src/HOL/Library/Polynomial.thy
 author eberlm Tue, 05 Jan 2016 20:23:49 +0100 changeset 62067 0fd850943901 parent 62065 1546a042e87b child 62072 bf3d9f113474 permissions -rw-r--r--
Fixed sectioning in HOL/Library/Polynomial
```
(*  Title:      HOL/Library/Polynomial.thy
Author:     Brian Huffman
Author:     Clemens Ballarin
Author:     Florian Haftmann
*)

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

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

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

definition cCons :: "'a::zero \<Rightarrow> 'a list \<Rightarrow> 'a list"  (infixr "##" 65)
where
"x ## xs = (if xs = [] \<and> x = 0 then [] else x # xs)"

lemma cCons_0_Nil_eq [simp]:
"0 ## [] = []"

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 \<noteq> 0 \<Longrightarrow> x ## xs = x # xs"

lemma strip_while_not_0_Cons_eq [simp]:
"strip_while (\<lambda>x. x = 0) (x # xs) = x ## strip_while (\<lambda>x. x = 0) xs"
proof (cases "x = 0")
case False then show ?thesis by simp
next
case True show ?thesis
proof (induct xs rule: rev_induct)
case Nil with True show ?case by simp
next
case (snoc y ys) then show ?case
by (cases "y = 0") (simp_all add: append_Cons [symmetric] del: append_Cons)
qed
qed

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

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

typedef (overloaded) 'a poly = "{f :: nat \<Rightarrow> 'a::zero. \<forall>\<^sub>\<infinity> n. f n = 0}"
morphisms coeff Abs_poly by (auto intro!: ALL_MOST)

setup_lifting type_definition_poly

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

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

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

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

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

lemma coeff_eq_0:
assumes "degree p < n"
shows "coeff p n = 0"
proof -
have "\<exists>n. \<forall>i>n. coeff p i = 0"
using MOST_coeff_eq_0 by (simp add: MOST_nat)
then have "\<forall>i>degree p. coeff p i = 0"
unfolding degree_def by (rule LeastI_ex)
with assms show ?thesis by simp
qed

lemma le_degree: "coeff p n \<noteq> 0 \<Longrightarrow> n \<le> degree p"
by (erule contrapos_np, rule coeff_eq_0, simp)

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

lemma less_degree_imp: "n < degree p \<Longrightarrow> \<exists>i>n. coeff p i \<noteq> 0"
unfolding degree_def by (drule not_less_Least, simp)

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

instantiation poly :: (zero) zero
begin

lift_definition zero_poly :: "'a poly"
is "\<lambda>_. 0" by (rule MOST_I) simp

instance ..

end

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

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

assumes "p \<noteq> 0"
shows "coeff p (degree p) \<noteq> 0"
proof (cases "degree p")
case 0
from \<open>p \<noteq> 0\<close> have "\<exists>n. coeff p n \<noteq> 0"
then obtain n where "coeff p n \<noteq> 0" ..
hence "n \<le> degree p" by (rule le_degree)
with \<open>coeff p n \<noteq> 0\<close> and \<open>degree p = 0\<close>
show "coeff p (degree p) \<noteq> 0" by simp
next
case (Suc n)
from \<open>degree p = Suc n\<close> have "n < degree p" by simp
hence "\<exists>i>n. coeff p i \<noteq> 0" by (rule less_degree_imp)
then obtain i where "n < i" and "coeff p i \<noteq> 0" by fast
from \<open>degree p = Suc n\<close> and \<open>n < i\<close> have "degree p \<le> i" by simp
also from \<open>coeff p i \<noteq> 0\<close> have "i \<le> degree p" by (rule le_degree)
finally have "degree p = i" .
with \<open>coeff p i \<noteq> 0\<close> show "coeff p (degree p) \<noteq> 0" by simp
qed

"coeff p (degree p) = 0 \<longleftrightarrow> p = 0"

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

lift_definition pCons :: "'a::zero \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
is "\<lambda>a p. case_nat a (coeff p)"
by (rule MOST_SucD) (simp add: MOST_coeff_eq_0)

lemmas coeff_pCons = pCons.rep_eq

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

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

lemma degree_pCons_le:
"degree (pCons a p) \<le> Suc (degree p)"
by (rule degree_le) (simp add: coeff_eq_0 coeff_pCons split: nat.split)

lemma degree_pCons_eq:
"p \<noteq> 0 \<Longrightarrow> degree (pCons a p) = Suc (degree p)"
apply (rule order_antisym [OF degree_pCons_le])
apply (rule le_degree, simp)
done

lemma degree_pCons_0:
"degree (pCons a 0) = 0"
apply (rule order_antisym [OF _ le0])
apply (rule degree_le, simp add: coeff_pCons split: nat.split)
done

lemma degree_pCons_eq_if [simp]:
"degree (pCons a p) = (if p = 0 then 0 else Suc (degree p))"
apply (cases "p = 0", simp_all)
apply (rule order_antisym [OF _ le0])
apply (rule degree_le, simp add: coeff_pCons split: nat.split)
apply (rule order_antisym [OF degree_pCons_le])
apply (rule le_degree, simp)
done

lemma pCons_0_0 [simp]:
"pCons 0 0 = 0"
by (rule poly_eqI) (simp add: coeff_pCons split: nat.split)

lemma pCons_eq_iff [simp]:
"pCons a p = pCons b q \<longleftrightarrow> a = b \<and> p = q"
proof safe
assume "pCons a p = pCons b q"
then have "coeff (pCons a p) 0 = coeff (pCons b q) 0" by simp
then show "a = b" by simp
next
assume "pCons a p = pCons b q"
then have "\<forall>n. coeff (pCons a p) (Suc n) =
coeff (pCons b q) (Suc n)" by simp
then show "p = q" by (simp add: poly_eq_iff)
qed

lemma pCons_eq_0_iff [simp]:
"pCons a p = 0 \<longleftrightarrow> a = 0 \<and> p = 0"
using pCons_eq_iff [of a p 0 0] by simp

lemma pCons_cases [cases type: poly]:
obtains (pCons) a q where "p = pCons a q"
proof
show "p = pCons (coeff p 0) (Abs_poly (\<lambda>n. coeff p (Suc n)))"
by transfer
(simp_all add: MOST_inj[where f=Suc and P="\<lambda>n. p n = 0" for p] fun_eq_iff Abs_poly_inverse
split: nat.split)
qed

lemma pCons_induct [case_names 0 pCons, induct type: poly]:
assumes zero: "P 0"
assumes pCons: "\<And>a p. a \<noteq> 0 \<or> p \<noteq> 0 \<Longrightarrow> P p \<Longrightarrow> P (pCons a p)"
shows "P p"
proof (induct p rule: measure_induct_rule [where f=degree])
case (less p)
obtain a q where "p = pCons a q" by (rule pCons_cases)
have "P q"
proof (cases "q = 0")
case True
then show "P q" by (simp add: zero)
next
case False
then have "degree (pCons a q) = Suc (degree q)"
by (rule degree_pCons_eq)
then have "degree q < degree p"
using \<open>p = pCons a q\<close> by simp
then show "P q"
by (rule less.hyps)
qed
have "P (pCons a q)"
proof (cases "a \<noteq> 0 \<or> q \<noteq> 0")
case True
with \<open>P q\<close> show ?thesis by (auto intro: pCons)
next
case False
with zero show ?thesis by simp
qed
then show ?case
using \<open>p = pCons a q\<close> by simp
qed

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

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

syntax
"_poly" :: "args \<Rightarrow> 'a poly"  ("[:(_):]")

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

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

primrec Poly :: "'a::zero list \<Rightarrow> 'a poly"
where
[code_post]: "Poly [] = 0"
| [code_post]: "Poly (a # as) = pCons a (Poly as)"

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

lemma Poly_eq_0:
"Poly as = 0 \<longleftrightarrow> (\<exists>n. as = replicate n 0)"
by (induct as) (auto simp add: Cons_replicate_eq)

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

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

lemma coeffs_eq_Nil [simp]:
"coeffs p = [] \<longleftrightarrow> p = 0"

lemma not_0_coeffs_not_Nil:
"p \<noteq> 0 \<Longrightarrow> coeffs p \<noteq> []"
by simp

lemma coeffs_0_eq_Nil [simp]:
"coeffs 0 = []"
by simp

lemma coeffs_pCons_eq_cCons [simp]:
"coeffs (pCons a p) = a ## coeffs p"
proof -
{ fix ms :: "nat list" and f :: "nat \<Rightarrow> 'a" and x :: "'a"
assume "\<forall>m\<in>set ms. m > 0"
then have "map (case_nat x f) ms = map f (map (\<lambda>n. n - 1) ms)"
by (induct ms) (auto split: nat.split)
}
note * = this
show ?thesis
by (simp add: coeffs_def * upt_conv_Cons coeff_pCons map_decr_upt del: upt_Suc)
qed

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

lemma coeffs_nth:
assumes "p \<noteq> 0" "n \<le> degree p"
shows   "coeffs p ! n = coeff p n"
using assms unfolding coeffs_def by (auto simp del: upt_Suc)

lemma not_0_cCons_eq [simp]:
"p \<noteq> 0 \<Longrightarrow> a ## coeffs p = a # coeffs p"

lemma Poly_coeffs [simp, code abstype]:
"Poly (coeffs p) = p"
by (induct p) auto

lemma coeffs_Poly [simp]:
"coeffs (Poly as) = strip_while (HOL.eq 0) as"
proof (induct as)
case Nil then show ?case by simp
next
case (Cons a as)
have "(\<forall>n. as \<noteq> replicate n 0) \<longleftrightarrow> (\<exists>a\<in>set as. a \<noteq> 0)"
using replicate_length_same [of as 0] by (auto dest: sym [of _ as])
with Cons show ?case by auto
qed

lemma last_coeffs_not_0:
"p \<noteq> 0 \<Longrightarrow> last (coeffs p) \<noteq> 0"
by (induct p) (auto simp add: cCons_def)

lemma strip_while_coeffs [simp]:
"strip_while (HOL.eq 0) (coeffs p) = coeffs p"
by (cases "p = 0") (auto dest: last_coeffs_not_0 intro: strip_while_not_last)

lemma coeffs_eq_iff:
"p = q \<longleftrightarrow> coeffs p = coeffs q" (is "?P \<longleftrightarrow> ?Q")
proof
assume ?P then show ?Q by simp
next
assume ?Q
then have "Poly (coeffs p) = Poly (coeffs q)" by simp
then show ?P by simp
qed

lemma coeff_Poly_eq:
"coeff (Poly xs) n = nth_default 0 xs n"
apply (induct xs arbitrary: n) apply simp_all
by (metis nat.case not0_implies_Suc nth_default_Cons_0 nth_default_Cons_Suc pCons.rep_eq)

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

lemma [code]:
"coeff p = nth_default 0 (coeffs p)"

lemma coeffs_eqI:
assumes coeff: "\<And>n. coeff p n = nth_default 0 xs n"
assumes zero: "xs \<noteq> [] \<Longrightarrow> last xs \<noteq> 0"
shows "coeffs p = xs"
proof -
from coeff have "p = Poly xs" by (simp add: poly_eq_iff coeff_Poly_eq)
with zero show ?thesis by simp (cases xs, simp_all)
qed

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

lemma length_coeffs_degree:
"p \<noteq> 0 \<Longrightarrow> length (coeffs p) = Suc (degree p)"
by (induct p) (auto simp add: cCons_def)

lemma [code abstract]:
"coeffs 0 = []"
by (fact coeffs_0_eq_Nil)

lemma [code abstract]:
"coeffs (pCons a p) = a ## coeffs p"
by (fact coeffs_pCons_eq_cCons)

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

definition
[code]: "HOL.equal (p::'a poly) q \<longleftrightarrow> HOL.equal (coeffs p) (coeffs q)"

instance
by standard (simp add: equal equal_poly_def coeffs_eq_iff)

end

lemma [code nbe]: "HOL.equal (p :: _ poly) p \<longleftrightarrow> True"
by (fact equal_refl)

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

lemma is_zero_null [code_abbrev]:
"is_zero p \<longleftrightarrow> p = 0"

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

definition fold_coeffs :: "('a::zero \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a poly \<Rightarrow> 'b \<Rightarrow> 'b"
where
"fold_coeffs f p = foldr f (coeffs p)"

lemma fold_coeffs_0_eq [simp]:
"fold_coeffs f 0 = id"

lemma fold_coeffs_pCons_eq [simp]:
"f 0 = id \<Longrightarrow> fold_coeffs f (pCons a p) = f a \<circ> fold_coeffs f p"
by (simp add: fold_coeffs_def cCons_def fun_eq_iff)

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

lemma fold_coeffs_pCons_coeff_not_0_eq [simp]:
"a \<noteq> 0 \<Longrightarrow> fold_coeffs f (pCons a p) = f a \<circ> fold_coeffs f p"

lemma fold_coeffs_pCons_not_0_0_eq [simp]:
"p \<noteq> 0 \<Longrightarrow> fold_coeffs f (pCons a p) = f a \<circ> fold_coeffs f p"

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

definition poly :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a"
where
"poly p = fold_coeffs (\<lambda>a f x. a + x * f x) p (\<lambda>x. 0)" \<comment> \<open>The Horner Schema\<close>

lemma poly_0 [simp]:
"poly 0 x = 0"

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

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

subsection \<open>Monomials\<close>

lift_definition monom :: "'a \<Rightarrow> nat \<Rightarrow> 'a::zero poly"
is "\<lambda>a m n. if m = n then a else 0"

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

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

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

lemma monom_eq_0 [simp]: "monom 0 n = 0"
by (rule poly_eqI) simp

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

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

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

lemma degree_monom_eq: "a \<noteq> 0 \<Longrightarrow> degree (monom a n) = n"
apply (rule order_antisym [OF degree_monom_le])
apply (rule le_degree, simp)
done

lemma coeffs_monom [code abstract]:
"coeffs (monom a n) = (if a = 0 then [] else replicate n 0 @ [a])"
by (induct n) (simp_all add: monom_0 monom_Suc)

lemma fold_coeffs_monom [simp]:
"a \<noteq> 0 \<Longrightarrow> fold_coeffs f (monom a n) = f 0 ^^ n \<circ> f a"
by (simp add: fold_coeffs_def coeffs_monom fun_eq_iff)

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

begin

lift_definition plus_poly :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
is "\<lambda>p q n. coeff p n + coeff q n"
proof -
fix q p :: "'a poly"
show "\<forall>\<^sub>\<infinity>n. coeff p n + coeff q n = 0"
using MOST_coeff_eq_0[of p] MOST_coeff_eq_0[of q] by eventually_elim simp
qed

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

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 \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
is "\<lambda>p q n. coeff p n - coeff q n"
proof -
fix q p :: "'a poly"
show "\<forall>\<^sub>\<infinity>n. coeff p n - coeff q n = 0"
using MOST_coeff_eq_0[of p] MOST_coeff_eq_0[of q] by eventually_elim simp
qed

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

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 \<Rightarrow> 'a poly"
is "\<lambda>p n. - coeff p n"
proof -
fix p :: "'a poly"
show "\<forall>\<^sub>\<infinity>n. - coeff p n = 0"
using MOST_coeff_eq_0 by simp
qed

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

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) \<le> max (degree p) (degree q)"
by (rule degree_le, auto simp add: coeff_eq_0)

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

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

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

"degree q < degree p \<Longrightarrow> 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) \<le> max (degree p) (degree q)"
using degree_add_le [where p=p and q="-q"]
by simp

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

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

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

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

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

lemma coeff_setsum: "coeff (\<Sum>x\<in>A. p x) i = (\<Sum>x\<in>A. coeff (p x) i)"
by (cases "finite A", induct set: finite, simp_all)

lemma monom_setsum: "monom (\<Sum>x\<in>A. a x) n = (\<Sum>x\<in>A. monom (a x) n)"
by (rule poly_eqI) (simp add: coeff_setsum)

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

lemma coeffs_plus_eq_plus_coeffs [code abstract]:
"coeffs (p + q) = plus_coeffs (coeffs p) (coeffs q)"
proof -
{ fix xs ys :: "'a list" and n
have "nth_default 0 (plus_coeffs xs ys) n = nth_default 0 xs n + nth_default 0 ys n"
proof (induct xs ys arbitrary: n rule: plus_coeffs.induct)
case (3 x xs y ys n)
then show ?case by (cases n) (auto simp add: cCons_def)
qed simp_all }
note * = this
{ fix xs ys :: "'a list"
assume "xs \<noteq> [] \<Longrightarrow> last xs \<noteq> 0" and "ys \<noteq> [] \<Longrightarrow> last ys \<noteq> 0"
moreover assume "plus_coeffs xs ys \<noteq> []"
ultimately have "last (plus_coeffs xs ys) \<noteq> 0"
proof (induct xs ys rule: plus_coeffs.induct)
case (3 x xs y ys) then show ?case by (auto simp add: cCons_def) metis
qed simp_all }
note ** = this
show ?thesis
apply (rule coeffs_eqI)
apply (rule **)
apply (auto dest: last_coeffs_not_0)
done
qed

lemma coeffs_uminus [code abstract]:
"coeffs (- p) = map (\<lambda>a. - a) (coeffs p)"
by (rule coeffs_eqI)
(simp_all add: not_0_coeffs_not_Nil last_map last_coeffs_not_0 nth_default_map_eq nth_default_coeffs_eq)

lemma [code]:
fixes p q :: "'a::ab_group_add poly"
shows "p - q = p + - q"

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

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

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

lemma poly_setsum: "poly (\<Sum>k\<in>A. p k) x = (\<Sum>k\<in>A. poly (p k) x)"
by (induct A rule: infinite_finite_induct) simp_all

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

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

lift_definition smult :: "'a::comm_semiring_0 \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
is "\<lambda>a p n. a * coeff p n"
proof -
fix a :: 'a and p :: "'a poly" show "\<forall>\<^sub>\<infinity> i. a * coeff p i = 0"
using MOST_coeff_eq_0[of p] by eventually_elim simp
qed

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

lemma degree_smult_le: "degree (smult a p) \<le> degree p"
by (rule degree_le, simp add: coeff_eq_0)

lemma smult_smult [simp]: "smult a (smult b p) = smult (a * b) p"
by (rule poly_eqI, simp add: mult.assoc)

lemma smult_0_right [simp]: "smult a 0 = 0"
by (rule poly_eqI, simp)

lemma smult_0_left [simp]: "smult 0 p = 0"
by (rule poly_eqI, simp)

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

"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 degree_smult_eq [simp]:
fixes a :: "'a::idom"
shows "degree (smult a p) = (if a = 0 then 0 else degree p)"
by (cases "a = 0", simp, simp add: degree_def)

lemma smult_eq_0_iff [simp]:
fixes a :: "'a::idom"
shows "smult a p = 0 \<longleftrightarrow> a = 0 \<or> p = 0"

lemma coeffs_smult [code abstract]:
fixes p :: "'a::idom poly"
shows "coeffs (smult a p) = (if a = 0 then [] else map (Groups.times a) (coeffs p))"
by (rule coeffs_eqI)
(auto simp add: not_0_coeffs_not_Nil last_map last_coeffs_not_0 nth_default_map_eq nth_default_coeffs_eq)

instantiation poly :: (comm_semiring_0) comm_semiring_0
begin

definition
"p * q = fold_coeffs (\<lambda>a p. smult a q + pCons 0 p) p 0"

lemma mult_poly_0_left: "(0::'a poly) * q = 0"

lemma mult_pCons_left [simp]:
"pCons a p * q = smult a q + pCons 0 (p * q)"
by (cases "p = 0 \<and> a = 0") (auto simp add: times_poly_def)

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

lemma mult_pCons_right [simp]:
"p * pCons a q = smult a p + pCons 0 (p * q)"

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

instance poly :: (comm_semiring_0_cancel) comm_semiring_0_cancel ..

lemma coeff_mult:
"coeff (p * q) n = (\<Sum>i\<le>n. coeff p i * coeff q (n-i))"
proof (induct p arbitrary: n)
case 0 show ?case by simp
next
case (pCons a p n) thus ?case
by (cases n, simp, simp add: setsum_atMost_Suc_shift
del: setsum_atMost_Suc)
qed

lemma degree_mult_le: "degree (p * q) \<le> degree p + degree q"
apply (rule degree_le)
apply (induct p)
apply simp
apply (simp add: coeff_eq_0 coeff_pCons split: nat.split)
done

lemma mult_monom: "monom a m * monom b n = monom (a * b) (m + n)"

instantiation poly :: (comm_semiring_1) comm_semiring_1
begin

definition one_poly_def: "1 = pCons 1 0"

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

end

instance poly :: (comm_ring) comm_ring ..

instance poly :: (comm_ring_1) comm_ring_1 ..

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

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

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) \<le> degree p * n"
by (induct n) (auto intro: order_trans degree_mult_le)

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

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

lemma poly_1 [simp]:
"poly 1 x = 1"

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

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

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

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

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 \<Longrightarrow> p dvd smult a q"
proof -
assume "p dvd q"
then obtain k where "q = p * k" ..
then have "smult a q = p * smult a k" by simp
then show "p dvd smult a q" ..
qed

lemma dvd_smult_cancel:
fixes a :: "'a::field"
shows "p dvd smult a q \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> p dvd q"
by (drule dvd_smult [where a="inverse a"]) simp

lemma dvd_smult_iff:
fixes a :: "'a::field"
shows "a \<noteq> 0 \<Longrightarrow> p dvd smult a q \<longleftrightarrow> p dvd q"
by (safe elim!: dvd_smult dvd_smult_cancel)

lemma smult_dvd_cancel:
"smult a p dvd q \<Longrightarrow> p dvd q"
proof -
assume "smult a p dvd q"
then obtain k where "q = smult a p * k" ..
then have "q = p * smult a k" by simp
then show "p dvd q" ..
qed

lemma smult_dvd:
fixes a :: "'a::field"
shows "p dvd q \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> smult a p dvd q"
by (rule smult_dvd_cancel [where a="inverse a"]) simp

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

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

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

instance poly :: (idom) idom
proof
fix p q :: "'a poly"
assume "p \<noteq> 0" and "q \<noteq> 0"
have "coeff (p * q) (degree p + degree q) =
coeff p (degree p) * coeff q (degree q)"
by (rule coeff_mult_degree_sum)
also have "coeff p (degree p) * coeff q (degree q) \<noteq> 0"
using \<open>p \<noteq> 0\<close> and \<open>q \<noteq> 0\<close> by simp
finally have "\<exists>n. coeff (p * q) n \<noteq> 0" ..
thus "p * q \<noteq> 0" by (simp add: poly_eq_iff)
qed

lemma degree_mult_eq:
fixes p q :: "'a::idom poly"
shows "\<lbrakk>p \<noteq> 0; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree (p * q) = degree p + degree q"
apply (rule order_antisym [OF degree_mult_le le_degree])
done

lemma degree_mult_right_le:
fixes p q :: "'a::idom poly"
assumes "q \<noteq> 0"
shows "degree p \<le> degree (p * q)"
using assms by (cases "p = 0") (simp_all add: degree_mult_eq)

lemma coeff_degree_mult:
fixes p q :: "'a::idom poly"
shows "coeff (p * q) (degree (p * q)) =
coeff q (degree q) * coeff p (degree p)"
by (cases "p = 0 \<or> q = 0") (auto simp add: degree_mult_eq coeff_mult_degree_sum)

lemma dvd_imp_degree_le:
fixes p q :: "'a::idom poly"
shows "\<lbrakk>p dvd q; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree p \<le> degree q"
by (erule dvdE, simp add: degree_mult_eq)

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

definition pos_poly :: "'a::linordered_idom poly \<Rightarrow> bool"
where
"pos_poly p \<longleftrightarrow> 0 < coeff p (degree p)"

lemma pos_poly_pCons:
"pos_poly (pCons a p) \<longleftrightarrow> pos_poly p \<or> (p = 0 \<and> 0 < a)"
unfolding pos_poly_def by simp

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

lemma pos_poly_add: "\<lbrakk>pos_poly p; pos_poly q\<rbrakk> \<Longrightarrow> pos_poly (p + q)"
apply (induct p arbitrary: q, simp)
done

lemma pos_poly_mult: "\<lbrakk>pos_poly p; pos_poly q\<rbrakk> \<Longrightarrow> pos_poly (p * q)"
unfolding pos_poly_def
apply (subgoal_tac "p \<noteq> 0 \<and> q \<noteq> 0")
apply auto
done

lemma pos_poly_total: "p = 0 \<or> pos_poly p \<or> pos_poly (- p)"
by (induct p) (auto simp add: pos_poly_pCons)

lemma last_coeffs_eq_coeff_degree:
"p \<noteq> 0 \<Longrightarrow> last (coeffs p) = coeff p (degree p)"

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

instantiation poly :: (linordered_idom) linordered_idom
begin

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

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

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

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

instance
proof
fix x y z :: "'a poly"
show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
unfolding less_eq_poly_def less_poly_def
apply safe
apply simp
apply simp
done
show "x \<le> x"
unfolding less_eq_poly_def by simp
show "x \<le> y \<Longrightarrow> y \<le> z \<Longrightarrow> x \<le> z"
unfolding less_eq_poly_def
apply safe
done
show "x \<le> y \<Longrightarrow> y \<le> x \<Longrightarrow> x = y"
unfolding less_eq_poly_def
apply safe
apply simp
done
show "x \<le> y \<Longrightarrow> z + x \<le> z + y"
unfolding less_eq_poly_def
apply safe
done
show "x \<le> y \<or> y \<le> x"
unfolding less_eq_poly_def
using pos_poly_total [of "x - y"]
by auto
show "x < y \<Longrightarrow> 0 < z \<Longrightarrow> z * x < z * y"
unfolding less_poly_def
by (simp add: right_diff_distrib [symmetric] pos_poly_mult)
show "\<bar>x\<bar> = (if x < 0 then - x else x)"
by (rule abs_poly_def)
show "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
by (rule sgn_poly_def)
qed

end

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

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

text \<open>
Synthetic division is simply division by the linear polynomial @{term "x - c"}.
\<close>

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

definition synthetic_div :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly"
where
"synthetic_div p c = fst (synthetic_divmod p c)"

lemma synthetic_divmod_0 [simp]:
"synthetic_divmod 0 c = (0, 0)"

lemma synthetic_divmod_pCons [simp]:
"synthetic_divmod (pCons a p) c = (\<lambda>(q, r). (pCons r q, a + c * r)) (synthetic_divmod p c)"
by (cases "p = 0 \<and> a = 0") (auto simp add: synthetic_divmod_def)

lemma synthetic_div_0 [simp]:
"synthetic_div 0 c = 0"
unfolding synthetic_div_def by simp

lemma synthetic_div_unique_lemma: "smult c p = pCons a p \<Longrightarrow> p = 0"
by (induct p arbitrary: a) simp_all

lemma snd_synthetic_divmod:
"snd (synthetic_divmod p c) = poly p c"
by (induct p, simp, simp add: split_def)

lemma synthetic_div_pCons [simp]:
"synthetic_div (pCons a p) c = pCons (poly p c) (synthetic_div p c)"
unfolding synthetic_div_def

lemma synthetic_div_eq_0_iff:
"synthetic_div p c = 0 \<longleftrightarrow> degree p = 0"
by (induct p, simp, case_tac p, simp)

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

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

lemma synthetic_div_unique:
"p + smult c q = pCons r q \<Longrightarrow> r = poly p c \<and> q = synthetic_div p c"
apply (induct p arbitrary: q r)
apply (simp, frule synthetic_div_unique_lemma, simp)
apply (case_tac q, force)
done

lemma synthetic_div_correct':
fixes c :: "'a::comm_ring_1"
shows "[:-c, 1:] * synthetic_div p c + [:poly p c:] = p"
using synthetic_div_correct [of p c]

lemma poly_eq_0_iff_dvd:
fixes c :: "'a::idom"
shows "poly p c = 0 \<longleftrightarrow> [:-c, 1:] dvd p"
proof
assume "poly p c = 0"
with synthetic_div_correct' [of c p]
have "p = [:-c, 1:] * synthetic_div p c" by simp
then show "[:-c, 1:] dvd p" ..
next
assume "[:-c, 1:] dvd p"
then obtain k where "p = [:-c, 1:] * k" by (rule dvdE)
then show "poly p c = 0" by simp
qed

lemma dvd_iff_poly_eq_0:
fixes c :: "'a::idom"
shows "[:c, 1:] dvd p \<longleftrightarrow> poly p (-c) = 0"

lemma poly_roots_finite:
fixes p :: "'a::idom poly"
shows "p \<noteq> 0 \<Longrightarrow> finite {x. poly p x = 0}"
proof (induct n \<equiv> "degree p" arbitrary: p)
case (0 p)
then obtain a where "a \<noteq> 0" and "p = [:a:]"
by (cases p, simp split: if_splits)
then show "finite {x. poly p x = 0}" by simp
next
case (Suc n p)
show "finite {x. poly p x = 0}"
proof (cases "\<exists>x. poly p x = 0")
case False
then show "finite {x. poly p x = 0}" by simp
next
case True
then obtain a where "poly p a = 0" ..
then have "[:-a, 1:] dvd p" by (simp only: poly_eq_0_iff_dvd)
then obtain k where k: "p = [:-a, 1:] * k" ..
with \<open>p \<noteq> 0\<close> have "k \<noteq> 0" by auto
with k have "degree p = Suc (degree k)"
by (simp add: degree_mult_eq del: mult_pCons_left)
with \<open>Suc n = degree p\<close> have "n = degree k" by simp
then have "finite {x. poly k x = 0}" using \<open>k \<noteq> 0\<close> by (rule Suc.hyps)
then have "finite (insert a {x. poly k x = 0})" by simp
then show "finite {x. poly p x = 0}"
by (simp add: k Collect_disj_eq del: mult_pCons_left)
qed
qed

lemma poly_eq_poly_eq_iff:
fixes p q :: "'a::{idom,ring_char_0} poly"
shows "poly p = poly q \<longleftrightarrow> p = q" (is "?P \<longleftrightarrow> ?Q")
proof
assume ?Q then show ?P by simp
next
{ fix p :: "'a::{idom,ring_char_0} poly"
have "poly p = poly 0 \<longleftrightarrow> p = 0"
apply (cases "p = 0", simp_all)
apply (drule poly_roots_finite)
done
} note this [of "p - q"]
moreover assume ?P
ultimately show ?Q by auto
qed

lemma poly_all_0_iff_0:
fixes p :: "'a::{ring_char_0, idom} poly"
shows "(\<forall>x. poly p x = 0) \<longleftrightarrow> p = 0"
by (auto simp add: poly_eq_poly_eq_iff [symmetric])

subsection \<open>Long division of polynomials\<close>

definition pdivmod_rel :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> bool"
where
"pdivmod_rel x y q r \<longleftrightarrow>
x = q * y + r \<and> (if y = 0 then q = 0 else r = 0 \<or> degree r < degree y)"

lemma pdivmod_rel_0:
"pdivmod_rel 0 y 0 0"
unfolding pdivmod_rel_def by simp

lemma pdivmod_rel_by_0:
"pdivmod_rel x 0 0 x"
unfolding pdivmod_rel_def by simp

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

lemma pdivmod_rel_pCons:
assumes rel: "pdivmod_rel x y q r"
assumes y: "y \<noteq> 0"
assumes b: "b = coeff (pCons a r) (degree y) / coeff y (degree y)"
shows "pdivmod_rel (pCons a x) y (pCons b q) (pCons a r - smult b y)"
(is "pdivmod_rel ?x y ?q ?r")
proof -
have x: "x = q * y + r" and r: "r = 0 \<or> degree r < degree y"
using assms unfolding pdivmod_rel_def by simp_all

have 1: "?x = ?q * y + ?r"
using b x by simp

have 2: "?r = 0 \<or> degree ?r < degree y"
proof (rule eq_zero_or_degree_less)
show "degree ?r \<le> degree y"
proof (rule degree_diff_le)
show "degree (pCons a r) \<le> degree y"
using r by auto
show "degree (smult b y) \<le> degree y"
by (rule degree_smult_le)
qed
next
show "coeff ?r (degree y) = 0"
using \<open>y \<noteq> 0\<close> unfolding b by simp
qed

from 1 2 show ?thesis
unfolding pdivmod_rel_def
using \<open>y \<noteq> 0\<close> by simp
qed

lemma pdivmod_rel_exists: "\<exists>q r. pdivmod_rel x y q r"
apply (cases "y = 0")
apply (fast intro!: pdivmod_rel_by_0)
apply (induct x)
apply (fast intro!: pdivmod_rel_0)
apply (fast intro!: pdivmod_rel_pCons)
done

lemma pdivmod_rel_unique:
assumes 1: "pdivmod_rel x y q1 r1"
assumes 2: "pdivmod_rel x y q2 r2"
shows "q1 = q2 \<and> r1 = r2"
proof (cases "y = 0")
assume "y = 0" with assms show ?thesis
next
assume [simp]: "y \<noteq> 0"
from 1 have q1: "x = q1 * y + r1" and r1: "r1 = 0 \<or> degree r1 < degree y"
unfolding pdivmod_rel_def by simp_all
from 2 have q2: "x = q2 * y + r2" and r2: "r2 = 0 \<or> degree r2 < degree y"
unfolding pdivmod_rel_def by simp_all
from q1 q2 have q3: "(q1 - q2) * y = r2 - r1"
from r1 r2 have r3: "(r2 - r1) = 0 \<or> degree (r2 - r1) < degree y"
by (auto intro: degree_diff_less)

show "q1 = q2 \<and> r1 = r2"
proof (rule ccontr)
assume "\<not> (q1 = q2 \<and> r1 = r2)"
with q3 have "q1 \<noteq> q2" and "r1 \<noteq> r2" by auto
with r3 have "degree (r2 - r1) < degree y" by simp
also have "degree y \<le> degree (q1 - q2) + degree y" by simp
also have "\<dots> = degree ((q1 - q2) * y)"
using \<open>q1 \<noteq> q2\<close> by (simp add: degree_mult_eq)
also have "\<dots> = degree (r2 - r1)"
using q3 by simp
finally have "degree (r2 - r1) < degree (r2 - r1)" .
then show "False" by simp
qed
qed

lemma pdivmod_rel_0_iff: "pdivmod_rel 0 y q r \<longleftrightarrow> q = 0 \<and> r = 0"
by (auto dest: pdivmod_rel_unique intro: pdivmod_rel_0)

lemma pdivmod_rel_by_0_iff: "pdivmod_rel x 0 q r \<longleftrightarrow> q = 0 \<and> r = x"
by (auto dest: pdivmod_rel_unique intro: pdivmod_rel_by_0)

lemmas pdivmod_rel_unique_div = pdivmod_rel_unique [THEN conjunct1]

lemmas pdivmod_rel_unique_mod = pdivmod_rel_unique [THEN conjunct2]

instantiation poly :: (field) ring_div
begin

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

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

lemma div_poly_eq:
"pdivmod_rel x y q r \<Longrightarrow> x div y = q"
unfolding div_poly_def
by (fast elim: pdivmod_rel_unique_div)

lemma mod_poly_eq:
"pdivmod_rel x y q r \<Longrightarrow> x mod y = r"
unfolding mod_poly_def
by (fast elim: pdivmod_rel_unique_mod)

lemma pdivmod_rel:
"pdivmod_rel x y (x div y) (x mod y)"
proof -
from pdivmod_rel_exists
obtain q r where "pdivmod_rel x y q r" by fast
thus ?thesis
qed

instance
proof
fix x y :: "'a poly"
show "x div y * y + x mod y = x"
using pdivmod_rel [of x y]
next
fix x :: "'a poly"
have "pdivmod_rel x 0 0 x"
by (rule pdivmod_rel_by_0)
thus "x div 0 = 0"
by (rule div_poly_eq)
next
fix y :: "'a poly"
have "pdivmod_rel 0 y 0 0"
by (rule pdivmod_rel_0)
thus "0 div y = 0"
by (rule div_poly_eq)
next
fix x y z :: "'a poly"
assume "y \<noteq> 0"
hence "pdivmod_rel (x + z * y) y (z + x div y) (x mod y)"
using pdivmod_rel [of x y]
thus "(x + z * y) div y = z + x div y"
by (rule div_poly_eq)
next
fix x y z :: "'a poly"
assume "x \<noteq> 0"
show "(x * y) div (x * z) = y div z"
proof (cases "y \<noteq> 0 \<and> z \<noteq> 0")
have "\<And>x::'a poly. pdivmod_rel x 0 0 x"
by (rule pdivmod_rel_by_0)
then have [simp]: "\<And>x::'a poly. x div 0 = 0"
by (rule div_poly_eq)
have "\<And>x::'a poly. pdivmod_rel 0 x 0 0"
by (rule pdivmod_rel_0)
then have [simp]: "\<And>x::'a poly. 0 div x = 0"
by (rule div_poly_eq)
case False then show ?thesis by auto
next
case True then have "y \<noteq> 0" and "z \<noteq> 0" by auto
with \<open>x \<noteq> 0\<close>
have "\<And>q r. pdivmod_rel y z q r \<Longrightarrow> pdivmod_rel (x * y) (x * z) q (x * r)"
by (auto simp add: pdivmod_rel_def algebra_simps)
moreover from pdivmod_rel have "pdivmod_rel y z (y div z) (y mod z)" .
ultimately have "pdivmod_rel (x * y) (x * z) (y div z) (x * (y mod z))" .
then show ?thesis by (simp add: div_poly_eq)
qed
qed

end

lemma is_unit_monom_0:
fixes a :: "'a::field"
assumes "a \<noteq> 0"
shows "is_unit (monom a 0)"
proof
from assms show "1 = monom a 0 * monom (1 / a) 0"
qed

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

lemma is_unit_iff_degree:
assumes "p \<noteq> 0"
shows "is_unit p \<longleftrightarrow> degree p = 0" (is "?P \<longleftrightarrow> ?Q")
proof
assume ?Q
then obtain a where "p = [:a:]" by (rule degree_eq_zeroE)
with assms show ?P by (simp add: is_unit_triv)
next
assume ?P
then obtain q where "q \<noteq> 0" "p * q = 1" ..
then have "degree (p * q) = degree 1"
by simp
with \<open>p \<noteq> 0\<close> \<open>q \<noteq> 0\<close> have "degree p + degree q = 0"
then show ?Q by simp
qed

lemma is_unit_pCons_iff:
"is_unit (pCons a p) \<longleftrightarrow> p = 0 \<and> a \<noteq> 0" (is "?P \<longleftrightarrow> ?Q")
by (cases "p = 0") (auto simp add: is_unit_triv is_unit_iff_degree)

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

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

instantiation poly :: (field) normalization_semidom
begin

definition normalize_poly :: "'a poly \<Rightarrow> 'a poly"
where "normalize_poly p = smult (1 / coeff p (degree p)) p"

definition unit_factor_poly :: "'a poly \<Rightarrow> 'a poly"
where "unit_factor_poly p = monom (coeff p (degree p)) 0"

instance
proof
fix p :: "'a poly"
show "unit_factor p * normalize p = p"
(simp only: mult_smult_left [symmetric] smult_monom, simp)
next
show "normalize 0 = (0::'a poly)"
next
show "unit_factor 0 = (0::'a poly)"
next
fix p :: "'a poly"
assume "is_unit p"
then obtain a where "p = monom a 0" and "a \<noteq> 0"
by (rule is_unit_polyE)
then show "normalize p = 1"
by (auto simp add: normalize_poly_def smult_monom degree_monom_eq)
next
fix p q :: "'a poly"
assume "q \<noteq> 0"
from \<open>q \<noteq> 0\<close> have "is_unit (monom (coeff q (degree q)) 0)"
by (auto intro: is_unit_monom_0)
then show "is_unit (unit_factor q)"
next
fix p q :: "'a poly"
have "monom (coeff (p * q) (degree (p * q))) 0 =
monom (coeff p (degree p)) 0 * monom (coeff q (degree q)) 0"
then show "unit_factor (p * q) =
unit_factor p * unit_factor q"
qed

end

lemma degree_mod_less:
"y \<noteq> 0 \<Longrightarrow> x mod y = 0 \<or> degree (x mod y) < degree y"
using pdivmod_rel [of x y]
unfolding pdivmod_rel_def by simp

lemma div_poly_less: "degree x < degree y \<Longrightarrow> x div y = 0"
proof -
assume "degree x < degree y"
hence "pdivmod_rel x y 0 x"
thus "x div y = 0" by (rule div_poly_eq)
qed

lemma mod_poly_less: "degree x < degree y \<Longrightarrow> x mod y = x"
proof -
assume "degree x < degree y"
hence "pdivmod_rel x y 0 x"
thus "x mod y = x" by (rule mod_poly_eq)
qed

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

lemma div_smult_left: "(smult a x) div y = smult a (x div y)"
by (rule div_poly_eq, rule pdivmod_rel_smult_left, rule pdivmod_rel)

lemma mod_smult_left: "(smult a x) mod y = smult a (x mod y)"
by (rule mod_poly_eq, rule pdivmod_rel_smult_left, rule pdivmod_rel)

lemma poly_div_minus_left [simp]:
fixes x y :: "'a::field poly"
shows "(- x) div y = - (x div y)"
using div_smult_left [of "- 1::'a"] by simp

lemma poly_mod_minus_left [simp]:
fixes x y :: "'a::field poly"
shows "(- x) mod y = - (x mod y)"
using mod_smult_left [of "- 1::'a"] by simp

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:
"\<lbrakk>a \<noteq> 0; pdivmod_rel x y q r\<rbrakk>
\<Longrightarrow> pdivmod_rel x (smult a y) (smult (inverse a) q) r"
unfolding pdivmod_rel_def by simp

lemma div_smult_right:
"a \<noteq> 0 \<Longrightarrow> x div (smult a y) = smult (inverse a) (x div y)"
by (rule div_poly_eq, erule pdivmod_rel_smult_right, rule pdivmod_rel)

lemma mod_smult_right: "a \<noteq> 0 \<Longrightarrow> x mod (smult a y) = x mod y"
by (rule mod_poly_eq, erule pdivmod_rel_smult_right, rule pdivmod_rel)

lemma poly_div_minus_right [simp]:
fixes x y :: "'a::field poly"
shows "x div (- y) = - (x div y)"
using div_smult_right [of "- 1::'a"] by (simp add: nonzero_inverse_minus_eq)

lemma poly_mod_minus_right [simp]:
fixes x y :: "'a::field poly"
shows "x mod (- y) = x mod y"
using mod_smult_right [of "- 1::'a"] by simp

lemma pdivmod_rel_mult:
"\<lbrakk>pdivmod_rel x y q r; pdivmod_rel q z q' r'\<rbrakk>
\<Longrightarrow> pdivmod_rel x (y * z) q' (y * r' + r)"
apply (cases "z = 0", simp add: pdivmod_rel_def)
apply (cases "y = 0", simp add: pdivmod_rel_by_0_iff pdivmod_rel_0_iff)
apply (cases "r = 0")
apply (cases "r' = 0")
apply (simp add: pdivmod_rel_def 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 \<noteq> 0"
defines b: "b \<equiv> coeff (pCons a (x mod y)) (degree y) / coeff y (degree y)"
shows "(pCons a x) mod y = (pCons a (x mod y) - smult b y)"
unfolding b
apply (rule mod_poly_eq)
apply (rule pdivmod_rel_pCons [OF pdivmod_rel y refl])
done

definition pdivmod :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<times> 'a poly"
where
"pdivmod p q = (p div q, p mod q)"

lemma div_poly_code [code]:
"p div q = fst (pdivmod p q)"

lemma mod_poly_code [code]:
"p mod q = snd (pdivmod p 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 [code]:
"pdivmod p q = (if q = 0 then (0, p)
else fold_coeffs (\<lambda>a (s, r).
let b = coeff (pCons a r) (degree q) / coeff q (degree q)
in (pCons b s, pCons a r - smult b q)
) p (0, 0))"
apply (cases "q = 0")
apply (rule sym)
apply (induct p)
apply (case_tac "a = 0 \<and> p = 0")
done

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

definition order :: "'a::idom \<Rightarrow> 'a poly \<Rightarrow> nat"
where
"order a p = (LEAST n. \<not> [:-a, 1:] ^ Suc n dvd p)"

lemma coeff_linear_power:
fixes a :: "'a::comm_semiring_1"
shows "coeff ([:a, 1:] ^ n) n = 1"
apply (induct n, simp_all)
apply (subst coeff_eq_0)
apply (auto intro: le_less_trans degree_power_le)
done

lemma degree_linear_power:
fixes a :: "'a::comm_semiring_1"
shows "degree ([:a, 1:] ^ n) = n"
apply (rule order_antisym)
apply (rule ord_le_eq_trans [OF degree_power_le], simp)
apply (rule le_degree, simp add: coeff_linear_power)
done

lemma order_1: "[:-a, 1:] ^ order a p dvd p"
apply (cases "p = 0", simp)
apply (cases "order a p", simp)
apply (subgoal_tac "nat < (LEAST n. \<not> [:-a, 1:] ^ Suc n dvd p)")
apply (drule not_less_Least, simp)
apply (fold order_def, simp)
done

lemma order_2: "p \<noteq> 0 \<Longrightarrow> \<not> [:-a, 1:] ^ Suc (order a p) dvd p"
unfolding order_def
apply (rule LeastI_ex)
apply (rule_tac x="degree p" in exI)
apply (rule notI)
apply (drule (1) dvd_imp_degree_le)
apply (simp only: degree_linear_power)
done

lemma order:
"p \<noteq> 0 \<Longrightarrow> [:-a, 1:] ^ order a p dvd p \<and> \<not> [:-a, 1:] ^ Suc (order a p) dvd p"
by (rule conjI [OF order_1 order_2])

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

lemma order_root: "poly p a = 0 \<longleftrightarrow> p = 0 \<or> order a p \<noteq> 0"
apply (cases "p = 0", simp_all)
apply (rule iffI)
apply (metis order_2 not_gr0 poly_eq_0_iff_dvd power_0 power_Suc_0 power_one_right)
unfolding poly_eq_0_iff_dvd
apply (metis dvd_power dvd_trans order_1)
done

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

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

instantiation poly :: (field) gcd
begin

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

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

declare gcd_poly.simps [simp del]

definition lcm_poly :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
where
"lcm_poly a b = a * b div smult (coeff a (degree a) * coeff b (degree b)) (gcd a b)"

instance ..

end

lemma
fixes x y :: "_ poly"
shows poly_gcd_dvd1 [iff]: "gcd x y dvd x"
and poly_gcd_dvd2 [iff]: "gcd x y dvd y"
apply (induct x y rule: gcd_poly.induct)
apply (fastforce simp add: smult_dvd_iff dest: inverse_zero_imp_zero)
apply (blast dest: dvd_mod_imp_dvd)
done

lemma poly_gcd_greatest:
fixes k x y :: "_ poly"
shows "k dvd x \<Longrightarrow> k dvd y \<Longrightarrow> k dvd gcd x y"
by (induct x y rule: gcd_poly.induct)

lemma dvd_poly_gcd_iff [iff]:
fixes k x y :: "_ poly"
shows "k dvd gcd x y \<longleftrightarrow> k dvd x \<and> k dvd y"
by (auto intro!: poly_gcd_greatest intro: dvd_trans [of _ "gcd x y"])

lemma poly_gcd_monic:
fixes x y :: "_ poly"
shows "coeff (gcd x y) (degree (gcd x y)) =
(if x = 0 \<and> y = 0 then 0 else 1)"
by (induct x y rule: gcd_poly.induct)

lemma poly_gcd_zero_iff [simp]:
fixes x y :: "_ poly"
shows "gcd x y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
by (simp only: dvd_0_left_iff [symmetric] dvd_poly_gcd_iff)

lemma poly_gcd_0_0 [simp]:
"gcd (0::_ poly) 0 = 0"
by simp

lemma poly_dvd_antisym:
fixes p q :: "'a::idom poly"
assumes coeff: "coeff p (degree p) = coeff q (degree q)"
assumes dvd1: "p dvd q" and dvd2: "q dvd p" shows "p = q"
proof (cases "p = 0")
case True with coeff show "p = q" by simp
next
case False with coeff have "q \<noteq> 0" by auto
have degree: "degree p = degree q"
using \<open>p dvd q\<close> \<open>q dvd p\<close> \<open>p \<noteq> 0\<close> \<open>q \<noteq> 0\<close>
by (intro order_antisym dvd_imp_degree_le)

from \<open>p dvd q\<close> obtain a where a: "q = p * a" ..
with \<open>q \<noteq> 0\<close> have "a \<noteq> 0" by auto
with degree a \<open>p \<noteq> 0\<close> have "degree a = 0"
with coeff a show "p = q"
by (cases a, auto split: if_splits)
qed

lemma poly_gcd_unique:
fixes d x y :: "_ poly"
assumes dvd1: "d dvd x" and dvd2: "d dvd y"
and greatest: "\<And>k. k dvd x \<Longrightarrow> k dvd y \<Longrightarrow> k dvd d"
and monic: "coeff d (degree d) = (if x = 0 \<and> y = 0 then 0 else 1)"
shows "gcd x y = d"
proof -
have "coeff (gcd x y) (degree (gcd x y)) = coeff d (degree d)"
moreover have "gcd x y dvd d"
using poly_gcd_dvd1 poly_gcd_dvd2 by (rule greatest)
moreover have "d dvd gcd x y"
using dvd1 dvd2 by (rule poly_gcd_greatest)
ultimately show ?thesis
by (rule poly_dvd_antisym)
qed

interpretation gcd_poly: abel_semigroup "gcd :: _ poly \<Rightarrow> _"
proof
fix x y z :: "'a poly"
show "gcd (gcd x y) z = gcd x (gcd y z)"
by (rule poly_gcd_unique) (auto intro: dvd_trans simp add: poly_gcd_monic)
show "gcd x y = gcd y x"
by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)
qed

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

lemmas poly_gcd_ac = poly_gcd_assoc poly_gcd_commute poly_gcd_left_commute

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

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

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

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

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

subsection \<open>Additional induction rules on polynomials\<close>

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

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

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

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

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

definition pcompose :: "'a::comm_semiring_0 poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
where
"pcompose p q = fold_coeffs (\<lambda>a c. [:a:] + q * c) p 0"

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

lemma pcompose_pCons:
"pcompose (pCons a p) q = [:a:] + q * pcompose p q"
by (cases "p = 0 \<and> a = 0") (auto simp add: pcompose_def)

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

lemma degree_pcompose_le:
"degree (pcompose p q) \<le> degree p * degree q"
apply (induct p, simp)
apply (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 "\<dots> + 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_minus:
fixes p r :: "'a :: comm_ring poly"
shows "pcompose (-p) r = -pcompose p r"
by (induction p) (simp_all add: pcompose_pCons)

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

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)

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

lemma degree_mult_eq_0:
fixes p q:: "'a :: idom poly"
shows "degree (p*q) = 0 \<longleftrightarrow> p=0 \<or> q=0 \<or> (p\<noteq>0 \<and> q\<noteq>0 \<and> degree p =0 \<and> degree q =0)"

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

lemma pcompose_0':"pcompose p 0=[:coeff p 0:]"
apply (induct p)
done

lemma degree_pcompose:
fixes p q:: "'a::idom poly"
shows "degree(pcompose p q) = degree p * degree q"
proof (induct p)
case 0
thus ?case by auto
next
case (pCons a p)
have "degree (q * pcompose p q) = 0 \<Longrightarrow> ?case"
proof (cases "p=0")
case True
thus ?thesis by auto
next
case False assume "degree (q * pcompose p q) = 0"
hence "degree q=0 \<or> pcompose p q=0" by (auto simp add:degree_mult_eq_0)
moreover have "\<lbrakk>pcompose p q=0;degree q\<noteq>0\<rbrakk> \<Longrightarrow> False" using pCons.hyps(2) `p\<noteq>0`
proof -
assume "pcompose p q=0" "degree q\<noteq>0"
hence "degree p=0" using pCons.hyps(2) by auto
then obtain a1 where "p=[:a1:]"
by (metis degree_pCons_eq_if old.nat.distinct(2) pCons_cases)
thus False using `pcompose p q=0` `p\<noteq>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 "... \<ge> 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 \<Longrightarrow> ?case"
proof -
assume asm:"0 < degree (q * pcompose p q)"
hence "p\<noteq>0" "q\<noteq>0" "pcompose p q\<noteq>0" by auto
have "degree (pcompose (pCons a p) q) = degree ( q * pcompose p q)"
unfolding pcompose_pCons
using degree_add_eq_right[of "[:a:]" ] asm by auto
thus ?thesis
using pCons.hyps(2) degree_mult_eq[OF `q\<noteq>0` `pcompose p q\<noteq>0`] by auto
qed
ultimately show ?case by blast
qed

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

definition lead_coeff:: "'a::zero poly \<Rightarrow> 'a" where
"lead_coeff p= coeff p (degree p)"

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

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

assumes "degree p < degree q"

fixes p q:: "'a::idom 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 \<Longrightarrow> ?case"
proof -
assume "degree ( q * pcompose p q) = 0"
hence "pcompose p q = 0" by (metis assms degree_0 degree_mult_eq_0 neq0_conv)
hence "p=0" using pcompose_eq_0[OF _ `degree q > 0`] by simp
thus ?thesis by auto
qed
moreover have "degree ( q * pcompose p q) > 0 \<Longrightarrow> ?case"
proof -
assume "degree ( q * pcompose p q) > 0"
hence "lead_coeff (pcompose (pCons a p) q) =lead_coeff ( q * pcompose p q)"
using pCons.hyps(2) lead_coeff_mult[of q "pcompose p q"] by simp
also have "... = lead_coeff p * lead_coeff q ^ (degree p + 1)"
by auto
finally show ?thesis by auto
qed
ultimately show ?case by blast
qed

"lead_coeff (smult c p :: 'a :: idom poly) = c * lead_coeff p"
proof -
have "smult c p = [:c:] * p" by simp
finally show ?thesis .
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