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Datei: Sprache: Isabelle

Untersuchung Isabelle^©

(*  Title:      HOL/Computational_Algebra/Polynomial.thy
    Author:     Brian Huffman
    Author:     Clemens Ballarin
    Author:     Amine Chaieb
    Author:     Florian Haftmann
*)

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

theory Polynomial
imports
  Complex_Main
  "HOL-Library.More_List"
  "HOL-Library.Infinite_Set"
  Factorial_Ring
begin

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

definition cCons :: "'a::zero \ 'a list \ 'a list" (infixr "##" 65)
  where "x ## xs = (if xs = [] \ 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 \ 0 \ x ## xs = x # xs"
  by (simp add: cCons_def)

lemma strip_while_not_0_Cons_eq [simp]:
  "strip_while (\x. x = 0) (x # xs) = x ## strip_while (\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 \ 'a::zero. \\<^sub>\ n. f n = 0}"
  morphisms coeff Abs_poly
  by (auto intro!: ALL_MOST)

setup_lifting type_definition_poly

lemma poly_eq_iff: "p = q \ (\n. coeff p n = coeff q n)"
  by (simp add: coeff_inject [symmetric] fun_eq_iff)

lemma poly_eqI: "(\n. coeff p n = coeff q n) \ p = q"
  by (simp add: poly_eq_iff)

lemma MOST_coeff_eq_0: "\\<^sub>\ n. coeff p n = 0"
  using coeff [of p] by simp

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

definition degree :: "'a::zero poly \ nat"
  where "degree p = (LEAST n. \i>n. coeff p i = 0)"

lemma coeff_eq_0:
  assumes "degree p < n"
  shows "coeff p n = 0"
proof -
  have "\n. \i>n. coeff p i = 0"
    using MOST_coeff_eq_0 by (simp add: MOST_nat)
  then have "\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 \ 0 \ n \ degree p"
  by (erule contrapos_np, rule coeff_eq_0, simp)

lemma degree_le: "\i>n. coeff p i = 0 \ degree p \ n"
  unfolding degree_def by (erule Least_le)

lemma less_degree_imp: "n < degree p \ \i>n. coeff p i \ 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 "\_. 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 \ 0"
  shows "coeff p (degree p) \ 0"
proof (cases "degree p")
  case 0
  from \<open>p \<noteq> 0\<close> obtain n where "coeff p n \<noteq> 0"
    by (auto simp add: poly_eq_iff)
  then have "n \ 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
  then have "\i>n. coeff p i \ 0"
    by (rule less_degree_imp)
  then obtain i where "n < i" and "coeff p i \ 0"
    by blast
  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 \ p = 0"
  by (cases "p = 0") (simp_all add: leading_coeff_neq_0)

lemma eq_zero_or_degree_less:
  assumes "degree p \ n" and "coeff p n = 0"
  shows "p = 0 \ 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)
  from \<open>degree p \<le> n\<close> have "\<forall>i>n. coeff p i = 0"
    by (simp add: coeff_eq_0)
  with \<open>coeff p n = 0\<close> have "\<forall>i\<ge>n. coeff p i = 0"
    by (simp add: le_less)
  with \<open>n = Suc m\<close> have "\<forall>i>m. coeff p i = 0"
    by (simp add: less_eq_Suc_le)
  then have "degree p \ m"
    by (rule degree_le)
  with \<open>n = Suc m\<close> have "degree p < n"
    by (simp add: less_Suc_eq_le)
  then show ?thesis ..
qed

lemma coeff_0_degree_minus_1: "coeff rrr dr = 0 \ degree rrr \ dr \ degree rrr \ dr - 1"
  using eq_zero_or_degree_less by fastforce

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

lift_definition pCons :: "'a::zero \ 'a poly \ 'a poly"
  is "\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) \ Suc (degree p)"
  by (rule degree_le) (simp add: coeff_eq_0 coeff_pCons split: nat.split)

lemma degree_pCons_eq: "p \ 0 \ degree (pCons a p) = Suc (degree p)"
  by (simp add: degree_pCons_le le_antisym le_degree)

lemma degree_pCons_0: "degree (pCons a 0) = 0"
proof -
  have "degree (pCons a 0) \ Suc 0"
    by (metis (no_types) degree_0 degree_pCons_le)
  then show ?thesis
    by (metis coeff_0 coeff_pCons_Suc degree_0 eq_zero_or_degree_less less_Suc0)
qed

lemma degree_pCons_eq_if [simp]: "degree (pCons a p) = (if p = 0 then 0 else Suc (degree p))"
  by (simp add: degree_pCons_0 degree_pCons_eq)

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 \ a = b \ 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 "coeff (pCons a p) (Suc n) = coeff (pCons b q) (Suc n)" for n
    by simp
  then show "p = q"
    by (simp add: poly_eq_iff)
qed

lemma pCons_eq_0_iff [simp]: "pCons a p = 0 \ a = 0 \ 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 (\n. coeff p (Suc n)))"
    by transfer
      (simp_all add: MOST_inj[where f=Suc and P="\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: "\a p. a \ 0 \ p \ 0 \ P p \ 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)
    with \<open>p = pCons a q\<close> have "degree q < degree p"
      by simp
    then show "P q"
      by (rule less.hyps)
  qed
  have "P (pCons a q)"
  proof (cases "a \ 0 \ q \ 0")
    case True
    with \<open>P q\<close> show ?thesis by (auto intro: pCons)
  next
    case False
    with zero show ?thesis by simp
  qed
  with \<open>p = pCons a q\<close> show ?case
    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
  then show thesis ..
qed

subsection \<open>Quickcheck generator for polynomials\<close>

quickcheck_generator poly constructors: "0 :: _ poly", pCons

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

syntax "_poly" :: "args \ 'a poly" ("[:(_):]")
translations
  "[:x, xs:]" \<rightleftharpoons> "CONST pCons x [:xs:]"
  "[:x:]" \<rightleftharpoons> "CONST pCons x 0"
  "[:x:]" \<leftharpoondown> "CONST pCons x (_constrain 0 t)"

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

primrec Poly :: "'a::zero list \ '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 \ (\n. as = replicate n 0)"
  by (induct as) (auto simp add: Cons_replicate_eq)

lemma Poly_append_replicate_zero [simp]: "Poly (as @ replicate n 0) = Poly as"
  by (induct as) simp_all

lemma Poly_snoc_zero [simp]: "Poly (as @ [0]) = Poly as"
  using Poly_append_replicate_zero [of as 1] by simp

lemma Poly_cCons_eq_pCons_Poly [simp]: "Poly (a ## p) = pCons a (Poly p)"
  by (simp add: cCons_def)

lemma Poly_on_rev_starting_with_0 [simp]: "hd as = 0 \ Poly (rev (tl as)) = Poly (rev as)"
  by (cases as) simp_all

lemma degree_Poly: "degree (Poly xs) \ length xs"
  by (induct xs) simp_all

lemma coeff_Poly_eq [simp]: "coeff (Poly xs) = nth_default 0 xs"
  by (induct xs) (simp_all add: fun_eq_iff coeff_pCons split: nat.splits)

definition coeffs :: "'a poly \ 'a::zero list"
  where "coeffs p = (if p = 0 then [] else map (\i. coeff p i) [0 ..< Suc (degree p)])"

lemma coeffs_eq_Nil [simp]: "coeffs p = [] \ p = 0"
  by (simp add: coeffs_def)

lemma not_0_coeffs_not_Nil: "p \ 0 \ coeffs p \ []"
  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 -
  have *: "\m\set ms. m > 0 \ map (case_nat x f) ms = map f (map (\n. n - 1) ms)"
    for ms :: "nat list" and f :: "nat \ 'a" and x :: "'a"
    by (induct ms) (auto split: nat.split)
  show ?thesis
    by (simp add: * coeffs_def upt_conv_Cons coeff_pCons map_decr_upt del: upt_Suc)
qed

lemma length_coeffs: "p \ 0 \ length (coeffs p) = degree p + 1"
  by (simp add: coeffs_def)

lemma coeffs_nth: "p \ 0 \ n \ degree p \ coeffs p ! n = coeff p n"
  by (auto simp: coeffs_def simp del: upt_Suc)

lemma coeff_in_coeffs: "p \ 0 \ n \ degree p \ coeff p n \ set (coeffs p)"
  using coeffs_nth [of p n, symmetric] by (simp add: length_coeffs)

lemma not_0_cCons_eq [simp]: "p \ 0 \ 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)
  from replicate_length_same [of as 0] have "(\n. as \ replicate n 0) \ (\a\set as. a \ 0)"
    by (auto dest: sym [of _ as])
  with Cons show ?case by auto
qed

lemma no_trailing_coeffs [simp]:
  "no_trailing (HOL.eq 0) (coeffs p)"
  by (induct p)  auto

lemma strip_while_coeffs [simp]:
  "strip_while (HOL.eq 0) (coeffs p) = coeffs p"
  by simp

lemma coeffs_eq_iff: "p = q \ coeffs p = coeffs q"
  (is "?P \ ?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 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: "\n. coeff p n = nth_default 0 xs n"
  assumes zero: "no_trailing (HOL.eq 0) xs"
  shows "coeffs p = xs"
proof -
  from coeff have "p = Poly xs"
    by (simp add: poly_eq_iff)
  with zero show ?thesis by simp
qed

lemma degree_eq_length_coeffs [code]: "degree p = length (coeffs p) - 1"
  by (simp add: coeffs_def)

lemma length_coeffs_degree: "p \ 0 \ length (coeffs p) = Suc (degree p)"
  by (induct p) (auto simp: 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)

lemma set_coeffs_subset_singleton_0_iff [simp]:
  "set (coeffs p) \ {0} \ p = 0"
  by (auto simp add: coeffs_def intro: classical)

lemma set_coeffs_not_only_0 [simp]:
  "set (coeffs p) \ {0}"
  by (auto simp add: set_eq_subset)

lemma forall_coeffs_conv:
  "(\n. P (coeff p n)) \ (\c \ set (coeffs p). P c)" if "P 0"
  using that by (auto simp add: coeffs_def)
    (metis atLeastLessThan_iff coeff_eq_0 not_less_iff_gr_or_eq zero_le)

instantiation poly :: ("{zero, equal}") equal
begin

definition [code]: "HOL.equal (p::'a poly) q \ 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 \ True"
  by (fact equal_refl)

definition is_zero :: "'a::zero poly \ bool"
  where [code]: "is_zero p \ List.null (coeffs p)"

lemma is_zero_null [code_abbrev]: "is_zero p \ p = 0"
  by (simp add: is_zero_def null_def)

subsubsection \<open>Reconstructing the polynomial from the list\<close>
  \<comment> \<open>contributed by Sebastiaan J.C. Joosten and René Thiemann\<close>

definition poly_of_list :: "'a::comm_monoid_add list \ 'a poly"
  where [simp]: "poly_of_list = Poly"

lemma poly_of_list_impl [code abstract]: "coeffs (poly_of_list as) = strip_while (HOL.eq 0) as"
  by simp

subsection \<open>Fold combinator for polynomials\<close>

definition fold_coeffs :: "('a::zero \ 'b \ 'b) \ 'a poly \ 'b \ '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 \ fold_coeffs f (pCons a p) = f a \ 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 \ 0 \ fold_coeffs f (pCons a p) = f a \ fold_coeffs f p"
  by (simp add: fold_coeffs_def)

lemma fold_coeffs_pCons_not_0_0_eq [simp]:
  "p \ 0 \ fold_coeffs f (pCons a p) = f a \ fold_coeffs f p"
  by (simp add: fold_coeffs_def)

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

definition poly :: \<open>'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a\<close>
  where \<open>poly p a = horner_sum id a (coeffs p)\<close>

lemma poly_eq_fold_coeffs:
  \<open>poly p = fold_coeffs (\<lambda>a f x. a + x * f x) p (\<lambda>x. 0)\<close>
  by (induction p) (auto simp add: fun_eq_iff poly_def)

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 \ a = 0") (auto simp add: poly_def)

lemma poly_altdef: "poly p x = (\i\degree p. coeff p i * x ^ i)"
  for x :: "'a::{comm_semiring_0,semiring_1}"
proof (induction p rule: pCons_induct)
  case 0
  then show ?case
    by simp
next
  case (pCons a p)
  show ?case
  proof (cases "p = 0")
    case True
    then show ?thesis by simp
  next
    case False
    let ?p' = "pCons a p"
    note poly_pCons[of a p x]
    also note pCons.IH
    also have "a + x * (\i\degree p. coeff p i * x ^ i) =
        coeff ?p' 0 * x^0 + (\i\degree p. coeff ?p' (Suc i) * x^Suc i)"
      by (simp add: field_simps sum_distrib_left coeff_pCons)
    also note sum.atMost_Suc_shift[symmetric]
    also note degree_pCons_eq[OF \<open>p \<noteq> 0\<close>, of a, symmetric]
    finally show ?thesis .
  qed
qed

lemma poly_0_coeff_0: "poly p 0 = coeff p 0"
  by (cases p) (auto simp: poly_altdef)

subsection \<open>Monomials\<close>

lift_definition monom :: "'a \ nat \ 'a::zero poly"
  is "\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 \ a = 0"
  by (simp add: poly_eq_iff)

lemma monom_eq_iff [simp]: "monom a n = monom b n \ a = b"
  by (simp add: poly_eq_iff)

lemma degree_monom_le: "degree (monom a n) \ n"
  by (rule degree_le, simp)

lemma degree_monom_eq: "a \ 0 \ degree (monom a n) = n"
  by (metis coeff_monom leading_coeff_0_iff)

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 \ 0 \ fold_coeffs f (monom a n) = f 0 ^^ n \ f a"
  by (simp add: fold_coeffs_def coeffs_monom fun_eq_iff)

lemma poly_monom: "poly (monom a n) x = a * x ^ n"
  for a x :: "'a::comm_semiring_1"
  by (cases "a = 0", simp_all) (induct n, simp_all add: mult.left_commute poly_eq_fold_coeffs)

lemma monom_eq_iff': "monom c n = monom d m \ c = d \ (c = 0 \ n = m)"
  by (auto simp: poly_eq_iff)

lemma monom_eq_const_iff: "monom c n = [:d:] \ c = d \ (c = 0 \ n = 0)"
  using monom_eq_iff'[of c n d 0] by (simp add: monom_0)

subsection \<open>Leading coefficient\<close>

abbreviation lead_coeff:: "'a::zero poly \ 'a"
  where "lead_coeff p \ coeff p (degree p)"

lemma lead_coeff_pCons[simp]:
  "p \ 0 \ lead_coeff (pCons a p) = lead_coeff p"
  "p = 0 \ lead_coeff (pCons a p) = a"
  by auto

lemma lead_coeff_monom [simp]: "lead_coeff (monom c n) = c"
  by (cases "c = 0") (simp_all add: degree_monom_eq)

lemma last_coeffs_eq_coeff_degree:
  "last (coeffs p) = lead_coeff p" if "p \ 0"
  using that by (simp add: coeffs_def)


subsection \<open>Addition and subtraction\<close>

instantiation poly :: (comm_monoid_add) comm_monoid_add
begin

lift_definition plus_poly :: "'a poly \ 'a poly \ 'a poly"
  is "\p q n. coeff p n + coeff q n"
proof -
  fix q p :: "'a poly"
  show "\\<^sub>\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 \ 'a poly \ 'a poly"
  is "\p q n. coeff p n - coeff q n"
proof -
  fix q p :: "'a poly"
  show "\\<^sub>\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 \ 'a poly"
  is "\p n. - coeff p n"
proof -
  fix p :: "'a poly"
  show "\\<^sub>\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) \ max (degree p) (degree q)"
  by (rule degree_le) (auto simp add: coeff_eq_0)

lemma degree_add_le: "degree p \ n \ degree q \ n \ degree (p + q) \ n"
  by (auto intro: order_trans degree_add_le_max)

lemma degree_add_less: "degree p < n \ degree q < n \ degree (p + q) < n"
  by (auto intro: le_less_trans degree_add_le_max)

lemma degree_add_eq_right: assumes "degree p < degree q" shows "degree (p + q) = degree q"
proof (cases "q = 0")
  case False
  show ?thesis
  proof (rule order_antisym)
    show "degree (p + q) \ degree q"
      by (simp add: assms degree_add_le order.strict_implies_order)
    show "degree q \ degree (p + q)"
      by (simp add: False assms coeff_eq_0 le_degree)
  qed
qed (use assms in auto)

lemma degree_add_eq_left: "degree q < degree p \ 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"
  by (simp add: degree_def)

lemma lead_coeff_add_le: "degree p < degree q \ lead_coeff (p + q) = lead_coeff q"
  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)

lemma degree_diff_le_max: "degree (p - q) \ max (degree p) (degree q)"
  for p q :: "'a::ab_group_add poly"
  using degree_add_le [where p=p and q="-q"] by simp

lemma degree_diff_le: "degree p \ n \ degree q \ n \ degree (p - q) \ n"
  for p q :: "'a::ab_group_add poly"
  using degree_add_le [of p n "- q"] by simp

lemma degree_diff_less: "degree p < n \ degree q < n \ degree (p - q) < n"
  for p q :: "'a::ab_group_add poly"
  using 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_sum: "coeff (\x\A. p x) i = (\x\A. coeff (p x) i)"
  by (induct A rule: infinite_finite_induct) simp_all

lemma monom_sum: "monom (\x\A. a x) n = (\x\A. monom (a x) n)"
  by (rule poly_eqI) (simp add: coeff_sum)

fun plus_coeffs :: "'a::comm_monoid_add list \ 'a list \ '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 -
  have *: "nth_default 0 (plus_coeffs xs ys) n = nth_default 0 xs n + nth_default 0 ys n"
    for xs ys :: "'a list" and 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
  have **: "no_trailing (HOL.eq 0) (plus_coeffs xs ys)"
    if "no_trailing (HOL.eq 0) xs" and "no_trailing (HOL.eq 0) ys"
    for xs ys :: "'a list"
    using that by (induct xs ys rule: plus_coeffs.induct) (simp_all add: cCons_def)
  show ?thesis
    by (rule coeffs_eqI) (auto simp add: * nth_default_coeffs_eq intro: **)
qed

lemma coeffs_uminus [code abstract]:
  "coeffs (- p) = map uminus (coeffs p)"
proof -
  have eq_0: "HOL.eq 0 \ uminus = HOL.eq (0::'a)"
    by (simp add: fun_eq_iff)
  show ?thesis
    by (rule coeffs_eqI) (simp_all add: nth_default_map_eq nth_default_coeffs_eq no_trailing_map eq_0)
qed

lemma [code]: "p - q = p + - q"
  for p q :: "'a::ab_group_add poly"
  by (fact diff_conv_add_uminus)

lemma poly_add [simp]: "poly (p + q) x = poly p x + poly q x"
proof (induction p arbitrary: q)
  case (pCons a p)
  then show ?case
    by (cases q) (simp add: algebra_simps)
qed auto

lemma poly_minus [simp]: "poly (- p) x = - poly p x"
  for x :: "'a::comm_ring"
  by (induct p) simp_all

lemma poly_diff [simp]: "poly (p - q) x = poly p x - poly q x"
  for x :: "'a::comm_ring"
  using poly_add [of p "- q" x] by simp

lemma poly_sum: "poly (\k\A. p k) x = (\k\A. poly (p k) x)"
  by (induct A rule: infinite_finite_induct) simp_all

lemma degree_sum_le: "finite S \ (\p. p \ S \ degree (f p) \ n) \ degree (sum f S) \ n"
proof (induct S rule: finite_induct)
  case empty
  then show ?case by simp
next
  case (insert p S)
  then have "degree (sum f S) \ n" "degree (f p) \ n"
    by auto
  then show ?case
    unfolding sum.insert[OF insert(1-2)] by (metis degree_add_le)
qed

lemma poly_as_sum_of_monoms':
  assumes "degree p \ n"
  shows "(\i\n. monom (coeff p i) i) = p"
proof -
  have eq: "\i. {..n} \ {i} = (if i \ n then {i} else {})"
    by auto
  from assms show ?thesis
    by (simp add: poly_eq_iff coeff_sum coeff_eq_0 sum.If_cases eq
        if_distrib[where f="\x. x * a" for a])
qed

lemma poly_as_sum_of_monoms: "(\i\degree p. monom (coeff p i) i) = p"
  by (intro poly_as_sum_of_monoms' order_refl)

lemma Poly_snoc: "Poly (xs @ [x]) = Poly xs + monom x (length xs)"
  by (induct 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 \ 'a poly \ 'a poly"
  is "\a p n. a * coeff p n"
proof -
  fix a :: 'a and p :: "'a poly"
  show "\\<^sub>\ 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) \ 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 (- p) = - smult a p"
  for a :: "'a::comm_ring"
  by (rule poly_eqI) simp

lemma smult_minus_left [simp]: "smult (- a) p = - smult a p"
  for a :: "'a::comm_ring"
  by (rule poly_eqI) simp

lemma smult_diff_right: "smult a (p - q) = smult a p - smult a q"
  for a :: "'a::comm_ring"
  by (rule poly_eqI) (simp add: algebra_simps)

lemma smult_diff_left: "smult (a - b) p = smult a p - smult b p"
  for a b :: "'a::comm_ring"
  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_all add: monom_0 monom_Suc)

lemma smult_Poly: "smult c (Poly xs) = Poly (map ((*) c) xs)"
  by (auto simp: poly_eq_iff nth_default_def)

lemma degree_smult_eq [simp]: "degree (smult a p) = (if a = 0 then 0 else degree p)"
  for a :: "'a::{comm_semiring_0,semiring_no_zero_divisors}"
  by (cases "a = 0") (simp_all add: degree_def)

lemma smult_eq_0_iff [simp]: "smult a p = 0 \ a = 0 \ p = 0"
  for a :: "'a::{comm_semiring_0,semiring_no_zero_divisors}"
  by (simp add: poly_eq_iff)

lemma coeffs_smult [code abstract]:
  "coeffs (smult a p) = (if a = 0 then [] else map (Groups.times a) (coeffs p))"
  for p :: "'a::{comm_semiring_0,semiring_no_zero_divisors} poly"
proof -
  have eq_0: "HOL.eq 0 \ times a = HOL.eq (0::'a)" if "a \ 0"
    using that by (simp add: fun_eq_iff)
  show ?thesis
    by (rule coeffs_eqI) (auto simp add: no_trailing_map nth_default_map_eq nth_default_coeffs_eq eq_0)
qed

lemma smult_eq_iff:
  fixes b :: "'a :: field"
  assumes "b \ 0"
  shows "smult a p = smult b q \ smult (a / b) p = q"
    (is "?lhs \ ?rhs")
proof
  assume ?lhs
  also from assms have "smult (inverse b) \ = q"
    by simp
  finally show ?rhs
    by (simp add: field_simps)
next
  assume ?rhs
  with assms show ?lhs by auto
qed

instantiation poly :: (comm_semiring_0) comm_semiring_0
begin

definition "p * q = fold_coeffs (\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 \ a = 0") (auto simp add: times_poly_def)

lemma mult_poly_0_right: "p * (0::'a poly) = 0"
  by (induct p) (simp_all add: mult_poly_0_left)

lemma mult_pCons_right [simp]: "p * pCons a q = smult a p + pCons 0 (p * q)"
  by (induct p) (simp_all add: mult_poly_0_left 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_all add: mult_poly_0 smult_add_right)

lemma mult_smult_right [simp]: "p * smult a q = smult a (p * q)"
  by (induct q) (simp_all add: mult_poly_0 smult_add_right)

lemma mult_poly_add_left: "(p + q) * r = p * r + q * r"
  for p q r :: "'a poly"
  by (induct r) (simp_all add: mult_poly_0 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_all add: mult_poly_0 mult_poly_add_left)
  show "p * q = q * p"
    by (induct p) (simp_all add: mult_poly_0)
qed

end

lemma coeff_mult_degree_sum:
  "coeff (p * q) (degree p + degree q) = coeff p (degree p) * coeff q (degree q)"
  by (induct p) (simp_all add: coeff_eq_0)

instance poly :: ("{comm_semiring_0,semiring_no_zero_divisors}") semiring_no_zero_divisors
proof
  fix p q :: "'a poly"
  assume "p \ 0" and "q \ 0"
  have "coeff (p * q) (degree p + degree q) = coeff p (degree p) * coeff q (degree q)"
    by (rule coeff_mult_degree_sum)
  also from \<open>p \<noteq> 0\<close> \<open>q \<noteq> 0\<close> have "coeff p (degree p) * coeff q (degree q) \<noteq> 0"
    by simp
  finally have "\n. coeff (p * q) n \ 0" ..
  then show "p * q \ 0"
    by (simp add: poly_eq_iff)
qed

instance poly :: (comm_semiring_0_cancel) comm_semiring_0_cancel ..

lemma coeff_mult: "coeff (p * q) n = (\i\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)
  then show ?case
    by (cases n) (simp_all add: sum.atMost_Suc_shift del: sum.atMost_Suc)
qed

lemma degree_mult_le: "degree (p * q) \ degree p + degree q"
proof (rule degree_le)
  show "\i>degree p + degree q. coeff (p * q) i = 0"
    by (induct p) (simp_all add: coeff_eq_0 coeff_pCons split: nat.split)
qed

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

lift_definition one_poly :: "'a poly"
  is "\n. of_bool (n = 0)"
  by (rule MOST_SucD) simp

lemma coeff_1 [simp]:
  "coeff 1 n = of_bool (n = 0)"
  by (simp add: one_poly.rep_eq)

lemma one_pCons:
  "1 = [:1:]"
  by (simp add: poly_eq_iff coeff_pCons split: nat.splits)

lemma pCons_one:
  "[:1:] = 1"
  by (simp add: one_pCons)

instance
  by standard (simp_all add: one_pCons)

end

lemma poly_1 [simp]:
  "poly 1 x = 1"
  by (simp add: one_pCons)

lemma one_poly_eq_simps [simp]:
  "1 = [:1:] \ True"
  "[:1:] = 1 \ True"
  by (simp_all add: one_pCons)

lemma degree_1 [simp]:
  "degree 1 = 0"
  by (simp add: one_pCons)

lemma coeffs_1_eq [simp, code abstract]:
  "coeffs 1 = [1]"
  by (simp add: one_pCons)

lemma smult_one [simp]:
  "smult c 1 = [:c:]"
  by (simp add: one_pCons)

lemma monom_eq_1 [simp]:
  "monom 1 0 = 1"
  by (simp add: monom_0 one_pCons)

lemma monom_eq_1_iff:
  "monom c n = 1 \ c = 1 \ n = 0"
  using monom_eq_const_iff [of c n 1] by auto

lemma monom_altdef:
  "monom c n = smult c ([:0, 1:] ^ n)"
  by (induct n) (simp_all add: monom_0 monom_Suc)

instance poly :: ("{comm_semiring_1,semiring_1_no_zero_divisors}") semiring_1_no_zero_divisors ..
instance poly :: (comm_ring) comm_ring ..
instance poly :: (comm_ring_1) comm_ring_1 ..
instance poly :: (comm_ring_1) comm_semiring_1_cancel ..

lemma degree_power_le: "degree (p ^ n) \ degree p * n"
  by (induct n) (auto intro: order_trans degree_mult_le)

lemma coeff_0_power: "coeff (p ^ n) 0 = coeff p 0 ^ n"
  by (induct n) (simp_all add: coeff_mult)

lemma poly_smult [simp]: "poly (smult a p) x = a * poly p x"
  by (induct p) (simp_all add: algebra_simps)

lemma poly_mult [simp]: "poly (p * q) x = poly p x * poly q x"
  by (induct p) (simp_all add: algebra_simps)

lemma poly_power [simp]: "poly (p ^ n) x = poly p x ^ n"
  for p :: "'a::comm_semiring_1 poly"
  by (induct n) simp_all

lemma poly_prod: "poly (\k\A. p k) x = (\k\A. poly (p k) x)"
  by (induct A rule: infinite_finite_induct) simp_all

lemma degree_prod_sum_le: "finite S \ degree (prod f S) \ sum (degree \ f) S"
proof (induct S rule: finite_induct)
  case empty
  then show ?case by simp
next
  case (insert a S)
  show ?case
    unfolding prod.insert[OF insert(1-2)] sum.insert[OF insert(1-2)]
    by (rule le_trans[OF degree_mult_le]) (use insert in auto)
qed

lemma coeff_0_prod_list: "coeff (prod_list xs) 0 = prod_list (map (\p. coeff p 0) xs)"
  by (induct xs) (simp_all add: coeff_mult)

lemma coeff_monom_mult: "coeff (monom c n * p) k = (if k < n then 0 else c * coeff p (k - n))"
proof -
  have "coeff (monom c n * p) k = (\i\k. (if n = i then c else 0) * coeff p (k - i))"
    by (simp add: coeff_mult)
  also have "\ = (\i\k. (if n = i then c * coeff p (k - i) else 0))"
    by (intro sum.cong) simp_all
  also have "\ = (if k < n then 0 else c * coeff p (k - n))"
    by simp
  finally show ?thesis .
qed

lemma monom_1_dvd_iff': "monom 1 n dvd p \ (\k
proof
  assume "monom 1 n dvd p"
  then obtain r where "p = monom 1 n * r"
    by (rule dvdE)
  then show "\k
    by (simp add: coeff_mult)
next
  assume zero: "(\k
  define r where "r = Abs_poly (\k. coeff p (k + n))"
  have "\\<^sub>\k. coeff p (k + n) = 0"
    by (subst cofinite_eq_sequentially, subst eventually_sequentially_seg,
        subst cofinite_eq_sequentially [symmetric]) transfer
  then have coeff_r [simp]: "coeff r k = coeff p (k + n)" for k
    unfolding r_def by (subst poly.Abs_poly_inverse) simp_all
  have "p = monom 1 n * r"
    by (rule poly_eqI, subst coeff_monom_mult) (simp_all add: zero)
  then show "monom 1 n dvd p" by simp
qed

subsection \<open>Mapping polynomials\<close>

definition map_poly :: "('a :: zero \ 'b :: zero) \ 'a poly \ 'b poly"
  where "map_poly f p = Poly (map f (coeffs p))"

lemma map_poly_0 [simp]: "map_poly f 0 = 0"
  by (simp add: map_poly_def)

lemma map_poly_1: "map_poly f 1 = [:f 1:]"
  by (simp add: map_poly_def)

lemma map_poly_1' [simp]: "f 1 = 1 \ map_poly f 1 = 1"
  by (simp add: map_poly_def one_pCons)

lemma coeff_map_poly:
  assumes "f 0 = 0"
  shows "coeff (map_poly f p) n = f (coeff p n)"
  by (auto simp: assms map_poly_def nth_default_def coeffs_def not_less Suc_le_eq coeff_eq_0
      simp del: upt_Suc)

lemma coeffs_map_poly [code abstract]:
  "coeffs (map_poly f p) = strip_while ((=) 0) (map f (coeffs p))"
  by (simp add: map_poly_def)

lemma coeffs_map_poly':
  assumes "\x. x \ 0 \ f x \ 0"
  shows "coeffs (map_poly f p) = map f (coeffs p)"
  using assms
  by (auto simp add: coeffs_map_poly strip_while_idem_iff
    last_coeffs_eq_coeff_degree no_trailing_unfold last_map)

lemma set_coeffs_map_poly:
  "(\x. f x = 0 \ x = 0) \ set (coeffs (map_poly f p)) = f ` set (coeffs p)"
  by (simp add: coeffs_map_poly')

lemma degree_map_poly:
  assumes "\x. x \ 0 \ f x \ 0"
  shows "degree (map_poly f p) = degree p"
  by (simp add: degree_eq_length_coeffs coeffs_map_poly' assms)

lemma map_poly_eq_0_iff:
  assumes "f 0 = 0" "\x. x \ set (coeffs p) \ x \ 0 \ f x \ 0"
  shows "map_poly f p = 0 \ p = 0"
proof -
  have "(coeff (map_poly f p) n = 0) = (coeff p n = 0)" for n
  proof -
    have "coeff (map_poly f p) n = f (coeff p n)"
      by (simp add: coeff_map_poly assms)
    also have "\ = 0 \ coeff p n = 0"
    proof (cases "n < length (coeffs p)")
      case True
      then have "coeff p n \ set (coeffs p)"
        by (auto simp: coeffs_def simp del: upt_Suc)
      with assms show "f (coeff p n) = 0 \ coeff p n = 0"
        by auto
    next
      case False
      then show ?thesis
        by (auto simp: assms length_coeffs nth_default_coeffs_eq [symmetric] nth_default_def)
    qed
    finally show ?thesis .
  qed
  then show ?thesis by (auto simp: poly_eq_iff)
qed

lemma map_poly_smult:
  assumes "f 0 = 0""\c x. f (c * x) = f c * f x"
  shows "map_poly f (smult c p) = smult (f c) (map_poly f p)"
  by (intro poly_eqI) (simp_all add: assms coeff_map_poly)

lemma map_poly_pCons:
  assumes "f 0 = 0"
  shows "map_poly f (pCons c p) = pCons (f c) (map_poly f p)"
  by (intro poly_eqI) (simp_all add: assms coeff_map_poly coeff_pCons split: nat.splits)

lemma map_poly_map_poly:
  assumes "f 0 = 0" "g 0 = 0"
  shows "map_poly f (map_poly g p) = map_poly (f \ g) p"
  by (intro poly_eqI) (simp add: coeff_map_poly assms)

lemma map_poly_id [simp]: "map_poly id p = p"
  by (simp add: map_poly_def)

lemma map_poly_id' [simp]: "map_poly (\x. x) p = p"
  by (simp add: map_poly_def)

lemma map_poly_cong:
  assumes "(\x. x \ set (coeffs p) \ f x = g x)"
  shows "map_poly f p = map_poly g p"
proof -
  from assms have "map f (coeffs p) = map g (coeffs p)"
    by (intro map_cong) simp_all
  then show ?thesis
    by (simp only: coeffs_eq_iff coeffs_map_poly)
qed

lemma map_poly_monom: "f 0 = 0 \ map_poly f (monom c n) = monom (f c) n"
  by (intro poly_eqI) (simp_all add: coeff_map_poly)

lemma map_poly_idI:
  assumes "\x. x \ set (coeffs p) \ f x = x"
  shows "map_poly f p = p"
  using map_poly_cong[OF assms, of _ id] by simp

lemma map_poly_idI':
  assumes "\x. x \ set (coeffs p) \ f x = x"
  shows "p = map_poly f p"
  using map_poly_cong[OF assms, of _ id] by simp

lemma smult_conv_map_poly: "smult c p = map_poly (\x. c * x) p"
  by (intro poly_eqI) (simp_all add: coeff_map_poly)

subsection \<open>Conversions\<close>

lemma of_nat_poly:
  "of_nat n = [:of_nat n:]"
  by (induct n) (simp_all add: one_pCons)

lemma of_nat_monom:
  "of_nat n = monom (of_nat n) 0"
  by (simp add: of_nat_poly monom_0)

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

lemma lead_coeff_of_nat [simp]:
  "lead_coeff (of_nat n) = of_nat n"
  by (simp add: of_nat_poly)

lemma of_int_poly:
  "of_int k = [:of_int k:]"
  by (simp only: of_int_of_nat of_nat_poly) simp

lemma of_int_monom:
  "of_int k = monom (of_int k) 0"
  by (simp add: of_int_poly monom_0)

lemma degree_of_int [simp]:
  "degree (of_int k) = 0"
  by (simp add: of_int_poly)

lemma lead_coeff_of_int [simp]:
  "lead_coeff (of_int k) = of_int k"
  by (simp add: of_int_poly)

lemma numeral_poly: "numeral n = [:numeral n:]"
proof -
  have "numeral n = of_nat (numeral n)"
    by simp
  also have "\ = [:of_nat (numeral n):]"
    by (simp add: of_nat_poly)
  finally show ?thesis
    by simp
qed

lemma numeral_monom:
  "numeral n = monom (numeral n) 0"
  by (simp add: numeral_poly monom_0)

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

lemma lead_coeff_numeral [simp]:
  "lead_coeff (numeral n) = numeral n"
  by (simp add: numeral_poly)

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

lemma dvd_smult:
  assumes "p dvd q"
  shows "p dvd smult a q"
proof -
  from assms 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: "p dvd smult a q \ a \ 0 \ p dvd q"
  for a :: "'a::field"
  by (drule dvd_smult [where a="inverse a"]) simp

lemma dvd_smult_iff: "a \ 0 \ p dvd smult a q \ p dvd q"
  for a :: "'a::field"
  by (safe elim!: dvd_smult dvd_smult_cancel)

lemma smult_dvd_cancel:
  assumes "smult a p dvd q"
  shows "p dvd q"
proof -
  from assms 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: "p dvd q \ a \ 0 \ smult a p dvd q"
  for a :: "'a::field"
  by (rule smult_dvd_cancel [where a="inverse a"]) simp

lemma smult_dvd_iff: "smult a p dvd q \ (if a = 0 then q = 0 else p dvd q)"
  for a :: "'a::field"
  by (auto elim: smult_dvd smult_dvd_cancel)

lemma is_unit_smult_iff: "smult c p dvd 1 \ c dvd 1 \ p dvd 1"
proof -
  have "smult c p = [:c:] * p" by simp
  also have "\ dvd 1 \ c dvd 1 \ p dvd 1"
  proof safe
    assume *: "[:c:] * p dvd 1"
    then show "p dvd 1"
      by (rule dvd_mult_right)
    from * obtain q where q: "1 = [:c:] * p * q"
      by (rule dvdE)
    have "c dvd c * (coeff p 0 * coeff q 0)"
      by simp
    also have "\ = coeff ([:c:] * p * q) 0"
      by (simp add: mult.assoc coeff_mult)
    also note q [symmetric]
    finally have "c dvd coeff 1 0" .
    then show "c dvd 1" by simp
  next
    assume "c dvd 1" "p dvd 1"
    from this(1) obtain d where "1 = c * d"
      by (rule dvdE)
    then have "1 = [:c:] * [:d:]"
      by (simp add: one_pCons ac_simps)
    then have "[:c:] dvd 1"
      by (rule dvdI)
    from mult_dvd_mono[OF this \<open>p dvd 1\<close>] show "[:c:] * p dvd 1"
      by simp
  qed
  finally show ?thesis .
qed

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

instance poly :: (idom) idom ..

instance poly :: ("{ring_char_0, comm_ring_1}") ring_char_0
  by standard (auto simp add: of_nat_poly intro: injI)

lemma degree_mult_eq: "p \ 0 \ q \ 0 \ degree (p * q) = degree p + degree q"
  for p q :: "'a::{comm_semiring_0,semiring_no_zero_divisors} poly"
  by (rule order_antisym [OF degree_mult_le le_degree]) (simp add: coeff_mult_degree_sum)

lemma degree_prod_eq_sum_degree:
  fixes A :: "'a set"
  and f :: "'a \ 'b::idom poly"
  assumes f0: "\i\A. f i \ 0"
  shows "degree (\i\A. (f i)) = (\i\A. degree (f i))"
  using assms
  by (induction A rule: infinite_finite_induct) (auto simp: degree_mult_eq)

lemma degree_mult_eq_0:
  "degree (p * q) = 0 \ p = 0 \ q = 0 \ (p \ 0 \ q \ 0 \ degree p = 0 \ degree q = 0)"
  for p q :: "'a::{comm_semiring_0,semiring_no_zero_divisors} poly"
  by (auto simp: degree_mult_eq)

lemma degree_power_eq: "p \ 0 \ degree ((p :: 'a :: idom poly) ^ n) = n * degree p"
  by (induction n) (simp_all add: degree_mult_eq)

lemma degree_mult_right_le:
  fixes p q :: "'a::{comm_semiring_0,semiring_no_zero_divisors} poly"
  assumes "q \ 0"
  shows "degree p \ degree (p * q)"
  using assms by (cases "p = 0") (simp_all add: degree_mult_eq)

lemma coeff_degree_mult: "coeff (p * q) (degree (p * q)) = coeff q (degree q) * coeff p (degree p)"
  for p q :: "'a::{comm_semiring_0,semiring_no_zero_divisors} poly"
  by (cases "p = 0 \ q = 0") (auto simp: degree_mult_eq coeff_mult_degree_sum mult_ac)

lemma dvd_imp_degree_le: "p dvd q \ q \ 0 \ degree p \ degree q"
  for p q :: "'a::{comm_semiring_1,semiring_no_zero_divisors} poly"
  by (erule dvdE, hypsubst, subst degree_mult_eq) auto

lemma divides_degree:
  fixes p q :: "'a ::{comm_semiring_1,semiring_no_zero_divisors} poly"
  assumes "p dvd q"
  shows "degree p \ degree q \ q = 0"
  by (metis dvd_imp_degree_le assms)

lemma const_poly_dvd_iff:
  fixes c :: "'a::{comm_semiring_1,semiring_no_zero_divisors}"
  shows "[:c:] dvd p \ (\n. c dvd coeff p n)"
proof (cases "c = 0 \ p = 0")
  case True
  then show ?thesis
    by (auto intro!: poly_eqI)
next
  case False
  show ?thesis
  proof
    assume "[:c:] dvd p"
    then show "\n. c dvd coeff p n"
      by (auto elim!: dvdE simp: coeffs_def)
  next
    assume *: "\n. c dvd coeff p n"
    define mydiv where "mydiv x y = (SOME z. x = y * z)" for x y :: 'a
    have mydiv: "x = y * mydiv x y" if "y dvd x" for x y
      using that unfolding mydiv_def dvd_def by (rule someI_ex)
    define q where "q = Poly (map (\a. mydiv a c) (coeffs p))"
    from False * have "p = q * [:c:]"
      by (intro poly_eqI)
        (auto simp: q_def nth_default_def not_less length_coeffs_degree coeffs_nth
          intro!: coeff_eq_0 mydiv)
    then show "[:c:] dvd p"
      by (simp only: dvd_triv_right)
  qed
qed

lemma const_poly_dvd_const_poly_iff [simp]: "[:a:] dvd [:b:] \ a dvd b"
  for a b :: "'a::{comm_semiring_1,semiring_no_zero_divisors}"
  by (subst const_poly_dvd_iff) (auto simp: coeff_pCons split: nat.splits)

lemma lead_coeff_mult: "lead_coeff (p * q) = lead_coeff p * lead_coeff q"
  for p q :: "'a::{comm_semiring_0, semiring_no_zero_divisors} poly"
  by (cases "p = 0 \ q = 0") (auto simp: coeff_mult_degree_sum degree_mult_eq)

lemma lead_coeff_prod: "lead_coeff (prod f A) = (\x\A. lead_coeff (f x))"
  for f :: "'a \ 'b::{comm_semiring_1, semiring_no_zero_divisors} poly"
  by (induction A rule: infinite_finite_induct) (auto simp: lead_coeff_mult)

lemma lead_coeff_smult: "lead_coeff (smult c p) = c * lead_coeff p"
  for p :: "'a::{comm_semiring_0,semiring_no_zero_divisors} poly"
proof -
  have "smult c p = [:c:] * p" by simp
  also have "lead_coeff \ = 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

lemma lead_coeff_power: "lead_coeff (p ^ n) = lead_coeff p ^ n"
  for p :: "'a::{comm_semiring_1,semiring_no_zero_divisors} poly"
  by (induct n) (simp_all add: lead_coeff_mult)

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

definition pos_poly :: "'a::linordered_semidom poly \ bool"
  where "pos_poly p \ 0 < coeff p (degree p)"

lemma pos_poly_pCons: "pos_poly (pCons a p) \ pos_poly p \ (p = 0 \ 0 < a)"
  by (simp add: pos_poly_def)

lemma not_pos_poly_0 [simp]: "\ pos_poly 0"
  by (simp add: pos_poly_def)

lemma pos_poly_add: "pos_poly p \ pos_poly q \ pos_poly (p + q)"
proof (induction p arbitrary: q)
  case (pCons a p)
  then show ?case
    by (cases q; force simp add: pos_poly_pCons add_pos_pos)
qed auto

lemma pos_poly_mult: "pos_poly p \ pos_poly q \ pos_poly (p * q)"
  by (simp add: pos_poly_def coeff_degree_mult)

lemma pos_poly_total: "p = 0 \ pos_poly p \ pos_poly (- p)"
  for p :: "'a::linordered_idom poly"
  by (induct p) (auto simp: pos_poly_pCons)

lemma pos_poly_coeffs [code]: "pos_poly p \ (let as = coeffs p in as \ [] \ last as > 0)"
  (is "?lhs \ ?rhs")
proof
  assume ?rhs
  then show ?lhs
    by (auto simp add: pos_poly_def last_coeffs_eq_coeff_degree)
next
  assume ?lhs
  then have *: "0 < coeff p (degree p)"
    by (simp add: pos_poly_def)
  then have "p \ 0"
    by auto
  with * show ?rhs
    by (simp add: last_coeffs_eq_coeff_degree)
qed

instantiation poly :: (linordered_idom) linordered_idom
begin

definition "x < y \ pos_poly (y - x)"

definition "x \ y \ x = y \ pos_poly (y - x)"

definition "\x::'a poly\ = (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 \ x \ y \ \ y \ x"
    unfolding less_eq_poly_def less_poly_def
    using pos_poly_add by force
  then show "x \ y \ y \ x \ x = y"
    using less_eq_poly_def less_poly_def by force
  show "x \ x"
    by (simp add: less_eq_poly_def)
  show "x \ y \ y \ z \ x \ z"
    using less_eq_poly_def pos_poly_add by fastforce
  show "x \ y \ z + x \ z + y"
    by (simp add: less_eq_poly_def)
  show "x \ y \ y \ x"
    unfolding less_eq_poly_def
    using pos_poly_total [of "x - y"]
    by auto
  show "x < y \ 0 < z \ z * x < z * y"
    by (simp add: less_poly_def right_diff_distrib [symmetric] pos_poly_mult)
  show "\x\ = (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>

subsubsection \<open>Synthetic division\<close>

text \<open>Synthetic division is simply division by the linear polynomial \<^term>\<open>x - c\<close>.\<close>

definition synthetic_divmod :: "'a::comm_semiring_0 poly \ 'a \ 'a poly \ 'a"
  where "synthetic_divmod p c = fold_coeffs (\a (q, r). (pCons r q, a + c * r)) p (0, 0)"

definition synthetic_div :: "'a::comm_semiring_0 poly \ 'a \ '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 = (\(q, r). (pCons r q, a + c * r)) (synthetic_divmod p c)"
  by (cases "p = 0 \ a = 0") (auto simp add: synthetic_divmod_def)

lemma synthetic_div_0 [simp]: "synthetic_div 0 c = 0"
  by (simp add: synthetic_div_def)

lemma synthetic_div_unique_lemma: "smult c p = pCons a p \ 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_all add: split_def)

lemma synthetic_div_pCons [simp]:
  "synthetic_div (pCons a p) c = pCons (poly p c) (synthetic_div p c)"
  by (simp add: synthetic_div_def split_def snd_synthetic_divmod)

lemma synthetic_div_eq_0_iff: "synthetic_div p c = 0 \ degree p = 0"
proof (induct p)
  case 0
  then show ?case by simp
next
  case (pCons a p)
  then show ?case by (cases p) simp
qed

lemma degree_synthetic_div: "degree (synthetic_div p c) = degree p - 1"
  by (induct p) (simp_all 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 \ r = poly p c \ q = synthetic_div p c"
proof (induction p arbitrary: q r)
  case 0
  then show ?case
    using synthetic_div_unique_lemma by fastforce
next
  case (pCons a p)
  then show ?case
    by (cases q; force)
qed

lemma synthetic_div_correct': "[:-c, 1:] * synthetic_div p c + [:poly p c:] = p"
  for c :: "'a::comm_ring_1"
  using synthetic_div_correct [of p c] by (simp add: algebra_simps)

subsubsection \<open>Polynomial roots\<close>

lemma poly_eq_0_iff_dvd: "poly p c = 0 \ [:- c, 1:] dvd p"
  (is "?lhs \ ?rhs")
  for c :: "'a::comm_ring_1"
proof
  assume ?lhs
  with synthetic_div_correct' [of c p] have "p = [:-c, 1:] * synthetic_div p c" by simp
  then show ?rhs ..
next
  assume ?rhs
  then obtain k where "p = [:-c, 1:] * k" by (rule dvdE)
  then show ?lhs by simp
qed

lemma dvd_iff_poly_eq_0: "[:c, 1:] dvd p \ poly p (- c) = 0"
  for c :: "'a::comm_ring_1"
  by (simp add: poly_eq_0_iff_dvd)

lemma poly_roots_finite: "p \ 0 \ finite {x. poly p x = 0}"
  for p :: "'a::{comm_ring_1,ring_no_zero_divisors} poly"
proof (induct n \<equiv> "degree p" arbitrary: p)
  case 0
  then obtain a where "a \ 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)
  show "finite {x. poly p x = 0}"
  proof (cases "\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
    from this \<open>k \<noteq> 0\<close> have "finite {x. poly k x = 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 Collect_disj_eq del: mult_pCons_left)
  qed
qed

lemma poly_eq_poly_eq_iff: "poly p = poly q \ p = q"
  (is "?lhs \ ?rhs")
  for p q :: "'a::{comm_ring_1,ring_no_zero_divisors,ring_char_0} poly"
proof
  assume ?rhs
  then show ?lhs by simp
next
  assume ?lhs
  have "poly p = poly 0 \ p = 0" for p :: "'a poly"
  proof (cases "p = 0")
    case False
    then show ?thesis
      by (auto simp add: infinite_UNIV_char_0 dest: poly_roots_finite)
  qed auto
  from \<open>?lhs\<close> and this [of "p - q"] show ?rhs
    by auto
qed

lemma poly_all_0_iff_0: "(\x. poly p x = 0) \ p = 0"
  for p :: "'a::{ring_char_0,comm_ring_1,ring_no_zero_divisors} poly"
  by (auto simp add: poly_eq_poly_eq_iff [symmetric])

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

definition order :: "'a::idom \ 'a poly \ nat"
  where "order a p = (LEAST n. \ [:-a, 1:] ^ Suc n dvd p)"

lemma coeff_linear_power: "coeff ([:a, 1:] ^ n) n = 1"
  for a :: "'a::comm_semiring_1"
proof (induct n)
  case (Suc n)
  have "degree ([:a, 1:] ^ n) \ 1 * n"
    by (metis One_nat_def degree_pCons_eq_if degree_power_le one_neq_zero one_pCons)
  then have "coeff ([:a, 1:] ^ n) (Suc n) = 0"
    by (simp add: coeff_eq_0)
  then show ?case
    using Suc.hyps by fastforce
qed auto

lemma degree_linear_power: "degree ([:a, 1:] ^ n) = n"
  for a :: "'a::comm_semiring_1"
proof (rule order_antisym)
  show "degree ([:a, 1:] ^ n) \ n"
    by (metis One_nat_def degree_pCons_eq_if degree_power_le mult.left_neutral one_neq_zero one_pCons)
qed (simp add: coeff_linear_power le_degree)

lemma order_1: "[:-a, 1:] ^ order a p dvd p"
proof (cases "p = 0")
  case False
  show ?thesis
  proof (cases "order a p")
    case (Suc n)
    then show ?thesis
      by (metis lessI not_less_Least order_def)
  qed auto
qed auto

lemma order_2:
  assumes "p \ 0"
  shows "\ [:-a, 1:] ^ Suc (order a p) dvd p"
proof -
  have False if "[:- a, 1:] ^ Suc (degree p) dvd p"
    using dvd_imp_degree_le [OF that]
    by (metis Suc_n_not_le_n assms degree_linear_power)
  then show ?thesis
    unfolding order_def
    by (metis (no_types, lifting) LeastI)
qed

lemma order: "p \ 0 \ [:-a, 1:] ^ order a p dvd p \ \ [:-a, 1:] ^ Suc (order a p) dvd p"
  by (rule conjI [OF order_1 order_2])

lemma order_degree:
  assumes p: "p \ 0"
  shows "order a p \ degree p"
proof -
  have "order a p = degree ([:-a, 1:] ^ order a p)"
    by (simp only: degree_linear_power)
  also from order_1 p have "\ \ degree p"
    by (rule dvd_imp_degree_le)
  finally show ?thesis .
qed

lemma order_root: "poly p a = 0 \ p = 0 \ order a p \ 0" (is "?lhs = ?rhs")
proof
  show "?lhs \ ?rhs"
    by (metis One_nat_def order_2 poly_eq_0_iff_dvd power_one_right)
  show "?rhs \ ?lhs"
    by (meson dvd_power dvd_trans neq0_conv order_1 poly_0 poly_eq_0_iff_dvd)
qed

lemma order_0I: "poly p a \ 0 \ order a p = 0"
  by (subst (asm) order_root) auto

lemma order_unique_lemma:
  fixes p :: "'a::idom poly"
  assumes "[:-a, 1:] ^ n dvd p" "\ [:-a, 1:] ^ Suc n dvd p"
  shows "order a p = n"
  unfolding Polynomial.order_def
  by (metis (mono_tags, lifting) Least_equality assms not_less_eq_eq power_le_dvd)

lemma order_mult:
  assumes "p * q \ 0" shows "order a (p * q) = order a p + order a q"
proof -
  define i where "i \ order a p"
  define j where "j \ order a q"
  define t where "t \ [:-a, 1:]"
  have t_dvd_iff: "\u. t dvd u \ poly u a = 0"
    by (simp add: t_def dvd_iff_poly_eq_0)
  have dvd: "t ^ i dvd p" "t ^ j dvd q" and "\ t ^ Suc i dvd p" "\ t ^ Suc j dvd q"
    using assms i_def j_def order_1 order_2 t_def by auto
  then have "\ t ^ Suc(i + j) dvd p * q"
    by (elim dvdE) (simp add: power_add t_dvd_iff)
  moreover have "t ^ (i + j) dvd p * q"
    using dvd by (simp add: mult_dvd_mono power_add)
  ultimately show "order a (p * q) = i + j"
    using order_unique_lemma t_def by blast
qed

lemma order_smult:
  assumes "c \ 0"
  shows "order x (smult c p) = order x p"
proof (cases "p = 0")
  case True
  then show ?thesis
    by simp
next
  case False
  have "smult c p = [:c:] * p" by simp
  also from assms False have "order x \ = order x [:c:] + order x p"
    by (subst order_mult) simp_all
  also have "order x [:c:] = 0"
    by (rule order_0I) (use assms in auto)
  finally show ?thesis
    by simp
qed

text \<open>Next three lemmas contributed by Wenda Li\<close>
lemma order_1_eq_0 [simp]:"order x 1 = 0"
  by (metis order_root poly_1 zero_neq_one)

lemma order_uminus[simp]: "order x (-p) = order x p"
  by (metis neg_equal_0_iff_equal order_smult smult_1_left smult_minus_left)

lemma order_power_n_n: "order a ([:-a,1:]^n)=n"
proof (induct n) (*might be proved more concisely using nat_less_induct*)
  case 0
  then show ?case
    by (metis order_root poly_1 power_0 zero_neq_one)
next
  case (Suc n)
  have "order a ([:- a, 1:] ^ Suc n) = order a ([:- a, 1:] ^ n) + order a [:-a,1:]"
    by (metis (no_types, hide_lams) One_nat_def add_Suc_right monoid_add_class.add.right_neutral
      one_neq_zero order_mult pCons_eq_0_iff power_add power_eq_0_iff power_one_right)
  moreover have "order a [:-a,1:] = 1"
    unfolding order_def
  proof (rule Least_equality, rule notI)
    assume "[:- a, 1:] ^ Suc 1 dvd [:- a, 1:]"
    then have "degree ([:- a, 1:] ^ Suc 1) \ degree ([:- a, 1:])"
      by (rule dvd_imp_degree_le) auto
    then show False
      by auto
  next
    fix y
    assume *: "\ [:- a, 1:] ^ Suc y dvd [:- a, 1:]"
    show "1 \ y"
    proof (rule ccontr)
      assume "\ 1 \ y"
      then have "y = 0" by auto
      then have "[:- a, 1:] ^ Suc y dvd [:- a, 1:]" by auto
      with * show False by auto
    qed
  qed
  ultimately show ?case
    using Suc by auto
qed

lemma order_0_monom [simp]: "c \ 0 \ order 0 (monom c n) = n"
  using order_power_n_n[of 0 n] by (simp add: monom_altdef order_smult)

lemma dvd_imp_order_le: "q \ 0 \ p dvd q \ Polynomial.order a p \ Polynomial.order a q"
  by (auto simp: order_mult elim: dvdE)

text \<open>Now justify the standard squarefree decomposition, i.e. \<open>f / gcd f f'\<close>.\<close>

lemma order_divides: "[:-a, 1:] ^ n dvd p \ p = 0 \ n \ order a p"
  by (meson dvd_0_right not_less_eq_eq order_1 order_2 power_le_dvd)

lemma order_decomp:
  assumes "p \ 0"
  shows "\q. p = [:- a, 1:] ^ order a p * q \ \ [:- a, 1:] dvd q"
proof -
  from assms have *: "[:- a, 1:] ^ order a p dvd p"
    and **: "\ [:- a, 1:] ^ Suc (order a p) dvd p"
    by (auto dest: order)
  from * obtain q where q: "p = [:- a, 1:] ^ order a p * q" ..
  with ** have "\ [:- a, 1:] ^ Suc (order a p) dvd [:- a, 1:] ^ order a p * q"
    by simp
  then have "\ [:- a, 1:] ^ order a p * [:- a, 1:] dvd [:- a, 1:] ^ order a p * q"
    by simp
  with idom_class.dvd_mult_cancel_left [of "[:- a, 1:] ^ order a p" "[:- a, 1:]" q]
  have "\ [:- a, 1:] dvd q" by auto
  with q show ?thesis by blast
qed

lemma monom_1_dvd_iff: "p \ 0 \ monom 1 n dvd p \ n \ order 0 p"
  using order_divides[of 0 n p] by (simp add: monom_altdef)

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"
    and "\p. (\a. P a \ poly p a \ 0) \ Q p"
    and "\a p. P a \ Q p \ 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")
    case True
    with assms(1) show ?thesis by simp
  next
    case False
    show ?thesis
    proof (cases "\a. P a \ poly p a = 0")
      case False
      then show ?thesis by (intro assms(2)) blast
    next
      case True
      then obtain a where a: "P a" "poly p a = 0"
        by blast
      then have "-[:-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 False have "q \ 0" by auto
      have "degree p = Suc (degree q)"
        by (subst q, subst degree_mult_eq) (simp_all add: \<open>q \<noteq> 0\<close>)
      then have "Q q" by (intro less) simp
      with a(1) have "Q ([:a, -1:] * q)"
        by (rule assms(3))
      with q show ?thesis by simp
    qed
  qed
qed

lemma dropWhile_replicate_append:
  "dropWhile ((=) a) (replicate n a @ ys) = dropWhile ((=) a) ys"
  by (induct n) simp_all

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