+
+lemma fls_of_nat_nth: "of_nat n $$ k = (if k=0 then of_nat n else 0)"
+  by (simp add: fls_of_nat)
+
+lemma fls_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: fls_of_nat)
+
+lemma fls_subdegree_of_nat [simp]: "fls_subdegree (of_nat n) = 0"
+  by (simp add: fls_of_nat)
+
+lemma fls_shift_of_nat_nth:
+  "fls_shift k (of_nat a) $$ n = (if n=-k then of_nat a else 0)"
+  by (simp add: fls_of_nat fls_shift_const_nth)
+
+lemma fls_base_factor_of_nat [simp]:
+  "fls_base_factor (of_nat n :: 'a::semiring_1 fls) = (of_nat n :: 'a fls)"
+  by (simp add: fls_of_nat)
+
+lemma fls_regpart_of_nat [simp]: "fls_regpart (of_nat n) = (of_nat n :: 'a::semiring_1 fps)"
+  by (simp add: fls_of_nat fps_of_nat)
+
+lemma fls_prpart_of_nat [simp]: "fls_prpart (of_nat n) = 0"
+  by (simp add: fls_prpart_eq0_iff)
+
+lemma fls_base_factor_to_fps_of_nat:
+  "fls_base_factor_to_fps (of_nat n) = (of_nat n :: 'a::semiring_1 fps)"
+  by simp
+
+lemma fps_to_fls_of_nat:
+  "fps_to_fls (of_nat n) = (of_nat n :: 'a::semiring_1 fls)"
+proof -
+  have "fps_to_fls (of_nat n) = fps_to_fls (fps_const (of_nat n))"
+    by (simp add: fps_of_nat)
+  thus ?thesis by (simp add: fls_of_nat)
+qed
+
+instance fls :: (comm_semiring_1) comm_semiring_1
+  by standard simp
+
+instance fls :: (ring) ring ..
+
+instance fls :: (comm_ring) comm_ring ..
+
+instance fls :: (ring_1) ring_1 ..
+
+lemma fls_of_int_nonneg: "(of_int (int n) :: 'a::ring_1 fls) = fls_const (of_int (int n))"
+  by (induct n) (auto intro: fls_eqI)
+
+lemma fls_of_int: "(of_int i :: 'a::ring_1 fls) = fls_const (of_int i)"
+proof (induct i)
+  case (neg i)
+  have "of_int (int (Suc i)) = fls_const (of_int (int (Suc i)) :: 'a)"
+    using fls_of_int_nonneg[of "Suc i"] by simp
+  hence "- of_int (int (Suc i)) = - fls_const (of_int (int (Suc i)) :: 'a)"
+    by simp
+  thus ?case by (simp add: fls_const_uminus[symmetric])
+qed (rule fls_of_int_nonneg)
+
+lemma fls_of_int_nth: "of_int n $$ k = (if k=0 then of_int n else 0)"
+  by (simp add: fls_of_int)
+
+lemma fls_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: fls_of_int)
+
+lemma fls_subdegree_of_int [simp]: "fls_subdegree (of_int i) = 0"
+  by (simp add: fls_of_int)
+
+lemma fls_shift_of_int_nth:
+  "fls_shift k (of_int i) $$ n = (if n=-k then of_int i else 0)"
+  by (simp add: fls_of_int_nth)
+
+lemma fls_base_factor_of_int [simp]:
+  "fls_base_factor (of_int i :: 'a::ring_1 fls) = (of_int i :: 'a fls)"
+  by (simp add: fls_of_int)
+
+lemma fls_regpart_of_int [simp]:
+  "fls_regpart (of_int i) = (of_int i :: 'a::ring_1 fps)"
+  by (simp add: fls_of_int fps_of_int)
+
+lemma fls_prpart_of_int [simp]: "fls_prpart (of_int n) = 0"
+  by (simp add: fls_prpart_eq0_iff)
+
+lemma fls_base_factor_to_fps_of_int:
+  "fls_base_factor_to_fps (of_int i) = (of_int i :: 'a::ring_1 fps)"
+  by simp
+
+lemma fps_to_fls_of_int:
+  "fps_to_fls (of_int i) = (of_int i :: 'a::ring_1 fls)"
+proof -
+  have "fps_to_fls (of_int i) = fps_to_fls (fps_const (of_int i))"
+    by (simp add: fps_of_int)
+  thus ?thesis by (simp add: fls_of_int)
+qed
+
+instance fls :: (comm_ring_1) comm_ring_1 ..
+
+instance fls :: (semiring_no_zero_divisors) semiring_no_zero_divisors
+proof
+  fix a b :: "'a fls"
+  assume "a \<noteq> 0" and "b \<noteq> 0"
+  hence "(a * b) $$ (fls_subdegree a + fls_subdegree b) \<noteq> 0" by simp
+  thus "a * b \<noteq> 0" using fls_nonzeroI by fast
+qed
+
+instance fls :: (semiring_1_no_zero_divisors) semiring_1_no_zero_divisors ..
+
+instance fls :: (ring_no_zero_divisors) ring_no_zero_divisors ..
+
+instance fls :: (ring_1_no_zero_divisors) ring_1_no_zero_divisors ..
+
+instance fls :: (idom) idom ..
+
+
+subsubsection \<open>Powers\<close>
+
+lemma fls_pow_subdegree_ge:
+  "f^n \<noteq> 0 \<Longrightarrow> fls_subdegree (f^n) \<ge> n * fls_subdegree f"
+proof (induct n)
+  case (Suc n) thus ?case
+    using fls_mult_subdegree_ge[of f "f^n"] by (fastforce simp: algebra_simps)
+qed simp
+
+lemma fls_pow_nth_below_subdegree:
+  "k < n * fls_subdegree f \<Longrightarrow> (f^n) $$ k = 0"
+  using fls_pow_subdegree_ge[of f n] by (cases "f^n = 0") auto
+
+lemma fls_pow_base [simp]:
+  "(f ^ n) $$ (n * fls_subdegree f) = (f $$ fls_subdegree f) ^ n"
+proof (induct n)
+  case (Suc n)
+  show ?case
+  proof (cases "Suc n * fls_subdegree f < fls_subdegree f + fls_subdegree (f^n)")
+    case True with Suc show ?thesis
+      by (simp_all add: fls_times_nth_eq0 distrib_right)
+  next
+    case False
+    from False have
+      "{0..int n * fls_subdegree f - fls_subdegree (f ^ n)} =
+        insert 0 {1..int n * fls_subdegree f - fls_subdegree (f ^ n)}"
+      by (auto simp: algebra_simps)
+    with False Suc show ?thesis
+      by (simp add: algebra_simps fls_times_nth(4) fls_pow_nth_below_subdegree)
+  qed
+qed simp
+
+lemma fls_pow_subdegree_eqI:
+  "(f $$ fls_subdegree f) ^ n \<noteq> 0 \<Longrightarrow> fls_subdegree (f^n) = n * fls_subdegree f"
+  using fls_pow_nth_below_subdegree by (fastforce intro: fls_subdegree_eqI)
+
+lemma fls_unit_base_subdegree_power:
+  "x * f $$ fls_subdegree f = 1 \<Longrightarrow> fls_subdegree (f ^ n) = n * fls_subdegree f"
+  "f $$ fls_subdegree f * y = 1 \<Longrightarrow> fls_subdegree (f ^ n) = n * fls_subdegree f"
+proof-
+  show "x * f $$ fls_subdegree f = 1 \<Longrightarrow> fls_subdegree (f ^ n) = n * fls_subdegree f"
+    using left_right_inverse_power[of x "f $$ fls_subdegree f" n]
+    by    (auto intro: fls_pow_subdegree_eqI)
+  show "f $$ fls_subdegree f * y = 1 \<Longrightarrow> fls_subdegree (f ^ n) = n * fls_subdegree f"
+    using left_right_inverse_power[of "f $$ fls_subdegree f" y n]
+    by    (auto intro: fls_pow_subdegree_eqI)
+qed
+
+lemma fls_base_dvd1_subdegree_power:
+  "f $$ fls_subdegree f dvd 1 \<Longrightarrow> fls_subdegree (f ^ n) = n * fls_subdegree f"
+  using fls_unit_base_subdegree_power unfolding dvd_def by auto
+
+lemma fls_pow_subdegree_ge0:
+  assumes "fls_subdegree f \<ge> 0"
+  shows   "fls_subdegree (f^n) \<ge> 0"
+proof (cases "f^n = 0")
+  case False
+  moreover from assms have "int n * fls_subdegree f \<ge> 0" by simp
+  ultimately show ?thesis using fls_pow_subdegree_ge by fastforce
+qed simp
+
+lemma fls_subdegree_pow:
+  fixes   f :: "'a::semiring_1_no_zero_divisors fls"
+  shows   "fls_subdegree (f ^ n) = n * fls_subdegree f"
+proof (cases "f=0")
+  case False thus ?thesis by (induct n) (simp_all add: algebra_simps)
+qed (cases "n=0", auto simp: zero_power)
+
+lemma fls_shifted_pow:
+  "(fls_shift m f) ^ n = fls_shift (n*m) (f ^ n)"
+  by (induct n) (simp_all add: fls_times_both_shifted_simp algebra_simps)
+
+lemma fls_pow_conv_fps_pow:
+  assumes "fls_subdegree f \<ge> 0"
+  shows   "f ^ n = fps_to_fls ( (fls_regpart f) ^ n )"
+proof (induct n)
+  case (Suc n) with assms show ?case
+    using fls_pow_subdegree_ge0[of f n]
+    by (simp add: fls_times_conv_fps_times)
+qed simp
+
+lemma fls_pow_conv_regpart:
+  "fls_subdegree f \<ge> 0 \<Longrightarrow> fls_regpart (f ^ n) = (fls_regpart f) ^ n"
+  using fls_pow_subdegree_ge0[of f n] fls_pow_conv_fps_pow[of f n]
+  by    (intro fps_to_fls_eq_imp_fps_eq) simp
+
+text \<open>These two lemmas show that shifting 1 is equivalent to powers of the implied variable.\<close>
+
+lemma fls_X_power_conv_shift_1: "fls_X ^ n = fls_shift (-n) 1"
+  by (simp add: fls_X_conv_shift_1 fls_shifted_pow)
+
+lemma fls_X_inv_power_conv_shift_1: "fls_X_inv ^ n = fls_shift n 1"
+  by (simp add: fls_X_inv_conv_shift_1 fls_shifted_pow)
+
+abbreviation "fls_X_intpow \<equiv> (\<lambda>i. fls_shift (-i) 1)"
+\<comment> \<open>
+  Unifies @{term fls_X} and @{term fls_X_inv} so that @{term "fls_X_intpow"} returns the equivalent
+  of the implied variable raised to the supplied integer argument of @{term "fls_X_intpow"}, whether
+  positive or negative.
+\<close>
+
+lemma fls_X_intpow_nonzero[simp]: "(fls_X_intpow i :: 'a::zero_neq_one fls) \<noteq> 0"
+  by (simp add: fls_shift_eq0_iff)
+
+lemma fls_X_intpow_power: "(fls_X_intpow i) ^ n = fls_X_intpow (n * i)"
+  by (simp add: fls_shifted_pow)
+
+lemma fls_X_power_nth [simp]: "fls_X ^ n $$ k = (if k=n then 1 else 0)"
+  by (simp add: fls_X_power_conv_shift_1)
+
+lemma fls_X_inv_power_nth [simp]: "fls_X_inv ^ n $$ k = (if k=-n then 1 else 0)"
+  by (simp add: fls_X_inv_power_conv_shift_1)
+
+lemma fls_X_pow_nonzero[simp]: "(fls_X ^ n :: 'a :: semiring_1 fls) \<noteq> 0"
+proof
+  assume "(fls_X ^ n :: 'a fls) = 0"
+  hence "(fls_X ^ n :: 'a fls) $$ n = 0" by simp
+  thus False by simp
+qed
+
+lemma fls_X_inv_pow_nonzero[simp]: "(fls_X_inv ^ n :: 'a :: semiring_1 fls) \<noteq> 0"
+proof
+  assume "(fls_X_inv ^ n :: 'a fls) = 0"
+  hence "(fls_X_inv ^ n :: 'a fls) $$ -n = 0" by simp
+  thus False by simp
+qed
+
+lemma fls_subdegree_fls_X_pow [simp]: "fls_subdegree (fls_X ^ n) = n"
+  by (intro fls_subdegree_eqI) (simp_all add: fls_X_power_conv_shift_1)
+
+lemma fls_subdegree_fls_X_inv_pow [simp]: "fls_subdegree (fls_X_inv ^ n) = -n"
+  by (intro fls_subdegree_eqI) (simp_all add: fls_X_inv_power_conv_shift_1)
+
+lemma fls_subdegree_fls_X_intpow [simp]:
+  "fls_subdegree ((fls_X_intpow i) :: 'a::zero_neq_one fls) = i"
+  by simp
+
+lemma fls_X_pow_conv_fps_X_pow: "fls_regpart (fls_X ^ n) = fps_X ^ n"
+  by (simp add: fls_pow_conv_regpart)
+
+lemma fls_X_inv_pow_regpart: "n > 0 \<Longrightarrow> fls_regpart (fls_X_inv ^ n) = 0"
+  by (auto intro: fps_ext simp: fls_X_inv_power_conv_shift_1)
+
+lemma fls_X_intpow_regpart:
+  "fls_regpart (fls_X_intpow i) = (if i\<ge>0 then fps_X ^ nat i else 0)"
+  using fls_X_pow_conv_fps_X_pow[of "nat i"]
+        fls_regpart_shift_conv_fps_shift[of "-i" 1]
+  by    (auto simp: fls_X_power_conv_shift_1 fps_shift_one)
+
+lemma fls_X_power_times_conv_shift:
+  "fls_X ^ n * f = fls_shift (-int n) f" "f * fls_X ^ n = fls_shift (-int n) f"
+  using fls_times_both_shifted_simp[of "-int n" 1 0 f]
+        fls_times_both_shifted_simp[of 0 f "-int n" 1]
+  by    (simp_all add: fls_X_power_conv_shift_1)
+
+lemma fls_X_inv_power_times_conv_shift:
+  "fls_X_inv ^ n * f = fls_shift (int n) f" "f * fls_X_inv ^ n = fls_shift (int n) f"
+  using fls_times_both_shifted_simp[of "int n" 1 0 f]
+        fls_times_both_shifted_simp[of 0 f "int n" 1]
+  by    (simp_all add: fls_X_inv_power_conv_shift_1)
+
+lemma fls_X_intpow_times_conv_shift:
+  fixes f :: "'a::semiring_1 fls"
+  shows "fls_X_intpow i * f = fls_shift (-i) f" "f * fls_X_intpow i = fls_shift (-i) f"
+  by    (simp_all add: fls_shifted_times_simps)
+
+lemmas fls_X_power_times_comm     = trans_sym[OF fls_X_power_times_conv_shift]
+lemmas fls_X_inv_power_times_comm = trans_sym[OF fls_X_inv_power_times_conv_shift]
+
+lemma fls_X_intpow_times_comm:
+  fixes f :: "'a::semiring_1 fls"
+  shows "fls_X_intpow i * f = f * fls_X_intpow i"
+  by    (simp add: fls_X_intpow_times_conv_shift)
+
+lemma fls_X_intpow_times_fls_X_intpow:
+  "(fls_X_intpow i :: 'a::semiring_1 fls) * fls_X_intpow j = fls_X_intpow (i+j)"
+  by (simp add: fls_times_both_shifted_simp)
+
+lemma fls_X_intpow_diff_conv_times:
+  "fls_X_intpow (i-j) = (fls_X_intpow i :: 'a::semiring_1 fls) * fls_X_intpow (-j)"
+  using fls_X_intpow_times_fls_X_intpow[of i "-j",symmetric] by simp
+
+lemma fls_mult_fls_X_power_nonzero:
+  assumes "f \<noteq> 0"
+  shows   "fls_X ^ n * f \<noteq> 0" "f * fls_X ^ n \<noteq> 0"
+  by      (auto simp: fls_X_power_times_conv_shift fls_shift_eq0_iff assms)
+
+lemma fls_mult_fls_X_inv_power_nonzero:
+  assumes "f \<noteq> 0"
+  shows   "fls_X_inv ^ n * f \<noteq> 0" "f * fls_X_inv ^ n \<noteq> 0"
+  by      (auto simp: fls_X_inv_power_times_conv_shift fls_shift_eq0_iff assms)
+
+lemma fls_mult_fls_X_intpow_nonzero:
+  fixes f :: "'a::semiring_1 fls"
+  assumes "f \<noteq> 0"
+  shows   "fls_X_intpow i * f \<noteq> 0" "f * fls_X_intpow i \<noteq> 0"
+  by      (auto simp: fls_X_intpow_times_conv_shift fls_shift_eq0_iff assms)
+
+lemma fls_subdegree_mult_fls_X_power:
+  assumes "f \<noteq> 0"
+  shows   "fls_subdegree (fls_X ^ n * f) = fls_subdegree f + n"
+  and     "fls_subdegree (f * fls_X ^ n) = fls_subdegree f + n"
+  by      (auto simp: fls_X_power_times_conv_shift assms)
+
+lemma fls_subdegree_mult_fls_X_inv_power:
+  assumes "f \<noteq> 0"
+  shows   "fls_subdegree (fls_X_inv ^ n * f) = fls_subdegree f - n"
+  and     "fls_subdegree (f * fls_X_inv ^ n) = fls_subdegree f - n"
+  by      (auto simp: fls_X_inv_power_times_conv_shift assms)
+
+lemma fls_subdegree_mult_fls_X_intpow:
+  fixes   f :: "'a::semiring_1 fls"
+  assumes "f \<noteq> 0"
+  shows   "fls_subdegree (fls_X_intpow i * f) = fls_subdegree f + i"
+  and     "fls_subdegree (f * fls_X_intpow i) = fls_subdegree f + i"
+  by      (auto simp: fls_X_intpow_times_conv_shift assms)
+
+lemma fls_X_shift:
+  "fls_shift (-int n) fls_X = fls_X ^ Suc n"
+  "fls_shift (int (Suc n)) fls_X = fls_X_inv ^ n"
+  using fls_X_power_conv_shift_1[of "Suc n", symmetric]
+  by    (simp_all add: fls_X_conv_shift_1 fls_X_inv_power_conv_shift_1)
+
+lemma fls_X_inv_shift:
+  "fls_shift (int n) fls_X_inv = fls_X_inv ^ Suc n"
+  "fls_shift (- int (Suc n)) fls_X_inv = fls_X ^ n"
+  using fls_X_inv_power_conv_shift_1[of "Suc n", symmetric]
+  by    (simp_all add: fls_X_inv_conv_shift_1 fls_X_power_conv_shift_1)
+
+lemma fls_X_power_base_factor: "fls_base_factor (fls_X ^ n) = 1"
+  by (simp add: fls_X_power_conv_shift_1)
+
+lemma fls_X_inv_power_base_factor: "fls_base_factor (fls_X_inv ^ n) = 1"
+  by (simp add: fls_X_inv_power_conv_shift_1)
+
+lemma fls_X_intpow_base_factor: "fls_base_factor (fls_X_intpow i) = 1"
+  using fls_base_factor_shift[of "-i" 1] by simp
+
+lemma fls_base_factor_mult_fls_X_power:
+  shows "fls_base_factor (fls_X ^ n * f) = fls_base_factor f"
+  and   "fls_base_factor (f * fls_X ^ n) = fls_base_factor f"
+  using fls_base_factor_shift[of "-int n" f]
+  by    (auto simp: fls_X_power_times_conv_shift)
+
+lemma fls_base_factor_mult_fls_X_inv_power:
+  shows "fls_base_factor (fls_X_inv ^ n * f) = fls_base_factor f"
+  and   "fls_base_factor (f * fls_X_inv ^ n) = fls_base_factor f"
+  using fls_base_factor_shift[of "int n" f]
+  by    (auto simp: fls_X_inv_power_times_conv_shift)
+
+lemma fls_base_factor_mult_fls_X_intpow:
+  fixes f :: "'a::semiring_1 fls"
+  shows "fls_base_factor (fls_X_intpow i * f) = fls_base_factor f"
+  and   "fls_base_factor (f * fls_X_intpow i) = fls_base_factor f"
+  using fls_base_factor_shift[of "-i" f]
+  by    (auto simp: fls_X_intpow_times_conv_shift)
+
+lemma fls_X_power_base_factor_to_fps: "fls_base_factor_to_fps (fls_X ^ n) = 1"
+proof-
+  define X where "X \<equiv> fls_X :: 'a::semiring_1 fls"
+  hence "fls_base_factor (X ^ n) = 1" using fls_X_power_base_factor by simp
+  thus "fls_base_factor_to_fps (X^n) = 1" by simp
+qed  
+
+lemma fls_X_inv_power_base_factor_to_fps: "fls_base_factor_to_fps (fls_X_inv ^ n) = 1"
+proof-
+  define iX where "iX \<equiv> fls_X_inv :: 'a::semiring_1 fls"
+  hence "fls_base_factor (iX ^ n) = 1" using fls_X_inv_power_base_factor by simp
+  thus "fls_base_factor_to_fps (iX^n) = 1" by simp
+qed  
+
+lemma fls_X_intpow_base_factor_to_fps: "fls_base_factor_to_fps (fls_X_intpow i) = 1"
+proof-
+  define f :: "'a fls" where "f \<equiv> fls_X_intpow i"
+  moreover have "fls_base_factor (fls_X_intpow i) = 1" by (rule fls_X_intpow_base_factor)
+  ultimately have "fls_base_factor f = 1" by simp
+  thus "fls_base_factor_to_fps f = 1" by simp
+qed
+
+lemma fls_base_factor_X_power_decompose:
+  fixes f :: "'a::semiring_1 fls"
+  shows "f = fls_base_factor f * fls_X_intpow (fls_subdegree f)"
+  and   "f = fls_X_intpow (fls_subdegree f) * fls_base_factor f"
+  by    (simp_all add: fls_times_both_shifted_simp)
+
+lemma fls_normalized_product_of_inverses:
+  assumes "f * g = 1"
+  shows   "fls_base_factor f * fls_base_factor g =
+            fls_X ^ (nat (-(fls_subdegree f+fls_subdegree g)))"
+  and     "fls_base_factor f * fls_base_factor g =
+            fls_X_intpow (-(fls_subdegree f+fls_subdegree g))"
+  using   fls_mult_subdegree_ge[of f g]
+          fls_times_base_factor_conv_shifted_times[of f g]
+  by      (simp_all add: assms fls_X_power_conv_shift_1 algebra_simps)
+
+lemma fls_fps_normalized_product_of_inverses:
+  assumes "f * g = 1"
+  shows   "fls_base_factor_to_fps f * fls_base_factor_to_fps g =
+            fps_X ^ (nat (-(fls_subdegree f+fls_subdegree g)))"
+  using fls_times_conv_regpart[of "fls_base_factor f" "fls_base_factor g"]
+        fls_base_factor_subdegree[of f] fls_base_factor_subdegree[of g]
+        fls_normalized_product_of_inverses(1)[OF assms]
+  by    (force simp: fls_X_pow_conv_fps_X_pow)
+
+
+subsubsection \<open>Inverses\<close>
+
+\<comment> \<open>See lemma fls_left_inverse\<close> 
+abbreviation fls_left_inverse ::
+  "'a::{comm_monoid_add,uminus,times} fls \<Rightarrow> 'a \<Rightarrow> 'a fls"
+  where
+  "fls_left_inverse f x \<equiv>
+    fls_shift (fls_subdegree f) (fps_to_fls (fps_left_inverse (fls_base_factor_to_fps f) x))"
+
+\<comment> \<open>See lemma fls_right_inverse\<close> 
+abbreviation fls_right_inverse ::
+  "'a::{comm_monoid_add,uminus,times} fls \<Rightarrow> 'a \<Rightarrow> 'a fls"
+  where
+  "fls_right_inverse f y \<equiv>
+    fls_shift (fls_subdegree f) (fps_to_fls (fps_right_inverse (fls_base_factor_to_fps f) y))"
+
+instantiation fls :: ("{comm_monoid_add,uminus,times,inverse}") inverse
+begin
+  definition fls_divide_def:
+    "f div g =
+      fls_shift (fls_subdegree g - fls_subdegree f) (
+        fps_to_fls ((fls_base_factor_to_fps f) div (fls_base_factor_to_fps g))
+      )
+    "
+  definition fls_inverse_def:
+    "inverse f = fls_shift (fls_subdegree f) (fps_to_fls (inverse (fls_base_factor_to_fps f)))"
+  instance ..
+end
+
+lemma fls_inverse_def':
+  "inverse f = fls_right_inverse f (inverse (f $$ fls_subdegree f))"
+  by (simp add: fls_inverse_def fps_inverse_def)
+
+lemma fls_lr_inverse_base:
+  "fls_left_inverse f x $$ (-fls_subdegree f) = x"
+  "fls_right_inverse f y $$ (-fls_subdegree f) = y"
+  by auto
+
+lemma fls_inverse_base:
+  "f \<noteq> 0 \<Longrightarrow> inverse f $$ (-fls_subdegree f) = inverse (f $$ fls_subdegree f)"
+  by (simp add: fls_inverse_def')
+
+lemma fls_lr_inverse_starting0:
+  fixes f :: "'a::{comm_monoid_add,mult_zero,uminus} fls"
+  and   g :: "'b::{ab_group_add,mult_zero} fls"
+  shows "fls_left_inverse f 0 = 0"
+  and   "fls_right_inverse g 0 = 0"
+  by    (simp_all add: fps_lr_inverse_starting0)
+
+lemma fls_lr_inverse_eq0_imp_starting0:
+  "fls_left_inverse f x = 0 \<Longrightarrow> x = 0"
+  "fls_right_inverse f x = 0 \<Longrightarrow> x = 0"
+proof-
+  assume "fls_left_inverse f x = 0"
+  hence "fps_left_inverse (fls_base_factor_to_fps f) x = 0"
+    using fls_shift_eq_iff fps_to_fls_eq_zero_iff by fastforce
+  thus "x = 0" using fps_lr_inverse_eq0_imp_starting0(1) by fast
+next
+  assume "fls_right_inverse f x = 0"
+  hence "fps_right_inverse (fls_base_factor_to_fps f) x = 0"
+    using fls_shift_eq_iff fps_to_fls_eq_zero_iff by fastforce
+  thus "x = 0" using fps_lr_inverse_eq0_imp_starting0(2) by fast
+qed
+
+lemma fls_lr_inverse_eq_0_iff:
+  fixes x :: "'a::{comm_monoid_add,mult_zero,uminus}"
+  and   y :: "'b::{ab_group_add,mult_zero}"
+  shows "fls_left_inverse f x = 0 \<longleftrightarrow> x = 0"
+  and   "fls_right_inverse g y = 0 \<longleftrightarrow> y = 0"
+  using fls_lr_inverse_starting0 fls_lr_inverse_eq0_imp_starting0
+  by    auto
+
+lemma fls_inverse_eq_0_iff':
+  fixes f :: "'a::{ab_group_add,inverse,mult_zero} fls"
+  shows "inverse f = 0 \<longleftrightarrow> (inverse (f $$ fls_subdegree f) = 0)"
+  using fls_lr_inverse_eq_0_iff(2)[of f "inverse (f $$ fls_subdegree f)"]
+  by    (simp add: fls_inverse_def')
+
+lemma fls_inverse_eq_0_iff[simp]:
+  "inverse f = (0:: ('a::division_ring) fls) \<longleftrightarrow> f $$ fls_subdegree f = 0"
+  using fls_inverse_eq_0_iff'[of f] by (cases "f=0") auto
+
+lemmas fls_inverse_eq_0' = iffD2[OF fls_inverse_eq_0_iff']
+lemmas fls_inverse_eq_0  = iffD2[OF fls_inverse_eq_0_iff]
+
+lemma fls_lr_inverse_const:
+  fixes a :: "'a::{ab_group_add,mult_zero}"
+  and   b :: "'b::{comm_monoid_add,mult_zero,uminus}"
+  shows "fls_left_inverse (fls_const a) x = fls_const x"
+  and   "fls_right_inverse (fls_const b) y = fls_const y"
+  by    (simp_all add: fps_const_lr_inverse)
+
+lemma fls_inverse_const:
+  fixes a :: "'a::{comm_monoid_add,inverse,mult_zero,uminus}"
+  shows "inverse (fls_const a) = fls_const (inverse a)"
+  using fls_lr_inverse_const(2)
+  by    (auto simp: fls_inverse_def')
+
+lemma fls_lr_inverse_of_nat:
+  fixes x :: "'a::{ring_1,mult_zero}"
+  and   y :: "'b::{semiring_1,uminus}"
+  shows "fls_left_inverse (of_nat n) x = fls_const x"
+  and   "fls_right_inverse (of_nat n) y = fls_const y"
+  using fls_lr_inverse_const
+  by    (auto simp: fls_of_nat)
+
+lemma fls_inverse_of_nat:
+  "inverse (of_nat n :: 'a :: {semiring_1,inverse,uminus} fls) = fls_const (inverse (of_nat n))"
+  by (simp add: fls_inverse_const fls_of_nat)
+
+lemma fls_lr_inverse_of_int:
+  fixes x :: "'a::{ring_1,mult_zero}"
+  shows "fls_left_inverse (of_int n) x = fls_const x"
+  and   "fls_right_inverse (of_int n) x = fls_const x"
+  using fls_lr_inverse_const
+  by    (auto simp: fls_of_int)
+
+lemma fls_inverse_of_int:
+  "inverse (of_int n :: 'a :: {ring_1,inverse,uminus} fls) = fls_const (inverse (of_int n))"
+  by      (simp add: fls_inverse_const fls_of_int)
+
+lemma fls_lr_inverse_zero:
+  fixes x :: "'a::{ab_group_add,mult_zero}"
+  and   y :: "'b::{comm_monoid_add,mult_zero,uminus}"
+  shows "fls_left_inverse 0 x = fls_const x"
+  and   "fls_right_inverse 0 y = fls_const y"
+  using fls_lr_inverse_const[of 0]
+  by    auto
+
+lemma fls_inverse_zero_conv_fls_const:
+  "inverse (0::'a::{comm_monoid_add,mult_zero,uminus,inverse} fls) = fls_const (inverse 0)"
+  using fls_lr_inverse_zero(2)[of "inverse (0::'a)"] by (simp add: fls_inverse_def')
+
+lemma fls_inverse_zero':
+  assumes "inverse (0::'a::{comm_monoid_add,inverse,mult_zero,uminus}) = 0"
+  shows   "inverse (0::'a fls) = 0"
+  by      (simp add: fls_inverse_zero_conv_fls_const assms)
+
+lemma fls_inverse_zero [simp]: "inverse (0::'a::division_ring fls) = 0"
+  by (rule fls_inverse_zero'[OF inverse_zero])
+
+lemma fls_inverse_base2:
+  fixes f :: "'a::{comm_monoid_add,mult_zero,uminus,inverse} fls"
+  shows "inverse f $$ (-fls_subdegree f) = inverse (f $$ fls_subdegree f)"
+  by    (cases "f=0") (simp_all add: fls_inverse_zero_conv_fls_const fls_inverse_def')
+
+lemma fls_lr_inverse_one:
+  fixes x :: "'a::{ab_group_add,mult_zero,one}"
+  and   y :: "'b::{comm_monoid_add,mult_zero,uminus,one}"
+  shows "fls_left_inverse 1 x = fls_const x"
+  and   "fls_right_inverse 1 y = fls_const y"
+  using fls_lr_inverse_const[of 1]
+  by    auto
+
+lemma fls_lr_inverse_one_one:
+  "fls_left_inverse 1 1 =
+    (1::'a::{ab_group_add,mult_zero,one} fls)"
+  "fls_right_inverse 1 1 =
+    (1::'b::{comm_monoid_add,mult_zero,uminus,one} fls)"
+  using fls_lr_inverse_one[of 1] by auto
+
+lemma fls_inverse_one:
+  assumes "inverse (1::'a::{comm_monoid_add,inverse,mult_zero,uminus,one}) = 1"
+  shows   "inverse (1::'a fls) = 1"
+  using   assms fls_lr_inverse_one_one(2)
+  by      (simp add: fls_inverse_def')
+
+lemma fls_left_inverse_delta:
+  fixes   b :: "'a::{ab_group_add,mult_zero}"
+  assumes "b \<noteq> 0"
+  shows   "fls_left_inverse (Abs_fls (\<lambda>n. if n=a then b else 0)) x =
+            Abs_fls (\<lambda>n. if n=-a then x else 0)"
+proof (intro fls_eqI)
+  fix n from assms show
+    "fls_left_inverse (Abs_fls (\<lambda>n. if n=a then b else 0)) x $$ n
+      = Abs_fls (\<lambda>n. if n = - a then x else 0) $$ n"
+    using fls_base_factor_to_fps_delta[of a b]
+          fls_lr_inverse_const(1)[of b]
+          fls_shift_const
+    by    simp
+qed
+
+lemma fls_right_inverse_delta:
+  fixes   b :: "'a::{comm_monoid_add,mult_zero,uminus}"
+  assumes "b \<noteq> 0"
+  shows   "fls_right_inverse (Abs_fls (\<lambda>n. if n=a then b else 0)) x =
+            Abs_fls (\<lambda>n. if n=-a then x else 0)"
+proof (intro fls_eqI)
+  fix n from assms show
+    "fls_right_inverse (Abs_fls (\<lambda>n. if n=a then b else 0)) x $$ n
+      = Abs_fls (\<lambda>n. if n = - a then x else 0) $$ n"
+    using fls_base_factor_to_fps_delta[of a b]
+          fls_lr_inverse_const(2)[of b]
+          fls_shift_const
+    by    simp
+qed
+
+lemma fls_inverse_delta_nonzero:
+  fixes   b :: "'a::{comm_monoid_add,inverse,mult_zero,uminus}"
+  assumes "b \<noteq> 0"
+  shows   "inverse (Abs_fls (\<lambda>n. if n=a then b else 0)) =
+            Abs_fls (\<lambda>n. if n=-a then inverse b else 0)"
+  using   assms fls_nonzeroI[of "Abs_fls (\<lambda>n. if n=a then b else 0)" a]
+  by      (simp add: fls_inverse_def' fls_right_inverse_delta[symmetric])
+
+lemma fls_inverse_delta:
+  fixes   b :: "'a::division_ring"
+  shows   "inverse (Abs_fls (\<lambda>n. if n=a then b else 0)) =
+            Abs_fls (\<lambda>n. if n=-a then inverse b else 0)"
+  by      (cases "b=0") (simp_all add: fls_inverse_delta_nonzero)
+
+lemma fls_lr_inverse_X:
+  fixes x :: "'a::{ab_group_add,mult_zero,zero_neq_one}"
+  and   y :: "'b::{comm_monoid_add,uminus,mult_zero,zero_neq_one}"
+  shows "fls_left_inverse fls_X x = fls_shift 1 (fls_const x)"
+  and   "fls_right_inverse fls_X y = fls_shift 1 (fls_const y)"
+  using fls_lr_inverse_one(1)[of x] fls_lr_inverse_one(2)[of y]
+  by    auto
+
+lemma fls_lr_inverse_X':
+  fixes x :: "'a::{ab_group_add,mult_zero,zero_neq_one,monoid_mult}"
+  and   y :: "'b::{comm_monoid_add,uminus,mult_zero,zero_neq_one,monoid_mult}"
+  shows "fls_left_inverse fls_X x = fls_const x * fls_X_inv"
+  and   "fls_right_inverse fls_X y = fls_const y * fls_X_inv"
+  using fls_lr_inverse_X(1)[of x] fls_lr_inverse_X(2)[of y]
+  by    (simp_all add: fls_X_inv_times_conv_shift(2))
+
+lemma fls_inverse_X':
+  assumes "inverse 1 = (1::'a::{comm_monoid_add,inverse,mult_zero,uminus,zero_neq_one})"
+  shows   "inverse (fls_X::'a fls) = fls_X_inv"
+  using   assms fls_lr_inverse_X(2)[of "1::'a"]
+  by      (simp add: fls_inverse_def' fls_X_inv_conv_shift_1)
+
+lemma fls_inverse_X: "inverse (fls_X::'a::division_ring fls) = fls_X_inv"
+  by (simp add: fls_inverse_X')
+
+lemma fls_lr_inverse_X_inv:
+  fixes x :: "'a::{ab_group_add,mult_zero,zero_neq_one}"
+  and   y :: "'b::{comm_monoid_add,uminus,mult_zero,zero_neq_one}"
+  shows "fls_left_inverse fls_X_inv x = fls_shift (-1) (fls_const x)"
+  and   "fls_right_inverse fls_X_inv y = fls_shift (-1) (fls_const y)"
+  using fls_lr_inverse_one(1)[of x] fls_lr_inverse_one(2)[of y]
+  by    auto
+
+lemma fls_lr_inverse_X_inv':
+  fixes x :: "'a::{ab_group_add,mult_zero,zero_neq_one,monoid_mult}"
+  and   y :: "'b::{comm_monoid_add,uminus,mult_zero,zero_neq_one,monoid_mult}"
+  shows "fls_left_inverse fls_X_inv x = fls_const x * fls_X"
+  and   "fls_right_inverse fls_X_inv y = fls_const y * fls_X"
+  using fls_lr_inverse_X_inv(1)[of x] fls_lr_inverse_X_inv(2)[of y]
+  by    (simp_all add: fls_X_times_conv_shift(2))
+
+lemma fls_inverse_X_inv':
+  assumes "inverse 1 = (1::'a::{comm_monoid_add,inverse,mult_zero,uminus,zero_neq_one})"
+  shows   "inverse (fls_X_inv::'a fls) = fls_X"
+  using   assms fls_lr_inverse_X_inv(2)[of "1::'a"]
+  by      (simp add: fls_inverse_def' fls_X_conv_shift_1)
+
+lemma fls_inverse_X_inv: "inverse (fls_X_inv::'a::division_ring fls) = fls_X"
+  by (simp add: fls_inverse_X_inv')
+
+lemma fls_lr_inverse_subdegree:
+  assumes "x \<noteq> 0"
+  shows   "fls_subdegree (fls_left_inverse f x) = - fls_subdegree f"
+  and     "fls_subdegree (fls_right_inverse f x) = - fls_subdegree f"
+  by      (auto intro: fls_subdegree_eqI simp: assms)
+
+lemma fls_inverse_subdegree':
+  "inverse (f $$ fls_subdegree f) \<noteq> 0 \<Longrightarrow> fls_subdegree (inverse f) = - fls_subdegree f"
+  using fls_lr_inverse_subdegree(2)[of "inverse (f $$ fls_subdegree f)"]
+  by    (simp add: fls_inverse_def')
+
+lemma fls_inverse_subdegree [simp]:
+  fixes f :: "'a::division_ring fls"
+  shows "fls_subdegree (inverse f) = - fls_subdegree f"
+  by    (cases "f=0")
+        (auto intro: fls_inverse_subdegree' simp: nonzero_imp_inverse_nonzero)
+
+lemma fls_inverse_subdegree_base_nonzero:
+  assumes "f \<noteq> 0" "inverse (f $$ fls_subdegree f) \<noteq> 0"
+  shows   "inverse f $$ (fls_subdegree (inverse f)) = inverse (f $$ fls_subdegree f)"
+  using   assms fls_inverse_subdegree'[of f] fls_inverse_base[of f]
+  by      simp
+
+lemma fls_inverse_subdegree_base:
+  fixes f :: "'a::{ab_group_add,inverse,mult_zero} fls"
+  shows "inverse f $$ (fls_subdegree (inverse f)) = inverse (f $$ fls_subdegree f)"
+  using fls_inverse_eq_0_iff'[of f] fls_inverse_subdegree_base_nonzero[of f]
+  by    (cases "f=0 \<or> inverse (f $$ fls_subdegree f) = 0")
+        (auto simp: fls_inverse_zero_conv_fls_const)
+
+lemma fls_lr_inverse_subdegree_0:
+  assumes "fls_subdegree f = 0"
+  shows   "fls_subdegree (fls_left_inverse f x) \<ge> 0"
+  and     "fls_subdegree (fls_right_inverse f x) \<ge> 0"
+  using   fls_subdegree_ge0I[of "fls_left_inverse f x"]
+          fls_subdegree_ge0I[of "fls_right_inverse f x"]
+  by      (auto simp: assms)
+
+lemma fls_inverse_subdegree_0:
+  "fls_subdegree f = 0 \<Longrightarrow> fls_subdegree (inverse f) \<ge> 0"
+  using fls_lr_inverse_subdegree_0(2)[of f] by (simp add: fls_inverse_def')
+
+lemma fls_lr_inverse_shift_nonzero:
+  fixes   f :: "'a::{comm_monoid_add,mult_zero,uminus} fls"
+  assumes "f \<noteq> 0"
+  shows   "fls_left_inverse (fls_shift m f) x = fls_shift (-m) (fls_left_inverse f x)"
+  and     "fls_right_inverse (fls_shift m f) x = fls_shift (-m) (fls_right_inverse f x)"
+  using   assms fls_base_factor_to_fps_shift[of m f] fls_shift_subdegree
+  by      auto
+
+lemma fls_inverse_shift_nonzero:
+  fixes   f :: "'a::{comm_monoid_add,inverse,mult_zero,uminus} fls"
+  assumes "f \<noteq> 0"
+  shows   "inverse (fls_shift m f) = fls_shift (-m) (inverse f)"
+  using   assms fls_lr_inverse_shift_nonzero(2)[of f m "inverse (f $$ fls_subdegree f)"]
+  by      (simp add: fls_inverse_def')
+
+lemma fls_inverse_shift:
+  fixes f :: "'a::division_ring fls"
+  shows "inverse (fls_shift m f) = fls_shift (-m) (inverse f)"
+  using fls_inverse_shift_nonzero
+  by    (cases "f=0") simp_all
+
+lemma fls_left_inverse_base_factor:
+  fixes   x :: "'a::{ab_group_add,mult_zero}"
+  assumes "x \<noteq> 0"
+  shows   "fls_left_inverse (fls_base_factor f) x = fls_base_factor (fls_left_inverse f x)"
+  using   assms fls_lr_inverse_zero(1)[of x] fls_lr_inverse_subdegree(1)[of x]
+  by      (cases "f=0") auto
+
+lemma fls_right_inverse_base_factor:
+  fixes   y :: "'a::{comm_monoid_add,mult_zero,uminus}"
+  assumes "y \<noteq> 0"
+  shows   "fls_right_inverse (fls_base_factor f) y = fls_base_factor (fls_right_inverse f y)"
+  using   assms fls_lr_inverse_zero(2)[of y] fls_lr_inverse_subdegree(2)[of y]
+  by      (cases "f=0") auto
+
+lemma fls_inverse_base_factor':
+  fixes   f :: "'a::{comm_monoid_add,inverse,mult_zero,uminus} fls"
+  assumes "inverse (f $$ fls_subdegree f) \<noteq> 0"
+  shows   "inverse (fls_base_factor f) = fls_base_factor (inverse f)"
+  by      (cases "f=0")
+          (simp_all add:
+            assms fls_inverse_shift_nonzero fls_inverse_subdegree'
+            fls_inverse_zero_conv_fls_const
+          )
+
+lemma fls_inverse_base_factor:
+  fixes f :: "'a::{ab_group_add,inverse,mult_zero} fls"
+  shows "inverse (fls_base_factor f) = fls_base_factor (inverse f)"
+  using fls_base_factor_base[of f] fls_inverse_eq_0_iff'[of f]
+        fls_inverse_eq_0_iff'[of "fls_base_factor f"] fls_inverse_base_factor'[of f]
+  by    (cases "inverse (f $$ fls_subdegree f) = 0") simp_all
+
+lemma fls_lr_inverse_regpart:
+  assumes "fls_subdegree f = 0"
+  shows   "fls_regpart (fls_left_inverse f x) = fps_left_inverse (fls_regpart f) x"
+  and     "fls_regpart (fls_right_inverse f y) = fps_right_inverse (fls_regpart f) y"
+  using   assms
+  by      auto
+
+lemma fls_inverse_regpart:
+  assumes "fls_subdegree f = 0"
+  shows   "fls_regpart (inverse f) = inverse (fls_regpart f)"
+  by      (simp add: assms fls_inverse_def)
+
+lemma fls_base_factor_to_fps_left_inverse:
+  fixes   x :: "'a::{ab_group_add,mult_zero}"
+  shows   "fls_base_factor_to_fps (fls_left_inverse f x) =
+            fps_left_inverse (fls_base_factor_to_fps f) x"
+  using   fls_left_inverse_base_factor[of x f] fls_base_factor_subdegree[of f]
+  by      (cases "x=0") (simp_all add: fls_lr_inverse_starting0(1) fps_lr_inverse_starting0(1))
+
+lemma fls_base_factor_to_fps_right_inverse_nonzero:
+  fixes   y :: "'a::{comm_monoid_add,mult_zero,uminus}"
+  assumes "y \<noteq> 0"
+  shows   "fls_base_factor_to_fps (fls_right_inverse f y) =
+            fps_right_inverse (fls_base_factor_to_fps f) y"
+  using   assms fls_right_inverse_base_factor[of y f]
+          fls_base_factor_subdegree[of f]
+  by      simp
+
+lemma fls_base_factor_to_fps_right_inverse:
+  fixes   y :: "'a::{ab_group_add,mult_zero}"
+  shows   "fls_base_factor_to_fps (fls_right_inverse f y) =
+            fps_right_inverse (fls_base_factor_to_fps f) y"
+  using   fls_base_factor_to_fps_right_inverse_nonzero[of y f]
+  by      (cases "y=0") (simp_all add: fls_lr_inverse_starting0(2) fps_lr_inverse_starting0(2))
+
+lemma fls_base_factor_to_fps_inverse_nonzero:
+  fixes   f :: "'a::{comm_monoid_add,inverse,mult_zero,uminus} fls"
+  assumes "inverse (f $$ fls_subdegree f) \<noteq> 0"
+  shows   "fls_base_factor_to_fps (inverse f) = inverse (fls_base_factor_to_fps f)"
+  using   assms fls_base_factor_to_fps_right_inverse_nonzero
+  by      (simp add: fls_inverse_def' fps_inverse_def)
+
+lemma fls_base_factor_to_fps_inverse:
+  fixes f :: "'a::{ab_group_add,inverse,mult_zero} fls"
+  shows "fls_base_factor_to_fps (inverse f) = inverse (fls_base_factor_to_fps f)"
+  using fls_base_factor_to_fps_right_inverse
+  by    (simp add: fls_inverse_def' fps_inverse_def)
+
+lemma fls_lr_inverse_fps_to_fls:
+  assumes "subdegree f = 0"
+  shows   "fls_left_inverse (fps_to_fls f) x = fps_to_fls (fps_left_inverse f x)"
+  and     "fls_right_inverse (fps_to_fls f) x = fps_to_fls (fps_right_inverse f x)"
+  using   assms fls_base_factor_to_fps_to_fls[of f]
+  by      (simp_all add: fls_subdegree_fls_to_fps)
+
+lemma fls_inverse_fps_to_fls:
+  "subdegree f = 0 \<Longrightarrow> inverse (fps_to_fls f) = fps_to_fls (inverse f)"
+  using nth_subdegree_nonzero[of f]
+  by  (cases "f=0")
+      (auto simp add:
+        fps_to_fls_nonzeroI fls_inverse_def' fls_subdegree_fls_to_fps fps_inverse_def
+        fls_lr_inverse_fps_to_fls(2)
+      )
+
+lemma fls_lr_inverse_X_power:
+  fixes x :: "'a::ring_1"
+  and   y :: "'b::{semiring_1,uminus}"
+  shows "fls_left_inverse (fls_X ^ n) x = fls_shift n (fls_const x)"
+  and   "fls_right_inverse (fls_X ^ n) y = fls_shift n (fls_const y)"
+  using fls_lr_inverse_one(1)[of x] fls_lr_inverse_one(2)[of y]
+  by    (simp_all add: fls_X_power_conv_shift_1)
+
+lemma fls_lr_inverse_X_power':
+  fixes x :: "'a::ring_1"
+  and   y :: "'b::{semiring_1,uminus}"
+  shows "fls_left_inverse (fls_X ^ n) x = fls_const x * fls_X_inv ^ n"
+  and   "fls_right_inverse (fls_X ^ n) y = fls_const y * fls_X_inv ^ n"
+  using fls_lr_inverse_X_power(1)[of n x] fls_lr_inverse_X_power(2)[of n y]
+  by    (simp_all add: fls_X_inv_power_times_conv_shift(2))
+
+lemma fls_inverse_X_power':
+  assumes "inverse 1 = (1::'a::{semiring_1,uminus,inverse})"
+  shows   "inverse ((fls_X ^ n)::'a fls) = fls_X_inv ^ n"
+  using   fls_lr_inverse_X_power'(2)[of n 1]
+  by      (simp add: fls_inverse_def' assms )
+
+lemma fls_inverse_X_power:
+  "inverse ((fls_X::'a::division_ring fls) ^ n) = fls_X_inv ^ n"
+  by (simp add: fls_inverse_X_power')
+
+lemma fls_lr_inverse_X_inv_power:
+  fixes x :: "'a::ring_1"
+  and   y :: "'b::{semiring_1,uminus}"
+  shows "fls_left_inverse (fls_X_inv ^ n) x = fls_shift (-n) (fls_const x)"
+  and   "fls_right_inverse (fls_X_inv ^ n) y = fls_shift (-n) (fls_const y)"
+  using fls_lr_inverse_one(1)[of x] fls_lr_inverse_one(2)[of y]
+  by    (simp_all add: fls_X_inv_power_conv_shift_1)
+
+lemma fls_lr_inverse_X_inv_power':
+  fixes x :: "'a::ring_1"
+  and   y :: "'b::{semiring_1,uminus}"
+  shows "fls_left_inverse (fls_X_inv ^ n) x = fls_const x * fls_X ^ n"
+  and   "fls_right_inverse (fls_X_inv ^ n) y = fls_const y * fls_X ^ n"
+  using fls_lr_inverse_X_inv_power(1)[of n x] fls_lr_inverse_X_inv_power(2)[of n y]
+  by    (simp_all add: fls_X_power_times_conv_shift(2))
+
+lemma fls_inverse_X_inv_power':
+  assumes "inverse 1 = (1::'a::{semiring_1,uminus,inverse})"
+  shows   "inverse ((fls_X_inv ^ n)::'a fls) = fls_X ^ n"
+  using   fls_lr_inverse_X_inv_power'(2)[of n 1]
+  by      (simp add: fls_inverse_def' assms)
+
+lemma fls_inverse_X_inv_power:
+  "inverse ((fls_X_inv::'a::division_ring fls) ^ n) = fls_X ^ n"
+  by (simp add: fls_inverse_X_inv_power')
+
+lemma fls_lr_inverse_X_intpow:
+  fixes x :: "'a::ring_1"
+  and   y :: "'b::{semiring_1,uminus}"
+  shows "fls_left_inverse (fls_X_intpow i) x = fls_shift i (fls_const x)"
+  and   "fls_right_inverse (fls_X_intpow i) y = fls_shift i (fls_const y)"
+  using fls_lr_inverse_one(1)[of x] fls_lr_inverse_one(2)[of y]
+  by    auto
+
+lemma fls_lr_inverse_X_intpow':
+  fixes x :: "'a::ring_1"
+  and   y :: "'b::{semiring_1,uminus}"
+  shows "fls_left_inverse (fls_X_intpow i) x = fls_const x * fls_X_intpow (-i)"
+  and   "fls_right_inverse (fls_X_intpow i) y = fls_const y * fls_X_intpow (-i)"
+  using fls_lr_inverse_X_intpow(1)[of i x] fls_lr_inverse_X_intpow(2)[of i y]
+  by    (simp_all add: fls_shifted_times_simps(1))
+
+lemma fls_inverse_X_intpow':
+  assumes "inverse 1 = (1::'a::{semiring_1,uminus,inverse})"
+  shows   "inverse (fls_X_intpow i :: 'a fls) = fls_X_intpow (-i)"
+  using   fls_lr_inverse_X_intpow'(2)[of i 1]
+  by      (simp add: fls_inverse_def' assms)
+
+lemma fls_inverse_X_intpow:
+  "inverse (fls_X_intpow i :: 'a::division_ring fls) = fls_X_intpow (-i)"
+  by (simp add: fls_inverse_X_intpow')
+
+lemma fls_left_inverse:
+  fixes   f :: "'a::ring_1 fls"
+  assumes "x * f $$ fls_subdegree f = 1"
+  shows   "fls_left_inverse f x * f = 1"
+proof-
+  from assms have "x \<noteq> 0" "x * (fls_base_factor_to_fps f$0) = 1" by auto
+  thus ?thesis
+    using fls_base_factor_to_fps_left_inverse[of f x]
+          fls_lr_inverse_subdegree(1)[of x] fps_left_inverse
+    by    (fastforce simp: fls_times_def)
+qed
+
+lemma fls_right_inverse:
+  fixes   f :: "'a::ring_1 fls"
+  assumes "f $$ fls_subdegree f * y = 1"
+  shows   "f * fls_right_inverse f y = 1"
+proof-
+  from assms have "y \<noteq> 0" "(fls_base_factor_to_fps f$0) * y = 1" by auto
+  thus ?thesis
+    using fls_base_factor_to_fps_right_inverse[of f y]
+          fls_lr_inverse_subdegree(2)[of y] fps_right_inverse
+    by    (fastforce simp: fls_times_def)
+qed
+
+\<comment> \<open>
+  It is possible in a ring for an element to have a left inverse but not a right inverse, or
+  vice versa. But when an element has both, they must be the same.
+\<close>
+lemma fls_left_inverse_eq_fls_right_inverse:
+  fixes   f :: "'a::ring_1 fls"
+  assumes "x * f $$ fls_subdegree f = 1" "f $$ fls_subdegree f * y = 1"
+  \<comment> \<open>These assumptions imply x equals y, but no need to assume that.\<close>
+  shows   "fls_left_inverse f x = fls_right_inverse f y"
+  using   assms
+  by      (simp add: fps_left_inverse_eq_fps_right_inverse)
+
+lemma fls_left_inverse_eq_inverse:
+  fixes   f :: "'a::division_ring fls"
+  shows   "fls_left_inverse f (inverse (f $$ fls_subdegree f)) = inverse f"
+proof (cases "f=0")
+  case True
+  hence "fls_left_inverse f (inverse (f $$ fls_subdegree f)) = fls_const (0::'a)"
+    by (simp add: fls_lr_inverse_zero(1)[symmetric])
+  with True show ?thesis by simp
+next
+  case False thus ?thesis
+    using fls_left_inverse_eq_fls_right_inverse[of "inverse (f $$ fls_subdegree f)"]
+    by    (auto simp add: fls_inverse_def')
+qed
+
+lemma fls_right_inverse_eq_inverse:
+  fixes f :: "'a::division_ring fls"
+  shows "fls_right_inverse f (inverse (f $$ fls_subdegree f)) = inverse f"
+proof (cases "f=0")
+  case True
+  hence "fls_right_inverse f (inverse (f $$ fls_subdegree f)) = fls_const (0::'a)"
+    by (simp add: fls_lr_inverse_zero(2)[symmetric])
+  with True show ?thesis by simp
+qed (simp add: fls_inverse_def')
+
+lemma fls_left_inverse_eq_fls_right_inverse_comm:
+  fixes   f :: "'a::comm_ring_1 fls"
+  assumes "x * f $$ fls_subdegree f = 1"
+  shows   "fls_left_inverse f x = fls_right_inverse f x"
+  using   assms fls_left_inverse_eq_fls_right_inverse[of x f x]
+  by      (simp add: mult.commute)
+
+lemma fls_left_inverse':
+  fixes   f :: "'a::ring_1 fls"
+  assumes "x * f $$ fls_subdegree f = 1" "f $$ fls_subdegree f * y = 1"
+  \<comment> \<open>These assumptions imply x equals y, but no need to assume that.\<close>
+  shows   "fls_right_inverse f y * f = 1"
+  using   assms fls_left_inverse_eq_fls_right_inverse[of x f y] fls_left_inverse[of x f]
+  by      simp
+
+lemma fls_right_inverse':
+  fixes   f :: "'a::ring_1 fls"
+  assumes "x * f $$ fls_subdegree f = 1" "f $$ fls_subdegree f * y = 1"
+  \<comment> \<open>These assumptions imply x equals y, but no need to assume that.\<close>
+  shows   "f * fls_left_inverse f x = 1"
+  using   assms fls_left_inverse_eq_fls_right_inverse[of x f y] fls_right_inverse[of f y]
+  by      simp
+
+lemma fls_mult_left_inverse_base_factor:
+  fixes   f :: "'a::ring_1 fls"
+  assumes "x * (f $$ fls_subdegree f) = 1"
+  shows   "fls_left_inverse (fls_base_factor f) x * f = fls_X_intpow (fls_subdegree f)"
+  using   assms fls_base_factor_to_fps_base_factor[of f] fls_base_factor_subdegree[of f]
+          fls_shifted_times_simps(2)[of "-fls_subdegree f" "fls_left_inverse f x" f]
+          fls_left_inverse[of x f]
+  by      simp
+
+lemma fls_mult_right_inverse_base_factor:
+  fixes   f :: "'a::ring_1 fls"
+  assumes "(f $$ fls_subdegree f) * y = 1"
+  shows   "f * fls_right_inverse (fls_base_factor f) y = fls_X_intpow (fls_subdegree f)"
+  using   assms fls_base_factor_to_fps_base_factor[of f] fls_base_factor_subdegree[of f]
+          fls_shifted_times_simps(1)[of f "-fls_subdegree f" "fls_right_inverse f y"]
+          fls_right_inverse[of f y]
+  by      simp
+
+lemma fls_mult_inverse_base_factor:
+  fixes   f :: "'a::division_ring fls"
+  assumes "f \<noteq> 0"
+  shows   "f * inverse (fls_base_factor f) = fls_X_intpow (fls_subdegree f)"
+  using   fls_mult_right_inverse_base_factor[of f "inverse (f $$ fls_subdegree f)"]
+          fls_base_factor_base[of f]
+  by      (simp add: assms fls_right_inverse_eq_inverse[symmetric])
+
+lemma fls_left_inverse_idempotent_ring1:
+  fixes   f :: "'a::ring_1 fls"
+  assumes "x * f $$ fls_subdegree f = 1" "y * x = 1"
+  \<comment> \<open>These assumptions imply y equals f $$ fls_subdegree f, but no need to assume that.\<close>
+  shows   "fls_left_inverse (fls_left_inverse f x) y = f"
+proof-
+  from assms(1) have
+    "fls_left_inverse (fls_left_inverse f x) y * fls_left_inverse f x * f =
+      fls_left_inverse (fls_left_inverse f x) y"
+    using fls_left_inverse[of x f]
+    by    (simp add: mult.assoc)
+  moreover have
+    "fls_left_inverse (fls_left_inverse f x) y * fls_left_inverse f x = 1"
+    using assms fls_lr_inverse_subdegree(1)[of x f] fls_lr_inverse_base(1)[of f x]
+    by    (fastforce intro: fls_left_inverse)
+  ultimately show ?thesis by simp
+qed
+
+lemma fls_left_inverse_idempotent_comm_ring1:
+  fixes   f :: "'a::comm_ring_1 fls"
+  assumes "x * f $$ fls_subdegree f = 1"
+  shows   "fls_left_inverse (fls_left_inverse f x) (f $$ fls_subdegree f) = f"
+  using   assms fls_left_inverse_idempotent_ring1[of x f "f $$ fls_subdegree f"]
+  by      (simp add: mult.commute)
+
+lemma fls_right_inverse_idempotent_ring1:
+  fixes   f :: "'a::ring_1 fls"
+  assumes "f $$ fls_subdegree f * x = 1" "x * y = 1"
+  \<comment> \<open>These assumptions imply y equals f $$ fls_subdegree f, but no need to assume that.\<close>
+  shows   "fls_right_inverse (fls_right_inverse f x) y = f"
+proof-
+  from assms(1) have
+    "f * (fls_right_inverse f x * fls_right_inverse (fls_right_inverse f x) y) =
+      fls_right_inverse (fls_right_inverse f x) y"
+    using fls_right_inverse [of f] 
+    by (simp add: mult.assoc[symmetric])
+  moreover have
+    "fls_right_inverse f x * fls_right_inverse (fls_right_inverse f x) y = 1"
+    using assms fls_lr_inverse_subdegree(2)[of x f] fls_lr_inverse_base(2)[of f x]
+    by    (fastforce intro: fls_right_inverse)
+  ultimately show ?thesis by simp
+qed
+
+lemma fls_right_inverse_idempotent_comm_ring1:
+  fixes   f :: "'a::comm_ring_1 fls"
+  assumes "f $$ fls_subdegree f * x = 1"
+  shows   "fls_right_inverse (fls_right_inverse f x) (f $$ fls_subdegree f) = f"
+  using   assms fls_right_inverse_idempotent_ring1[of f x "f $$ fls_subdegree f"]
+  by      (simp add: mult.commute)
+
+lemma fls_lr_inverse_unique_ring1:
+  fixes   f g :: "'a :: ring_1 fls"
+  assumes fg: "f * g = 1" "g $$ fls_subdegree g * f $$ fls_subdegree f = 1"
+  shows   "fls_left_inverse g (f $$ fls_subdegree f) = f"
+  and     "fls_right_inverse f (g $$ fls_subdegree g) = g"
+proof-
+
+  have "f $$ fls_subdegree f * g $$ fls_subdegree g \<noteq> 0"
+  proof
+    assume "f $$ fls_subdegree f * g $$ fls_subdegree g = 0"
+    hence "f $$ fls_subdegree f * (g $$ fls_subdegree g * f $$ fls_subdegree f) = 0"
+      by (simp add: mult.assoc[symmetric])
+    with fg(2) show False by simp
+  qed
+  with fg(1) have subdeg_sum: "fls_subdegree f + fls_subdegree g = 0"
+    using fls_mult_nonzero_base_subdegree_eq[of f g] by simp
+  hence subdeg_sum':
+    "fls_subdegree f = -fls_subdegree g" "fls_subdegree g = -fls_subdegree f"
+    by auto
+
+  from fg(1) have f_ne_0: "f\<noteq>0" by auto
+  moreover have
+    "fps_left_inverse (fls_base_factor_to_fps g) (fls_regpart (fls_shift (-fls_subdegree g) f)$0)
+      = fls_regpart (fls_shift (-fls_subdegree g) f)"
+  proof (intro fps_lr_inverse_unique_ring1(1))
+    from fg(1) show
+      "fls_regpart (fls_shift (-fls_subdegree g) f) * fls_base_factor_to_fps g = 1"
+      using f_ne_0 fls_times_conv_regpart[of "fls_shift (-fls_subdegree g) f" "fls_base_factor g"]
+            fls_base_factor_subdegree[of g]
+      by    (simp add: fls_times_both_shifted_simp subdeg_sum)
+    from fg(2) show
+      "fls_base_factor_to_fps g $ 0 * fls_regpart (fls_shift (-fls_subdegree g) f) $ 0 = 1"
+      by (simp add: subdeg_sum'(2))
+  qed
+  ultimately show "fls_left_inverse g (f $$ fls_subdegree f) = f"
+    by (simp add: subdeg_sum'(2))
+
+  from fg(1) have g_ne_0: "g\<noteq>0" by auto
+  moreover have
+    "fps_right_inverse (fls_base_factor_to_fps f) (fls_regpart (fls_shift (-fls_subdegree f) g)$0)
+      = fls_regpart (fls_shift (-fls_subdegree f) g)"
+  proof (intro fps_lr_inverse_unique_ring1(2))
+    from fg(1) show
+      "fls_base_factor_to_fps f * fls_regpart (fls_shift (-fls_subdegree f) g) = 1"
+      using g_ne_0 fls_times_conv_regpart[of "fls_base_factor f" "fls_shift (-fls_subdegree f) g"]
+            fls_base_factor_subdegree[of f]
+      by    (simp add: fls_times_both_shifted_simp subdeg_sum add.commute)
+    from fg(2) show
+      "fls_regpart (fls_shift (-fls_subdegree f) g) $ 0 * fls_base_factor_to_fps f $ 0 = 1"
+      by (simp add: subdeg_sum'(1))
+  qed
+  ultimately show "fls_right_inverse f (g $$ fls_subdegree g) = g"
+    by (simp add: subdeg_sum'(2))
+
+qed
+
+lemma fls_lr_inverse_unique_divring:
+  fixes   f g :: "'a ::division_ring fls"
+  assumes fg: "f * g = 1"
+  shows   "fls_left_inverse g (f $$ fls_subdegree f) = f"
+  and     "fls_right_inverse f (g $$ fls_subdegree g) = g"
+proof-
+  from fg have "f \<noteq>0" "g \<noteq> 0" by auto
+  with fg have "fls_subdegree f + fls_subdegree g = 0" using fls_subdegree_mult by force
+  with fg have "f $$ fls_subdegree f * g $$ fls_subdegree g = 1"
+    using fls_times_base[of f g] by simp
+  hence "g $$ fls_subdegree g * f $$ fls_subdegree f = 1"
+    using inverse_unique[of "f $$ fls_subdegree f"] left_inverse[of "f $$ fls_subdegree f"]
+    by    force
+  thus
+    "fls_left_inverse g (f $$ fls_subdegree f) = f"
+    "fls_right_inverse f (g $$ fls_subdegree g) = g"
+    using fg fls_lr_inverse_unique_ring1
+    by    auto
+qed
+
+lemma fls_lr_inverse_minus:
+  fixes f :: "'a::ring_1 fls"
+  shows "fls_left_inverse (-f) (-x) = - fls_left_inverse f x"
+  and   "fls_right_inverse (-f) (-x) = - fls_right_inverse f x"
+  by (simp_all add: fps_lr_inverse_minus)
+
+lemma fls_inverse_minus [simp]: "inverse (-f) = -inverse (f :: 'a :: division_ring fls)"
+  using fls_lr_inverse_minus(2)[of f] by (simp add: fls_inverse_def')
+
+lemma fls_lr_inverse_mult_ring1:
+  fixes   f g :: "'a::ring_1 fls"
+  assumes x: "x * f $$ fls_subdegree f = 1" "f $$ fls_subdegree f * x = 1"
+  and     y: "y * g $$ fls_subdegree g = 1" "g $$ fls_subdegree g * y = 1"
+  shows   "fls_left_inverse (f * g) (y*x) = fls_left_inverse g y * fls_left_inverse f x"
+  and     "fls_right_inverse (f * g) (y*x) = fls_right_inverse g y * fls_right_inverse f x"
+proof-
+  from x(1) y(2) have "x * (f $$ fls_subdegree f * g $$ fls_subdegree g) * y = 1"
+    by (simp add: mult.assoc)
+  hence base_prod: "f $$ fls_subdegree f * g $$ fls_subdegree g \<noteq> 0" by auto
+  hence subdegrees: "fls_subdegree (f*g) = fls_subdegree f + fls_subdegree g"
+    using fls_mult_nonzero_base_subdegree_eq[of f g] by simp
+
+  have norm:
+    "fls_base_factor_to_fps (f * g) = fls_base_factor_to_fps f * fls_base_factor_to_fps g"
+    using base_prod fls_base_factor_to_fps_mult'[of f g] by simp
+
+  have
+    "fls_left_inverse (f * g) (y*x) =
+      fls_shift (fls_subdegree (f * g)) (
+        fps_to_fls (
+          fps_left_inverse (fls_base_factor_to_fps f * fls_base_factor_to_fps g) (y*x)
+        )
+      )
+    "
+    using norm
+    by    simp
+  thus "fls_left_inverse (f * g) (y*x) = fls_left_inverse g y * fls_left_inverse f x"
+    using x y
+          fps_lr_inverse_mult_ring1(1)[of
+            x "fls_base_factor_to_fps f" y "fls_base_factor_to_fps g"
+          ]
+    by    (simp add:
+            fls_times_both_shifted_simp fls_times_fps_to_fls subdegrees algebra_simps
+          )
+
+  have
+    "fls_right_inverse (f * g) (y*x) =
+      fls_shift (fls_subdegree (f * g)) (
+        fps_to_fls (
+          fps_right_inverse (fls_base_factor_to_fps f * fls_base_factor_to_fps g) (y*x)
+        )
+      )
+    "
+    using norm
+    by    simp
+  thus "fls_right_inverse (f * g) (y*x) = fls_right_inverse g y * fls_right_inverse f x"
+    using x y
+          fps_lr_inverse_mult_ring1(2)[of
+            x "fls_base_factor_to_fps f" y "fls_base_factor_to_fps g"
+          ]
+    by    (simp add:
+            fls_times_both_shifted_simp fls_times_fps_to_fls subdegrees algebra_simps
+          )
+
+qed
+
+lemma fls_lr_inverse_power_ring1:
+  fixes   f :: "'a::ring_1 fls"
+  assumes x: "x * f $$ fls_subdegree f = 1" "f $$ fls_subdegree f * x = 1"
+  shows   "fls_left_inverse (f ^ n) (x ^ n) = (fls_left_inverse f x) ^ n"
+          "fls_right_inverse (f ^ n) (x ^ n) = (fls_right_inverse f x) ^ n"
+proof-
+
+  show "fls_left_inverse (f ^ n) (x ^ n) = (fls_left_inverse f x) ^ n"
+  proof (induct n)
+    case 0 show ?case using fls_lr_inverse_one(1)[of 1] by simp
+  next
+    case (Suc n) with assms show ?case
+      using fls_lr_inverse_mult_ring1(1)[of x f "x^n" "f^n"]
+      by    (simp add:
+              power_Suc2[symmetric] fls_unit_base_subdegree_power(1) left_right_inverse_power
+            )
+  qed
+
+  show "fls_right_inverse (f ^ n) (x ^ n) = (fls_right_inverse f x) ^ n"
+  proof (induct n)
+    case 0 show ?case using fls_lr_inverse_one(2)[of 1] by simp
+  next
+    case (Suc n) with assms show ?case
+      using fls_lr_inverse_mult_ring1(2)[of x f "x^n" "f^n"]
+      by    (simp add:
+              power_Suc2[symmetric] fls_unit_base_subdegree_power(1) left_right_inverse_power
+            )
+  qed
+
+qed
+
+lemma fls_divide_convert_times_inverse:
+  fixes   f g :: "'a::{comm_monoid_add,inverse,mult_zero,uminus} fls"
+  shows   "f / g = f * inverse g"
+  using fls_base_factor_to_fps_subdegree[of g] fps_to_fls_base_factor_to_fps[of f]
+        fls_times_both_shifted_simp[of "-fls_subdegree f" "fls_base_factor f"]
+  by    (simp add:
+          fls_divide_def fps_divide_unit' fls_times_fps_to_fls
+          fls_conv_base_factor_shift_subdegree fls_inverse_def
+        )
+
+instance fls :: (division_ring) division_ring
+proof
+  fix a b :: "'a fls"
+  show "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
+    using fls_left_inverse'[of "inverse (a $$ fls_subdegree a)" a]
+    by    (simp add: fls_inverse_def')
+  show "a \<noteq> 0 \<Longrightarrow> a * inverse a = 1"
+    using fls_right_inverse[of a]
+    by    (simp add: fls_inverse_def')
+  show "a / b = a * inverse b" using fls_divide_convert_times_inverse by fast
+  show "inverse (0::'a fls) = 0" by simp
+qed
+
+lemma fls_lr_inverse_mult_divring:
+  fixes   f g   :: "'a::division_ring fls"
+  and     df dg :: int
+  defines "df \<equiv> fls_subdegree f"
+  and     "dg \<equiv> fls_subdegree g"
+  shows   "fls_left_inverse (f*g) (inverse ((f*g)$$(df+dg))) =
+            fls_left_inverse g (inverse (g$$dg)) * fls_left_inverse f (inverse (f$$df))"
+  and     "fls_right_inverse (f*g) (inverse ((f*g)$$(df+dg))) =
+            fls_right_inverse g (inverse (g$$dg)) * fls_right_inverse f (inverse (f$$df))"
+proof -
+  show
+    "fls_left_inverse (f*g) (inverse ((f*g)$$(df+dg))) =
+      fls_left_inverse g (inverse (g$$dg)) * fls_left_inverse f (inverse (f$$df))"
+  proof (cases "f=0 \<or> g=0")
+    case True thus ?thesis
+      using fls_lr_inverse_zero(1)[of "inverse (0::'a)"] by (auto simp add: assms)
+  next
+    case False thus ?thesis
+      using fls_left_inverse_eq_inverse[of "f*g"] nonzero_inverse_mult_distrib[of f g]
+            fls_left_inverse_eq_inverse[of g] fls_left_inverse_eq_inverse[of f]
+      by    (simp add: assms)
+  qed
+  show
+    "fls_right_inverse (f*g) (inverse ((f*g)$$(df+dg))) =
+      fls_right_inverse g (inverse (g$$dg)) * fls_right_inverse f (inverse (f$$df))"
+  proof (cases "f=0 \<or> g=0")
+    case True thus ?thesis
+      using fls_lr_inverse_zero(2)[of "inverse (0::'a)"] by (auto simp add: assms)
+  next
+    case False thus ?thesis
+      using fls_inverse_def'[of "f*g"] nonzero_inverse_mult_distrib[of f g]
+            fls_inverse_def'[of g] fls_inverse_def'[of f]
+      by    (simp add: assms)
+  qed
+qed
+
+lemma fls_lr_inverse_power_divring:
+  fixes f :: "'a::division_ring fls"
+  shows "fls_left_inverse (f ^ n) ((inverse (f $$ fls_subdegree f)) ^ n) =
+          (fls_left_inverse f (inverse (f $$ fls_subdegree f))) ^ n"
+        "fls_right_inverse (f ^ n) ((inverse (f $$ fls_subdegree f)) ^ n) =
+          (fls_right_inverse f (inverse (f $$ fls_subdegree f))) ^ n"
+proof -
+  have
+    "fls_right_inverse (f ^ n) ((inverse (f $$ fls_subdegree f)) ^ n) =
+      inverse f ^ n"
+    "fls_left_inverse (f ^ n) ((inverse (f $$ fls_subdegree f)) ^ n) =
+      inverse f ^ n"
+    using fls_left_inverse_eq_inverse[of "f^n"] fls_right_inverse_eq_inverse[of "f^n"]
+    by    (auto simp add: divide_simps fls_subdegree_pow)
+  thus
+    "fls_left_inverse (f ^ n) ((inverse (f $$ fls_subdegree f)) ^ n) =
+      (fls_left_inverse f (inverse (f $$ fls_subdegree f))) ^ n"
+    "fls_right_inverse (f ^ n) ((inverse (f $$ fls_subdegree f)) ^ n) =
+      (fls_right_inverse f (inverse (f $$ fls_subdegree f))) ^ n"
+    using fls_left_inverse_eq_inverse[of f] fls_right_inverse_eq_inverse[of f]
+    by    auto
+qed
+
+instance fls :: (field) field
+  by (standard, simp_all add: field_simps)
+
+
+subsubsection \<open>Division\<close>
+
+lemma fls_divide_nth_below:
+  fixes f g :: "'a::{comm_monoid_add,uminus,times,inverse} fls"
+  shows "n < fls_subdegree f - fls_subdegree g \<Longrightarrow> (f div g) $$ n = 0"
+  by    (simp add: fls_divide_def)
+
+lemma fls_divide_nth_base:
+  fixes f g :: "'a::division_ring fls"
+  shows
+    "(f div g) $$ (fls_subdegree f - fls_subdegree g) =
+      f $$ fls_subdegree f / g $$ fls_subdegree g"
+  using fps_divide_nth_0'[of "fls_base_factor_to_fps g" "fls_base_factor_to_fps f"]
+        fls_base_factor_to_fps_subdegree[of g]
+  by    (simp add: fls_divide_def)
+
+lemma fls_div_zero [simp]:
+  "0 div (g :: 'a :: {comm_monoid_add,inverse,mult_zero,uminus} fls) = 0"
+  by (simp add: fls_divide_def)
+
+lemma fls_div_by_zero:
+  fixes   g :: "'a::{comm_monoid_add,inverse,mult_zero,uminus} fls"
+  assumes "inverse (0::'a) = 0"
+  shows   "g div 0 = 0"
+  by      (simp add: fls_divide_def assms fps_div_by_zero')
+
+lemma fls_divide_times:
+  fixes f g :: "'a::{semiring_0,inverse,uminus} fls"
+  shows "(f * g) / h = f * (g / h)"
+  by    (simp add: fls_divide_convert_times_inverse mult.assoc)
+
+lemma fls_divide_times2:
+  fixes f g :: "'a::{comm_semiring_0,inverse,uminus} fls"
+  shows "(f * g) / h = (f / h) * g"
+  using fls_divide_times[of g f h]
+  by    (simp add: mult.commute)
+
+lemma fls_divide_subdegree_ge:
+  fixes   f g :: "'a::{comm_monoid_add,uminus,times,inverse} fls"
+  assumes "f / g \<noteq> 0"
+  shows   "fls_subdegree (f / g) \<ge> fls_subdegree f - fls_subdegree g"
+  using   assms fls_divide_nth_below
+  by      (intro fls_subdegree_geI) simp
+
+lemma fls_divide_subdegree:
+  fixes   f g :: "'a::division_ring fls"
+  assumes "f \<noteq> 0" "g \<noteq> 0"
+  shows   "fls_subdegree (f / g) = fls_subdegree f - fls_subdegree g"
+proof (intro antisym)
+  from assms have "f $$ fls_subdegree f / g $$ fls_subdegree g \<noteq> 0" by (simp add: field_simps)
+  thus "fls_subdegree (f/g) \<le> fls_subdegree f - fls_subdegree g"
+    using fls_divide_nth_base[of f g] by (intro fls_subdegree_leI) simp
+  from assms have "f / g \<noteq> 0" by (simp add: field_simps)
+  thus "fls_subdegree (f/g) \<ge> fls_subdegree f - fls_subdegree g"
+    using fls_divide_subdegree_ge by fast
+qed
+
+lemma fls_divide_shift_numer_nonzero:
+  fixes   f g :: "'a :: {comm_monoid_add,inverse,times,uminus} fls"
+  assumes "f \<noteq> 0"
+  shows   "fls_shift m f / g = fls_shift m (f/g)"
+  using   assms fls_base_factor_to_fps_shift[of m f]
+  by      (simp add: fls_divide_def algebra_simps)
+
+lemma fls_divide_shift_numer:
+  fixes f g :: "'a :: {comm_monoid_add,inverse,mult_zero,uminus} fls"
+  shows "fls_shift m f / g = fls_shift m (f/g)"
+  using fls_divide_shift_numer_nonzero
+  by    (cases "f=0") auto
+
+lemma fls_divide_shift_denom_nonzero:
+  fixes   f g :: "'a :: {comm_monoid_add,inverse,times,uminus} fls"
+  assumes "g \<noteq> 0"
+  shows   "f / fls_shift m g = fls_shift (-m) (f/g)"
+  using   assms fls_base_factor_to_fps_shift[of m g]
+  by      (simp add: fls_divide_def algebra_simps)
+
+lemma fls_divide_shift_denom:
+  fixes   f g :: "'a :: division_ring fls"
+  shows   "f / fls_shift m g = fls_shift (-m) (f/g)"
+  using   fls_divide_shift_denom_nonzero
+  by      (cases "g=0") auto
+
+lemma fls_divide_shift_both_nonzero:
+  fixes   f g :: "'a :: {comm_monoid_add,inverse,times,uminus} fls"
+  assumes "f \<noteq> 0" "g \<noteq> 0"
+  shows   "fls_shift n f / fls_shift m g = fls_shift (n-m) (f/g)"
+  by      (simp add: assms fls_divide_shift_numer_nonzero fls_divide_shift_denom_nonzero)
+
+lemma fls_divide_shift_both [simp]:
+  fixes   f g :: "'a :: division_ring fls"
+  shows   "fls_shift n f / fls_shift m g = fls_shift (n-m) (f/g)"
+  using   fls_divide_shift_both_nonzero
+  by      (cases "f=0 \<or> g=0") auto
+
+lemma fls_divide_base_factor_numer:
+  "fls_base_factor f / g = fls_shift (fls_subdegree f) (f/g)"
+  using fls_base_factor_to_fps_base_factor[of f]
+        fls_base_factor_subdegree[of f]
+  by    (simp add: fls_divide_def algebra_simps)
+
+lemma fls_divide_base_factor_denom:
+  "f / fls_base_factor g = fls_shift (-fls_subdegree g) (f/g)"
+  using fls_base_factor_to_fps_base_factor[of g]
+        fls_base_factor_subdegree[of g]
+  by    (simp add: fls_divide_def)
+
+lemma fls_divide_base_factor':
+  "fls_base_factor f / fls_base_factor g = fls_shift (fls_subdegree f - fls_subdegree g) (f/g)"
+  using fls_divide_base_factor_numer[of f "fls_base_factor g"]
+        fls_divide_base_factor_denom[of f g]
+  by    simp
+
+lemma fls_divide_base_factor:
+  fixes f g :: "'a :: division_ring fls"
+  shows "fls_base_factor f / fls_base_factor g = fls_base_factor (f/g)"
+  using fls_divide_subdegree[of f g] fls_divide_base_factor'
+  by    fastforce
+
+lemma fls_divide_regpart:
+  fixes   f g :: "'a::{inverse,comm_monoid_add,uminus,mult_zero} fls"
+  assumes "fls_subdegree f \<ge> 0" "fls_subdegree g \<ge> 0"
+  shows   "fls_regpart (f / g) = fls_regpart f / fls_regpart g"
+proof -
+  have deg0:
+    "\<And>g. fls_subdegree g = 0 \<Longrightarrow>
+      fls_regpart (f / g) = fls_regpart f / fls_regpart g"
+    by  (simp add:
+          assms(1) fls_divide_convert_times_inverse fls_inverse_subdegree_0
+          fls_times_conv_regpart fls_inverse_regpart fls_regpart_subdegree_conv fps_divide_unit'
+        )
+  show ?thesis
+  proof (cases "fls_subdegree g = 0")
+    case False
+    hence "fls_base_factor g \<noteq> 0" using fls_base_factor_nonzero[of g] by force
+    with assms(2) show ?thesis
+      using fls_divide_shift_denom_nonzero[of "fls_base_factor g" f "-fls_subdegree g"]
+            fps_shift_fls_regpart_conv_fls_shift[of
+              "nat (fls_subdegree g)" "f / fls_base_factor g"
+            ]
+            fls_base_factor_subdegree[of g] deg0
+            fls_regpart_subdegree_conv[of g] fps_unit_factor_fls_regpart[of g]
+      by    (simp add:
+              fls_conv_base_factor_shift_subdegree fls_regpart_subdegree_conv fps_divide_def
+            )
+  qed (rule deg0)
+qed
+
+lemma fls_divide_fls_base_factor_to_fps':
+  fixes f g :: "'a::{comm_monoid_add,uminus,inverse,mult_zero} fls"
+  shows
+    "fls_base_factor_to_fps f / fls_base_factor_to_fps g =
+      fls_regpart (fls_shift (fls_subdegree f - fls_subdegree g) (f / g))"
+  using fls_base_factor_subdegree[of f] fls_base_factor_subdegree[of g]
+        fls_divide_regpart[of "fls_base_factor f" "fls_base_factor g"]
+        fls_divide_base_factor'[of f g]
+    by  simp
+
+lemma fls_divide_fls_base_factor_to_fps:
+  fixes f g :: "'a::division_ring fls"
+  shows "fls_base_factor_to_fps f / fls_base_factor_to_fps g = fls_base_factor_to_fps (f / g)"
+  using fls_divide_fls_base_factor_to_fps' fls_divide_subdegree[of f g]
+  by    fastforce
+
+lemma fls_divide_fps_to_fls:
+  fixes f g :: "'a::{inverse,ab_group_add,mult_zero} fps"
+  assumes "subdegree f \<ge> subdegree g"
+  shows   "fps_to_fls f / fps_to_fls g = fps_to_fls (f/g)"
+proof-
+  have 1:
+    "fps_to_fls f / fps_to_fls g =
+      fls_shift (int (subdegree g)) (fps_to_fls (f * inverse (unit_factor g)))"
+    using fls_base_factor_to_fps_to_fls[of f] fls_base_factor_to_fps_to_fls[of g]
+          fls_subdegree_fls_to_fps[of f] fls_subdegree_fls_to_fps[of g]
+          fps_divide_def[of "unit_factor f" "unit_factor g"]
+          fls_times_fps_to_fls[of "unit_factor f" "inverse (unit_factor g)"]
+          fls_shifted_times_simps(2)[of "-int (subdegree f)" "fps_to_fls (unit_factor f)"]
+          fls_times_fps_to_fls[of f "inverse (unit_factor g)"]
+    by    (simp add: fls_divide_def)
+  with assms show ?thesis
+    using fps_mult_subdegree_ge[of f "inverse (unit_factor g)"]
+          fps_shift_to_fls[of "subdegree g" "f * inverse (unit_factor g)"]
+    by    (cases "f * inverse (unit_factor g) = 0") (simp_all add: fps_divide_def)
+qed
+
+lemma fls_divide_1':
+  fixes   f :: "'a::{comm_monoid_add,inverse,mult_zero,uminus,zero_neq_one,monoid_mult} fls"
+  assumes "inverse (1::'a) = 1"
+  shows   "f / 1 = f"
+  using   assms fls_conv_base_factor_to_fps_shift_subdegree[of f]
+  by      (simp add: fls_divide_def fps_divide_1')
+
+lemma fls_divide_1 [simp]: "a / 1 = (a::'a::division_ring fls)"
+  by (rule fls_divide_1'[OF inverse_1])
+
+lemma fls_const_divide_const:
+  fixes x y :: "'a::division_ring"
+  shows "fls_const x / fls_const y = fls_const (x/y)"
+  by    (simp add: fls_divide_def fls_base_factor_to_fps_const fps_const_divide)
+
+lemma fls_divide_X':
+  fixes   f :: "'a::{comm_monoid_add,inverse,mult_zero,uminus,zero_neq_one,monoid_mult} fls"
+  assumes "inverse (1::'a) = 1"
+  shows   "f / fls_X = fls_shift 1 f"
+proof-
+  from assms have
+    "f / fls_X =
+      fls_shift 1 (fls_shift (-fls_subdegree f) (fps_to_fls (fls_base_factor_to_fps f)))"
+    by (simp add: fls_divide_def fps_divide_1')
+  also have "\<dots> = fls_shift 1 f"
+    using fls_conv_base_factor_to_fps_shift_subdegree[of f]
+    by simp
+  finally show ?thesis by simp
+qed
+
+lemma fls_divide_X [simp]:
+  fixes f :: "'a::division_ring fls"
+  shows "f / fls_X = fls_shift 1 f"
+  by    (rule fls_divide_X'[OF inverse_1])
+
+lemma fls_divide_X_power':
+  fixes   f :: "'a::{semiring_1,inverse,uminus} fls"
+  assumes "inverse (1::'a) = 1"
+  shows   "f / (fls_X ^ n) = fls_shift n f"
+proof-
+  have "fls_base_factor_to_fps ((fls_X::'a fls) ^ n) = 1" by (rule fls_X_power_base_factor_to_fps)
+  with assms have
+    "f / (fls_X ^ n) =
+      fls_shift n (fls_shift (-fls_subdegree f) (fps_to_fls (fls_base_factor_to_fps f)))"
+    by (simp add: fls_divide_def fps_divide_1')
+  also have "\<dots> = fls_shift n f"
+    using fls_conv_base_factor_to_fps_shift_subdegree[of f] by simp
+  finally show ?thesis by simp
+qed
+
+lemma fls_divide_X_power [simp]:
+  fixes f :: "'a::division_ring fls"
+  shows "f / (fls_X ^ n) = fls_shift n f"
+  by    (rule fls_divide_X_power'[OF inverse_1])
+
+lemma fls_divide_X_inv':
+  fixes   f :: "'a::{comm_monoid_add,inverse,mult_zero,uminus,zero_neq_one,monoid_mult} fls"
+  assumes "inverse (1::'a) = 1"
+  shows   "f / fls_X_inv = fls_shift (-1) f"
+proof-
+  from assms have
+    "f / fls_X_inv =
+      fls_shift (-1) (fls_shift (-fls_subdegree f) (fps_to_fls (fls_base_factor_to_fps f)))"
+    by (simp add: fls_divide_def fps_divide_1' algebra_simps)
+  also have "\<dots> = fls_shift (-1) f"
+    using fls_conv_base_factor_to_fps_shift_subdegree[of f]
+    by simp
+  finally show ?thesis by simp
+qed
+
+lemma fls_divide_X_inv [simp]:
+  fixes f :: "'a::division_ring fls"
+  shows "f / fls_X_inv = fls_shift (-1) f"
+  by    (rule fls_divide_X_inv'[OF inverse_1])
+
+lemma fls_divide_X_inv_power':
+  fixes   f :: "'a::{semiring_1,inverse,uminus} fls"
+  assumes "inverse (1::'a) = 1"
+  shows   "f / (fls_X_inv ^ n) = fls_shift (-int n) f"
+proof-
+  have "fls_base_factor_to_fps ((fls_X_inv::'a fls) ^ n) = 1"
+    by (rule fls_X_inv_power_base_factor_to_fps)
+  with assms have
+    "f / (fls_X_inv ^ n) =
+      fls_shift (-int n + -fls_subdegree f) (fps_to_fls (fls_base_factor_to_fps f))"
+    by (simp add: fls_divide_def fps_divide_1')
+  also have
+    "\<dots> = fls_shift (-int n) (fls_shift (-fls_subdegree f) (fps_to_fls (fls_base_factor_to_fps f)))"
+    by (simp add: add.commute)
+  also have "\<dots> = fls_shift (-int n) f"
+    using fls_conv_base_factor_to_fps_shift_subdegree[of f] by simp
+  finally show ?thesis by simp
+qed
+
+lemma fls_divide_X_inv_power [simp]:
+  fixes f :: "'a::division_ring fls"
+  shows "f / (fls_X_inv ^ n) = fls_shift (-int n) f"
+  by    (rule fls_divide_X_inv_power'[OF inverse_1])
+
+lemma fls_divide_X_intpow':
+  fixes   f :: "'a::{semiring_1,inverse,uminus} fls"
+  assumes "inverse (1::'a) = 1"
+  shows   "f / (fls_X_intpow i) = fls_shift i f"
+  using   assms
+  by      (simp add: fls_divide_shift_denom_nonzero fls_divide_1')
+
+lemma fls_divide_X_intpow_conv_times':
+  fixes   f :: "'a::{semiring_1,inverse,uminus} fls"
+  assumes "inverse (1::'a) = 1"
+  shows   "f / (fls_X_intpow i) = f * fls_X_intpow (-i)"
+  using   assms fls_X_intpow_times_conv_shift(2)[of f "-i"]
+  by      (simp add: fls_divide_X_intpow')
+
+lemma fls_divide_X_intpow:
+  fixes f :: "'a::division_ring fls"
+  shows "f / (fls_X_intpow i) = fls_shift i f"
+  by    (rule fls_divide_X_intpow'[OF inverse_1])
+
+lemma fls_divide_X_intpow_conv_times:
+  fixes f :: "'a::division_ring fls"
+  shows "f / (fls_X_intpow i) = f * fls_X_intpow (-i)"
+  by    (rule fls_divide_X_intpow_conv_times'[OF inverse_1])
+
+lemma fls_X_intpow_div_fls_X_intpow_semiring1:
+  assumes "inverse (1::'a::{semiring_1,inverse,uminus}) = 1"
+  shows   "(fls_X_intpow i :: 'a fls) / fls_X_intpow j = fls_X_intpow (i-j)"
+  by      (simp add: assms fls_divide_shift_both_nonzero fls_divide_1')
+
+lemma fls_X_intpow_div_fls_X_intpow:
+  "(fls_X_intpow i :: 'a::division_ring fls) / fls_X_intpow j = fls_X_intpow (i-j)"
+  by (rule fls_X_intpow_div_fls_X_intpow_semiring1[OF inverse_1])
+
+lemma fls_divide_add:
+  fixes   f g h :: "'a::{semiring_0,inverse,uminus} fls"
+  shows   "(f + g) / h = f / h + g / h"
+  by      (simp add: fls_divide_convert_times_inverse algebra_simps)
+
+lemma fls_divide_diff:
+  fixes f g h :: "'a::{ring,inverse} fls"
+  shows "(f - g) / h = f / h - g / h"
+  by    (simp add: fls_divide_convert_times_inverse algebra_simps)
+
+lemma fls_divide_uminus:
+  fixes f g h :: "'a::{ring,inverse} fls"
+  shows "(- f) / g = - (f / g)"
+  by    (simp add: fls_divide_convert_times_inverse)
+
+lemma fls_divide_uminus':
+  fixes f g h :: "'a::division_ring fls"
+  shows "f / (- g) = - (f / g)"
+  by    (simp add: fls_divide_convert_times_inverse)
+
+
+subsubsection \<open>Units\<close>
+
+lemma fls_is_left_unit_iff_base_is_left_unit:
+  fixes f :: "'a :: ring_1_no_zero_divisors fls"
+  shows "(\<exists>g. 1 = f * g) \<longleftrightarrow> (\<exists>k. 1 = f $$ fls_subdegree f * k)"
+proof
+  assume "\<exists>g. 1 = f * g"
+  then obtain g where "1 = f * g" by fast
+  hence "1 = (f $$ fls_subdegree f) * (g $$ fls_subdegree g)"
+    using fls_subdegree_mult[of f g] fls_times_base[of f g] by fastforce
+  thus "\<exists>k. 1 = f $$ fls_subdegree f * k" by fast
+next
+  assume "\<exists>k. 1 = f $$ fls_subdegree f * k"
+  then obtain k where "1 = f $$ fls_subdegree f * k" by fast
+  hence "1 = f * fls_right_inverse f k"
+    using fls_right_inverse by simp
+  thus "\<exists>g. 1 = f * g" by fast
+qed
+
+lemma fls_is_right_unit_iff_base_is_right_unit:
+  fixes f :: "'a :: ring_1_no_zero_divisors fls"
+  shows "(\<exists>g. 1 = g * f) \<longleftrightarrow> (\<exists>k. 1 = k * f $$ fls_subdegree f)"
+proof
+  assume "\<exists>g. 1 = g * f"
+  then obtain g where "1 = g * f" by fast
+  hence "1 = (g $$ fls_subdegree g) * (f $$ fls_subdegree f)"
+    using fls_subdegree_mult[of g f] fls_times_base[of g f] by fastforce
+  thus "\<exists>k. 1 = k * f $$ fls_subdegree f" by fast
+next
+  assume "\<exists>k. 1 = k * f $$ fls_subdegree f"
+  then obtain k where "1 = k * f $$ fls_subdegree f" by fast
+  hence "1 = fls_left_inverse f k * f"
+    using fls_left_inverse by simp
+  thus "\<exists>g. 1 = g * f" by fast
+qed
+
+
+subsection \<open>Formal differentiation and integration\<close>
+
+
+subsubsection \<open>Derivative definition and basic properties\<close>
+
+definition "fls_deriv f = Abs_fls (\<lambda>n. of_int (n+1) * f$$(n+1))"
+
+lemma fls_deriv_nth[simp]: "fls_deriv f $$ n = of_int (n+1) * f$$(n+1)"
+proof-
+  obtain N where "\<forall>n<N. f$$n = 0" by (elim fls_nth_vanishes_belowE)
+  hence "\<forall>n<N-1. of_int (n+1) * f$$(n+1) = 0" by auto
+  thus ?thesis using nth_Abs_fls_lower_bound unfolding fls_deriv_def by simp
+qed
+
+lemma fls_deriv_residue: "fls_deriv f $$ -1 = 0"
+  by simp
+
+lemma fls_deriv_const[simp]: "fls_deriv (fls_const x) = 0"
+proof (intro fls_eqI)
+  fix n show "fls_deriv (fls_const x) $$ n = 0$$n"
+    by (cases "n+1=0") auto
+qed
+
+lemma fls_deriv_of_nat[simp]: "fls_deriv (of_nat n) = 0"
+  by (simp add: fls_of_nat)
+
+lemma fls_deriv_of_int[simp]: "fls_deriv (of_int i) = 0"
+  by (simp add: fls_of_int)
+
+lemma fls_deriv_zero[simp]: "fls_deriv 0 = 0"
+  using fls_deriv_const[of 0] by simp
+
+lemma fls_deriv_one[simp]: "fls_deriv 1 = 0"
+  using fls_deriv_const[of 1] by simp
+
+lemma fls_deriv_subdegree':
+  assumes "of_int (fls_subdegree f) * f $$ fls_subdegree f \<noteq> 0"
+  shows   "fls_subdegree (fls_deriv f) = fls_subdegree f - 1"
+  by      (auto intro: fls_subdegree_eqI simp: assms)
+
+lemma fls_deriv_subdegree0:
+  assumes "fls_subdegree f = 0"
+  shows   "fls_subdegree (fls_deriv f) \<ge> 0"
+proof (cases "fls_deriv f = 0")
+  case False
+  show ?thesis
+  proof (intro fls_subdegree_geI, rule False)
+    fix k :: int assume "k < 0"
+    with assms show "fls_deriv f $$ k = 0" by (cases "k=-1") auto
+  qed
+qed simp
+
+lemma fls_subdegree_deriv':
+  fixes   f :: "'a::ring_1_no_zero_divisors fls"
+  assumes "(of_int (fls_subdegree f) :: 'a) \<noteq> 0"
+  shows   "fls_subdegree (fls_deriv f) = fls_subdegree f - 1"
+  using   assms nth_fls_subdegree_zero_iff[of f]
+  by      (auto intro: fls_deriv_subdegree')
+
+lemma fls_subdegree_deriv:
+  fixes   f :: "'a::{ring_1_no_zero_divisors,ring_char_0} fls"
+  assumes "fls_subdegree f \<noteq> 0"
+  shows   "fls_subdegree (fls_deriv f) = fls_subdegree f - 1"
+  by      (auto intro: fls_subdegree_deriv' simp: assms)
+
+text \<open>
+  Shifting is like multiplying by a power of the implied variable, and so satisfies a product-like
+  rule.
+\<close>
+
+lemma fls_deriv_shift:
+  "fls_deriv (fls_shift n f) = of_int (-n) * fls_shift (n+1) f + fls_shift n (fls_deriv f)"
+  by (intro fls_eqI) (simp flip: fls_shift_fls_shift add: algebra_simps)
+
+lemma fls_deriv_X [simp]: "fls_deriv fls_X = 1"
+  by (intro fls_eqI) simp
+
+lemma fls_deriv_X_inv [simp]: "fls_deriv fls_X_inv = - (fls_X_inv\<^sup>2)"
+proof-
+  have "fls_deriv fls_X_inv = - (fls_shift 2 1)"
+    by (simp add: fls_X_inv_conv_shift_1 fls_deriv_shift)
+  thus ?thesis by (simp add: fls_X_inv_power_conv_shift_1)
+qed
+
+lemma fls_deriv_delta:
+  "fls_deriv (Abs_fls (\<lambda>n. if n=m then c else 0)) =
+    Abs_fls (\<lambda>n. if n=m-1 then of_int m * c else 0)"
+proof-
+  have
+    "fls_deriv (Abs_fls (\<lambda>n. if n=m then c else 0)) = fls_shift (1-m) (fls_const (of_int m * c))"
+    using fls_deriv_shift[of "-m" "fls_const c"]
+    by    (simp
+            add: fls_shift_const fls_of_int fls_shifted_times_simps(1)[symmetric]
+            fls_const_mult_const[symmetric]
+            del: fls_const_mult_const
+          )
+  thus ?thesis by (simp add: fls_shift_const)
+qed
+
+lemma fls_deriv_base_factor:
+  "fls_deriv (fls_base_factor f) =
+    of_int (-fls_subdegree f) * fls_shift (fls_subdegree f + 1) f +
+    fls_shift (fls_subdegree f) (fls_deriv f)"
+  by (simp add: fls_deriv_shift)
+
+lemma fls_regpart_deriv: "fls_regpart (fls_deriv f) = fps_deriv (fls_regpart f)"
+proof (intro fps_ext)
+  fix n
+  have  1: "(of_nat n :: 'a) + 1 = of_nat (n+1)"
+  and   2: "int n + 1 = int (n + 1)"
+    by  auto
+  show "fls_regpart (fls_deriv f) $ n = fps_deriv (fls_regpart f) $ n" by (simp add: 1 2)
+qed
+
+lemma fls_prpart_deriv:
+  fixes f :: "'a :: {comm_ring_1,ring_no_zero_divisors} fls"
+  \<comment> \<open>Commutivity and no zero divisors are required by the definition of @{const pderiv}.\<close>
+  shows "fls_prpart (fls_deriv f) = - pCons 0 (pCons 0 (pderiv (fls_prpart f)))"
+proof (intro poly_eqI)
+  fix n
+  show
+    "coeff (fls_prpart (fls_deriv f)) n =
+      coeff (- pCons 0 (pCons 0 (pderiv (fls_prpart f)))) n"
+  proof (cases n)
+    case (Suc m)
+    hence n: "n = Suc m" by fast
+    show ?thesis
+    proof (cases m)
+      case (Suc k)
+      with n have
+        "coeff (- pCons 0 (pCons 0 (pderiv (fls_prpart f)))) n =
+          - coeff (pderiv (fls_prpart f)) k"
+        by (simp flip: coeff_minus)
+      with Suc n show ?thesis by (simp add: coeff_pderiv algebra_simps)
+    qed (simp add: n)
+  qed simp
+qed
+
+lemma pderiv_fls_prpart:
+  "pderiv (fls_prpart f) = - poly_shift 2 (fls_prpart (fls_deriv f))"
+  by (intro poly_eqI) (simp add: coeff_pderiv coeff_poly_shift algebra_simps)
+
+lemma fls_deriv_fps_to_fls: "fls_deriv (fps_to_fls f) = fps_to_fls (fps_deriv f)"
+proof (intro fls_eqI)
+  fix n
+  show "fls_deriv (fps_to_fls f) $$ n  = fps_to_fls (fps_deriv f) $$ n"
+  proof (cases "n\<ge>0")
+    case True
+    from True have 1: "nat (n + 1) = nat n + 1" by simp
+    from True have 2: "(of_int (n + 1) :: 'a) = of_nat (nat (n+1))" by simp
+    from True show ?thesis using arg_cong[OF 2, of "\<lambda>x. x * f $ (nat n+1)"] by (simp add: 1)
+  next
+    case False thus ?thesis by (cases "n=-1") auto
+  qed
+qed
+
+
+subsubsection \<open>Algebra rules of the derivative\<close>
+
+lemma fls_deriv_add [simp]: "fls_deriv (f+g) = fls_deriv f + fls_deriv g"
+  by (auto intro: fls_eqI simp: algebra_simps)
+
+lemma fls_deriv_sub [simp]: "fls_deriv (f-g) = fls_deriv f - fls_deriv g"
+  by (auto intro: fls_eqI simp: algebra_simps)
+
+lemma fls_deriv_neg [simp]: "fls_deriv (-f) = - fls_deriv f"
+  using fls_deriv_sub[of 0 f] by simp
+
+lemma fls_deriv_mult [simp]:
+  "fls_deriv (f*g) = f * fls_deriv g + fls_deriv f * g"
+proof-
+  define df dg :: int
+    where "df \<equiv> fls_subdegree f"
+    and   "dg \<equiv> fls_subdegree g"
+  define uf ug :: "'a fls"
+    where "uf \<equiv> fls_base_factor f"
+    and   "ug \<equiv> fls_base_factor g"
+  have
+    "f * fls_deriv g =
+      of_int dg * fls_shift (1 - dg) (f * ug) + fls_shift (-dg) (f * fls_deriv ug)"
+    "fls_deriv f * g =
+      of_int df * fls_shift (1 - df) (uf * g) + fls_shift (-df) (fls_deriv uf * g)"
+    using fls_deriv_shift[of "-df" uf] fls_deriv_shift[of "-dg" ug]
+          mult_of_int_commute[of dg f]
+          mult.assoc[of "of_int dg" f]
+          fls_shifted_times_simps(1)[of f "1 - dg" ug]
+          fls_shifted_times_simps(1)[of f "-dg" "fls_deriv ug"]
+          fls_shifted_times_simps(2)[of "1 - df" uf g]
+          fls_shifted_times_simps(2)[of "-df" "fls_deriv uf" g]
+    by (auto simp add: algebra_simps df_def dg_def uf_def ug_def)
+  moreover have
+    "fls_deriv (f*g) =
+      ( of_int dg * fls_shift (1 - dg) (f * ug) + fls_shift (-dg) (f * fls_deriv ug) ) +
+      ( of_int df * fls_shift (1 - df) (uf * g) + fls_shift (-df) (fls_deriv uf * g) )
+    "
+    using fls_deriv_shift[of
+            "- (df + dg)" "fps_to_fls (fls_base_factor_to_fps f * fls_base_factor_to_fps g)"
+          ]
+          fls_deriv_fps_to_fls[of "fls_base_factor_to_fps f * fls_base_factor_to_fps g"]
+          fps_deriv_mult[of "fls_base_factor_to_fps f" "fls_base_factor_to_fps g"]
+          distrib_right[of
+            "of_int df" "of_int dg"
+            "fls_shift (1 - (df + dg)) (
+              fps_to_fls (fls_base_factor_to_fps f * fls_base_factor_to_fps g)
+            )"
+          ]
+          fls_times_conv_fps_times[of uf ug]
+          fls_base_factor_subdegree[of f] fls_base_factor_subdegree[of g]
+          fls_regpart_deriv[of ug]
+          fls_times_conv_fps_times[of uf "fls_deriv ug"]
+          fls_deriv_subdegree0[of ug]
+          fls_regpart_deriv[of uf]
+          fls_times_conv_fps_times[of "fls_deriv uf" ug]
+          fls_deriv_subdegree0[of uf]
+          fls_shifted_times_simps(1)[of uf "-dg" ug]
+          fls_shifted_times_simps(1)[of "fls_deriv uf" "-dg" ug]
+          fls_shifted_times_simps(2)[of "-df" uf ug]
+          fls_shifted_times_simps(2)[of "-df" uf "fls_deriv ug"]
+    by (simp add: fls_times_def algebra_simps df_def dg_def uf_def ug_def)
+  ultimately show ?thesis by simp
+qed
+
+lemma fls_deriv_mult_const_left:
+  "fls_deriv (fls_const c * f) = fls_const c * fls_deriv f"
+  by simp
+
+lemma fls_deriv_linear:
+  "fls_deriv (fls_const a * f + fls_const b * g) =
+    fls_const a * fls_deriv f + fls_const b * fls_deriv g"
+  by simp
+
+lemma fls_deriv_mult_const_right:
+  "fls_deriv (f * fls_const c) = fls_deriv f * fls_const c"
+  by simp
+
+lemma fls_deriv_linear2:
+  "fls_deriv (f * fls_const a + g * fls_const b) =
+    fls_deriv f * fls_const a + fls_deriv g * fls_const b"
+  by simp
+
+lemma fls_deriv_sum:
+  "fls_deriv (sum f S) = sum (\<lambda>i. fls_deriv (f i)) S"
+proof (cases "finite S")
+  case True show ?thesis
+    by (induct rule: finite_induct [OF True]) simp_all
+qed simp
+
+lemma fls_deriv_power:
+  fixes f :: "'a::comm_ring_1 fls"
+  shows "fls_deriv (f^n) = of_nat n * f^(n-1) * fls_deriv f"
+proof (cases n)
+  case (Suc m)
+  have "fls_deriv (f^Suc m) = of_nat (Suc m) * f^m * fls_deriv f"
+    by (induct m) (simp_all add: algebra_simps)
+  with Suc show ?thesis by simp
+qed simp
+
+lemma fls_deriv_X_power:
+  "fls_deriv (fls_X ^ n) = of_nat n * fls_X ^ (n-1)"
+proof (cases n)
+  case (Suc m)
+  have "fls_deriv (fls_X^Suc m) = of_nat (Suc m) * fls_X^m"
+    by (induct m) (simp_all add: mult_of_nat_commute algebra_simps)
+  with Suc show ?thesis by simp
+qed simp
+
+lemma fls_deriv_X_inv_power:
+  "fls_deriv (fls_X_inv ^ n) = - of_nat n * fls_X_inv ^ (Suc n)"
+proof (cases n)
+  case (Suc m)
+  define iX :: "'a fls" where "iX \<equiv> fls_X_inv"
+  have "fls_deriv (iX ^ Suc m) = - of_nat (Suc m) * iX ^ (Suc (Suc m))"
+  proof (induct m)
+    case (Suc m)
+    have "- of_nat (Suc m + 1) * iX ^ Suc (Suc (Suc m)) =
+            iX * (-of_nat (Suc m) * iX ^ Suc (Suc m)) +
+                  - (iX ^ 2 * iX ^ Suc m)"
+      using distrib_right[of "-of_nat (Suc m)" "-(1::'a fls)" "fls_X_inv ^ Suc (Suc (Suc m))"]
+      by (simp add: algebra_simps mult_of_nat_commute power2_eq_square Suc iX_def)
+    thus ?case using Suc by (simp add: iX_def)
+  qed (simp add: numeral_2_eq_2 iX_def)
+  with Suc show ?thesis by (simp add: iX_def)
+qed simp
+
+lemma fls_deriv_X_intpow:
+  "fls_deriv (fls_X_intpow i) = of_int i * fls_X_intpow (i-1)"
+  by (simp add: fls_deriv_shift)
+
+lemma fls_deriv_lr_inverse:
+  assumes "x * f $$ fls_subdegree f = 1" "f $$ fls_subdegree f * y = 1"
+  \<comment> \<open>These assumptions imply x equals y, but no need to assume that.\<close>
+  shows   "fls_deriv (fls_left_inverse f x) =
+            - fls_left_inverse f x * fls_deriv f * fls_left_inverse f x"
+  and     "fls_deriv (fls_right_inverse f y) =
+            - fls_right_inverse f y * fls_deriv f * fls_right_inverse f y"
+proof-
+
+  define L where "L \<equiv> fls_left_inverse f x"
+  hence "fls_deriv (L * f) = 0" using fls_left_inverse[OF assms(1)] by simp
+  with assms show "fls_deriv L = - L * fls_deriv f * L"
+    using fls_right_inverse'[OF assms]
+    by    (simp add: minus_unique mult.assoc L_def)
+
+  define R where "R \<equiv> fls_right_inverse f y"
+  hence "fls_deriv (f * R) = 0" using fls_right_inverse[OF assms(2)] by simp
+  hence 1: "f * fls_deriv R + fls_deriv f * R = 0" by simp
+  have "R * f * fls_deriv R = - R * fls_deriv f * R"
+    using iffD2[OF eq_neg_iff_add_eq_0, OF 1] by (simp add: mult.assoc)
+  thus "fls_deriv R = - R * fls_deriv f * R"
+    using fls_left_inverse'[OF assms] by (simp add: R_def)
+
+qed
+
+lemma fls_deriv_lr_inverse_comm:
+  fixes   x y :: "'a::comm_ring_1"
+  assumes "x * f $$ fls_subdegree f = 1"
+  shows   "fls_deriv (fls_left_inverse f x) = - fls_deriv f * (fls_left_inverse f x)\<^sup>2"
+  and     "fls_deriv (fls_right_inverse f x) = - fls_deriv f * (fls_right_inverse f x)\<^sup>2"
+  using   assms fls_deriv_lr_inverse[of x f x]
+  by      (simp_all add: mult.commute power2_eq_square)
+
+lemma fls_inverse_deriv_divring:
+  fixes a :: "'a::division_ring fls"
+  shows "fls_deriv (inverse a) = - inverse a * fls_deriv a * inverse a"
+proof (cases "a=0")
+  case False thus ?thesis
+    using fls_deriv_lr_inverse(2)[of
+            "inverse (a $$ fls_subdegree a)" a "inverse (a $$ fls_subdegree a)"
+          ]
+    by    (auto simp add: fls_inverse_def')
+qed simp
+
+lemma fls_inverse_deriv:
+  fixes a :: "'a::field fls"
+  shows "fls_deriv (inverse a) = - fls_deriv a * (inverse a)\<^sup>2"
+  by    (simp add: fls_inverse_deriv_divring power2_eq_square)
+
+lemma fls_inverse_deriv':
+  fixes a :: "'a::field fls"
+  shows "fls_deriv (inverse a) = - fls_deriv a / a\<^sup>2"
+  using fls_inverse_deriv[of a]
+  by    (simp add: field_simps)
+
+
+subsubsection \<open>Equality of derivatives\<close>
+
+lemma fls_deriv_eq_0_iff:
+  "fls_deriv f = 0 \<longleftrightarrow> f = fls_const (f$$0 :: 'a::{ring_1_no_zero_divisors,ring_char_0})"
+proof
+  assume f: "fls_deriv f = 0"
+  show "f = fls_const (f$$0)"
+  proof (intro fls_eqI)
+    fix n
+    from f have "of_int n * f$$ n = 0" using fls_deriv_nth[of f "n-1"] by simp
+    thus "f$$n = fls_const (f$$0) $$ n" by (cases "n=0") auto
+  qed
+next
+  show "f = fls_const (f$$0) \<Longrightarrow> fls_deriv f = 0" using fls_deriv_const[of "f$$0"] by simp
+qed
+
+lemma fls_deriv_eq_iff:
+  fixes f g :: "'a::{ring_1_no_zero_divisors,ring_char_0} fls"
+  shows "fls_deriv f = fls_deriv g \<longleftrightarrow> (f = fls_const(f$$0 - g$$0) + g)"
+proof -
+  have "fls_deriv f = fls_deriv g \<longleftrightarrow> fls_deriv (f - g) = 0"
+    by simp
+  also have "\<dots> \<longleftrightarrow> f - g = fls_const ((f - g) $$ 0)"
+    unfolding fls_deriv_eq_0_iff ..
+  finally show ?thesis
+    by (simp add: field_simps)
+qed
+
+lemma fls_deriv_eq_iff_ex:
+  fixes f g :: "'a::{ring_1_no_zero_divisors,ring_char_0} fls"
+  shows "(fls_deriv f = fls_deriv g) \<longleftrightarrow> (\<exists>c. f = fls_const c + g)"
+  by    (auto simp: fls_deriv_eq_iff)
+
+
+subsubsection \<open>Residues\<close>
+
+definition fls_residue_def[simp]: "fls_residue f \<equiv> f $$ -1"
+
+lemma fls_residue_deriv: "fls_residue (fls_deriv f) = 0"
+  by simp
+
+lemma fls_residue_add: "fls_residue (f+g) = fls_residue f + fls_residue g"
+  by simp
+
+lemma fls_residue_times_deriv:
+  "fls_residue (fls_deriv f * g) = - fls_residue (f * fls_deriv g)"
+  using fls_residue_deriv[of "f*g"] minus_unique[of "fls_residue (f * fls_deriv g)"]
+  by    simp
+
+lemma fls_residue_power_series: "fls_subdegree f \<ge> 0 \<Longrightarrow> fls_residue f = 0"
+  by simp
+
+lemma fls_residue_fls_X_intpow:
+  "fls_residue (fls_X_intpow i) = (if i=-1 then 1 else 0)"
+  by simp
+
+lemma fls_residue_shift_nth:
+  fixes f :: "'a::semiring_1 fls"
+  shows "f$$n = fls_residue (fls_X_intpow (-n-1) * f)"
+  by    (simp add: fls_shifted_times_transfer)
+
+lemma fls_residue_fls_const_times:
+  fixes f :: "'a::{comm_monoid_add, mult_zero} fls"
+  shows "fls_residue (fls_const c * f) = c * fls_residue f"
+  and   "fls_residue (f * fls_const c) = fls_residue f * c"
+  by    simp_all
+
+lemma fls_residue_of_int_times:
+  fixes f :: "'a::ring_1 fls"
+  shows "fls_residue (of_int i * f) = of_int i * fls_residue f"
+  and   "fls_residue (f * of_int i) = fls_residue f * of_int i"
+  by    (simp_all add: fls_residue_fls_const_times fls_of_int)
+
+lemma fls_residue_deriv_times_lr_inverse_eq_subdegree:
+  fixes   f g :: "'a::ring_1 fls"
+  assumes "y * (f $$ fls_subdegree f) = 1" "(f $$ fls_subdegree f) * y = 1"
+  shows   "fls_residue (fls_deriv f * fls_right_inverse f y)  = of_int (fls_subdegree f)"
+  and     "fls_residue (fls_deriv f * fls_left_inverse f y)   = of_int (fls_subdegree f)"
+  and     "fls_residue (fls_left_inverse f y * fls_deriv f)   = of_int (fls_subdegree f)"
+  and     "fls_residue (fls_right_inverse f y * fls_deriv f)  = of_int (fls_subdegree f)"
+proof-
+  define df :: int where "df \<equiv> fls_subdegree f"
+  define B X :: "'a fls"
+    where "B \<equiv> fls_base_factor f"
+    and   "X \<equiv> (fls_X_intpow df :: 'a fls)"
+  define D L R :: "'a fls"
+    where "D \<equiv> fls_deriv B"
+    and   "L \<equiv> fls_left_inverse B y"
+    and   "R \<equiv> fls_right_inverse B y"
+  have intpow_diff: "fls_X_intpow (df - 1) = X * fls_X_inv"
+    using fls_X_intpow_diff_conv_times[of df 1] by (simp add: X_def fls_X_inv_conv_shift_1)
+ 
+
+  show "fls_residue (fls_deriv f * fls_right_inverse f y) = of_int df"
+  proof-
+    have subdegree_DR: "fls_subdegree (D * R) \<ge> 0"
+      using fls_base_factor_subdegree[of f] fls_base_factor_subdegree[of "fls_right_inverse f y"]
+            assms(1) fls_right_inverse_base_factor[of y f] fls_mult_subdegree_ge_0[of D R]
+      by    (force simp: fls_deriv_subdegree0 D_def R_def B_def)
+    have decomp: "f = X * B"
+      unfolding X_def B_def df_def by (rule fls_base_factor_X_power_decompose(2)[of f])
+    hence "fls_deriv f = X * D + of_int df * X * fls_X_inv * B"
+      using intpow_diff fls_deriv_mult[of X B]
+      by    (simp add: fls_deriv_X_intpow X_def B_def D_def mult.assoc)
+    moreover from assms have "fls_right_inverse (X * B) y = R * fls_right_inverse X 1"
+      using fls_base_factor_base[of f] fls_lr_inverse_mult_ring1(2)[of 1 X]
+      by    (simp add: X_def B_def R_def)
+    ultimately have
+      "fls_deriv f * fls_right_inverse f y =
+        (D + of_int df * fls_X_inv * B) * R * (X * fls_right_inverse X 1)"
+      by (simp add: decomp algebra_simps X_def fls_X_intpow_times_comm)
+    also have "\<dots> = D * R + of_int df * fls_X_inv"
+      using fls_right_inverse[of X 1]
+            assms fls_base_factor_base[of f] fls_right_inverse[of B y]
+      by    (simp add: X_def distrib_right mult.assoc B_def R_def)
+    finally show ?thesis using subdegree_DR by simp
+  qed
+
+  with assms show "fls_residue (fls_deriv f * fls_left_inverse f y) = of_int df"
+    using fls_left_inverse_eq_fls_right_inverse[of y f] by simp
+
+  show "fls_residue (fls_left_inverse f y * fls_deriv f) = of_int df"
+  proof-
+    have subdegree_LD: "fls_subdegree (L * D) \<ge> 0"
+      using fls_base_factor_subdegree[of f] fls_base_factor_subdegree[of "fls_left_inverse f y"]
+            assms(1) fls_left_inverse_base_factor[of y f] fls_mult_subdegree_ge_0[of L D]
+      by    (force simp: fls_deriv_subdegree0 D_def L_def B_def)
+    have decomp: "f = B * X"
+      unfolding X_def B_def df_def by (rule fls_base_factor_X_power_decompose(1)[of f])
+    hence "fls_deriv f = D * X + B * of_int df * X * fls_X_inv"
+      using intpow_diff fls_deriv_mult[of B X]
+      by    (simp add: fls_deriv_X_intpow X_def D_def B_def mult.assoc)
+    moreover from assms have "fls_left_inverse (B * X) y = fls_left_inverse X 1 * L"
+      using fls_base_factor_base[of f] fls_lr_inverse_mult_ring1(1)[of _ _ 1 X]
+      by    (simp add: X_def B_def L_def)
+    ultimately have
+      "fls_left_inverse f y * fls_deriv f =
+        fls_left_inverse X 1 * X * L * (D + B * (of_int df * fls_X_inv))"
+      by (simp add: decomp algebra_simps X_def fls_X_intpow_times_comm)
+    also have "\<dots> = L * D + of_int df * fls_X_inv"
+      using assms fls_left_inverse[of 1 X] fls_base_factor_base[of f] fls_left_inverse[of y B]
+       by   (simp add: X_def distrib_left mult.assoc[symmetric] L_def B_def)
+    finally show ?thesis using subdegree_LD by simp
+  qed
+
+  with assms show "fls_residue (fls_right_inverse f y * fls_deriv f) = of_int df"
+    using fls_left_inverse_eq_fls_right_inverse[of y f] by simp
+
+qed
+
+lemma fls_residue_deriv_times_inverse_eq_subdegree:
+  fixes f g :: "'a::division_ring fls"
+  shows "fls_residue (fls_deriv f * inverse f) = of_int (fls_subdegree f)"
+  and   "fls_residue (inverse f * fls_deriv f) = of_int (fls_subdegree f)"
+proof-
+  show "fls_residue (fls_deriv f * inverse f) = of_int (fls_subdegree f)"
+    using fls_residue_deriv_times_lr_inverse_eq_subdegree(1)[of _ f]
+    by    (cases "f=0") (auto simp: fls_inverse_def')
+  show "fls_residue (inverse f * fls_deriv f) = of_int (fls_subdegree f)"
+    using fls_residue_deriv_times_lr_inverse_eq_subdegree(4)[of _ f]
+    by    (cases "f=0") (auto simp: fls_inverse_def')
+qed
+
+
+subsubsection \<open>Integral definition and basic properties\<close>
+
+\<comment> \<open>To incorporate a constant of integration, just add an fps_const.\<close>
+definition fls_integral :: "'a::{ring_1,inverse} fls \<Rightarrow> 'a fls"
+  where "fls_integral a = Abs_fls (\<lambda>n. if n=0 then 0 else inverse (of_int n) * a$$(n - 1))"
+
+lemma fls_integral_nth [simp]:
+  "fls_integral a $$ n = (if n=0 then 0 else inverse (of_int n) * a$$(n-1))"
+proof-
+  define F where "F \<equiv> (\<lambda>n. if n=0 then 0 else inverse (of_int n) * a$$(n - 1))"
+  obtain N where "\<forall>n<N. a$$n = 0" by (elim fls_nth_vanishes_belowE)
+  hence "\<forall>n<N. F n = 0" by (auto simp add: F_def)
+  thus ?thesis using nth_Abs_fls_lower_bound[of N F] unfolding fls_integral_def F_def by simp
+qed
+
+lemma fls_integral_conv_fps_zeroth_integral:
+  assumes "fls_subdegree a \<ge> 0"
+  shows   "fls_integral a = fps_to_fls (fps_integral0 (fls_regpart a))"
+proof (rule fls_eqI)
+  fix n
+  show "fls_integral a $$ n = fps_to_fls (fps_integral0 (fls_regpart a)) $$ n"
+  proof (cases "n>0")
+    case False with assms show ?thesis by simp
+  next
+    case True
+    hence "int ((nat n) - 1) = n - 1" by simp
+    with True show ?thesis by (simp add: fps_integral_def)
+  qed
+qed
+
+lemma fls_integral_zero [simp]: "fls_integral 0 = 0"
+  by (intro fls_eqI) simp
+
+lemma fls_integral_const':
+  fixes   x :: "'a::{ring_1,inverse}"
+  assumes "inverse (1::'a) = 1"
+  shows   "fls_integral (fls_const x) = fls_const x * fls_X"
+  by      (intro fls_eqI) (simp add: assms)
+
+lemma fls_integral_const:
+  fixes x :: "'a::division_ring"
+  shows "fls_integral (fls_const x) = fls_const x * fls_X"
+  by    (rule fls_integral_const'[OF inverse_1])
+
+lemma fls_integral_of_nat':
+  assumes "inverse (1::'a::{ring_1,inverse}) = 1"
+  shows   "fls_integral (of_nat n :: 'a fls) = of_nat n * fls_X"
+  by      (simp add: assms fls_integral_const' fls_of_nat)
+
+lemma fls_integral_of_nat:
+  "fls_integral (of_nat n :: 'a::division_ring fls) = of_nat n * fls_X"
+  by (rule fls_integral_of_nat'[OF inverse_1])
+
+lemma fls_integral_of_int':
+  assumes "inverse (1::'a::{ring_1,inverse}) = 1"
+  shows   "fls_integral (of_int i :: 'a fls) = of_int i * fls_X"
+  by      (simp add: assms fls_integral_const' fls_of_int)
+
+lemma fls_integral_of_int:
+  "fls_integral (of_int i :: 'a::division_ring fls) = of_int i * fls_X"
+  by (rule fls_integral_of_int'[OF inverse_1])
+
+lemma fls_integral_one':
+  assumes "inverse (1::'a::{ring_1,inverse}) = 1"
+  shows   "fls_integral (1::'a fls) = fls_X"
+  using   fls_integral_const'[of 1]
+  by      (force simp: assms)
+
+lemma fls_integral_one: "fls_integral (1::'a::division_ring fls) = fls_X"
+  by (rule fls_integral_one'[OF inverse_1])
+
+lemma fls_subdegree_integral_ge:
+  "fls_integral f \<noteq> 0 \<Longrightarrow> fls_subdegree (fls_integral f) \<ge> fls_subdegree f + 1"
+  by (intro fls_subdegree_geI) simp_all
+
+lemma fls_subdegree_integral:
+  fixes   f :: "'a::{division_ring,ring_char_0} fls"
+  assumes "f \<noteq> 0" "fls_subdegree f \<noteq> -1"
+  shows   "fls_subdegree (fls_integral f) = fls_subdegree f + 1"
+  using   assms of_int_0_eq_iff[of "fls_subdegree f + 1"] fls_subdegree_integral_ge
+  by      (intro fls_subdegree_eqI) simp_all
+
+lemma fls_integral_X [simp]:
+  "fls_integral (fls_X::'a::{ring_1,inverse} fls) =
+    fls_const (inverse (of_int 2)) * fls_X\<^sup>2"
+proof (intro fls_eqI)
+  fix n
+  show "fls_integral (fls_X::'a fls) $$ n = (fls_const (inverse (of_int 2)) * fls_X\<^sup>2) $$ n"
+    using arg_cong[OF fls_X_power_nth, of "\<lambda>x. inverse (of_int 2) * x", of 2 n, symmetric]
+    by    (auto simp add: )
+qed
+
+lemma fls_integral_X_power:
+  "fls_integral (fls_X ^ n ::'a :: {ring_1,inverse} fls) =
+    fls_const (inverse (of_nat (Suc n))) * fls_X ^ Suc n"
+proof (intro fls_eqI)
+  fix k
+  have "(fls_X :: 'a fls) ^ Suc n $$ k = (if k=Suc n then 1 else 0)"
+    by (rule fls_X_power_nth)
+  thus 
+    "fls_integral ((fls_X::'a fls) ^ n) $$ k =
+      (fls_const (inverse (of_nat (Suc n))) * (fls_X::'a fls) ^ Suc n) $$ k"
+    by simp
+qed
+
+lemma fls_integral_X_power_char0:
+  "fls_integral (fls_X ^ n :: 'a :: {ring_char_0,inverse} fls) =
+    inverse (of_nat (Suc n)) * fls_X ^ Suc n"
+proof -
+  have "(of_nat (Suc n) :: 'a) \<noteq> 0" by (rule of_nat_neq_0)
+  hence "fls_const (inverse (of_nat (Suc n) :: 'a)) = inverse (fls_const (of_nat (Suc n)))"
+    by (simp add: fls_inverse_const)
+  moreover have
+    "fls_integral ((fls_X::'a fls) ^ n) = fls_const (inverse (of_nat (Suc n))) * fls_X ^ Suc n"
+    by (rule fls_integral_X_power)
+  ultimately show ?thesis by (simp add: fls_of_nat)
+qed
+
+lemma fls_integral_X_inv [simp]: "fls_integral (fls_X_inv::'a::{ring_1,inverse} fls) = 0"
+  by (intro fls_eqI) simp
+
+lemma fls_integral_X_inv_power:
+  assumes "n \<ge> 2"
+  shows
+    "fls_integral (fls_X_inv ^ n :: 'a :: {ring_1,inverse} fls) =
+      fls_const (inverse (of_int (1 - int n))) * fls_X_inv ^ (n-1)"
+proof (rule fls_eqI)
+  fix k show
+    "fls_integral (fls_X_inv ^ n :: 'a fls) $$ k=
+      (fls_const (inverse (of_int (1 - int n))) * fls_X_inv ^ (n-1)) $$ k"
+  proof (cases "k=0")
+    case True with assms show ?thesis by simp
+  next
+    case False
+    from assms have "int (n-1) = int n - 1" by simp
+    hence
+      "(fls_const (inverse (of_int (1 - int n))) * (fls_X_inv:: 'a fls) ^ (n-1)) $$ k =
+      (if k = 1 - int n then inverse (of_int k) else 0)"
+      by (simp add: fls_X_inv_power_times_conv_shift(2))
+    with False show ?thesis by (simp add: algebra_simps)
+  qed
+qed
+
+lemma fls_integral_X_inv_power_char0:
+  assumes "n \<ge> 2"
+  shows
+    "fls_integral (fls_X_inv ^ n :: 'a :: {ring_char_0,inverse} fls) =
+      inverse (of_int (1 - int n)) * fls_X_inv ^ (n-1)"
+proof-
+  from assms have "(of_int (1 - int n) :: 'a) \<noteq> 0" by simp
+  hence
+    "fls_const (inverse (of_int (1 - int n) :: 'a)) = inverse (fls_const (of_int (1 - int n)))"
+    by (simp add: fls_inverse_const)
+  moreover have
+    "fls_integral (fls_X_inv ^ n :: 'a fls) =
+      fls_const (inverse (of_int (1 - int n))) * fls_X_inv ^ (n-1)"
+    using assms by (rule fls_integral_X_inv_power)
+  ultimately show ?thesis by (simp add: fls_of_int)
+qed
+
+lemma fls_integral_X_inv_power':
+  assumes "n \<ge> 1"
+  shows
+    "fls_integral (fls_X_inv ^ n :: 'a :: division_ring fls) =
+      - fls_const (inverse (of_nat (n-1))) * fls_X_inv ^ (n-1)"
+proof (cases "n = 1")
+  case False
+  with assms have n: "n \<ge> 2" by simp
+  hence
+    "fls_integral (fls_X_inv ^ n :: 'a fls) =
+      fls_const (inverse (- of_nat (nat (int n - 1)))) * fls_X_inv ^ (n-1)"
+    by (simp add: fls_integral_X_inv_power)
+  moreover from n have "nat (int n - 1) = n - 1" by simp
+  ultimately show ?thesis
+    using inverse_minus_eq[of "of_nat (n-1) :: 'a"] by simp
+qed simp
+
+lemma fls_integral_X_inv_power_char0':
+  assumes "n \<ge> 1"
+  shows
+    "fls_integral (fls_X_inv ^ n :: 'a :: {division_ring,ring_char_0} fls) =
+      - inverse (of_nat (n-1)) * fls_X_inv ^ (n-1)"
+proof (cases "n=1")
+  case False with assms show ?thesis
+    by (simp add: fls_integral_X_inv_power' fls_inverse_const fls_of_nat)
+qed simp    
+
+lemma fls_integral_delta:
+  assumes "m \<noteq> -1"
+  shows
+    "fls_integral (Abs_fls (\<lambda>n. if n=m then c else 0)) =
+      Abs_fls (\<lambda>n. if n=m+1 then inverse (of_int (m+1)) * c else 0)"
+  using   assms
+  by      (intro fls_eqI) auto
+
+lemma fls_regpart_integral:
+  "fls_regpart (fls_integral f) = fps_integral0 (fls_regpart f)"
+proof (rule fps_ext)
+  fix n
+  show "fls_regpart (fls_integral f) $ n = fps_integral0 (fls_regpart f) $ n"
+    by (cases n) (simp_all add: fps_integral_def)
+qed
+
+lemma fls_integral_fps_to_fls:
+  "fls_integral (fps_to_fls f) = fps_to_fls (fps_integral0 f)"
+proof (intro fls_eqI)
+  fix n :: int
+  show "fls_integral (fps_to_fls f) $$ n = fps_to_fls (fps_integral0 f) $$ n"
+  proof (cases "n<1")
+    case True thus ?thesis by simp
+  next
+    case False
+    hence "nat (n-1) = nat n - 1" by simp
+    with False show ?thesis by (cases "nat n") auto
+  qed
+qed
+
+
+subsubsection \<open>Algebra rules of the integral\<close>
+
+lemma fls_integral_add [simp]: "fls_integral (f+g) = fls_integral f + fls_integral g"
+  by (intro fls_eqI) (simp add: algebra_simps)
+
+lemma fls_integral_sub [simp]: "fls_integral (f-g) = fls_integral f - fls_integral g"
+  by (intro fls_eqI) (simp add: algebra_simps)
+
+lemma fls_integral_neg [simp]: "fls_integral (-f) = - fls_integral f"
+  using fls_integral_sub[of 0 f] by simp
+
+lemma fls_integral_mult_const_left:
+  "fls_integral (fls_const c * f) = fls_const c * fls_integral (f :: 'a::division_ring fls)"
+  by (intro fls_eqI) (simp add: mult.assoc mult_inverse_of_int_commute)
+
+lemma fls_integral_mult_const_left_comm:
+  fixes f :: "'a::{comm_ring_1,inverse} fls"
+  shows "fls_integral (fls_const c * f) = fls_const c * fls_integral f"
+  by (intro fls_eqI) (simp add: mult.assoc mult.commute)
+
+lemma fls_integral_linear:
+  fixes f g :: "'a::division_ring fls"
+  shows
+    "fls_integral (fls_const a * f + fls_const b * g) =
+      fls_const a * fls_integral f + fls_const b * fls_integral g"
+  by    (simp add: fls_integral_mult_const_left)
+
+lemma fls_integral_linear_comm:
+  fixes f g :: "'a::{comm_ring_1,inverse} fls"
+  shows
+    "fls_integral (fls_const a * f + fls_const b * g) =
+      fls_const a * fls_integral f + fls_const b * fls_integral g"
+  by    (simp add: fls_integral_mult_const_left_comm)
+
+lemma fls_integral_mult_const_right:
+  "fls_integral (f * fls_const c) = fls_integral f * fls_const c"
+  by (intro fls_eqI) (simp add: mult.assoc)
+
+lemma fls_integral_linear2:
+    "fls_integral (f * fls_const a + g * fls_const b) =
+      fls_integral f * fls_const a + fls_integral g * fls_const b"
+  by    (simp add: fls_integral_mult_const_right)
+
+lemma fls_integral_sum:
+  "fls_integral (sum f S) = sum (\<lambda>i. fls_integral (f i)) S"
+proof (cases "finite S")
+  case True show ?thesis
+    by (induct rule: finite_induct [OF True]) simp_all
+qed simp
+
+
+subsubsection \<open>Derivatives of integrals and vice versa\<close>
+
+lemma fls_integral_fls_deriv:
+  fixes a :: "'a::{division_ring,ring_char_0} fls"
+  shows "fls_integral (fls_deriv a) + fls_const (a$$0) = a"
+  by    (intro fls_eqI) (simp add: mult.assoc[symmetric])
+
+lemma fls_deriv_fls_integral:
+  fixes   a :: "'a::{division_ring,ring_char_0} fls"
+  assumes "fls_residue a = 0"
+  shows   "fls_deriv (fls_integral a) = a"
+proof (intro fls_eqI)
+  fix n :: int
+  show "fls_deriv (fls_integral a) $$ n = a $$ n"
+  proof (cases "n=-1")
+    case True with assms show ?thesis by simp
+  next
+    case False
+    hence "(of_int (n+1) :: 'a) \<noteq> 0" using of_int_eq_0_iff[of "n+1"] by simp
+    hence "(of_int (n+1) :: 'a) * inverse (of_int (n+1) :: 'a) = (1::'a)"
+      using of_int_eq_0_iff[of "n+1"] by simp
+    moreover have
+      "fls_deriv (fls_integral a) $$ n =
+        (if n=-1 then 0 else of_int (n+1) * inverse (of_int (n+1)) * a$$n)"
+      by (simp add: mult.assoc)
+    ultimately show ?thesis
+      by (simp add: False)
+  qed
+qed
+
+text \<open>Series with zero residue are precisely the derivatives.\<close>
+
+lemma fls_residue_nonzero_ex_antiderivative:
+  fixes   f :: "'a::{division_ring,ring_char_0} fls"
+  assumes "fls_residue f = 0"
+  shows   "\<exists>F. fls_deriv F = f"
+  using   assms fls_deriv_fls_integral
+  by      auto
+
+lemma fls_ex_antiderivative_residue_nonzero:
+  assumes "\<exists>F. fls_deriv F = f"
+  shows   "fls_residue f = 0"
+  using   assms fls_residue_deriv
+  by      auto
+
+lemma fls_residue_nonzero_ex_anitderivative_iff:
+  fixes f :: "'a::{division_ring,ring_char_0} fls"
+  shows "fls_residue f = 0 \<longleftrightarrow> (\<exists>F. fls_deriv F = f)"
+  using fls_residue_nonzero_ex_antiderivative fls_ex_antiderivative_residue_nonzero
+  by    fast
+
+
+subsection \<open>Topology\<close>
+
+instantiation fls :: (group_add) metric_space
+begin
+
+definition
+  dist_fls_def:
+    "dist (a :: 'a fls) b =
+      (if a = b
+        then 0
+        else if fls_subdegree (a-b) \<ge> 0
+          then inverse (2 ^ nat (fls_subdegree (a-b)))
+          else 2 ^ nat (-fls_subdegree (a-b))
+      )"
+
+lemma dist_fls_ge0: "dist (a :: 'a fls) b \<ge> 0"
+  by (simp add: dist_fls_def)
+
+definition uniformity_fls_def [code del]:
+  "(uniformity :: ('a fls \<times> 'a fls) filter) = (INF e:{0 <..}. principal {(x, y). dist x y < e})"
+
+definition open_fls_def' [code del]:
+  "open (U :: 'a fls set) \<longleftrightarrow> (\<forall>x\<in>U. eventually (\<lambda>(x', y). x' = x \<longrightarrow> y \<in> U) uniformity)"
+
+lemma dist_fls_sym: "dist (a :: 'a fls) b = dist b a"
+  by  (cases "a\<noteq>b", cases "fls_subdegree (a-b) \<ge> 0")
+      (simp_all add: fls_subdegree_minus_sym dist_fls_def)
+
+context
+begin
+
+private lemma instance_helper:
+  fixes   a b c :: "'a fls"
+  assumes neq: "a\<noteq>b" "a\<noteq>c"
+  and     dist_ineq: "dist a b > dist a c"
+  shows   "fls_subdegree (a - b) < fls_subdegree (a - c)"
+proof (
+  cases "fls_subdegree (a-b) \<ge> 0" "fls_subdegree (a-c) \<ge> 0"
+  rule: case_split[case_product case_split]
+)
+  case True_True with neq dist_ineq show ?thesis by (simp add: dist_fls_def)
+next
+  case False_True with dist_ineq show ?thesis by (simp add: dist_fls_def)
+next
+  case False_False with neq dist_ineq show ?thesis by (simp add: dist_fls_def)
+next
+  case True_False
+  with neq
+    have "(1::real) > 2 ^ (nat (fls_subdegree (a-b)) + nat (-fls_subdegree (a-c)))"
+    and  "nat (fls_subdegree (a-b)) + nat (-fls_subdegree (a-c)) =
+            nat (fls_subdegree (a-b) - fls_subdegree (a-c))"
+    using dist_ineq
+    by    (simp_all add: dist_fls_def field_simps power_add)
+  hence "\<not> (1::real) < 2 ^ (nat (fls_subdegree (a-b) - fls_subdegree (a-c)))" by simp
+  hence "\<not> (0 < nat (fls_subdegree (a - b) - fls_subdegree (a - c)))" by auto
+  hence "fls_subdegree (a - b) \<le> fls_subdegree (a - c)" by simp
+  with True_False show ?thesis by simp
+qed
+
+instance
+proof
+  show th: "dist a b = 0 \<longleftrightarrow> a = b" for a b :: "'a fls"
+    by (simp add: dist_fls_def split: if_split_asm)
+  then have th'[simp]: "dist a a = 0" for a :: "'a fls" by simp
+
+  fix a b c :: "'a fls"
+  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_fls_def)
+  next
+    case 2
+    then show ?thesis
+      by (cases "c = a") (simp_all add: th dist_fls_sym)
+  next
+    case neq: 3
+    have False if "dist a b > dist a c + dist b c"
+    proof -
+      from neq have "dist a b > 0" "dist b c > 0" "dist a c > 0" by (simp_all add: dist_fls_def)
+      with that have dist_ineq: "dist a b > dist a c" "dist a b > dist b c" by simp_all
+      have "fls_subdegree (a - b) < fls_subdegree (a - c)"
+      and  "fls_subdegree (a - b) < fls_subdegree (b - c)"
+        using instance_helper[of a b c] instance_helper[of b a c] neq dist_ineq
+        by    (simp_all add: dist_fls_sym fls_subdegree_minus_sym)
+      hence "(a - c) $$ fls_subdegree (a - b) = 0" and "(b - c) $$ fls_subdegree (a - b) = 0"
+        by  (simp_all only: fls_eq0_below_subdegree)
+      hence "(a - b) $$ fls_subdegree (a - b) = 0" by simp
+      moreover from neq have "(a - b) $$ fls_subdegree (a - b) \<noteq> 0"
+        by (intro nth_fls_subdegree_nonzero) simp
+      ultimately show False by contradiction
+    qed
+    thus ?thesis by (auto simp: not_le[symmetric])
+  qed
+qed (rule open_fls_def' uniformity_fls_def)+
+
+end
+end
+
+lemma open_fls_def:
+  "open (S :: 'a::group_add fls set) = (\<forall>a \<in> S. \<exists>r. r >0 \<and> {y. dist y a < r} \<subseteq> S)"
+  unfolding open_dist subset_eq by simp
+
+
+subsection \<open>Notation bundle\<close>
+
+no_notation fls_nth (infixl "$$" 75)
+
+bundle fls_notation
+begin
+notation fls_nth (infixl "$$" 75)
+end
+
+end
\ No newline at end of file
--- a/src/HOL/Computational_Algebra/Formal_Power_Series.thy	Mon Feb 04 15:39:37 2019 +0100
+++ b/src/HOL/Computational_Algebra/Formal_Power_Series.thy	Mon Feb 04 17:19:04 2019 +0100
@@ -1,5 +1,7 @@
 (*  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 M√ľnchen
 *)
 
 section \<open>A formalization of formal power series\<close>
@@ -22,6 +24,8 @@
 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)
 
@@ -40,6 +44,31 @@
 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)"
@@ -76,6 +105,9 @@
 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)))"
@@ -88,6 +120,19 @@
 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"
+  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]
@@ -102,385 +147,11 @@
   shows "x * (if b then y else 0) = (if b then x * y else 0)"
   by simp
 
-lemma cond_value_iff: "f (if b then x else y) = (if b then f x else f y)"
-  by auto
-
-
-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
-
-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) semigroup_mult
-proof
-  fix a b c :: "'a fps"
-  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
-
-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) ab_semigroup_mult
-proof
-  fix a b :: "'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
-
-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 :: (semiring_1) monoid_mult
-proof
-  fix a :: "'a fps"
-  show "1 * a = a"
-    by (simp add: fps_ext fps_mult_nth mult_delta_left sum.delta)
-  show "a * 1 = a"
-    by (simp add: fps_ext fps_mult_nth mult_delta_right sum.delta')
-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)
-
-instance fps :: (semiring_0) semiring
-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)
-qed
-
-instance fps :: (semiring_0) semiring_0
-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
-
-instance fps :: (semiring_0_cancel) semiring_0_cancel ..
-
-instance fps :: (semiring_1) semiring_1 ..
-
-
-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_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_eq_iff: "f = g \<longleftrightarrow> (\<forall>n. f $ n = g $n)"
-  by (rule expand_fps_eq)
-
-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_1_eq_1 [simp]: "fps_const 1 = 1"
-  by (simp add: fps_ext)
-
-lemma fps_const_neg [simp]: "- (fps_const (c::'a::ring)) = 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_sub [simp]: "fps_const (c::'a::group_add) - fps_const d = fps_const (c - d)"
-  by (simp add: fps_ext)
-
-lemma fps_const_mult[simp]: "fps_const (c::'a::ring) * fps_const d = fps_const (c * d)"
-  by (simp add: fps_eq_iff fps_mult_nth sum.neutral)
-
-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_mult_left: "fps_const (c::'a::semiring_0) * f = Abs_fps (\<lambda>n. c * f$n)"
-  unfolding fps_eq_iff fps_mult_nth
-  by (simp add: fps_const_def mult_delta_left sum.delta)
-
-lemma fps_const_mult_right: "f * fps_const (c::'a::semiring_0) = Abs_fps (\<lambda>n. f$n * c)"
-  unfolding fps_eq_iff fps_mult_nth
-  by (simp add: fps_const_def mult_delta_right sum.delta')
-
-lemma fps_mult_left_const_nth [simp]: "(fps_const (c::'a::semiring_1) * f)$n = c* f$n"
-  by (simp add: fps_mult_nth mult_delta_left sum.delta)
-
-lemma fps_mult_right_const_nth [simp]: "(f * fps_const (c::'a::semiring_1))$n = f$n * c"
-  by (simp add: fps_mult_nth mult_delta_right sum.delta')
-
-
-subsection \<open>Formal power series form an integral domain\<close>
-
-instance fps :: (ring) ring ..
-
-instance fps :: (ring_1) ring_1
-  by (intro_classes, auto simp add: distrib_right)
-
-instance fps :: (comm_ring_1) comm_ring_1
-  by (intro_classes, auto simp add: distrib_right)
-
-instance fps :: (ring_no_zero_divisors) ring_no_zero_divisors
-proof
-  fix a b :: "'a fps"
-  assume "a \<noteq> 0" and "b \<noteq> 0"
-  then obtain i j where i: "a $ i \<noteq> 0" "\<forall>k<i. a $ k = 0" and j: "b $ j \<noteq> 0" "\<forall>k<j. b $ k =0"
-    unfolding fps_nonzero_nth_minimal
-    by blast+
-  have "(a * b) $ (i + j) = (\<Sum>k=0..i+j. a $ k * b $ (i + j - k))"
-    by (rule fps_mult_nth)
-  also have "\<dots> = (a $ i * b $ (i + j - i)) + (\<Sum>k\<in>{0..i+j} - {i}. a $ k * b $ (i + j - k))"
-    by (rule sum.remove) simp_all
-  also have "(\<Sum>k\<in>{0..i+j}-{i}. a $ k * b $ (i + j - k)) = 0"
-  proof (rule sum.neutral [rule_format])
-    fix k assume "k \<in> {0..i+j} - {i}"
-    then have "k < i \<or> i+j-k < j"
-      by auto
-    then show "a $ k * b $ (i + j - k) = 0"
-      using i j by auto
-  qed
-  also have "a $ i * b $ (i + j - i) + 0 = a $ i * b $ j"
-    by simp
-  also have "a $ i * b $ j \<noteq> 0"
-    using i j by simp
-  finally have "(a*b) $ (i+j) \<noteq> 0" .
-  then show "a * b \<noteq> 0"
-    unfolding fps_nonzero_nth by blast
-qed
-
-instance fps :: (ring_1_no_zero_divisors) ring_1_no_zero_divisors ..
-
-instance fps :: (idom) idom ..
-
-lemma 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])
-
-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 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 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 
-
-
-subsection \<open>The efps_Xtractor series fps_X\<close>
-
-lemma minus_one_power_iff: "(- (1::'a::comm_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 fps_X_mult_nth [simp]:
-  "(fps_X * (f :: 'a::semiring_1 fps)) $n = (if n = 0 then 0 else f $ (n - 1))"
-proof (cases "n = 0")
-  case False
-  have "(fps_X * f) $n = (\<Sum>i = 0..n. fps_X $ i * f $ (n - i))"
-    by (simp add: fps_mult_nth)
-  also have "\<dots> = f $ (n - 1)"
-    using False by (simp add: fps_X_def mult_delta_left sum.delta)
-  finally show ?thesis
-    using False by simp
-next
-  case True
-  then show ?thesis
-    by (simp add: fps_mult_nth fps_X_def)
-qed
-
-lemma fps_X_mult_right_nth[simp]:
-  "((a::'a::semiring_1 fps) * fps_X) $ n = (if n = 0 then 0 else a $ (n - 1))"
-proof -
-  have "(a * fps_X) $ n = (\<Sum>i = 0..n. a $ i * (if n - i = Suc 0 then 1 else 0))"
-    by (simp add: fps_times_def fps_X_def)
-  also have "\<dots> = (\<Sum>i = 0..n. if i = n - 1 then if n = 0 then 0 else a $ i else 0)"
-    by (intro sum.cong) auto
-  also have "\<dots> = (if n = 0 then 0 else a $ (n - 1))" by (simp add: sum.delta)
-  finally show ?thesis .
-qed
-
-lemma fps_mult_fps_X_commute: "fps_X * (a :: 'a :: semiring_1 fps) = a * fps_X" 
-  by (simp add: fps_eq_iff)
-
-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::'a::comm_ring_1)"
-  by (simp add: fps_X_power_iff)
-
-lemma fps_X_power_mult_nth: "(fps_X^k * (f :: 'a::comm_ring_1 fps)) $n = (if n < k then 0 else f $ (n - k))"
-  apply (induct k arbitrary: n)
-  apply simp
-  unfolding power_Suc mult.assoc
-  apply (case_tac n)
-  apply auto
-  done
-
-lemma fps_X_power_mult_right_nth:
-    "((f :: 'a::comm_ring_1 fps) * fps_X^k) $n = (if n < k then 0 else f $ (n - k))"
-  by (metis fps_X_power_mult_nth mult.commute)
-
-
-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 :: 'a :: {semiring_1,zero_neq_one} fps) \<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 :: ('a :: {comm_ring_1}) fps) ^ 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
+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>
@@ -543,21 +214,12 @@
   "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 :: ('a :: zero_neq_one) fps) = 0"
-  by (auto intro!: subdegreeI)
-
-lemma subdegree_fps_X [simp]: "subdegree (fps_X :: ('a :: zero_neq_one) fps) = 1"
-  by (auto intro!: subdegreeI simp: fps_X_def)
-
-lemma subdegree_fps_const [simp]: "subdegree (fps_const c) = 0"
-  by (cases "c = 0") (auto intro!: subdegreeI)
-
-lemma subdegree_numeral [simp]: "subdegree (numeral n) = 0"
-  by (simp add: numeral_fps_const)
+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")
@@ -569,6 +231,119 @@
 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_head_Suc)
+
 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"
@@ -583,100 +358,660 @@
     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 add: sum.delta)
+  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 :: ('a :: {ring_no_zero_divisors}) fps) * g) = subdegree f + subdegree g"
-proof (rule subdegreeI)
-  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
+  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
-  also have "... = f $ subdegree f * g $ subdegree g" by (simp add: sum.delta)
-  also from assms have "... \<noteq> 0" by auto
-  finally show "(f * g) $ (subdegree f + subdegree g) \<noteq> 0" .
-next
-  fix m assume m: "m < subdegree f + subdegree g"
-  have "(f * g) $ m = (\<Sum>i=0..m. f$i * g$(m-i))" by (simp add: fps_mult_nth)
-  also have "... = (\<Sum>i=0..m. 0)"
-  proof (rule sum.cong)
-    fix i assume "i \<in> {0..m}"
-    with m have "i < subdegree f \<or> m - i < subdegree g" by auto
-    thus "f$i * g$(m-i) = 0" by (elim disjE) 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_head_Suc)
+      qed
+    qed    
+    thus "c = 0 \<or> a = b" by fast
   qed auto
-  finally show "(f * g) $ m = 0" by simp
-qed
+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_head_Suc[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 :: ring_1_no_zero_divisors) fps) ^ n) = n * subdegree f"
+  "subdegree ((f :: ('a :: semiring_1_no_zero_divisors) fps) ^ n) = n * subdegree f"
   by (cases "f = 0"; induction n) simp_all
 
-lemma subdegree_uminus [simp]:
-  "subdegree (-(f::('a::group_add) fps)) = subdegree f"
-  by (simp add: subdegree_def)
-
-lemma subdegree_minus_commute [simp]:
-  "subdegree (f-(g::('a::group_add) fps)) = subdegree (g - f)"
-proof -
-  have "f - g = -(g - f)" by simp
-  also have "subdegree ... = subdegree (g - f)" by (simp only: subdegree_uminus)
-  finally show ?thesis .
-qed
-
-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_geI)
-  from assms show "f + g \<noteq> 0" by (subst (asm) eq_neg_iff_add_eq_0)
-next
-  fix i assume "i < min (subdegree f) (subdegree g)"
-  hence "f $ i = 0" and "g $ i = 0" by auto
-  thus "(f + g) $ i = 0" by force
-qed
-
-lemma subdegree_add_eq1:
+
+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_head_Suc)
+  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"
-  assumes "subdegree f < subdegree (g :: ('a :: {group_add}) fps)"
-  shows   "subdegree (f + g) = subdegree f"
-proof (rule antisym[OF subdegree_leI])
-  from assms show "subdegree (f + g) \<ge> subdegree f"
-    by (intro order.trans[OF min.boundedI subdegree_add_ge]) auto
-  from assms have "f $ subdegree f \<noteq> 0" "g $ subdegree f = 0" by auto
-  thus "(f + g) $ subdegree f \<noteq> 0" by simp
-qed
-
-lemma subdegree_add_eq2:
-  assumes "g \<noteq> 0"
-  assumes "subdegree g < subdegree (f :: ('a :: {ab_group_add}) fps)"
-  shows   "subdegree (f + g) = subdegree g"
-  using subdegree_add_eq1[OF assms] by (simp add: add.commute)
-
-lemma subdegree_diff_eq1:
+  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_head_Suc)
+
+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"
-  assumes "subdegree f < subdegree (g :: ('a :: {ab_group_add}) fps)"
-  shows   "subdegree (f - g) = subdegree f"
-  using subdegree_add_eq1[of f "-g"] assms by (simp add: add.commute)
-
-lemma subdegree_diff_eq2:
-  assumes "g \<noteq> 0"
-  assumes "subdegree g < subdegree (f :: ('a :: {ab_group_add}) fps)"
-  shows   "subdegree (f - g) = subdegree g"
-  using subdegree_add_eq2[of "-g" f] assms by (simp add: add.commute)
-
-lemma subdegree_diff_ge [simp]:
-  assumes "f \<noteq> (g :: ('a :: {group_add}) fps)"
-  shows   "subdegree (f - g) \<ge> min (subdegree f) (subdegree g)"
-  using assms subdegree_add_ge[of f "-g"] by simp
-
-
+  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>
@@ -702,65 +1037,287 @@
 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) ::'a::comm_ring_1 fps)"
-  by (intro fps_ext) (auto simp: fps_shift_def )
-
-lemma fps_shift_times_fps_X_power:
-  "n \<le> subdegree f \<Longrightarrow> fps_shift n f * fps_X ^ n = (f :: 'a :: comm_ring_1 fps)"
-  by (intro fps_ext) (auto simp: 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 :: 'a :: comm_ring_1 fps)"
-  by (intro fps_ext) (auto simp: 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 :: 'a :: comm_ring_1 fps)"
-  by (intro fps_ext) (auto simp: fps_X_power_mult_right_nth nth_less_subdegree_zero)
+  "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 :: 'a :: comm_ring_1 fps) - n"
-  by (cases "f = 0") (force intro: nth_less_subdegree_zero subdegreeI)+
-
-lemma subdegree_decompose:
-  "f = fps_shift (subdegree f) f * fps_X ^ subdegree (f :: ('a :: comm_ring_1) fps)"
-  by (rule fps_ext) (auto simp: fps_X_power_mult_right_nth)
-
-lemma subdegree_decompose':
-  "n \<le> subdegree (f :: ('a :: comm_ring_1) fps) \<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)
+  "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_ring_1} fps)"
-  shows   "fps_shift n (h*g) = h * fps_shift n g"
-proof -
-  from assms have "g = fps_shift n g * fps_X^n" by (rule subdegree_decompose')
-  also have "h * ... = (h * fps_shift n g) * fps_X^n" by simp
-  also have "fps_shift n ... = h * fps_shift n g" by simp
-  finally show ?thesis .
+  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_head_Suc[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_ring_1} fps)"
+  assumes "n \<le> subdegree (g :: 'b :: comm_semiring_0 fps)"
   shows   "fps_shift n (g*h) = h * fps_shift n g"
-  by (subst mult.commute, subst fps_shift_mult) (simp_all add: assms)
-
-lemma nth_subdegree_zero_iff [simp]: "f $ subdegree f = 0 \<longleftrightarrow> f = 0"
-  by (cases "f = 0") auto
+  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)"
 
@@ -774,7 +1331,7 @@
   proof (cases "f = 0")
     assume "f \<noteq> 0"
     with A have "n \<le> subdegree f"
-      by (intro subdegree_geI) (auto simp: fps_eq_iff split: if_split_asm)
+      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)
@@ -795,13 +1352,28 @@
   by (simp add: numeral_fps_const fps_cutoff_fps_const)
 
 lemma fps_shift_cutoff:
-  "fps_shift n (f :: ('a :: comm_ring_1) fps) * fps_X^n + fps_cutoff n f = f"
+  "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 :: (comm_ring_1) dist
+instantiation fps :: ("{minus,zero}") dist
 begin
 
 definition
@@ -810,14 +1382,11 @@
 lemma dist_fps_ge0: "dist (a :: 'a fps) b \<ge> 0"
   by (simp add: dist_fps_def)
 
-lemma dist_fps_sym: "dist (a :: 'a fps) b = dist b a"
-  by (simp add: dist_fps_def)
-
 instance ..
 
 end
 
-instantiation fps :: (comm_ring_1) metric_space
+instantiation fps :: (group_add) metric_space
 begin
 
 definition uniformity_fps_def [code del]:
@@ -826,6 +1395,9 @@
 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"
@@ -863,9 +1435,9 @@
 
 end
 
-declare uniformity_Abort[where 'a="'a :: comm_ring_1 fps", code]
-
-lemma open_fps_def: "open (S :: 'a::comm_ring_1 fps set) = (\<forall>a \<in> S. \<exists>r. r >0 \<and> {y. dist y a < r} \<subseteq> S)"
+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>
@@ -902,9 +1474,8 @@
     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$n else 0::'a::comm_ring_1)"
-  by (auto simp add: fps_sum_nth cond_value_iff cong del: if_weak_cong)
+lemma fps_sum_rep_nth: "(sum (\<lambda>i. fps_const(a$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")
@@ -948,121 +1519,463 @@
 qed
 
 
-subsection \<open>Inverses of formal power series\<close>
+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
 
-fun natfun_inverse:: "'a fps \<Rightarrow> nat \<Rightarrow> 'a"
-where
-  "natfun_inverse f 0 = inverse (f$0)"
-| "natfun_inverse f n = - inverse (f$0) * sum (\<lambda>i. f$i * natfun_inverse f (n - i)) {1..n}"
-
-definition fps_inverse_def: "inverse f = (if f $ 0 = 0 then 0 else Abs_fps (natfun_inverse f))"
-
-definition fps_divide_def:
-  "f div g = (if g = 0 then 0 else
-     let n = subdegree g; h = fps_shift n g
-     in  fps_shift n (f * inverse h))"
+\<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::{comm_monoid_add,inverse,times,uminus} fps) = 0"
-  by (simp add: fps_ext fps_inverse_def)
-
-lemma fps_inverse_one [simp]: "inverse (1 :: 'a::{division_ring,zero_neq_one} fps) = 1"
-  apply (auto simp add: expand_fps_eq fps_inverse_def)
-  apply (case_tac n)
-  apply auto
-  done
+  "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_head_Suc 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 f0: "f$0 \<noteq> (0::'a::field)"
-  shows "inverse f * f = 1"
-proof -
-  have c: "inverse f * f = f * inverse f"
-    by (simp add: mult.commute)
-  from f0 have ifn: "\<And>n. inverse f $ n = natfun_inverse f n"
-    by (simp add: fps_inverse_def)
-  from f0 have th0: "(inverse f * f) $ 0 = 1"
-    by (simp add: fps_mult_nth fps_inverse_def)
-  have "(inverse f * f)$n = 0" if np: "n > 0" for n
-  proof -
-    from np have eq: "{0..n} = {0} \<union> {1 .. n}"
-      by auto
-    have d: "{0} \<inter> {1 .. n} = {}"
-      by auto
-    from f0 np have th0: "- (inverse f $ n) =
-      (sum (\<lambda>i. f$i * natfun_inverse f (n - i)) {1..n}) / (f$0)"
-      by (cases n) (simp_all add: divide_inverse fps_inverse_def)
-    from th0[symmetric, unfolded nonzero_divide_eq_eq[OF f0]]
-    have th1: "sum (\<lambda>i. f$i * natfun_inverse f (n - i)) {1..n} = - (f$0) * (inverse f)$n"
-      by (simp add: field_simps)
-    have "(f * inverse f) $ n = (\<Sum>i = 0..n. f $i * natfun_inverse f (n - i))"
-      unfolding fps_mult_nth ifn ..
-    also have "\<dots> = f$0 * natfun_inverse f n + (\<Sum>i = 1..n. f$i * natfun_inverse f (n-i))"
-      by (simp add: eq)
-    also have "\<dots> = 0"
-      unfolding th1 ifn by simp
-    finally show ?thesis unfolding c .
-  qed
-  with th0 show ?thesis
-    by (simp add: fps_eq_iff)
-qed
-
-lemma fps_inverse_0_iff[simp]: "(inverse f) $ 0 = (0::'a::division_ring) \<longleftrightarrow> f $ 0 = 0"
-  by (simp add: fps_inverse_def nonzero_imp_inverse_nonzero)
-
-lemma fps_inverse_nth_0 [simp]: "inverse f $ 0 = inverse (f $ 0 :: 'a :: division_ring)"
-  by (simp add: fps_inverse_def)
-
-lemma fps_inverse_eq_0_iff[simp]: "inverse f = (0:: ('a::division_ring) fps) \<longleftrightarrow> f $ 0 = 0"
-proof
-  assume A: "inverse f = 0"
-  have "0 = inverse f $ 0" by (subst A) simp
-  thus "f $ 0 = 0" by simp
-qed (simp add: fps_inverse_def)
+  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]:
-  assumes f0: "f$0 \<noteq> (0::'a::field)"
-  shows "inverse (inverse f) = f"
-proof -
-  from f0 have if0: "inverse f $ 0 \<noteq> 0" by simp
-  from inverse_mult_eq_1[OF f0] inverse_mult_eq_1[OF if0]
-  have "inverse f * f = inverse f * inverse (inverse f)"
-    by (simp add: ac_simps)
-  then show ?thesis
-    using f0 unfolding mult_cancel_left by 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_head_Suc)
+        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:
-  assumes fg: "(f :: 'a :: field fps) * g = 1"
+  fixes   f g :: "'a :: division_ring fps"
+  assumes fg: "f * g = 1"
   shows   "inverse f = g"
 proof -
-  have f0: "f $ 0 \<noteq> 0"
-  proof
-    assume "f $ 0 = 0"
-    hence "0 = (f * g) $ 0" by simp
-    also from fg have "(f * g) $ 0 = 1" by simp
-    finally show False by simp
-  qed
-  from inverse_mult_eq_1[OF this] fg
-  have th0: "inverse f * f = g * f"
-    by (simp add: ac_simps)
-  then show ?thesis
-    using f0
-    unfolding mult_cancel_right
-    by (auto simp add: expand_fps_eq)
-qed
-
-lemma fps_inverse_eq_0: "f$0 = 0 \<Longrightarrow> inverse (f :: 'a :: division_ring fps) = 0"
-  by simp
-  
+  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"
@@ -1088,43 +2001,648 @@
     unfolding th1
     apply (simp add: sum.union_disjoint[OF f d, unfolded eq[symmetric]] del: One_nat_def)
     unfolding th2
-    apply (simp add: sum.delta)
+    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"
-proof (cases "f$0 = 0 \<or> g$0 = 0")
-  assume "\<not>(f$0 = 0 \<or> g$0 = 0)"
-  hence [simp]: "f$0 \<noteq> 0" "g$0 \<noteq> 0" by simp_all
-  show ?thesis
-  proof (rule fps_inverse_unique)
-    have "f * g * (inverse f * inverse g) = (inverse f * f) * (inverse g * g)" by simp
-    also have "... = 1" by (subst (1 2) inverse_mult_eq_1) simp_all
-    finally show "f * g * (inverse f * inverse g) = 1" .
+  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_head_Suc)
+        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_head_Suc)
+
+  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 A: "f$0 = 0 \<or> g$0 = 0"
-  hence "inverse (f * g) = 0" by simp
-  also from A have "... = inverse f * inverse g" by auto
-  finally show "inverse (f * g) = inverse f * inverse g" .
-qed
-
-
-lemma fps_inverse_gp: "inverse (Abs_fps(\<lambda>n. (1::'a::field))) =
-    Abs_fps (\<lambda>n. if n= 0 then 1 else if n=1 then - 1 else 0)"
-  apply (rule fps_inverse_unique)
-  apply (simp_all add: fps_eq_iff fps_mult_nth sum_zero_lemma)
-  done
-
-lemma subdegree_inverse [simp]: "subdegree (inverse (f::'a::field fps)) = 0"
-proof (cases "f$0 = 0")
-  assume nz: "f$0 \<noteq> 0"
-  hence "subdegree (inverse f) + subdegree f = subdegree (inverse f * f)"
-    by (subst subdegree_mult) auto
-  also from nz have "subdegree f = 0" by (simp add: subdegree_eq_0_iff)
-  also from nz have "inverse f * f = 1" by (rule inverse_mult_eq_1)
-  finally show "subdegree (inverse f) = 0" by simp
-qed (simp_all add: fps_inverse_def)
+  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
@@ -1137,204 +2655,272 @@
   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"
-  by (rule dvd_trans, subst fps_is_unit_iff) simp_all
-
-instantiation fps :: (field) normalization_semidom
-begin
-
-definition fps_unit_factor_def [simp]:
-  "unit_factor f = fps_shift (subdegree f) f"
-
-definition fps_normalize_def [simp]:
-  "normalize f = (if f = 0 then 0 else fps_X ^ subdegree f)"
-
-instance proof
-  fix f :: "'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"
-  proof (cases "f = 0 \<or> g = 0")
-    assume "\<not>(f = 0 \<or> g = 0)"
-    thus "unit_factor (f * g) = unit_factor f * unit_factor g"
-    unfolding fps_unit_factor_def
-      by (auto simp: fps_shift_fps_shift fps_shift_mult fps_shift_mult_right)
-  qed auto
-next
-  fix f g :: "'a fps"
-  assume "g \<noteq> 0"
-  then have "f * (fps_shift (subdegree g) g * inverse (fps_shift (subdegree g) g)) = f"
-    by (metis add_cancel_right_left fps_shift_nth inverse_mult_eq_1 mult.commute mult_cancel_left2 nth_subdegree_nonzero)
-  then have "fps_shift (subdegree g) (g * (f * inverse (fps_shift (subdegree g) g))) = f"
-    by (simp add: fps_shift_mult_right mult.commute)
-  with \<open>g \<noteq> 0\<close> show "f * g / g = f"
-    by (simp add: fps_divide_def Let_def ac_simps)
-qed (auto simp add: fps_divide_def Let_def)
-
-end
-
-instantiation fps :: (field) idom_modulo
-begin
-
-definition fps_mod_def:
-  "f mod g = (if g = 0 then f else
-     let n = subdegree g; h = fps_shift n g
-     in  fps_cutoff n (f * inverse h) * h)"
-
-lemma fps_mod_eq_zero:
-  assumes "g \<noteq> 0" and "subdegree f \<ge> subdegree g"
-  shows   "f mod g = 0"
-  using assms by (cases "f = 0") (auto simp: fps_cutoff_zero_iff fps_mod_def Let_def)
-
-lemma fps_times_divide_eq:
-  assumes "g \<noteq> 0" and "subdegree f \<ge> subdegree (g :: 'a fps)"
-  shows   "f div g * g = f"
-proof (cases "f = 0")
-  assume nz: "f \<noteq> 0"
-  define n where "n = subdegree g"
-  define h where "h = fps_shift n g"
-  from assms have [simp]: "h $ 0 \<noteq> 0" unfolding h_def by (simp add: n_def)
-
-  from assms nz have "f div g * g = fps_shift n (f * inverse h) * g"
-    by (simp add: fps_divide_def Let_def h_def n_def)
-  also have "... = fps_shift n (f * inverse h) * fps_X^n * h" unfolding h_def n_def
-    by (subst subdegree_decompose[of g]) simp
-  also have "fps_shift n (f * inverse h) * fps_X^n = f * inverse h"
-    by (rule fps_shift_times_fps_X_power) (simp_all add: nz assms n_def)
-  also have "... * h = f * (inverse h * h)" by simp
-  also have "inverse h * h = 1" by (rule inverse_mult_eq_1) simp
-  finally show ?thesis by simp
-qed (simp_all add: fps_divide_def Let_def)
-
-lemma
-  assumes "g$0 \<noteq> 0"
-  shows   fps_divide_unit: "f div g = f * inverse g" and fps_mod_unit [simp]: "f mod g = 0"
-proof -
-  from assms have [simp]: "subdegree g = 0" by (simp add: subdegree_eq_0_iff)
-  from assms show "f div g = f * inverse g"
-    by (auto simp: fps_divide_def Let_def subdegree_eq_0_iff)
-  from assms show "f mod g = 0" by (intro fps_mod_eq_zero) auto
-qed
-
-instance proof
-  fix f g :: "'a fps"
-  define n where "n = subdegree g"
-  define h where "h = fps_shift n g"
-
-  show "f div g * g + f mod g = f"
-  proof (cases "g = 0 \<or> f = 0")
-    assume "\<not>(g = 0 \<or> f = 0)"
-    hence nz [simp]: "f \<noteq> 0" "g \<noteq> 0" by simp_all
-    show ?thesis
-    proof (rule disjE[OF le_less_linear])
-      assume "subdegree f \<ge> subdegree g"
-      with nz show ?thesis by (simp add: fps_mod_eq_zero fps_times_divide_eq)
-    next
-      assume "subdegree f < subdegree g"
-      have g_decomp: "g = h * fps_X^n" unfolding h_def n_def by (rule subdegree_decompose)
-      have "f div g * g + f mod g =
-              fps_shift n (f * inverse h) * g + fps_cutoff n (f * inverse h) * h"
-        by (simp add: fps_mod_def fps_divide_def Let_def n_def h_def)
-      also have "... = h * (fps_shift n (f * inverse h) * fps_X^n + fps_cutoff n (f * inverse h))"
-        by (subst g_decomp) (simp add: algebra_simps)
-      also have "... = f * (inverse h * h)"
-        by (subst fps_shift_cutoff) simp
-      also have "inverse h * h = 1" by (rule inverse_mult_eq_1) (simp add: h_def n_def)
-      finally show ?thesis by simp
-    qed
-  qed (auto simp: fps_mod_def fps_divide_def Let_def)
-qed
-
-end
-
-lemma subdegree_mod:
-  assumes "f \<noteq> 0" "subdegree f < subdegree g"
-  shows   "subdegree (f mod g) = subdegree f"
-proof (cases "f div g * g = 0")
-  assume "f div g * g \<noteq> 0"
-  hence [simp]: "f div g \<noteq> 0" "g \<noteq> 0" by auto
-  from div_mult_mod_eq[of f g] have "f mod g = f - f div g * g" by (simp add: algebra_simps)
-  also from assms have "subdegree ... = subdegree f"
-    by (intro subdegree_diff_eq1) simp_all
-  finally show ?thesis .
-next
-  assume zero: "f div g * g = 0"
-  from div_mult_mod_eq[of f g] have "f mod g = f - f div g * g" by (simp add: algebra_simps)
-  also note zero
-  finally show ?thesis by simp
-qed
-
-lemma fps_divide_nth_0 [simp]: "g $ 0 \<noteq> 0 \<Longrightarrow> (f div g) $ 0 = f $ 0 / (g $ 0 :: _ :: field)"
-  by (simp add: fps_divide_unit divide_inverse)
-
+  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 :: 'a :: idom fps) dvd g \<Longrightarrow> g \<noteq> 0 \<Longrightarrow> subdegree f \<le> subdegree g"
-  by (auto elim: dvdE)
+  "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 have "g mod f = 0"
-    by (simp add: fps_mod_def Let_def fps_cutoff_zero_iff)
-  thus "f dvd g" by (simp add: dvd_eq_mod_eq_0)
+  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 fps_shift_altdef:
-  "fps_shift n f = (f :: 'a :: field fps) div fps_X^n"
-  by (simp add: fps_divide_def)
-  
-lemma fps_div_fps_X_power_nth: "((f :: 'a :: field fps) div fps_X^n) $ k = f $ (k + n)"
-  by (simp add: fps_shift_altdef [symmetric])
-
-lemma fps_div_fps_X_nth: "((f :: 'a :: field fps) div fps_X) $ k = f $ Suc k"
-  using fps_div_fps_X_power_nth[of f 1] by simp
-
-lemma fps_const_inverse: "inverse (fps_const (a::'a::field)) = fps_const (inverse a)"
-  by (cases "a \<noteq> 0", rule fps_inverse_unique) (auto simp: fps_eq_iff)
-
-lemma fps_const_divide: "fps_const (x :: _ :: field) / fps_const y = fps_const (x / y)"
-  by (cases "y = 0") (simp_all add: fps_divide_unit fps_const_inverse divide_inverse)
-
-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 fps_numeral_divide_divide:
-  "x / numeral b / numeral c = (x / numeral (b * c) :: 'a :: field fps)"
-  by (cases "numeral b = (0::'a)"; cases "numeral c = (0::'a)")
-      (simp_all add: fps_divide_unit fps_inverse_mult [symmetric] numeral_fps_const numeral_mult 
-                del: numeral_mult [symmetric])
-
-lemma fps_numeral_mult_divide:
-  "numeral b * x / numeral c = (numeral b / numeral c * x :: 'a :: field fps)"
-  by (cases "numeral c = (0::'a)") (simp_all add: fps_divide_unit numeral_fps_const)
-
-lemmas fps_numeral_simps = 
-  fps_numeral_divide_divide fps_numeral_mult_divide inverse_fps_numeral neg_numeral_fps_const
+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:
-  assumes "q dvd p"
-  shows   "subdegree ((p :: 'a :: field fps) div q) = subdegree p - subdegree q"
-proof (cases "p = 0")
-  case False
-  from assms have "p = p div q * q" by simp
-  also from assms False have "subdegree \<dots> = subdegree (p div q) + subdegree q"
-    by (intro subdegree_mult) (auto simp: dvd_div_eq_0_iff)
-  finally show ?thesis by simp
-qed simp_all
+  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 :: 'a :: field fps) div q) = subdegree p"
-  using assms by (subst subdegree_div) simp_all
+  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
+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"
+  show "unit_factor (f * g) = unit_factor f * unit_factor g"
+    using fps_unit_factor_mult by simp
+  show "unit_factor f * normalize f = f"
+    by (simp add: fps_shift_times_fps_X_power)
+qed (simp_all add: fps_divide_def Let_def)
+
+end
 
 
 subsection \<open>Formal power series form a Euclidean ring\<close>
@@ -1345,80 +2931,22 @@
 definition fps_euclidean_size_def:
   "euclidean_size f = (if f = 0 then 0 else 2 ^ subdegree f)"
 
-context
-begin
-
-private lemma fps_divide_cancel_aux1:
-  assumes "h$0 \<noteq> (0 :: 'a :: field)"
-  shows   "(h * f) div (h * g) = f div g"
-proof (cases "g = 0")
-  assume "g \<noteq> 0"
-  from assms have "h \<noteq> 0" by auto
-  note nz [simp] = \<open>g \<noteq> 0\<close> \<open>h \<noteq> 0\<close>
-  from assms have [simp]: "subdegree h = 0" by (simp add: subdegree_eq_0_iff)
-
-  have "(h * f) div (h * g) =
-          fps_shift (subdegree g) (h * f * inverse (fps_shift (subdegree g) (h*g)))"
-    by (simp add: fps_divide_def Let_def)
-  also have "h * f * inverse (fps_shift (subdegree g) (h*g)) =
-               (inverse h * h) * f * inverse (fps_shift (subdegree g) g)"
-    by (subst fps_shift_mult) (simp_all add: algebra_simps fps_inverse_mult)
-  also from assms have "inverse h * h = 1" by (rule inverse_mult_eq_1)
-  finally show "(h * f) div (h * g) = f div g" by (simp_all add: fps_divide_def Let_def)
-qed (simp_all add: fps_divide_def)
-
-private lemma fps_divide_cancel_aux2:
-  "(f * fps_X^m) div (g * fps_X^m) = f div (g :: 'a :: field fps)"
-proof (cases "g = 0")
-  assume [simp]: "g \<noteq> 0"
-  have "(f * fps_X^m) div (g * fps_X^m) =
-          fps_shift (subdegree g + m) (f*inverse (fps_shift (subdegree g + m) (g*fps_X^m))*fps_X^m)"
-    by (simp add: fps_divide_def Let_def algebra_simps)
-  also have "... = f div g"
-    by (simp add: fps_shift_times_fps_X_power'' fps_divide_def Let_def)
-  finally show ?thesis .
-qed (simp_all add: fps_divide_def)
-
 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") (auto simp: fps_euclidean_size_def)
+    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
-  show "(h * f) div (h * g) = f div g" if "h \<noteq> 0"
-    for f g h :: "'a fps"
-  proof -
-    define m where "m = subdegree h"
-    define h' where "h' = fps_shift m h"
-    have h_decomp: "h = h' * fps_X ^ m" unfolding h'_def m_def by (rule subdegree_decompose)
-    from \<open>h \<noteq> 0\<close> have [simp]: "h'$0 \<noteq> 0" by (simp add: h'_def m_def)
-    have "(h * f) div (h * g) = (h' * f * fps_X^m) div (h' * g * fps_X^m)"
-      by (simp add: h_decomp algebra_simps)
-    also have "... = f div g"
-      by (simp add: fps_divide_cancel_aux1 fps_divide_cancel_aux2)
-    finally show ?thesis .
-  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"
-    if "h \<noteq> 0" for f g h :: "'a fps"
-  proof -
-    define n h' where dfs: "n = subdegree h" "h' = fps_shift n h"
-    have "(f + g * h) div h = fps_shift n (f * inverse h') + fps_shift n (g * (h * inverse h'))"
-      by (simp add: fps_divide_def Let_def dfs [symmetric] algebra_simps fps_shift_add that)
-    also have "h * inverse h' = (inverse h' * h') * fps_X^n"
-      by (subst subdegree_decompose) (simp_all add: dfs)
-    also have "... = fps_X^n"
-      by (subst inverse_mult_eq_1) (simp_all add: dfs that)
-    also have "fps_shift n (g * fps_X^n) = g" by simp
-    also have "fps_shift n (f * inverse h') = f div h"
-      by (simp add: fps_divide_def Let_def dfs)
-    finally show ?thesis by simp
-  qed
-qed (simp_all add: fps_euclidean_size_def)
-
-end
+    by (simp add: fps_divide_add fps_divide_times_eq)
+qed (simp add: fps_euclidean_size_def)
 
 end
 
@@ -1441,11 +2969,11 @@
   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") (auto simp: fps_dvd_iff)
+    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 :: 'a :: field fps) g =
+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
@@ -1460,11 +2988,11 @@
   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") (auto simp: fps_dvd_iff)
+    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 :: 'a :: field fps) g =
+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)
 
@@ -1480,11 +3008,11 @@
   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) (auto simp: fps_dvd_iff[symmetric])
+    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 :: 'a :: field fps set) =
+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
 
@@ -1510,7 +3038,7 @@
 qed simp_all
 
 lemma fps_Lcm_altdef:
-  "Lcm (A :: 'a :: field fps set) =
+  "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)")
@@ -1533,70 +3061,65 @@
 
 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)"
+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 :: 'a :: {comm_ring_1, semiring_char_0})"
-  by (induction n arbitrary: f) (simp_all del: funpow.simps add: funpow_Suc_right algebra_simps)
-
-lemma fps_deriv_linear[simp]:
-  "fps_deriv (fps_const (a::'a::comm_semiring_1) * f + fps_const b * g) =
-    fps_const a * fps_deriv f + fps_const b * fps_deriv g"
-  unfolding fps_eq_iff fps_add_nth  fps_const_mult_left fps_deriv_nth by (simp add: field_simps)
+  "(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]:
-  fixes f :: "'a::comm_ring_1 fps"
-  shows "fps_deriv (f * g) = f * fps_deriv g + fps_deriv f * g"
-proof -
-  let ?D = "fps_deriv"
-  have "(f * ?D g + ?D f * g) $ n = ?D (f*g) $ n" for n
-  proof -
-    let ?Zn = "{0 ..n}"
-    let ?Zn1 = "{0 .. n + 1}"
-    let ?g = "\<lambda>i. of_nat (i+1) * g $ (i+1) * f $ (n - i) +
-        of_nat (i+1)* f $ (i+1) * g $ (n - i)"
-    let ?h = "\<lambda>i. of_nat i * g $ i * f $ ((n+1) - i) +
-        of_nat i* f $ i * g $ ((n + 1) - i)"
-    have s0: "sum (\<lambda>i. of_nat i * f $ i * g $ (n + 1 - i)) ?Zn1 =
-      sum (\<lambda>i. of_nat (n + 1 - i) * f $ (n + 1 - i) * g $ i) ?Zn1"
-       by (rule sum.reindex_bij_witness[where i="(-) (n + 1)" and j="(-) (n + 1)"]) auto
-    have s1: "sum (\<lambda>i. f $ i * g $ (n + 1 - i)) ?Zn1 =
-      sum (\<lambda>i. f $ (n + 1 - i) * g $ i) ?Zn1"
-       by (rule sum.reindex_bij_witness[where i="(-) (n + 1)" and j="(-) (n + 1)"]) auto
-    have "(f * ?D g + ?D f * g)$n = (?D g * f + ?D f * g)$n"
-      by (simp only: mult.commute)
-    also have "\<dots> = (\<Sum>i = 0..n. ?g i)"
-      by (simp add: fps_mult_nth sum.distrib[symmetric])
-    also have "\<dots> = sum ?h {0..n+1}"
-      by (rule sum.reindex_bij_witness_not_neutral
-            [where S'="{}" and T'="{0}" and j="Suc" and i="\<lambda>i. i - 1"]) auto
-    also have "\<dots> = (fps_deriv (f * g)) $ n"
-      apply (simp only: fps_deriv_nth fps_mult_nth sum.distrib)
-      unfolding s0 s1
-      unfolding sum.distrib[symmetric] sum_distrib_left
-      apply (rule sum.cong)
-      apply (auto simp add: of_nat_diff field_simps)
-      done
-    finally show ?thesis .
+  "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_head_Suc 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
-  then show ?thesis
-    unfolding fps_eq_iff by auto
+  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_head_Suc)
+  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::comm_ring_1 fps)) = - (fps_deriv f)"
+  "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:: 'a::comm_ring_1 fps) + g) = fps_deriv f + fps_deriv g"
-  using fps_deriv_linear[of 1 f 1 g] by simp
+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::comm_ring_1 fps) - g) = fps_deriv f - fps_deriv g"
+  "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"
@@ -1605,25 +3128,33 @@
 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::'a::comm_ring_1) * f) = fps_const c * fps_deriv f"
+  "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 )
+  by (simp add: fps_deriv_def fps_eq_iff)
 
 lemma fps_deriv_mult_const_right[simp]:
-  "fps_deriv (f * fps_const (c::'a::comm_ring_1)) = fps_deriv f * fps_const c"
+  "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 :: 'a::comm_ring_1 fps)) S"
+  "fps_deriv (sum f S) = sum (\<lambda>i. fps_deriv (f i)) S"
 proof (cases "finite S")
   case False
   then show ?thesis by simp
@@ -1633,35 +3164,28 @@
 qed
 
 lemma fps_deriv_eq_0_iff [simp]:
-  "fps_deriv f = 0 \<longleftrightarrow> f = fps_const (f$0 :: 'a::{idom,semiring_char_0})"
-  (is "?lhs \<longleftrightarrow> ?rhs")
+  "fps_deriv f = 0 \<longleftrightarrow> f = fps_const (f$0 :: 'a::{semiring_no_zero_divisors,semiring_char_0})"
 proof
-  show ?lhs if ?rhs
-  proof -
-    from that have "fps_deriv f = fps_deriv (fps_const (f$0))"
-      by simp
-    then show ?thesis
-      by simp
+  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
-  show ?rhs if ?lhs
-  proof -
-    from that have "\<forall>n. (fps_deriv f)$n = 0"
-      by simp
-    then have "\<forall>n. f$(n+1) = 0"
-      by (simp del: of_nat_Suc of_nat_add One_nat_def)
-    then show ?thesis
-      apply (clarsimp simp add: fps_eq_iff fps_const_def)
-      apply (erule_tac x="n - 1" in allE)
-      apply simp
-      done
-  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 :: "'a::{idom,semiring_char_0} fps"
+  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 ..
@@ -1670,8 +3194,9 @@
 qed
 
 lemma fps_deriv_eq_iff_ex:
-  "(fps_deriv f = fps_deriv g) \<longleftrightarrow> (\<exists>c::'a::{idom,semiring_char_0}. f = fps_const c + g)"
-  by (auto simp: 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> (\<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"
@@ -1683,20 +3208,20 @@
   by (induct n arbitrary: f) auto
 
 lemma fps_nth_deriv_linear[simp]:
-  "fps_nth_deriv n (fps_const (a::'a::comm_semiring_1) * f + fps_const b * g) =
+  "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 simp add: fps_nth_deriv_commute)
+  by (induct n arbitrary: f g) auto
 
 lemma fps_nth_deriv_neg[simp]:
-  "fps_nth_deriv n (- (f :: 'a::comm_ring_1 fps)) = - (fps_nth_deriv n f)"
+  "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::comm_ring_1 fps) + g) = fps_nth_deriv n f + fps_nth_deriv n g"
+  "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::comm_ring_1 fps) - g) = fps_nth_deriv n f - fps_nth_deriv n g"
+  "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"
@@ -1710,15 +3235,15 @@
   by (cases n) simp_all
 
 lemma fps_nth_deriv_mult_const_left[simp]:
-  "fps_nth_deriv n (fps_const (c::'a::comm_ring_1) * f) = fps_const c * fps_nth_deriv n f"
+  "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::'a::comm_ring_1)) = fps_nth_deriv n f * fps_const c"
-  using fps_nth_deriv_linear[of n "c" f 0 0] by (simp add: mult.commute)
+  "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::comm_ring_1 fps)) S"
+  "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
@@ -1729,152 +3254,55 @@
 
 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) (auto simp add: field_simps)
-
-
-subsection \<open>Powers\<close>
-
-lemma fps_power_zeroth_eq_one: "a$0 =1 \<Longrightarrow> a^n $ 0 = (1::'a::semiring_1)"
-  by (induct n) (auto simp add: expand_fps_eq fps_mult_nth)
-
-lemma fps_power_first_eq: "(a :: 'a::comm_ring_1 fps) $ 0 =1 \<Longrightarrow> a^n $ 1 = of_nat n * a$1"
-proof (induct n)
-  case 0
-  then show ?case by simp
-next
-  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: field_simps)
-qed
-
-lemma startsby_one_power:"a $ 0 = (1::'a::comm_ring_1) \<Longrightarrow> a^n $ 0 = 1"
-  by (induct n) (auto simp add: fps_mult_nth)
-
-lemma startsby_zero_power:"a $0 = (0::'a::comm_ring_1) \<Longrightarrow> n > 0 \<Longrightarrow> a^n $0 = 0"
-  by (induct n) (auto simp add: fps_mult_nth)
-
-lemma startsby_power:"a $0 = (v::'a::comm_ring_1) \<Longrightarrow> a^n $0 = v^n"
-  by (induct n) (auto simp add: fps_mult_nth)
-
-lemma startsby_zero_power_iff[simp]: "a^n $0 = (0::'a::idom) \<longleftrightarrow> n \<noteq> 0 \<and> a$0 = 0"
-  apply (rule iffI)
-  apply (induct n)
-  apply (auto simp add: fps_mult_nth)
-  apply (rule startsby_zero_power, simp_all)
-  done
-
-lemma startsby_zero_power_prefix:
-  assumes a0: "a $ 0 = (0::'a::idom)"
-  shows "\<forall>n < k. a ^ k $ n = 0"
-  using a0
-proof (induct k rule: nat_less_induct)
-  fix k
-  assume H: "\<forall>m<k. a $0 =  0 \<longrightarrow> (\<forall>n<m. a ^ m $ n = 0)" and a0: "a $ 0 = 0"
-  show "\<forall>m<k. a ^ k $ m = 0"
-  proof (cases k)
-    case 0
-    then show ?thesis by simp
-  next
-    case (Suc l)
-    have "a^k $ m = 0" if mk: "m < k" for m
-    proof (cases "m = 0")
-      case True
-      then show ?thesis
-        using startsby_zero_power[of a k] Suc a0 by simp
-    next
-      case False
-      have "a ^k $ m = (a^l * a) $m"
-        by (simp add: Suc mult.commute)
-      also have "\<dots> = (\<Sum>i = 0..m. a ^ l $ i * a $ (m - i))"
-        by (simp add: fps_mult_nth)
-      also have "\<dots> = 0"
-        apply (rule sum.neutral)
-        apply auto
-        apply (case_tac "x = m")
-        using a0 apply simp
-        apply (rule H[rule_format])
-        using a0 Suc mk apply auto
-        done
-      finally show ?thesis .
-    qed
-    then show ?thesis by blast
-  qed
-qed
-
-lemma startsby_zero_sum_depends:
-  assumes a0: "a $0 = (0::'a::idom)"
-    and kn: "n \<ge> k"
-  shows "sum (\<lambda>i. (a ^ i)$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
-
-lemma startsby_zero_power_nth_same:
-  assumes a0: "a$0 = (0::'a::idom)"
-  shows "a^n $ n = (a$1) ^ n"
-proof (induct n)
-  case 0
-  then show ?case by simp
-next
-  case (Suc n)
-  have "a ^ Suc n $ (Suc n) = (a^n * a)$(Suc n)"
-    by (simp add: field_simps)
-  also have "\<dots> = sum (\<lambda>i. a^n$i * a $ (Suc n - i)) {0.. Suc n}"
-    by (simp add: fps_mult_nth)
-  also have "\<dots> = sum (\<lambda>i. a^n$i * a $ (Suc n - i)) {n .. Suc n}"
-    apply (rule sum.mono_neutral_right)
-    apply simp
-    apply clarsimp
-    apply clarsimp
-    apply (rule startsby_zero_power_prefix[rule_format, OF a0])
-    apply arith
-    done
-  also have "\<dots> = a^n $ n * a$1"
-    using a0 by simp
-  finally show ?case
-    using Suc.hyps by simp
-qed
-
-lemma fps_inverse_power:
-  fixes a :: "'a::field fps"
-  shows "inverse (a^n) = inverse a ^ n"
-  by (induction n) (simp_all add: fps_inverse_mult)
-
-lemma fps_deriv_power:
-  "fps_deriv (a ^ n) = fps_const (of_nat n :: 'a::comm_ring_1) * fps_deriv a * a ^ (n - 1)"
-  apply (induct n)
-  apply (auto simp add: field_simps fps_const_add[symmetric] simp del: fps_const_add)
-  apply (case_tac n)
-  apply (auto simp add: field_simps)
-  done
+  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 a0: "a$0 \<noteq> 0"
-  shows "fps_deriv (inverse a) = - fps_deriv a * (inverse a)\<^sup>2"
-proof -
-  from inverse_mult_eq_1[OF a0]
-  have "fps_deriv (inverse a * a) = 0" by simp
-  then have "inverse a * fps_deriv a + fps_deriv (inverse a) * a = 0"
-    by simp
-  then have "inverse a * (inverse a * fps_deriv a + fps_deriv (inverse a) * a) = 0"
-    by simp
-  with inverse_mult_eq_1[OF a0]
-  have "(inverse a)\<^sup>2 * fps_deriv a + fps_deriv (inverse a) = 0"
-    unfolding power2_eq_square
-    apply (simp add: field_simps)
-    apply (simp add: mult.assoc[symmetric])
-    done
-  then have "(inverse a)\<^sup>2 * fps_deriv a + fps_deriv (inverse a) - fps_deriv a * (inverse a)\<^sup>2 =
-      0 - fps_deriv a * (inverse a)\<^sup>2"
-    by simp
-  then show "fps_deriv (inverse a) = - fps_deriv a * (inverse a)\<^sup>2"
-    by (simp add: field_simps)
-qed
+  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"
@@ -1883,18 +3311,7 @@
   using fps_inverse_deriv[OF a0] a0
   by (simp add: fps_divide_unit power2_eq_square fps_inverse_mult)
 
-lemma inverse_mult_eq_1':
-  assumes f0: "f$0 \<noteq> (0::'a::field)"
-  shows "f * inverse f = 1"
-  by (metis mult.commute inverse_mult_eq_1 f0)
-
-lemma fps_inverse_minus [simp]: "inverse (-f) = -inverse (f :: 'a :: field fps)"
-  by (cases "f$0 = 0") (auto intro: fps_inverse_unique simp: inverse_mult_eq_1' fps_inverse_eq_0)
-  
-lemma divide_fps_const [simp]: "f / fps_const (c :: 'a :: field) = fps_const (inverse c) * f"
-  by (cases "c = 0") (simp_all add: fps_divide_unit fps_const_inverse)
-
-(* FIfps_XME: The last part of this proof should go through by simp once we have a proper
+(* 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)"
@@ -1906,58 +3323,367 @@
   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") (auto simp: eq_divide_imp)
-qed
-
-lemma fps_inverse_gp': "inverse (Abs_fps (\<lambda>n. 1::'a::field)) = 1 - fps_X"
-  by (simp add: fps_inverse_gp fps_eq_iff fps_X_def)
-
-lemma fps_one_over_one_minus_fps_X_squared:
-  "inverse ((1 - fps_X)^2 :: 'a :: field fps) = Abs_fps (\<lambda>n. of_nat (n+1))"
-proof -
-  have "inverse ((1 - fps_X)^2 :: 'a fps) = fps_deriv (inverse (1 - fps_X))"
-    by (subst fps_inverse_deriv) (simp_all add: fps_inverse_power)
-  also have "inverse (1 - fps_X :: 'a fps) = Abs_fps (\<lambda>_. 1)"
-    by (subst fps_inverse_gp' [symmetric]) simp
-  also have "fps_deriv \<dots> = Abs_fps (\<lambda>n. of_nat (n + 1))"
-    by (simp add: fps_deriv_def)
-  finally show ?thesis .
+  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
 
-lemma fps_inverse_fps_X_plus1: "inverse (1 + fps_X) = Abs_fps (\<lambda>n. (- (1::'a::field)) ^ n)"
-  (is "_ = ?r")
-proof -
-  have eq: "(1 + fps_X) * ?r = 1"
-    unfolding minus_one_power_iff
-    by (auto simp add: field_simps fps_eq_iff)
-  show ?thesis
-    by (auto simp add: eq intro: fps_inverse_unique)
-qed
+
+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_head_Suc)
+  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
+
+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_head_Suc[of 0 "Suc n" "\<lambda>i. a $ i * a ^ n $ (Suc n - i)"]
+          sum_head_Suc[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::field_char_0 fps \<Rightarrow> 'a \<Rightarrow> 'a fps"
-  where "fps_integral a a0 = Abs_fps (\<lambda>n. if n = 0 then a0 else (a$(n - 1) / of_nat n))"
-
-lemma fps_deriv_fps_integral: "fps_deriv (fps_integral a a0) = a"
-  unfolding fps_integral_def fps_deriv_def
-  by (simp add: fps_eq_iff del: of_nat_Suc)
+definition fps_integral :: "'a::{semiring_1,inverse} fps \<Rightarrow> 'a \<Rightarrow> 'a fps"
+  where "fps_integral a a0 =
+          Abs_fps (\<lambda>n. if n=0 then a0 else inverse (of_nat n) * a$(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"
-  (is "?l = ?r")
-proof -
-  have "fps_deriv ?l = fps_deriv ?r"
-    by (simp add: fps_deriv_fps_integral)
-  moreover have "?l$0 = ?r$0"
-    by (simp add: fps_integral_def)
-  ultimately show ?thesis
-    unfolding fps_deriv_eq_iff by auto
+  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
 
 
@@ -1973,10 +3699,10 @@
   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 sum.delta')
+  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 sum.delta)
+  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
@@ -1985,7 +3711,7 @@
   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 sum.delta not_le)
+  by (simp add: fps_eq_iff fps_compose_def mult_delta_left not_le)
 
 
 subsection \<open>Rules from Herbert Wilf's Generatingfunctionology\<close>
@@ -2070,56 +3796,35 @@
 subsubsection \<open>Rule 5 --- summation and "division" by (1 - fps_X)\<close>
 
 lemma fps_divide_fps_X_minus1_sum_lemma:
-  "a = ((1::'a::comm_ring_1 fps) - fps_X) * Abs_fps (\<lambda>n. sum (\<lambda>i. a $ i) {0..n})"
-proof -
-  let ?sa = "Abs_fps (\<lambda>n. sum (\<lambda>i. a $ i) {0..n})"
-  have th0: "\<And>i. (1 - (fps_X::'a fps)) $ i = (if i = 0 then 1 else if i = 1 then - 1 else 0)"
-    by simp
-  have "a$n = ((1 - fps_X) * ?sa) $ n" for n
-  proof (cases "n = 0")
-    case True
-    then show ?thesis
-      by (simp add: fps_mult_nth)
-  next
-    case False
-    then have u: "{0} \<union> ({1} \<union> {2..n}) = {0..n}" "{1} \<union> {2..n} = {1..n}"
-      "{0..n - 1} \<union> {n} = {0..n}"
-      by (auto simp: set_eq_iff)
-    have d: "{0} \<inter> ({1} \<union> {2..n}) = {}" "{1} \<inter> {2..n} = {}" "{0..n - 1} \<inter> {n} = {}"
-      using False by simp_all
-    have f: "finite {0}" "finite {1}" "finite {2 .. n}"
-      "finite {0 .. n - 1}" "finite {n}" by simp_all
-    have "((1 - fps_X) * ?sa) $ n = sum (\<lambda>i. (1 - fps_X)$ i * ?sa $ (n - i)) {0 .. n}"
-      by (simp add: fps_mult_nth)
-    also have "\<dots> = a$n"
-      unfolding th0
-      unfolding sum.union_disjoint[OF f(1) finite_UnI[OF f(2,3)] d(1), unfolded u(1)]
-      unfolding sum.union_disjoint[OF f(2) f(3) d(2)]
-      apply (simp)
-      unfolding sum.union_disjoint[OF f(4,5) d(3), unfolded u(3)]
-      apply simp
-      done
-    finally show ?thesis
-      by simp
-  qed
-  then show ?thesis
-    unfolding fps_eq_iff by blast
+  "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_head_Suc[of 0 "Suc m" "\<lambda>i. f $ i * g $ (Suc m - i)"]
+            sum_head_Suc[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::field fps) - fps_X) = Abs_fps (\<lambda>n. sum (\<lambda>i. a $ i) {0..n})"
-proof -
-  let ?fps_X = "1 - (fps_X::'a fps)"
-  have th0: "?fps_X $ 0 \<noteq> 0"
-    by simp
-  have "a /?fps_X = ?fps_X *  Abs_fps (\<lambda>n::nat. sum (($) a) {0..n}) * inverse ?fps_X"
-    using fps_divide_fps_X_minus1_sum_lemma[of a, symmetric] th0
-    by (simp add: fps_divide_def mult.assoc)
-  also have "\<dots> = (inverse ?fps_X * ?fps_X) * Abs_fps (\<lambda>n::nat. sum (($) a) {0..n}) "
-    by (simp add: ac_simps)
-  finally show ?thesis
-    by (simp add: inverse_mult_eq_1[OF th0])
-qed
+  "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
@@ -2292,7 +3997,7 @@
       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)
+    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"
@@ -2373,22 +4078,7 @@
     (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)
 
-lemma fps_nth_power_0:
-  fixes m :: nat
-    and a :: "'a::comm_ring_1 fps"
-  shows "(a ^m)$0 = (a$0) ^ m"
-proof (cases m)
-  case 0
-  then show ?thesis by simp
-next
-  case (Suc n)
-  then have c: "m = card {0..n}" by simp
-  have "(a ^m)$0 = prod (\<lambda>i. a$0) {0..n}"
-    by (simp add: Suc fps_power_nth del: replicate.simps power_Suc)
-  also have "\<dots> = (a$0) ^ m"
-   unfolding c by (rule prod_constant)
- finally show ?thesis .
-qed
+lemmas fps_nth_power_0 = fps_power_zeroth
 
 lemma natpermute_max_card:
   assumes n0: "n \<noteq> 0"
@@ -2495,7 +4185,7 @@
         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) (auto simp: natpermute_def)
+        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" 
@@ -2609,12 +4299,14 @@
   {
     show "wf ?R" by auto
   next
-    fix r k a n xs i
+    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 (auto simp add: in_set_conv_nth natpermute_def)
+        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
@@ -2637,7 +4329,9 @@
       apply (metis not_less)
       done
   next
-    fix r k a n
+    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
@@ -2868,7 +4562,7 @@
           have False if c: "n \<le> xs ! i"
           proof -
             from xs i have "xs ! i \<noteq> n"
-              by (auto simp add: in_set_conv_nth natpermute_def)
+              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
@@ -2938,15 +4632,15 @@
     by metis
 qed
 
-lemma fps_deriv_radical:
+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)"
+    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 = "(fps_const (of_nat (Suc k)) * ?r ^ k)"
+  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"
@@ -2957,7 +4651,7 @@
   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)
+    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"
@@ -2965,6 +4659,15 @@
   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"
@@ -3035,9 +4738,6 @@
 qed
 *)
 
-lemma fps_divide_1 [simp]: "(a :: 'a::field fps) / 1 = a"
-  by (fact div_by_1)
-
 lemma radical_divide:
   fixes a :: "'a::field_char_0 fps"
   assumes kp: "k > 0"
@@ -3114,11 +4814,11 @@
       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)$(n - j)) {0..n})) {1.. Suc n}"
       apply (rule sum.mono_neutral_right)
-      apply (auto simp add: mult_delta_left sum.delta not_le)
+      apply (auto simp add: mult_delta_left not_le)
       done
     also have "\<dots> = sum (\<lambda>i. of_nat (i + 1) * a$(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]) (auto simp add: mult.assoc)
+      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}" .
 
@@ -3148,33 +4848,16 @@
   then show ?thesis by (simp add: fps_eq_iff)
 qed
 
-lemma fps_mult_fps_X_plus_1_nth:
-  "((1+fps_X)*a) $n = (if n = 0 then (a$n :: 'a::comm_ring_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$n :: 'a::comm_ring_1) else a$n + a$(n - 1))"
-    by (simp add: Suc)
-  finally show ?thesis .
-qed
-
 
 subsection \<open>Finite FPS (i.e. polynomials) and fps_X\<close>
 
 lemma fps_poly_sum_fps_X:
-  assumes "\<forall>i > n. a$i = (0::'a::comm_ring_1)"
+  assumes "\<forall>i > n. a$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 sum.delta' assms)
+    by (simp add: mult_delta_right assms)
   then show ?thesis
     unfolding fps_eq_iff by blast
 qed
@@ -3269,13 +4952,13 @@
   done
 
 lemma fps_compose_1[simp]: "1 oo a = 1"
-  by (simp add: fps_eq_iff fps_compose_nth mult_delta_left sum.delta)
+  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 (auto simp add: fps_eq_iff fps_compose_nth power_0_left sum.neutral)
+  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)
@@ -3312,7 +4995,7 @@
     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 (auto simp add: subset_eq)
+  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
@@ -3451,7 +5134,7 @@
   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 sum.delta)
+  by (simp add: fps_eq_iff fps_compose_nth mult_delta_left)
 
 lemma fps_inverse_compose:
   assumes b0: "(b$0 :: 'a::field) = 0"
@@ -3498,9 +5181,6 @@
     fps_compose_1 fps_compose_sub_distrib fps_X_fps_compose_startby0[OF a0] ..
 qed
 
-lemma fps_const_power [simp]: "fps_const (c::'a::ring_1) ^ n = fps_const (c^n)"
-  by (induct n) auto
-
 lemma fps_compose_radical:
   assumes b0: "b$0 = (0::'a::field_char_0)"
     and ra0: "r (Suc k) (a$0) ^ Suc k = a$0"
@@ -3527,7 +5207,7 @@
 
 lemma fps_const_mult_apply_right:
   "(a oo b) * fps_const (c::'a::comm_semiring_1) = (fps_const c * a) oo b"
-  by (auto simp add: fps_const_mult_apply_left mult.commute)
+  by (simp add: fps_const_mult_apply_left mult.commute)
 
 lemma fps_compose_assoc:
   assumes c0: "c$0 = (0::'a::idom)"
@@ -3577,7 +5257,7 @@
     next
       case 2
       then show ?thesis
-        by (simp add: fps_compose_nth mult_delta_left sum.delta)
+        by (simp add: fps_compose_nth mult_delta_left)
     qed
   qed
   then show ?thesis
@@ -3666,7 +5346,7 @@
   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 (auto simp add: fps_ginv_def)
+    apply (simp_all add: fps_ginv_def)
     done
   then have "?r b (?r c a) oo c = b oo a"
     unfolding fps_ginv[OF a0 a1] .
@@ -3675,7 +5355,7 @@
   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 (auto simp add: fps_inv_def)
+    apply (simp_all add: fps_inv_def)
     done
   then show ?thesis
     unfolding fps_inv_right[OF c0 c1] by simp
@@ -3713,7 +5393,7 @@
 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 sum.delta' cong: if_cong)
+                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)"
@@ -3809,7 +5489,7 @@
 qed
 
 lemma fps_exp_power_mult: "(fps_exp (c::'a::field_char_0))^n = fps_exp (of_nat n * c)"
-  by (induct n) (auto simp add: field_simps fps_exp_add_mult)
+  by (induct n) (simp_all add: field_simps fps_exp_add_mult)
 
 lemma radical_fps_exp:
   assumes r: "r (Suc k) 1 = 1"
@@ -3845,7 +5525,7 @@
   "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 (auto simp: fps_eq_iff)
+  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] 
@@ -3868,7 +5548,7 @@
 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 (auto simp add: fps_eq_iff)
+  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)"
@@ -3927,7 +5607,7 @@
 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) (auto simp: fps_ln_def of_nat_diff)
+    by (intro fps_ext) (simp add: fps_ln_def of_nat_diff)
   thus ?thesis by simp
 qed
 
@@ -4338,7 +6018,7 @@
   by (intro fps_ext) (auto simp: fps_sin_def elim!: oddE)
 
 lemma fps_cos_0 [simp]: "fps_cos 0 = 1"
-  by (intro fps_ext) (auto simp: fps_cos_def)
+  by (intro fps_ext) (simp add: fps_cos_def)
 
 lemma fps_sin_deriv:
   "fps_deriv (fps_sin c) = fps_const c * fps_cos c"
@@ -4354,7 +6034,7 @@
     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)"
+    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)
@@ -4379,7 +6059,7 @@
     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)))"
+    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)"
@@ -4405,7 +6085,7 @@
   then have "?lhs = fps_const (?lhs $ 0)"
     unfolding fps_deriv_eq_0_iff .
   also have "\<dots> = 1"
-    by (auto simp add: fps_eq_iff numeral_2_eq_2 fps_mult_nth fps_cos_def fps_sin_def)
+    by (simp add: fps_eq_iff numeral_2_eq_2 fps_mult_nth fps_cos_def fps_sin_def)
   finally show ?thesis .
 qed
 
@@ -4489,12 +6169,6 @@
   apply (simp only: ac_simps)
   done
 
-lemma mult_nth_0 [simp]: "(a * b) $ 0 = a $ 0 * b $ 0"
-  by (simp add: fps_mult_nth)
-
-lemma mult_nth_1 [simp]: "(a * b) $ 1 = a $ 0 * b $ 1 + a $ 1 * b $ 0"
-  by (simp add: fps_mult_nth)
-
 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
@@ -4510,10 +6184,10 @@
   done
 
 lemma fps_sin_even: "fps_sin (- c) = - fps_sin c"
-  by (auto simp add: fps_eq_iff fps_sin_def)
+  by (simp add: fps_eq_iff fps_sin_def)
 
 lemma fps_cos_odd: "fps_cos (- c) = fps_cos c"
-  by (auto simp add: fps_eq_iff fps_cos_def)
+  by (simp add: fps_eq_iff fps_cos_def)
 
 definition "fps_tan c = fps_sin c / fps_cos c"
 
@@ -4533,7 +6207,7 @@
   finally show ?thesis by simp
 qed
 
-text \<open>Connection to \<^const>\<open>fps_exp\<close> over the complex numbers --- Euler and de Moivre.\<close>
+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")
@@ -4558,19 +6232,6 @@
 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_const_minus: "fps_const (c::'a::group_add) - fps_const d = fps_const (c - d)"
-  by (fact fps_const_sub)
-
-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_deriv_of_int [simp]: "fps_deriv (of_int n) = 0"
-  by (simp add: fps_of_int [symmetric])
-
-lemma fps_numeral_fps_const: "numeral i = fps_const (numeral i :: 'a::comm_ring_1)"
-  by (fact numeral_fps_const) (* FIfps_XME: duplicate *)
-
 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"
@@ -4623,7 +6284,7 @@
 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) (auto simp add: foldl_mult_start)
+  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) /
@@ -4638,8 +6299,8 @@
   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 (auto simp add: fps_eq_iff pochhammer_fact[symmetric]
-      fps_compose_nth power_mult_distrib cond_value_iff sum.delta' cong