(*  Title:      HOL/Library/Polynomial.thy

     Author:     Brian Huffman

     Author:     Clemens Ballarin

     Author:     Florian Haftmann

 *)

 section \<open>Polynomials as type over a ring structure\<close>

 theory Polynomial

 imports Main "~~/src/HOL/GCD" "~~/src/HOL/Library/More_List" "~~/src/HOL/Library/Infinite_Set"

 begin

 subsection \<open>Auxiliary: operations for lists (later) representing coefficients\<close>

 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 \<open>Definition of type \<open>poly\<close>\<close>

 typedef (overloaded) 'a poly = "{f :: nat \<Rightarrow> 'a::zero. \<forall>\<^sub>\<infinity> n. f n = 0}"

   morphisms coeff Abs_poly by (auto intro!: ALL_MOST)

 setup_lifting 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)

 lemma poly_eqI: "(\<And>n. coeff p n = coeff q n) \<Longrightarrow> p = q"

   by (simp add: poly_eq_iff)

 lemma MOST_coeff_eq_0: "\<forall>\<^sub>\<infinity> n. coeff p n = 0"

   using coeff [of p] by simp

 subsection \<open>Degree of a polynomial\<close>

 definition degree :: "'a::zero poly \<Rightarrow> nat"

 where

   "degree p = (LEAST n. \<forall>i>n. coeff p i = 0)"

 lemma coeff_eq_0:

   assumes "degree p < n"

   shows "coeff p n = 0"

 proof -

   have "\<exists>n. \<forall>i>n. coeff p i = 0"

     using MOST_coeff_eq_0 by (simp add: MOST_nat)

   then have "\<forall>i>degree p. coeff p i = 0"

     unfolding degree_def by (rule LeastI_ex)

   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)

 lemma degree_le: "\<forall>i>n. coeff p i = 0 \<Longrightarrow> degree p \<le> n"

   unfolding degree_def by (erule Least_le)

 lemma less_degree_imp: "n < degree p \<Longrightarrow> \<exists>i>n. coeff p i \<noteq> 0"

   unfolding degree_def by (drule not_less_Least, simp)

 subsection \<open>The zero polynomial\<close>

 instantiation poly :: (zero) zero

 begin

 lift_definition zero_poly :: "'a poly"

   is "\<lambda>_. 0" by (rule MOST_I) simp

 instance ..

 end

 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"

 proof (cases "degree p")

   case 0

   from \<open>p \<noteq> 0\<close> have "\<exists>n. coeff p n \<noteq> 0"

     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 \<open>coeff p n \<noteq> 0\<close> and \<open>degree p = 0\<close>

   show "coeff p (degree p) \<noteq> 0" by simp

 next

   case (Suc n)

   from \<open>degree p = Suc n\<close> have "n < degree p" by simp

   hence "\<exists>i>n. coeff p i \<noteq> 0" by (rule less_degree_imp)

   then obtain i where "n < i" and "coeff p i \<noteq> 0" by fast

   from \<open>degree p = Suc n\<close> and \<open>n < i\<close> have "degree p \<le> i" by simp

   also from \<open>coeff p i \<noteq> 0\<close> have "i \<le> degree p" by (rule le_degree)

   finally have "degree p = i" .

   with \<open>coeff p i \<noteq> 0\<close> show "coeff p (degree p) \<noteq> 0" by simp

 qed

 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 \<open>List-style constructor for polynomials\<close>

 lift_definition pCons :: "'a::zero \<Rightarrow> 'a poly \<Rightarrow> 'a poly"

   is "\<lambda>a p. case_nat a (coeff p)"

   by (rule MOST_SucD) (simp add: MOST_coeff_eq_0)

 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 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

 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]:

   "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

   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: 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 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 transfer

        (simp_all add: MOST_inj[where f=Suc and P="\<lambda>n. p n = 0" for p] fun_eq_iff Abs_poly_inverse

                  split: nat.split)

 qed

 lemma pCons_induct [case_names 0 pCons, induct type: poly]:

   assumes zero: "P 0"

   assumes pCons: "\<And>a p. a \<noteq> 0 \<or> p \<noteq> 0 \<Longrightarrow> P p \<Longrightarrow> P (pCons a p)"

   shows "P p"

 proof (induct p rule: measure_induct_rule [where f=degree])

   case (less p)

   obtain a q where "p = pCons a q" by (rule pCons_cases)

   have "P q"

   proof (cases "q = 0")

     case True

     then show "P q" by (simp add: zero)

   next

     case False

     then have "degree (pCons a q) = Suc (degree q)"

       by (rule degree_pCons_eq)

     then have "degree q < degree p"

       using \<open>p = pCons a q\<close> by simp

     then show "P q"

       by (rule less.hyps)

   qed

   have "P (pCons a q)"

   proof (cases "a \<noteq> 0 \<or> q \<noteq> 0")

     case True

     with \<open>P q\<close> show ?thesis by (auto intro: pCons)

   next

     case False

     with zero show ?thesis by simp

   qed

   then show ?case

     using \<open>p = pCons a q\<close> by simp

 qed

 lemma degree_eq_zeroE:

   fixes p :: "'a::zero poly"

   assumes "degree p = 0"

   obtains a where "p = pCons a 0"

 proof -

   obtain a q where p: "p = pCons a q" by (cases p)

   with assms have "q = 0" by (cases "q = 0") simp_all

   with p have "p = pCons a 0" by simp

   with that show thesis .

 qed

 subsection \<open>List-style syntax for polynomials\<close>

 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 \<open>Representation of polynomials by lists of coefficients\<close>

 primrec Poly :: "'a::zero list \<Rightarrow> 'a poly"

 where

   [code_post]: "Poly [] = 0"

 | [code_post]: "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)

 lemma degree_Poly: "degree (Poly xs) \<le> length xs"

   by (induction xs) simp_all

 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 (case_nat x f) ms = map f (map (\<lambda>n. n - 1) ms)"

       by (induct ms) (auto split: nat.split)

   }

   note * = this

   show ?thesis

     by (simp add: coeffs_def * upt_conv_Cons coeff_pCons map_decr_upt del: upt_Suc)

 qed

 lemma length_coeffs: "p \<noteq> 0 \<Longrightarrow> length (coeffs p) = degree p + 1"

   by (simp add: coeffs_def)

 lemma coeffs_nth:

   assumes "p \<noteq> 0" "n \<le> degree p"

   shows   "coeffs p ! n = coeff p n"

   using assms unfolding coeffs_def by (auto simp del: upt_Suc)

 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) auto

 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 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

   by standard (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 \<open>Fold combinator for polynomials\<close>

 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 \<open>Canonical morphism on polynomials -- evaluation\<close>

 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)" \<comment> \<open>The Horner Schema\<close>

 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)

 lemma poly_altdef: 

   "poly p (x :: 'a :: {comm_semiring_0, semiring_1}) = (\<Sum>i\<le>degree p. coeff p i * x ^ i)"

 proof (induction p rule: pCons_induct)

   case (pCons a p)

     show ?case

     proof (cases "p = 0")

       case False

       let ?p' = "pCons a p"

       note poly_pCons[of a p x]

       also note pCons.IH

       also have "a + x * (\<Sum>i\<le>degree p. coeff p i * x ^ i) =

                  coeff ?p' 0 * x^0 + (\<Sum>i\<le>degree p. coeff ?p' (Suc i) * x^Suc i)"

           by (simp add: field_simps setsum_right_distrib coeff_pCons)

       also note setsum_atMost_Suc_shift[symmetric]

       also note degree_pCons_eq[OF \<open>p \<noteq> 0\<close>, of a, symmetric]

       finally show ?thesis .

    qed simp

 qed simp

 subsection \<open>Monomials\<close>

 lift_definition monom :: "'a \<Rightarrow> nat \<Rightarrow> 'a::zero poly"

   is "\<lambda>a m n. if m = n then a else 0"

   by (simp add: MOST_iff_cofinite)

 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_eqI) simp

 lemma monom_eq_0_iff [simp]: "monom a n = 0 \<longleftrightarrow> a = 0"

   by (simp add: poly_eq_iff)

 lemma monom_eq_iff [simp]: "monom a n = monom b n \<longleftrightarrow> a = b"

   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 \<open>Addition and subtraction\<close>

 instantiation poly :: (comm_monoid_add) comm_monoid_add

 begin

 lift_definition plus_poly :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"

   is "\<lambda>p q n. coeff p n + coeff q n"

 proof -

   fix q p :: "'a poly"

   show "\<forall>\<^sub>\<infinity>n. coeff p n + coeff q n = 0"

     using MOST_coeff_eq_0[of p] MOST_coeff_eq_0[of q] by eventually_elim simp

 qed

 lemma coeff_add [simp]: "coeff (p + q) n = coeff p n + coeff q n"

   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: poly_eq_iff add.assoc)

   show "p + q = q + p"

     by (simp add: poly_eq_iff add.commute)

   show "0 + p = p"

     by (simp add: poly_eq_iff)

 qed

 end

 instantiation poly :: (cancel_comm_monoid_add) cancel_comm_monoid_add

 begin

 lift_definition minus_poly :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"

   is "\<lambda>p q n. coeff p n - coeff q n"

 proof -

   fix q p :: "'a poly"

   show "\<forall>\<^sub>\<infinity>n. coeff p n - coeff q n = 0"

     using MOST_coeff_eq_0[of p] MOST_coeff_eq_0[of q] by eventually_elim simp

 qed

 lemma coeff_diff [simp]: "coeff (p - q) n = coeff p n - coeff q n"

   by (simp add: minus_poly.rep_eq)

 instance

 proof

   fix p q r :: "'a poly"

   show "p + q - p = q"

     by (simp add: poly_eq_iff)

   show "p - q - r = p - (q + r)"

     by (simp add: poly_eq_iff diff_diff_eq)

 qed

 end

 instantiation poly :: (ab_group_add) ab_group_add

 begin

 lift_definition uminus_poly :: "'a poly \<Rightarrow> 'a poly"

   is "\<lambda>p n. - coeff p n"

 proof -

   fix p :: "'a poly"

   show "\<forall>\<^sub>\<infinity>n. - coeff p n = 0"

     using MOST_coeff_eq_0 by simp

 qed

 lemma coeff_minus [simp]: "coeff (- p) n = - coeff p n"

   by (simp add: uminus_poly.rep_eq)

 instance

 proof

   fix p q :: "'a poly"

   show "- p + p = 0"

     by (simp add: poly_eq_iff)

   show "p - q = p + - q"

     by (simp add: poly_eq_iff)

 qed

 end

 lemma add_pCons [simp]:

   "pCons a p + pCons b q = pCons (a + b) (p + q)"

   by (rule poly_eqI, simp add: coeff_pCons split: nat.split)

 lemma minus_pCons [simp]:

   "- pCons a p = pCons (- a) (- p)"

   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_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:

   "\<lbrakk>degree p \<le> n; degree q \<le> n\<rbrakk> \<Longrightarrow> degree (p + q) \<le> n"

   by (auto intro: order_trans degree_add_le_max)

 lemma degree_add_less:

   "\<lbrakk>degree p < n; degree q < n\<rbrakk> \<Longrightarrow> degree (p + q) < n"

   by (auto intro: le_less_trans degree_add_le_max)

 lemma degree_add_eq_right:

   "degree p < degree q \<Longrightarrow> degree (p + q) = degree q"

   apply (cases "q = 0", simp)

   apply (rule order_antisym)

   apply (simp add: degree_add_le)

   apply (rule le_degree)

   apply (simp add: coeff_eq_0)

   done

 lemma degree_add_eq_left:

   "degree q < degree p \<Longrightarrow> degree (p + q) = degree p"

   using degree_add_eq_right [of q p]

   by (simp add: add.commute)

 lemma degree_minus [simp]:

   "degree (- p) = degree p"

   unfolding degree_def by simp

 lemma degree_diff_le_max:

   fixes p q :: "'a :: ab_group_add poly"

   shows "degree (p - q) \<le> max (degree p) (degree q)"

   using degree_add_le [where p=p and q="-q"]

   by simp

 lemma degree_diff_le:

   fixes p q :: "'a :: ab_group_add poly"

   assumes "degree p \<le> n" and "degree q \<le> n"

   shows "degree (p - q) \<le> n"

   using assms degree_add_le [of p n "- q"] by simp

 lemma degree_diff_less:

   fixes p q :: "'a :: ab_group_add poly"

   assumes "degree p < n" and "degree q < n"

   shows "degree (p - q) < n"

   using assms degree_add_less [of p n "- q"] by simp

 lemma add_monom: "monom a n + monom b n = monom (a + b) n"

   by (rule poly_eqI) simp

 lemma diff_monom: "monom a n - monom b n = monom (a - b) n"

   by (rule poly_eqI) simp

 lemma minus_monom: "- monom a n = monom (-a) n"

   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_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 (fact diff_conv_add_uminus)

 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"

   using poly_add [of p "- q" x] by simp

 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

 lemma Poly_snoc: "Poly (xs @ [x]) = Poly xs + monom x (length xs)"

   by (induction xs) (simp_all add: monom_0 monom_Suc)

 subsection \<open>Multiplication by a constant, polynomial multiplication and the unit polynomial\<close>

 lift_definition smult :: "'a::comm_semiring_0 \<Rightarrow> 'a poly \<Rightarrow> 'a poly"

   is "\<lambda>a p n. a * coeff p n"

 proof -

   fix a :: 'a and p :: "'a poly" show "\<forall>\<^sub>\<infinity> i. a * coeff p i = 0"

     using MOST_coeff_eq_0[of p] by eventually_elim simp

 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_eqI, simp add: mult.assoc)

 lemma smult_0_right [simp]: "smult a 0 = 0"

   by (rule poly_eqI, simp)

 lemma smult_0_left [simp]: "smult 0 p = 0"

   by (rule poly_eqI, simp)

 lemma smult_1_left [simp]: "smult (1::'a::comm_semiring_1) p = p"

   by (rule poly_eqI, simp)

 lemma smult_add_right:

   "smult a (p + q) = smult a p + smult a q"

   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_eqI, simp add: algebra_simps)

 lemma smult_minus_right [simp]:

   "smult (a::'a::comm_ring) (- p) = - smult a p"

   by (rule poly_eqI, simp)

 lemma smult_minus_left [simp]:

   "smult (- a::'a::comm_ring) p = - smult a p"

   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_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_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_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]:

   fixes a :: "'a::idom"

   shows "degree (smult a p) = (if a = 0 then 0 else degree p)"

   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: 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

   "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"

   by (simp add: times_poly_def)

 lemma mult_pCons_left [simp]:

   "pCons a p * q = smult a q + pCons 0 (p * q)"

   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)

 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)

 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_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)

 instance

 proof

   fix p q r :: "'a poly"

   show 0: "0 * p = 0"

     by (rule mult_poly_0_left)

   show "p * 0 = 0"

     by (rule mult_poly_0_right)

   show "(p + q) * r = p * r + q * r"

     by (rule mult_poly_add_left)

   show "(p * q) * r = p * (q * r)"

     by (induct p, simp add: mult_poly_0, simp add: mult_poly_add_left)

   show "p * q = q * p"

     by (induct p, simp add: mult_poly_0, simp)

 qed

 end

 instance poly :: (comm_semiring_0_cancel) comm_semiring_0_cancel ..

 lemma coeff_mult:

   "coeff (p * q) n = (\<Sum>i\<le>n. coeff p i * coeff q (n-i))"

 proof (induct p arbitrary: n)

   case 0 show ?case by simp

 next

   case (pCons a p n) thus ?case

     by (cases n, simp, simp add: setsum_atMost_Suc_shift

                             del: setsum_atMost_Suc)

 qed

 lemma degree_mult_le: "degree (p * q) \<le> degree p + degree q"

 apply (rule degree_le)

 apply (induct p)

 apply simp

 apply (simp add: coeff_eq_0 coeff_pCons split: nat.split)

 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)

 instantiation poly :: (comm_semiring_1) comm_semiring_1

 begin

 definition one_poly_def: "1 = pCons 1 0"

 instance

 proof

   show "1 * p = p" for p :: "'a poly"

     unfolding one_poly_def by simp

   show "0 \<noteq> (1::'a poly)"

     unfolding one_poly_def by simp

 qed

 end

 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 monom_eq_1 [simp]:

   "monom 1 0 = 1"

   by (simp add: monom_0 one_poly_def)

 lemma degree_1 [simp]: "degree 1 = 0"

   unfolding one_poly_def

   by (rule degree_pCons_0)

 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 \<open>Conversions from natural numbers\<close>

 lemma of_nat_poly: "of_nat n = [:of_nat n :: 'a :: comm_semiring_1:]"

 proof (induction n)

   case (Suc n)

   hence "of_nat (Suc n) = 1 + (of_nat n :: 'a poly)" 

     by simp

   also have "(of_nat n :: 'a poly) = [: of_nat n :]" 

     by (subst Suc) (rule refl)

   also have "1 = [:1:]" by (simp add: one_poly_def)

   finally show ?case by (subst (asm) add_pCons) simp

 qed simp

 lemma degree_of_nat [simp]: "degree (of_nat n) = 0"

   by (simp add: of_nat_poly)

 lemma degree_numeral [simp]: "degree (numeral n) = 0"

   by (subst of_nat_numeral [symmetric], subst of_nat_poly) simp

 lemma numeral_poly: "numeral n = [:numeral n:]"

   by (subst of_nat_numeral [symmetric], subst of_nat_poly) simp

 subsection \<open>Lemmas about divisibility\<close>

 lemma dvd_smult: "p dvd q \<Longrightarrow> p dvd smult a q"

 proof -

   assume "p dvd q"

   then obtain k where "q = p * k" ..

   then have "smult a q = p * smult a k" by simp

   then show "p dvd smult a q" ..

 qed

 lemma dvd_smult_cancel:

   fixes a :: "'a::field"

   shows "p dvd smult a q \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> p dvd q"

   by (drule dvd_smult [where a="inverse a"]) simp

 lemma dvd_smult_iff:

   fixes a :: "'a::field"

   shows "a \<noteq> 0 \<Longrightarrow> p dvd smult a q \<longleftrightarrow> p dvd q"

   by (safe elim!: dvd_smult dvd_smult_cancel)

 lemma smult_dvd_cancel:

   "smult a p dvd q \<Longrightarrow> p dvd q"

 proof -

   assume "smult a p dvd q"

   then obtain k where "q = smult a p * k" ..

   then have "q = p * smult a k" by simp

   then show "p dvd q" ..

 qed

 lemma smult_dvd:

   fixes a :: "'a::field"

   shows "p dvd q \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> smult a p dvd q"

   by (rule smult_dvd_cancel [where a="inverse a"]) simp

 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)

 subsection \<open>Polynomials form an integral domain\<close>

 lemma coeff_mult_degree_sum:

   "coeff (p * q) (degree p + degree q) =

    coeff p (degree p) * coeff q (degree q)"

   by (induct p, simp, simp add: coeff_eq_0)

 instance poly :: (idom) idom

 proof

   fix p q :: "'a poly"

   assume "p \<noteq> 0" and "q \<noteq> 0"

   have "coeff (p * q) (degree p + degree q) =

         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 \<open>p \<noteq> 0\<close> and \<open>q \<noteq> 0\<close> by simp

   finally have "\<exists>n. coeff (p * q) n \<noteq> 0" ..

   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"

 apply (rule order_antisym [OF degree_mult_le le_degree])

 apply (simp add: coeff_mult_degree_sum)

 done

 lemma degree_mult_right_le:

   fixes p q :: "'a::idom poly"

   assumes "q \<noteq> 0"

   shows "degree p \<le> degree (p * q)"

   using assms by (cases "p = 0") (simp_all add: degree_mult_eq)

 lemma coeff_degree_mult:

   fixes p q :: "'a::idom poly"

   shows "coeff (p * q) (degree (p * q)) =

     coeff q (degree q) * coeff p (degree p)"

   by (cases "p = 0 \<or> q = 0") (auto simp add: degree_mult_eq coeff_mult_degree_sum)

 lemma dvd_imp_degree_le:

   fixes p q :: "'a::idom poly"

   shows "\<lbrakk>p dvd q; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree p \<le> degree q"

   by (erule dvdE, simp add: degree_mult_eq)

 subsection \<open>Polynomials form an ordered integral domain\<close>

 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)"

   unfolding pos_poly_def by simp

 lemma not_pos_poly_0 [simp]: "\<not> pos_poly 0"

   unfolding pos_poly_def by simp

 lemma pos_poly_add: "\<lbrakk>pos_poly p; pos_poly q\<rbrakk> \<Longrightarrow> pos_poly (p + q)"

   apply (induct p arbitrary: q, simp)

   apply (case_tac q, force simp add: pos_poly_pCons add_pos_pos)

   done

 lemma pos_poly_mult: "\<lbrakk>pos_poly p; pos_poly q\<rbrakk> \<Longrightarrow> pos_poly (p * q)"

   unfolding pos_poly_def

   apply (subgoal_tac "p \<noteq> 0 \<and> q \<noteq> 0")

   apply (simp add: degree_mult_eq coeff_mult_degree_sum)

   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

   "x < y \<longleftrightarrow> pos_poly (y - x)"

 definition

   "x \<le> y \<longleftrightarrow> x = y \<or> pos_poly (y - x)"

 definition

   "\<bar>x::'a poly\<bar> = (if x < 0 then - x else x)"

 definition

   "sgn (x::'a poly) = (if x = 0 then 0 else if 0 < x then 1 else - 1)"

 instance

 proof

   fix x y z :: "'a poly"

   show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"

     unfolding less_eq_poly_def less_poly_def

     apply safe

     apply simp

     apply (drule (1) pos_poly_add)

     apply simp

     done

   show "x \<le> x"

     unfolding less_eq_poly_def by simp

   show "x \<le> y \<Longrightarrow> y \<le> z \<Longrightarrow> x \<le> z"

     unfolding less_eq_poly_def

     apply safe

     apply (drule (1) pos_poly_add)

     apply (simp add: algebra_simps)

     done

   show "x \<le> y \<Longrightarrow> y \<le> x \<Longrightarrow> x = y"

     unfolding less_eq_poly_def

     apply safe

     apply (drule (1) pos_poly_add)

     apply simp

     done

   show "x \<le> y \<Longrightarrow> z + x \<le> z + y"

     unfolding less_eq_poly_def

     apply safe

     apply (simp add: algebra_simps)

     done

   show "x \<le> y \<or> y \<le> x"

     unfolding less_eq_poly_def

     using pos_poly_total [of "x - y"]

     by auto

   show "x < y \<Longrightarrow> 0 < z \<Longrightarrow> z * x < z * y"

     unfolding less_poly_def

     by (simp add: right_diff_distrib [symmetric] pos_poly_mult)

   show "\<bar>x\<bar> = (if x < 0 then - x else x)"

     by (rule abs_poly_def)

   show "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"

     by (rule sgn_poly_def)

 qed

 end

 text \<open>TODO: Simplification rules for comparisons\<close>

 subsection \<open>Synthetic division and polynomial roots\<close>

 text \<open>

   Synthetic division is simply division by the linear polynomial @{term "x - c"}.

 \<close>

 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 \<open>p \<noteq> 0\<close> 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 \<open>Suc n = degree p\<close> have "n = degree k" by simp

     then have "finite {x. poly k x = 0}" using \<open>k \<noteq> 0\<close> 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 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 \<open>Long division of polynomials\<close>

 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:

   "pdivmod_rel 0 y 0 0"

   unfolding pdivmod_rel_def by simp

 lemma pdivmod_rel_by_0:

   "pdivmod_rel x 0 0 x"

   unfolding pdivmod_rel_def by simp

 lemma eq_zero_or_degree_less:

   assumes "degree p \<le> n" and "coeff p n = 0"

   shows "p = 0 \<or> degree p < n"

 proof (cases n)

   case 0

   with \<open>degree p \<le> n\<close> and \<open>coeff p n = 0\<close>

   have "coeff p (degree p) = 0" by simp

   then have "p = 0" by simp

   then show ?thesis ..

 next

   case (Suc m)

   have "\<forall>i>n. coeff p i = 0"

     using \<open>degree p \<le> n\<close> by (simp add: coeff_eq_0)

   then have "\<forall>i\<ge>n. coeff p i = 0"

     using \<open>coeff p n = 0\<close> by (simp add: le_less)

   then have "\<forall>i>m. coeff p i = 0"

     using \<open>n = Suc m\<close> by (simp add: less_eq_Suc_le)

   then have "degree p \<le> m"

     by (rule degree_le)

   then have "degree p < n"

     using \<open>n = Suc m\<close> by (simp add: less_Suc_eq_le)

   then show ?thesis ..

 qed

 lemma pdivmod_rel_pCons:

   assumes rel: "pdivmod_rel x y q r"

   assumes y: "y \<noteq> 0"

   assumes b: "b = coeff (pCons a r) (degree y) / coeff y (degree y)"

   shows "pdivmod_rel (pCons a x) y (pCons b q) (pCons a r - smult b y)"

     (is "pdivmod_rel ?x y ?q ?r")

 proof -

   have x: "x = q * y + r" and r: "r = 0 \<or> degree r < degree y"

     using assms unfolding pdivmod_rel_def by simp_all

   have 1: "?x = ?q * y + ?r"

     using b x by simp

   have 2: "?r = 0 \<or> degree ?r < degree y"

   proof (rule eq_zero_or_degree_less)

     show "degree ?r \<le> degree y"

     proof (rule degree_diff_le)

       show "degree (pCons a r) \<le> degree y"

         using r by auto

       show "degree (smult b y) \<le> degree y"

         by (rule degree_smult_le)

     qed

   next

     show "coeff ?r (degree y) = 0"

       using \<open>y \<noteq> 0\<close> unfolding b by simp

   qed

   from 1 2 show ?thesis

     unfolding pdivmod_rel_def

     using \<open>y \<noteq> 0\<close> by simp

 qed

 lemma pdivmod_rel_exists: "\<exists>q r. pdivmod_rel x y q r"

 apply (cases "y = 0")

 apply (fast intro!: pdivmod_rel_by_0)

 apply (induct x)

 apply (fast intro!: pdivmod_rel_0)

 apply (fast intro!: pdivmod_rel_pCons)

 done

 lemma pdivmod_rel_unique:

   assumes 1: "pdivmod_rel x y q1 r1"

   assumes 2: "pdivmod_rel x y q2 r2"

   shows "q1 = q2 \<and> r1 = r2"

 proof (cases "y = 0")

   assume "y = 0" with assms show ?thesis

     by (simp add: pdivmod_rel_def)

 next

   assume [simp]: "y \<noteq> 0"

   from 1 have q1: "x = q1 * y + r1" and r1: "r1 = 0 \<or> degree r1 < degree y"

     unfolding pdivmod_rel_def by simp_all

   from 2 have q2: "x = q2 * y + r2" and r2: "r2 = 0 \<or> degree r2 < degree y"

     unfolding pdivmod_rel_def by simp_all

   from q1 q2 have q3: "(q1 - q2) * y = r2 - r1"

     by (simp add: algebra_simps)

   from r1 r2 have r3: "(r2 - r1) = 0 \<or> degree (r2 - r1) < degree y"

     by (auto intro: degree_diff_less)

   show "q1 = q2 \<and> r1 = r2"

   proof (rule ccontr)

     assume "\<not> (q1 = q2 \<and> r1 = r2)"

     with q3 have "q1 \<noteq> q2" and "r1 \<noteq> r2" by auto

     with r3 have "degree (r2 - r1) < degree y" by simp

     also have "degree y \<le> degree (q1 - q2) + degree y" by simp

     also have "\<dots> = degree ((q1 - q2) * y)"

       using \<open>q1 \<noteq> q2\<close> by (simp add: degree_mult_eq)

     also have "\<dots> = degree (r2 - r1)"

       using q3 by simp

     finally have "degree (r2 - r1) < degree (r2 - r1)" .

     then show "False" by simp

   qed

 qed

 lemma pdivmod_rel_0_iff: "pdivmod_rel 0 y q r \<longleftrightarrow> q = 0 \<and> r = 0"

 by (auto dest: pdivmod_rel_unique intro: pdivmod_rel_0)

 lemma pdivmod_rel_by_0_iff: "pdivmod_rel x 0 q r \<longleftrightarrow> q = 0 \<and> r = x"

 by (auto dest: pdivmod_rel_unique intro: pdivmod_rel_by_0)

 lemmas pdivmod_rel_unique_div = pdivmod_rel_unique [THEN conjunct1]

 lemmas pdivmod_rel_unique_mod = pdivmod_rel_unique [THEN conjunct2]

 instantiation poly :: (field) ring_div

 begin

 definition divide_poly where

   div_poly_def: "x div y = (THE q. \<exists>r. pdivmod_rel x y q r)"

 definition mod_poly where

   "x mod y = (THE r. \<exists>q. pdivmod_rel x y q r)"

 lemma div_poly_eq:

   "pdivmod_rel x y q r \<Longrightarrow> x div y = q"

 unfolding div_poly_def

 by (fast elim: pdivmod_rel_unique_div)

 lemma mod_poly_eq:

   "pdivmod_rel x y q r \<Longrightarrow> x mod y = r"

 unfolding mod_poly_def

 by (fast elim: pdivmod_rel_unique_mod)

 lemma pdivmod_rel:

   "pdivmod_rel x y (x div y) (x mod y)"

 proof -

   from pdivmod_rel_exists

     obtain q r where "pdivmod_rel x y q r" by fast

   thus ?thesis

     by (simp add: div_poly_eq mod_poly_eq)

 qed

 instance

 proof

   fix x y :: "'a poly"

   show "x div y * y + x mod y = x"

     using pdivmod_rel [of x y]

     by (simp add: pdivmod_rel_def)

 next

   fix x :: "'a poly"

   have "pdivmod_rel x 0 0 x"

     by (rule pdivmod_rel_by_0)

   thus "x div 0 = 0"

     by (rule div_poly_eq)

 next

   fix y :: "'a poly"

   have "pdivmod_rel 0 y 0 0"

     by (rule pdivmod_rel_0)

   thus "0 div y = 0"

     by (rule div_poly_eq)

 next

   fix x y z :: "'a poly"

   assume "y \<noteq> 0"

   hence "pdivmod_rel (x + z * y) y (z + x div y) (x mod y)"

     using pdivmod_rel [of x y]

     by (simp add: pdivmod_rel_def distrib_right)

   thus "(x + z * y) div y = z + x div y"

     by (rule div_poly_eq)

 next

   fix x y z :: "'a poly"

   assume "x \<noteq> 0"

   show "(x * y) div (x * z) = y div z"

   proof (cases "y \<noteq> 0 \<and> z \<noteq> 0")

     have "\<And>x::'a poly. pdivmod_rel x 0 0 x"

       by (rule pdivmod_rel_by_0)

     then have [simp]: "\<And>x::'a poly. x div 0 = 0"

       by (rule div_poly_eq)

     have "\<And>x::'a poly. pdivmod_rel 0 x 0 0"

       by (rule pdivmod_rel_0)

     then have [simp]: "\<And>x::'a poly. 0 div x = 0"

       by (rule div_poly_eq)

     case False then show ?thesis by auto

   next

     case True then have "y \<noteq> 0" and "z \<noteq> 0" by auto

     with \<open>x \<noteq> 0\<close>

     have "\<And>q r. pdivmod_rel y z q r \<Longrightarrow> pdivmod_rel (x * y) (x * z) q (x * r)"

       by (auto simp add: pdivmod_rel_def algebra_simps)

         (rule classical, simp add: degree_mult_eq)

     moreover from pdivmod_rel have "pdivmod_rel y z (y div z) (y mod z)" .

     ultimately have "pdivmod_rel (x * y) (x * z) (y div z) (x * (y mod z))" .

     then show ?thesis by (simp add: div_poly_eq)

   qed

 qed

 end

 lemma is_unit_monom_0:

   fixes a :: "'a::field"

   assumes "a \<noteq> 0"

   shows "is_unit (monom a 0)"

 proof

   from assms show "1 = monom a 0 * monom (1 / a) 0"

     by (simp add: mult_monom)

 qed

 lemma is_unit_triv:

   fixes a :: "'a::field"

   assumes "a \<noteq> 0"

   shows "is_unit [:a:]"

   using assms by (simp add: is_unit_monom_0 monom_0 [symmetric])

 lemma is_unit_iff_degree:

   assumes "p \<noteq> 0"

   shows "is_unit p \<longleftrightarrow> degree p = 0" (is "?P \<longleftrightarrow> ?Q")

 proof

   assume ?Q

   then obtain a where "p = [:a:]" by (rule degree_eq_zeroE)

   with assms show ?P by (simp add: is_unit_triv)

 next

   assume ?P

   then obtain q where "q \<noteq> 0" "p * q = 1" ..

   then have "degree (p * q) = degree 1"

     by simp

   with \<open>p \<noteq> 0\<close> \<open>q \<noteq> 0\<close> have "degree p + degree q = 0"

     by (simp add: degree_mult_eq)

   then show ?Q by simp

 qed

 lemma is_unit_pCons_iff:

   "is_unit (pCons a p) \<longleftrightarrow> p = 0 \<and> a \<noteq> 0" (is "?P \<longleftrightarrow> ?Q")

   by (cases "p = 0") (auto simp add: is_unit_triv is_unit_iff_degree)

 lemma is_unit_monom_trival:

   fixes p :: "'a::field poly"

   assumes "is_unit p"

   shows "monom (coeff p (degree p)) 0 = p"

   using assms by (cases p) (simp_all add: monom_0 is_unit_pCons_iff)

 lemma is_unit_polyE:

   assumes "is_unit p"

   obtains a where "p = monom a 0" and "a \<noteq> 0"

 proof -

   obtain a q where "p = pCons a q" by (cases p)

   with assms have "p = [:a:]" and "a \<noteq> 0"

     by (simp_all add: is_unit_pCons_iff)

   with that show thesis by (simp add: monom_0)

 qed

 instantiation poly :: (field) normalization_semidom

 begin

 definition normalize_poly :: "'a poly \<Rightarrow> 'a poly"

   where "normalize_poly p = smult (1 / coeff p (degree p)) p"

 definition unit_factor_poly :: "'a poly \<Rightarrow> 'a poly"

   where "unit_factor_poly p = monom (coeff p (degree p)) 0"

 instance

 proof

   fix p :: "'a poly"

   show "unit_factor p * normalize p = p"

     by (simp add: normalize_poly_def unit_factor_poly_def)

       (simp only: mult_smult_left [symmetric] smult_monom, simp)

 next

   show "normalize 0 = (0::'a poly)"

     by (simp add: normalize_poly_def)

 next

   show "unit_factor 0 = (0::'a poly)"

     by (simp add: unit_factor_poly_def)

 next

   fix p :: "'a poly"

   assume "is_unit p"

   then obtain a where "p = monom a 0" and "a \<noteq> 0"

     by (rule is_unit_polyE)

   then show "normalize p = 1"

     by (auto simp add: normalize_poly_def smult_monom degree_monom_eq)

 next

   fix p q :: "'a poly"

   assume "q \<noteq> 0"

   from \<open>q \<noteq> 0\<close> have "is_unit (monom (coeff q (degree q)) 0)"

     by (auto intro: is_unit_monom_0)

   then show "is_unit (unit_factor q)"

     by (simp add: unit_factor_poly_def)

 next

   fix p q :: "'a poly"

   have "monom (coeff (p * q) (degree (p * q))) 0 =

     monom (coeff p (degree p)) 0 * monom (coeff q (degree q)) 0"

     by (simp add: monom_0 coeff_degree_mult)

   then show "unit_factor (p * q) =

     unit_factor p * unit_factor q"

     by (simp add: unit_factor_poly_def)

 qed

 end

 lemma degree_mod_less:

   "y \<noteq> 0 \<Longrightarrow> x mod y = 0 \<or> degree (x mod y) < degree y"

   using pdivmod_rel [of x y]

   unfolding pdivmod_rel_def by simp

 lemma div_poly_less: "degree x < degree y \<Longrightarrow> x div y = 0"

 proof -

   assume "degree x < degree y"

   hence "pdivmod_rel x y 0 x"

     by (simp add: pdivmod_rel_def)

   thus "x div y = 0" by (rule div_poly_eq)

 qed

 lemma mod_poly_less: "degree x < degree y \<Longrightarrow> x mod y = x"

 proof -

   assume "degree x < degree y"

   hence "pdivmod_rel x y 0 x"

     by (simp add: pdivmod_rel_def)

   thus "x mod y = x" by (rule mod_poly_eq)

 qed

 lemma pdivmod_rel_smult_left:

   "pdivmod_rel x y q r

     \<Longrightarrow> pdivmod_rel (smult a x) y (smult a q) (smult a r)"

   unfolding pdivmod_rel_def by (simp add: smult_add_right)

 lemma div_smult_left: "(smult a x) div y = smult a (x div y)"

   by (rule div_poly_eq, rule pdivmod_rel_smult_left, rule pdivmod_rel)

 lemma mod_smult_left: "(smult a x) mod y = smult a (x mod y)"

   by (rule mod_poly_eq, rule pdivmod_rel_smult_left, rule pdivmod_rel)

 lemma poly_div_minus_left [simp]:

   fixes x y :: "'a::field poly"

   shows "(- x) div y = - (x div y)"

   using div_smult_left [of "- 1::'a"] by simp

 lemma poly_mod_minus_left [simp]:

   fixes x y :: "'a::field poly"

   shows "(- x) mod y = - (x mod y)"

   using mod_smult_left [of "- 1::'a"] by simp

 lemma pdivmod_rel_add_left:

   assumes "pdivmod_rel x y q r"

   assumes "pdivmod_rel x' y q' r'"

   shows "pdivmod_rel (x + x') y (q + q') (r + r')"

   using assms unfolding pdivmod_rel_def

   by (auto simp add: algebra_simps degree_add_less)

 lemma poly_div_add_left:

   fixes x y z :: "'a::field poly"

   shows "(x + y) div z = x div z + y div z"

   using pdivmod_rel_add_left [OF pdivmod_rel pdivmod_rel]

   by (rule div_poly_eq)

 lemma poly_mod_add_left:

   fixes x y z :: "'a::field poly"

   shows "(x + y) mod z = x mod z + y mod z"

   using pdivmod_rel_add_left [OF pdivmod_rel pdivmod_rel]

   by (rule mod_poly_eq)

 lemma poly_div_diff_left:

   fixes x y z :: "'a::field poly"

   shows "(x - y) div z = x div z - y div z"

   by (simp only: diff_conv_add_uminus poly_div_add_left poly_div_minus_left)

 lemma poly_mod_diff_left:

   fixes x y z :: "'a::field poly"

   shows "(x - y) mod z = x mod z - y mod z"

   by (simp only: diff_conv_add_uminus poly_mod_add_left poly_mod_minus_left)

 lemma pdivmod_rel_smult_right:

   "\<lbrakk>a \<noteq> 0; pdivmod_rel x y q r\<rbrakk>

     \<Longrightarrow> pdivmod_rel x (smult a y) (smult (inverse a) q) r"

   unfolding pdivmod_rel_def by simp

 lemma div_smult_right:

   "a \<noteq> 0 \<Longrightarrow> x div (smult a y) = smult (inverse a) (x div y)"

   by (rule div_poly_eq, erule pdivmod_rel_smult_right, rule pdivmod_rel)

 lemma mod_smult_right: "a \<noteq> 0 \<Longrightarrow> x mod (smult a y) = x mod y"

   by (rule mod_poly_eq, erule pdivmod_rel_smult_right, rule pdivmod_rel)

 lemma poly_div_minus_right [simp]:

   fixes x y :: "'a::field poly"

   shows "x div (- y) = - (x div y)"

   using div_smult_right [of "- 1::'a"] by (simp add: nonzero_inverse_minus_eq)

 lemma poly_mod_minus_right [simp]:

   fixes x y :: "'a::field poly"

   shows "x mod (- y) = x mod y"

   using mod_smult_right [of "- 1::'a"] by simp

 lemma pdivmod_rel_mult:

   "\<lbrakk>pdivmod_rel x y q r; pdivmod_rel q z q' r'\<rbrakk>

     \<Longrightarrow> pdivmod_rel x (y * z) q' (y * r' + r)"

 apply (cases "z = 0", simp add: pdivmod_rel_def)

 apply (cases "y = 0", simp add: pdivmod_rel_by_0_iff pdivmod_rel_0_iff)

 apply (cases "r = 0")

 apply (cases "r' = 0")

 apply (simp add: pdivmod_rel_def)

 apply (simp add: pdivmod_rel_def field_simps degree_mult_eq)

 apply (cases "r' = 0")

 apply (simp add: pdivmod_rel_def degree_mult_eq)

 apply (simp add: pdivmod_rel_def field_simps)

 apply (simp add: degree_mult_eq degree_add_less)

 done

 lemma poly_div_mult_right:

   fixes x y z :: "'a::field poly"

   shows "x div (y * z) = (x div y) div z"

   by (rule div_poly_eq, rule pdivmod_rel_mult, (rule pdivmod_rel)+)

 lemma poly_mod_mult_right:

   fixes x y z :: "'a::field poly"

   shows "x mod (y * z) = y * (x div y mod z) + x mod y"

   by (rule mod_poly_eq, rule pdivmod_rel_mult, (rule pdivmod_rel)+)

 lemma mod_pCons:

   fixes a and x

   assumes y: "y \<noteq> 0"

   defines b: "b \<equiv> coeff (pCons a (x mod y)) (degree y) / coeff y (degree y)"

   shows "(pCons a x) mod y = (pCons a (x mod y) - smult b y)"

 unfolding b

 apply (rule mod_poly_eq)

 apply (rule pdivmod_rel_pCons [OF pdivmod_rel y refl])

 done

 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 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

 subsection \<open>Order of polynomial roots\<close>

 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"

   shows "coeff ([:a, 1:] ^ n) n = 1"

 apply (induct n, simp_all)

 apply (subst coeff_eq_0)

 apply (auto intro: le_less_trans degree_power_le)

 done

 lemma degree_linear_power:

   fixes a :: "'a::comm_semiring_1"

   shows "degree ([:a, 1:] ^ n) = n"

 apply (rule order_antisym)

 apply (rule ord_le_eq_trans [OF degree_power_le], simp)

 apply (rule le_degree, simp add: coeff_linear_power)

 done

 lemma order_1: "[:-a, 1:] ^ order a p dvd p"

 apply (cases "p = 0", simp)

 apply (cases "order a p", simp)

 apply (subgoal_tac "nat < (LEAST n. \<not> [:-a, 1:] ^ Suc n dvd p)")

 apply (drule not_less_Least, simp)

 apply (fold order_def, simp)

 done

 lemma order_2: "p \<noteq> 0 \<Longrightarrow> \<not> [:-a, 1:] ^ Suc (order a p) dvd p"

 unfolding order_def

 apply (rule LeastI_ex)

 apply (rule_tac x="degree p" in exI)

 apply (rule notI)

 apply (drule (1) dvd_imp_degree_le)

 apply (simp only: degree_linear_power)

 done

 lemma order:

   "p \<noteq> 0 \<Longrightarrow> [:-a, 1:] ^ order a p dvd p \<and> \<not> [:-a, 1:] ^ Suc (order a p) dvd p"

 by (rule conjI [OF order_1 order_2])

 lemma order_degree:

   assumes p: "p \<noteq> 0"

   shows "order a p \<le> degree p"

 proof -

   have "order a p = degree ([:-a, 1:] ^ order a p)"

     by (simp only: degree_linear_power)

   also have "\<dots> \<le> degree p"

     using order_1 p by (rule dvd_imp_degree_le)

   finally show ?thesis .

 qed

 lemma order_root: "poly p a = 0 \<longleftrightarrow> p = 0 \<or> order a p \<noteq> 0"

 apply (cases "p = 0", simp_all)

 apply (rule iffI)

 apply (metis order_2 not_gr0 poly_eq_0_iff_dvd power_0 power_Suc_0 power_one_right)

 unfolding poly_eq_0_iff_dvd

 apply (metis dvd_power dvd_trans order_1)

 done

 lemma order_0I: "poly p a \<noteq> 0 \<Longrightarrow> order a p = 0"

   by (subst (asm) order_root) auto

 subsection \<open>GCD of polynomials\<close>

 instantiation poly :: (field) gcd

 begin

 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]

 definition lcm_poly :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"

 where

   "lcm_poly a b = a * b div smult (coeff a (degree a) * coeff b (degree b)) (gcd a b)"

 instance ..

 end

 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 (auto intro!: poly_gcd_greatest intro: dvd_trans [of _ "gcd x y"])

 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 \<open>p dvd q\<close> \<open>q dvd p\<close> \<open>p \<noteq> 0\<close> \<open>q \<noteq> 0\<close>

     by (intro order_antisym dvd_imp_degree_le)

   from \<open>p dvd q\<close> obtain a where a: "q = p * a" ..

   with \<open>q \<noteq> 0\<close> have "a \<noteq> 0" by auto

   with degree a \<open>p \<noteq> 0\<close> 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 \<open>Additional induction rules on polynomials\<close>

 text \<open>

   An induction rule for induction over the roots of a polynomial with a certain property. 

   (e.g. all positive roots)

 \<close>

 lemma poly_root_induct [case_names 0 no_roots root]:

   fixes p :: "'a :: idom poly"

   assumes "Q 0"

   assumes "\<And>p. (\<And>a. P a \<Longrightarrow> poly p a \<noteq> 0) \<Longrightarrow> Q p"

   assumes "\<And>a p. P a \<Longrightarrow> Q p \<Longrightarrow> Q ([:a, -1:] * p)"

   shows   "Q p"

 proof (induction "degree p" arbitrary: p rule: less_induct)

   case (less p)

   show ?case

   proof (cases "p = 0")

     assume nz: "p \<noteq> 0"

     show ?case

     proof (cases "\<exists>a. P a \<and> poly p a = 0")

       case False

       thus ?thesis by (intro assms(2)) blast

     next

       case True

       then obtain a where a: "P a" "poly p a = 0" 

         by blast

       hence "-[:-a, 1:] dvd p" 

         by (subst minus_dvd_iff) (simp add: poly_eq_0_iff_dvd)

       then obtain q where q: "p = [:a, -1:] * q" by (elim dvdE) simp

       with nz have q_nz: "q \<noteq> 0" by auto

       have "degree p = Suc (degree q)"

         by (subst q, subst degree_mult_eq) (simp_all add: q_nz)

       hence "Q q" by (intro less) simp

       from a(1) and this have "Q ([:a, -1:] * q)" 

         by (rule assms(3))

       with q show ?thesis by simp

     qed

   qed (simp add: assms(1))

 qed

 lemma dropWhile_replicate_append: 

   "dropWhile (op= a) (replicate n a @ ys) = dropWhile (op= a) ys"

   by (induction n) simp_all

 lemma Poly_append_replicate_0: "Poly (xs @ replicate n 0) = Poly xs"

   by (subst coeffs_eq_iff) (simp_all add: strip_while_def dropWhile_replicate_append)

 text \<open>

   An induction rule for simultaneous induction over two polynomials, 

   prepending one coefficient in each step.

 \<close>

 lemma poly_induct2 [case_names 0 pCons]:

   assumes "P 0 0" "\<And>a p b q. P p q \<Longrightarrow> P (pCons a p) (pCons b q)"

   shows   "P p q"

 proof -

   def n \<equiv> "max (length (coeffs p)) (length (coeffs q))"

   def xs \<equiv> "coeffs p @ (replicate (n - length (coeffs p)) 0)"

   def ys \<equiv> "coeffs q @ (replicate (n - length (coeffs q)) 0)"

   have "length xs = length ys" 

     by (simp add: xs_def ys_def n_def)

   hence "P (Poly xs) (Poly ys)" 

     by (induction rule: list_induct2) (simp_all add: assms)

   also have "Poly xs = p" 

     by (simp add: xs_def Poly_append_replicate_0)

   also have "Poly ys = q" 

     by (simp add: ys_def Poly_append_replicate_0)

   finally show ?thesis .

 qed

 subsection \<open>Composition of polynomials\<close>

 definition pcompose :: "'a::comm_semiring_0 poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"

 where

   "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 pcompose_add:

   fixes p q r :: "'a :: {comm_semiring_0, ab_semigroup_add} poly"

   shows "pcompose (p + q) r = pcompose p r + pcompose q r"

 proof (induction p q rule: poly_induct2)

   case (pCons a p b q)

   have "pcompose (pCons a p + pCons b q) r = 

           [:a + b:] + r * pcompose p r + r * pcompose q r"

     by (simp_all add: pcompose_pCons pCons.IH algebra_simps)

   also have "[:a + b:] = [:a:] + [:b:]" by simp

   also have "\<dots> + r * pcompose p r + r * pcompose q r = 

                  pcompose (pCons a p) r + pcompose (pCons b q) r"

     by (simp only: pcompose_pCons add_ac)

   finally show ?case .

 qed simp

 lemma pcompose_minus:

   fixes p r :: "'a :: comm_ring poly"

   shows "pcompose (-p) r = -pcompose p r"

   by (induction p) (simp_all add: pcompose_pCons)

 lemma pcompose_diff:

   fixes p q r :: "'a :: comm_ring poly"

   shows "pcompose (p - q) r = pcompose p r - pcompose q r"

   using pcompose_add[of p "-q"] by (simp add: pcompose_minus)

 lemma pcompose_smult:

   fixes p r :: "'a :: comm_semiring_0 poly"

   shows "pcompose (smult a p) r = smult a (pcompose p r)"

   by (induction p) 

      (simp_all add: pcompose_pCons pcompose_add smult_add_right)

 lemma pcompose_mult:

   fixes p q r :: "'a :: comm_semiring_0 poly"

   shows "pcompose (p * q) r = pcompose p r * pcompose q r"

   by (induction p arbitrary: q)

      (simp_all add: pcompose_add pcompose_smult pcompose_pCons algebra_simps)

 lemma pcompose_assoc: 

   "pcompose p (pcompose q r :: 'a :: comm_semiring_0 poly ) =

      pcompose (pcompose p q) r"

   by (induction p arbitrary: q) 

      (simp_all add: pcompose_pCons pcompose_add pcompose_mult)

 (* The remainder of this section and the next were contributed by Wenda Li *)

 lemma degree_mult_eq_0:

   fixes p q:: "'a :: idom poly"

   shows "degree (p*q) = 0 \<longleftrightarrow> p=0 \<or> q=0 \<or> (p\<noteq>0 \<and> q\<noteq>0 \<and> degree p =0 \<and> degree q =0)"

 by (auto simp add:degree_mult_eq)

 lemma pcompose_const[simp]:"pcompose [:a:] q = [:a:]" by (subst pcompose_pCons,simp) 

 lemma pcompose_0':"pcompose p 0=[:coeff p 0:]"

   apply (induct p)

   apply (auto simp add:pcompose_pCons)

 done

 lemma degree_pcompose:

   fixes p q:: "'a::idom poly"

   shows "degree(pcompose p q) = degree p * degree q"

 proof (induct p)

   case 0

   thus ?case by auto

 next

   case (pCons a p)

   have "degree (q * pcompose p q) = 0 \<Longrightarrow> ?case" 

     proof (cases "p=0")

       case True

       thus ?thesis by auto

     next

       case False assume "degree (q * pcompose p q) = 0"

       hence "degree q=0 \<or> pcompose p q=0" by (auto simp add:degree_mult_eq_0)

       moreover have "\<lbrakk>pcompose p q=0;degree q\<noteq>0\<rbrakk> \<Longrightarrow> False" using pCons.hyps(2) \<open>p\<noteq>0\<close> 

         proof -

           assume "pcompose p q=0" "degree q\<noteq>0"

           hence "degree p=0" using pCons.hyps(2) by auto

           then obtain a1 where "p=[:a1:]"

             by (metis degree_pCons_eq_if old.nat.distinct(2) pCons_cases)

           thus False using \<open>pcompose p q=0\<close> \<open>p\<noteq>0\<close> by auto

         qed

       ultimately have "degree (pCons a p) * degree q=0" by auto

       moreover have "degree (pcompose (pCons a p) q) = 0" 

         proof -

           have" 0 = max (degree [:a:]) (degree (q*pcompose p q))"

             using \<open>degree (q * pcompose p q) = 0\<close> by simp

           also have "... \<ge> degree ([:a:] + q * pcompose p q)"

             by (rule degree_add_le_max)

           finally show ?thesis by (auto simp add:pcompose_pCons)

         qed

       ultimately show ?thesis by simp

     qed

   moreover have "degree (q * pcompose p q)>0 \<Longrightarrow> ?case" 

     proof -

       assume asm:"0 < degree (q * pcompose p q)"

       hence "p\<noteq>0" "q\<noteq>0" "pcompose p q\<noteq>0" by auto

       have "degree (pcompose (pCons a p) q) = degree ( q * pcompose p q)"

         unfolding pcompose_pCons

         using degree_add_eq_right[of "[:a:]" ] asm by auto       

       thus ?thesis 

         using pCons.hyps(2) degree_mult_eq[OF \<open>q\<noteq>0\<close> \<open>pcompose p q\<noteq>0\<close>] by auto

     qed

   ultimately show ?case by blast

 qed

 lemma pcompose_eq_0:

   fixes p q:: "'a::idom poly"

   assumes "pcompose p q=0" "degree q>0" 

   shows "p=0"

 proof -

   have "degree p=0" using assms degree_pcompose[of p q] by auto

   then obtain a where "p=[:a:]" 

     by (metis degree_pCons_eq_if gr0_conv_Suc neq0_conv pCons_cases)

   hence "a=0" using assms(1) by auto

   thus ?thesis using \<open>p=[:a:]\<close> by simp

 qed

 subsection \<open>Leading coefficient\<close>

 definition lead_coeff:: "'a::zero poly \<Rightarrow> 'a" where

   "lead_coeff p= coeff p (degree p)"

 lemma lead_coeff_pCons[simp]:

     "p\<noteq>0 \<Longrightarrow>lead_coeff (pCons a p) = lead_coeff p"

     "p=0 \<Longrightarrow> lead_coeff (pCons a p) = a"

 unfolding lead_coeff_def by auto

 lemma lead_coeff_0[simp]:"lead_coeff 0 =0" 

   unfolding lead_coeff_def by auto

 lemma lead_coeff_mult:

    fixes p q::"'a ::idom poly"

    shows "lead_coeff (p * q) = lead_coeff p * lead_coeff q"

 by (unfold lead_coeff_def,cases "p=0 \<or> q=0",auto simp add:coeff_mult_degree_sum degree_mult_eq)

 lemma lead_coeff_add_le:

   assumes "degree p < degree q"

   shows "lead_coeff (p+q) = lead_coeff q" 

 using assms unfolding lead_coeff_def

 by (metis coeff_add coeff_eq_0 monoid_add_class.add.left_neutral degree_add_eq_right)

 lemma lead_coeff_minus:

   "lead_coeff (-p) = - lead_coeff p"

 by (metis coeff_minus degree_minus lead_coeff_def)

 lemma lead_coeff_comp:

   fixes p q:: "'a::idom poly"

   assumes "degree q > 0" 

   shows "lead_coeff (pcompose p q) = lead_coeff p * lead_coeff q ^ (degree p)"

 proof (induct p)

   case 0

   thus ?case unfolding lead_coeff_def by auto

 next

   case (pCons a p)

   have "degree ( q * pcompose p q) = 0 \<Longrightarrow> ?case"

     proof -

       assume "degree ( q * pcompose p q) = 0"

       hence "pcompose p q = 0" by (metis assms degree_0 degree_mult_eq_0 neq0_conv)

       hence "p=0" using pcompose_eq_0[OF _ \<open>degree q > 0\<close>] by simp

       thus ?thesis by auto

     qed

   moreover have "degree ( q * pcompose p q) > 0 \<Longrightarrow> ?case" 

     proof -

       assume "degree ( q * pcompose p q) > 0"

       hence "lead_coeff (pcompose (pCons a p) q) =lead_coeff ( q * pcompose p q)"

         by (auto simp add:pcompose_pCons lead_coeff_add_le)

       also have "... = lead_coeff q * (lead_coeff p * lead_coeff q ^ degree p)"

         using pCons.hyps(2) lead_coeff_mult[of q "pcompose p q"] by simp

       also have "... = lead_coeff p * lead_coeff q ^ (degree p + 1)"

         by auto

       finally show ?thesis by auto

     qed

   ultimately show ?case by blast

 qed

 lemma lead_coeff_smult: 

   "lead_coeff (smult c p :: 'a :: idom poly) = c * lead_coeff p"

 proof -

   have "smult c p = [:c:] * p" by simp

   also have "lead_coeff \<dots> = c * lead_coeff p" 

     by (subst lead_coeff_mult) simp_all

   finally show ?thesis .

 qed

 lemma lead_coeff_1 [simp]: "lead_coeff 1 = 1"

   by (simp add: lead_coeff_def)

 lemma lead_coeff_of_nat [simp]:

   "lead_coeff (of_nat n) = (of_nat n :: 'a :: {comm_semiring_1,semiring_char_0})"

   by (induction n) (simp_all add: lead_coeff_def of_nat_poly)

 lemma lead_coeff_numeral [simp]: 

   "lead_coeff (numeral n) = numeral n"

   unfolding lead_coeff_def

   by (subst of_nat_numeral [symmetric], subst of_nat_poly) simp

 lemma lead_coeff_power: 

   "lead_coeff (p ^ n :: 'a :: idom poly) = lead_coeff p ^ n"

   by (induction n) (simp_all add: lead_coeff_mult)

 lemma lead_coeff_nonzero: "p \<noteq> 0 \<Longrightarrow> lead_coeff p \<noteq> 0"

   by (simp add: lead_coeff_def)

 no_notation cCons (infixr "##" 65)

 end