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

lemma almost_everywhere_zero_Suc:
assumes "almost_everywhere_zero f"
shows "almost_everywhere_zero (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. nat_case a (coeff p)"
using coeff_almost_everywhere_zero by (rule almost_everywhere_zero_nat_case)

lemmas coeff_pCons = pCons.rep_eq

lemma coeff_pCons_0 [simp]:
"coeff (pCons a p) 0 = a"
by transfer simp

lemma coeff_pCons_Suc [simp]:
"coeff (pCons a p) (Suc n) = coeff p n"
by (simp add: coeff_pCons)

lemma 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. 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
then have "P (pCons a q)"
by (rule pCons)
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
"Poly [] = 0"
| "Poly (a # as) = pCons a (Poly as)"

lemma Poly_replicate_0 [simp]:
"Poly (replicate n 0) = 0"
by (induct n) simp_all

lemma Poly_eq_0:
"Poly as = 0 <-> (∃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 (nat_case x f) ms = map f (map (λn. n - 1) ms)"
by (induct ms) (auto, metis Suc_pred' nat_case_Suc) }
note * = this
show ?thesis
by (simp add: coeffs_def * upt_conv_Cons coeff_pCons map_decr_upt One_nat_def del: upt_Suc)
qed

lemma not_0_cCons_eq [simp]:
"p ≠ 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) (simp_all add: cCons_def)

lemma coeffs_Poly [simp]:
"coeffs (Poly as) = strip_while (HOL.eq 0) as"
proof (induct as)
case Nil then show ?case by simp
next
case (Cons a as)
have "(∀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_0 nat_case_Suc not0_implies_Suc nth_default_Cons_0 nth_default_Cons_Suc pCons.rep_eq)

lemma nth_default_coeffs_eq:
"nth_default 0 (coeffs p) = coeff p"
by (simp add: fun_eq_iff coeff_Poly_eq [symmetric])

lemma [code]:
"coeff p = nth_default 0 (coeffs p)"
by (simp add: nth_default_coeffs_eq)

lemma coeffs_eqI:
assumes coeff: "!!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 diff_minus)
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 add: diff_minus)

lemma degree_diff_le:
"[|degree p ≤ n; degree q ≤ n|] ==> degree (p - q) ≤ n"
by (simp add: diff_minus degree_add_le)

lemma degree_diff_less:
"[|degree p < n; degree q < n|] ==> degree (p - q) < n"
by (simp add: diff_minus degree_add_less)

lemma add_monom: "monom a n + monom b n = monom (a + b) n"
by (rule poly_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 simp

lemma poly_add [simp]: "poly (p + q) x = poly p x + poly q x"
apply (induct p arbitrary: q, simp)
apply (case_tac q, simp, simp add: algebra_simps)
done

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

lemma poly_diff [simp]:
fixes x :: "'a::comm_ring"
shows "poly (p - q) x = poly p x - poly q x"
by (simp add: diff_minus)

lemma poly_setsum: "poly (∑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 mult_pos_pos)
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 uminus_add_conv_diff Collect_disj_eq
del: mult_pCons_left)
qed
qed

lemma poly_eq_poly_eq_iff:
fixes p q :: "'a::{idom,ring_char_0} poly"
shows "poly p = poly q <-> 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 del: minus_one) (* FIXME *)

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 del: minus_one) (* FIXME *)

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 del: minus_one) (* FIXME *)

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 del: minus_one) (* FIXME *)

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 (rule ccontr, simp)
apply (frule order_2 [where a=a], simp)
apply (simp add: poly_eq_0_iff_dvd)
apply (simp add: poly_eq_0_iff_dvd)
apply (simp only: order_def)
apply (drule not_less_Least, simp)
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