src/HOL/Library/Polynomial.thy

 (*  Title:      HOL/Library/Polynomial.thy
     Author:     Brian Huffman
     Author:     Clemens Ballarin
+    Author:     Florian Haftmann
 *)
 
-header {* Univariate Polynomials *}
+header {* Polynomials as type over a ring structure *}
 
 theory Polynomial
-imports Main
+imports Main GCD
 begin
 
+subsection {* Auxiliary: operations for lists (later) representing coefficients *}
+
+definition strip_while :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list"
+where
+  "strip_while P = rev \<circ> dropWhile P \<circ> rev"
+
+lemma strip_while_Nil [simp]:
+  "strip_while P [] = []"
+  by (simp add: strip_while_def)
+
+lemma strip_while_append [simp]:
+  "\<not> P x \<Longrightarrow> strip_while P (xs @ [x]) = xs @ [x]"
+  by (simp add: strip_while_def)
+
+lemma strip_while_append_rec [simp]:
+  "P x \<Longrightarrow> strip_while P (xs @ [x]) = strip_while P xs"
+  by (simp add: strip_while_def)
+
+lemma strip_while_Cons [simp]:
+  "\<not> P x \<Longrightarrow> strip_while P (x # xs) = x # strip_while P xs"
+  by (induct xs rule: rev_induct) (simp_all add: strip_while_def)
+
+lemma strip_while_eq_Nil [simp]:
+  "strip_while P xs = [] \<longleftrightarrow> (\<forall>x\<in>set xs. P x)"
+  by (simp add: strip_while_def)
+
+lemma strip_while_eq_Cons_rec:
+  "strip_while P (x # xs) = x # strip_while P xs \<longleftrightarrow> \<not> (P x \<and> (\<forall>x\<in>set xs. P x))"
+  by (induct xs rule: rev_induct) (simp_all add: strip_while_def)
+
+lemma strip_while_not_last [simp]:
+  "\<not> P (last xs) \<Longrightarrow> strip_while P xs = xs"
+  by (cases xs rule: rev_cases) simp_all
+
+lemma split_strip_while_append:
+  fixes xs :: "'a list"
+  obtains ys zs :: "'a list"
+  where "strip_while P xs = ys" and "\<forall>x\<in>set zs. P x" and "xs = ys @ zs"
+proof (rule that)
+  show "strip_while P xs = strip_while P xs" ..
+  show "\<forall>x\<in>set (rev (takeWhile P (rev xs))). P x" by (simp add: takeWhile_eq_all_conv [symmetric])
+  have "rev xs = rev (strip_while P xs @ rev (takeWhile P (rev xs)))"
+    by (simp add: strip_while_def)
+  then show "xs = strip_while P xs @ rev (takeWhile P (rev xs))"
+    by (simp only: rev_is_rev_conv)
+qed
+
+
+definition nth_default :: "'a \<Rightarrow> 'a list \<Rightarrow> nat \<Rightarrow> 'a"
+where
+  "nth_default x xs n = (if n < length xs then xs ! n else x)"
+
+lemma nth_default_Nil [simp]:
+  "nth_default y [] n = y"
+  by (simp add: nth_default_def)
+
+lemma nth_default_Cons_0 [simp]:
+  "nth_default y (x # xs) 0 = x"
+  by (simp add: nth_default_def)
+
+lemma nth_default_Cons_Suc [simp]:
+  "nth_default y (x # xs) (Suc n) = nth_default y xs n"
+  by (simp add: nth_default_def)
+
+lemma nth_default_map_eq:
+  "f y = x \<Longrightarrow> nth_default x (map f xs) n = f (nth_default y xs n)"
+  by (simp add: nth_default_def)
+
+lemma nth_default_strip_while_eq [simp]:
+  "nth_default x (strip_while (HOL.eq x) xs) n = nth_default x xs n"
+proof -
+  from split_strip_while_append obtain ys zs
+    where "strip_while (HOL.eq x) xs = ys" and "\<forall>z\<in>set zs. x = z" and "xs = ys @ zs" by blast
+  then show ?thesis by (simp add: nth_default_def not_less nth_append)
+qed
+
+
+definition cCons :: "'a::zero \<Rightarrow> 'a list \<Rightarrow> 'a list"  (infixr "##" 65)
+where
+  "x ## xs = (if xs = [] \<and> x = 0 then [] else x # xs)"
+
+lemma cCons_0_Nil_eq [simp]:
+  "0 ## [] = []"
+  by (simp add: cCons_def)
+
+lemma cCons_Cons_eq [simp]:
+  "x ## y # ys = x # y # ys"
+  by (simp add: cCons_def)
+
+lemma cCons_append_Cons_eq [simp]:
+  "x ## xs @ y # ys = x # xs @ y # ys"
+  by (simp add: cCons_def)
+
+lemma cCons_not_0_eq [simp]:
+  "x \<noteq> 0 \<Longrightarrow> x ## xs = x # xs"
+  by (simp add: cCons_def)
+
+lemma strip_while_not_0_Cons_eq [simp]:
+  "strip_while (\<lambda>x. x = 0) (x # xs) = x ## strip_while (\<lambda>x. x = 0) xs"
+proof (cases "x = 0")
+  case False then show ?thesis by simp
+next
+  case True show ?thesis
+  proof (induct xs rule: rev_induct)
+    case Nil with True show ?case by simp
+  next
+    case (snoc y ys) then show ?case
+      by (cases "y = 0") (simp_all add: append_Cons [symmetric] del: append_Cons)
+  qed
+qed
+
+lemma tl_cCons [simp]:
+  "tl (x ## xs) = xs"
+  by (simp add: cCons_def)
+
+
+subsection {* Almost everywhere zero functions *}
+
+definition almost_everywhere_zero :: "(nat \<Rightarrow> 'a::zero) \<Rightarrow> bool"
+where
+  "almost_everywhere_zero f \<longleftrightarrow> (\<exists>n. \<forall>i>n. f i = 0)"
+
+lemma almost_everywhere_zeroI:
+  "(\<And>i. i > n \<Longrightarrow> f i = 0) \<Longrightarrow> almost_everywhere_zero f"
+  by (auto simp add: almost_everywhere_zero_def)
+
+lemma almost_everywhere_zeroE:
+  assumes "almost_everywhere_zero f"
+  obtains n where "\<And>i. i > n \<Longrightarrow> f i = 0"
+proof -
+  from assms have "\<exists>n. \<forall>i>n. f i = 0" by (simp add: almost_everywhere_zero_def)
+  then obtain n where "\<And>i. i > n \<Longrightarrow> f i = 0" by blast
+  with that show thesis .
+qed
+
+lemma almost_everywhere_zero_nat_case:
+  assumes "almost_everywhere_zero f"
+  shows "almost_everywhere_zero (nat_case a f)"
+  using assms
+  by (auto intro!: almost_everywhere_zeroI elim!: almost_everywhere_zeroE split: nat.split)
+    blast
+
+lemma almost_everywhere_zero_Suc:
+  assumes "almost_everywhere_zero f"
+  shows "almost_everywhere_zero (\<lambda>n. f (Suc n))"
+proof -
+  from assms obtain n where "\<And>i. i > n \<Longrightarrow> f i = 0" by (erule almost_everywhere_zeroE)
+  then have "\<And>i. i > n \<Longrightarrow> f (Suc i) = 0" by auto
+  then show ?thesis by (rule almost_everywhere_zeroI)
+qed
+
+
 subsection {* Definition of type @{text poly} *}
 
-definition "Poly = {f::nat \<Rightarrow> 'a::zero. \<exists>n. \<forall>i>n. f i = 0}"
-
-typedef 'a poly = "Poly :: (nat => 'a::zero) set"
+typedef 'a poly = "{f :: nat \<Rightarrow> 'a::zero. almost_everywhere_zero f}"
   morphisms coeff Abs_poly
-  unfolding Poly_def by auto
-
-(* FIXME should be named poly_eq_iff *)
-lemma expand_poly_eq: "p = q \<longleftrightarrow> (\<forall>n. coeff p n = coeff q n)"
+  unfolding almost_everywhere_zero_def by auto
+
+setup_lifting (no_code) type_definition_poly
+
+lemma poly_eq_iff: "p = q \<longleftrightarrow> (\<forall>n. coeff p n = coeff q n)"
   by (simp add: coeff_inject [symmetric] fun_eq_iff)
 
-(* FIXME should be named poly_eqI *)
-lemma poly_ext: "(\<And>n. coeff p n = coeff q n) \<Longrightarrow> p = q"
-  by (simp add: expand_poly_eq)
+lemma poly_eqI: "(\<And>n. coeff p n = coeff q n) \<Longrightarrow> p = q"
+  by (simp add: poly_eq_iff)
+
+lemma coeff_almost_everywhere_zero:
+  "almost_everywhere_zero (coeff p)"
+  using coeff [of p] by simp
 
 
 subsection {* Degree of a polynomial *}
 
-definition
-  degree :: "'a::zero poly \<Rightarrow> nat" where
+definition degree :: "'a::zero poly \<Rightarrow> nat"
+where
   "degree p = (LEAST n. \<forall>i>n. coeff p i = 0)"
 
-lemma coeff_eq_0: "degree p < n \<Longrightarrow> coeff p n = 0"
+lemma coeff_eq_0:
+  assumes "degree p < n"
+  shows "coeff p n = 0"
 proof -
-  have "coeff p \<in> Poly"
-    by (rule coeff)
-  hence "\<exists>n. \<forall>i>n. coeff p i = 0"
-    unfolding Poly_def by simp
-  hence "\<forall>i>degree p. coeff p i = 0"
+  from coeff_almost_everywhere_zero
+  have "\<exists>n. \<forall>i>n. coeff p i = 0" by (blast intro: almost_everywhere_zeroE)
+  then have "\<forall>i>degree p. coeff p i = 0"
     unfolding degree_def by (rule LeastI_ex)
-  moreover assume "degree p < n"
-  ultimately show ?thesis by simp
+  with assms show ?thesis by simp
 qed
 
 lemma le_degree: "coeff p n \<noteq> 0 \<Longrightarrow> n \<le> degree p"
   by (erule contrapos_np, rule coeff_eq_0, simp)
 

 subsection {* The zero polynomial *}
 
 instantiation poly :: (zero) zero
 begin
 
-definition
-  zero_poly_def: "0 = Abs_poly (\<lambda>n. 0)"
+lift_definition zero_poly :: "'a poly"
+  is "\<lambda>_. 0" by (rule almost_everywhere_zeroI) simp
 
 instance ..
+
 end
 
-lemma coeff_0 [simp]: "coeff 0 n = 0"
-  unfolding zero_poly_def
-  by (simp add: Abs_poly_inverse Poly_def)
-
-lemma degree_0 [simp]: "degree 0 = 0"
+lemma coeff_0 [simp]:
+  "coeff 0 n = 0"
+  by transfer rule
+
+lemma degree_0 [simp]:
+  "degree 0 = 0"
   by (rule order_antisym [OF degree_le le0]) simp
 
 lemma leading_coeff_neq_0:
-  assumes "p \<noteq> 0" shows "coeff p (degree p) \<noteq> 0"
+  assumes "p \<noteq> 0"
+  shows "coeff p (degree p) \<noteq> 0"
 proof (cases "degree p")
   case 0
   from `p \<noteq> 0` have "\<exists>n. coeff p n \<noteq> 0"
-    by (simp add: expand_poly_eq)
+    by (simp add: poly_eq_iff)
   then obtain n where "coeff p n \<noteq> 0" ..
   hence "n \<le> degree p" by (rule le_degree)
   with `coeff p n \<noteq> 0` and `degree p = 0`
   show "coeff p (degree p) \<noteq> 0" by simp
 next

   also from `coeff p i \<noteq> 0` have "i \<le> degree p" by (rule le_degree)
   finally have "degree p = i" .
   with `coeff p i \<noteq> 0` show "coeff p (degree p) \<noteq> 0" by simp
 qed
 
-lemma leading_coeff_0_iff [simp]: "coeff p (degree p) = 0 \<longleftrightarrow> p = 0"
+lemma leading_coeff_0_iff [simp]:
+  "coeff p (degree p) = 0 \<longleftrightarrow> p = 0"
   by (cases "p = 0", simp, simp add: leading_coeff_neq_0)
 
 
 subsection {* List-style constructor for polynomials *}
 
-definition
-  pCons :: "'a::zero \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
-where
-  "pCons a p = Abs_poly (nat_case a (coeff p))"
-
-syntax
-  "_poly" :: "args \<Rightarrow> 'a poly"  ("[:(_):]")
-
-translations
-  "[:x, xs:]" == "CONST pCons x [:xs:]"
-  "[:x:]" == "CONST pCons x 0"
-  "[:x:]" <= "CONST pCons x (_constrain 0 t)"
-
-lemma Poly_nat_case: "f \<in> Poly \<Longrightarrow> nat_case a f \<in> Poly"
-  unfolding Poly_def by (auto split: nat.split)
-
-lemma coeff_pCons:
-  "coeff (pCons a p) = nat_case a (coeff p)"
-  unfolding pCons_def
-  by (simp add: Abs_poly_inverse Poly_nat_case coeff)
-
-lemma coeff_pCons_0 [simp]: "coeff (pCons a p) 0 = a"
+lift_definition pCons :: "'a::zero \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
+  is "\<lambda>a p. nat_case a (coeff p)"
+  using coeff_almost_everywhere_zero by (rule almost_everywhere_zero_nat_case)
+
+lemmas coeff_pCons = pCons.rep_eq
+
+lemma coeff_pCons_0 [simp]:
+  "coeff (pCons a p) 0 = a"
+  by transfer simp
+
+lemma coeff_pCons_Suc [simp]:
+  "coeff (pCons a p) (Suc n) = coeff p n"
   by (simp add: coeff_pCons)
 
-lemma coeff_pCons_Suc [simp]: "coeff (pCons a p) (Suc n) = coeff p n"
-  by (simp add: coeff_pCons)
-
-lemma degree_pCons_le: "degree (pCons a p) \<le> Suc (degree p)"
-by (rule degree_le, simp add: coeff_eq_0 coeff_pCons split: nat.split)
+lemma degree_pCons_le:
+  "degree (pCons a p) \<le> Suc (degree p)"
+  by (rule degree_le) (simp add: coeff_eq_0 coeff_pCons split: nat.split)
 
 lemma degree_pCons_eq:
   "p \<noteq> 0 \<Longrightarrow> degree (pCons a p) = Suc (degree p)"
-apply (rule order_antisym [OF degree_pCons_le])
-apply (rule le_degree, simp)
-done
-
-lemma degree_pCons_0: "degree (pCons a 0) = 0"
-apply (rule order_antisym [OF _ le0])
-apply (rule degree_le, simp add: coeff_pCons split: nat.split)
-done
+  apply (rule order_antisym [OF degree_pCons_le])
+  apply (rule le_degree, simp)
+  done
+
+lemma degree_pCons_0:
+  "degree (pCons a 0) = 0"
+  apply (rule order_antisym [OF _ le0])
+  apply (rule degree_le, simp add: coeff_pCons split: nat.split)
+  done
 
 lemma degree_pCons_eq_if [simp]:
   "degree (pCons a p) = (if p = 0 then 0 else Suc (degree p))"
-apply (cases "p = 0", simp_all)
-apply (rule order_antisym [OF _ le0])
-apply (rule degree_le, simp add: coeff_pCons split: nat.split)
-apply (rule order_antisym [OF degree_pCons_le])
-apply (rule le_degree, simp)
-done
-
-lemma pCons_0_0 [simp, code_post]: "pCons 0 0 = 0"
-by (rule poly_ext, simp add: coeff_pCons split: nat.split)
+  apply (cases "p = 0", simp_all)
+  apply (rule order_antisym [OF _ le0])
+  apply (rule degree_le, simp add: coeff_pCons split: nat.split)
+  apply (rule order_antisym [OF degree_pCons_le])
+  apply (rule le_degree, simp)
+  done
+
+lemma pCons_0_0 [simp]:
+  "pCons 0 0 = 0"
+  by (rule poly_eqI) (simp add: coeff_pCons split: nat.split)
 
 lemma pCons_eq_iff [simp]:
   "pCons a p = pCons b q \<longleftrightarrow> a = b \<and> p = q"
-proof (safe)
+proof safe
   assume "pCons a p = pCons b q"
   then have "coeff (pCons a p) 0 = coeff (pCons b q) 0" by simp
   then show "a = b" by simp
 next
   assume "pCons a p = pCons b q"
   then have "\<forall>n. coeff (pCons a p) (Suc n) =
                  coeff (pCons b q) (Suc n)" by simp
-  then show "p = q" by (simp add: expand_poly_eq)
-qed
-
-lemma pCons_eq_0_iff [simp]: "pCons a p = 0 \<longleftrightarrow> a = 0 \<and> p = 0"
+  then show "p = q" by (simp add: poly_eq_iff)
+qed
+
+lemma pCons_eq_0_iff [simp]:
+  "pCons a p = 0 \<longleftrightarrow> a = 0 \<and> p = 0"
   using pCons_eq_iff [of a p 0 0] by simp
-
-lemma Poly_Suc: "f \<in> Poly \<Longrightarrow> (\<lambda>n. f (Suc n)) \<in> Poly"
-  unfolding Poly_def
-  by (clarify, rule_tac x=n in exI, simp)
 
 lemma pCons_cases [cases type: poly]:
   obtains (pCons) a q where "p = pCons a q"
 proof
   show "p = pCons (coeff p 0) (Abs_poly (\<lambda>n. coeff p (Suc n)))"
-    by (rule poly_ext)
-       (simp add: Abs_poly_inverse Poly_Suc coeff coeff_pCons
-             split: nat.split)
+    by transfer
+      (simp add: Abs_poly_inverse almost_everywhere_zero_Suc fun_eq_iff split: nat.split)
 qed
 
 lemma pCons_induct [case_names 0 pCons, induct type: poly]:
   assumes zero: "P 0"
   assumes pCons: "\<And>a p. P p \<Longrightarrow> P (pCons a p)"

   then show ?case
     using `p = pCons a q` by simp
 qed
 
 
-subsection {* Recursion combinator for polynomials *}
-
-function
-  poly_rec :: "'b \<Rightarrow> ('a::zero \<Rightarrow> 'a poly \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a poly \<Rightarrow> 'b"
+subsection {* List-style syntax for polynomials *}
+
+syntax
+  "_poly" :: "args \<Rightarrow> 'a poly"  ("[:(_):]")
+
+translations
+  "[:x, xs:]" == "CONST pCons x [:xs:]"
+  "[:x:]" == "CONST pCons x 0"
+  "[:x:]" <= "CONST pCons x (_constrain 0 t)"
+
+
+subsection {* Representation of polynomials by lists of coefficients *}
+
+primrec Poly :: "'a::zero list \<Rightarrow> 'a poly"
 where
-  poly_rec_pCons_eq_if [simp del]:
-    "poly_rec z f (pCons a p) = f a p (if p = 0 then z else poly_rec z f p)"
-by (case_tac x, rename_tac q, case_tac q, auto)
-
-termination poly_rec
-by (relation "measure (degree \<circ> snd \<circ> snd)", simp)
-   (simp add: degree_pCons_eq)
-
-lemma poly_rec_0:
-  "f 0 0 z = z \<Longrightarrow> poly_rec z f 0 = z"
-  using poly_rec_pCons_eq_if [of z f 0 0] by simp
-
-lemma poly_rec_pCons:
-  "f 0 0 z = z \<Longrightarrow> poly_rec z f (pCons a p) = f a p (poly_rec z f p)"
-  by (simp add: poly_rec_pCons_eq_if poly_rec_0)
+  "Poly [] = 0"
+| "Poly (a # as) = pCons a (Poly as)"
+
+lemma Poly_replicate_0 [simp]:
+  "Poly (replicate n 0) = 0"
+  by (induct n) simp_all
+
+lemma Poly_eq_0:
+  "Poly as = 0 \<longleftrightarrow> (\<exists>n. as = replicate n 0)"
+  by (induct as) (auto simp add: Cons_replicate_eq)
+
+definition coeffs :: "'a poly \<Rightarrow> 'a::zero list"
+where
+  "coeffs p = (if p = 0 then [] else map (\<lambda>i. coeff p i) [0 ..< Suc (degree p)])"
+
+lemma coeffs_eq_Nil [simp]:
+  "coeffs p = [] \<longleftrightarrow> p = 0"
+  by (simp add: coeffs_def)
+
+lemma not_0_coeffs_not_Nil:
+  "p \<noteq> 0 \<Longrightarrow> coeffs p \<noteq> []"
+  by simp
+
+lemma coeffs_0_eq_Nil [simp]:
+  "coeffs 0 = []"
+  by simp
+
+lemma coeffs_pCons_eq_cCons [simp]:
+  "coeffs (pCons a p) = a ## coeffs p"
+proof -
+  { fix ms :: "nat list" and f :: "nat \<Rightarrow> 'a" and x :: "'a"
+    assume "\<forall>m\<in>set ms. m > 0"
+    then have "map (nat_case x f) ms = map f (map (\<lambda>n. n - 1) ms)"
+      by (induct ms) (auto, metis Suc_pred' nat_case_Suc) }
+  note * = this
+  show ?thesis
+    by (simp add: coeffs_def * upt_conv_Cons coeff_pCons map_decr_upt One_nat_def del: upt_Suc)
+qed
+
+lemma not_0_cCons_eq [simp]:
+  "p \<noteq> 0 \<Longrightarrow> a ## coeffs p = a # coeffs p"
+  by (simp add: cCons_def)
+
+lemma Poly_coeffs [simp, code abstype]:
+  "Poly (coeffs p) = p"
+  by (induct p) (simp_all add: cCons_def)
+
+lemma coeffs_Poly [simp]:
+  "coeffs (Poly as) = strip_while (HOL.eq 0) as"
+proof (induct as)
+  case Nil then show ?case by simp
+next
+  case (Cons a as)
+  have "(\<forall>n. as \<noteq> replicate n 0) \<longleftrightarrow> (\<exists>a\<in>set as. a \<noteq> 0)"
+    using replicate_length_same [of as 0] by (auto dest: sym [of _ as])
+  with Cons show ?case by auto
+qed
+
+lemma last_coeffs_not_0:
+  "p \<noteq> 0 \<Longrightarrow> last (coeffs p) \<noteq> 0"
+  by (induct p) (auto simp add: cCons_def)
+
+lemma strip_while_coeffs [simp]:
+  "strip_while (HOL.eq 0) (coeffs p) = coeffs p"
+  by (cases "p = 0") (auto dest: last_coeffs_not_0 intro: strip_while_not_last)
+
+lemma coeffs_eq_iff:
+  "p = q \<longleftrightarrow> coeffs p = coeffs q" (is "?P \<longleftrightarrow> ?Q")
+proof
+  assume ?P then show ?Q by simp
+next
+  assume ?Q
+  then have "Poly (coeffs p) = Poly (coeffs q)" by simp
+  then show ?P by simp
+qed
+
+lemma coeff_Poly_eq:
+  "coeff (Poly xs) n = nth_default 0 xs n"
+  apply (induct xs arbitrary: n) apply simp_all
+  by (metis nat_case_0 nat_case_Suc not0_implies_Suc nth_default_Cons_0 nth_default_Cons_Suc pCons.rep_eq)
+
+lemma nth_default_coeffs_eq:
+  "nth_default 0 (coeffs p) = coeff p"
+  by (simp add: fun_eq_iff coeff_Poly_eq [symmetric])
+
+lemma [code]:
+  "coeff p = nth_default 0 (coeffs p)"
+  by (simp add: nth_default_coeffs_eq)
+
+lemma coeffs_eqI:
+  assumes coeff: "\<And>n. coeff p n = nth_default 0 xs n"
+  assumes zero: "xs \<noteq> [] \<Longrightarrow> last xs \<noteq> 0"
+  shows "coeffs p = xs"
+proof -
+  from coeff have "p = Poly xs" by (simp add: poly_eq_iff coeff_Poly_eq)
+  with zero show ?thesis by simp (cases xs, simp_all)
+qed
+
+lemma degree_eq_length_coeffs [code]:
+  "degree p = length (coeffs p) - 1"
+  by (simp add: coeffs_def)
+
+lemma length_coeffs_degree:
+  "p \<noteq> 0 \<Longrightarrow> length (coeffs p) = Suc (degree p)"
+  by (induct p) (auto simp add: cCons_def)
+
+lemma [code abstract]:
+  "coeffs 0 = []"
+  by (fact coeffs_0_eq_Nil)
+
+lemma [code abstract]:
+  "coeffs (pCons a p) = a ## coeffs p"
+  by (fact coeffs_pCons_eq_cCons)
+
+instantiation poly :: ("{zero, equal}") equal
+begin
+
+definition
+  [code]: "HOL.equal (p::'a poly) q \<longleftrightarrow> HOL.equal (coeffs p) (coeffs q)"
+
+instance proof
+qed (simp add: equal equal_poly_def coeffs_eq_iff)
+
+end
+
+lemma [code nbe]:
+  "HOL.equal (p :: _ poly) p \<longleftrightarrow> True"
+  by (fact equal_refl)
+
+definition is_zero :: "'a::zero poly \<Rightarrow> bool"
+where
+  [code]: "is_zero p \<longleftrightarrow> List.null (coeffs p)"
+
+lemma is_zero_null [code_abbrev]:
+  "is_zero p \<longleftrightarrow> p = 0"
+  by (simp add: is_zero_def null_def)
+
+
+subsection {* Fold combinator for polynomials *}
+
+definition fold_coeffs :: "('a::zero \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a poly \<Rightarrow> 'b \<Rightarrow> 'b"
+where
+  "fold_coeffs f p = foldr f (coeffs p)"
+
+lemma fold_coeffs_0_eq [simp]:
+  "fold_coeffs f 0 = id"
+  by (simp add: fold_coeffs_def)
+
+lemma fold_coeffs_pCons_eq [simp]:
+  "f 0 = id \<Longrightarrow> fold_coeffs f (pCons a p) = f a \<circ> fold_coeffs f p"
+  by (simp add: fold_coeffs_def cCons_def fun_eq_iff)
+
+lemma fold_coeffs_pCons_0_0_eq [simp]:
+  "fold_coeffs f (pCons 0 0) = id"
+  by (simp add: fold_coeffs_def)
+
+lemma fold_coeffs_pCons_coeff_not_0_eq [simp]:
+  "a \<noteq> 0 \<Longrightarrow> fold_coeffs f (pCons a p) = f a \<circ> fold_coeffs f p"
+  by (simp add: fold_coeffs_def)
+
+lemma fold_coeffs_pCons_not_0_0_eq [simp]:
+  "p \<noteq> 0 \<Longrightarrow> fold_coeffs f (pCons a p) = f a \<circ> fold_coeffs f p"
+  by (simp add: fold_coeffs_def)
+
+
+subsection {* Canonical morphism on polynomials -- evaluation *}
+
+definition poly :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a"
+where
+  "poly p = fold_coeffs (\<lambda>a f x. a + x * f x) p (\<lambda>x. 0)" -- {* The Horner Schema *}
+
+lemma poly_0 [simp]:
+  "poly 0 x = 0"
+  by (simp add: poly_def)
+
+lemma poly_pCons [simp]:
+  "poly (pCons a p) x = a + x * poly p x"
+  by (cases "p = 0 \<and> a = 0") (auto simp add: poly_def)
 
 
 subsection {* Monomials *}
 
-definition
-  monom :: "'a \<Rightarrow> nat \<Rightarrow> 'a::zero poly" where
-  "monom a m = Abs_poly (\<lambda>n. if m = n then a else 0)"
-
-lemma coeff_monom [simp]: "coeff (monom a m) n = (if m=n then a else 0)"
-  unfolding monom_def
-  by (subst Abs_poly_inverse, auto simp add: Poly_def)
-
-lemma monom_0: "monom a 0 = pCons a 0"
-  by (rule poly_ext, simp add: coeff_pCons split: nat.split)
-
-lemma monom_Suc: "monom a (Suc n) = pCons 0 (monom a n)"
-  by (rule poly_ext, simp add: coeff_pCons split: nat.split)
+lift_definition monom :: "'a \<Rightarrow> nat \<Rightarrow> 'a::zero poly"
+  is "\<lambda>a m n. if m = n then a else 0"
+  by (auto intro!: almost_everywhere_zeroI)
+
+lemma coeff_monom [simp]:
+  "coeff (monom a m) n = (if m = n then a else 0)"
+  by transfer rule
+
+lemma monom_0:
+  "monom a 0 = pCons a 0"
+  by (rule poly_eqI) (simp add: coeff_pCons split: nat.split)
+
+lemma monom_Suc:
+  "monom a (Suc n) = pCons 0 (monom a n)"
+  by (rule poly_eqI) (simp add: coeff_pCons split: nat.split)
 
 lemma monom_eq_0 [simp]: "monom 0 n = 0"
-  by (rule poly_ext) simp
+  by (rule poly_eqI) simp
 
 lemma monom_eq_0_iff [simp]: "monom a n = 0 \<longleftrightarrow> a = 0"
-  by (simp add: expand_poly_eq)
+  by (simp add: poly_eq_iff)
 
 lemma monom_eq_iff [simp]: "monom a n = monom b n \<longleftrightarrow> a = b"
-  by (simp add: expand_poly_eq)
+  by (simp add: poly_eq_iff)
 
 lemma degree_monom_le: "degree (monom a n) \<le> n"
   by (rule degree_le, simp)
 
 lemma degree_monom_eq: "a \<noteq> 0 \<Longrightarrow> degree (monom a n) = n"
   apply (rule order_antisym [OF degree_monom_le])
   apply (rule le_degree, simp)
   done
 
+lemma coeffs_monom [code abstract]:
+  "coeffs (monom a n) = (if a = 0 then [] else replicate n 0 @ [a])"
+  by (induct n) (simp_all add: monom_0 monom_Suc)
+
+lemma fold_coeffs_monom [simp]:
+  "a \<noteq> 0 \<Longrightarrow> fold_coeffs f (monom a n) = f 0 ^^ n \<circ> f a"
+  by (simp add: fold_coeffs_def coeffs_monom fun_eq_iff)
+
+lemma poly_monom:
+  fixes a x :: "'a::{comm_semiring_1}"
+  shows "poly (monom a n) x = a * x ^ n"
+  by (cases "a = 0", simp_all)
+    (induct n, simp_all add: mult.left_commute poly_def)
+
 
 subsection {* Addition and subtraction *}
 
 instantiation poly :: (comm_monoid_add) comm_monoid_add
 begin
 
-definition
-  plus_poly_def:
-    "p + q = Abs_poly (\<lambda>n. coeff p n + coeff q n)"
-
-lemma Poly_add:
-  fixes f g :: "nat \<Rightarrow> 'a"
-  shows "\<lbrakk>f \<in> Poly; g \<in> Poly\<rbrakk> \<Longrightarrow> (\<lambda>n. f n + g n) \<in> Poly"
-  unfolding Poly_def
-  apply (clarify, rename_tac m n)
-  apply (rule_tac x="max m n" in exI, simp)
-  done
+lift_definition plus_poly :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
+  is "\<lambda>p q n. coeff p n + coeff q n"
+proof (rule almost_everywhere_zeroI) 
+  fix q p :: "'a poly" and i
+  assume "max (degree q) (degree p) < i"
+  then show "coeff p i + coeff q i = 0"
+    by (simp add: coeff_eq_0)
+qed
 
 lemma coeff_add [simp]:
   "coeff (p + q) n = coeff p n + coeff q n"
-  unfolding plus_poly_def
-  by (simp add: Abs_poly_inverse coeff Poly_add)
+  by (simp add: plus_poly.rep_eq)
 
 instance proof
   fix p q r :: "'a poly"
   show "(p + q) + r = p + (q + r)"
-    by (simp add: expand_poly_eq add_assoc)
+    by (simp add: poly_eq_iff add_assoc)
   show "p + q = q + p"
-    by (simp add: expand_poly_eq add_commute)
+    by (simp add: poly_eq_iff add_commute)
   show "0 + p = p"
-    by (simp add: expand_poly_eq)
+    by (simp add: poly_eq_iff)
 qed
 
 end
 
 instance poly :: (cancel_comm_monoid_add) cancel_comm_monoid_add
 proof
   fix p q r :: "'a poly"
   assume "p + q = p + r" thus "q = r"
-    by (simp add: expand_poly_eq)
+    by (simp add: poly_eq_iff)
 qed
 
 instantiation poly :: (ab_group_add) ab_group_add
 begin
 
-definition
-  uminus_poly_def:
-    "- p = Abs_poly (\<lambda>n. - coeff p n)"
-
-definition
-  minus_poly_def:
-    "p - q = Abs_poly (\<lambda>n. coeff p n - coeff q n)"
-
-lemma Poly_minus:
-  fixes f :: "nat \<Rightarrow> 'a"
-  shows "f \<in> Poly \<Longrightarrow> (\<lambda>n. - f n) \<in> Poly"
-  unfolding Poly_def by simp
-
-lemma Poly_diff:
-  fixes f g :: "nat \<Rightarrow> 'a"
-  shows "\<lbrakk>f \<in> Poly; g \<in> Poly\<rbrakk> \<Longrightarrow> (\<lambda>n. f n - g n) \<in> Poly"
-  unfolding diff_minus by (simp add: Poly_add Poly_minus)
+lift_definition uminus_poly :: "'a poly \<Rightarrow> 'a poly"
+  is "\<lambda>p n. - coeff p n"
+proof (rule almost_everywhere_zeroI)
+  fix p :: "'a poly" and i
+  assume "degree p < i"
+  then show "- coeff p i = 0"
+    by (simp add: coeff_eq_0)
+qed
+
+lift_definition minus_poly :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
+  is "\<lambda>p q n. coeff p n - coeff q n"
+proof (rule almost_everywhere_zeroI) 
+  fix q p :: "'a poly" and i
+  assume "max (degree q) (degree p) < i"
+  then show "coeff p i - coeff q i = 0"
+    by (simp add: coeff_eq_0)
+qed
 
 lemma coeff_minus [simp]: "coeff (- p) n = - coeff p n"
-  unfolding uminus_poly_def
-  by (simp add: Abs_poly_inverse coeff Poly_minus)
+  by (simp add: uminus_poly.rep_eq)
 
 lemma coeff_diff [simp]:
   "coeff (p - q) n = coeff p n - coeff q n"
-  unfolding minus_poly_def
-  by (simp add: Abs_poly_inverse coeff Poly_diff)
+  by (simp add: minus_poly.rep_eq)
 
 instance proof
   fix p q :: "'a poly"
   show "- p + p = 0"
-    by (simp add: expand_poly_eq)
+    by (simp add: poly_eq_iff)
   show "p - q = p + - q"
-    by (simp add: expand_poly_eq diff_minus)
+    by (simp add: poly_eq_iff diff_minus)
 qed
 
 end
 
 lemma add_pCons [simp]:
   "pCons a p + pCons b q = pCons (a + b) (p + q)"
-  by (rule poly_ext, simp add: coeff_pCons split: nat.split)
+  by (rule poly_eqI, simp add: coeff_pCons split: nat.split)
 
 lemma minus_pCons [simp]:
   "- pCons a p = pCons (- a) (- p)"
-  by (rule poly_ext, simp add: coeff_pCons split: nat.split)
+  by (rule poly_eqI, simp add: coeff_pCons split: nat.split)
 
 lemma diff_pCons [simp]:
   "pCons a p - pCons b q = pCons (a - b) (p - q)"
-  by (rule poly_ext, simp add: coeff_pCons split: nat.split)
+  by (rule poly_eqI, simp add: coeff_pCons split: nat.split)
 
 lemma degree_add_le_max: "degree (p + q) \<le> max (degree p) (degree q)"
   by (rule degree_le, auto simp add: coeff_eq_0)
 
 lemma degree_add_le:

 lemma degree_diff_less:
   "\<lbrakk>degree p < n; degree q < n\<rbrakk> \<Longrightarrow> degree (p - q) < n"
   by (simp add: diff_minus degree_add_less)
 
 lemma add_monom: "monom a n + monom b n = monom (a + b) n"
-  by (rule poly_ext) simp
+  by (rule poly_eqI) simp
 
 lemma diff_monom: "monom a n - monom b n = monom (a - b) n"
-  by (rule poly_ext) simp
+  by (rule poly_eqI) simp
 
 lemma minus_monom: "- monom a n = monom (-a) n"
-  by (rule poly_ext) simp
+  by (rule poly_eqI) simp
 
 lemma coeff_setsum: "coeff (\<Sum>x\<in>A. p x) i = (\<Sum>x\<in>A. coeff (p x) i)"
   by (cases "finite A", induct set: finite, simp_all)
 
 lemma monom_setsum: "monom (\<Sum>x\<in>A. a x) n = (\<Sum>x\<in>A. monom (a x) n)"
-  by (rule poly_ext) (simp add: coeff_setsum)
-
-
-subsection {* Multiplication by a constant *}
-
-definition
-  smult :: "'a::comm_semiring_0 \<Rightarrow> 'a poly \<Rightarrow> 'a poly" where
-  "smult a p = Abs_poly (\<lambda>n. a * coeff p n)"
-
-lemma Poly_smult:
-  fixes f :: "nat \<Rightarrow> 'a::comm_semiring_0"
-  shows "f \<in> Poly \<Longrightarrow> (\<lambda>n. a * f n) \<in> Poly"
-  unfolding Poly_def
-  by (clarify, rule_tac x=n in exI, simp)
-
-lemma coeff_smult [simp]: "coeff (smult a p) n = a * coeff p n"
-  unfolding smult_def
-  by (simp add: Abs_poly_inverse Poly_smult coeff)
+  by (rule poly_eqI) (simp add: coeff_setsum)
+
+fun plus_coeffs :: "'a::comm_monoid_add list \<Rightarrow> 'a list \<Rightarrow> 'a list"
+where
+  "plus_coeffs xs [] = xs"
+| "plus_coeffs [] ys = ys"
+| "plus_coeffs (x # xs) (y # ys) = (x + y) ## plus_coeffs xs ys"
+
+lemma coeffs_plus_eq_plus_coeffs [code abstract]:
+  "coeffs (p + q) = plus_coeffs (coeffs p) (coeffs q)"
+proof -
+  { fix xs ys :: "'a list" and n
+    have "nth_default 0 (plus_coeffs xs ys) n = nth_default 0 xs n + nth_default 0 ys n"
+    proof (induct xs ys arbitrary: n rule: plus_coeffs.induct)
+      case (3 x xs y ys n) then show ?case by (cases n) (auto simp add: cCons_def)
+    qed simp_all }
+  note * = this
+  { fix xs ys :: "'a list"
+    assume "xs \<noteq> [] \<Longrightarrow> last xs \<noteq> 0" and "ys \<noteq> [] \<Longrightarrow> last ys \<noteq> 0"
+    moreover assume "plus_coeffs xs ys \<noteq> []"
+    ultimately have "last (plus_coeffs xs ys) \<noteq> 0"
+    proof (induct xs ys rule: plus_coeffs.induct)
+      case (3 x xs y ys) then show ?case by (auto simp add: cCons_def) metis
+    qed simp_all }
+  note ** = this
+  show ?thesis
+    apply (rule coeffs_eqI)
+    apply (simp add: * nth_default_coeffs_eq)
+    apply (rule **)
+    apply (auto dest: last_coeffs_not_0)
+    done
+qed
+
+lemma coeffs_uminus [code abstract]:
+  "coeffs (- p) = map (\<lambda>a. - a) (coeffs p)"
+  by (rule coeffs_eqI)
+    (simp_all add: not_0_coeffs_not_Nil last_map last_coeffs_not_0 nth_default_map_eq nth_default_coeffs_eq)
+
+lemma [code]:
+  fixes p q :: "'a::ab_group_add poly"
+  shows "p - q = p + - q"
+  by simp
+
+lemma poly_add [simp]: "poly (p + q) x = poly p x + poly q x"
+  apply (induct p arbitrary: q, simp)
+  apply (case_tac q, simp, simp add: algebra_simps)
+  done
+
+lemma poly_minus [simp]:
+  fixes x :: "'a::comm_ring"
+  shows "poly (- p) x = - poly p x"
+  by (induct p) simp_all
+
+lemma poly_diff [simp]:
+  fixes x :: "'a::comm_ring"
+  shows "poly (p - q) x = poly p x - poly q x"
+  by (simp add: diff_minus)
+
+lemma poly_setsum: "poly (\<Sum>k\<in>A. p k) x = (\<Sum>k\<in>A. poly (p k) x)"
+  by (induct A rule: infinite_finite_induct) simp_all
+
+
+subsection {* Multiplication by a constant, polynomial multiplication and the unit polynomial *}
+
+lift_definition smult :: "'a::comm_semiring_0 \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
+  is "\<lambda>a p n. a * coeff p n"
+proof (rule almost_everywhere_zeroI)
+  fix a :: 'a and p :: "'a poly" and i
+  assume "degree p < i"
+  then show "a * coeff p i = 0"
+    by (simp add: coeff_eq_0)
+qed
+
+lemma coeff_smult [simp]:
+  "coeff (smult a p) n = a * coeff p n"
+  by (simp add: smult.rep_eq)
 
 lemma degree_smult_le: "degree (smult a p) \<le> degree p"
   by (rule degree_le, simp add: coeff_eq_0)
 
 lemma smult_smult [simp]: "smult a (smult b p) = smult (a * b) p"
-  by (rule poly_ext, simp add: mult_assoc)
+  by (rule poly_eqI, simp add: mult_assoc)
 
 lemma smult_0_right [simp]: "smult a 0 = 0"
-  by (rule poly_ext, simp)
+  by (rule poly_eqI, simp)
 
 lemma smult_0_left [simp]: "smult 0 p = 0"
-  by (rule poly_ext, simp)
+  by (rule poly_eqI, simp)
 
 lemma smult_1_left [simp]: "smult (1::'a::comm_semiring_1) p = p"
-  by (rule poly_ext, simp)
+  by (rule poly_eqI, simp)
 
 lemma smult_add_right:
   "smult a (p + q) = smult a p + smult a q"
-  by (rule poly_ext, simp add: algebra_simps)
+  by (rule poly_eqI, simp add: algebra_simps)
 
 lemma smult_add_left:
   "smult (a + b) p = smult a p + smult b p"
-  by (rule poly_ext, simp add: algebra_simps)
+  by (rule poly_eqI, simp add: algebra_simps)
 
 lemma smult_minus_right [simp]:
   "smult (a::'a::comm_ring) (- p) = - smult a p"
-  by (rule poly_ext, simp)
+  by (rule poly_eqI, simp)
 
 lemma smult_minus_left [simp]:
   "smult (- a::'a::comm_ring) p = - smult a p"
-  by (rule poly_ext, simp)
+  by (rule poly_eqI, simp)
 
 lemma smult_diff_right:
   "smult (a::'a::comm_ring) (p - q) = smult a p - smult a q"
-  by (rule poly_ext, simp add: algebra_simps)
+  by (rule poly_eqI, simp add: algebra_simps)
 
 lemma smult_diff_left:
   "smult (a - b::'a::comm_ring) p = smult a p - smult b p"
-  by (rule poly_ext, simp add: algebra_simps)
+  by (rule poly_eqI, simp add: algebra_simps)
 
 lemmas smult_distribs =
   smult_add_left smult_add_right
   smult_diff_left smult_diff_right
 
 lemma smult_pCons [simp]:
   "smult a (pCons b p) = pCons (a * b) (smult a p)"
-  by (rule poly_ext, simp add: coeff_pCons split: nat.split)
+  by (rule poly_eqI, simp add: coeff_pCons split: nat.split)
 
 lemma smult_monom: "smult a (monom b n) = monom (a * b) n"
   by (induct n, simp add: monom_0, simp add: monom_Suc)
 
 lemma degree_smult_eq [simp]:

   by (cases "a = 0", simp, simp add: degree_def)
 
 lemma smult_eq_0_iff [simp]:
   fixes a :: "'a::idom"
   shows "smult a p = 0 \<longleftrightarrow> a = 0 \<or> p = 0"
-  by (simp add: expand_poly_eq)
-
-
-subsection {* Multiplication of polynomials *}
-
-(* TODO: move to Set_Interval.thy *)
-lemma setsum_atMost_Suc_shift:
-  fixes f :: "nat \<Rightarrow> 'a::comm_monoid_add"
-  shows "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))"
-proof (induct n)
-  case 0 show ?case by simp
-next
-  case (Suc n) note IH = this
-  have "(\<Sum>i\<le>Suc (Suc n). f i) = (\<Sum>i\<le>Suc n. f i) + f (Suc (Suc n))"
-    by (rule setsum_atMost_Suc)
-  also have "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))"
-    by (rule IH)
-  also have "f 0 + (\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) =
-             f 0 + ((\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)))"
-    by (rule add_assoc)
-  also have "(\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) = (\<Sum>i\<le>Suc n. f (Suc i))"
-    by (rule setsum_atMost_Suc [symmetric])
-  finally show ?case .
-qed
+  by (simp add: poly_eq_iff)
+
+lemma coeffs_smult [code abstract]:
+  fixes p :: "'a::idom poly"
+  shows "coeffs (smult a p) = (if a = 0 then [] else map (Groups.times a) (coeffs p))"
+  by (rule coeffs_eqI)
+    (auto simp add: not_0_coeffs_not_Nil last_map last_coeffs_not_0 nth_default_map_eq nth_default_coeffs_eq)
 
 instantiation poly :: (comm_semiring_0) comm_semiring_0
 begin
 
 definition
-  times_poly_def:
-    "p * q = poly_rec 0 (\<lambda>a p pq. smult a q + pCons 0 pq) p"
+  "p * q = fold_coeffs (\<lambda>a p. smult a q + pCons 0 p) p 0"
 
 lemma mult_poly_0_left: "(0::'a poly) * q = 0"
-  unfolding times_poly_def by (simp add: poly_rec_0)
+  by (simp add: times_poly_def)
 
 lemma mult_pCons_left [simp]:
   "pCons a p * q = smult a q + pCons 0 (p * q)"
-  unfolding times_poly_def by (simp add: poly_rec_pCons)
+  by (cases "p = 0 \<and> a = 0") (auto simp add: times_poly_def)
 
 lemma mult_poly_0_right: "p * (0::'a poly) = 0"
-  by (induct p, simp add: mult_poly_0_left, simp)
+  by (induct p) (simp add: mult_poly_0_left, simp)
 
 lemma mult_pCons_right [simp]:
   "p * pCons a q = smult a p + pCons 0 (p * q)"
-  by (induct p, simp add: mult_poly_0_left, simp add: algebra_simps)
+  by (induct p) (simp add: mult_poly_0_left, simp add: algebra_simps)
 
 lemmas mult_poly_0 = mult_poly_0_left mult_poly_0_right
 
-lemma mult_smult_left [simp]: "smult a p * q = smult a (p * q)"
-  by (induct p, simp add: mult_poly_0, simp add: smult_add_right)
-
-lemma mult_smult_right [simp]: "p * smult a q = smult a (p * q)"
-  by (induct q, simp add: mult_poly_0, simp add: smult_add_right)
+lemma mult_smult_left [simp]:
+  "smult a p * q = smult a (p * q)"
+  by (induct p) (simp add: mult_poly_0, simp add: smult_add_right)
+
+lemma mult_smult_right [simp]:
+  "p * smult a q = smult a (p * q)"
+  by (induct q) (simp add: mult_poly_0, simp add: smult_add_right)
 
 lemma mult_poly_add_left:
   fixes p q r :: "'a poly"
   shows "(p + q) * r = p * r + q * r"
-  by (induct r, simp add: mult_poly_0,
-                simp add: smult_distribs algebra_simps)
+  by (induct r) (simp add: mult_poly_0, simp add: smult_distribs algebra_simps)
 
 instance proof
   fix p q r :: "'a poly"
   show 0: "0 * p = 0"
     by (rule mult_poly_0_left)

 done
 
 lemma mult_monom: "monom a m * monom b n = monom (a * b) (m + n)"
   by (induct m, simp add: monom_0 smult_monom, simp add: monom_Suc)
 
-
-subsection {* The unit polynomial and exponentiation *}
-
 instantiation poly :: (comm_semiring_1) comm_semiring_1
 begin
 
-definition
-  one_poly_def:
-    "1 = pCons 1 0"
+definition one_poly_def:
+  "1 = pCons 1 0"
 
 instance proof
   fix p :: "'a poly" show "1 * p = p"
-    unfolding one_poly_def
-    by simp
+    unfolding one_poly_def by simp
 next
   show "0 \<noteq> (1::'a poly)"
     unfolding one_poly_def by simp
 qed
 
 end
 
 instance poly :: (comm_semiring_1_cancel) comm_semiring_1_cancel ..
+
+instance poly :: (comm_ring) comm_ring ..
+
+instance poly :: (comm_ring_1) comm_ring_1 ..
 
 lemma coeff_1 [simp]: "coeff 1 n = (if n = 0 then 1 else 0)"
   unfolding one_poly_def
   by (simp add: coeff_pCons split: nat.split)
 
 lemma degree_1 [simp]: "degree 1 = 0"
   unfolding one_poly_def
   by (rule degree_pCons_0)
 
-text {* Lemmas about divisibility *}
+lemma coeffs_1_eq [simp, code abstract]:
+  "coeffs 1 = [1]"
+  by (simp add: one_poly_def)
+
+lemma degree_power_le:
+  "degree (p ^ n) \<le> degree p * n"
+  by (induct n) (auto intro: order_trans degree_mult_le)
+
+lemma poly_smult [simp]:
+  "poly (smult a p) x = a * poly p x"
+  by (induct p, simp, simp add: algebra_simps)
+
+lemma poly_mult [simp]:
+  "poly (p * q) x = poly p x * poly q x"
+  by (induct p, simp_all, simp add: algebra_simps)
+
+lemma poly_1 [simp]:
+  "poly 1 x = 1"
+  by (simp add: one_poly_def)
+
+lemma poly_power [simp]:
+  fixes p :: "'a::{comm_semiring_1} poly"
+  shows "poly (p ^ n) x = poly p x ^ n"
+  by (induct n) simp_all
+
+
+subsection {* Lemmas about divisibility *}
 
 lemma dvd_smult: "p dvd q \<Longrightarrow> p dvd smult a q"
 proof -
   assume "p dvd q"
   then obtain k where "q = p * k" ..

 
 lemma smult_dvd_iff:
   fixes a :: "'a::field"
   shows "smult a p dvd q \<longleftrightarrow> (if a = 0 then q = 0 else p dvd q)"
   by (auto elim: smult_dvd smult_dvd_cancel)
-
-lemma degree_power_le: "degree (p ^ n) \<le> degree p * n"
-by (induct n, simp, auto intro: order_trans degree_mult_le)
-
-instance poly :: (comm_ring) comm_ring ..
-
-instance poly :: (comm_ring_1) comm_ring_1 ..
 
 
 subsection {* Polynomials form an integral domain *}
 
 lemma coeff_mult_degree_sum:

         coeff p (degree p) * coeff q (degree q)"
     by (rule coeff_mult_degree_sum)
   also have "coeff p (degree p) * coeff q (degree q) \<noteq> 0"
     using `p \<noteq> 0` and `q \<noteq> 0` by simp
   finally have "\<exists>n. coeff (p * q) n \<noteq> 0" ..
-  thus "p * q \<noteq> 0" by (simp add: expand_poly_eq)
+  thus "p * q \<noteq> 0" by (simp add: poly_eq_iff)
 qed
 
 lemma degree_mult_eq:
   fixes p q :: "'a::idom poly"
   shows "\<lbrakk>p \<noteq> 0; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree (p * q) = degree p + degree q"

   by (erule dvdE, simp add: degree_mult_eq)
 
 
 subsection {* Polynomials form an ordered integral domain *}
 
-definition
-  pos_poly :: "'a::linordered_idom poly \<Rightarrow> bool"
+definition pos_poly :: "'a::linordered_idom poly \<Rightarrow> bool"
 where
   "pos_poly p \<longleftrightarrow> 0 < coeff p (degree p)"
 
 lemma pos_poly_pCons:
   "pos_poly (pCons a p) \<longleftrightarrow> pos_poly p \<or> (p = 0 \<and> 0 < a)"

   apply auto
   done
 
 lemma pos_poly_total: "p = 0 \<or> pos_poly p \<or> pos_poly (- p)"
 by (induct p) (auto simp add: pos_poly_pCons)
+
+lemma last_coeffs_eq_coeff_degree:
+  "p \<noteq> 0 \<Longrightarrow> last (coeffs p) = coeff p (degree p)"
+  by (simp add: coeffs_def)
+
+lemma pos_poly_coeffs [code]:
+  "pos_poly p \<longleftrightarrow> (let as = coeffs p in as \<noteq> [] \<and> last as > 0)" (is "?P \<longleftrightarrow> ?Q")
+proof
+  assume ?Q then show ?P by (auto simp add: pos_poly_def last_coeffs_eq_coeff_degree)
+next
+  assume ?P then have *: "0 < coeff p (degree p)" by (simp add: pos_poly_def)
+  then have "p \<noteq> 0" by auto
+  with * show ?Q by (simp add: last_coeffs_eq_coeff_degree)
+qed
 
 instantiation poly :: (linordered_idom) linordered_idom
 begin
 
 definition

 end
 
 text {* TODO: Simplification rules for comparisons *}
 
 
+subsection {* Synthetic division and polynomial roots *}
+
+text {*
+  Synthetic division is simply division by the linear polynomial @{term "x - c"}.
+*}
+
+definition synthetic_divmod :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly \<times> 'a"
+where
+  "synthetic_divmod p c = fold_coeffs (\<lambda>a (q, r). (pCons r q, a + c * r)) p (0, 0)"
+
+definition synthetic_div :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly"
+where
+  "synthetic_div p c = fst (synthetic_divmod p c)"
+
+lemma synthetic_divmod_0 [simp]:
+  "synthetic_divmod 0 c = (0, 0)"
+  by (simp add: synthetic_divmod_def)
+
+lemma synthetic_divmod_pCons [simp]:
+  "synthetic_divmod (pCons a p) c = (\<lambda>(q, r). (pCons r q, a + c * r)) (synthetic_divmod p c)"
+  by (cases "p = 0 \<and> a = 0") (auto simp add: synthetic_divmod_def)
+
+lemma synthetic_div_0 [simp]:
+  "synthetic_div 0 c = 0"
+  unfolding synthetic_div_def by simp
+
+lemma synthetic_div_unique_lemma: "smult c p = pCons a p \<Longrightarrow> p = 0"
+by (induct p arbitrary: a) simp_all
+
+lemma snd_synthetic_divmod:
+  "snd (synthetic_divmod p c) = poly p c"
+  by (induct p, simp, simp add: split_def)
+
+lemma synthetic_div_pCons [simp]:
+  "synthetic_div (pCons a p) c = pCons (poly p c) (synthetic_div p c)"
+  unfolding synthetic_div_def
+  by (simp add: split_def snd_synthetic_divmod)
+
+lemma synthetic_div_eq_0_iff:
+  "synthetic_div p c = 0 \<longleftrightarrow> degree p = 0"
+  by (induct p, simp, case_tac p, simp)
+
+lemma degree_synthetic_div:
+  "degree (synthetic_div p c) = degree p - 1"
+  by (induct p, simp, simp add: synthetic_div_eq_0_iff)
+
+lemma synthetic_div_correct:
+  "p + smult c (synthetic_div p c) = pCons (poly p c) (synthetic_div p c)"
+  by (induct p) simp_all
+
+lemma synthetic_div_unique:
+  "p + smult c q = pCons r q \<Longrightarrow> r = poly p c \<and> q = synthetic_div p c"
+apply (induct p arbitrary: q r)
+apply (simp, frule synthetic_div_unique_lemma, simp)
+apply (case_tac q, force)
+done
+
+lemma synthetic_div_correct':
+  fixes c :: "'a::comm_ring_1"
+  shows "[:-c, 1:] * synthetic_div p c + [:poly p c:] = p"
+  using synthetic_div_correct [of p c]
+  by (simp add: algebra_simps)
+
+lemma poly_eq_0_iff_dvd:
+  fixes c :: "'a::idom"
+  shows "poly p c = 0 \<longleftrightarrow> [:-c, 1:] dvd p"
+proof
+  assume "poly p c = 0"
+  with synthetic_div_correct' [of c p]
+  have "p = [:-c, 1:] * synthetic_div p c" by simp
+  then show "[:-c, 1:] dvd p" ..
+next
+  assume "[:-c, 1:] dvd p"
+  then obtain k where "p = [:-c, 1:] * k" by (rule dvdE)
+  then show "poly p c = 0" by simp
+qed
+
+lemma dvd_iff_poly_eq_0:
+  fixes c :: "'a::idom"
+  shows "[:c, 1:] dvd p \<longleftrightarrow> poly p (-c) = 0"
+  by (simp add: poly_eq_0_iff_dvd)
+
+lemma poly_roots_finite:
+  fixes p :: "'a::idom poly"
+  shows "p \<noteq> 0 \<Longrightarrow> finite {x. poly p x = 0}"
+proof (induct n \<equiv> "degree p" arbitrary: p)
+  case (0 p)
+  then obtain a where "a \<noteq> 0" and "p = [:a:]"
+    by (cases p, simp split: if_splits)
+  then show "finite {x. poly p x = 0}" by simp
+next
+  case (Suc n p)
+  show "finite {x. poly p x = 0}"
+  proof (cases "\<exists>x. poly p x = 0")
+    case False
+    then show "finite {x. poly p x = 0}" by simp
+  next
+    case True
+    then obtain a where "poly p a = 0" ..
+    then have "[:-a, 1:] dvd p" by (simp only: poly_eq_0_iff_dvd)
+    then obtain k where k: "p = [:-a, 1:] * k" ..
+    with `p \<noteq> 0` have "k \<noteq> 0" by auto
+    with k have "degree p = Suc (degree k)"
+      by (simp add: degree_mult_eq del: mult_pCons_left)
+    with `Suc n = degree p` have "n = degree k" by simp
+    then have "finite {x. poly k x = 0}" using `k \<noteq> 0` by (rule Suc.hyps)
+    then have "finite (insert a {x. poly k x = 0})" by simp
+    then show "finite {x. poly p x = 0}"
+      by (simp add: k uminus_add_conv_diff Collect_disj_eq
+               del: mult_pCons_left)
+  qed
+qed
+
+lemma poly_eq_poly_eq_iff:
+  fixes p q :: "'a::{idom,ring_char_0} poly"
+  shows "poly p = poly q \<longleftrightarrow> p = q" (is "?P \<longleftrightarrow> ?Q")
+proof
+  assume ?Q then show ?P by simp
+next
+  { fix p :: "'a::{idom,ring_char_0} poly"
+    have "poly p = poly 0 \<longleftrightarrow> p = 0"
+      apply (cases "p = 0", simp_all)
+      apply (drule poly_roots_finite)
+      apply (auto simp add: infinite_UNIV_char_0)
+      done
+  } note this [of "p - q"]
+  moreover assume ?P
+  ultimately show ?Q by auto
+qed
+
+lemma poly_all_0_iff_0:
+  fixes p :: "'a::{ring_char_0, idom} poly"
+  shows "(\<forall>x. poly p x = 0) \<longleftrightarrow> p = 0"
+  by (auto simp add: poly_eq_poly_eq_iff [symmetric])
+
+
 subsection {* Long division of polynomials *}
 
-definition
-  pdivmod_rel :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> bool"
+definition pdivmod_rel :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> bool"
 where
   "pdivmod_rel x y q r \<longleftrightarrow>
     x = q * y + r \<and> (if y = 0 then q = 0 else r = 0 \<or> degree r < degree y)"
 
 lemma pdivmod_rel_0:

 unfolding b
 apply (rule mod_poly_eq)
 apply (rule pdivmod_rel_pCons [OF pdivmod_rel y refl])
 done
 
-
-subsection {* GCD of polynomials *}
-
-function
-  poly_gcd :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly" where
-  "poly_gcd x 0 = smult (inverse (coeff x (degree x))) x"
-| "y \<noteq> 0 \<Longrightarrow> poly_gcd x y = poly_gcd y (x mod y)"
-by auto
-
-termination poly_gcd
-by (relation "measure (\<lambda>(x, y). if y = 0 then 0 else Suc (degree y))")
-   (auto dest: degree_mod_less)
-
-declare poly_gcd.simps [simp del]
-
-lemma poly_gcd_dvd1 [iff]: "poly_gcd x y dvd x"
-  and poly_gcd_dvd2 [iff]: "poly_gcd x y dvd y"
-  apply (induct x y rule: poly_gcd.induct)
-  apply (simp_all add: poly_gcd.simps)
-  apply (fastforce simp add: smult_dvd_iff dest: inverse_zero_imp_zero)
-  apply (blast dest: dvd_mod_imp_dvd)
+definition pdivmod :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<times> 'a poly"
+where
+  "pdivmod p q = (p div q, p mod q)"
+
+lemma div_poly_code [code]: 
+  "p div q = fst (pdivmod p q)"
+  by (simp add: pdivmod_def)
+
+lemma mod_poly_code [code]:
+  "p mod q = snd (pdivmod p q)"
+  by (simp add: pdivmod_def)
+
+lemma pdivmod_0:
+  "pdivmod 0 q = (0, 0)"
+  by (simp add: pdivmod_def)
+
+lemma pdivmod_pCons:
+  "pdivmod (pCons a p) q =
+    (if q = 0 then (0, pCons a p) else
+      (let (s, r) = pdivmod p q;
+           b = coeff (pCons a r) (degree q) / coeff q (degree q)
+        in (pCons b s, pCons a r - smult b q)))"
+  apply (simp add: pdivmod_def Let_def, safe)
+  apply (rule div_poly_eq)
+  apply (erule pdivmod_rel_pCons [OF pdivmod_rel _ refl])
+  apply (rule mod_poly_eq)
+  apply (erule pdivmod_rel_pCons [OF pdivmod_rel _ refl])
   done
 
-lemma poly_gcd_greatest: "k dvd x \<Longrightarrow> k dvd y \<Longrightarrow> k dvd poly_gcd x y"
-  by (induct x y rule: poly_gcd.induct)
-     (simp_all add: poly_gcd.simps dvd_mod dvd_smult)
-
-lemma dvd_poly_gcd_iff [iff]:
-  "k dvd poly_gcd x y \<longleftrightarrow> k dvd x \<and> k dvd y"
-  by (blast intro!: poly_gcd_greatest intro: dvd_trans)
-
-lemma poly_gcd_monic:
-  "coeff (poly_gcd x y) (degree (poly_gcd x y)) =
-    (if x = 0 \<and> y = 0 then 0 else 1)"
-  by (induct x y rule: poly_gcd.induct)
-     (simp_all add: poly_gcd.simps nonzero_imp_inverse_nonzero)
-
-lemma poly_gcd_zero_iff [simp]:
-  "poly_gcd x y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
-  by (simp only: dvd_0_left_iff [symmetric] dvd_poly_gcd_iff)
-
-lemma poly_gcd_0_0 [simp]: "poly_gcd 0 0 = 0"
-  by simp
-
-lemma poly_dvd_antisym:
-  fixes p q :: "'a::idom poly"
-  assumes coeff: "coeff p (degree p) = coeff q (degree q)"
-  assumes dvd1: "p dvd q" and dvd2: "q dvd p" shows "p = q"
-proof (cases "p = 0")
-  case True with coeff show "p = q" by simp
-next
-  case False with coeff have "q \<noteq> 0" by auto
-  have degree: "degree p = degree q"
-    using `p dvd q` `q dvd p` `p \<noteq> 0` `q \<noteq> 0`
-    by (intro order_antisym dvd_imp_degree_le)
-
-  from `p dvd q` obtain a where a: "q = p * a" ..
-  with `q \<noteq> 0` have "a \<noteq> 0" by auto
-  with degree a `p \<noteq> 0` have "degree a = 0"
-    by (simp add: degree_mult_eq)
-  with coeff a show "p = q"
-    by (cases a, auto split: if_splits)
-qed
-
-lemma poly_gcd_unique:
-  assumes dvd1: "d dvd x" and dvd2: "d dvd y"
-    and greatest: "\<And>k. k dvd x \<Longrightarrow> k dvd y \<Longrightarrow> k dvd d"
-    and monic: "coeff d (degree d) = (if x = 0 \<and> y = 0 then 0 else 1)"
-  shows "poly_gcd x y = d"
-proof -
-  have "coeff (poly_gcd x y) (degree (poly_gcd x y)) = coeff d (degree d)"
-    by (simp_all add: poly_gcd_monic monic)
-  moreover have "poly_gcd x y dvd d"
-    using poly_gcd_dvd1 poly_gcd_dvd2 by (rule greatest)
-  moreover have "d dvd poly_gcd x y"
-    using dvd1 dvd2 by (rule poly_gcd_greatest)
-  ultimately show ?thesis
-    by (rule poly_dvd_antisym)
-qed
-
-interpretation poly_gcd: abel_semigroup poly_gcd
-proof
-  fix x y z :: "'a poly"
-  show "poly_gcd (poly_gcd x y) z = poly_gcd x (poly_gcd y z)"
-    by (rule poly_gcd_unique) (auto intro: dvd_trans simp add: poly_gcd_monic)
-  show "poly_gcd x y = poly_gcd y x"
-    by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)
-qed
-
-lemmas poly_gcd_assoc = poly_gcd.assoc
-lemmas poly_gcd_commute = poly_gcd.commute
-lemmas poly_gcd_left_commute = poly_gcd.left_commute
-
-lemmas poly_gcd_ac = poly_gcd_assoc poly_gcd_commute poly_gcd_left_commute
-
-lemma poly_gcd_1_left [simp]: "poly_gcd 1 y = 1"
-by (rule poly_gcd_unique) simp_all
-
-lemma poly_gcd_1_right [simp]: "poly_gcd x 1 = 1"
-by (rule poly_gcd_unique) simp_all
-
-lemma poly_gcd_minus_left [simp]: "poly_gcd (- x) y = poly_gcd x y"
-by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)
-
-lemma poly_gcd_minus_right [simp]: "poly_gcd x (- y) = poly_gcd x y"
-by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)
-
-
-subsection {* Evaluation of polynomials *}
-
-definition
-  poly :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a" where
-  "poly = poly_rec (\<lambda>x. 0) (\<lambda>a p f x. a + x * f x)"
-
-lemma poly_0 [simp]: "poly 0 x = 0"
-  unfolding poly_def by (simp add: poly_rec_0)
-
-lemma poly_pCons [simp]: "poly (pCons a p) x = a + x * poly p x"
-  unfolding poly_def by (simp add: poly_rec_pCons)
-
-lemma poly_1 [simp]: "poly 1 x = 1"
-  unfolding one_poly_def by simp
-
-lemma poly_monom:
-  fixes a x :: "'a::{comm_semiring_1}"
-  shows "poly (monom a n) x = a * x ^ n"
-  by (induct n, simp add: monom_0, simp add: monom_Suc power_Suc mult_ac)
-
-lemma poly_add [simp]: "poly (p + q) x = poly p x + poly q x"
-  apply (induct p arbitrary: q, simp)
-  apply (case_tac q, simp, simp add: algebra_simps)
+lemma pdivmod_fold_coeffs [code]:
+  "pdivmod p q = (if q = 0 then (0, p)
+    else fold_coeffs (\<lambda>a (s, r).
+      let b = coeff (pCons a r) (degree q) / coeff q (degree q)
+      in (pCons b s, pCons a r - smult b q)
+   ) p (0, 0))"
+  apply (cases "q = 0")
+  apply (simp add: pdivmod_def)
+  apply (rule sym)
+  apply (induct p)
+  apply (simp_all add: pdivmod_0 pdivmod_pCons)
+  apply (case_tac "a = 0 \<and> p = 0")
+  apply (auto simp add: pdivmod_def)
   done
 
-lemma poly_minus [simp]:
-  fixes x :: "'a::comm_ring"
-  shows "poly (- p) x = - poly p x"
-  by (induct p, simp_all)
-
-lemma poly_diff [simp]:
-  fixes x :: "'a::comm_ring"
-  shows "poly (p - q) x = poly p x - poly q x"
-  by (simp add: diff_minus)
-
-lemma poly_setsum: "poly (\<Sum>k\<in>A. p k) x = (\<Sum>k\<in>A. poly (p k) x)"
-  by (cases "finite A", induct set: finite, simp_all)
-
-lemma poly_smult [simp]: "poly (smult a p) x = a * poly p x"
-  by (induct p, simp, simp add: algebra_simps)
-
-lemma poly_mult [simp]: "poly (p * q) x = poly p x * poly q x"
-  by (induct p, simp_all, simp add: algebra_simps)
-
-lemma poly_power [simp]:
-  fixes p :: "'a::{comm_semiring_1} poly"
-  shows "poly (p ^ n) x = poly p x ^ n"
-  by (induct n, simp, simp add: power_Suc)
-
-
-subsection {* Synthetic division *}
-
-text {*
-  Synthetic division is simply division by the
-  linear polynomial @{term "x - c"}.
-*}
-
-definition
-  synthetic_divmod :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly \<times> 'a"
-where
-  "synthetic_divmod p c =
-    poly_rec (0, 0) (\<lambda>a p (q, r). (pCons r q, a + c * r)) p"
-
-definition
-  synthetic_div :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly"
-where
-  "synthetic_div p c = fst (synthetic_divmod p c)"
-
-lemma synthetic_divmod_0 [simp]:
-  "synthetic_divmod 0 c = (0, 0)"
-  unfolding synthetic_divmod_def
-  by (simp add: poly_rec_0)
-
-lemma synthetic_divmod_pCons [simp]:
-  "synthetic_divmod (pCons a p) c =
-    (\<lambda>(q, r). (pCons r q, a + c * r)) (synthetic_divmod p c)"
-  unfolding synthetic_divmod_def
-  by (simp add: poly_rec_pCons)
-
-lemma snd_synthetic_divmod: "snd (synthetic_divmod p c) = poly p c"
-  by (induct p, simp, simp add: split_def)
-
-lemma synthetic_div_0 [simp]: "synthetic_div 0 c = 0"
-  unfolding synthetic_div_def by simp
-
-lemma synthetic_div_pCons [simp]:
-  "synthetic_div (pCons a p) c = pCons (poly p c) (synthetic_div p c)"
-  unfolding synthetic_div_def
-  by (simp add: split_def snd_synthetic_divmod)
-
-lemma synthetic_div_eq_0_iff:
-  "synthetic_div p c = 0 \<longleftrightarrow> degree p = 0"
-  by (induct p, simp, case_tac p, simp)
-
-lemma degree_synthetic_div:
-  "degree (synthetic_div p c) = degree p - 1"
-  by (induct p, simp, simp add: synthetic_div_eq_0_iff)
-
-lemma synthetic_div_correct:
-  "p + smult c (synthetic_div p c) = pCons (poly p c) (synthetic_div p c)"
-  by (induct p) simp_all
-
-lemma synthetic_div_unique_lemma: "smult c p = pCons a p \<Longrightarrow> p = 0"
-by (induct p arbitrary: a) simp_all
-
-lemma synthetic_div_unique:
-  "p + smult c q = pCons r q \<Longrightarrow> r = poly p c \<and> q = synthetic_div p c"
-apply (induct p arbitrary: q r)
-apply (simp, frule synthetic_div_unique_lemma, simp)
-apply (case_tac q, force)
-done
-
-lemma synthetic_div_correct':
-  fixes c :: "'a::comm_ring_1"
-  shows "[:-c, 1:] * synthetic_div p c + [:poly p c:] = p"
-  using synthetic_div_correct [of p c]
-  by (simp add: algebra_simps)
-
-lemma poly_eq_0_iff_dvd:
-  fixes c :: "'a::idom"
-  shows "poly p c = 0 \<longleftrightarrow> [:-c, 1:] dvd p"
-proof
-  assume "poly p c = 0"
-  with synthetic_div_correct' [of c p]
-  have "p = [:-c, 1:] * synthetic_div p c" by simp
-  then show "[:-c, 1:] dvd p" ..
-next
-  assume "[:-c, 1:] dvd p"
-  then obtain k where "p = [:-c, 1:] * k" by (rule dvdE)
-  then show "poly p c = 0" by simp
-qed
-
-lemma dvd_iff_poly_eq_0:
-  fixes c :: "'a::idom"
-  shows "[:c, 1:] dvd p \<longleftrightarrow> poly p (-c) = 0"
-  by (simp add: poly_eq_0_iff_dvd)
-
-lemma poly_roots_finite:
-  fixes p :: "'a::idom poly"
-  shows "p \<noteq> 0 \<Longrightarrow> finite {x. poly p x = 0}"
-proof (induct n \<equiv> "degree p" arbitrary: p)
-  case (0 p)
-  then obtain a where "a \<noteq> 0" and "p = [:a:]"
-    by (cases p, simp split: if_splits)
-  then show "finite {x. poly p x = 0}" by simp
-next
-  case (Suc n p)
-  show "finite {x. poly p x = 0}"
-  proof (cases "\<exists>x. poly p x = 0")
-    case False
-    then show "finite {x. poly p x = 0}" by simp
-  next
-    case True
-    then obtain a where "poly p a = 0" ..
-    then have "[:-a, 1:] dvd p" by (simp only: poly_eq_0_iff_dvd)
-    then obtain k where k: "p = [:-a, 1:] * k" ..
-    with `p \<noteq> 0` have "k \<noteq> 0" by auto
-    with k have "degree p = Suc (degree k)"
-      by (simp add: degree_mult_eq del: mult_pCons_left)
-    with `Suc n = degree p` have "n = degree k" by simp
-    then have "finite {x. poly k x = 0}" using `k \<noteq> 0` by (rule Suc.hyps)
-    then have "finite (insert a {x. poly k x = 0})" by simp
-    then show "finite {x. poly p x = 0}"
-      by (simp add: k uminus_add_conv_diff Collect_disj_eq
-               del: mult_pCons_left)
-  qed
-qed
-
-lemma poly_zero:
-  fixes p :: "'a::{idom,ring_char_0} poly"
-  shows "poly p = poly 0 \<longleftrightarrow> p = 0"
-apply (cases "p = 0", simp_all)
-apply (drule poly_roots_finite)
-apply (auto simp add: infinite_UNIV_char_0)
-done
-
-lemma poly_eq_iff:
-  fixes p q :: "'a::{idom,ring_char_0} poly"
-  shows "poly p = poly q \<longleftrightarrow> p = q"
-  using poly_zero [of "p - q"]
-  by (simp add: fun_eq_iff)
-
-
-subsection {* Composition of polynomials *}
-
-definition
-  pcompose :: "'a::comm_semiring_0 poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
-where
-  "pcompose p q = poly_rec 0 (\<lambda>a _ c. [:a:] + q * c) p"
-
-lemma pcompose_0 [simp]: "pcompose 0 q = 0"
-  unfolding pcompose_def by (simp add: poly_rec_0)
-
-lemma pcompose_pCons:
-  "pcompose (pCons a p) q = [:a:] + q * pcompose p q"
-  unfolding pcompose_def by (simp add: poly_rec_pCons)
-
-lemma poly_pcompose: "poly (pcompose p q) x = poly p (poly q x)"
-  by (induct p) (simp_all add: pcompose_pCons)
-
-lemma degree_pcompose_le:
-  "degree (pcompose p q) \<le> degree p * degree q"
-apply (induct p, simp)
-apply (simp add: pcompose_pCons, clarify)
-apply (rule degree_add_le, simp)
-apply (rule order_trans [OF degree_mult_le], simp)
-done
-
 
 subsection {* Order of polynomial roots *}
 
-definition
-  order :: "'a::idom \<Rightarrow> 'a poly \<Rightarrow> nat"
+definition order :: "'a::idom \<Rightarrow> 'a poly \<Rightarrow> nat"
 where
   "order a p = (LEAST n. \<not> [:-a, 1:] ^ Suc n dvd p)"
 
 lemma coeff_linear_power:
   fixes a :: "'a::comm_semiring_1"

 apply (simp only: order_def)
 apply (drule not_less_Least, simp)
 done
 
 
-subsection {* Configuration of the code generator *}
-
-code_datatype "0::'a::zero poly" pCons
-
-quickcheck_generator poly constructors: "0::'a::zero poly", pCons
-
-instantiation poly :: ("{zero, equal}") equal
+subsection {* GCD of polynomials *}
+
+instantiation poly :: (field) gcd
 begin
 
-definition
-  "HOL.equal (p::'a poly) q \<longleftrightarrow> p = q"
-
-instance proof
-qed (rule equal_poly_def)
+function gcd_poly :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
+where
+  "gcd (x::'a poly) 0 = smult (inverse (coeff x (degree x))) x"
+| "y \<noteq> 0 \<Longrightarrow> gcd (x::'a poly) y = gcd y (x mod y)"
+by auto
+
+termination "gcd :: _ poly \<Rightarrow> _"
+by (relation "measure (\<lambda>(x, y). if y = 0 then 0 else Suc (degree y))")
+   (auto dest: degree_mod_less)
+
+declare gcd_poly.simps [simp del]
+
+instance ..
 
 end
 
-lemma eq_poly_code [code]:
-  "HOL.equal (0::_ poly) (0::_ poly) \<longleftrightarrow> True"
-  "HOL.equal (0::_ poly) (pCons b q) \<longleftrightarrow> HOL.equal 0 b \<and> HOL.equal 0 q"
-  "HOL.equal (pCons a p) (0::_ poly) \<longleftrightarrow> HOL.equal a 0 \<and> HOL.equal p 0"
-  "HOL.equal (pCons a p) (pCons b q) \<longleftrightarrow> HOL.equal a b \<and> HOL.equal p q"
-  by (simp_all add: equal)
-
-lemma [code nbe]:
-  "HOL.equal (p :: _ poly) p \<longleftrightarrow> True"
-  by (fact equal_refl)
-
-lemmas coeff_code [code] =
-  coeff_0 coeff_pCons_0 coeff_pCons_Suc
-
-lemmas degree_code [code] =
-  degree_0 degree_pCons_eq_if
-
-lemmas monom_poly_code [code] =
-  monom_0 monom_Suc
-
-lemma add_poly_code [code]:
-  "0 + q = (q :: _ poly)"
-  "p + 0 = (p :: _ poly)"
-  "pCons a p + pCons b q = pCons (a + b) (p + q)"
-by simp_all
-
-lemma minus_poly_code [code]:
-  "- 0 = (0 :: _ poly)"
-  "- pCons a p = pCons (- a) (- p)"
-by simp_all
-
-lemma diff_poly_code [code]:
-  "0 - q = (- q :: _ poly)"
-  "p - 0 = (p :: _ poly)"
-  "pCons a p - pCons b q = pCons (a - b) (p - q)"
-by simp_all
-
-lemmas smult_poly_code [code] =
-  smult_0_right smult_pCons
-
-lemma mult_poly_code [code]:
-  "0 * q = (0 :: _ poly)"
-  "pCons a p * q = smult a q + pCons 0 (p * q)"
-by simp_all
-
-lemmas poly_code [code] =
-  poly_0 poly_pCons
-
-lemmas synthetic_divmod_code [code] =
-  synthetic_divmod_0 synthetic_divmod_pCons
-
-text {* code generator setup for div and mod *}
-
-definition
-  pdivmod :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<times> 'a poly"
+lemma
+  fixes x y :: "_ poly"
+  shows poly_gcd_dvd1 [iff]: "gcd x y dvd x"
+    and poly_gcd_dvd2 [iff]: "gcd x y dvd y"
+  apply (induct x y rule: gcd_poly.induct)
+  apply (simp_all add: gcd_poly.simps)
+  apply (fastforce simp add: smult_dvd_iff dest: inverse_zero_imp_zero)
+  apply (blast dest: dvd_mod_imp_dvd)
+  done
+
+lemma poly_gcd_greatest:
+  fixes k x y :: "_ poly"
+  shows "k dvd x \<Longrightarrow> k dvd y \<Longrightarrow> k dvd gcd x y"
+  by (induct x y rule: gcd_poly.induct)
+     (simp_all add: gcd_poly.simps dvd_mod dvd_smult)
+
+lemma dvd_poly_gcd_iff [iff]:
+  fixes k x y :: "_ poly"
+  shows "k dvd gcd x y \<longleftrightarrow> k dvd x \<and> k dvd y"
+  by (blast intro!: poly_gcd_greatest intro: dvd_trans)
+
+lemma poly_gcd_monic:
+  fixes x y :: "_ poly"
+  shows "coeff (gcd x y) (degree (gcd x y)) =
+    (if x = 0 \<and> y = 0 then 0 else 1)"
+  by (induct x y rule: gcd_poly.induct)
+     (simp_all add: gcd_poly.simps nonzero_imp_inverse_nonzero)
+
+lemma poly_gcd_zero_iff [simp]:
+  fixes x y :: "_ poly"
+  shows "gcd x y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
+  by (simp only: dvd_0_left_iff [symmetric] dvd_poly_gcd_iff)
+
+lemma poly_gcd_0_0 [simp]:
+  "gcd (0::_ poly) 0 = 0"
+  by simp
+
+lemma poly_dvd_antisym:
+  fixes p q :: "'a::idom poly"
+  assumes coeff: "coeff p (degree p) = coeff q (degree q)"
+  assumes dvd1: "p dvd q" and dvd2: "q dvd p" shows "p = q"
+proof (cases "p = 0")
+  case True with coeff show "p = q" by simp
+next
+  case False with coeff have "q \<noteq> 0" by auto
+  have degree: "degree p = degree q"
+    using `p dvd q` `q dvd p` `p \<noteq> 0` `q \<noteq> 0`
+    by (intro order_antisym dvd_imp_degree_le)
+
+  from `p dvd q` obtain a where a: "q = p * a" ..
+  with `q \<noteq> 0` have "a \<noteq> 0" by auto
+  with degree a `p \<noteq> 0` have "degree a = 0"
+    by (simp add: degree_mult_eq)
+  with coeff a show "p = q"
+    by (cases a, auto split: if_splits)
+qed
+
+lemma poly_gcd_unique:
+  fixes d x y :: "_ poly"
+  assumes dvd1: "d dvd x" and dvd2: "d dvd y"
+    and greatest: "\<And>k. k dvd x \<Longrightarrow> k dvd y \<Longrightarrow> k dvd d"
+    and monic: "coeff d (degree d) = (if x = 0 \<and> y = 0 then 0 else 1)"
+  shows "gcd x y = d"
+proof -
+  have "coeff (gcd x y) (degree (gcd x y)) = coeff d (degree d)"
+    by (simp_all add: poly_gcd_monic monic)
+  moreover have "gcd x y dvd d"
+    using poly_gcd_dvd1 poly_gcd_dvd2 by (rule greatest)
+  moreover have "d dvd gcd x y"
+    using dvd1 dvd2 by (rule poly_gcd_greatest)
+  ultimately show ?thesis
+    by (rule poly_dvd_antisym)
+qed
+
+interpretation gcd_poly!: abel_semigroup "gcd :: _ poly \<Rightarrow> _"
+proof
+  fix x y z :: "'a poly"
+  show "gcd (gcd x y) z = gcd x (gcd y z)"
+    by (rule poly_gcd_unique) (auto intro: dvd_trans simp add: poly_gcd_monic)
+  show "gcd x y = gcd y x"
+    by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)
+qed
+
+lemmas poly_gcd_assoc = gcd_poly.assoc
+lemmas poly_gcd_commute = gcd_poly.commute
+lemmas poly_gcd_left_commute = gcd_poly.left_commute
+
+lemmas poly_gcd_ac = poly_gcd_assoc poly_gcd_commute poly_gcd_left_commute
+
+lemma poly_gcd_1_left [simp]: "gcd 1 y = (1 :: _ poly)"
+by (rule poly_gcd_unique) simp_all
+
+lemma poly_gcd_1_right [simp]: "gcd x 1 = (1 :: _ poly)"
+by (rule poly_gcd_unique) simp_all
+
+lemma poly_gcd_minus_left [simp]: "gcd (- x) y = gcd x (y :: _ poly)"
+by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)
+
+lemma poly_gcd_minus_right [simp]: "gcd x (- y) = gcd x (y :: _ poly)"
+by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)
+
+lemma poly_gcd_code [code]:
+  "gcd x y = (if y = 0 then smult (inverse (coeff x (degree x))) x else gcd y (x mod (y :: _ poly)))"
+  by (simp add: gcd_poly.simps)
+
+
+subsection {* Composition of polynomials *}
+
+definition pcompose :: "'a::comm_semiring_0 poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
 where
-  "pdivmod x y = (x div y, x mod y)"
-
-lemma div_poly_code [code]: "x div y = fst (pdivmod x y)"
-  unfolding pdivmod_def by simp
-
-lemma mod_poly_code [code]: "x mod y = snd (pdivmod x y)"
-  unfolding pdivmod_def by simp
-
-lemma pdivmod_0 [code]: "pdivmod 0 y = (0, 0)"
-  unfolding pdivmod_def by simp
-
-lemma pdivmod_pCons [code]:
-  "pdivmod (pCons a x) y =
-    (if y = 0 then (0, pCons a x) else
-      (let (q, r) = pdivmod x y;
-           b = coeff (pCons a r) (degree y) / coeff y (degree y)
-        in (pCons b q, pCons a r - smult b y)))"
-apply (simp add: pdivmod_def Let_def, safe)
-apply (rule div_poly_eq)
-apply (erule pdivmod_rel_pCons [OF pdivmod_rel _ refl])
-apply (rule mod_poly_eq)
-apply (erule pdivmod_rel_pCons [OF pdivmod_rel _ refl])
+  "pcompose p q = fold_coeffs (\<lambda>a c. [:a:] + q * c) p 0"
+
+lemma pcompose_0 [simp]:
+  "pcompose 0 q = 0"
+  by (simp add: pcompose_def)
+
+lemma pcompose_pCons:
+  "pcompose (pCons a p) q = [:a:] + q * pcompose p q"
+  by (cases "p = 0 \<and> a = 0") (auto simp add: pcompose_def)
+
+lemma poly_pcompose:
+  "poly (pcompose p q) x = poly p (poly q x)"
+  by (induct p) (simp_all add: pcompose_pCons)
+
+lemma degree_pcompose_le:
+  "degree (pcompose p q) \<le> degree p * degree q"
+apply (induct p, simp)
+apply (simp add: pcompose_pCons, clarify)
+apply (rule degree_add_le, simp)
+apply (rule order_trans [OF degree_mult_le], simp)
 done
 
-lemma poly_gcd_code [code]:
-  "poly_gcd x y =
-    (if y = 0 then smult (inverse (coeff x (degree x))) x
-              else poly_gcd y (x mod y))"
-  by (simp add: poly_gcd.simps)
+
+no_notation cCons (infixr "##" 65)
 
 end
+