Theory Polynomial

theory Polynomial
imports GCD
(*  Title:      HOL/Library/Polynomial.thy
    Author:     Brian Huffman
    Author:     Clemens Ballarin
    Author:     Florian Haftmann
*)

header {* Polynomials as type over a ring structure *}

theory Polynomial
imports Main GCD
begin

subsection {* Auxiliary: operations for lists (later) representing coefficients *}

definition strip_while :: "('a => bool) => 'a list => 'a list"
where
  "strip_while P = rev o dropWhile P o rev"

lemma strip_while_Nil [simp]:
  "strip_while P [] = []"
  by (simp add: strip_while_def)

lemma strip_while_append [simp]:
  "¬ P x ==> strip_while P (xs @ [x]) = xs @ [x]"
  by (simp add: strip_while_def)

lemma strip_while_append_rec [simp]:
  "P x ==> strip_while P (xs @ [x]) = strip_while P xs"
  by (simp add: strip_while_def)

lemma strip_while_Cons [simp]:
  "¬ P x ==> strip_while P (x # xs) = x # strip_while P xs"
  by (induct xs rule: rev_induct) (simp_all add: strip_while_def)

lemma strip_while_eq_Nil [simp]:
  "strip_while P xs = [] <-> (∀x∈set xs. P x)"
  by (simp add: strip_while_def)

lemma strip_while_eq_Cons_rec:
  "strip_while P (x # xs) = x # strip_while P xs <-> ¬ (P x ∧ (∀x∈set xs. P x))"
  by (induct xs rule: rev_induct) (simp_all add: strip_while_def)

lemma strip_while_not_last [simp]:
  "¬ P (last xs) ==> strip_while P xs = xs"
  by (cases xs rule: rev_cases) simp_all

lemma split_strip_while_append:
  fixes xs :: "'a list"
  obtains ys zs :: "'a list"
  where "strip_while P xs = ys" and "∀x∈set zs. P x" and "xs = ys @ zs"
proof (rule that)
  show "strip_while P xs = strip_while P xs" ..
  show "∀x∈set (rev (takeWhile P (rev xs))). P x" by (simp add: takeWhile_eq_all_conv [symmetric])
  have "rev xs = rev (strip_while P xs @ rev (takeWhile P (rev xs)))"
    by (simp add: strip_while_def)
  then show "xs = strip_while P xs @ rev (takeWhile P (rev xs))"
    by (simp only: rev_is_rev_conv)
qed


definition nth_default :: "'a => 'a list => nat => 'a"
where
  "nth_default x xs n = (if n < length xs then xs ! n else x)"

lemma nth_default_Nil [simp]:
  "nth_default y [] n = y"
  by (simp add: nth_default_def)

lemma nth_default_Cons_0 [simp]:
  "nth_default y (x # xs) 0 = x"
  by (simp add: nth_default_def)

lemma nth_default_Cons_Suc [simp]:
  "nth_default y (x # xs) (Suc n) = nth_default y xs n"
  by (simp add: nth_default_def)

lemma nth_default_map_eq:
  "f y = x ==> nth_default x (map f xs) n = f (nth_default y xs n)"
  by (simp add: nth_default_def)

lemma nth_default_strip_while_eq [simp]:
  "nth_default x (strip_while (HOL.eq x) xs) n = nth_default x xs n"
proof -
  from split_strip_while_append obtain ys zs
    where "strip_while (HOL.eq x) xs = ys" and "∀z∈set zs. x = z" and "xs = ys @ zs" by blast
  then show ?thesis by (simp add: nth_default_def not_less nth_append)
qed


definition cCons :: "'a::zero => '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 (λx. x = 0) (x # xs) = x ## strip_while (λx. x = 0) xs"
proof (cases "x = 0")
  case False then show ?thesis by simp
next
  case True show ?thesis
  proof (induct xs rule: rev_induct)
    case Nil with True show ?case by simp
  next
    case (snoc y ys) then show ?case
      by (cases "y = 0") (simp_all add: append_Cons [symmetric] del: append_Cons)
  qed
qed

lemma tl_cCons [simp]:
  "tl (x ## xs) = xs"
  by (simp add: cCons_def)


subsection {* Almost everywhere zero functions *}

definition almost_everywhere_zero :: "(nat => 'a::zero) => bool"
where
  "almost_everywhere_zero f <-> (∃n. ∀i>n. f i = 0)"

lemma almost_everywhere_zeroI:
  "(!!i. i > n ==> f i = 0) ==> almost_everywhere_zero f"
  by (auto simp add: almost_everywhere_zero_def)

lemma almost_everywhere_zeroE:
  assumes "almost_everywhere_zero f"
  obtains n where "!!i. i > n ==> f i = 0"
proof -
  from assms have "∃n. ∀i>n. f i = 0" by (simp add: almost_everywhere_zero_def)
  then obtain n where "!!i. i > n ==> f i = 0" by blast
  with that show thesis .
qed

lemma almost_everywhere_zero_case_nat:
  assumes "almost_everywhere_zero f"
  shows "almost_everywhere_zero (case_nat a f)"
  using assms
  by (auto intro!: almost_everywhere_zeroI elim!: almost_everywhere_zeroE split: nat.split)
    blast

lemma almost_everywhere_zero_Suc:
  assumes "almost_everywhere_zero f"
  shows "almost_everywhere_zero (λn. f (Suc n))"
proof -
  from assms obtain n where "!!i. i > n ==> f i = 0" by (erule almost_everywhere_zeroE)
  then have "!!i. i > n ==> f (Suc i) = 0" by auto
  then show ?thesis by (rule almost_everywhere_zeroI)
qed


subsection {* Definition of type @{text poly} *}

typedef 'a poly = "{f :: nat => 'a::zero. almost_everywhere_zero f}"
  morphisms coeff Abs_poly
  unfolding almost_everywhere_zero_def by auto

setup_lifting (no_code) 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 coeff_almost_everywhere_zero:
  "almost_everywhere_zero (coeff p)"
  using coeff [of p] by simp


subsection {* Degree of a polynomial *}

definition degree :: "'a::zero poly => 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 -
  from coeff_almost_everywhere_zero
  have "∃n. ∀i>n. coeff p i = 0" by (blast intro: almost_everywhere_zeroE)
  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 {* The zero polynomial *}

instantiation poly :: (zero) zero
begin

lift_definition zero_poly :: "'a poly"
  is "λ_. 0" by (rule almost_everywhere_zeroI) 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 `p ≠ 0` have "∃n. coeff p n ≠ 0"
    by (simp add: poly_eq_iff)
  then obtain n where "coeff p n ≠ 0" ..
  hence "n ≤ degree p" by (rule le_degree)
  with `coeff p n ≠ 0` and `degree p = 0`
  show "coeff p (degree p) ≠ 0" by simp
next
  case (Suc n)
  from `degree p = Suc n` have "n < degree p" by simp
  hence "∃i>n. coeff p i ≠ 0" by (rule less_degree_imp)
  then obtain i where "n < i" and "coeff p i ≠ 0" by fast
  from `degree p = Suc n` and `n < i` have "degree p ≤ i" by simp
  also from `coeff p i ≠ 0` have "i ≤ degree p" by (rule le_degree)
  finally have "degree p = i" .
  with `coeff p i ≠ 0` show "coeff p (degree p) ≠ 0" by simp
qed

lemma leading_coeff_0_iff [simp]:
  "coeff p (degree p) = 0 <-> p = 0"
  by (cases "p = 0", simp, simp add: leading_coeff_neq_0)


subsection {* List-style constructor for polynomials *}

lift_definition pCons :: "'a::zero => 'a poly => 'a poly"
  is "λa p. case_nat a (coeff p)"
  using coeff_almost_everywhere_zero by (rule almost_everywhere_zero_case_nat)

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)"
  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 <-> 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 "∀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 <-> 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 (λn. coeff p (Suc n)))"
    by transfer
      (simp add: Abs_poly_inverse almost_everywhere_zero_Suc fun_eq_iff split: nat.split)
qed

lemma pCons_induct [case_names 0 pCons, induct type: poly]:
  assumes zero: "P 0"
  assumes pCons: "!!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)
    then have "degree q < degree p"
      using `p = pCons a q` 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 `P q` show ?thesis by (auto intro: pCons)
  next
    case False
    with zero show ?thesis by simp
  qed
  then show ?case
    using `p = pCons a q` by simp
qed


subsection {* List-style syntax for polynomials *}

syntax
  "_poly" :: "args => 'a poly"  ("[:(_):]")

translations
  "[:x, xs:]" == "CONST pCons x [:xs:]"
  "[:x:]" == "CONST pCons x 0"
  "[:x:]" <= "CONST pCons x (_constrain 0 t)"


subsection {* Representation of polynomials by lists of coefficients *}

primrec Poly :: "'a::zero list => '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)

definition coeffs :: "'a poly => 'a::zero list"
where
  "coeffs p = (if p = 0 then [] else map (λi. 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 -
  { fix ms :: "nat list" and f :: "nat => 'a" and x :: "'a"
    assume "∀m∈set ms. m > 0"
    then have "map (case_nat x f) ms = map f (map (λn. n - 1) ms)"
      by (induct ms) (auto, metis Suc_pred' nat.case(2)) }
  note * = this
  show ?thesis
    by (simp add: coeffs_def * upt_conv_Cons coeff_pCons map_decr_upt One_nat_def del: upt_Suc)
qed

lemma not_0_cCons_eq [simp]:
  "p ≠ 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)
  have "(∀n. as ≠ replicate n 0) <-> (∃a∈set as. a ≠ 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 ≠ 0 ==> last (coeffs p) ≠ 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 <-> 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 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: "!!n. coeff p n = nth_default 0 xs n"
  assumes zero: "xs ≠ [] ==> last xs ≠ 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 ≠ 0 ==> 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 <-> HOL.equal (coeffs p) (coeffs q)"

instance proof
qed (simp add: equal equal_poly_def coeffs_eq_iff)

end

lemma [code nbe]:
  "HOL.equal (p :: _ poly) p <-> 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)


subsection {* Fold combinator for polynomials *}

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 o 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 o 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 o fold_coeffs f p"
  by (simp add: fold_coeffs_def)


subsection {* Canonical morphism on polynomials -- evaluation *}

definition poly :: "'a::comm_semiring_0 poly => 'a => 'a"
where
  "poly p = fold_coeffs (λa f x. a + x * f x) p (λx. 0)" -- {* The Horner Schema *}

lemma poly_0 [simp]:
  "poly 0 x = 0"
  by (simp add: poly_def)

lemma poly_pCons [simp]:
  "poly (pCons a p) x = a + x * poly p x"
  by (cases "p = 0 ∧ a = 0") (auto simp add: poly_def)


subsection {* Monomials *}

lift_definition monom :: "'a => nat => 'a::zero poly"
  is "λa m n. if m = n then a else 0"
  by (auto intro!: almost_everywhere_zeroI)

lemma coeff_monom [simp]:
  "coeff (monom a m) n = (if m = n then a else 0)"
  by transfer rule

lemma monom_0:
  "monom a 0 = pCons a 0"
  by (rule poly_eqI) (simp add: coeff_pCons split: nat.split)

lemma monom_Suc:
  "monom a (Suc n) = pCons 0 (monom a n)"
  by (rule poly_eqI) (simp add: coeff_pCons split: nat.split)

lemma monom_eq_0 [simp]: "monom 0 n = 0"
  by (rule poly_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"
  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 ≠ 0 ==> fold_coeffs f (monom a n) = f 0 ^^ n o f a"
  by (simp add: fold_coeffs_def coeffs_monom fun_eq_iff)

lemma poly_monom:
  fixes a x :: "'a::{comm_semiring_1}"
  shows "poly (monom a n) x = a * x ^ n"
  by (cases "a = 0", simp_all)
    (induct n, simp_all add: mult.left_commute poly_def)


subsection {* Addition and subtraction *}

instantiation poly :: (comm_monoid_add) comm_monoid_add
begin

lift_definition plus_poly :: "'a poly => 'a poly => 'a poly"
  is "λp q n. coeff p n + coeff q n"
proof (rule almost_everywhere_zeroI) 
  fix q p :: "'a poly" and i
  assume "max (degree q) (degree p) < i"
  then show "coeff p i + coeff q i = 0"
    by (simp add: coeff_eq_0)
qed

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

instance poly :: (cancel_comm_monoid_add) cancel_comm_monoid_add
proof
  fix p q r :: "'a poly"
  assume "p + q = p + r" thus "q = r"
    by (simp add: poly_eq_iff)
qed

instantiation poly :: (ab_group_add) ab_group_add
begin

lift_definition uminus_poly :: "'a poly => 'a poly"
  is "λp n. - coeff p n"
proof (rule almost_everywhere_zeroI)
  fix p :: "'a poly" and i
  assume "degree p < i"
  then show "- coeff p i = 0"
    by (simp add: coeff_eq_0)
qed

lift_definition minus_poly :: "'a poly => 'a poly => 'a poly"
  is "λp q n. coeff p n - coeff q n"
proof (rule almost_everywhere_zeroI) 
  fix q p :: "'a poly" and i
  assume "max (degree q) (degree p) < i"
  then show "coeff p i - coeff q i = 0"
    by (simp add: coeff_eq_0)
qed

lemma coeff_minus [simp]: "coeff (- p) n = - coeff p n"
  by (simp add: uminus_poly.rep_eq)

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 :: "'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:
  "degree p < degree q ==> 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 ==> 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: "degree (p - q) ≤ max (degree p) (degree q)"
  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"
  using degree_add_le [of p n "- q"] by simp

lemma degree_diff_less:
  "[|degree p < n; degree q < n|] ==> degree (p - q) < n"
  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_setsum: "coeff (∑x∈A. p x) i = (∑x∈A. coeff (p x) i)"
  by (cases "finite A", induct set: finite, simp_all)

lemma monom_setsum: "monom (∑x∈A. a x) n = (∑x∈A. monom (a x) n)"
  by (rule poly_eqI) (simp add: coeff_setsum)

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 -
  { 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 ≠ [] ==> last xs ≠ 0" and "ys ≠ [] ==> last ys ≠ 0"
    moreover assume "plus_coeffs xs ys ≠ []"
    ultimately have "last (plus_coeffs xs ys) ≠ 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 (λa. - 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 ab_add_uminus_conv_diff)

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 (∑k∈A. p k) x = (∑k∈A. poly (p k) x)"
  by (induct A rule: infinite_finite_induct) simp_all


subsection {* Multiplication by a constant, polynomial multiplication and the unit polynomial *}

lift_definition smult :: "'a::comm_semiring_0 => 'a poly => 'a poly"
  is "λa p n. a * coeff p n"
proof (rule almost_everywhere_zeroI)
  fix a :: 'a and p :: "'a poly" and i
  assume "degree p < i"
  then show "a * coeff p i = 0"
    by (simp add: coeff_eq_0)
qed

lemma coeff_smult [simp]:
  "coeff (smult a p) n = a * coeff p n"
  by (simp add: smult.rep_eq)

lemma degree_smult_le: "degree (smult a p) ≤ 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 <-> a = 0 ∨ 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 (λa 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 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 = (∑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) thus ?case
    by (cases n, simp, simp add: setsum_atMost_Suc_shift
                            del: setsum_atMost_Suc)
qed

lemma degree_mult_le: "degree (p * q) ≤ 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
  fix p :: "'a poly" show "1 * p = p"
    unfolding one_poly_def by simp
next
  show "0 ≠ (1::'a poly)"
    unfolding one_poly_def by simp
qed

end

instance poly :: (comm_semiring_1_cancel) comm_semiring_1_cancel ..

instance poly :: (comm_ring) comm_ring ..

instance poly :: (comm_ring_1) comm_ring_1 ..

lemma coeff_1 [simp]: "coeff 1 n = (if n = 0 then 1 else 0)"
  unfolding one_poly_def
  by (simp add: coeff_pCons split: nat.split)

lemma degree_1 [simp]: "degree 1 = 0"
  unfolding one_poly_def
  by (rule degree_pCons_0)

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

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

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

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

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

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


subsection {* Lemmas about divisibility *}

lemma dvd_smult: "p dvd q ==> 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 ==> a ≠ 0 ==> p dvd q"
  by (drule dvd_smult [where a="inverse a"]) simp

lemma dvd_smult_iff:
  fixes a :: "'a::field"
  shows "a ≠ 0 ==> p dvd smult a q <-> p dvd q"
  by (safe elim!: dvd_smult dvd_smult_cancel)

lemma smult_dvd_cancel:
  "smult a p dvd q ==> 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 ==> a ≠ 0 ==> 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 <-> (if a = 0 then q = 0 else p dvd q)"
  by (auto elim: smult_dvd smult_dvd_cancel)


subsection {* Polynomials form an integral domain *}

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 ≠ 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 have "coeff p (degree p) * coeff q (degree q) ≠ 0"
    using `p ≠ 0` and `q ≠ 0` by simp
  finally have "∃n. coeff (p * q) n ≠ 0" ..
  thus "p * q ≠ 0" by (simp add: poly_eq_iff)
qed

lemma degree_mult_eq:
  fixes p q :: "'a::idom poly"
  shows "[|p ≠ 0; q ≠ 0|] ==> 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 dvd_imp_degree_le:
  fixes p q :: "'a::idom poly"
  shows "[|p dvd q; q ≠ 0|] ==> degree p ≤ degree q"
  by (erule dvdE, simp add: degree_mult_eq)


subsection {* Polynomials form an ordered integral domain *}

definition pos_poly :: "'a::linordered_idom poly => 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)"
  unfolding pos_poly_def by simp

lemma not_pos_poly_0 [simp]: "¬ pos_poly 0"
  unfolding pos_poly_def by simp

lemma pos_poly_add: "[|pos_poly p; pos_poly q|] ==> 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: "[|pos_poly p; pos_poly q|] ==> pos_poly (p * q)"
  unfolding pos_poly_def
  apply (subgoal_tac "p ≠ 0 ∧ q ≠ 0")
  apply (simp add: degree_mult_eq coeff_mult_degree_sum)
  apply auto
  done

lemma pos_poly_total: "p = 0 ∨ pos_poly p ∨ pos_poly (- p)"
by (induct p) (auto simp add: pos_poly_pCons)

lemma last_coeffs_eq_coeff_degree:
  "p ≠ 0 ==> last (coeffs p) = coeff p (degree p)"
  by (simp add: coeffs_def)

lemma pos_poly_coeffs [code]:
  "pos_poly p <-> (let as = coeffs p in as ≠ [] ∧ last as > 0)" (is "?P <-> ?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 ≠ 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 <-> pos_poly (y - x)"

definition
  "x ≤ y <-> x = y ∨ pos_poly (y - x)"

definition
  "abs (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 :: "'a poly"
  show "x < y <-> x ≤ y ∧ ¬ y ≤ x"
    unfolding less_eq_poly_def less_poly_def
    apply safe
    apply simp
    apply (drule (1) pos_poly_add)
    apply simp
    done
next
  fix x :: "'a poly" show "x ≤ x"
    unfolding less_eq_poly_def by simp
next
  fix x y z :: "'a poly"
  assume "x ≤ y" and "y ≤ z" thus "x ≤ z"
    unfolding less_eq_poly_def
    apply safe
    apply (drule (1) pos_poly_add)
    apply (simp add: algebra_simps)
    done
next
  fix x y :: "'a poly"
  assume "x ≤ y" and "y ≤ x" thus "x = y"
    unfolding less_eq_poly_def
    apply safe
    apply (drule (1) pos_poly_add)
    apply simp
    done
next
  fix x y z :: "'a poly"
  assume "x ≤ y" thus "z + x ≤ z + y"
    unfolding less_eq_poly_def
    apply safe
    apply (simp add: algebra_simps)
    done
next
  fix x y :: "'a poly"
  show "x ≤ y ∨ y ≤ x"
    unfolding less_eq_poly_def
    using pos_poly_total [of "x - y"]
    by auto
next
  fix x y z :: "'a poly"
  assume "x < y" and "0 < z"
  thus "z * x < z * y"
    unfolding less_poly_def
    by (simp add: right_diff_distrib [symmetric] pos_poly_mult)
next
  fix x :: "'a poly"
  show "¦x¦ = (if x < 0 then - x else x)"
    by (rule abs_poly_def)
next
  fix x :: "'a poly"
  show "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
    by (rule sgn_poly_def)
qed

end

text {* TODO: Simplification rules for comparisons *}


subsection {* Synthetic division and polynomial roots *}

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

definition synthetic_divmod :: "'a::comm_semiring_0 poly => 'a => 'a poly × 'a"
where
  "synthetic_divmod p c = fold_coeffs (λa (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 = (λ(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"
  unfolding synthetic_div_def by simp

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, 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 <-> 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 ==> r = poly p c ∧ 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 <-> [:-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 <-> poly p (-c) = 0"
  by (simp add: poly_eq_0_iff_dvd)

lemma poly_roots_finite:
  fixes p :: "'a::idom poly"
  shows "p ≠ 0 ==> finite {x. poly p x = 0}"
proof (induct n  "degree p" arbitrary: p)
  case (0 p)
  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 p)
  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 `p ≠ 0` have "k ≠ 0" by auto
    with k have "degree p = Suc (degree k)"
      by (simp add: degree_mult_eq del: mult_pCons_left)
    with `Suc n = degree p` have "n = degree k" by simp
    then have "finite {x. poly k x = 0}" using `k ≠ 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:
  fixes p q :: "'a::{idom,ring_char_0} poly"
  shows "poly p = poly q <-> p = q" (is "?P <-> ?Q")
proof
  assume ?Q then show ?P by simp
next
  { fix p :: "'a::{idom,ring_char_0} poly"
    have "poly p = poly 0 <-> 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 "(∀x. poly p x = 0) <-> p = 0"
  by (auto simp add: poly_eq_poly_eq_iff [symmetric])


subsection {* Long division of polynomials *}

definition pdivmod_rel :: "'a::field poly => 'a poly => 'a poly => 'a poly => bool"
where
  "pdivmod_rel x y q r <->
    x = q * y + r ∧ (if y = 0 then q = 0 else r = 0 ∨ 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 ≤ n" and "coeff p n = 0"
  shows "p = 0 ∨ degree p < n"
proof (cases n)
  case 0
  with `degree p ≤ n` and `coeff p n = 0`
  have "coeff p (degree p) = 0" by simp
  then have "p = 0" by simp
  then show ?thesis ..
next
  case (Suc m)
  have "∀i>n. coeff p i = 0"
    using `degree p ≤ n` by (simp add: coeff_eq_0)
  then have "∀i≥n. coeff p i = 0"
    using `coeff p n = 0` by (simp add: le_less)
  then have "∀i>m. coeff p i = 0"
    using `n = Suc m` by (simp add: less_eq_Suc_le)
  then have "degree p ≤ m"
    by (rule degree_le)
  then have "degree p < n"
    using `n = Suc m` 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 ≠ 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 ∨ 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 ∨ degree ?r < degree y"
  proof (rule eq_zero_or_degree_less)
    show "degree ?r ≤ degree y"
    proof (rule degree_diff_le)
      show "degree (pCons a r) ≤ degree y"
        using r by auto
      show "degree (smult b y) ≤ degree y"
        by (rule degree_smult_le)
    qed
  next
    show "coeff ?r (degree y) = 0"
      using `y ≠ 0` unfolding b by simp
  qed

  from 1 2 show ?thesis
    unfolding pdivmod_rel_def
    using `y ≠ 0` by simp
qed

lemma pdivmod_rel_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 ∧ r1 = r2"
proof (cases "y = 0")
  assume "y = 0" with assms show ?thesis
    by (simp add: pdivmod_rel_def)
next
  assume [simp]: "y ≠ 0"
  from 1 have q1: "x = q1 * y + r1" and r1: "r1 = 0 ∨ degree r1 < degree y"
    unfolding pdivmod_rel_def by simp_all
  from 2 have q2: "x = q2 * y + r2" and r2: "r2 = 0 ∨ 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 ∨ degree (r2 - r1) < degree y"
    by (auto intro: degree_diff_less)

  show "q1 = q2 ∧ r1 = r2"
  proof (rule ccontr)
    assume "¬ (q1 = q2 ∧ r1 = r2)"
    with q3 have "q1 ≠ q2" and "r1 ≠ r2" by auto
    with r3 have "degree (r2 - r1) < degree y" by simp
    also have "degree y ≤ degree (q1 - q2) + degree y" by simp
    also have "… = degree ((q1 - q2) * y)"
      using `q1 ≠ q2` by (simp add: degree_mult_eq)
    also have "… = 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 <-> q = 0 ∧ r = 0"
by (auto dest: pdivmod_rel_unique intro: pdivmod_rel_0)

lemma pdivmod_rel_by_0_iff: "pdivmod_rel x 0 q r <-> q = 0 ∧ 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 div_poly where
  "x div y = (THE q. ∃r. pdivmod_rel x y q r)"

definition mod_poly where
  "x mod y = (THE r. ∃q. pdivmod_rel x y q r)"

lemma div_poly_eq:
  "pdivmod_rel x y q r ==> 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 ==> 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 ≠ 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 ≠ 0"
  show "(x * y) div (x * z) = y div z"
  proof (cases "y ≠ 0 ∧ z ≠ 0")
    have "!!x::'a poly. pdivmod_rel x 0 0 x"
      by (rule pdivmod_rel_by_0)
    then have [simp]: "!!x::'a poly. x div 0 = 0"
      by (rule div_poly_eq)
    have "!!x::'a poly. pdivmod_rel 0 x 0 0"
      by (rule pdivmod_rel_0)
    then have [simp]: "!!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 ≠ 0" and "z ≠ 0" by auto
    with `x ≠ 0`
    have "!!q r. pdivmod_rel y z q r ==> 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 degree_mod_less:
  "y ≠ 0 ==> x mod y = 0 ∨ 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 ==> 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 ==> 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
    ==> 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: distrib 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:
  "[|a ≠ 0; pdivmod_rel x y q r|]
    ==> pdivmod_rel x (smult a y) (smult (inverse a) q) r"
  unfolding pdivmod_rel_def by simp

lemma div_smult_right:
  "a ≠ 0 ==> 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 ≠ 0 ==> 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:
  "[|pdivmod_rel x y q r; pdivmod_rel q z q' r'|]
    ==> 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 ≠ 0"
  defines b: "b ≡ 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 => 'a poly => 'a poly × '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 (λa (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 ∧ p = 0")
  apply (auto simp add: pdivmod_def)
  done


subsection {* Order of polynomial roots *}

definition order :: "'a::idom => 'a poly => nat"
where
  "order a p = (LEAST n. ¬ [:-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. ¬ [:-a, 1:] ^ Suc n dvd p)")
apply (drule not_less_Least, simp)
apply (fold order_def, simp)
done

lemma order_2: "p ≠ 0 ==> ¬ [:-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 ≠ 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 have "… ≤ degree p"
    using order_1 p by (rule dvd_imp_degree_le)
  finally show ?thesis .
qed

lemma order_root: "poly p a = 0 <-> p = 0 ∨ order a p ≠ 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


subsection {* GCD of polynomials *}

instantiation poly :: (field) gcd
begin

function gcd_poly :: "'a::field poly => 'a poly => 'a poly"
where
  "gcd (x::'a poly) 0 = smult (inverse (coeff x (degree x))) x"
| "y ≠ 0 ==> gcd (x::'a poly) y = gcd y (x mod y)"
by auto

termination "gcd :: _ poly => _"
by (relation "measure (λ(x, y). if y = 0 then 0 else Suc (degree y))")
   (auto dest: degree_mod_less)

declare gcd_poly.simps [simp del]

instance ..

end

lemma
  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 ==> k dvd y ==> 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 <-> k dvd x ∧ k dvd y"
  by (blast intro!: poly_gcd_greatest intro: dvd_trans)

lemma poly_gcd_monic:
  fixes x y :: "_ poly"
  shows "coeff (gcd x y) (degree (gcd x y)) =
    (if x = 0 ∧ 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 <-> x = 0 ∧ 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 ≠ 0" by auto
  have degree: "degree p = degree q"
    using `p dvd q` `q dvd p` `p ≠ 0` `q ≠ 0`
    by (intro order_antisym dvd_imp_degree_le)

  from `p dvd q` obtain a where a: "q = p * a" ..
  with `q ≠ 0` have "a ≠ 0" by auto
  with degree a `p ≠ 0` have "degree a = 0"
    by (simp add: degree_mult_eq)
  with coeff a show "p = q"
    by (cases a, auto split: if_splits)
qed

lemma poly_gcd_unique:
  fixes d x y :: "_ poly"
  assumes dvd1: "d dvd x" and dvd2: "d dvd y"
    and greatest: "!!k. k dvd x ==> k dvd y ==> k dvd d"
    and monic: "coeff d (degree d) = (if x = 0 ∧ 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 => _"
proof
  fix x y z :: "'a poly"
  show "gcd (gcd x y) z = gcd x (gcd y z)"
    by (rule poly_gcd_unique) (auto intro: dvd_trans simp add: poly_gcd_monic)
  show "gcd x y = gcd y x"
    by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)
qed

lemmas poly_gcd_assoc = gcd_poly.assoc
lemmas poly_gcd_commute = gcd_poly.commute
lemmas poly_gcd_left_commute = gcd_poly.left_commute

lemmas poly_gcd_ac = poly_gcd_assoc poly_gcd_commute poly_gcd_left_commute

lemma poly_gcd_1_left [simp]: "gcd 1 y = (1 :: _ poly)"
by (rule poly_gcd_unique) simp_all

lemma poly_gcd_1_right [simp]: "gcd x 1 = (1 :: _ poly)"
by (rule poly_gcd_unique) simp_all

lemma poly_gcd_minus_left [simp]: "gcd (- x) y = gcd x (y :: _ poly)"
by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)

lemma poly_gcd_minus_right [simp]: "gcd x (- y) = gcd x (y :: _ poly)"
by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)

lemma poly_gcd_code [code]:
  "gcd x y = (if y = 0 then smult (inverse (coeff x (degree x))) x else gcd y (x mod (y :: _ poly)))"
  by (simp add: gcd_poly.simps)


subsection {* Composition of polynomials *}

definition pcompose :: "'a::comm_semiring_0 poly => 'a poly => 'a poly"
where
  "pcompose p q = fold_coeffs (λa 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 ∧ 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) ≤ 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


no_notation cCons (infixr "##" 65)

end