# Theory Polynomial

theory Polynomial
imports Deriv More_List Infinite_Set
```(*  Title:      HOL/Library/Polynomial.thy
Author:     Brian Huffman
Author:     Clemens Ballarin
Author:     Amine Chaieb
Author:     Florian Haftmann
*)

section ‹Polynomials as type over a ring structure›

theory Polynomial
imports Main "~~/src/HOL/Deriv" "~~/src/HOL/Library/More_List"
"~~/src/HOL/Library/Infinite_Set"
begin

subsection ‹Auxiliary: operations for lists (later) representing coefficients›

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 ## [] = []"

lemma cCons_Cons_eq [simp]:
"x ## y # ys = x # y # ys"

lemma cCons_append_Cons_eq [simp]:
"x ## xs @ y # ys = x # xs @ y # ys"

lemma cCons_not_0_eq [simp]:
"x ≠ 0 ⟹ x ## xs = x # xs"

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"

subsection ‹Definition of type ‹poly››

typedef (overloaded) 'a poly = "{f :: nat ⇒ 'a::zero. ∀⇩∞ n. f n = 0}"
morphisms coeff Abs_poly
by (auto intro!: ALL_MOST)

setup_lifting type_definition_poly

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

lemma poly_eqI: "(⋀n. coeff p n = coeff q n) ⟹ p = q"

lemma MOST_coeff_eq_0: "∀⇩∞ n. coeff p n = 0"
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 -
have "∃n. ∀i>n. coeff p i = 0"
using MOST_coeff_eq_0 by (simp add: MOST_nat)
then have "∀i>degree p. coeff p i = 0"
unfolding degree_def by (rule LeastI_ex)
with assms show ?thesis by simp
qed

lemma le_degree: "coeff p n ≠ 0 ⟹ n ≤ degree p"
by (erule contrapos_np, rule coeff_eq_0, simp)

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

lemma less_degree_imp: "n < degree p ⟹ ∃i>n. coeff p i ≠ 0"
unfolding degree_def by (drule not_less_Least, simp)

subsection ‹The zero polynomial›

instantiation poly :: (zero) zero
begin

lift_definition zero_poly :: "'a poly"
is "λ_. 0" by (rule MOST_I) simp

instance ..

end

lemma coeff_0 [simp]:
"coeff 0 n = 0"
by transfer rule

lemma degree_0 [simp]:
"degree 0 = 0"
by (rule order_antisym [OF degree_le le0]) simp

assumes "p ≠ 0"
shows "coeff p (degree p) ≠ 0"
proof (cases "degree p")
case 0
from ‹p ≠ 0› have "∃n. coeff p n ≠ 0"
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

"coeff p (degree p) = 0 ⟷ p = 0"

subsection ‹List-style constructor for polynomials›

lift_definition pCons :: "'a::zero ⇒ 'a poly ⇒ 'a poly"
is "λa p. case_nat a (coeff p)"
by (rule MOST_SucD) (simp add: MOST_coeff_eq_0)

lemmas coeff_pCons = pCons.rep_eq

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

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

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_all add: MOST_inj[where f=Suc and P="λn. p n = 0" for p] fun_eq_iff Abs_poly_inverse
split: nat.split)
qed

lemma pCons_induct [case_names 0 pCons, induct type: poly]:
assumes zero: "P 0"
assumes pCons: "⋀a p. a ≠ 0 ∨ p ≠ 0 ⟹ P p ⟹ P (pCons a p)"
shows "P p"
proof (induct p rule: measure_induct_rule [where f=degree])
case (less p)
obtain a q where "p = pCons a q" by (rule pCons_cases)
have "P q"
proof (cases "q = 0")
case True
then show "P q" by (simp add: zero)
next
case False
then have "degree (pCons a q) = Suc (degree q)"
by (rule degree_pCons_eq)
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

lemma degree_eq_zeroE:
fixes p :: "'a::zero poly"
assumes "degree p = 0"
obtains a where "p = pCons a 0"
proof -
obtain a q where p: "p = pCons a q" by (cases p)
with assms have "q = 0" by (cases "q = 0") simp_all
with p have "p = pCons a 0" by simp
with that show thesis .
qed

subsection ‹Quickcheck generator for polynomials›

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

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)

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

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

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

lemma Poly_on_rev_starting_with_0 [simp]:
assumes "hd as = 0"
shows "Poly (rev (tl as)) = Poly (rev as)"
using assms by (cases as) simp_all

lemma degree_Poly: "degree (Poly xs) ≤ length xs"
by (induction xs) simp_all

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

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

lemma coeffs_eq_Nil [simp]:
"coeffs p = [] ⟷ p = 0"

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 split: nat.split)
}
note * = this
show ?thesis
by (simp add: coeffs_def * upt_conv_Cons coeff_pCons map_decr_upt del: upt_Suc)
qed

lemma length_coeffs: "p ≠ 0 ⟹ length (coeffs p) = degree p + 1"

lemma coeffs_nth:
assumes "p ≠ 0" "n ≤ degree p"
shows   "coeffs p ! n = coeff p n"
using assms unfolding coeffs_def by (auto simp del: upt_Suc)

lemma not_0_cCons_eq [simp]:
"p ≠ 0 ⟹ a ## coeffs p = a # coeffs p"

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 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)"

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)
with zero show ?thesis by simp (cases xs, simp_all)
qed

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

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
by standard (simp add: equal equal_poly_def coeffs_eq_iff)

end

lemma [code nbe]: "HOL.equal (p :: _ poly) p ⟷ True"
by (fact equal_refl)

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

lemma is_zero_null [code_abbrev]:
"is_zero p ⟷ p = 0"

subsubsection ‹Reconstructing the polynomial from the list›
― ‹contributed by Sebastiaan J.C. Joosten and RenÃ© Thiemann›

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

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

subsection ‹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"

lemma fold_coeffs_pCons_eq [simp]:
"f 0 = id ⟹ fold_coeffs f (pCons a p) = f a ∘ fold_coeffs f p"
by (simp add: fold_coeffs_def cCons_def fun_eq_iff)

lemma fold_coeffs_pCons_0_0_eq [simp]:
"fold_coeffs f (pCons 0 0) = id"

lemma fold_coeffs_pCons_coeff_not_0_eq [simp]:
"a ≠ 0 ⟹ fold_coeffs f (pCons a p) = f a ∘ fold_coeffs f p"

lemma fold_coeffs_pCons_not_0_0_eq [simp]:
"p ≠ 0 ⟹ fold_coeffs f (pCons a p) = f a ∘ fold_coeffs f p"

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"

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

lemma poly_altdef:
"poly p (x :: 'a :: {comm_semiring_0, semiring_1}) = (∑i≤degree p. coeff p i * x ^ i)"
proof (induction p rule: pCons_induct)
case (pCons a p)
show ?case
proof (cases "p = 0")
case False
let ?p' = "pCons a p"
note poly_pCons[of a p x]
also note pCons.IH
also have "a + x * (∑i≤degree p. coeff p i * x ^ i) =
coeff ?p' 0 * x^0 + (∑i≤degree p. coeff ?p' (Suc i) * x^Suc i)"
by (simp add: field_simps sum_distrib_left coeff_pCons)
also note sum_atMost_Suc_shift[symmetric]
also note degree_pCons_eq[OF ‹p ≠ 0›, of a, symmetric]
finally show ?thesis .
qed simp
qed simp

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

subsection ‹Monomials›

lift_definition monom :: "'a ⇒ nat ⇒ 'a::zero poly"
is "λa m n. if m = n then a else 0"

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"

lemma monom_eq_iff [simp]: "monom a n = monom b n ⟷ a = b"

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 ∘ 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)

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

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

begin

lift_definition plus_poly :: "'a poly ⇒ 'a poly ⇒ 'a poly"
is "λp q n. coeff p n + coeff q n"
proof -
fix q p :: "'a poly"
show "∀⇩∞n. coeff p n + coeff q n = 0"
using MOST_coeff_eq_0[of p] MOST_coeff_eq_0[of q] by eventually_elim simp
qed

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

instance
proof
fix p q r :: "'a poly"
show "(p + q) + r = p + (q + r)"
show "p + q = q + p"
show "0 + p = p"
qed

end

begin

lift_definition minus_poly :: "'a poly ⇒ 'a poly ⇒ 'a poly"
is "λp q n. coeff p n - coeff q n"
proof -
fix q p :: "'a poly"
show "∀⇩∞n. coeff p n - coeff q n = 0"
using MOST_coeff_eq_0[of p] MOST_coeff_eq_0[of q] by eventually_elim simp
qed

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

instance
proof
fix p q r :: "'a poly"
show "p + q - p = q"
show "p - q - r = p - (q + r)"
qed

end

begin

lift_definition uminus_poly :: "'a poly ⇒ 'a poly"
is "λp n. - coeff p n"
proof -
fix p :: "'a poly"
show "∀⇩∞n. - coeff p n = 0"
using MOST_coeff_eq_0 by simp
qed

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

instance
proof
fix p q :: "'a poly"
show "- p + p = 0"
show "p - q = p + - q"
qed

end

"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)

"⟦degree p ≤ n; degree q ≤ n⟧ ⟹ degree (p + q) ≤ n"

"⟦degree p < n; degree q < n⟧ ⟹ degree (p + q) < n"

"degree p < degree q ⟹ degree (p + q) = degree q"
apply (cases "q = 0", simp)
apply (rule order_antisym)
apply (rule le_degree)
done

"degree q < degree p ⟹ degree (p + q) = degree p"

lemma degree_minus [simp]:
"degree (- p) = degree p"
unfolding degree_def by simp

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

lemma degree_diff_le:
fixes p q :: "'a :: ab_group_add poly"
assumes "degree p ≤ n" and "degree q ≤ n"
shows "degree (p - q) ≤ n"
using assms degree_add_le [of p n "- q"] by simp

lemma degree_diff_less:
fixes p q :: "'a :: ab_group_add poly"
assumes "degree p < n" and "degree q < n"
shows "degree (p - q) < n"
using assms degree_add_less [of p n "- q"] by simp

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

lemma diff_monom: "monom a n - monom b n = monom (a - b) n"
by (rule poly_eqI) simp

lemma minus_monom: "- monom a n = monom (-a) n"
by (rule poly_eqI) simp

lemma coeff_sum: "coeff (∑x∈A. p x) i = (∑x∈A. coeff (p x) i)"
by (cases "finite A", induct set: finite, simp_all)

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

fun plus_coeffs :: "'a::comm_monoid_add list ⇒ 'a list ⇒ 'a list"
where
"plus_coeffs xs [] = xs"
| "plus_coeffs [] ys = ys"
| "plus_coeffs (x # xs) (y # ys) = (x + y) ## plus_coeffs xs ys"

lemma coeffs_plus_eq_plus_coeffs [code abstract]:
"coeffs (p + q) = plus_coeffs (coeffs p) (coeffs q)"
proof -
{ 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 (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"

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

lemma degree_sum_le: "finite S ⟹ (⋀ p . p ∈ S ⟹ degree (f p) ≤ n)
⟹ degree (sum f S) ≤ n"
proof (induct S rule: finite_induct)
case (insert p S)
hence "degree (sum f S) ≤ n" "degree (f p) ≤ n" by auto
thus ?case unfolding sum.insert[OF insert(1-2)] by (metis degree_add_le)
qed simp

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

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

lemma Poly_snoc: "Poly (xs @ [x]) = Poly xs + monom x (length xs)"
by (induction xs) (simp_all add: monom_0 monom_Suc)

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 -
fix a :: 'a and p :: "'a poly" show "∀⇩∞ i. a * coeff p i = 0"
using MOST_coeff_eq_0[of p] by eventually_elim simp
qed

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

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)

"smult a (p + q) = smult a p + smult a q"
by (rule poly_eqI, simp add: algebra_simps)

"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_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"

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

lemma degree_smult_eq [simp]:
fixes a :: "'a::{comm_semiring_0,semiring_no_zero_divisors}"
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::{comm_semiring_0,semiring_no_zero_divisors}"
shows "smult a p = 0 ⟷ a = 0 ∨ p = 0"

lemma coeffs_smult [code abstract]:
fixes p :: "'a::{comm_semiring_0,semiring_no_zero_divisors} 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"

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)"

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)"

lemma mult_smult_right [simp]:
"p * smult a q = smult a (p * q)"

fixes p q r :: "'a poly"
shows "(p + q) * r = p * r + q * r"

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"
show "(p * q) * r = p * (q * r)"
show "p * q = q * p"
by (induct p, simp add: mult_poly_0, simp)
qed

end

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

instance poly :: ("{comm_semiring_0,semiring_no_zero_divisors}") semiring_no_zero_divisors
proof
fix p q :: "'a poly"
assume "p ≠ 0" and "q ≠ 0"
have "coeff (p * q) (degree p + degree q) =
coeff p (degree p) * coeff q (degree q)"
by (rule coeff_mult_degree_sum)
also 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

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: sum_atMost_Suc_shift
del: sum_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)"

instantiation poly :: (comm_semiring_1) comm_semiring_1
begin

definition one_poly_def: "1 = pCons 1 0"

instance
proof
show "1 * p = p" for p :: "'a poly"
unfolding one_poly_def by simp
show "0 ≠ (1::'a poly)"
unfolding one_poly_def by simp
qed

end

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

instance poly :: (comm_ring) comm_ring ..

instance poly :: (comm_ring_1) comm_ring_1 ..

instance poly :: (comm_ring_1) comm_semiring_1_cancel ..

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

lemma monom_eq_1 [simp]:
"monom 1 0 = 1"

lemma monom_eq_1_iff: "monom c n = 1 ⟷ c = 1 ∧ n = 0"
using monom_eq_const_iff[of c n 1] by (auto simp: one_poly_def)

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

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

lemma coeffs_1_eq [simp, code abstract]:
"coeffs 1 = [1]"

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"

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

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

lemma degree_prod_sum_le: "finite S ⟹ degree (prod f S) ≤ sum (degree o f) S"
proof (induct S rule: finite_induct)
case (insert a S)
show ?case unfolding prod.insert[OF insert(1-2)] sum.insert[OF insert(1-2)]
by (rule le_trans[OF degree_mult_le], insert insert, auto)
qed simp

subsection ‹Conversions from natural numbers›

lemma of_nat_poly: "of_nat n = [:of_nat n :: 'a :: comm_semiring_1:]"
proof (induction n)
case (Suc n)
hence "of_nat (Suc n) = 1 + (of_nat n :: 'a poly)"
by simp
also have "(of_nat n :: 'a poly) = [: of_nat n :]"
by (subst Suc) (rule refl)
also have "1 = [:1:]" by (simp add: one_poly_def)
finally show ?case by (subst (asm) add_pCons) simp
qed simp

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

lemma degree_numeral [simp]: "degree (numeral n) = 0"
by (subst of_nat_numeral [symmetric], subst of_nat_poly) simp

lemma numeral_poly: "numeral n = [:numeral n:]"
by (subst of_nat_numeral [symmetric], subst of_nat_poly) simp

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›

instance poly :: (idom) idom ..

lemma degree_mult_eq:
fixes p q :: "'a::{comm_semiring_0,semiring_no_zero_divisors} poly"
shows "⟦p ≠ 0; q ≠ 0⟧ ⟹ degree (p * q) = degree p + degree q"
apply (rule order_antisym [OF degree_mult_le le_degree])
done

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

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

lemma dvd_imp_degree_le:
fixes p q :: "'a::{comm_semiring_1,semiring_no_zero_divisors} poly"
shows "⟦p dvd q; q ≠ 0⟧ ⟹ degree p ≤ degree q"
by (erule dvdE, hypsubst, subst degree_mult_eq) auto

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

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

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

subsection ‹Polynomials form an ordered integral domain›

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

lemma pos_poly_pCons:
"pos_poly (pCons a p) ⟷ pos_poly p ∨ (p = 0 ∧ 0 < a)"
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)
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 auto
done

lemma pos_poly_total: "(p :: 'a :: linordered_idom poly) = 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)"

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

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

instance
proof
fix x y z :: "'a poly"
show "x < y ⟷ x ≤ y ∧ ¬ y ≤ x"
unfolding less_eq_poly_def less_poly_def
apply safe
apply simp
apply simp
done
show "x ≤ x"
unfolding less_eq_poly_def by simp
show "x ≤ y ⟹ y ≤ z ⟹ x ≤ z"
unfolding less_eq_poly_def
apply safe
done
show "x ≤ y ⟹ y ≤ x ⟹ x = y"
unfolding less_eq_poly_def
apply safe
apply simp
done
show "x ≤ y ⟹ z + x ≤ z + y"
unfolding less_eq_poly_def
apply safe
done
show "x ≤ y ∨ y ≤ x"
unfolding less_eq_poly_def
using pos_poly_total [of "x - y"]
by auto
show "x < y ⟹ 0 < z ⟹ z * x < z * y"
unfolding less_poly_def
by (simp add: right_diff_distrib [symmetric] pos_poly_mult)
show "¦x¦ = (if x < 0 then - x else x)"
by (rule abs_poly_def)
show "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
by (rule sgn_poly_def)
qed

end

text ‹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)"

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

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

lemma degree_synthetic_div:
"degree (synthetic_div p c) = degree p - 1"
by (induct p) (simp_all add: synthetic_div_eq_0_iff)

lemma synthetic_div_correct:
"p + smult c (synthetic_div p c) = pCons (poly p c) (synthetic_div p c)"
by (induct p) simp_all

lemma synthetic_div_unique:
"p + smult c q = pCons r q ⟹ r = poly p c ∧ q = synthetic_div p c"
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]

lemma poly_eq_0_iff_dvd:
fixes c :: "'a::{comm_ring_1}"
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::{comm_ring_1}"
shows "[:c, 1:] dvd p ⟷ poly p (-c) = 0"

lemma poly_roots_finite:
fixes p :: "'a::{comm_ring_1,ring_no_zero_divisors} 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::{comm_ring_1,ring_no_zero_divisors,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 poly"
have "poly p = poly 0 ⟷ p = 0"
apply (cases "p = 0", simp_all)
apply (drule poly_roots_finite)
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, comm_ring_1,ring_no_zero_divisors} 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
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"
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]

subsection‹Pseudo-Division and Division of Polynomials›

fun pseudo_divmod_main :: "'a :: comm_ring_1  ⇒ 'a poly ⇒ 'a poly ⇒ 'a poly
⇒ nat ⇒ nat ⇒ 'a poly × 'a poly" where
"pseudo_divmod_main lc q r d dr (Suc n) = (let
rr = smult lc r;
qq = coeff r dr;
rrr = rr - monom qq n * d;
qqq = smult lc q + monom qq n
in pseudo_divmod_main lc qqq rrr d (dr - 1) n)"
| "pseudo_divmod_main lc q r d dr 0 = (q,r)"

definition pseudo_divmod :: "'a :: comm_ring_1 poly ⇒ 'a poly ⇒ 'a poly × 'a poly" where
"pseudo_divmod p q ≡ if q = 0 then (0,p) else
pseudo_divmod_main (coeff q (degree q)) 0 p q (degree p) (1 + length (coeffs p) - length (coeffs q))"

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

lemma pseudo_divmod_main: assumes d: "d ≠ 0" "lc = coeff d (degree d)"
and *: "degree r ≤ dr" "pseudo_divmod_main lc q r d dr n = (q',r')"
"n = 1 + dr - degree d ∨ dr = 0 ∧ n = 0 ∧ r = 0"
shows "(r' = 0 ∨ degree r' < degree d) ∧ smult (lc^n) (d * q + r) = d * q' + r'"
using *
proof (induct n arbitrary: q r dr)
case (Suc n q r dr)
let ?rr = "smult lc r"
let ?qq = "coeff r dr"
define b where [simp]: "b = monom ?qq n"
let ?rrr = "?rr - b * d"
let ?qqq = "smult lc q + b"
note res = Suc(3)
from res[unfolded pseudo_divmod_main.simps[of lc q] Let_def]
have res: "pseudo_divmod_main lc ?qqq ?rrr d (dr - 1) n = (q',r')"
by (simp del: pseudo_divmod_main.simps)
have dr: "dr = n + degree d" using Suc(4) by auto
have "coeff (b * d) dr = coeff b n * coeff d (degree d)"
proof (cases "?qq = 0")
case False
hence n: "n = degree b" by (simp add: degree_monom_eq)
show ?thesis unfolding n dr by (simp add: coeff_mult_degree_sum)
qed auto
also have "… = lc * coeff b n" unfolding d by simp
finally have "coeff (b * d) dr = lc * coeff b n" .
moreover have "coeff ?rr dr = lc * coeff r dr" by simp
ultimately have c0: "coeff ?rrr dr = 0" by auto
have dr: "dr = n + degree d" using Suc(4) by auto
have deg_rr: "degree ?rr ≤ dr" using Suc(2)
using degree_smult_le dual_order.trans by blast
have deg_bd: "degree (b * d) ≤ dr"
unfolding dr
by(rule order.trans[OF degree_mult_le], auto simp: degree_monom_le)
have "degree ?rrr ≤ dr"
using degree_diff_le[OF deg_rr deg_bd] by auto
with c0 have deg_rrr: "degree ?rrr ≤ (dr - 1)" by (rule coeff_0_degree_minus_1)
have "n = 1 + (dr - 1) - degree d ∨ dr - 1 = 0 ∧ n = 0 ∧ ?rrr = 0"
proof (cases dr)
case 0
with Suc(4) have 0: "dr = 0" "n = 0" "degree d = 0" by auto
with deg_rrr have "degree ?rrr = 0" by simp
hence "∃ a. ?rrr = [: a :]" by (metis degree_pCons_eq_if old.nat.distinct(2) pCons_cases)
from this obtain a where rrr: "?rrr = [:a:]" by auto
show ?thesis unfolding 0 using c0 unfolding rrr 0 by simp
qed (insert Suc(4), auto)
note IH = Suc(1)[OF deg_rrr res this]
show ?case
proof (intro conjI)
show "r' = 0 ∨ degree r' < degree d" using IH by blast
show "smult (lc ^ Suc n) (d * q + r) = d * q' + r'"
unfolding IH[THEN conjunct2,symmetric]
qed
qed auto

lemma pseudo_divmod:
assumes g: "g ≠ 0" and *: "pseudo_divmod f g = (q,r)"
shows "smult (coeff g (degree g) ^ (Suc (degree f) - degree g)) f = g * q + r" (is ?A)
and "r = 0 ∨ degree r < degree g" (is ?B)
proof -
from *[unfolded pseudo_divmod_def Let_def]
have "pseudo_divmod_main (coeff g (degree g)) 0 f g (degree f) (1 + length (coeffs f) - length (coeffs g)) = (q, r)" by (auto simp: g)
note main = pseudo_divmod_main[OF _ _ _ this, OF g refl le_refl]
have "1 + length (coeffs f) - length (coeffs g) = 1 + degree f - degree g ∨
degree f = 0 ∧ 1 + length (coeffs f) - length (coeffs g) = 0 ∧ f = 0" using g
by (cases "f = 0"; cases "coeffs g", auto simp: degree_eq_length_coeffs)
note main = main[OF this]
from main show "r = 0 ∨ degree r < degree g" by auto
show "smult (coeff g (degree g) ^ (Suc (degree f) - degree g)) f = g * q + r"
by (subst main[THEN conjunct2, symmetric], simp add: degree_eq_length_coeffs,
insert g, cases "f = 0"; cases "coeffs g", auto)
qed

definition "pseudo_mod_main lc r d dr n = snd (pseudo_divmod_main lc 0 r d dr n)"

lemma snd_pseudo_divmod_main:
"snd (pseudo_divmod_main lc q r d dr n) = snd (pseudo_divmod_main lc q' r d dr n)"
by (induct n arbitrary: q q' lc r d dr; simp add: Let_def)

definition pseudo_mod
:: "'a :: {comm_ring_1,semiring_1_no_zero_divisors} poly ⇒ 'a poly ⇒ 'a poly" where
"pseudo_mod f g = snd (pseudo_divmod f g)"

lemma pseudo_mod:
fixes f g
defines "r ≡ pseudo_mod f g"
assumes g: "g ≠ 0"
shows "∃ a q. a ≠ 0 ∧ smult a f = g * q + r" "r = 0 ∨ degree r < degree g"
proof -
let ?cg = "coeff g (degree g)"
let ?cge = "?cg ^ (Suc (degree f) - degree g)"
define a where "a = ?cge"
obtain q where pdm: "pseudo_divmod f g = (q,r)" using r_def[unfolded pseudo_mod_def]
by (cases "pseudo_divmod f g", auto)
from pseudo_divmod[OF g pdm] have id: "smult a f = g * q + r" and "r = 0 ∨ degree r < degree g"
unfolding a_def by auto
show "r = 0 ∨ degree r < degree g" by fact
from g have "a ≠ 0" unfolding a_def by auto
thus "∃ a q. a ≠ 0 ∧ smult a f = g * q + r" using id by auto
qed

instantiation poly :: (idom_divide) idom_divide
begin

fun divide_poly_main :: "'a ⇒ 'a poly ⇒ 'a poly ⇒ 'a poly
⇒ nat ⇒ nat ⇒ 'a poly" where
"divide_poly_main lc q r d dr (Suc n) = (let cr = coeff r dr; a = cr div lc; mon = monom a n in
if False ∨ a * lc = cr then (* False ∨ is only because of problem in function-package *)
divide_poly_main
lc
(q + mon)
(r - mon * d)
d (dr - 1) n else 0)"
| "divide_poly_main lc q r d dr 0 = q"

definition divide_poly :: "'a poly ⇒ 'a poly ⇒ 'a poly" where
"divide_poly f g = (if g = 0 then 0 else
divide_poly_main (coeff g (degree g)) 0 f g (degree f) (1 + length (coeffs f) - length (coeffs g)))"

lemma divide_poly_main:
assumes d: "d ≠ 0" "lc = coeff d (degree d)"
and *: "degree (d * r) ≤ dr" "divide_poly_main lc q (d * r) d dr n = q'"
"n = 1 + dr - degree d ∨ dr = 0 ∧ n = 0 ∧ d * r = 0"
shows "q' = q + r"
using *
proof (induct n arbitrary: q r dr)
case (Suc n q r dr)
let ?rr = "d * r"
let ?a = "coeff ?rr dr"
let ?qq = "?a div lc"
define b where [simp]: "b = monom ?qq n"
let ?rrr =  "d * (r - b)"
let ?qqq = "q + b"
note res = Suc(3)
have dr: "dr = n + degree d" using Suc(4) by auto
have lc: "lc ≠ 0" using d by auto
have "coeff (b * d) dr = coeff b n * coeff d (degree d)"
proof (cases "?qq = 0")
case False
hence n: "n = degree b" by (simp add: degree_monom_eq)
show ?thesis unfolding n dr by (simp add: coeff_mult_degree_sum)
qed simp
also have "… = lc * coeff b n" unfolding d by simp
finally have c2: "coeff (b * d) dr = lc * coeff b n" .
have rrr: "?rrr = ?rr - b * d" by (simp add: field_simps)
have c1: "coeff (d * r) dr = lc * coeff r n"
proof (cases "degree r = n")
case True
thus ?thesis using Suc(2) unfolding dr using coeff_mult_degree_sum[of d r] d by (auto simp: ac_simps)
next
case False
have "degree r ≤ n" using dr Suc(2) by auto
with False have r_n: "degree r < n" by auto
hence right: "lc * coeff r n = 0" by (simp add: coeff_eq_0)
have "coeff (d * r) dr = coeff (d * r) (degree d + n)" unfolding dr by (simp add: ac_simps)
also have "… = 0" using r_n
coeff_mult_degree_sum degree_mult_le dr le_eq_less_or_eq)
finally show ?thesis unfolding right .
qed
have c0: "coeff ?rrr dr = 0"
and id: "lc * (coeff (d * r) dr div lc) = coeff (d * r) dr" unfolding rrr coeff_diff c2
unfolding b_def coeff_monom coeff_smult c1 using lc by auto
from res[unfolded divide_poly_main.simps[of lc q] Let_def] id
have res: "divide_poly_main lc ?qqq ?rrr d (dr - 1) n = q'"
by (simp del: divide_poly_main.simps add: field_simps)
note IH = Suc(1)[OF _ res]
have dr: "dr = n + degree d" using Suc(4) by auto
have deg_rr: "degree ?rr ≤ dr" using Suc(2) by auto
have deg_bd: "degree (b * d) ≤ dr"
unfolding dr b_def by (rule order.trans[OF degree_mult_le], auto simp: degree_monom_le)
have "degree ?rrr ≤ dr" unfolding rrr by (rule degree_diff_le[OF deg_rr deg_bd])
with c0 have deg_rrr: "degree ?rrr ≤ (dr - 1)" by (rule coeff_0_degree_minus_1)
have "n = 1 + (dr - 1) - degree d ∨ dr - 1 = 0 ∧ n = 0 ∧ ?rrr = 0"
proof (cases dr)
case 0
with Suc(4) have 0: "dr = 0" "n = 0" "degree d = 0" by auto
with deg_rrr have "degree ?rrr = 0" by simp
from degree_eq_zeroE[OF this] obtain a where rrr: "?rrr = [:a:]" by metis
show ?thesis unfolding 0 using c0 unfolding rrr 0 by simp
qed (insert Suc(4), auto)
note IH = IH[OF deg_rrr this]
show ?case using IH by simp
next
case (0 q r dr)
show ?case
proof (cases "r = 0")
case True
thus ?thesis using 0 by auto
next
case False
have "degree (d * r) = degree d + degree r" using d False
by (subst degree_mult_eq, auto)
thus ?thesis using 0 d by auto
qed
qed

lemma fst_pseudo_divmod_main_as_divide_poly_main:
assumes d: "d ≠ 0"
defines lc: "lc ≡ coeff d (degree d)"
shows "fst (pseudo_divmod_main lc q r d dr n) = divide_poly_main lc (smult (lc^n) q) (smult (lc^n) r) d dr n"
proof(induct n arbitrary: q r dr)
case 0 then show ?case by simp
next
case (Suc n)
note lc0 = leading_coeff_neq_0[OF d, folded lc]
then have "pseudo_divmod_main lc q r d dr (Suc n) =
pseudo_divmod_main lc (smult lc q + monom (coeff r dr) n)
(smult lc r - monom (coeff r dr) n * d) d (dr - 1) n"
also have "fst ... = divide_poly_main lc
(smult (lc^n) (smult lc q + monom (coeff r dr) n))
(smult (lc^n) (smult lc r - monom (coeff r dr) n * d))
d (dr - 1) n"
unfolding Suc[unfolded divide_poly_main.simps Let_def]..
also have "... = divide_poly_main lc (smult (lc ^ Suc n) q)
(smult (lc ^ Suc n) r) d dr (Suc n)"
unfolding smult_monom smult_distribs mult_smult_left[symmetric]
using lc0 by (simp add: Let_def ac_simps)
finally show ?case.
qed

lemma divide_poly_main_0: "divide_poly_main 0 0 r d dr n = 0"
proof (induct n arbitrary: r d dr)
case (Suc n r d dr)
show ?case unfolding divide_poly_main.simps[of _ _ r] Let_def
by (simp add: Suc del: divide_poly_main.simps)
qed simp

lemma divide_poly:
assumes g: "g ≠ 0"
shows "(f * g) div g = (f :: 'a poly)"
proof -
have "divide_poly_main (coeff g (degree g)) 0 (g * f) g (degree (g * f)) (1 + length (coeffs (g * f)) - length (coeffs  g))
= (f * g) div g" unfolding divide_poly_def Let_def by (simp add: ac_simps)
note main = divide_poly_main[OF g refl le_refl this]
{
fix f :: "'a poly"
assume "f ≠ 0"
hence "length (coeffs f) = Suc (degree f)" unfolding degree_eq_length_coeffs by auto
} note len = this
have "(f * g) div g = 0 + f"
proof (rule main, goal_cases)
case 1
show ?case
proof (cases "f = 0")
case True
with g show ?thesis by (auto simp: degree_eq_length_coeffs)
next
case False
with g have fg: "g * f ≠ 0" by auto
show ?thesis unfolding len[OF fg] len[OF g] by auto
qed
qed
thus ?thesis by simp
qed

lemma divide_poly_0: "f div 0 = (0 :: 'a poly)"
by (simp add: divide_poly_def Let_def divide_poly_main_0)

instance by (standard, auto simp: divide_poly divide_poly_0)
end

instance poly :: (idom_divide) algebraic_semidom ..

subsubsection‹Division in Field Polynomials›

text‹
This part connects the above result to the division of field polynomials.
Mainly imported from Isabelle 2016.
›

lemma pseudo_divmod_field:
assumes g: "(g::'a::field poly) ≠ 0" and *: "pseudo_divmod f g = (q,r)"
defines "c ≡ coeff g (degree g) ^ (Suc (degree f) - degree g)"
shows "f = g * smult (1/c) q + smult (1/c) r"
proof -
from leading_coeff_neq_0[OF g] have c0: "c ≠ 0" unfolding c_def by auto
from pseudo_divmod(1)[OF g *, folded c_def]
have "smult c f = g * q + r" by auto
also have "smult (1/c) ... = g * smult (1/c) q + smult (1/c) r" by (simp add: smult_add_right)
finally show ?thesis using c0 by auto
qed

lemma divide_poly_main_field:
assumes d: "(d::'a::field poly) ≠ 0"
defines lc: "lc ≡ coeff d (degree d)"
shows "divide_poly_main lc q r d dr n = fst (pseudo_divmod_main lc (smult ((1/lc)^n) q) (smult ((1/lc)^n) r) d dr n)"
unfolding lc
by(subst fst_pseudo_divmod_main_as_divide_poly_main, auto simp: d power_one_over)

lemma divide_poly_field:
fixes f g :: "'a :: field poly"
defines "f' ≡ smult ((1 / coeff g (degree g)) ^ (Suc (degree f) - degree g)) f"
shows "(f::'a::field poly) div g = fst (pseudo_divmod f' g)"
proof (cases "g = 0")
case True show ?thesis
unfolding divide_poly_def pseudo_divmod_def Let_def f'_def True by (simp add: divide_poly_main_0)
next
case False
from leading_coeff_neq_0[OF False] have "degree f' = degree f" unfolding f'_def by auto
then show ?thesis
using length_coeffs_degree[of f'] length_coeffs_degree[of f]
unfolding divide_poly_def pseudo_divmod_def Let_def
divide_poly_main_field[OF False]
length_coeffs_degree[OF False]
f'_def
by force
qed

instantiation poly :: (field) ring_div
begin

definition modulo_poly where
mod_poly_def: "f mod g ≡
if g = 0 then f
else pseudo_mod (smult ((1/coeff g (degree g)) ^ (Suc (degree f) - degree g)) f) g"

lemma pdivmod_rel: "pdivmod_rel (x::'a::field poly) y (x div y) (x mod y)"
unfolding pdivmod_rel_def
proof (intro conjI)
show "x = x div y * y + x mod y"
proof(cases "y = 0")
case True show ?thesis by(simp add: True divide_poly_def divide_poly_0 mod_poly_def)
next
case False
then have "pseudo_divmod (smult ((1 / coeff y (degree y)) ^ (Suc (degree x) - degree y)) x) y =
(x div y, x mod y)"
unfolding divide_poly_field mod_poly_def pseudo_mod_def by simp
from pseudo_divmod[OF False this]
show ?thesis using False
qed
show "if y = 0 then x div y = 0 else x mod y = 0 ∨ degree (x mod y) < degree y"
proof (cases "y = 0")
case True then show ?thesis by auto
next
case False
with pseudo_mod[OF this] show ?thesis unfolding mod_poly_def by simp
qed
qed

lemma div_poly_eq:
"pdivmod_rel (x::'a::field poly) y q r ⟹ x div y = q"
by(rule pdivmod_rel_unique_div[OF pdivmod_rel])

lemma mod_poly_eq:
"pdivmod_rel (x::'a::field poly) y q r ⟹ x mod y = r"
by (rule pdivmod_rel_unique_mod[OF pdivmod_rel])

instance
proof
fix x y :: "'a poly"
from pdivmod_rel[of x y,unfolded pdivmod_rel_def]
show "x div y * y + x mod y = x" by auto
next
fix x :: "'a poly"
show "x div 0 = 0" by simp
next
fix y :: "'a poly"
show "0 div y = 0" by simp
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]
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)
moreover from pdivmod_rel have "pdivmod_rel y z (y div z) (y mod z)" .
ultimately have "pdivmod_rel (x * y) (x * z) (y div z) (x * (y mod z))" .
then show ?thesis by (simp add: div_poly_eq)
qed
qed

end

lemma is_unit_monom_0:
fixes a :: "'a::field"
assumes "a ≠ 0"
shows "is_unit (monom a 0)"
proof
from assms show "1 = monom a 0 * monom (inverse a) 0"
qed

lemma is_unit_triv:
fixes a :: "'a::field"
assumes "a ≠ 0"
shows "is_unit [:a:]"
using assms by (simp add: is_unit_monom_0 monom_0 [symmetric])

lemma is_unit_iff_degree:
assumes "p ≠ (0 :: _ :: field poly)"
shows "is_unit p ⟷ degree p = 0" (is "?P ⟷ ?Q")
proof
assume ?Q
then obtain a where "p = [:a:]" by (rule degree_eq_zeroE)
with assms show ?P by (simp add: is_unit_triv)
next
assume ?P
then obtain q where "q ≠ 0" "p * q = 1" ..
then have "degree (p * q) = degree 1"
by simp
with ‹p ≠ 0› ‹q ≠ 0› have "degree p + degree q = 0"
then show ?Q by simp
qed

lemma is_unit_pCons_iff:
"is_unit (pCons (a::_::field) p) ⟷ p = 0 ∧ a ≠ 0"
by (cases "p = 0") (auto simp add: is_unit_triv is_unit_iff_degree)

lemma is_unit_monom_trival:
fixes p :: "'a::field poly"
assumes "is_unit p"
shows "monom (coeff p (degree p)) 0 = p"
using assms by (cases p) (simp_all add: monom_0 is_unit_pCons_iff)

lemma is_unit_polyE:
assumes "is_unit (p::_::field poly)"
obtains a where "p = monom a 0" and "a ≠ 0"
proof -
obtain a q where "p = pCons a q" by (cases p)
with assms have "p = [:a:]" and "a ≠ 0"
with that show thesis by (simp add: monom_0)
qed

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::'a::field poly) < degree y ⟹ x div y = 0"
proof -
assume "degree x < degree y"
hence "pdivmod_rel x y 0 x"
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"
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)"

lemma div_smult_left: "(smult (a::'a::field) 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

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

fixes x y z :: "'a::field poly"
shows "(x + y) div z = x div z + y div z"
by (rule div_poly_eq)

fixes x y z :: "'a::field poly"
shows "(x + y) mod z = x mod z + y mod z"
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"

lemma poly_mod_diff_left:
fixes x y z :: "'a::field poly"
shows "(x - y) mod z = x mod z - y mod z"

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::'a::field) ≠ 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 field_simps degree_mult_eq)
apply (cases "r' = 0")
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 pdivmod_0:
"pdivmod 0 q = (0, 0)"

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:
"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 (rule sym)
apply (induct p)
apply (case_tac "a = 0 ∧ p = 0")
done

subsection ‹List-based versions for fast implementation›
(* Subsection by:
Sebastiaan Joosten
*)
fun minus_poly_rev_list :: "'a :: group_add list ⇒ 'a list ⇒ 'a list" where
"minus_poly_rev_list (x # xs) (y # ys) = (x - y) # (minus_poly_rev_list xs ys)"
| "minus_poly_rev_list xs [] = xs"
| "minus_poly_rev_list [] (y # ys) = []"

fun pseudo_divmod_main_list :: "'a::comm_ring_1 ⇒ 'a list ⇒ 'a list ⇒ 'a list
⇒ nat ⇒ 'a list × 'a list" where
"pseudo_divmod_main_list lc q r d (Suc n) = (let
rr = map (op * lc) r;
a = hd r;
qqq = cCons a (map (op * lc) q);
rrr = tl (if a = 0 then rr else minus_poly_rev_list rr (map (op * a) d))
in pseudo_divmod_main_list lc qqq rrr d n)"
| "pseudo_divmod_main_list lc q r d 0 = (q,r)"

fun pseudo_mod_main_list :: "'a::comm_ring_1 ⇒ 'a list ⇒ 'a list
⇒ nat ⇒ 'a list" where
"pseudo_mod_main_list lc r d (Suc n) = (let
rr = map (op * lc) r;
a = hd r;
rrr = tl (if a = 0 then rr else minus_poly_rev_list rr (map (op * a) d))
in pseudo_mod_main_list lc rrr d n)"
| "pseudo_mod_main_list lc r d 0 = r"

fun divmod_poly_one_main_list :: "'a::comm_ring_1 list ⇒ 'a list ⇒ 'a list
⇒ nat ⇒ 'a list × 'a list" where
"divmod_poly_one_main_list q r d (Suc n) = (let
a = hd r;
qqq = cCons a q;
rr = tl (if a = 0 then r else minus_poly_rev_list r (map (op * a) d))
in divmod_poly_one_main_list qqq rr d n)"
| "divmod_poly_one_main_list q r d 0 = (q,r)"

fun mod_poly_one_main_list :: "'a::comm_ring_1 list ⇒ 'a list
⇒ nat ⇒ 'a list" where
"mod_poly_one_main_list r d (Suc n) = (let
a = hd r;
rr = tl (if a = 0 then r else minus_poly_rev_list r (map (op * a) d))
in mod_poly_one_main_list rr d n)"
| "mod_poly_one_main_list r d 0 = r"

definition pseudo_divmod_list :: "'a::comm_ring_1 list ⇒ 'a list ⇒ 'a list × 'a list" where
"pseudo_divmod_list p q =
(if q = [] then ([],p) else
(let rq = rev q;
(qu,re) = pseudo_divmod_main_list (hd rq) [] (rev p) rq (1 + length p - length q) in
(qu,rev re)))"

definition pseudo_mod_list :: "'a::comm_ring_1 list ⇒ 'a list ⇒ 'a list" where
"pseudo_mod_list p q =
(if q = [] then p else
(let rq = rev q;
re = pseudo_mod_main_list (hd rq) (rev p) rq (1 + length p - length q) in
(rev re)))"

lemma minus_zero_does_nothing:
"(minus_poly_rev_list x (map (op * 0) y)) = (x :: 'a :: ring list)"
by(induct x y rule: minus_poly_rev_list.induct, auto)

lemma length_minus_poly_rev_list[simp]:
"length (minus_poly_rev_list xs ys) = length xs"
by(induct xs ys rule: minus_poly_rev_list.induct, auto)

lemma if_0_minus_poly_rev_list:
"(if a = 0 then x else minus_poly_rev_list x (map (op * a) y))
= minus_poly_rev_list x (map (op * (a :: 'a :: ring)) y)"

lemma Poly_append:
"Poly ((a::'a::comm_semiring_1 list) @ b) = Poly a + monom 1 (length a) * Poly b"
by (induct a,auto simp: monom_0 monom_Suc)

lemma minus_poly_rev_list: "length p ≥ length (q :: 'a :: comm_ring_1 list) ⟹
Poly (rev (minus_poly_rev_list (rev p) (rev q)))
= Poly p - monom 1 (length p - length q) * Poly q"
proof (induct "rev p" "rev q" arbitrary: p q rule: minus_poly_rev_list.induct)
case (1 x xs y ys)
have "length (rev q) ≤ length (rev p)" using 1 by simp
from this[folded 1(2,3)] have ys_xs:"length ys ≤ length xs" by simp
hence a:"Poly (rev (minus_poly_rev_list xs ys)) =
Poly (rev xs) - monom 1 (length xs - length ys) * Poly (rev ys)"
by(subst "1.hyps"(1)[of "rev xs" "rev ys", unfolded rev_rev_ident length_rev],auto)
have "Poly p - monom 1 (length p - length q) * Poly q
= Poly (rev (rev p)) - monom 1 (length (rev (rev p)) - length (rev (rev q))) * Poly (rev (rev q))"
by simp
also have "… = Poly (rev (x # xs)) - monom 1 (length (x # xs) - length (y # ys)) * Poly (rev (y # ys))"
unfolding 1(2,3) by simp
also have "… = Poly (rev xs) + monom x (length xs) -
(monom 1 (length xs - length ys) * Poly (rev ys) + monom y (length xs))" using ys_xs
by (simp add:Poly_append distrib_left mult_monom smult_monom)
also have "… = Poly (rev (minus_poly_rev_list xs ys)) + monom (x - y) (length xs)"
unfolding a diff_monom[symmetric] by(simp)
finally show ?case
unfolding 1(2,3)[symmetric] by (simp add: smult_monom Poly_append)
qed auto

lemma smult_monom_mult: "smult a (monom b n * f) = monom (a * b) n * f"
using smult_monom [of a _ n] by (metis mult_smult_left)

"length d ≤ length r ⟹ d≠[] ⟹
hd (minus_poly_rev_list (map (op * (last d :: 'a :: comm_ring)) r) (map (op * (hd r)) (rev d))) = 0"
proof(induct r)
case (Cons a rs)
thus ?case by(cases "rev d", simp_all add:ac_simps)
qed simp

lemma Poly_map: "Poly (map (op * a) p) = smult a (Poly p)"
proof (induct p)
case(Cons x xs) thus ?case by (cases "Poly xs = 0",auto)
qed simp

lemma last_coeff_is_hd: "xs ≠ [] ⟹ coeff (Poly xs) (length xs - 1) = hd (rev xs)"
by (simp_all add: hd_conv_nth rev_nth nth_default_nth nth_append)

lemma pseudo_divmod_main_list_invar :
and lc:"last d = lc"
and dNonempty:"d ≠ []"
and "pseudo_divmod_main_list lc q (rev r) (rev d) n = (q',rev r')"
and "n = (1 + length r - length d)"
shows
"pseudo_divmod_main lc (monom 1 n * Poly q) (Poly r) (Poly d) (length r - 1) n =
(Poly q', Poly r')"
using assms(4-)
proof(induct "n" arbitrary: r q)
case (Suc n r q)
have ifCond: "¬ Suc (length r) ≤ length d" using Suc.prems by simp
have rNonempty:"r ≠ []"
using ifCond dNonempty using Suc_leI length_greater_0_conv list.size(3) by fastforce
let ?a = "(hd (rev r))"
let ?rr = "map (op * lc) (rev r)"
let ?rrr = "rev (tl (minus_poly_rev_list ?rr (map (op * ?a) (rev d))))"
let ?qq = "cCons ?a (map (op * lc) q)"
have n: "n = (1 + length r - length d - 1)"
using ifCond Suc(3) by simp
have rr_val:"(length ?rrr) = (length r - 1)" using ifCond by auto
hence rr_smaller: "(1 + length r - length d - 1) = (1 + length ?rrr - length d)"
using rNonempty ifCond unfolding One_nat_def by auto
from ifCond have id: "Suc (length r) - length d = Suc (length r - length d)" by auto
have "pseudo_divmod_main_list lc ?qq (rev ?rrr) (rev d) (1 + length r - length d - 1) = (q', rev r')"
using Suc.prems ifCond by (simp add:Let_def if_0_minus_poly_rev_list id)
hence v:"pseudo_divmod_main_list lc ?qq (rev ?rrr) (rev d) n = (q', rev r')"
using n by auto
have sucrr:"Suc (length r) - length d = Suc (length r - length d)"
using Suc_diff_le ifCond not_less_eq_eq by blast
have n_ok : "n = 1 + (length ?rrr) - length d" using Suc(3) rNonempty by simp
have cong: "⋀ x1 x2 x3 x4 y1 y2 y3 y4. x1 = y1 ⟹ x2 = y2 ⟹ x3 = y3 ⟹ x4 = y4 ⟹
pseudo_divmod_main lc x1 x2 x3 x4 n = pseudo_divmod_main lc y1 y2 y3 y4 n" by simp
have hd_rev:"coeff (Poly r) (length r - Suc 0) = hd (rev r)"
using last_coeff_is_hd[OF rNonempty] by simp
show ?case unfolding Suc.hyps(1)[OF v n_ok, symmetric] pseudo_divmod_main.simps Let_def
proof (rule cong[OF _ _ refl], goal_cases)
case 1
show ?case unfolding monom_Suc hd_rev[symmetric]
next
case 2
show ?case
proof (subst Poly_on_rev_starting_with_0, goal_cases)
show "hd (minus_poly_rev_list (map (op * lc) (rev r)) (map (op * (hd (rev r))) (rev d))) = 0"
by (fold lc, subst head_minus_poly_rev_list, insert ifCond dNonempty,auto)
from ifCond have "length d ≤ length r" by simp
then show "smult lc (Poly r) - monom (coeff (Poly r) (length r - 1)) n * Poly d =
Poly (rev (minus_poly_rev_list (map (op * lc) (rev r)) (map (op * (hd (rev r))) (rev d))))"
by (fold rev_map) (auto simp add: n smult_monom_mult Poly_map hd_rev [symmetric]
minus_poly_rev_list)
qed
qed simp
qed simp

lemma pseudo_divmod_impl[code]: "pseudo_divmod (f::'a::comm_ring_1 poly) g =
map_prod poly_of_list poly_of_list (pseudo_divmod_list (coeffs f) (coeffs g))"
proof (cases "g=0")
case False
hence coeffs_g_nonempty:"(coeffs g) ≠ []" by simp
hence lastcoeffs:"last (coeffs g) = coeff g (degree g)"
by (simp add: hd_rev last_coeffs_eq_coeff_degree not_0_coeffs_not_Nil)
obtain q r where qr: "pseudo_divmod_main_list
(last (coeffs g)) (rev [])
(rev (coeffs f)) (rev (coeffs g))
(1 + length (coeffs f) -
length (coeffs g)) = (q,rev (rev r))"  by force
then have qr': "pseudo_divmod_main_list
(hd (rev (coeffs g))) []
(rev (coeffs f)) (rev (coeffs g))
(1 + length (coeffs f) -
length (coeffs g)) = (q,r)" using hd_rev[OF coeffs_g_nonempty] by(auto)
from False have cg: "(coeffs g = []) = False" by auto
have last_non0:"last (coeffs g) ≠ 0" using False by (simp add:last_coeffs_not_0)
show ?thesis
unfolding pseudo_divmod_def pseudo_divmod_list_def Let_def qr' map_prod_def split cg if_False
pseudo_divmod_main_list_invar[OF last_non0 _ _ qr,unfolded lastcoeffs,simplified,symmetric,OF False]
poly_of_list_def
using False by (auto simp: degree_eq_length_coeffs)
next
case True
show ?thesis unfolding True unfolding pseudo_divmod_def pseudo_divmod_list_def
by auto
qed

lemma pseudo_mod_main_list: "snd (pseudo_divmod_main_list l q
xs ys n) = pseudo_mod_main_list l xs ys n"
by (induct n arbitrary: l q xs ys, auto simp: Let_def)

lemma pseudo_mod_impl[code]: "pseudo_mod f g =
poly_of_list (pseudo_mod_list (coeffs f) (coeffs g))"
proof -
have snd_case: "⋀ f g p. snd ((λ (x,y). (f x, g y)) p) = g (snd p)"
by auto
show ?thesis
unfolding pseudo_mod_def pseudo_divmod_impl pseudo_divmod_list_def
pseudo_mod_list_def Let_def
qed

(* *************** *)
subsubsection ‹Improved Code-Equations for Polynomial (Pseudo) Division›

lemma pdivmod_pdivmodrel: "pdivmod_rel p q r s = (pdivmod p q = (r, s))"
by (metis pdivmod_def pdivmod_rel pdivmod_rel_unique prod.sel)

lemma pdivmod_via_pseudo_divmod: "pdivmod f g = (if g = 0 then (0,f)
else let
ilc = inverse (coeff g (degree g));
h = smult ilc g;
(q,r) = pseudo_divmod f h
in (smult ilc q, r))" (is "?l = ?r")
proof (cases "g = 0")
case False
define lc where "lc = inverse (coeff g (degree g))"
define h where "h = smult lc g"
from False have h1: "coeff h (degree h) = 1" and lc: "lc ≠ 0" unfolding h_def lc_def by auto
hence h0: "h ≠ 0" by auto
obtain q r where p: "pseudo_divmod f h = (q,r)" by force
from False have id: "?r = (smult lc q, r)"
unfolding Let_def h_def[symmetric] lc_def[symmetric] p by auto
from pseudo_divmod[OF h0 p, unfolded h1]
have f: "f = h * q + r" and r: "r = 0 ∨ degree r < degree h" by auto
have "pdivmod_rel f h q r" unfolding pdivmod_rel_def using f r h0 by auto
hence "pdivmod f h = (q,r)" by (simp add: pdivmod_pdivmodrel)
hence "pdivmod f g = (smult lc q, r)"
unfolding pdivmod_def h_def div_smult_right[OF lc] mod_smult_right[OF lc]
using lc by auto
with id show ?thesis by auto
qed (auto simp: pdivmod_def)

lemma pdivmod_via_pseudo_divmod_list: "pdivmod f g = (let
cg = coeffs g
in if cg = [] then (0,f)
else let
cf = coeffs f;
ilc = inverse (last cg);
ch = map (op * ilc) cg;
(q,r) = pseudo_divmod_main_list 1 [] (rev cf) (rev ch) (1 + length cf - length cg)
in (poly_of_list (map (op * ilc) q), poly_of_list (rev r)))"
proof -
note d = pdivmod_via_pseudo_divmod
pseudo_divmod_impl pseudo_divmod_list_def
show ?thesis
proof (cases "g = 0")
case True thus ?thesis unfolding d by auto
next
case False
define ilc where "ilc = inverse (coeff g (degree g))"
from False have ilc: "ilc ≠ 0" unfolding ilc_def by auto
with False have id: "(g = 0) = False" "(coeffs g = []) = False"
"last (coeffs g) = coeff g (degree g)"
"(coeffs (smult ilc g) = []) = False"
by (auto simp: last_coeffs_eq_coeff_degree)
have id2: "hd (rev (coeffs (smult ilc g))) = 1"
by (subst hd_rev, insert id ilc, auto simp: coeffs_smult, subst last_map, auto simp: id ilc_def)
have id3: "length (coeffs (smult ilc g)) = length (coeffs g)"
"rev (coeffs (smult ilc g)) = rev (map (op * ilc) (coeffs g))" unfolding coeffs_smult using ilc by auto
obtain q r where pair: "pseudo_divmod_main_list 1 [] (rev (coeffs f)) (rev (map (op * ilc) (coeffs g)))
(1 + length (coeffs f) - length (coeffs g)) = (q,r)" by force
show ?thesis unfolding d Let_def id if_False ilc_def[symmetric] map_prod_def[symmetric] id2
unfolding id3 pair map_prod_def split by (auto simp: Poly_map)
qed
qed

lemma pseudo_divmod_main_list_1: "pseudo_divmod_main_list 1 = divmod_poly_one_main_list"
proof (intro ext, goal_cases)
case (1 q r d n)
{
fix xs :: "'a list"
have "map (op * 1) xs = xs" by (induct xs, auto)
} note [simp] = this
show ?case by (induct n arbitrary: q r d, auto simp: Let_def)
qed

fun divide_poly_main_list :: "'a::idom_divide ⇒ 'a list ⇒ 'a list ⇒ 'a list
⇒ nat ⇒ 'a list" where
"divide_poly_main_list lc q r d (Suc n) = (let
cr = hd r
in if cr = 0 then divide_poly_main_list lc (cCons cr q) (tl r) d n else let
a = cr div lc;
qq = cCons a q;
rr = minus_poly_rev_list r (map (op * a) d)
in if hd rr = 0 then divide_poly_main_list lc qq (tl rr) d n else [])"
| "divide_poly_main_list lc q r d 0 = q"

lemma divide_poly_main_list_simp[simp]: "divide_poly_main_list lc q r d (Suc n) = (let
cr = hd r;
a = cr div lc;
qq = cCons a q;
rr = minus_poly_rev_list r (map (op * a) d)
in if hd rr = 0 then divide_poly_main_list lc qq (tl rr) d n else [])"

declare divide_poly_main_list.simps(1)[simp del]

definition divide_poly_list :: "'a::idom_divide poly ⇒ 'a poly ⇒ 'a poly" where
"divide_poly_list f g =
(let cg = coeffs g
in if cg = [] then g
else let cf = coeffs f; cgr = rev cg
in poly_of_list (divide_poly_main_list (hd cgr) [] (rev cf) cgr (1 + length cf - length cg)))"

lemmas pdivmod_via_divmod_list[code] = pdivmod_via_pseudo_divmod_list[unfolded pseudo_divmod_main_list_1]

lemma mod_poly_one_main_list: "snd (divmod_poly_one_main_list q r d n) = mod_poly_one_main_list r d n"
by  (induct n arbitrary: q r d, auto simp: Let_def)

lemma mod_poly_code[code]: "f mod g =
(let cg = coeffs g
in if cg = [] then f
else let cf = coeffs f; ilc = inverse (last cg); ch = map (op * ilc) cg;
r = mod_poly_one_main_list (rev cf) (rev ch) (1 + length cf - length cg)
in poly_of_list (rev r))" (is "?l = ?r")
proof -
have "?l = snd (pdivmod f g)" unfolding pdivmod_def by simp
also have "… = ?r" unfolding pdivmod_via_divmod_list Let_def
mod_poly_one_main_list[symmetric, of _ _ _ Nil] by (auto split: prod.splits)
finally show ?thesis .
qed

definition div_field_poly_impl :: "'a :: field poly ⇒ 'a poly ⇒ 'a poly" where
"div_field_poly_impl f g = (
let cg = coeffs g
in if cg = [] then 0
else let cf = coeffs f; ilc = inverse (last cg); ch = map (op * ilc) cg;
q = fst (divmod_poly_one_main_list [] (rev cf) (rev ch) (1 + length cf - length cg))
in poly_of_list ((map (op * ilc) q)))"

text ‹We do not declare the following lemma as code equation, since then polynomial division
on non-fields will no longer be executable. However, a code-unfold is possible, since
‹div_field_poly_impl› is a bit more efficient than the generic polynomial division.›
lemma div_field_poly_impl[code_unfold]: "op div = div_field_poly_impl"
proof (intro ext)
fix f g :: "'a poly"
have "f div g = fst (pdivmod f g)" unfolding pdivmod_def by simp
also have "… = div_field_poly_impl f g" unfolding
div_field_poly_impl_def pdivmod_via_divmod_list Let_def by (auto split: prod.splits)
finally show "f div g =  div_field_poly_impl f g" .
qed

lemma divide_poly_main_list:
assumes lc0: "lc ≠ 0"
and lc:"last d = lc"
and d:"d ≠ []"
and "n = (1 + length r - length d)"
shows
"Poly (divide_poly_main_list lc q (rev r) (rev d) n) =
divide_poly_main lc (monom 1 n * Poly q) (Poly r) (Poly d) (length r - 1) n"
using assms(4-)
proof(induct "n" arbitrary: r q)
case (Suc n r q)
have ifCond: "¬ Suc (length r) ≤ length d" using Suc.prems by simp
have r: "r ≠ []"
using ifCond d using Suc_leI length_greater_0_conv list.size(3) by fastforce
then obtain rr lcr where r: "r = rr @ [lcr]" by (cases r rule: rev_cases, auto)
from d lc obtain dd where d: "d = dd @ [lc]"
by (cases d rule: rev_cases, auto)
from Suc(2) ifCond have n: "n = 1 + length rr - length d" by (auto simp: r)
from ifCond have len: "length dd ≤ length rr" by (simp add: r d)
show ?case
proof (cases "lcr div lc * lc = lcr")
case False
thus ?thesis unfolding Suc(2)[symmetric] using r d
by (auto simp add: Let_def nth_default_append)
next
case True
hence id:
"?thesis = (Poly (divide_poly_main_list lc (cCons (lcr div lc) q)
(rev (rev (minus_poly_rev_list (rev rr) (rev (map (op * (lcr div lc)) dd))))) (rev d) n) =
divide_poly_main lc
(monom 1 (Suc n) * Poly q + monom (lcr div lc) n)
(Poly r - monom (lcr div lc) n * Poly d)
(Poly d) (length rr - 1) n)"
using r d
by (cases r rule: rev_cases; cases "d" rule: rev_cases;
auto simp add: Let_def rev_map nth_default_append)
have cong: "⋀ x1 x2 x3 x4 y1 y2 y3 y4. x1 = y1 ⟹ x2 = y2 ⟹ x3 = y3 ⟹ x4 = y4 ⟹
divide_poly_main lc x1 x2 x3 x4 n = divide_poly_main lc y1 y2 y3 y4 n" by simp
show ?thesis unfolding id
proof (subst Suc(1), simp add: n,
subst minus_poly_rev_list, force simp: len, rule cong[OF _ _ refl], goal_cases)
case 2
have "monom lcr (length rr) = monom (lcr div lc) (length rr - length dd) * monom lc (length dd)"
by (simp add: mult_monom len True)
thus ?case unfolding r d Poly_append n ring_distribs
by (auto simp: Poly_map smult_monom smult_monom_mult)
qed (auto simp: len monom_Suc smult_monom)
qed
qed simp

lemma divide_poly_list[code]: "f div g = divide_poly_list f g"
proof -
note d = divide_poly_def divide_poly_list_def
show ?thesis
proof (cases "g = 0")
case True
show ?thesis unfolding d True by auto
next
case False
then obtain cg lcg where cg: "coeffs g = cg @ [lcg]" by (cases "coeffs g" rule: rev_cases, auto)
with False have id: "(g = 0) = False" "(cg @ [lcg] = []) = False" by auto
from cg False have lcg: "coeff g (degree g) = lcg"
using last_coeffs_eq_coeff_degree last_snoc by force
with False have lcg0: "lcg ≠ 0" by auto
from cg have ltp: "Poly (cg @ [lcg]) = g"
using Poly_coeffs [of g] by auto
show ?thesis unfolding d cg Let_def id if_False poly_of_list_def
by (subst divide_poly_main_list, insert False cg lcg0, auto simp: lcg ltp,
qed
qed

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

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

subsection ‹Additional induction rules on polynomials›

text ‹
An induction rule for induction over the roots of a polynomial with a certain property.
(e.g. all positive roots)
›
lemma poly_root_induct [case_names 0 no_roots root]:
fixes p :: "'a :: idom poly"
assumes "Q 0"
assumes "⋀p. (⋀a. P a ⟹ poly p a ≠ 0) ⟹ Q p"
assumes "⋀a p. P a ⟹ Q p ⟹ Q ([:a, -1:] * p)"
shows   "Q p"
proof (induction "degree p" arbitrary: p rule: less_induct)
case (less p)
show ?case
proof (cases "p = 0")
assume nz: "p ≠ 0"
show ?case
proof (cases "∃a. P a ∧ poly p a = 0")
case False
thus ?thesis by (intro assms(2)) blast
next
case True
then obtain a where a: "P a" "poly p a = 0"
by blast
hence "-[:-a, 1:] dvd p"
by (subst minus_dvd_iff) (simp add: poly_eq_0_iff_dvd)
then obtain q where q: "p = [:a, -1:] * q" by (elim dvdE) simp
with nz have q_nz: "q ≠ 0" by auto
have "degree p = Suc (degree q)"
by (subst q, subst degree_mult_eq) (simp_all add: q_nz)
hence "Q q" by (intro less) simp
from a(1) and this have "Q ([:a, -1:] * q)"
by (rule assms(3))
with q show ?thesis by simp
qed
qed

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

lemma Poly_append_replicate_0: "Poly (xs @ replicate n 0) = Poly xs"
by (subst coeffs_eq_iff) (simp_all add: strip_while_def dropWhile_replicate_append)

text ‹
An induction rule for simultaneous induction over two polynomials,
prepending one coefficient in each step.
›
lemma poly_induct2 [case_names 0 pCons]:
assumes "P 0 0" "⋀a p b q. P p q ⟹ P (pCons a p) (pCons b q)"
shows   "P p q"
proof -
define n where "n = max (length (coeffs p)) (length (coeffs q))"
define xs where "xs = coeffs p @ (replicate (n - length (coeffs p)) 0)"
define ys where "ys = coeffs q @ (replicate (n - length (coeffs q)) 0)"
have "length xs = length ys"
by (simp add: xs_def ys_def n_def)
hence "P (Poly xs) (Poly ys)"
by (induction rule: list_induct2) (simp_all add: assms)
also have "Poly xs = p"
also have "Poly ys = q"
finally show ?thesis .
qed

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"

notation pcompose (infixl "∘⇩p" 71)

lemma pcompose_0 [simp]:
"pcompose 0 q = 0"

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 pcompose_1:
fixes p :: "'a :: comm_semiring_1 poly"
shows "pcompose 1 p = 1"
unfolding one_poly_def by (auto simp: pcompose_pCons)

lemma poly_pcompose:
"poly (pcompose p q) x = poly p (poly q x)"
by (induct p) (simp_all add: pcompose_pCons)

lemma degree_pcompose_le:
"degree (pcompose p q) ≤ degree p * degree q"
apply (induct p, simp)
apply (rule order_trans [OF degree_mult_le], simp)
done

fixes p q r :: "'a :: {comm_semiring_0, ab_semigroup_add} poly"
shows "pcompose (p + q) r = pcompose p r + pcompose q r"
proof (induction p q rule: poly_induct2)
case (pCons a p b q)
have "pcompose (pCons a p + pCons b q) r =
[:a + b:] + r * pcompose p r + r * pcompose q r"
by (simp_all add: pcompose_pCons pCons.IH algebra_simps)
also have "[:a + b:] = [:a:] + [:b:]" by simp
also have "… + r * pcompose p r + r * pcompose q r =
pcompose (pCons a p) r + pcompose (pCons b q) r"
finally show ?case .
qed simp

lemma pcompose_uminus:
fixes p r :: "'a :: comm_ring poly"
shows "pcompose (-p) r = -pcompose p r"
by (induction p) (simp_all add: pcompose_pCons)

lemma pcompose_diff:
fixes p q r :: "'a :: comm_ring poly"
shows "pcompose (p - q) r = pcompose p r - pcompose q r"

lemma pcompose_smult:
fixes p r :: "'a :: comm_semiring_0 poly"
shows "pcompose (smult a p) r = smult a (pcompose p r)"
by (induction p)

lemma pcompose_mult:
fixes p q r :: "'a :: comm_semiring_0 poly"
shows "pcompose (p * q) r = pcompose p r * pcompose q r"
by (induction p arbitrary: q)

lemma pcompose_assoc:
"pcompose p (pcompose q r :: 'a :: comm_semiring_0 poly ) =
pcompose (pcompose p q) r"
by (induction p arbitrary: q)

lemma pcompose_idR[simp]:
fixes p :: "'a :: comm_semiring_1 poly"
shows "pcompose p [: 0, 1 :] = p"
by (induct p; simp add: pcompose_pCons)

lemma pcompose_sum: "pcompose (sum f A) p = sum (λi. pcompose (f i) p) A"
by (cases "finite A", induction rule: finite_induct)

lemma pcompose_prod: "pcompose (prod f A) p = prod (λi. pcompose (f i) p) A"
by (cases "finite A", induction rule: finite_induct)

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

lemma degree_mult_eq_0:
fixes p q:: "'a :: {comm_semiring_0,semiring_no_zero_divisors} poly"
shows "degree (p*q) = 0 ⟷ p=0 ∨ q=0 ∨ (p≠0 ∧ q≠0 ∧ degree p =0 ∧ degree q =0)"

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

lemma pcompose_0': "pcompose p 0 = [:coeff p 0:]"
by (induct p) (auto simp add:pcompose_pCons)

lemma degree_pcompose:
fixes p q:: "'a::{comm_semiring_0,semiring_no_zero_divisors} poly"
shows "degree (pcompose p q) = degree p * degree q"
proof (induct p)
case 0
thus ?case by auto
next
case (pCons a p)
have "degree (q * pcompose p q) = 0 ⟹ ?case"
proof (cases "p=0")
case True
thus ?thesis by auto
next
case False assume "degree (q * pcompose p q) = 0"
hence "degree q=0 ∨ pcompose p q=0" by (auto simp add: degree_mult_eq_0)
moreover have "⟦pcompose p q=0;degree q≠0⟧ ⟹ False" using pCons.hyps(2) ‹p≠0›
proof -
assume "pcompose p q=0" "degree q≠0"
hence "degree p=0" using pCons.hyps(2) by auto
then obtain a1 where "p=[:a1:]"
by (metis degree_pCons_eq_if old.nat.distinct(2) pCons_cases)
thus False using ‹pcompose p q=0› ‹p≠0› by auto
qed
ultimately have "degree (pCons a p) * degree q=0" by auto
moreover have "degree (pcompose (pCons a p) q) = 0"
proof -
have" 0 = max (degree [:a:]) (degree (q*pcompose p q))"
using ‹degree (q * pcompose p q) = 0› by simp
also have "... ≥ degree ([:a:] + q * pcompose p q)"
finally show ?thesis by (auto simp add:pcompose_pCons)
qed
ultimately show ?thesis by simp
qed
moreover have "degree (q * pcompose p q)>0 ⟹ ?case"
proof -
assume asm:"0 < degree (q * pcompose p q)"
hence "p≠0" "q≠0" "pcompose p q≠0" by auto
have "degree (pcompose (pCons a p) q) = degree ( q * pcompose p q)"
unfolding pcompose_pCons
using degree_add_eq_right[of "[:a:]" ] asm by auto
thus ?thesis
using pCons.hyps(2) degree_mult_eq[OF ‹q≠0› ‹pcompose p q≠0›] by auto
qed
ultimately show ?case by blast
qed

lemma pcompose_eq_0:
fixes p q:: "'a :: {comm_semiring_0,semiring_no_zero_divisors} poly"
assumes "pcompose p q = 0" "degree q > 0"
shows "p = 0"
proof -
have "degree p=0" using assms degree_pcompose[of p q] by auto
then obtain a where "p=[:a:]"
by (metis degree_pCons_eq_if gr0_conv_Suc neq0_conv pCons_cases)
hence "a=0" using assms(1) by auto
thus ?thesis using ‹p=[:a:]› by simp
qed

definition lead_coeff:: "'a::zero poly ⇒ 'a" where
"lead_coeff p= coeff p (degree p)"

"p=0 ⟹ lead_coeff (pCons a p) = a"

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

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

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

assumes "degree p < degree q"

"lead_coeff (smult c p :: 'a :: {comm_semiring_0,semiring_no_zero_divisors} poly) = c * lead_coeff p"
proof -
have "smult c p = [:c:] * p" by simp
finally show ?thesis .
qed

fixes p q:: "'a::{comm_semiring_1,semiring_no_zero_divisors} poly"
assumes "degree q > 0"
proof (induct p)
case 0
thus ?case unfolding lead_coeff_def by auto
next
case (pCons a p)
have "degree ( q * pcompose p q) = 0 ⟹ ?case"
proof -
assume "degree ( q * pcompose p q) = 0"
hence "pcompose p q = 0" by (metis assms degree_0 degree_mult_eq_0 neq0_conv)
hence "p=0" using pcompose_eq_0[OF _ ‹degree q > 0›] by simp
thus ?thesis by auto
qed
moreover have "degree ( q * pcompose p q) > 0 ⟹ ?case"
proof -
assume "degree ( q * pcompose p q) > 0"
hence "lead_coeff (pcompose (pCons a p) q) =lead_coeff ( q * pcompose p q)"
using pCons.hyps(2) lead_coeff_mult[of q "pcompose p q"] by simp
also have "... = lead_coeff p * lead_coeff q ^ (degree p + 1)"
by (auto simp: mult_ac)
finally show ?thesis by auto
qed
ultimately show ?case by blast
qed

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

"lead_coeff (numeral n) = numeral n"
by (subst of_nat_numeral [symmetric], subst of_nat_poly) simp

"lead_coeff (p ^ n :: 'a :: {comm_semiring_1,semiring_no_zero_divisors} poly) = lead_coeff p ^ n"

subsection ‹Shifting polynomials›

definition poly_shift :: "nat ⇒ ('a::zero) poly ⇒ 'a poly" where
"poly_shift n p = Abs_poly (λi. coeff p (i + n))"

lemma nth_default_drop: "nth_default x (drop n xs) m = nth_default x xs (m + n)"

lemma nth_default_take: "nth_default x (take n xs) m = (if m < n then nth_default x xs m else x)"

lemma coeff_poly_shift: "coeff (poly_shift n p) i = coeff p (i + n)"
proof -
from MOST_coeff_eq_0[of p] obtain m where "∀k>m. coeff p k = 0" by (auto simp: MOST_nat)
hence "∀k>m. coeff p (k + n) = 0" by auto
hence "∀⇩∞k. coeff p (k + n) = 0" by (auto simp: MOST_nat)
thus ?thesis by (simp add: poly_shift_def poly.Abs_poly_inverse)
qed

lemma poly_shift_id [simp]: "poly_shift 0 = (λx. x)"
by (simp add: poly_eq_iff fun_eq_iff coeff_poly_shift)

lemma poly_shift_0 [simp]: "poly_shift n 0 = 0"

lemma poly_shift_1: "poly_shift n 1 = (if n = 0 then 1 else 0)"

lemma poly_shift_monom: "poly_shift n (monom c m) = (if m ≥ n then monom c (m - n) else 0)"
by (auto simp add: poly_eq_iff coeff_poly_shift)

lemma coeffs_shift_poly [code abstract]: "coeffs (poly_shift n p) = drop n (coeffs p)"
proof (cases "p = 0")
case False
thus ?thesis
by (intro coeffs_eqI)
(simp_all add: coeff_poly_shift nth_default_drop last_coeffs_not_0 nth_default_coeffs_eq)
qed simp_all

subsection ‹Truncating polynomials›

definition poly_cutoff where
"poly_cutoff n p = Abs_poly (λk. if k < n then coeff p k else 0)"

lemma coeff_poly_cutoff: "coeff (poly_cutoff n p) k = (if k < n then coeff p k else 0)"
unfolding poly_cutoff_def
by (subst poly.Abs_poly_inverse) (auto simp: MOST_nat intro: exI[of _ n])

lemma poly_cutoff_0 [simp]: "poly_cutoff n 0 = 0"

lemma poly_cutoff_1 [simp]: "poly_cutoff n 1 = (if n = 0 then 0 else 1)"

lemma coeffs_poly_cutoff [code abstract]:
"coeffs (poly_cutoff n p) = strip_while (op = 0) (take n (coeffs p))"
proof (cases "strip_while (op = 0) (take n (coeffs p)) = []")
case True
hence "coeff (poly_cutoff n p) k = 0" for k
unfolding coeff_poly_cutoff
by (auto simp: nth_default_coeffs_eq [symmetric] nth_default_def set_conv_nth)
hence "poly_cutoff n p = 0" by (simp add: poly_eq_iff)
thus ?thesis by (subst True) simp_all
next
case False
have "no_trailing (op = 0) (strip_while (op = 0) (take n (coeffs p)))" by simp
with False have "last (strip_while (op = 0) (take n (coeffs p))) ≠ 0"
unfolding no_trailing_unfold by auto
thus ?thesis
by (intro coeffs_eqI)
(simp_all add: coeff_poly_cutoff last_coeffs_not_0 nth_default_take nth_default_coeffs_eq)
qed

subsection ‹Reflecting polynomials›

definition reflect_poly where
"reflect_poly p = Poly (rev (coeffs p))"

lemma coeffs_reflect_poly [code abstract]:
"coeffs (reflect_poly p) = rev (dropWhile (op = 0) (coeffs p))"
unfolding reflect_poly_def by simp

lemma reflect_poly_0 [simp]: "reflect_poly 0 = 0"

lemma reflect_poly_1 [simp]: "reflect_poly 1 = 1"

lemma coeff_reflect_poly:
"coeff (reflect_poly p) n = (if n > degree p then 0 else coeff p (degree p - n))"
by (cases "p = 0") (auto simp add: reflect_poly_def coeff_Poly_eq nth_default_def
rev_nth degree_eq_length_coeffs coeffs_nth not_less
dest: le_imp_less_Suc)

lemma coeff_0_reflect_poly_0_iff [simp]: "coeff (reflect_poly p) 0 = 0 ⟷ p = 0"

lemma reflect_poly_at_0_eq_0_iff [simp]: "poly (reflect_poly p) 0 = 0 ⟷ p = 0"

lemma reflect_poly_pCons':
"p ≠ 0 ⟹ reflect_poly (pCons c p) = reflect_poly p + monom c (Suc (degree p))"
by (intro poly_eqI)
(auto simp: coeff_reflect_poly coeff_pCons not_less Suc_diff_le split: nat.split)

lemma reflect_poly_const [simp]: "reflect_poly [:a:] = [:a:]"
by (cases "a = 0") (simp_all add: reflect_poly_def)

lemma poly_reflect_poly_nz:
"(x :: 'a :: field) ≠ 0 ⟹ poly (reflect_poly p) x = x ^ degree p * poly p (inverse x)"
by (induction rule: pCons_induct) (simp_all add: field_simps reflect_poly_pCons' poly_monom)

lemma coeff_0_reflect_poly [simp]: "coeff (reflect_poly p) 0 = lead_coeff p"

lemma poly_reflect_poly_0 [simp]: "poly (reflect_poly p) 0 = lead_coeff p"

lemma reflect_poly_reflect_poly [simp]: "coeff p 0 ≠ 0 ⟹ reflect_poly (reflect_poly p) = p"
by (cases p rule: pCons_cases) (simp add: reflect_poly_def )

lemma degree_reflect_poly_le: "degree (reflect_poly p) ≤ degree p"
by (simp add: degree_eq_length_coeffs coeffs_reflect_poly length_dropWhile_le diff_le_mono)

lemma reflect_poly_pCons:
"a ≠ 0 ⟹ reflect_poly (pCons a p) = Poly (rev (a # coeffs p))"
by (subst coeffs_eq_iff) (simp add: coeffs_reflect_poly)

lemma degree_reflect_poly_eq [simp]: "coeff p 0 ≠ 0 ⟹ degree (reflect_poly p) = degree p"
by (cases p rule: pCons_cases) (simp add: reflect_poly_pCons degree_eq_length_coeffs)

(* TODO: does this work with zero divisors as well? Probably not. *)
lemma reflect_poly_mult:
"reflect_poly (p * q) =
reflect_poly p * reflect_poly (q :: _ :: {comm_semiring_0,semiring_no_zero_divisors} poly)"
proof (cases "p = 0 ∨ q = 0")
case False
hence [simp]: "p ≠ 0" "q ≠ 0" by auto
show ?thesis
proof (rule poly_eqI)
fix i :: nat
show "coeff (reflect_poly (p * q)) i = coeff (reflect_poly p * reflect_poly q) i"
proof (cases "i ≤ degree (p * q)")
case True
def A ≡ "{..i} ∩ {i - degree q..degree p}"
def B ≡ "{..degree p} ∩ {degree p - i..degree (p*q) - i}"
let ?f = "λj. degree p - j"

from True have "coeff (reflect_poly (p * q)) i = coeff (p * q) (degree (p * q) - i)"
also have "… = (∑j≤degree (p * q) - i. coeff p j * coeff q (degree (p * q) - i - j))"
unfolding coeff_mult by simp
also have "… = (∑j∈B. coeff p j * coeff q (degree (p * q) - i - j))"
by (intro sum.mono_neutral_right) (auto simp: B_def degree_mult_eq not_le coeff_eq_0)
also from True have "… = (∑j∈A. coeff p (degree p - j) * coeff q (degree q - (i - j)))"
by (intro sum.reindex_bij_witness[of _ ?f ?f])
(auto simp: A_def B_def degree_mult_eq add_ac)
also have "… = (∑j≤i. if j ∈ {i - degree q..degree p} then
coeff p (degree p - j) * coeff q (degree q - (i - j)) else 0)"
by (subst sum.inter_restrict [symmetric]) (simp_all add: A_def)
also have "… = coeff (reflect_poly p * reflect_poly q) i"
by (fastforce simp: coeff_mult coeff_reflect_poly intro!: sum.cong)
finally show ?thesis .
qed (auto simp: coeff_mult coeff_reflect_poly coeff_eq_0 degree_mult_eq intro!: sum.neutral)
qed
qed auto

lemma reflect_poly_smult:
"reflect_poly (Polynomial.smult (c::'a::{comm_semiring_0,semiring_no_zero_divisors}) p) =
Polynomial.smult c (reflect_poly p)"
using reflect_poly_mult[of "[:c:]" p] by simp

lemma reflect_poly_power:
"reflect_poly (p ^ n :: 'a :: {comm_semiring_1,semiring_no_zero_divisors} poly) =
reflect_poly p ^ n"
by (induction n) (simp_all add: reflect_poly_mult)

lemma reflect_poly_prod:
"reflect_poly (prod (f :: _ ⇒ _ :: {comm_semiring_0,semiring_no_zero_divisors} poly) A) =
prod (λx. reflect_poly (f x)) A"
by (cases "finite A", induction rule: finite_induct) (simp_all add: reflect_poly_mult)

lemma reflect_poly_prod_list:
"reflect_poly (prod_list (xs :: _ :: {comm_semiring_0,semiring_no_zero_divisors} poly list)) =
prod_list (map reflect_poly xs)"
by (induction xs) (simp_all add: reflect_poly_mult)

lemma reflect_poly_Poly_nz:
"xs ≠ [] ⟹ last xs ≠ 0 ⟹ reflect_poly (Poly xs) = Poly (rev xs)"
unfolding reflect_poly_def coeffs_Poly by simp

lemmas reflect_poly_simps =
reflect_poly_0 reflect_poly_1 reflect_poly_const reflect_poly_smult reflect_poly_mult
reflect_poly_power reflect_poly_prod reflect_poly_prod_list

subsection ‹Derivatives of univariate polynomials›

function pderiv :: "('a :: {comm_semiring_1,semiring_no_zero_divisors}) poly ⇒ 'a poly"
where
"pderiv (pCons a p) = (if p = 0 then 0 else p + pCons 0 (pderiv p))"
by (auto intro: pCons_cases)

termination pderiv
by (relation "measure degree") simp_all

declare pderiv.simps[simp del]

lemma pderiv_0 [simp]:
"pderiv 0 = 0"
using pderiv.simps [of 0 0] by simp

lemma pderiv_pCons:
"pderiv (pCons a p) = p + pCons 0 (pderiv p)"

lemma pderiv_1 [simp]: "pderiv 1 = 0"
unfolding one_poly_def by (simp add: pderiv_pCons)

lemma pderiv_of_nat  [simp]: "pderiv (of_nat n) = 0"
and pderiv_numeral [simp]: "pderiv (numeral m) = 0"
by (simp_all add: of_nat_poly numeral_poly pderiv_pCons)

lemma coeff_pderiv: "coeff (pderiv p) n = of_nat (Suc n) * coeff p (Suc n)"
by (induct p arbitrary: n)
(auto simp add: pderiv_pCons coeff_pCons algebra_simps split: nat.split)

fun pderiv_coeffs_code
:: "('a :: {comm_semiring_1,semiring_no_zero_divisors}) ⇒ 'a list ⇒ 'a list" where
"pderiv_coeffs_code f (x # xs) = cCons (f * x) (pderiv_coeffs_code (f+1) xs)"
| "pderiv_coeffs_code f [] = []"

definition pderiv_coeffs ::
"'a :: {comm_semiring_1,semiring_no_zero_divisors} list ⇒ 'a list" where
"pderiv_coeffs xs = pderiv_coeffs_code 1 (tl xs)"

(* Efficient code for pderiv contributed by RenÃ© Thiemann and Akihisa Yamada *)
lemma pderiv_coeffs_code:
"nth_default 0 (pderiv_coeffs_code f xs) n = (f + of_nat n) * (nth_default 0 xs n)"
proof (induct xs arbitrary: f n)
case (Cons x xs f n)
show ?case
proof (cases n)
case 0
thus ?thesis by (cases "pderiv_coeffs_code (f + 1) xs = [] ∧ f * x = 0", auto simp: cCons_def)
next
case (Suc m) note n = this
show ?thesis
proof (cases "pderiv_coeffs_code (f + 1) xs = [] ∧ f * x = 0")
case False
hence "nth_default 0 (pderiv_coeffs_code f (x # xs)) n =
nth_default 0 (pderiv_coeffs_code (f + 1) xs) m"
by (auto simp: cCons_def n)
also have "… = (f + of_nat n) * (nth_default 0 xs m)"
finally show ?thesis by (simp add: n)
next
case True
{
fix g
have "pderiv_coeffs_code g xs = [] ⟹ g + of_nat m = 0 ∨ nth_default 0 xs m = 0"
proof (induct xs arbitrary: g m)
case (Cons x xs g)
from Cons(2) have empty: "pderiv_coeffs_code (g + 1) xs = []"
and g: "(g = 0 ∨ x = 0)"
by (auto simp: cCons_def split: if_splits)
note IH = Cons(1)[OF empty]
from IH[of m] IH[of "m - 1"] g
show ?case by (cases m, auto simp: field_simps)
qed simp
} note empty = this
from True have "nth_default 0 (pderiv_coeffs_code f (x # xs)) n = 0"
by (auto simp: cCons_def n)
moreover have "(f + of_nat n) * nth_default 0 (x # xs) n = 0" using True
by (simp add: n, insert empty[of "f+1"], auto simp: field_simps)
ultimately show ?thesis by simp
qed
qed
qed simp

lemma map_upt_Suc: "map f [0 ..< Suc n] = f 0 # map (λ i. f (Suc i)) [0 ..< n]"
by (induct n arbitrary: f, auto)

lemma coeffs_pderiv_code [code abstract]:
"coeffs (pderiv p) = pderiv_coeffs (coeffs p)" unfolding pderiv_coeffs_def
proof (rule coeffs_eqI, unfold pderiv_coeffs_code coeff_pderiv, goal_cases)
case (1 n)
have id: "coeff p (Suc n) = nth_default 0 (map (λi. coeff p (Suc i)) [0..<degree p]) n"
by (cases "n < degree p", auto simp: nth_default_def coeff_eq_0)
show ?case unfolding coeffs_def map_upt_Suc by (auto simp: id)
next
case 2
obtain n xs where id: "tl (coeffs p) = xs" "(1 :: 'a) = n" by auto
from 2 show ?case
unfolding id by (induct xs arbitrary: n, auto simp: cCons_def)
qed

context
assumes "SORT_CONSTRAINT('a::{comm_semiring_1,semiring_no_zero_divisors, semiring_char_0})"
begin

lemma pderiv_eq_0_iff:
"pderiv (p :: 'a poly) = 0 ⟷ degree p = 0"
apply (rule iffI)
apply (cases p, simp)
apply (simp add: poly_eq_iff coeff_pderiv del: of_nat_Suc)
apply (simp add: poly_eq_iff coeff_pderiv coeff_eq_0)
done

lemma degree_pderiv: "degree (pderiv (p :: 'a poly)) = degree p - 1"
apply (rule order_antisym [OF degree_le])
apply (cases "degree p", simp)
apply (rule le_degree)
apply (simp add: coeff_pderiv del: of_nat_Suc)
done

lemma not_dvd_pderiv:
assumes "degree (p :: 'a poly) ≠ 0"
shows "¬ p dvd pderiv p"
proof
assume dvd: "p dvd pderiv p"
then obtain q where p: "pderiv p = p * q" unfolding dvd_def by auto
from dvd have le: "degree p ≤ degree (pderiv p)"
by (simp add: assms dvd_imp_degree_le pderiv_eq_0_iff)
from this[unfolded degree_pderiv] assms show False by auto
qed

lemma dvd_pderiv_iff [simp]: "(p :: 'a poly) dvd pderiv p ⟷ degree p = 0"
using not_dvd_pderiv[of p] by (auto simp: pderiv_eq_0_iff [symmetric])

end

lemma pderiv_singleton [simp]: "pderiv [:a:] = 0"

lemma pderiv_add: "pderiv (p + q) = pderiv p + pderiv q"
by (rule poly_eqI, simp add: coeff_pderiv algebra_simps)

lemma pderiv_minus: "pderiv (- p :: 'a :: idom poly) = - pderiv p"
by (rule poly_eqI, simp add: coeff_pderiv algebra_simps)

lemma pderiv_diff: "pderiv ((p :: _ :: idom poly) - q) = pderiv p - pderiv q"
by (rule poly_eqI, simp add: coeff_pderiv algebra_simps)

lemma pderiv_smult: "pderiv (smult a p) = smult a (pderiv p)"
by (rule poly_eqI, simp add: coeff_pderiv algebra_simps)

lemma pderiv_mult: "pderiv (p * q) = p * pderiv q + q * pderiv p"
by (induct p) (auto simp: pderiv_add pderiv_smult pderiv_pCons algebra_simps)

lemma pderiv_power_Suc:
"pderiv (p ^ Suc n) = smult (of_nat (Suc n)) (p ^ n) * pderiv p"
apply (induct n)
apply simp
apply (subst power_Suc)
apply (subst pderiv_mult)
apply (erule ssubst)
apply (simp only: of_nat_Suc smult_add_left smult_1_left)
done

lemma pderiv_prod: "pderiv (prod f (as)) =
(∑a ∈ as. prod f (as - {a}) * pderiv (f a))"
proof (induct as rule: infinite_finite_induct)
case (insert a as)
hence id: "prod f (insert a as) = f a * prod f as"
"⋀ g. sum g (insert a as) = g a + sum g as"
"insert a as - {a} = as"
by auto
{
fix b
assume "b ∈ as"
hence id2: "insert a as - {b} = insert a (as - {b})" using ‹a ∉ as› by auto
have "prod f (insert a as - {b}) = f a * prod f (as - {b})"
unfolding id2
by (subst prod.insert, insert insert, auto)
} note id2 = this
show ?case
unfolding id pderiv_mult insert(3) sum_distrib_left
by (auto simp add: ac_simps id2 intro!: sum.cong)
qed auto

lemma DERIV_pow2: "DERIV (%x. x ^ Suc n) x :> real (Suc n) * (x ^ n)"
by (rule DERIV_cong, rule DERIV_pow, simp)
declare DERIV_pow2 [simp] DERIV_pow [simp]

lemma DERIV_add_const: "DERIV f x :> D ==>  DERIV (%x. a + f x :: 'a::real_normed_field) x :> D"
by (rule DERIV_cong, rule DERIV_add, auto)

lemma poly_DERIV [simp]: "DERIV (%x. poly p x) x :> poly (pderiv p) x"
by (induct p, auto intro!: derivative_eq_intros simp add: pderiv_pCons)

lemma continuous_on_poly [continuous_intros]:
fixes p :: "'a :: {real_normed_field} poly"
assumes "continuous_on A f"
shows   "continuous_on A (λx. poly p (f x))"
proof -
have "continuous_on A (λx. (∑i≤degree p. (f x) ^ i * coeff p i))"
by (intro continuous_intros assms)
also have "… = (λx. poly p (f x))" by (intro ext) (simp add: poly_altdef mult_ac)
finally show ?thesis .
qed

text‹Consequences of the derivative theorem above›

lemma poly_differentiable[simp]: "(%x. poly p x) differentiable (at x::real filter)"
apply (blast intro: poly_DERIV)
done

lemma poly_isCont[simp]: "isCont (%x. poly p x) (x::real)"
by (rule poly_DERIV [THEN DERIV_isCont])

lemma poly_IVT_pos: "[| a < b; poly p (a::real) < 0; 0 < poly p b |]
==> ∃x. a < x & x < b & (poly p x = 0)"
using IVT_objl [of "poly p" a 0 b]

lemma poly_IVT_neg: "[| (a::real) < b; 0 < poly p a; poly p b < 0 |]
==> ∃x. a < x & x < b & (poly p x = 0)"
by (insert poly_IVT_pos [where p = "- p" ]) simp

lemma poly_IVT:
fixes p::"real poly"
assumes "a<b" and "poly p a * poly p b < 0"
shows "∃x>a. x < b ∧ poly p x = 0"
by (metis assms(1) assms(2) less_not_sym mult_less_0_iff poly_IVT_neg poly_IVT_pos)

lemma poly_MVT: "(a::real) < b ==>
∃x. a < x & x < b & (poly p b - poly p a = (b - a) * poly (pderiv p) x)"
using MVT [of a b "poly p"]
apply auto
apply (rule_tac x = z in exI)
apply (auto simp add: mult_left_cancel poly_DERIV [THEN DERIV_unique])
done

lemma poly_MVT':
assumes "{min a b..max a b} ⊆ A"
shows   "∃x∈A. poly p b - poly p a = (b - a) * poly (pderiv p) (x::real)"
proof (cases a b rule: linorder_cases)
case less
from poly_MVT[OF less, of p] guess x by (elim exE conjE)
thus ?thesis by (intro bexI[of _ x]) (auto intro!: subsetD[OF assms])

next
case greater
from poly_MVT[OF greater, of p] guess x by (elim exE conjE)
thus ?thesis by (intro bexI[of _ x]) (auto simp: algebra_simps intro!: subsetD[OF assms])
qed (insert assms, auto)

lemma poly_pinfty_gt_lc:
fixes p :: "real poly"
shows "∃ n. ∀ x ≥ n. poly p x ≥ lead_coeff p"
using assms
proof (induct p)
case 0
then show ?case by auto
next
case (pCons a p)
from this(1) consider "a ≠ 0" "p = 0" | "p ≠ 0" by auto
then show ?case
proof cases
case 1
then show ?thesis by auto
next
case 2
with pCons obtain n1 where gte_lcoeff: "∀x≥n1. lead_coeff p ≤ poly p x"
by auto
from pCons(3) ‹p ≠ 0› have gt_0: "lead_coeff p > 0" by auto
define n where "n = max n1 (1 + ¦a¦ / lead_coeff p)"
have "lead_coeff (pCons a p) ≤ poly (pCons a p) x" if "n ≤ x" for x
proof -
from gte_lcoeff that have "lead_coeff p ≤ poly p x"
by (auto simp: n_def)
with gt_0 have "¦a¦ / lead_coeff p ≥ ¦a¦ / poly p x" and "poly p x > 0"
by (auto intro: frac_le)
with ‹n≤x›[unfolded n_def] have "x ≥ 1 + ¦a¦ / poly p x"
by auto
with ‹lead_coeff p ≤ poly p x› ‹poly p x > 0› ‹p ≠ 0›
show "lead_coeff (pCons a p) ≤ poly (pCons a p) x"
by (auto simp: field_simps)
qed
then show ?thesis by blast
qed
qed

subsection ‹Algebraic numbers›

text ‹
Algebraic numbers can be defined in two equivalent ways: all real numbers that are
roots of rational polynomials or of integer polynomials. The Algebraic-Numbers AFP entry
uses the rational definition, but we need the integer definition.

The equivalence is obvious since any rational polynomial can be multiplied with the
LCM of its coefficients, yielding an integer polynomial with the same roots.
›
subsection ‹Algebraic numbers›

definition algebraic :: "'a :: field_char_0 ⇒ bool" where
"algebraic x ⟷ (∃p. (∀i. coeff p i ∈ ℤ) ∧ p ≠ 0 ∧ poly p x = 0)"

lemma algebraicI:
assumes "⋀i. coeff p i ∈ ℤ" "p ≠ 0" "poly p x = 0"
shows   "algebraic x"
using assms unfolding algebraic_def by blast

lemma algebraicE:
assumes "algebraic x"
obtains p where "⋀i. coeff p i ∈ ℤ" "p ≠ 0" "poly p x = 0"
using assms unfolding algebraic_def by blast

lemma quotient_of_denom_pos': "snd (quotient_of x) > 0"
using quotient_of_denom_pos[OF surjective_pairing] .

lemma of_int_div_in_Ints:
"b dvd a ⟹ of_int a div of_int b ∈ (ℤ :: 'a :: ring_div set)"
proof (cases "of_int b = (0 :: 'a)")
assume "b dvd a" "of_int b ≠ (0::'a)"
then obtain c where "a = b * c" by (elim dvdE)
with ‹of_int b ≠ (0::'a)› show ?thesis by simp
qed auto

lemma of_int_divide_in_Ints:
"b dvd a ⟹ of_int a / of_int b ∈ (ℤ :: 'a :: field set)"
proof (cases "of_int b = (0 :: 'a)")
assume "b dvd a" "of_int b ≠ (0::'a)"
then obtain c where "a = b * c" by (elim dvdE)
with ‹of_int b ≠ (0::'a)› show ?thesis by simp
qed auto

lemma algebraic_altdef:
fixes p :: "'a :: field_char_0 poly"
shows "algebraic x ⟷ (∃p. (∀i. coeff p i ∈ ℚ) ∧ p ≠ 0 ∧ poly p x = 0)"
proof safe
fix p assume rat: "∀i. coeff p i ∈ ℚ" and root: "poly p x = 0" and nz: "p ≠ 0"
define cs where "cs = coeffs p"
from rat have "∀c∈range (coeff p). ∃c'. c = of_rat c'" unfolding Rats_def by blast
then obtain f where f: "coeff p i = of_rat (f (coeff p i))" for i
by (subst (asm) bchoice_iff) blast
define cs' where "cs' = map (quotient_of ∘ f) (coeffs p)"
define d where "d = Lcm (set (map snd cs'))"
define p' where "p' = smult (of_int d) p"

have "∀n. coeff p' n ∈ ℤ"
proof
fix n :: nat
show "coeff p' n ∈ ℤ"
proof (cases "n ≤ degree p")
case True
define c where "c = coeff p n"
define a where "a = fst (quotient_of (f (coeff p n)))"
define b where "b = snd (quotient_of (f (coeff p n)))"
have b_pos: "b > 0" unfolding b_def using quotient_of_denom_pos' by simp
have "coeff p' n = of_int d * coeff p n" by (simp add: p'_def)
also have "coeff p n = of_rat (of_int a / of_int b)" unfolding a_def b_def
by (subst quotient_of_div [of "f (coeff p n)", symmetric])
also have "of_int d * … = of_rat (of_int (a*d) / of_int b)"
also from nz True have "b ∈ snd ` set cs'" unfolding cs'_def
by (force simp: o_def b_def coeffs_def simp del: upt_Suc)
hence "b dvd (a * d)" unfolding d_def by simp
hence "of_int (a * d) / of_int b ∈ (ℤ :: rat set)"
by (rule of_int_divide_in_Ints)
hence "of_rat (of_int (a * d) / of_int b) ∈ ℤ" by (elim Ints_cases) auto
finally show ?thesis .
qed (auto simp: p'_def not_le coeff_eq_0)
qed

moreover have "set (map snd cs') ⊆ {0<..}"
unfolding cs'_def using quotient_of_denom_pos' by (auto simp: coeffs_def simp del: upt_Suc)
hence "d ≠ 0" unfolding d_def by (induction cs') simp_all
with nz have "p' ≠ 0" by (simp add: p'_def)
moreover from root have "poly p' x = 0" by (simp add: p'_def)
ultimately show "algebraic x" unfolding algebraic_def by blast
next
assume "algebraic x"
then obtain p where p: "coeff p i ∈ ℤ" "poly p x = 0" "p ≠ 0" for i
by (force simp: algebraic_def)
moreover have "coeff p i ∈ ℤ ⟹ coeff p i ∈ ℚ" for i by (elim Ints_cases) simp
ultimately show  "(∃p. (∀i. coeff p i ∈ ℚ) ∧ p ≠ 0 ∧ poly p x = 0)" by auto
qed

text‹Lemmas for Derivatives›

lemma order_unique_lemma:
fixes p :: "'a::idom poly"
assumes "[:-a, 1:] ^ n dvd p" "¬ [:-a, 1:] ^ Suc n dvd p"
shows "n = order a p"
unfolding Polynomial.order_def
apply (rule Least_equality [symmetric])
apply (fact assms)
apply (rule classical)
apply (erule notE)
unfolding not_less_eq_eq
using assms(1) apply (rule power_le_dvd)
apply assumption
done

lemma lemma_order_pderiv1:
"pderiv ([:- a, 1:] ^ Suc n * q) = [:- a, 1:] ^ Suc n * pderiv q +
smult (of_nat (Suc n)) (q * [:- a, 1:] ^ n)"
apply (simp only: pderiv_mult pderiv_power_Suc)
apply (simp del: power_Suc of_nat_Suc add: pderiv_pCons)
done

lemma lemma_order_pderiv:
fixes p :: "'a :: field_char_0 poly"
assumes n: "0 < n"
and pd: "pderiv p ≠ 0"
and pe: "p = [:- a, 1:] ^ n * q"
and nd: "~ [:- a, 1:] dvd q"
shows "n = Suc (order a (pderiv p))"
using n
proof -
have "pderiv ([:- a, 1:] ^ n * q) ≠ 0"
using assms by auto
obtain n' where "n = Suc n'" "0 < Suc n'" "pderiv ([:- a, 1:] ^ Suc n' * q) ≠ 0"
using assms by (cases n) auto
have *: "!!k l. k dvd k * pderiv q + smult (of_nat (Suc n')) l ⟹ k dvd l"
by (auto simp del: of_nat_Suc simp: dvd_add_right_iff dvd_smult_iff)
have "n' = order a (pderiv ([:- a, 1:] ^ Suc n' * q))"
proof (rule order_unique_lemma)
show "[:- a, 1:] ^ n' dvd pderiv ([:- a, 1:] ^ Suc n' * q)"
apply (subst lemma_order_pderiv1)
apply (metis dvdI dvd_mult2 power_Suc2)
apply (metis dvd_smult dvd_triv_right)
done
next
show "¬ [:- a, 1:] ^ Suc n' dvd pderiv ([:- a, 1:] ^ Suc n' * q)"
apply (subst lemma_order_pderiv1)
by (metis * nd dvd_mult_cancel_right power_not_zero pCons_eq_0_iff power_Suc zero_neq_one)
qed
then show ?thesis
by (metis ‹n = Suc n'› pe)
qed

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

lemma order_pderiv:
"⟦pderiv p ≠ 0; order a (p :: 'a :: field_char_0 poly) ≠ 0⟧ ⟹
(order a p = Suc (order a (pderiv p)))"
apply (case_tac "p = 0", simp)
apply (drule_tac a = a and p = p in order_decomp)
using neq0_conv
apply (blast intro: lemma_order_pderiv)
done

lemma order_mult: "p * q ≠ 0 ⟹ order a (p * q) = order a p + order a q"
proof -
define i where "i = order a p"
define j where "j = order a q"
define t where "t = [:-a, 1:]"
have t_dvd_iff: "⋀u. t dvd u ⟷ poly u a = 0"
unfolding t_def by (simp add: dvd_iff_poly_eq_0)
assume "p * q ≠ 0"
then show "order a (p * q) = i + j"
apply clarsimp
apply (drule order [where a=a and p=p, folded i_def t_def])
apply (drule order [where a=a and p=q, folded j_def t_def])
apply clarify
apply (erule dvdE)+
apply (rule order_unique_lemma [symmetric], fold t_def)
done
qed

lemma order_smult:
assumes "c ≠ 0"
shows "order x (smult c p) = order x p"
proof (cases "p = 0")
case False
have "smult c p = [:c:] * p" by simp
also from assms False have "order x … = order x [:c:] + order x p"
by (subst order_mult) simp_all
also from assms have "order x [:c:] = 0" by (intro order_0I) auto
finally show ?thesis by simp
qed simp

(* Next two lemmas contributed by Wenda Li *)
lemma order_1_eq_0 [simp]:"order x 1 = 0"
by (metis order_root poly_1 zero_neq_one)

lemma order_power_n_n: "order a ([:-a,1:]^n)=n"
proof (induct n) (*might be proved more concisely using nat_less_induct*)
case 0
thus ?case by (metis order_root poly_1 power_0 zero_neq_one)
next
case (Suc n)
have "order a ([:- a, 1:] ^ Suc n)=order a ([:- a, 1:] ^ n) + order a [:-a,1:]"
one_neq_zero order_mult pCons_eq_0_iff power_add power_eq_0_iff power_one_right)
moreover have "order a [:-a,1:]=1" unfolding order_def
proof (rule Least_equality,rule ccontr)
assume  "¬ ¬ [:- a, 1:] ^ Suc 1 dvd [:- a, 1:]"
hence "[:- a, 1:] ^ Suc 1 dvd [:- a, 1:]" by simp
hence "degree ([:- a, 1:] ^ Suc 1) ≤ degree ([:- a, 1:] )"
by (rule dvd_imp_degree_le,auto)
thus False by auto
next
fix y assume asm:"¬ [:- a, 1:] ^ Suc y dvd [:- a, 1:]"
show "1 ≤ y"
proof (rule ccontr)
assume "¬ 1 ≤ y"
hence "y=0" by auto
hence "[:- a, 1:] ^ Suc y dvd [:- a, 1:]" by auto
thus False using asm by auto
qed
qed
ultimately show ?case using Suc by auto
qed

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

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

text‹Now justify the standard squarefree decomposition, i.e. f / gcd(f,f').›

lemma order_divides: "[:-a, 1:] ^ n dvd p ⟷ p = 0 ∨ n ≤ order a p"
apply (cases "p = 0", auto)
apply (drule order_2 [where a=a and p=p])
apply (metis not_less_eq_eq power_le_dvd)
apply (erule power_le_dvd [OF order_1])
done

lemma monom_1_dvd_iff:
assumes "p ≠ 0"
shows   "monom 1 n dvd p ⟷ n ≤ Polynomial.order 0 p"
using assms order_divides[of 0 n p] by (simp add: monom_altdef)

lemma poly_squarefree_decomp_order:
assumes "pderiv (p :: 'a :: field_char_0 poly) ≠ 0"
and p: "p = q * d"
and p': "pderiv p = e * d"
and d: "d = r * p + s * pderiv p"
shows "order a q = (if order a p = 0 then 0 else 1)"
proof (rule classical)
assume 1: "order a q ≠ (if order a p = 0 then 0 else 1)"
from ‹pderiv p ≠ 0› have "p ≠ 0" by auto
with p have "order a p = order a q + order a d"
with 1 have "order a p ≠ 0" by (auto split: if_splits)
have "order a (pderiv p) = order a e + order a d"
using ‹pderiv p ≠ 0› ‹pderiv p = e * d› by (simp add: order_mult)
have "order a p = Suc (order a (pderiv p))"
using ‹pderiv p ≠ 0› ‹order a p ≠ 0› by (rule order_pderiv)
have "d ≠ 0" using ‹p ≠ 0› ‹p = q * d› by simp
have "([:-a, 1:] ^ (order a (pderiv p))) dvd d"
apply (rule dvd_mult)
apply (simp add: order_divides ‹p ≠ 0›
‹order a p = Suc (order a (pderiv p))›)
apply (rule dvd_mult)
done
then have "order a (pderiv p) ≤ order a d"
using ‹d ≠ 0› by (simp add: order_divides)
show ?thesis
using ‹order a p = order a q + order a d›
using ‹order a (pderiv p) = order a e + order a d›
using ‹order a p = Suc (order a (pderiv p))›
using ‹order a (pderiv p) ≤ order a d›
by auto
qed

lemma poly_squarefree_decomp_order2:
"⟦pderiv p ≠ (0 :: 'a :: field_char_0 poly);
p = q * d;
pderiv p = e * d;
d = r * p + s * pderiv p
⟧ ⟹ ∀a. order a q = (if order a p = 0 then 0 else 1)"
by (blast intro: poly_squarefree_decomp_order)

lemma order_pderiv2:
"⟦pderiv p ≠ 0; order a (p :: 'a :: field_char_0 poly) ≠ 0⟧
⟹ (order a (pderiv p) = n) = (order a p = Suc n)"
by (auto dest: order_pderiv)

definition
rsquarefree :: "'a::idom poly => bool" where
"rsquarefree p = (p ≠ 0 & (∀a. (order a p = 0) | (order a p = 1)))"

lemma pderiv_iszero: "pderiv p = 0 ⟹ ∃h. p = [:h :: 'a :: {semidom,semiring_char_0}:]"
by (cases p) (auto simp: pderiv_eq_0_iff split: if_splits)

lemma rsquarefree_roots:
fixes p :: "'a :: field_char_0 poly"
shows "rsquarefree p = (∀a. ¬(poly p a = 0 ∧ poly (pderiv p) a = 0))"
apply (case_tac "p = 0", simp, simp)
apply (case_tac "pderiv p = 0")
apply simp
apply (drule pderiv_iszero, clarsimp)
apply (metis coeff_0 coeff_pCons_0 degree_pCons_0 le0 le_antisym order_degree)
apply (force simp add: order_root order_pderiv2)
done

lemma poly_squarefree_decomp:
assumes "pderiv (p :: 'a :: field_char_0 poly) ≠ 0"
and "p = q * d"
and "pderiv p = e * d"
and "d = r * p + s * pderiv p"
shows "rsquarefree q & (∀a. (poly q a = 0) = (poly p a = 0))"
proof -
from ‹pderiv p ≠ 0› have "p ≠ 0" by auto
with ‹p = q * d› have "q ≠ 0" by simp
have "∀a. order a q = (if order a p = 0 then 0 else 1)"
using assms by (rule poly_squarefree_decomp_order2)
with ‹p ≠ 0› ‹q ≠ 0› show ?thesis
qed

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

lemma monom_1_dvd_iff':
"monom 1 n dvd p ⟷ (∀k<n. coeff p k = 0)"
proof
assume "monom 1 n dvd p"
then obtain r where r: "p = monom 1 n * r" by (elim dvdE)
thus "∀k<n. coeff p k = 0" by (simp add: coeff_mult)
next
assume zero: "(∀k<n. coeff p k = 0)"
define r where "r = Abs_poly (λk. coeff p (k + n))"
have "∀⇩∞k. coeff p (k + n) = 0"
by (subst cofinite_eq_sequentially, subst eventually_sequentially_seg,
subst cofinite_eq_sequentially [symmetric]) transfer
hence coeff_r [simp]: "coeff r k = coeff p (k + n)" for k unfolding r_def
by (subst poly.Abs_poly_inverse) simp_all
have "p = monom 1 n * r"
by (intro poly_eqI, subst coeff_monom_mult) (insert zero, simp_all)
thus "monom 1 n dvd p" by simp
qed

no_notation cCons (infixr "##" 65)

end
```