# Theory Polynomial

Up to index of Isabelle/HOL/HOL-Library

theory Polynomial
imports Main
`(*  Title:      HOL/Library/Polynomial.thy    Author:     Brian Huffman    Author:     Clemens Ballarin*)header {* Univariate Polynomials *}theory Polynomialimports Mainbeginsubsection {* Definition of type @{text poly} *}definition "Poly = {f::nat => 'a::zero. ∃n. ∀i>n. f i = 0}"typedef 'a poly = "Poly :: (nat => 'a::zero) set"  morphisms coeff Abs_poly  unfolding Poly_def by auto(* FIXME should be named poly_eq_iff *)lemma expand_poly_eq: "p = q <-> (∀n. coeff p n = coeff q n)"  by (simp add: coeff_inject [symmetric] fun_eq_iff)(* FIXME should be named poly_eqI *)lemma poly_ext: "(!!n. coeff p n = coeff q n) ==> p = q"  by (simp add: expand_poly_eq)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: "degree p < n ==> coeff p n = 0"proof -  have "coeff p ∈ Poly"    by (rule coeff)  hence "∃n. ∀i>n. coeff p i = 0"    unfolding Poly_def by simp  hence "∀i>degree p. coeff p i = 0"    unfolding degree_def by (rule LeastI_ex)  moreover assume "degree p < n"  ultimately show ?thesis by simpqedlemma 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) zerobegindefinition  zero_poly_def: "0 = Abs_poly (λn. 0)"instance ..endlemma coeff_0 [simp]: "coeff 0 n = 0"  unfolding zero_poly_def  by (simp add: Abs_poly_inverse Poly_def)lemma degree_0 [simp]: "degree 0 = 0"  by (rule order_antisym [OF degree_le le0]) simplemma leading_coeff_neq_0:  assumes "p ≠ 0" shows "coeff p (degree p) ≠ 0"proof (cases "degree p")  case 0  from `p ≠ 0` have "∃n. coeff p n ≠ 0"    by (simp add: expand_poly_eq)  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 simpnext  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 simpqedlemma leading_coeff_0_iff [simp]: "coeff p (degree p) = 0 <-> p = 0"  by (cases "p = 0", simp, simp add: leading_coeff_neq_0)subsection {* List-style constructor for polynomials *}definition  pCons :: "'a::zero => 'a poly => 'a poly"where  "pCons a p = Abs_poly (nat_case a (coeff p))"syntax  "_poly" :: "args => 'a poly"  ("[:(_):]")translations  "[:x, xs:]" == "CONST pCons x [:xs:]"  "[:x:]" == "CONST pCons x 0"  "[:x:]" <= "CONST pCons x (_constrain 0 t)"lemma Poly_nat_case: "f ∈ Poly ==> nat_case a f ∈ Poly"  unfolding Poly_def by (auto split: nat.split)lemma coeff_pCons:  "coeff (pCons a p) = nat_case a (coeff p)"  unfolding pCons_def  by (simp add: Abs_poly_inverse Poly_nat_case coeff)lemma coeff_pCons_0 [simp]: "coeff (pCons a p) 0 = a"  by (simp add: coeff_pCons)lemma coeff_pCons_Suc [simp]: "coeff (pCons a p) (Suc n) = coeff p n"  by (simp add: coeff_pCons)lemma degree_pCons_le: "degree (pCons a p) ≤ Suc (degree p)"by (rule degree_le, simp add: coeff_eq_0 coeff_pCons split: nat.split)lemma degree_pCons_eq:  "p ≠ 0 ==> degree (pCons a p) = Suc (degree p)"apply (rule order_antisym [OF degree_pCons_le])apply (rule le_degree, simp)donelemma 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)donelemma 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)donelemma pCons_0_0 [simp, code_post]: "pCons 0 0 = 0"by (rule poly_ext, 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 simpnext  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: expand_poly_eq)qedlemma pCons_eq_0_iff [simp]: "pCons a p = 0 <-> a = 0 ∧ p = 0"  using pCons_eq_iff [of a p 0 0] by simplemma Poly_Suc: "f ∈ Poly ==> (λn. f (Suc n)) ∈ Poly"  unfolding Poly_def  by (clarify, rule_tac x=n in exI, 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 (rule poly_ext)       (simp add: Abs_poly_inverse Poly_Suc coeff coeff_pCons             split: nat.split)qedlemma pCons_induct [case_names 0 pCons, induct type: poly]:  assumes zero: "P 0"  assumes pCons: "!!a p. P p ==> P (pCons a p)"  shows "P p"proof (induct p rule: measure_induct_rule [where f=degree])  case (less p)  obtain a q where "p = pCons a q" by (rule pCons_cases)  have "P q"  proof (cases "q = 0")    case True    then show "P q" by (simp add: zero)  next    case False    then have "degree (pCons a q) = Suc (degree q)"      by (rule degree_pCons_eq)    then have "degree q < degree p"      using `p = pCons a q` by simp    then show "P q"      by (rule less.hyps)  qed  then have "P (pCons a q)"    by (rule pCons)  then show ?case    using `p = pCons a q` by simpqedsubsection {* Recursion combinator for polynomials *}function  poly_rec :: "'b => ('a::zero => 'a poly => 'b => 'b) => 'a poly => 'b"where  poly_rec_pCons_eq_if [simp del]:    "poly_rec z f (pCons a p) = f a p (if p = 0 then z else poly_rec z f p)"by (case_tac x, rename_tac q, case_tac q, auto)termination poly_recby (relation "measure (degree o snd o snd)", simp)   (simp add: degree_pCons_eq)lemma poly_rec_0:  "f 0 0 z = z ==> poly_rec z f 0 = z"  using poly_rec_pCons_eq_if [of z f 0 0] by simplemma poly_rec_pCons:  "f 0 0 z = z ==> poly_rec z f (pCons a p) = f a p (poly_rec z f p)"  by (simp add: poly_rec_pCons_eq_if poly_rec_0)subsection {* Monomials *}definition  monom :: "'a => nat => 'a::zero poly" where  "monom a m = Abs_poly (λn. if m = n then a else 0)"lemma coeff_monom [simp]: "coeff (monom a m) n = (if m=n then a else 0)"  unfolding monom_def  by (subst Abs_poly_inverse, auto simp add: Poly_def)lemma monom_0: "monom a 0 = pCons a 0"  by (rule poly_ext, simp add: coeff_pCons split: nat.split)lemma monom_Suc: "monom a (Suc n) = pCons 0 (monom a n)"  by (rule poly_ext, simp add: coeff_pCons split: nat.split)lemma monom_eq_0 [simp]: "monom 0 n = 0"  by (rule poly_ext) simplemma monom_eq_0_iff [simp]: "monom a n = 0 <-> a = 0"  by (simp add: expand_poly_eq)lemma monom_eq_iff [simp]: "monom a n = monom b n <-> a = b"  by (simp add: expand_poly_eq)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)  donesubsection {* Addition and subtraction *}instantiation poly :: (comm_monoid_add) comm_monoid_addbegindefinition  plus_poly_def:    "p + q = Abs_poly (λn. coeff p n + coeff q n)"lemma Poly_add:  fixes f g :: "nat => 'a"  shows "[|f ∈ Poly; g ∈ Poly|] ==> (λn. f n + g n) ∈ Poly"  unfolding Poly_def  apply (clarify, rename_tac m n)  apply (rule_tac x="max m n" in exI, simp)  donelemma coeff_add [simp]:  "coeff (p + q) n = coeff p n + coeff q n"  unfolding plus_poly_def  by (simp add: Abs_poly_inverse coeff Poly_add)instance proof  fix p q r :: "'a poly"  show "(p + q) + r = p + (q + r)"    by (simp add: expand_poly_eq add_assoc)  show "p + q = q + p"    by (simp add: expand_poly_eq add_commute)  show "0 + p = p"    by (simp add: expand_poly_eq)qedendinstance poly :: (cancel_comm_monoid_add) cancel_comm_monoid_addproof  fix p q r :: "'a poly"  assume "p + q = p + r" thus "q = r"    by (simp add: expand_poly_eq)qedinstantiation poly :: (ab_group_add) ab_group_addbegindefinition  uminus_poly_def:    "- p = Abs_poly (λn. - coeff p n)"definition  minus_poly_def:    "p - q = Abs_poly (λn. coeff p n - coeff q n)"lemma Poly_minus:  fixes f :: "nat => 'a"  shows "f ∈ Poly ==> (λn. - f n) ∈ Poly"  unfolding Poly_def by simplemma Poly_diff:  fixes f g :: "nat => 'a"  shows "[|f ∈ Poly; g ∈ Poly|] ==> (λn. f n - g n) ∈ Poly"  unfolding diff_minus by (simp add: Poly_add Poly_minus)lemma coeff_minus [simp]: "coeff (- p) n = - coeff p n"  unfolding uminus_poly_def  by (simp add: Abs_poly_inverse coeff Poly_minus)lemma coeff_diff [simp]:  "coeff (p - q) n = coeff p n - coeff q n"  unfolding minus_poly_def  by (simp add: Abs_poly_inverse coeff Poly_diff)instance proof  fix p q :: "'a poly"  show "- p + p = 0"    by (simp add: expand_poly_eq)  show "p - q = p + - q"    by (simp add: expand_poly_eq diff_minus)qedendlemma add_pCons [simp]:  "pCons a p + pCons b q = pCons (a + b) (p + q)"  by (rule poly_ext, simp add: coeff_pCons split: nat.split)lemma minus_pCons [simp]:  "- pCons a p = pCons (- a) (- p)"  by (rule poly_ext, 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_ext, simp add: coeff_pCons split: nat.split)lemma degree_add_le_max: "degree (p + q) ≤ max (degree p) (degree q)"  by (rule degree_le, auto simp add: coeff_eq_0)lemma degree_add_le:  "[|degree p ≤ n; degree q ≤ n|] ==> degree (p + q) ≤ n"  by (auto intro: order_trans degree_add_le_max)lemma degree_add_less:  "[|degree p < n; degree q < n|] ==> degree (p + q) < n"  by (auto intro: le_less_trans degree_add_le_max)lemma degree_add_eq_right:  "degree p < degree q ==> degree (p + q) = degree q"  apply (cases "q = 0", simp)  apply (rule order_antisym)  apply (simp add: degree_add_le)  apply (rule le_degree)  apply (simp add: coeff_eq_0)  donelemma degree_add_eq_left:  "degree q < degree p ==> degree (p + q) = degree p"  using degree_add_eq_right [of q p]  by (simp add: add_commute)lemma degree_minus [simp]: "degree (- p) = degree p"  unfolding degree_def by simplemma degree_diff_le_max: "degree (p - q) ≤ max (degree p) (degree q)"  using degree_add_le [where p=p and q="-q"]  by (simp add: diff_minus)lemma degree_diff_le:  "[|degree p ≤ n; degree q ≤ n|] ==> degree (p - q) ≤ n"  by (simp add: diff_minus degree_add_le)lemma degree_diff_less:  "[|degree p < n; degree q < n|] ==> degree (p - q) < n"  by (simp add: diff_minus degree_add_less)lemma add_monom: "monom a n + monom b n = monom (a + b) n"  by (rule poly_ext) simplemma diff_monom: "monom a n - monom b n = monom (a - b) n"  by (rule poly_ext) simplemma minus_monom: "- monom a n = monom (-a) n"  by (rule poly_ext) simplemma coeff_setsum: "coeff (∑x∈A. p x) i = (∑x∈A. coeff (p x) i)"  by (cases "finite A", induct set: finite, simp_all)lemma monom_setsum: "monom (∑x∈A. a x) n = (∑x∈A. monom (a x) n)"  by (rule poly_ext) (simp add: coeff_setsum)subsection {* Multiplication by a constant *}definition  smult :: "'a::comm_semiring_0 => 'a poly => 'a poly" where  "smult a p = Abs_poly (λn. a * coeff p n)"lemma Poly_smult:  fixes f :: "nat => 'a::comm_semiring_0"  shows "f ∈ Poly ==> (λn. a * f n) ∈ Poly"  unfolding Poly_def  by (clarify, rule_tac x=n in exI, simp)lemma coeff_smult [simp]: "coeff (smult a p) n = a * coeff p n"  unfolding smult_def  by (simp add: Abs_poly_inverse Poly_smult coeff)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_ext, simp add: mult_assoc)lemma smult_0_right [simp]: "smult a 0 = 0"  by (rule poly_ext, simp)lemma smult_0_left [simp]: "smult 0 p = 0"  by (rule poly_ext, simp)lemma smult_1_left [simp]: "smult (1::'a::comm_semiring_1) p = p"  by (rule poly_ext, simp)lemma smult_add_right:  "smult a (p + q) = smult a p + smult a q"  by (rule poly_ext, simp add: algebra_simps)lemma smult_add_left:  "smult (a + b) p = smult a p + smult b p"  by (rule poly_ext, simp add: algebra_simps)lemma smult_minus_right [simp]:  "smult (a::'a::comm_ring) (- p) = - smult a p"  by (rule poly_ext, simp)lemma smult_minus_left [simp]:  "smult (- a::'a::comm_ring) p = - smult a p"  by (rule poly_ext, simp)lemma smult_diff_right:  "smult (a::'a::comm_ring) (p - q) = smult a p - smult a q"  by (rule poly_ext, simp add: algebra_simps)lemma smult_diff_left:  "smult (a - b::'a::comm_ring) p = smult a p - smult b p"  by (rule poly_ext, simp add: algebra_simps)lemmas smult_distribs =  smult_add_left smult_add_right  smult_diff_left smult_diff_rightlemma smult_pCons [simp]:  "smult a (pCons b p) = pCons (a * b) (smult a p)"  by (rule poly_ext, simp add: coeff_pCons split: nat.split)lemma smult_monom: "smult a (monom b n) = monom (a * b) n"  by (induct n, simp add: monom_0, simp add: monom_Suc)lemma degree_smult_eq [simp]:  fixes a :: "'a::idom"  shows "degree (smult a p) = (if a = 0 then 0 else degree p)"  by (cases "a = 0", simp, simp add: degree_def)lemma smult_eq_0_iff [simp]:  fixes a :: "'a::idom"  shows "smult a p = 0 <-> a = 0 ∨ p = 0"  by (simp add: expand_poly_eq)subsection {* Multiplication of polynomials *}(* TODO: move to Set_Interval.thy *)lemma setsum_atMost_Suc_shift:  fixes f :: "nat => 'a::comm_monoid_add"  shows "(∑i≤Suc n. f i) = f 0 + (∑i≤n. f (Suc i))"proof (induct n)  case 0 show ?case by simpnext  case (Suc n) note IH = this  have "(∑i≤Suc (Suc n). f i) = (∑i≤Suc n. f i) + f (Suc (Suc n))"    by (rule setsum_atMost_Suc)  also have "(∑i≤Suc n. f i) = f 0 + (∑i≤n. f (Suc i))"    by (rule IH)  also have "f 0 + (∑i≤n. f (Suc i)) + f (Suc (Suc n)) =             f 0 + ((∑i≤n. f (Suc i)) + f (Suc (Suc n)))"    by (rule add_assoc)  also have "(∑i≤n. f (Suc i)) + f (Suc (Suc n)) = (∑i≤Suc n. f (Suc i))"    by (rule setsum_atMost_Suc [symmetric])  finally show ?case .qedinstantiation poly :: (comm_semiring_0) comm_semiring_0begindefinition  times_poly_def:    "p * q = poly_rec 0 (λa p pq. smult a q + pCons 0 pq) p"lemma mult_poly_0_left: "(0::'a poly) * q = 0"  unfolding times_poly_def by (simp add: poly_rec_0)lemma mult_pCons_left [simp]:  "pCons a p * q = smult a q + pCons 0 (p * q)"  unfolding times_poly_def by (simp add: poly_rec_pCons)lemma mult_poly_0_right: "p * (0::'a poly) = 0"  by (induct p, simp add: mult_poly_0_left, simp)lemma mult_pCons_right [simp]:  "p * pCons a q = smult a p + pCons 0 (p * q)"  by (induct p, simp add: mult_poly_0_left, simp add: algebra_simps)lemmas mult_poly_0 = mult_poly_0_left mult_poly_0_rightlemma mult_smult_left [simp]: "smult a p * q = smult a (p * q)"  by (induct p, simp add: mult_poly_0, simp add: smult_add_right)lemma mult_smult_right [simp]: "p * smult a q = smult a (p * q)"  by (induct q, simp add: mult_poly_0, simp add: smult_add_right)lemma mult_poly_add_left:  fixes p q r :: "'a poly"  shows "(p + q) * r = p * r + q * r"  by (induct r, simp add: mult_poly_0,                simp add: smult_distribs algebra_simps)instance proof  fix p q r :: "'a poly"  show 0: "0 * p = 0"    by (rule mult_poly_0_left)  show "p * 0 = 0"    by (rule mult_poly_0_right)  show "(p + q) * r = p * r + q * r"    by (rule mult_poly_add_left)  show "(p * q) * r = p * (q * r)"    by (induct p, simp add: mult_poly_0, simp add: mult_poly_add_left)  show "p * q = q * p"    by (induct p, simp add: mult_poly_0, simp)qedendinstance 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 simpnext  case (pCons a p n) thus ?case    by (cases n, simp, simp add: setsum_atMost_Suc_shift                            del: setsum_atMost_Suc)qedlemma degree_mult_le: "degree (p * q) ≤ degree p + degree q"apply (rule degree_le)apply (induct p)apply simpapply (simp add: coeff_eq_0 coeff_pCons split: nat.split)donelemma mult_monom: "monom a m * monom b n = monom (a * b) (m + n)"  by (induct m, simp add: monom_0 smult_monom, simp add: monom_Suc)subsection {* The unit polynomial and exponentiation *}instantiation poly :: (comm_semiring_1) comm_semiring_1begindefinition  one_poly_def:    "1 = pCons 1 0"instance proof  fix p :: "'a poly" show "1 * p = p"    unfolding one_poly_def    by simpnext  show "0 ≠ (1::'a poly)"    unfolding one_poly_def by simpqedendinstance poly :: (comm_semiring_1_cancel) 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 degree_1 [simp]: "degree 1 = 0"  unfolding one_poly_def  by (rule degree_pCons_0)text {* Lemmas about divisibility *}lemma dvd_smult: "p dvd q ==> p dvd smult a q"proof -  assume "p dvd q"  then obtain k where "q = p * k" ..  then have "smult a q = p * smult a k" by simp  then show "p dvd smult a q" ..qedlemma 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"]) simplemma 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" ..qedlemma 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"]) simplemma 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)lemma degree_power_le: "degree (p ^ n) ≤ degree p * n"by (induct n, simp, auto intro: order_trans degree_mult_le)instance poly :: (comm_ring) comm_ring ..instance poly :: (comm_ring_1) comm_ring_1 ..subsection {* Polynomials form an integral domain *}lemma coeff_mult_degree_sum:  "coeff (p * q) (degree p + degree q) =   coeff p (degree p) * coeff q (degree q)"  by (induct p, simp, simp add: coeff_eq_0)instance poly :: (idom) idomproof  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: expand_poly_eq)qedlemma degree_mult_eq:  fixes p q :: "'a::idom poly"  shows "[|p ≠ 0; q ≠ 0|] ==> degree (p * q) = degree p + degree q"apply (rule order_antisym [OF degree_mult_le le_degree])apply (simp add: coeff_mult_degree_sum)donelemma dvd_imp_degree_le:  fixes p q :: "'a::idom poly"  shows "[|p dvd q; q ≠ 0|] ==> degree p ≤ degree q"  by (erule dvdE, simp add: degree_mult_eq)subsection {* Polynomials form an ordered integral domain *}definition  pos_poly :: "'a::linordered_idom poly => bool"where  "pos_poly p <-> 0 < coeff p (degree p)"lemma pos_poly_pCons:  "pos_poly (pCons a p) <-> pos_poly p ∨ (p = 0 ∧ 0 < a)"  unfolding pos_poly_def by simplemma not_pos_poly_0 [simp]: "¬ pos_poly 0"  unfolding pos_poly_def by simplemma pos_poly_add: "[|pos_poly p; pos_poly q|] ==> pos_poly (p + q)"  apply (induct p arbitrary: q, simp)  apply (case_tac q, force simp add: pos_poly_pCons add_pos_pos)  donelemma pos_poly_mult: "[|pos_poly p; pos_poly q|] ==> pos_poly (p * q)"  unfolding pos_poly_def  apply (subgoal_tac "p ≠ 0 ∧ q ≠ 0")  apply (simp add: degree_mult_eq coeff_mult_degree_sum mult_pos_pos)  apply auto  donelemma pos_poly_total: "p = 0 ∨ pos_poly p ∨ pos_poly (- p)"by (induct p) (auto simp add: pos_poly_pCons)instantiation poly :: (linordered_idom) linordered_idombegindefinition  "x < y <-> pos_poly (y - x)"definition  "x ≤ y <-> x = y ∨ pos_poly (y - x)"definition  "abs (x::'a poly) = (if x < 0 then - x else x)"definition  "sgn (x::'a poly) = (if x = 0 then 0 else if 0 < x then 1 else - 1)"instance proof  fix x y :: "'a poly"  show "x < y <-> x ≤ y ∧ ¬ y ≤ x"    unfolding less_eq_poly_def less_poly_def    apply safe    apply simp    apply (drule (1) pos_poly_add)    apply simp    donenext  fix x :: "'a poly" show "x ≤ x"    unfolding less_eq_poly_def by simpnext  fix x y z :: "'a poly"  assume "x ≤ y" and "y ≤ z" thus "x ≤ z"    unfolding less_eq_poly_def    apply safe    apply (drule (1) pos_poly_add)    apply (simp add: algebra_simps)    donenext  fix x y :: "'a poly"  assume "x ≤ y" and "y ≤ x" thus "x = y"    unfolding less_eq_poly_def    apply safe    apply (drule (1) pos_poly_add)    apply simp    donenext  fix x y z :: "'a poly"  assume "x ≤ y" thus "z + x ≤ z + y"    unfolding less_eq_poly_def    apply safe    apply (simp add: algebra_simps)    donenext  fix x y :: "'a poly"  show "x ≤ y ∨ y ≤ x"    unfolding less_eq_poly_def    using pos_poly_total [of "x - y"]    by autonext  fix x y z :: "'a poly"  assume "x < y" and "0 < z"  thus "z * x < z * y"    unfolding less_poly_def    by (simp add: right_diff_distrib [symmetric] pos_poly_mult)next  fix x :: "'a poly"  show "¦x¦ = (if x < 0 then - x else x)"    by (rule abs_poly_def)next  fix x :: "'a poly"  show "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"    by (rule sgn_poly_def)qedendtext {* TODO: Simplification rules for comparisons *}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 simplemma pdivmod_rel_by_0:  "pdivmod_rel x 0 0 x"  unfolding pdivmod_rel_def by simplemma 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 ..qedlemma 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 simpqedlemma 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)donelemma pdivmod_rel_unique:  assumes 1: "pdivmod_rel x y q1 r1"  assumes 2: "pdivmod_rel x y q2 r2"  shows "q1 = q2 ∧ r1 = r2"proof (cases "y = 0")  assume "y = 0" with assms show ?thesis    by (simp add: pdivmod_rel_def)next  assume [simp]: "y ≠ 0"  from 1 have q1: "x = q1 * y + r1" and r1: "r1 = 0 ∨ degree r1 < degree y"    unfolding pdivmod_rel_def by simp_all  from 2 have q2: "x = q2 * y + r2" and r2: "r2 = 0 ∨ degree r2 < degree y"    unfolding pdivmod_rel_def by simp_all  from q1 q2 have q3: "(q1 - q2) * y = r2 - r1"    by (simp add: algebra_simps)  from r1 r2 have r3: "(r2 - r1) = 0 ∨ degree (r2 - r1) < degree y"    by (auto intro: degree_diff_less)  show "q1 = q2 ∧ r1 = r2"  proof (rule ccontr)    assume "¬ (q1 = q2 ∧ r1 = r2)"    with q3 have "q1 ≠ q2" and "r1 ≠ r2" by auto    with r3 have "degree (r2 - r1) < degree y" by simp    also have "degree y ≤ degree (q1 - q2) + degree y" by simp    also have "… = degree ((q1 - q2) * y)"      using `q1 ≠ q2` by (simp add: degree_mult_eq)    also have "… = degree (r2 - r1)"      using q3 by simp    finally have "degree (r2 - r1) < degree (r2 - r1)" .    then show "False" by simp  qedqedlemma pdivmod_rel_0_iff: "pdivmod_rel 0 y q r <-> q = 0 ∧ r = 0"by (auto dest: pdivmod_rel_unique intro: pdivmod_rel_0)lemma pdivmod_rel_by_0_iff: "pdivmod_rel x 0 q r <-> q = 0 ∧ r = x"by (auto dest: pdivmod_rel_unique intro: pdivmod_rel_by_0)lemmas pdivmod_rel_unique_div = pdivmod_rel_unique [THEN conjunct1]lemmas pdivmod_rel_unique_mod = pdivmod_rel_unique [THEN conjunct2]instantiation poly :: (field) ring_divbegindefinition div_poly where  "x div y = (THE q. ∃r. pdivmod_rel x y q r)"definition mod_poly where  "x mod y = (THE r. ∃q. pdivmod_rel x y q r)"lemma div_poly_eq:  "pdivmod_rel x y q r ==> x div y = q"unfolding div_poly_defby (fast elim: pdivmod_rel_unique_div)lemma mod_poly_eq:  "pdivmod_rel x y q r ==> x mod y = r"unfolding mod_poly_defby (fast elim: pdivmod_rel_unique_mod)lemma pdivmod_rel:  "pdivmod_rel x y (x div y) (x mod y)"proof -  from pdivmod_rel_exists    obtain q r where "pdivmod_rel x y q r" by fast  thus ?thesis    by (simp add: div_poly_eq mod_poly_eq)qedinstance proof  fix x y :: "'a poly"  show "x div y * y + x mod y = x"    using pdivmod_rel [of x y]    by (simp add: pdivmod_rel_def)next  fix x :: "'a poly"  have "pdivmod_rel x 0 0 x"    by (rule pdivmod_rel_by_0)  thus "x div 0 = 0"    by (rule div_poly_eq)next  fix y :: "'a poly"  have "pdivmod_rel 0 y 0 0"    by (rule pdivmod_rel_0)  thus "0 div y = 0"    by (rule div_poly_eq)next  fix x y z :: "'a poly"  assume "y ≠ 0"  hence "pdivmod_rel (x + z * y) y (z + x div y) (x mod y)"    using pdivmod_rel [of x y]    by (simp add: pdivmod_rel_def distrib_right)  thus "(x + z * y) div y = z + x div y"    by (rule div_poly_eq)next  fix x y z :: "'a poly"  assume "x ≠ 0"  show "(x * y) div (x * z) = y div z"  proof (cases "y ≠ 0 ∧ z ≠ 0")    have "!!x::'a poly. pdivmod_rel x 0 0 x"      by (rule pdivmod_rel_by_0)    then have [simp]: "!!x::'a poly. x div 0 = 0"      by (rule div_poly_eq)    have "!!x::'a poly. pdivmod_rel 0 x 0 0"      by (rule pdivmod_rel_0)    then have [simp]: "!!x::'a poly. 0 div x = 0"      by (rule div_poly_eq)    case False then show ?thesis by auto  next    case True then have "y ≠ 0" and "z ≠ 0" by auto    with `x ≠ 0`    have "!!q r. pdivmod_rel y z q r ==> pdivmod_rel (x * y) (x * z) q (x * r)"      by (auto simp add: pdivmod_rel_def algebra_simps)        (rule classical, simp add: degree_mult_eq)    moreover from pdivmod_rel have "pdivmod_rel y z (y div z) (y mod z)" .    ultimately have "pdivmod_rel (x * y) (x * z) (y div z) (x * (y mod z))" .    then show ?thesis by (simp add: div_poly_eq)  qedqedendlemma 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 simplemma div_poly_less: "degree x < degree y ==> x div y = 0"proof -  assume "degree x < degree y"  hence "pdivmod_rel x y 0 x"    by (simp add: pdivmod_rel_def)  thus "x div y = 0" by (rule div_poly_eq)qedlemma mod_poly_less: "degree x < degree y ==> x mod y = x"proof -  assume "degree x < degree y"  hence "pdivmod_rel x y 0 x"    by (simp add: pdivmod_rel_def)  thus "x mod y = x" by (rule mod_poly_eq)qedlemma pdivmod_rel_smult_left:  "pdivmod_rel x y q r    ==> pdivmod_rel (smult a x) y (smult a q) (smult a r)"  unfolding pdivmod_rel_def by (simp add: smult_add_right)lemma div_smult_left: "(smult a x) div y = smult a (x div y)"  by (rule div_poly_eq, rule pdivmod_rel_smult_left, rule pdivmod_rel)lemma mod_smult_left: "(smult a x) mod y = smult a (x mod y)"  by (rule mod_poly_eq, rule pdivmod_rel_smult_left, rule pdivmod_rel)lemma poly_div_minus_left [simp]:  fixes x y :: "'a::field poly"  shows "(- x) div y = - (x div y)"  using div_smult_left [of "- 1::'a"] by (simp del: minus_one) (* FIXME *)lemma poly_mod_minus_left [simp]:  fixes x y :: "'a::field poly"  shows "(- x) mod y = - (x mod y)"  using mod_smult_left [of "- 1::'a"] by (simp del: minus_one) (* FIXME *)lemma pdivmod_rel_smult_right:  "[|a ≠ 0; pdivmod_rel x y q r|]    ==> pdivmod_rel x (smult a y) (smult (inverse a) q) r"  unfolding pdivmod_rel_def by simplemma div_smult_right:  "a ≠ 0 ==> x div (smult a y) = smult (inverse a) (x div y)"  by (rule div_poly_eq, erule pdivmod_rel_smult_right, rule pdivmod_rel)lemma mod_smult_right: "a ≠ 0 ==> x mod (smult a y) = x mod y"  by (rule mod_poly_eq, erule pdivmod_rel_smult_right, rule pdivmod_rel)lemma poly_div_minus_right [simp]:  fixes x y :: "'a::field poly"  shows "x div (- y) = - (x div y)"  using div_smult_right [of "- 1::'a"]  by (simp add: nonzero_inverse_minus_eq del: minus_one) (* FIXME *)lemma poly_mod_minus_right [simp]:  fixes x y :: "'a::field poly"  shows "x mod (- y) = x mod y"  using mod_smult_right [of "- 1::'a"] by (simp del: minus_one) (* FIXME *)lemma pdivmod_rel_mult:  "[|pdivmod_rel x y q r; pdivmod_rel q z q' r'|]    ==> pdivmod_rel x (y * z) q' (y * r' + r)"apply (cases "z = 0", simp add: pdivmod_rel_def)apply (cases "y = 0", simp add: pdivmod_rel_by_0_iff pdivmod_rel_0_iff)apply (cases "r = 0")apply (cases "r' = 0")apply (simp add: pdivmod_rel_def)apply (simp add: pdivmod_rel_def field_simps degree_mult_eq)apply (cases "r' = 0")apply (simp add: pdivmod_rel_def degree_mult_eq)apply (simp add: pdivmod_rel_def field_simps)apply (simp add: degree_mult_eq degree_add_less)donelemma 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 bapply (rule mod_poly_eq)apply (rule pdivmod_rel_pCons [OF pdivmod_rel y refl])donesubsection {* GCD of polynomials *}function  poly_gcd :: "'a::field poly => 'a poly => 'a poly" where  "poly_gcd x 0 = smult (inverse (coeff x (degree x))) x"| "y ≠ 0 ==> poly_gcd x y = poly_gcd y (x mod y)"by autotermination poly_gcdby (relation "measure (λ(x, y). if y = 0 then 0 else Suc (degree y))")   (auto dest: degree_mod_less)declare poly_gcd.simps [simp del]lemma poly_gcd_dvd1 [iff]: "poly_gcd x y dvd x"  and poly_gcd_dvd2 [iff]: "poly_gcd x y dvd y"  apply (induct x y rule: poly_gcd.induct)  apply (simp_all add: poly_gcd.simps)  apply (fastforce simp add: smult_dvd_iff dest: inverse_zero_imp_zero)  apply (blast dest: dvd_mod_imp_dvd)  donelemma poly_gcd_greatest: "k dvd x ==> k dvd y ==> k dvd poly_gcd x y"  by (induct x y rule: poly_gcd.induct)     (simp_all add: poly_gcd.simps dvd_mod dvd_smult)lemma dvd_poly_gcd_iff [iff]:  "k dvd poly_gcd x y <-> k dvd x ∧ k dvd y"  by (blast intro!: poly_gcd_greatest intro: dvd_trans)lemma poly_gcd_monic:  "coeff (poly_gcd x y) (degree (poly_gcd x y)) =    (if x = 0 ∧ y = 0 then 0 else 1)"  by (induct x y rule: poly_gcd.induct)     (simp_all add: poly_gcd.simps nonzero_imp_inverse_nonzero)lemma poly_gcd_zero_iff [simp]:  "poly_gcd x y = 0 <-> x = 0 ∧ y = 0"  by (simp only: dvd_0_left_iff [symmetric] dvd_poly_gcd_iff)lemma poly_gcd_0_0 [simp]: "poly_gcd 0 0 = 0"  by simplemma 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 simpnext  case False with coeff have "q ≠ 0" by auto  have degree: "degree p = degree q"    using `p dvd q` `q dvd p` `p ≠ 0` `q ≠ 0`    by (intro order_antisym dvd_imp_degree_le)  from `p dvd q` obtain a where a: "q = p * a" ..  with `q ≠ 0` have "a ≠ 0" by auto  with degree a `p ≠ 0` have "degree a = 0"    by (simp add: degree_mult_eq)  with coeff a show "p = q"    by (cases a, auto split: if_splits)qedlemma poly_gcd_unique:  assumes dvd1: "d dvd x" and dvd2: "d dvd y"    and greatest: "!!k. k dvd x ==> k dvd y ==> k dvd d"    and monic: "coeff d (degree d) = (if x = 0 ∧ y = 0 then 0 else 1)"  shows "poly_gcd x y = d"proof -  have "coeff (poly_gcd x y) (degree (poly_gcd x y)) = coeff d (degree d)"    by (simp_all add: poly_gcd_monic monic)  moreover have "poly_gcd x y dvd d"    using poly_gcd_dvd1 poly_gcd_dvd2 by (rule greatest)  moreover have "d dvd poly_gcd x y"    using dvd1 dvd2 by (rule poly_gcd_greatest)  ultimately show ?thesis    by (rule poly_dvd_antisym)qedinterpretation poly_gcd: abel_semigroup poly_gcdproof  fix x y z :: "'a poly"  show "poly_gcd (poly_gcd x y) z = poly_gcd x (poly_gcd y z)"    by (rule poly_gcd_unique) (auto intro: dvd_trans simp add: poly_gcd_monic)  show "poly_gcd x y = poly_gcd y x"    by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)qedlemmas poly_gcd_assoc = poly_gcd.assoclemmas poly_gcd_commute = poly_gcd.commutelemmas poly_gcd_left_commute = poly_gcd.left_commutelemmas poly_gcd_ac = poly_gcd_assoc poly_gcd_commute poly_gcd_left_commutelemma poly_gcd_1_left [simp]: "poly_gcd 1 y = 1"by (rule poly_gcd_unique) simp_alllemma poly_gcd_1_right [simp]: "poly_gcd x 1 = 1"by (rule poly_gcd_unique) simp_alllemma poly_gcd_minus_left [simp]: "poly_gcd (- x) y = poly_gcd x y"by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)lemma poly_gcd_minus_right [simp]: "poly_gcd x (- y) = poly_gcd x y"by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)subsection {* Evaluation of polynomials *}definition  poly :: "'a::comm_semiring_0 poly => 'a => 'a" where  "poly = poly_rec (λx. 0) (λa p f x. a + x * f x)"lemma poly_0 [simp]: "poly 0 x = 0"  unfolding poly_def by (simp add: poly_rec_0)lemma poly_pCons [simp]: "poly (pCons a p) x = a + x * poly p x"  unfolding poly_def by (simp add: poly_rec_pCons)lemma poly_1 [simp]: "poly 1 x = 1"  unfolding one_poly_def by simplemma poly_monom:  fixes a x :: "'a::{comm_semiring_1}"  shows "poly (monom a n) x = a * x ^ n"  by (induct n, simp add: monom_0, simp add: monom_Suc power_Suc mult_ac)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)  donelemma poly_minus [simp]:  fixes x :: "'a::comm_ring"  shows "poly (- p) x = - poly p x"  by (induct p, simp_all)lemma poly_diff [simp]:  fixes x :: "'a::comm_ring"  shows "poly (p - q) x = poly p x - poly q x"  by (simp add: diff_minus)lemma poly_setsum: "poly (∑k∈A. p k) x = (∑k∈A. poly (p k) x)"  by (cases "finite A", induct set: finite, simp_all)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_power [simp]:  fixes p :: "'a::{comm_semiring_1} poly"  shows "poly (p ^ n) x = poly p x ^ n"  by (induct n, simp, simp add: power_Suc)subsection {* Synthetic division *}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 =    poly_rec (0, 0) (λa p (q, r). (pCons r q, a + c * r)) p"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)"  unfolding synthetic_divmod_def  by (simp add: poly_rec_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)"  unfolding synthetic_divmod_def  by (simp add: poly_rec_pCons)lemma snd_synthetic_divmod: "snd (synthetic_divmod p c) = poly p c"  by (induct p, simp, simp add: split_def)lemma synthetic_div_0 [simp]: "synthetic_div 0 c = 0"  unfolding synthetic_div_def by simplemma synthetic_div_pCons [simp]:  "synthetic_div (pCons a p) c = pCons (poly p c) (synthetic_div p c)"  unfolding synthetic_div_def  by (simp add: split_def snd_synthetic_divmod)lemma synthetic_div_eq_0_iff:  "synthetic_div p c = 0 <-> degree p = 0"  by (induct p, simp, case_tac p, simp)lemma degree_synthetic_div:  "degree (synthetic_div p c) = degree p - 1"  by (induct p, simp, simp add: synthetic_div_eq_0_iff)lemma synthetic_div_correct:  "p + smult c (synthetic_div p c) = pCons (poly p c) (synthetic_div p c)"  by (induct p) simp_alllemma synthetic_div_unique_lemma: "smult c p = pCons a p ==> p = 0"by (induct p arbitrary: a) simp_alllemma 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)donelemma synthetic_div_correct':  fixes c :: "'a::comm_ring_1"  shows "[:-c, 1:] * synthetic_div p c + [:poly p c:] = p"  using synthetic_div_correct [of p c]  by (simp add: algebra_simps)lemma poly_eq_0_iff_dvd:  fixes c :: "'a::idom"  shows "poly p c = 0 <-> [:-c, 1:] dvd p"proof  assume "poly p c = 0"  with synthetic_div_correct' [of c p]  have "p = [:-c, 1:] * synthetic_div p c" by simp  then show "[:-c, 1:] dvd p" ..next  assume "[:-c, 1:] dvd p"  then obtain k where "p = [:-c, 1:] * k" by (rule dvdE)  then show "poly p c = 0" by simpqedlemma dvd_iff_poly_eq_0:  fixes c :: "'a::idom"  shows "[:c, 1:] dvd p <-> poly p (-c) = 0"  by (simp add: poly_eq_0_iff_dvd)lemma poly_roots_finite:  fixes p :: "'a::idom poly"  shows "p ≠ 0 ==> finite {x. poly p x = 0}"proof (induct n ≡ "degree p" arbitrary: p)  case (0 p)  then obtain a where "a ≠ 0" and "p = [:a:]"    by (cases p, simp split: if_splits)  then show "finite {x. poly p x = 0}" by simpnext  case (Suc n p)  show "finite {x. poly p x = 0}"  proof (cases "∃x. poly p x = 0")    case False    then show "finite {x. poly p x = 0}" by simp  next    case True    then obtain a where "poly p a = 0" ..    then have "[:-a, 1:] dvd p" by (simp only: poly_eq_0_iff_dvd)    then obtain k where k: "p = [:-a, 1:] * k" ..    with `p ≠ 0` have "k ≠ 0" by auto    with k have "degree p = Suc (degree k)"      by (simp add: degree_mult_eq del: mult_pCons_left)    with `Suc n = degree p` have "n = degree k" by simp    then have "finite {x. poly k x = 0}" using `k ≠ 0` by (rule Suc.hyps)    then have "finite (insert a {x. poly k x = 0})" by simp    then show "finite {x. poly p x = 0}"      by (simp add: k uminus_add_conv_diff Collect_disj_eq               del: mult_pCons_left)  qedqedlemma poly_zero:  fixes p :: "'a::{idom,ring_char_0} poly"  shows "poly p = poly 0 <-> p = 0"apply (cases "p = 0", simp_all)apply (drule poly_roots_finite)apply (auto simp add: infinite_UNIV_char_0)donelemma poly_eq_iff:  fixes p q :: "'a::{idom,ring_char_0} poly"  shows "poly p = poly q <-> p = q"  using poly_zero [of "p - q"]  by (simp add: fun_eq_iff)subsection {* Composition of polynomials *}definition  pcompose :: "'a::comm_semiring_0 poly => 'a poly => 'a poly"where  "pcompose p q = poly_rec 0 (λa _ c. [:a:] + q * c) p"lemma pcompose_0 [simp]: "pcompose 0 q = 0"  unfolding pcompose_def by (simp add: poly_rec_0)lemma pcompose_pCons:  "pcompose (pCons a p) q = [:a:] + q * pcompose p q"  unfolding pcompose_def by (simp add: poly_rec_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 (simp add: pcompose_pCons, clarify)apply (rule degree_add_le, simp)apply (rule order_trans [OF degree_mult_le], simp)donesubsection {* 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)donelemma 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)donelemma 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)donelemma order_2: "p ≠ 0 ==> ¬ [:-a, 1:] ^ Suc (order a p) dvd p"unfolding order_defapply (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)donelemma 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 .qedlemma order_root: "poly p a = 0 <-> p = 0 ∨ order a p ≠ 0"apply (cases "p = 0", simp_all)apply (rule iffI)apply (rule ccontr, simp)apply (frule order_2 [where a=a], simp)apply (simp add: poly_eq_0_iff_dvd)apply (simp add: poly_eq_0_iff_dvd)apply (simp only: order_def)apply (drule not_less_Least, simp)donesubsection {* Configuration of the code generator *}code_datatype "0::'a::zero poly" pConsquickcheck_generator poly constructors: "0::'a::zero poly", pConsinstantiation poly :: ("{zero, equal}") equalbegindefinition  "HOL.equal (p::'a poly) q <-> p = q"instance proofqed (rule equal_poly_def)endlemma eq_poly_code [code]:  "HOL.equal (0::_ poly) (0::_ poly) <-> True"  "HOL.equal (0::_ poly) (pCons b q) <-> HOL.equal 0 b ∧ HOL.equal 0 q"  "HOL.equal (pCons a p) (0::_ poly) <-> HOL.equal a 0 ∧ HOL.equal p 0"  "HOL.equal (pCons a p) (pCons b q) <-> HOL.equal a b ∧ HOL.equal p q"  by (simp_all add: equal)lemma [code nbe]:  "HOL.equal (p :: _ poly) p <-> True"  by (fact equal_refl)lemmas coeff_code [code] =  coeff_0 coeff_pCons_0 coeff_pCons_Suclemmas degree_code [code] =  degree_0 degree_pCons_eq_iflemmas monom_poly_code [code] =  monom_0 monom_Suclemma add_poly_code [code]:  "0 + q = (q :: _ poly)"  "p + 0 = (p :: _ poly)"  "pCons a p + pCons b q = pCons (a + b) (p + q)"by simp_alllemma minus_poly_code [code]:  "- 0 = (0 :: _ poly)"  "- pCons a p = pCons (- a) (- p)"by simp_alllemma diff_poly_code [code]:  "0 - q = (- q :: _ poly)"  "p - 0 = (p :: _ poly)"  "pCons a p - pCons b q = pCons (a - b) (p - q)"by simp_alllemmas smult_poly_code [code] =  smult_0_right smult_pConslemma mult_poly_code [code]:  "0 * q = (0 :: _ poly)"  "pCons a p * q = smult a q + pCons 0 (p * q)"by simp_alllemmas poly_code [code] =  poly_0 poly_pConslemmas synthetic_divmod_code [code] =  synthetic_divmod_0 synthetic_divmod_pConstext {* code generator setup for div and mod *}definition  pdivmod :: "'a::field poly => 'a poly => 'a poly × 'a poly"where  "pdivmod x y = (x div y, x mod y)"lemma div_poly_code [code]: "x div y = fst (pdivmod x y)"  unfolding pdivmod_def by simplemma mod_poly_code [code]: "x mod y = snd (pdivmod x y)"  unfolding pdivmod_def by simplemma pdivmod_0 [code]: "pdivmod 0 y = (0, 0)"  unfolding pdivmod_def by simplemma pdivmod_pCons [code]:  "pdivmod (pCons a x) y =    (if y = 0 then (0, pCons a x) else      (let (q, r) = pdivmod x y;           b = coeff (pCons a r) (degree y) / coeff y (degree y)        in (pCons b q, pCons a r - smult b y)))"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])donelemma poly_gcd_code [code]:  "poly_gcd x y =    (if y = 0 then smult (inverse (coeff x (degree x))) x              else poly_gcd y (x mod y))"  by (simp add: poly_gcd.simps)end`