(* Title: HOL/Computational_Algebra/Polynomial.thy Author: Brian Huffman Author: Clemens Ballarin Author: Amine Chaieb Author: Florian Haftmann *) section ‹Polynomials as type over a ring structure› theory Polynomial imports Complex_Main "HOL-Library.More_List" "HOL-Library.Infinite_Set" Primes begin context semidom_modulo begin lemma not_dvd_imp_mod_neq_0: ‹a mod b ≠ 0› if ‹¬ b dvd a› using that mod_0_imp_dvd [of a b] by blast end subsection ‹Auxiliary: operations for lists (later) representing coefficients› definition cCons :: "'a::zero ⇒ 'a list ⇒ 'a list" (infixr "##" 65) where "x ## xs = (if xs = [] ∧ x = 0 then [] else x # xs)" lemma cCons_0_Nil_eq [simp]: "0 ## [] = []" by (simp add: cCons_def) lemma cCons_Cons_eq [simp]: "x ## y # ys = x # y # ys" by (simp add: cCons_def) lemma cCons_append_Cons_eq [simp]: "x ## xs @ y # ys = x # xs @ y # ys" by (simp add: cCons_def) lemma cCons_not_0_eq [simp]: "x ≠ 0 ⟹ x ## xs = x # xs" by (simp add: cCons_def) lemma strip_while_not_0_Cons_eq [simp]: "strip_while (λx. x = 0) (x # xs) = x ## strip_while (λx. x = 0) xs" proof (cases "x = 0") case False then show ?thesis by simp next case True show ?thesis proof (induct xs rule: rev_induct) case Nil with True show ?case by simp next case (snoc y ys) then show ?case by (cases "y = 0") (simp_all add: append_Cons [symmetric] del: append_Cons) qed qed lemma tl_cCons [simp]: "tl (x ## xs) = xs" by (simp add: cCons_def) subsection ‹Definition of type ‹poly›› typedef (overloaded) 'a poly = "{f :: nat ⇒ 'a::zero. ∀⇩_{∞}n. f n = 0}" morphisms coeff Abs_poly by (auto intro!: ALL_MOST) setup_lifting type_definition_poly lemma poly_eq_iff: "p = q ⟷ (∀n. coeff p n = coeff q n)" by (simp add: coeff_inject [symmetric] fun_eq_iff) lemma poly_eqI: "(⋀n. coeff p n = coeff q n) ⟹ p = q" by (simp add: poly_eq_iff) lemma MOST_coeff_eq_0: "∀⇩_{∞}n. coeff p n = 0" using coeff [of p] by simp lemma coeff_Abs_poly: assumes "⋀i. i > n ⟹ f i = 0" shows "coeff (Abs_poly f) = f" proof (rule Abs_poly_inverse, clarify) have "eventually (λi. i > n) cofinite" by (auto simp: MOST_nat) thus "eventually (λi. f i = 0) cofinite" by eventually_elim (use assms in auto) qed subsection ‹Degree of a polynomial› definition degree :: "'a::zero poly ⇒ nat" where "degree p = (LEAST n. ∀i>n. coeff p i = 0)" lemma degree_cong: assumes "⋀i. coeff p i = 0 ⟷ coeff q i = 0" shows "degree p = degree q" proof - have "(λn. ∀i>n. poly.coeff p i = 0) = (λn. ∀i>n. poly.coeff q i = 0)" using assms by (auto simp: fun_eq_iff) thus ?thesis by (simp only: degree_def) qed lemma coeff_Abs_poly_If_le: "coeff (Abs_poly (λi. if i ≤ n then f i else 0)) = (λi. if i ≤ n then f i else 0)" proof (rule Abs_poly_inverse, clarify) have "eventually (λi. i > n) cofinite" by (auto simp: MOST_nat) thus "eventually (λi. (if i ≤ n then f i else 0) = 0) cofinite" by eventually_elim auto qed lemma coeff_eq_0: assumes "degree p < n" shows "coeff p n = 0" proof - have "∃n. ∀i>n. coeff p i = 0" using MOST_coeff_eq_0 by (simp add: MOST_nat) then have "∀i>degree p. coeff p i = 0" unfolding degree_def by (rule LeastI_ex) with assms show ?thesis by simp qed lemma le_degree: "coeff p n ≠ 0 ⟹ n ≤ degree p" by (erule contrapos_np, rule coeff_eq_0, simp) lemma degree_le: "∀i>n. coeff p i = 0 ⟹ degree p ≤ n" unfolding degree_def by (erule Least_le) lemma less_degree_imp: "n < degree p ⟹ ∃i>n. coeff p i ≠ 0" unfolding degree_def by (drule not_less_Least, simp) subsection ‹The zero polynomial› instantiation poly :: (zero) zero begin lift_definition zero_poly :: "'a poly" is "λ_. 0" by (rule MOST_I) simp instance .. end lemma coeff_0 [simp]: "coeff 0 n = 0" by transfer rule lemma degree_0 [simp]: "degree 0 = 0" by (rule order_antisym [OF degree_le le0]) simp lemma leading_coeff_neq_0: assumes "p ≠ 0" shows "coeff p (degree p) ≠ 0" proof (cases "degree p") case 0 from ‹p ≠ 0› obtain n where "coeff p n ≠ 0" by (auto simp add: poly_eq_iff) then have "n ≤ degree p" by (rule le_degree) with ‹coeff p n ≠ 0› and ‹degree p = 0› show "coeff p (degree p) ≠ 0" by simp next case (Suc n) from ‹degree p = Suc n› have "n < degree p" by simp then have "∃i>n. coeff p i ≠ 0" by (rule less_degree_imp) then obtain i where "n < i" and "coeff p i ≠ 0" by blast from ‹degree p = Suc n› and ‹n < i› have "degree p ≤ i" by simp also from ‹coeff p i ≠ 0› have "i ≤ degree p" by (rule le_degree) finally have "degree p = i" . with ‹coeff p i ≠ 0› show "coeff p (degree p) ≠ 0" by simp qed lemma leading_coeff_0_iff [simp]: "coeff p (degree p) = 0 ⟷ p = 0" by (cases "p = 0") (simp_all add: leading_coeff_neq_0) lemma degree_lessI: assumes "p ≠ 0 ∨ n > 0" "∀k≥n. coeff p k = 0" shows "degree p < n" proof (cases "p = 0") case False show ?thesis proof (rule ccontr) assume *: "¬(degree p < n)" define d where "d = degree p" from ‹p ≠ 0› have "coeff p d ≠ 0" by (auto simp: d_def) moreover have "coeff p d = 0" using assms(2) * by (auto simp: not_less) ultimately show False by contradiction qed qed (use assms in auto) lemma eq_zero_or_degree_less: assumes "degree p ≤ n" and "coeff p n = 0" shows "p = 0 ∨ degree p < n" proof (cases n) case 0 with ‹degree p ≤ n› and ‹coeff p n = 0› have "coeff p (degree p) = 0" by simp then have "p = 0" by simp then show ?thesis .. next case (Suc m) from ‹degree p ≤ n› have "∀i>n. coeff p i = 0" by (simp add: coeff_eq_0) with ‹coeff p n = 0› have "∀i≥n. coeff p i = 0" by (simp add: le_less) with ‹n = Suc m› have "∀i>m. coeff p i = 0" by (simp add: less_eq_Suc_le) then have "degree p ≤ m" by (rule degree_le) with ‹n = Suc m› have "degree p < n" by (simp add: less_Suc_eq_le) then show ?thesis .. qed lemma coeff_0_degree_minus_1: "coeff rrr dr = 0 ⟹ degree rrr ≤ dr ⟹ degree rrr ≤ dr - 1" using eq_zero_or_degree_less by fastforce subsection ‹List-style constructor for polynomials› lift_definition pCons :: "'a::zero ⇒ 'a poly ⇒ 'a poly" is "λa p. case_nat a (coeff p)" by (rule MOST_SucD) (simp add: MOST_coeff_eq_0) lemmas coeff_pCons = pCons.rep_eq lemma coeff_pCons': "poly.coeff (pCons c p) n = (if n = 0 then c else poly.coeff p (n - 1))" by transfer'(auto split: nat.splits) lemma coeff_pCons_0 [simp]: "coeff (pCons a p) 0 = a" by transfer simp lemma coeff_pCons_Suc [simp]: "coeff (pCons a p) (Suc n) = coeff p n" by (simp add: coeff_pCons) lemma degree_pCons_le: "degree (pCons a p) ≤ Suc (degree p)" by (rule degree_le) (simp add: coeff_eq_0 coeff_pCons split: nat.split) lemma degree_pCons_eq: "p ≠ 0 ⟹ degree (pCons a p) = Suc (degree p)" by (simp add: degree_pCons_le le_antisym le_degree) lemma degree_pCons_0: "degree (pCons a 0) = 0" proof - have "degree (pCons a 0) ≤ Suc 0" by (metis (no_types) degree_0 degree_pCons_le) then show ?thesis by (metis coeff_0 coeff_pCons_Suc degree_0 eq_zero_or_degree_less less_Suc0) qed lemma degree_pCons_eq_if [simp]: "degree (pCons a p) = (if p = 0 then 0 else Suc (degree p))" by (simp add: degree_pCons_0 degree_pCons_eq) lemma pCons_0_0 [simp]: "pCons 0 0 = 0" by (rule poly_eqI) (simp add: coeff_pCons split: nat.split) lemma pCons_eq_iff [simp]: "pCons a p = pCons b q ⟷ a = b ∧ p = q" proof safe assume "pCons a p = pCons b q" then have "coeff (pCons a p) 0 = coeff (pCons b q) 0" by simp then show "a = b" by simp next assume "pCons a p = pCons b q" then have "coeff (pCons a p) (Suc n) = coeff (pCons b q) (Suc n)" for n by simp then show "p = q" by (simp add: poly_eq_iff) qed lemma pCons_eq_0_iff [simp]: "pCons a p = 0 ⟷ a = 0 ∧ p = 0" using pCons_eq_iff [of a p 0 0] by simp lemma pCons_cases [cases type: poly]: obtains (pCons) a q where "p = pCons a q" proof show "p = pCons (coeff p 0) (Abs_poly (λn. coeff p (Suc n)))" by transfer (simp_all add: MOST_inj[where f=Suc and P="λn. p n = 0" for p] fun_eq_iff Abs_poly_inverse split: nat.split) qed lemma pCons_induct [case_names 0 pCons, induct type: poly]: assumes zero: "P 0" assumes pCons: "⋀a p. a ≠ 0 ∨ p ≠ 0 ⟹ P p ⟹ P (pCons a p)" shows "P p" proof (induct p rule: measure_induct_rule [where f=degree]) case (less p) obtain a q where "p = pCons a q" by (rule pCons_cases) have "P q" proof (cases "q = 0") case True then show "P q" by (simp add: zero) next case False then have "degree (pCons a q) = Suc (degree q)" by (rule degree_pCons_eq) with ‹p = pCons a q› have "degree q < degree p" by simp then show "P q" by (rule less.hyps) qed have "P (pCons a q)" proof (cases "a ≠ 0 ∨ q ≠ 0") case True with ‹P q› show ?thesis by (auto intro: pCons) next case False with zero show ?thesis by simp qed with ‹p = pCons a q› show ?case 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 then show thesis .. qed subsection ‹Quickcheck generator for polynomials› quickcheck_generator poly constructors: "0 :: _ poly", pCons subsection ‹List-style syntax for polynomials› syntax "_poly" :: "args ⇒ 'a poly" ("[:(_):]") translations "[:x, xs:]" ⇌ "CONST pCons x [:xs:]" "[:x:]" ⇌ "CONST pCons x 0" "[:x:]" ↽ "CONST pCons x (_constrain 0 t)" subsection ‹Representation of polynomials by lists of coefficients› primrec Poly :: "'a::zero list ⇒ 'a poly" where [code_post]: "Poly [] = 0" | [code_post]: "Poly (a # as) = pCons a (Poly as)" lemma Poly_replicate_0 [simp]: "Poly (replicate n 0) = 0" by (induct n) simp_all lemma Poly_eq_0: "Poly as = 0 ⟷ (∃n. as = replicate n 0)" by (induct as) (auto simp add: Cons_replicate_eq) lemma Poly_append_replicate_zero [simp]: "Poly (as @ replicate n 0) = Poly as" by (induct as) simp_all lemma Poly_snoc_zero [simp]: "Poly (as @ [0]) = Poly as" using Poly_append_replicate_zero [of as 1] by simp lemma Poly_cCons_eq_pCons_Poly [simp]: "Poly (a ## p) = pCons a (Poly p)" by (simp add: cCons_def) lemma Poly_on_rev_starting_with_0 [simp]: "hd as = 0 ⟹ Poly (rev (tl as)) = Poly (rev as)" by (cases as) simp_all lemma degree_Poly: "degree (Poly xs) ≤ length xs" by (induct xs) simp_all lemma coeff_Poly_eq [simp]: "coeff (Poly xs) = nth_default 0 xs" by (induct xs) (simp_all add: fun_eq_iff coeff_pCons split: nat.splits) definition coeffs :: "'a poly ⇒ 'a::zero list" where "coeffs p = (if p = 0 then [] else map (λi. coeff p i) [0 ..< Suc (degree p)])" lemma coeffs_eq_Nil [simp]: "coeffs p = [] ⟷ p = 0" by (simp add: coeffs_def) lemma not_0_coeffs_not_Nil: "p ≠ 0 ⟹ coeffs p ≠ []" by simp lemma coeffs_0_eq_Nil [simp]: "coeffs 0 = []" by simp lemma coeffs_pCons_eq_cCons [simp]: "coeffs (pCons a p) = a ## coeffs p" proof - have *: "∀m∈set ms. m > 0 ⟹ map (case_nat x f) ms = map f (map (λn. n - 1) ms)" for ms :: "nat list" and f :: "nat ⇒ 'a" and x :: "'a" by (induct ms) (auto split: nat.split) show ?thesis by (simp add: * coeffs_def upt_conv_Cons coeff_pCons map_decr_upt del: upt_Suc) qed lemma length_coeffs: "p ≠ 0 ⟹ length (coeffs p) = degree p + 1" by (simp add: coeffs_def) lemma coeffs_nth: "p ≠ 0 ⟹ n ≤ degree p ⟹ coeffs p ! n = coeff p n" by (auto simp: coeffs_def simp del: upt_Suc) lemma coeff_in_coeffs: "p ≠ 0 ⟹ n ≤ degree p ⟹ coeff p n ∈ set (coeffs p)" using coeffs_nth [of p n, symmetric] by (simp add: length_coeffs) lemma not_0_cCons_eq [simp]: "p ≠ 0 ⟹ a ## coeffs p = a # coeffs p" by (simp add: cCons_def) lemma Poly_coeffs [simp, code abstype]: "Poly (coeffs p) = p" by (induct p) auto lemma coeffs_Poly [simp]: "coeffs (Poly as) = strip_while (HOL.eq 0) as" proof (induct as) case Nil then show ?case by simp next case (Cons a as) from replicate_length_same [of as 0] have "(∀n. as ≠ replicate n 0) ⟷ (∃a∈set as. a ≠ 0)" by (auto dest: sym [of _ as]) with Cons show ?case by auto qed lemma no_trailing_coeffs [simp]: "no_trailing (HOL.eq 0) (coeffs p)" by (induct p) auto lemma strip_while_coeffs [simp]: "strip_while (HOL.eq 0) (coeffs p) = coeffs p" by simp lemma coeffs_eq_iff: "p = q ⟷ coeffs p = coeffs q" (is "?P ⟷ ?Q") proof assume ?P then show ?Q by simp next assume ?Q then have "Poly (coeffs p) = Poly (coeffs q)" by simp then show ?P by simp qed lemma nth_default_coeffs_eq: "nth_default 0 (coeffs p) = coeff p" by (simp add: fun_eq_iff coeff_Poly_eq [symmetric]) lemma [code]: "coeff p = nth_default 0 (coeffs p)" by (simp add: nth_default_coeffs_eq) lemma coeffs_eqI: assumes coeff: "⋀n. coeff p n = nth_default 0 xs n" assumes zero: "no_trailing (HOL.eq 0) xs" shows "coeffs p = xs" proof - from coeff have "p = Poly xs" by (simp add: poly_eq_iff) with zero show ?thesis by simp qed lemma degree_eq_length_coeffs [code]: "degree p = length (coeffs p) - 1" by (simp add: coeffs_def) lemma length_coeffs_degree: "p ≠ 0 ⟹ length (coeffs p) = Suc (degree p)" by (induct p) (auto simp: 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) lemma set_coeffs_subset_singleton_0_iff [simp]: "set (coeffs p) ⊆ {0} ⟷ p = 0" by (auto simp add: coeffs_def intro: classical) lemma set_coeffs_not_only_0 [simp]: "set (coeffs p) ≠ {0}" by (auto simp add: set_eq_subset) lemma forall_coeffs_conv: "(∀n. P (coeff p n)) ⟷ (∀c ∈ set (coeffs p). P c)" if "P 0" using that by (auto simp add: coeffs_def) (metis atLeastLessThan_iff coeff_eq_0 not_less_iff_gr_or_eq zero_le) instantiation poly :: ("{zero, equal}") equal begin definition [code]: "HOL.equal (p::'a poly) q ⟷ HOL.equal (coeffs p) (coeffs q)" instance by standard (simp add: equal equal_poly_def coeffs_eq_iff) end lemma [code nbe]: "HOL.equal (p :: _ poly) p ⟷ True" by (fact equal_refl) definition is_zero :: "'a::zero poly ⇒ bool" where [code]: "is_zero p ⟷ List.null (coeffs p)" lemma is_zero_null [code_abbrev]: "is_zero p ⟷ p = 0" by (simp add: is_zero_def null_def) text ‹Reconstructing the polynomial from the list› ― ‹contributed by Sebastiaan J.C. Joosten and René Thiemann› definition poly_of_list :: "'a::comm_monoid_add list ⇒ 'a poly" where [simp]: "poly_of_list = Poly" lemma poly_of_list_impl [code abstract]: "coeffs (poly_of_list as) = strip_while (HOL.eq 0) as" by simp subsection ‹Fold combinator for polynomials› definition fold_coeffs :: "('a::zero ⇒ 'b ⇒ 'b) ⇒ 'a poly ⇒ 'b ⇒ 'b" where "fold_coeffs f p = foldr f (coeffs p)" lemma fold_coeffs_0_eq [simp]: "fold_coeffs f 0 = id" by (simp add: fold_coeffs_def) lemma fold_coeffs_pCons_eq [simp]: "f 0 = id ⟹ fold_coeffs f (pCons a p) = f a ∘ fold_coeffs f p" by (simp add: fold_coeffs_def cCons_def fun_eq_iff) lemma fold_coeffs_pCons_0_0_eq [simp]: "fold_coeffs f (pCons 0 0) = id" by (simp add: fold_coeffs_def) lemma fold_coeffs_pCons_coeff_not_0_eq [simp]: "a ≠ 0 ⟹ fold_coeffs f (pCons a p) = f a ∘ fold_coeffs f p" by (simp add: fold_coeffs_def) lemma fold_coeffs_pCons_not_0_0_eq [simp]: "p ≠ 0 ⟹ fold_coeffs f (pCons a p) = f a ∘ fold_coeffs f p" by (simp add: fold_coeffs_def) subsection ‹Canonical morphism on polynomials -- evaluation› definition poly :: ‹'a::comm_semiring_0 poly ⇒ 'a ⇒ 'a› where ‹poly p a = horner_sum id a (coeffs p)› lemma poly_eq_fold_coeffs: ‹poly p = fold_coeffs (λa f x. a + x * f x) p (λx. 0)› by (induction p) (auto simp add: fun_eq_iff poly_def) lemma poly_0 [simp]: "poly 0 x = 0" by (simp add: poly_def) lemma poly_pCons [simp]: "poly (pCons a p) x = a + x * poly p x" by (cases "p = 0 ∧ a = 0") (auto simp add: poly_def) lemma poly_altdef: "poly p x = (∑i≤degree p. coeff p i * x ^ i)" for x :: "'a::{comm_semiring_0,semiring_1}" proof (induction p rule: pCons_induct) case 0 then show ?case by simp next case (pCons a p) show ?case proof (cases "p = 0") case True then show ?thesis by simp next case False let ?p' = "pCons a p" note poly_pCons[of a p x] also note pCons.IH also have "a + x * (∑i≤degree p. coeff p i * x ^ i) = coeff ?p' 0 * x^0 + (∑i≤degree p. coeff ?p' (Suc i) * x^Suc i)" by (simp add: field_simps sum_distrib_left coeff_pCons) also note sum.atMost_Suc_shift[symmetric] also note degree_pCons_eq[OF ‹p ≠ 0›, of a, symmetric] finally show ?thesis . qed qed lemma poly_0_coeff_0: "poly p 0 = coeff p 0" by (cases p) (auto simp: poly_altdef) subsection ‹Monomials› lift_definition monom :: "'a ⇒ nat ⇒ 'a::zero poly" is "λa m n. if m = n then a else 0" by (simp add: MOST_iff_cofinite) 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 = [:a:]" by (rule poly_eqI) (simp add: coeff_pCons split: nat.split) lemma monom_Suc: "monom a (Suc n) = pCons 0 (monom a n)" by (rule poly_eqI) (simp add: coeff_pCons split: nat.split) lemma monom_eq_0 [simp]: "monom 0 n = 0" by (rule poly_eqI) simp lemma monom_eq_0_iff [simp]: "monom a n = 0 ⟷ a = 0" by (simp add: poly_eq_iff) lemma monom_eq_iff [simp]: "monom a n = monom b n ⟷ a = b" by (simp add: poly_eq_iff) lemma degree_monom_le: "degree (monom a n) ≤ n" by (rule degree_le, simp) lemma degree_monom_eq: "a ≠ 0 ⟹ degree (monom a n) = n" by (metis coeff_monom leading_coeff_0_iff) lemma coeffs_monom [code abstract]: "coeffs (monom a n) = (if a = 0 then [] else replicate n 0 @ [a])" by (induct n) (simp_all add: monom_0 monom_Suc) lemma fold_coeffs_monom [simp]: "a ≠ 0 ⟹ fold_coeffs f (monom a n) = f 0 ^^ n ∘ f a" by (simp add: fold_coeffs_def coeffs_monom fun_eq_iff) lemma poly_monom: "poly (monom a n) x = a * x ^ n" for a x :: "'a::comm_semiring_1" by (cases "a = 0", simp_all) (induct n, simp_all add: mult.left_commute poly_eq_fold_coeffs) lemma monom_eq_iff': "monom c n = monom d m ⟷ c = d ∧ (c = 0 ∨ n = m)" by (auto simp: poly_eq_iff) lemma monom_eq_const_iff: "monom c n = [:d:] ⟷ c = d ∧ (c = 0 ∨ n = 0)" using monom_eq_iff'[of c n d 0] by (simp add: monom_0) subsection ‹Leading coefficient› abbreviation lead_coeff:: "'a::zero poly ⇒ 'a" where "lead_coeff p ≡ coeff p (degree p)" lemma lead_coeff_pCons[simp]: "p ≠ 0 ⟹ lead_coeff (pCons a p) = lead_coeff p" "p = 0 ⟹ lead_coeff (pCons a p) = a" by auto lemma lead_coeff_monom [simp]: "lead_coeff (monom c n) = c" by (cases "c = 0") (simp_all add: degree_monom_eq) lemma last_coeffs_eq_coeff_degree: "last (coeffs p) = lead_coeff p" if "p ≠ 0" using that by (simp add: coeffs_def) subsection ‹Addition and subtraction› instantiation poly :: (comm_monoid_add) comm_monoid_add begin lift_definition plus_poly :: "'a poly ⇒ 'a poly ⇒ 'a poly" is "λp q n. coeff p n + coeff q n" proof - fix q p :: "'a poly" show "∀⇩_{∞}n. coeff p n + coeff q n = 0" using MOST_coeff_eq_0[of p] MOST_coeff_eq_0[of q] by eventually_elim simp qed lemma coeff_add [simp]: "coeff (p + q) n = coeff p n + coeff q n" by (simp add: plus_poly.rep_eq) instance proof fix p q r :: "'a poly" show "(p + q) + r = p + (q + r)" by (simp add: poly_eq_iff add.assoc) show "p + q = q + p" by (simp add: poly_eq_iff add.commute) show "0 + p = p" by (simp add: poly_eq_iff) qed end instantiation poly :: (cancel_comm_monoid_add) cancel_comm_monoid_add begin lift_definition minus_poly :: "'a poly ⇒ 'a poly ⇒ 'a poly" is "λp q n. coeff p n - coeff q n" proof - fix q p :: "'a poly" show "∀⇩_{∞}n. coeff p n - coeff q n = 0" using MOST_coeff_eq_0[of p] MOST_coeff_eq_0[of q] by eventually_elim simp qed lemma coeff_diff [simp]: "coeff (p - q) n = coeff p n - coeff q n" by (simp add: minus_poly.rep_eq) instance proof fix p q r :: "'a poly" show "p + q - p = q" by (simp add: poly_eq_iff) show "p - q - r = p - (q + r)" by (simp add: poly_eq_iff diff_diff_eq) qed end instantiation poly :: (ab_group_add) ab_group_add begin lift_definition uminus_poly :: "'a poly ⇒ 'a poly" is "λp n. - coeff p n" proof - fix p :: "'a poly" show "∀⇩_{∞}n. - coeff p n = 0" using MOST_coeff_eq_0 by simp qed lemma coeff_minus [simp]: "coeff (- p) n = - coeff p n" by (simp add: uminus_poly.rep_eq) instance proof fix p q :: "'a poly" show "- p + p = 0" by (simp add: poly_eq_iff) show "p - q = p + - q" by (simp add: poly_eq_iff) qed end lemma add_pCons [simp]: "pCons a p + pCons b q = pCons (a + b) (p + q)" by (rule poly_eqI) (simp add: coeff_pCons split: nat.split) lemma minus_pCons [simp]: "- pCons a p = pCons (- a) (- p)" by (rule poly_eqI) (simp add: coeff_pCons split: nat.split) lemma diff_pCons [simp]: "pCons a p - pCons b q = pCons (a - b) (p - q)" by (rule poly_eqI) (simp add: coeff_pCons split: nat.split) lemma degree_add_le_max: "degree (p + q) ≤ max (degree p) (degree q)" by (rule degree_le) (auto simp add: coeff_eq_0) lemma degree_add_le: "degree p ≤ n ⟹ degree q ≤ n ⟹ degree (p + q) ≤ n" by (auto intro: order_trans degree_add_le_max) lemma degree_add_less: "degree p < n ⟹ degree q < n ⟹ degree (p + q) < n" by (auto intro: le_less_trans degree_add_le_max) lemma degree_add_eq_right: assumes "degree p < degree q" shows "degree (p + q) = degree q" proof (cases "q = 0") case False show ?thesis proof (rule order_antisym) show "degree (p + q) ≤ degree q" by (simp add: assms degree_add_le order.strict_implies_order) show "degree q ≤ degree (p + q)" by (simp add: False assms coeff_eq_0 le_degree) qed qed (use assms in auto) lemma degree_add_eq_left: "degree q < degree p ⟹ degree (p + q) = degree p" using degree_add_eq_right [of q p] by (simp add: add.commute) lemma degree_minus [simp]: "degree (- p) = degree p" by (simp add: degree_def) lemma lead_coeff_add_le: "degree p < degree q ⟹ lead_coeff (p + q) = lead_coeff q" by (metis coeff_add coeff_eq_0 monoid_add_class.add.left_neutral degree_add_eq_right) lemma lead_coeff_minus: "lead_coeff (- p) = - lead_coeff p" by (metis coeff_minus degree_minus) lemma degree_diff_le_max: "degree (p - q) ≤ max (degree p) (degree q)" for p q :: "'a::ab_group_add poly" using degree_add_le [where p=p and q="-q"] by simp lemma degree_diff_le: "degree p ≤ n ⟹ degree q ≤ n ⟹ degree (p - q) ≤ n" for p q :: "'a::ab_group_add poly" using degree_add_le [of p n "- q"] by simp lemma degree_diff_less: "degree p < n ⟹ degree q < n ⟹ degree (p - q) < n" for p q :: "'a::ab_group_add poly" using degree_add_less [of p n "- q"] by simp lemma add_monom: "monom a n + monom b n = monom (a + b) n" by (rule poly_eqI) simp lemma diff_monom: "monom a n - monom b n = monom (a - b) n" by (rule poly_eqI) simp lemma minus_monom: "- monom a n = monom (- a) n" by (rule poly_eqI) simp lemma coeff_sum: "coeff (∑x∈A. p x) i = (∑x∈A. coeff (p x) i)" by (induct A rule: infinite_finite_induct) simp_all lemma monom_sum: "monom (∑x∈A. a x) n = (∑x∈A. monom (a x) n)" by (rule poly_eqI) (simp add: coeff_sum) fun plus_coeffs :: "'a::comm_monoid_add list ⇒ 'a list ⇒ 'a list" where "plus_coeffs xs [] = xs" | "plus_coeffs [] ys = ys" | "plus_coeffs (x # xs) (y # ys) = (x + y) ## plus_coeffs xs ys" lemma coeffs_plus_eq_plus_coeffs [code abstract]: "coeffs (p + q) = plus_coeffs (coeffs p) (coeffs q)" proof - have *: "nth_default 0 (plus_coeffs xs ys) n = nth_default 0 xs n + nth_default 0 ys n" for xs ys :: "'a list" and 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 have **: "no_trailing (HOL.eq 0) (plus_coeffs xs ys)" if "no_trailing (HOL.eq 0) xs" and "no_trailing (HOL.eq 0) ys" for xs ys :: "'a list" using that by (induct xs ys rule: plus_coeffs.induct) (simp_all add: cCons_def) show ?thesis by (rule coeffs_eqI) (auto simp add: * nth_default_coeffs_eq intro: **) qed lemma coeffs_uminus [code abstract]: "coeffs (- p) = map uminus (coeffs p)" proof - have eq_0: "HOL.eq 0 ∘ uminus = HOL.eq (0::'a)" by (simp add: fun_eq_iff) show ?thesis by (rule coeffs_eqI) (simp_all add: nth_default_map_eq nth_default_coeffs_eq no_trailing_map eq_0) qed lemma [code]: "p - q = p + - q" for p q :: "'a::ab_group_add poly" by (fact diff_conv_add_uminus) lemma poly_add [simp]: "poly (p + q) x = poly p x + poly q x" proof (induction p arbitrary: q) case (pCons a p) then show ?case by (cases q) (simp add: algebra_simps) qed auto lemma poly_minus [simp]: "poly (- p) x = - poly p x" for x :: "'a::comm_ring" by (induct p) simp_all lemma poly_diff [simp]: "poly (p - q) x = poly p x - poly q x" for x :: "'a::comm_ring" using poly_add [of p "- q" x] by simp lemma poly_sum: "poly (∑k∈A. p k) x = (∑k∈A. poly (p k) x)" by (induct A rule: infinite_finite_induct) simp_all lemma poly_sum_list: "poly (∑p←ps. p) y = (∑p←ps. poly p y)" by (induction ps) auto lemma poly_sum_mset: "poly (∑x∈#A. p x) y = (∑x∈#A. poly (p x) y)" by (induction A) auto lemma degree_sum_le: "finite S ⟹ (⋀p. p ∈ S ⟹ degree (f p) ≤ n) ⟹ degree (sum f S) ≤ n" proof (induct S rule: finite_induct) case empty then show ?case by simp next case (insert p S) then have "degree (sum f S) ≤ n" "degree (f p) ≤ n" by auto then show ?case unfolding sum.insert[OF insert(1-2)] by (metis degree_add_le) qed lemma degree_sum_less: assumes "⋀x. x ∈ A ⟹ degree (f x) < n" "n > 0" shows "degree (sum f A) < n" using assms by (induction rule: infinite_finite_induct) (auto intro!: degree_add_less) lemma poly_as_sum_of_monoms': assumes "degree p ≤ n" shows "(∑i≤n. monom (coeff p i) i) = p" proof - have eq: "⋀i. {..n} ∩ {i} = (if i ≤ n then {i} else {})" by auto from assms show ?thesis by (simp add: poly_eq_iff coeff_sum coeff_eq_0 sum.If_cases eq if_distrib[where f="λx. x * a" for a]) qed lemma poly_as_sum_of_monoms: "(∑i≤degree p. monom (coeff p i) i) = p" by (intro poly_as_sum_of_monoms' order_refl) lemma Poly_snoc: "Poly (xs @ [x]) = Poly xs + monom x (length xs)" by (induct xs) (simp_all add: monom_0 monom_Suc) subsection ‹Multiplication by a constant, polynomial multiplication and the unit polynomial› lift_definition smult :: "'a::comm_semiring_0 ⇒ 'a poly ⇒ 'a poly" is "λa p n. a * coeff p n" proof - fix a :: 'a and p :: "'a poly" show "∀⇩_{∞}i. a * coeff p i = 0" using MOST_coeff_eq_0[of p] by eventually_elim simp qed lemma coeff_smult [simp]: "coeff (smult a p) n = a * coeff p n" by (simp add: smult.rep_eq) lemma degree_smult_le: "degree (smult a p) ≤ degree p" by (rule degree_le) (simp add: coeff_eq_0) lemma smult_smult [simp]: "smult a (smult b p) = smult (a * b) p" by (rule poly_eqI) (simp add: mult.assoc) lemma smult_0_right [simp]: "smult a 0 = 0" by (rule poly_eqI) simp lemma smult_0_left [simp]: "smult 0 p = 0" by (rule poly_eqI) simp lemma smult_1_left [simp]: "smult (1::'a::comm_semiring_1) p = p" by (rule poly_eqI) simp lemma smult_add_right: "smult a (p + q) = smult a p + smult a q" by (rule poly_eqI) (simp add: algebra_simps) lemma smult_add_left: "smult (a + b) p = smult a p + smult b p" by (rule poly_eqI) (simp add: algebra_simps) lemma smult_minus_right [simp]: "smult a (- p) = - smult a p" for a :: "'a::comm_ring" by (rule poly_eqI) simp lemma smult_minus_left [simp]: "smult (- a) p = - smult a p" for a :: "'a::comm_ring" by (rule poly_eqI) simp lemma smult_diff_right: "smult a (p - q) = smult a p - smult a q" for a :: "'a::comm_ring" by (rule poly_eqI) (simp add: algebra_simps) lemma smult_diff_left: "smult (a - b) p = smult a p - smult b p" for a b :: "'a::comm_ring" by (rule poly_eqI) (simp add: algebra_simps) lemmas smult_distribs = smult_add_left smult_add_right smult_diff_left smult_diff_right lemma smult_pCons [simp]: "smult a (pCons b p) = pCons (a * b) (smult a p)" by (rule poly_eqI) (simp add: coeff_pCons split: nat.split) lemma smult_monom: "smult a (monom b n) = monom (a * b) n" by (induct n) (simp_all add: monom_0 monom_Suc) lemma smult_Poly: "smult c (Poly xs) = Poly (map ((*) c) xs)" by (auto simp: poly_eq_iff nth_default_def) lemma degree_smult_eq [simp]: "degree (smult a p) = (if a = 0 then 0 else degree p)" for a :: "'a::{comm_semiring_0,semiring_no_zero_divisors}" by (cases "a = 0") (simp_all add: degree_def) lemma smult_eq_0_iff [simp]: "smult a p = 0 ⟷ a = 0 ∨ p = 0" for a :: "'a::{comm_semiring_0,semiring_no_zero_divisors}" by (simp add: poly_eq_iff) lemma coeffs_smult [code abstract]: "coeffs (smult a p) = (if a = 0 then [] else map (Groups.times a) (coeffs p))" for p :: "'a::{comm_semiring_0,semiring_no_zero_divisors} poly" proof - have eq_0: "HOL.eq 0 ∘ times a = HOL.eq (0::'a)" if "a ≠ 0" using that by (simp add: fun_eq_iff) show ?thesis by (rule coeffs_eqI) (auto simp add: no_trailing_map nth_default_map_eq nth_default_coeffs_eq eq_0) qed lemma smult_eq_iff: fixes b :: "'a :: field" assumes "b ≠ 0" shows "smult a p = smult b q ⟷ smult (a / b) p = q" (is "?lhs ⟷ ?rhs") proof assume ?lhs also from assms have "smult (inverse b) … = q" by simp finally show ?rhs by (simp add: field_simps) next assume ?rhs with assms show ?lhs by auto qed instantiation poly :: (comm_semiring_0) comm_semiring_0 begin definition "p * q = fold_coeffs (λa p. smult a q + pCons 0 p) p 0" lemma mult_poly_0_left: "(0::'a poly) * q = 0" by (simp add: times_poly_def) lemma mult_pCons_left [simp]: "pCons a p * q = smult a q + pCons 0 (p * q)" by (cases "p = 0 ∧ a = 0") (auto simp add: times_poly_def) lemma mult_poly_0_right: "p * (0::'a poly) = 0" by (induct p) (simp_all add: mult_poly_0_left) lemma mult_pCons_right [simp]: "p * pCons a q = smult a p + pCons 0 (p * q)" by (induct p) (simp_all add: mult_poly_0_left algebra_simps) lemmas mult_poly_0 = mult_poly_0_left mult_poly_0_right lemma mult_smult_left [simp]: "smult a p * q = smult a (p * q)" by (induct p) (simp_all add: mult_poly_0 smult_add_right) lemma mult_smult_right [simp]: "p * smult a q = smult a (p * q)" by (induct q) (simp_all add: mult_poly_0 smult_add_right) lemma mult_poly_add_left: "(p + q) * r = p * r + q * r" for p q r :: "'a poly" by (induct r) (simp_all add: mult_poly_0 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_all add: mult_poly_0 mult_poly_add_left) show "p * q = q * p" by (induct p) (simp_all add: mult_poly_0) qed end lemma coeff_mult_degree_sum: "coeff (p * q) (degree p + degree q) = coeff p (degree p) * coeff q (degree q)" by (induct p) (simp_all add: coeff_eq_0) instance poly :: ("{comm_semiring_0,semiring_no_zero_divisors}") semiring_no_zero_divisors proof fix p q :: "'a poly" assume "p ≠ 0" and "q ≠ 0" have "coeff (p * q) (degree p + degree q) = coeff p (degree p) * coeff q (degree q)" by (rule coeff_mult_degree_sum) also from ‹p ≠ 0› ‹q ≠ 0› have "coeff p (degree p) * coeff q (degree q) ≠ 0" by simp finally have "∃n. coeff (p * q) n ≠ 0" .. then show "p * q ≠ 0" by (simp add: poly_eq_iff) qed instance poly :: (comm_semiring_0_cancel) comm_semiring_0_cancel .. lemma coeff_mult: "coeff (p * q) n = (∑i≤n. coeff p i * coeff q (n-i))" proof (induct p arbitrary: n) case 0 show ?case by simp next case (pCons a p n) then show ?case by (cases n) (simp_all add: sum.atMost_Suc_shift del: sum.atMost_Suc) qed lemma coeff_mult_0: "coeff (p * q) 0 = coeff p 0 * coeff q 0" by (simp add: coeff_mult) lemma degree_mult_le: "degree (p * q) ≤ degree p + degree q" proof (rule degree_le) show "∀i>degree p + degree q. coeff (p * q) i = 0" by (induct p) (simp_all add: coeff_eq_0 coeff_pCons split: nat.split) qed lemma mult_monom: "monom a m * monom b n = monom (a * b) (m + n)" by (induct m) (simp add: monom_0 smult_monom, simp add: monom_Suc) instantiation poly :: (comm_semiring_1) comm_semiring_1 begin lift_definition one_poly :: "'a poly" is "λn. of_bool (n = 0)" by (rule MOST_SucD) simp lemma coeff_1 [simp]: "coeff 1 n = of_bool (n = 0)" by (simp add: one_poly.rep_eq) lemma one_pCons: "1 = [:1:]" by (simp add: poly_eq_iff coeff_pCons split: nat.splits) lemma pCons_one: "[:1:] = 1" by (simp add: one_pCons) instance by standard (simp_all add: one_pCons) end lemma poly_1 [simp]: "poly 1 x = 1" by (simp add: one_pCons) lemma one_poly_eq_simps [simp]: "1 = [:1:] ⟷ True" "[:1:] = 1 ⟷ True" by (simp_all add: one_pCons) lemma degree_1 [simp]: "degree 1 = 0" by (simp add: one_pCons) lemma coeffs_1_eq [simp, code abstract]: "coeffs 1 = [1]" by (simp add: one_pCons) lemma smult_one [simp]: "smult c 1 = [:c:]" by (simp add: one_pCons) lemma monom_eq_1 [simp]: "monom 1 0 = 1" by (simp add: monom_0 one_pCons) lemma monom_eq_1_iff: "monom c n = 1 ⟷ c = 1 ∧ n = 0" using monom_eq_const_iff [of c n 1] by auto lemma monom_altdef: "monom c n = smult c ([:0, 1:] ^ n)" by (induct n) (simp_all add: monom_0 monom_Suc) instance poly :: ("{comm_semiring_1,semiring_1_no_zero_divisors}") semiring_1_no_zero_divisors .. instance poly :: (comm_ring) comm_ring .. instance poly :: (comm_ring_1) comm_ring_1 .. instance poly :: (comm_ring_1) comm_semiring_1_cancel .. lemma prod_smult: "(∏x∈A. smult (c x) (p x)) = smult (prod c A) (prod p A)" by (induction A rule: infinite_finite_induct) (auto simp: mult_ac) lemma degree_power_le: "degree (p ^ n) ≤ degree p * n" by (induct n) (auto intro: order_trans degree_mult_le) lemma coeff_0_power: "coeff (p ^ n) 0 = coeff p 0 ^ n" by (induct n) (simp_all add: coeff_mult) lemma poly_smult [simp]: "poly (smult a p) x = a * poly p x" by (induct p) (simp_all add: algebra_simps) lemma poly_mult [simp]: "poly (p * q) x = poly p x * poly q x" by (induct p) (simp_all add: algebra_simps) lemma poly_power [simp]: "poly (p ^ n) x = poly p x ^ n" for p :: "'a::comm_semiring_1 poly" by (induct n) simp_all lemma poly_prod: "poly (∏k∈A. p k) x = (∏k∈A. poly (p k) x)" by (induct A rule: infinite_finite_induct) simp_all lemma poly_prod_list: "poly (∏p←ps. p) y = (∏p←ps. poly p y)" by (induction ps) auto lemma poly_prod_mset: "poly (∏x∈#A. p x) y = (∏x∈#A. poly (p x) y)" by (induction A) auto lemma poly_const_pow: "[: c :] ^ n = [: c ^ n :]" by (induction n) (auto simp: algebra_simps) lemma monom_power: "monom c n ^ k = monom (c ^ k) (n * k)" by (induction k) (auto simp: mult_monom) lemma degree_prod_sum_le: "finite S ⟹ degree (prod f S) ≤ sum (degree ∘ f) S" proof (induct S rule: finite_induct) case empty then show ?case by simp next case (insert a S) show ?case unfolding prod.insert[OF insert(1-2)] sum.insert[OF insert(1-2)] by (rule le_trans[OF degree_mult_le]) (use insert in auto) qed lemma coeff_0_prod_list: "coeff (prod_list xs) 0 = prod_list (map (λp. coeff p 0) xs)" by (induct xs) (simp_all add: coeff_mult) lemma coeff_monom_mult: "coeff (monom c n * p) k = (if k < n then 0 else c * coeff p (k - n))" proof - have "coeff (monom c n * p) k = (∑i≤k. (if n = i then c else 0) * coeff p (k - i))" by (simp add: coeff_mult) also have "… = (∑i≤k. (if n = i then c * coeff p (k - i) else 0))" by (intro sum.cong) simp_all also have "… = (if k < n then 0 else c * coeff p (k - n))" by simp finally show ?thesis . qed lemma monom_1_dvd_iff': "monom 1 n dvd p ⟷ (∀k<n. coeff p k = 0)" proof assume "monom 1 n dvd p" then obtain r where "p = monom 1 n * r" by (rule dvdE) then show "∀k<n. coeff p k = 0" by (simp add: coeff_mult) next assume zero: "(∀k<n. coeff p k = 0)" define r where "r = Abs_poly (λk. coeff p (k + n))" have "∀⇩_{∞}k. coeff p (k + n) = 0" by (subst cofinite_eq_sequentially, subst eventually_sequentially_seg, subst cofinite_eq_sequentially [symmetric]) transfer then have coeff_r [simp]: "coeff r k = coeff p (k + n)" for k unfolding r_def by (subst poly.Abs_poly_inverse) simp_all have "p = monom 1 n * r" by (rule poly_eqI, subst coeff_monom_mult) (simp_all add: zero) then show "monom 1 n dvd p" by simp qed subsection ‹Mapping polynomials› definition map_poly :: "('a :: zero ⇒ 'b :: zero) ⇒ 'a poly ⇒ 'b poly" where "map_poly f p = Poly (map f (coeffs p))" lemma map_poly_0 [simp]: "map_poly f 0 = 0" by (simp add: map_poly_def) lemma map_poly_1: "map_poly f 1 = [:f 1:]" by (simp add: map_poly_def) lemma map_poly_1' [simp]: "f 1 = 1 ⟹ map_poly f 1 = 1" by (simp add: map_poly_def one_pCons) lemma coeff_map_poly: assumes "f 0 = 0" shows "coeff (map_poly f p) n = f (coeff p n)" by (auto simp: assms map_poly_def nth_default_def coeffs_def not_less Suc_le_eq coeff_eq_0 simp del: upt_Suc) lemma coeffs_map_poly [code abstract]: "coeffs (map_poly f p) = strip_while ((=) 0) (map f (coeffs p))" by (simp add: map_poly_def) lemma coeffs_map_poly': assumes "⋀x. x ≠ 0 ⟹ f x ≠ 0" shows "coeffs (map_poly f p) = map f (coeffs p)" using assms by (auto simp add: coeffs_map_poly strip_while_idem_iff last_coeffs_eq_coeff_degree no_trailing_unfold last_map) lemma set_coeffs_map_poly: "(⋀x. f x = 0 ⟷ x = 0) ⟹ set (coeffs (map_poly f p)) = f ` set (coeffs p)" by (simp add: coeffs_map_poly') lemma degree_map_poly: assumes "⋀x. x ≠ 0 ⟹ f x ≠ 0" shows "degree (map_poly f p) = degree p" by (simp add: degree_eq_length_coeffs coeffs_map_poly' assms) lemma map_poly_eq_0_iff: assumes "f 0 = 0" "⋀x. x ∈ set (coeffs p) ⟹ x ≠ 0 ⟹ f x ≠ 0" shows "map_poly f p = 0 ⟷ p = 0" proof - have "(coeff (map_poly f p) n = 0) = (coeff p n = 0)" for n proof - have "coeff (map_poly f p) n = f (coeff p n)" by (simp add: coeff_map_poly assms) also have "… = 0 ⟷ coeff p n = 0" proof (cases "n < length (coeffs p)") case True then have "coeff p n ∈ set (coeffs p)" by (auto simp: coeffs_def simp del: upt_Suc) with assms show "f (coeff p n) = 0 ⟷ coeff p n = 0" by auto next case False then show ?thesis by (auto simp: assms length_coeffs nth_default_coeffs_eq [symmetric] nth_default_def) qed finally show ?thesis . qed then show ?thesis by (auto simp: poly_eq_iff) qed lemma map_poly_smult: assumes "f 0 = 0""⋀c x. f (c * x) = f c * f x" shows "map_poly f (smult c p) = smult (f c) (map_poly f p)" by (intro poly_eqI) (simp_all add: assms coeff_map_poly) lemma map_poly_pCons: assumes "f 0 = 0" shows "map_poly f (pCons c p) = pCons (f c) (map_poly f p)" by (intro poly_eqI) (simp_all add: assms coeff_map_poly coeff_pCons split: nat.splits) lemma map_poly_map_poly: assumes "f 0 = 0" "g 0 = 0" shows "map_poly f (map_poly g p) = map_poly (f ∘ g) p" by (intro poly_eqI) (simp add: coeff_map_poly assms) lemma map_poly_id [simp]: "map_poly id p = p" by (simp add: map_poly_def) lemma map_poly_id' [simp]: "map_poly (λx. x) p = p" by (simp add: map_poly_def) lemma map_poly_cong: assumes "(⋀x. x ∈ set (coeffs p) ⟹ f x = g x)" shows "map_poly f p = map_poly g p" proof - from assms have "map f (coeffs p) = map g (coeffs p)" by (intro map_cong) simp_all then show ?thesis by (simp only: coeffs_eq_iff coeffs_map_poly) qed lemma map_poly_monom: "f 0 = 0 ⟹ map_poly f (monom c n) = monom (f c) n" by (intro poly_eqI) (simp_all add: coeff_map_poly) lemma map_poly_idI: assumes "⋀x. x ∈ set (coeffs p) ⟹ f x = x" shows "map_poly f p = p" using map_poly_cong[OF assms, of _ id] by simp lemma map_poly_idI': assumes "⋀x. x ∈ set (coeffs p) ⟹ f x = x" shows "p = map_poly f p" using map_poly_cong[OF assms, of _ id] by simp lemma smult_conv_map_poly: "smult c p = map_poly (λx. c * x) p" by (intro poly_eqI) (simp_all add: coeff_map_poly) lemma poly_cnj: "cnj (poly p z) = poly (map_poly cnj p) (cnj z)" by (simp add: poly_altdef degree_map_poly coeff_map_poly) lemma poly_cnj_real: assumes "⋀n. poly.coeff p n ∈ ℝ" shows "cnj (poly p z) = poly p (cnj z)" proof - from assms have "map_poly cnj p = p" by (intro poly_eqI) (auto simp: coeff_map_poly Reals_cnj_iff) with poly_cnj[of p z] show ?thesis by simp qed lemma real_poly_cnj_root_iff: assumes "⋀n. poly.coeff p n ∈ ℝ" shows "poly p (cnj z) = 0 ⟷ poly p z = 0" proof - have "poly p (cnj z) = cnj (poly p z)" by (simp add: poly_cnj_real assms) also have "… = 0 ⟷ poly p z = 0" by simp finally show ?thesis . qed lemma sum_to_poly: "(∑x∈A. [:f x:]) = [:∑x∈A. f x:]" by (induction A rule: infinite_finite_induct) auto lemma diff_to_poly: "[:c:] - [:d:] = [:c - d:]" by (simp add: poly_eq_iff mult_ac) lemma mult_to_poly: "[:c:] * [:d:] = [:c * d:]" by (simp add: poly_eq_iff mult_ac) lemma prod_to_poly: "(∏x∈A. [:f x:]) = [:∏x∈A. f x:]" by (induction A rule: infinite_finite_induct) (auto simp: mult_to_poly mult_ac) lemma poly_map_poly_cnj [simp]: "poly (map_poly cnj p) x = cnj (poly p (cnj x))" by (induction p) (auto simp: map_poly_pCons) subsection ‹Conversions› lemma of_nat_poly: "of_nat n = [:of_nat n:]" by (induct n) (simp_all add: one_pCons) lemma of_nat_monom: "of_nat n = monom (of_nat n) 0" by (simp add: of_nat_poly monom_0) lemma degree_of_nat [simp]: "degree (of_nat n) = 0" by (simp add: of_nat_poly) lemma lead_coeff_of_nat [simp]: "lead_coeff (of_nat n) = of_nat n" by (simp add: of_nat_poly) lemma of_int_poly: "of_int k = [:of_int k:]" by (simp only: of_int_of_nat of_nat_poly) simp lemma of_int_monom: "of_int k = monom (of_int k) 0" by (simp add: of_int_poly monom_0) lemma degree_of_int [simp]: "degree (of_int k) = 0" by (simp add: of_int_poly) lemma lead_coeff_of_int [simp]: "lead_coeff (of_int k) = of_int k" by (simp add: of_int_poly) lemma poly_of_nat [simp]: "poly (of_nat n) x = of_nat n" by (simp add: of_nat_poly) lemma poly_of_int [simp]: "poly (of_int n) x = of_int n" by (simp add: of_int_poly) lemma poly_numeral [simp]: "poly (numeral n) x = numeral n" by (metis of_nat_numeral poly_of_nat) lemma numeral_poly: "numeral n = [:numeral n:]" proof - have "numeral n = of_nat (numeral n)" by simp also have "… = [:of_nat (numeral n):]" by (simp add: of_nat_poly) finally show ?thesis by simp qed lemma numeral_monom: "numeral n = monom (numeral n) 0" by (simp add: numeral_poly monom_0) lemma degree_numeral [simp]: "degree (numeral n) = 0" by (simp add: numeral_poly) lemma lead_coeff_numeral [simp]: "lead_coeff (numeral n) = numeral n" by (simp add: numeral_poly) lemma coeff_linear_poly_power: fixes c :: "'a :: semiring_1" assumes "i ≤ n" shows "coeff ([:a, b:] ^ n) i = of_nat (n choose i) * b ^ i * a ^ (n - i)" proof - have "[:a, b:] = monom b 1 + [:a:]" by (simp add: monom_altdef) also have "coeff (… ^ n) i = (∑k≤n. a^(n-k) * of_nat (n choose k) * (if k = i then b ^ k else 0))" by (subst binomial_ring) (simp add: coeff_sum of_nat_poly monom_power poly_const_pow mult_ac) also have "… = (∑k∈{i}. a ^ (n - i) * b ^ i * of_nat (n choose k))" using assms by (intro sum.mono_neutral_cong_right) (auto simp: mult_ac) finally show *: ?thesis by (simp add: mult_ac) qed subsection ‹Lemmas about divisibility› lemma dvd_smult: assumes "p dvd q" shows "p dvd smult a q" proof - from assms 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: "p dvd smult a q ⟹ a ≠ 0 ⟹ p dvd q" for a :: "'a::field" by (drule dvd_smult [where a="inverse a"]) simp lemma dvd_smult_iff: "a ≠ 0 ⟹ p dvd smult a q ⟷ p dvd q" for a :: "'a::field" by (safe elim!: dvd_smult dvd_smult_cancel) lemma smult_dvd_cancel: assumes "smult a p dvd q" shows "p dvd q" proof - from assms 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: "p dvd q ⟹ a ≠ 0 ⟹ smult a p dvd q" for a :: "'a::field" by (rule smult_dvd_cancel [where a="inverse a"]) simp lemma smult_dvd_iff: "smult a p dvd q ⟷ (if a = 0 then q = 0 else p dvd q)" for a :: "'a::field" by (auto elim: smult_dvd smult_dvd_cancel) lemma is_unit_smult_iff: "smult c p dvd 1 ⟷ c dvd 1 ∧ p dvd 1" proof - have "smult c p = [:c:] * p" by simp also have "… dvd 1 ⟷ c dvd 1 ∧ p dvd 1" proof safe assume *: "[:c:] * p dvd 1" then show "p dvd 1" by (rule dvd_mult_right) from * obtain q where q: "1 = [:c:] * p * q" by (rule dvdE) have "c dvd c * (coeff p 0 * coeff q 0)" by simp also have "… = coeff ([:c:] * p * q) 0" by (simp add: mult.assoc coeff_mult) also note q [symmetric] finally have "c dvd coeff 1 0" . then show "c dvd 1" by simp next assume "c dvd 1" "p dvd 1" from this(1) obtain d where "1 = c * d" by (rule dvdE) then have "1 = [:c:] * [:d:]" by (simp add: one_pCons ac_simps) then have "[:c:] dvd 1" by (rule dvdI) from mult_dvd_mono[OF this ‹p dvd 1›] show "[:c:] * p dvd 1" by simp qed finally show ?thesis . qed subsection ‹Polynomials form an integral domain› instance poly :: (idom) idom .. instance poly :: ("{ring_char_0, comm_ring_1}") ring_char_0 by standard (auto simp add: of_nat_poly intro: injI) lemma semiring_char_poly [simp]: "CHAR('a :: comm_semiring_1 poly) = CHAR('a)" by (rule CHAR_eqI) (auto simp: of_nat_poly of_nat_eq_0_iff_char_dvd) instance poly :: ("{semiring_prime_char,comm_semiring_1}") semiring_prime_char by (rule semiring_prime_charI) auto instance poly :: ("{comm_semiring_prime_char,comm_semiring_1}") comm_semiring_prime_char by standard instance poly :: ("{comm_ring_prime_char,comm_semiring_1}") comm_ring_prime_char by standard instance poly :: ("{idom_prime_char,comm_semiring_1}") idom_prime_char by standard lemma degree_mult_eq: "p ≠ 0 ⟹ q ≠ 0 ⟹ degree (p * q) = degree p + degree q" for p q :: "'a::{comm_semiring_0,semiring_no_zero_divisors} poly" by (rule order_antisym [OF degree_mult_le le_degree]) (simp add: coeff_mult_degree_sum) lemma degree_prod_sum_eq: "(⋀x. x ∈ A ⟹ f x ≠ 0) ⟹ degree (prod f A :: 'a :: idom poly) = (∑x∈A. degree (f x))" by (induction A rule: infinite_finite_induct) (auto simp: degree_mult_eq) lemma dvd_imp_degree: ‹degree x ≤ degree y› if ‹x dvd y› ‹x ≠ 0› ‹y ≠ 0› for x y :: ‹'a::{comm_semiring_1,semiring_no_zero_divisors} poly› proof - from ‹x dvd y› obtain z where ‹y = x * z› .. with ‹x ≠ 0› ‹y ≠ 0› show ?thesis by (simp add: degree_mult_eq) qed lemma degree_prod_eq_sum_degree: fixes A :: "'a set" and f :: "'a ⇒ 'b::idom poly" assumes f0: "∀i∈A. f i ≠ 0" shows "degree (∏i∈A. (f i)) = (∑i∈A. degree (f i))" using assms by (induction A rule: infinite_finite_induct) (auto simp: degree_mult_eq) lemma degree_mult_eq_0: "degree (p * q) = 0 ⟷ p = 0 ∨ q = 0 ∨ (p ≠ 0 ∧ q ≠ 0 ∧ degree p = 0 ∧ degree q = 0)" for p q :: "'a::{comm_semiring_0,semiring_no_zero_divisors} poly" by (auto simp: degree_mult_eq) lemma degree_power_eq: "p ≠ 0 ⟹ degree ((p :: 'a :: idom poly) ^ n) = n * degree p" by (induction n) (simp_all add: degree_mult_eq) lemma degree_mult_right_le: fixes p q :: "'a::{comm_semiring_0,semiring_no_zero_divisors} poly" assumes "q ≠ 0" shows "degree p ≤ degree (p * q)" using assms by (cases "p = 0") (simp_all add: degree_mult_eq) lemma coeff_degree_mult: "coeff (p * q) (degree (p * q)) = coeff q (degree q) * coeff p (degree p)" for p q :: "'a::{comm_semiring_0,semiring_no_zero_divisors} poly" by (cases "p = 0 ∨ q = 0") (auto simp: degree_mult_eq coeff_mult_degree_sum mult_ac) lemma dvd_imp_degree_le: "p dvd q ⟹ q ≠ 0 ⟹ degree p ≤ degree q" for p q :: "'a::{comm_semiring_1,semiring_no_zero_divisors} poly" by (erule dvdE, hypsubst, subst degree_mult_eq) auto lemma divides_degree: fixes p q :: "'a ::{comm_semiring_1,semiring_no_zero_divisors} poly" assumes "p dvd q" shows "degree p ≤ degree q ∨ q = 0" by (metis dvd_imp_degree_le assms) lemma const_poly_dvd_iff: fixes c :: "'a::{comm_semiring_1,semiring_no_zero_divisors}" shows "[:c:] dvd p ⟷ (∀n. c dvd coeff p n)" proof (cases "c = 0 ∨ p = 0") case True then show ?thesis by (auto intro!: poly_eqI) next case False show ?thesis proof assume "[:c:] dvd p" then show "∀n. c dvd coeff p n" by (auto simp: coeffs_def) next assume *: "∀n. c dvd coeff p n" define mydiv where "mydiv x y = (SOME z. x = y * z)" for x y :: 'a have mydiv: "x = y * mydiv x y" if "y dvd x" for x y using that unfolding mydiv_def dvd_def by (rule someI_ex) define q where "q = Poly (map (λa. mydiv a c) (coeffs p))" from False * have "p = q * [:c:]" by (intro poly_eqI) (auto simp: q_def nth_default_def not_less length_coeffs_degree coeffs_nth intro!: coeff_eq_0 mydiv) then show "[:c:] dvd p" by (simp only: dvd_triv_right) qed qed lemma const_poly_dvd_const_poly_iff [simp]: "[:a:] dvd [:b:] ⟷ a dvd b" for a b :: "'a::{comm_semiring_1,semiring_no_zero_divisors}" by (subst const_poly_dvd_iff) (auto simp: coeff_pCons split: nat.splits) lemma lead_coeff_mult: "lead_coeff (p * q) = lead_coeff p * lead_coeff q" for p q :: "'a::{comm_semiring_0, semiring_no_zero_divisors} poly" by (cases "p = 0 ∨ q = 0") (auto simp: coeff_mult_degree_sum degree_mult_eq) lemma lead_coeff_prod: "lead_coeff (prod f A) = (∏x∈A. lead_coeff (f x))" for f :: "'a ⇒ 'b::{comm_semiring_1, semiring_no_zero_divisors} poly" by (induction A rule: infinite_finite_induct) (auto simp: lead_coeff_mult) lemma lead_coeff_smult: "lead_coeff (smult c p) = c * lead_coeff p" for p :: "'a::{comm_semiring_0,semiring_no_zero_divisors} poly" proof - have "smult c p = [:c:] * p" by simp also have "lead_coeff … = c * lead_coeff p" by (subst lead_coeff_mult) simp_all finally show ?thesis . qed lemma lead_coeff_1 [simp]: "lead_coeff 1 = 1" by simp lemma lead_coeff_power: "lead_coeff (p ^ n) = lead_coeff p ^ n" for p :: "'a::{comm_semiring_1,semiring_no_zero_divisors} poly" by (induct n) (simp_all add: lead_coeff_mult) subsection ‹Polynomials form an ordered integral domain› definition pos_poly :: "'a::linordered_semidom poly ⇒ bool" where "pos_poly p ⟷ 0 < coeff p (degree p)" lemma pos_poly_pCons: "pos_poly (pCons a p)