(* 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" Factorial_Ring begin subsection ‹Auxiliary: operations for lists (later) representing coefficients› definition cCons :: "'a::zero ⇒ 'a list ⇒ 'a list" (infixr "##" 65) where "x ## xs = (if xs = [] ∧ x = 0 then [] else x # xs)" lemma cCons_0_Nil_eq [simp]: "0 ## [] = []" 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 subsection ‹Degree of a polynomial› definition degree :: "'a::zero poly ⇒ nat" where "degree p = (LEAST n. ∀i>n. coeff p i = 0)" lemma coeff_eq_0: assumes "degree p < n" shows "coeff p n = 0" proof - have "∃n. ∀i>n. coeff p i = 0" using MOST_coeff_eq_0 by (simp add: MOST_nat) then have "∀i>degree p. coeff p i = 0" unfolding degree_def by (rule LeastI_ex) with assms show ?thesis by simp qed lemma le_degree: "coeff p n ≠ 0 ⟹ n ≤ degree p" by (erule contrapos_np, rule coeff_eq_0, simp) lemma degree_le: "∀i>n. coeff p i = 0 ⟹ degree p ≤ n" unfolding degree_def by (erule Least_le) lemma less_degree_imp: "n < degree p ⟹ ∃i>n. coeff p i ≠ 0" unfolding degree_def by (drule not_less_Least, simp) subsection ‹The zero polynomial› instantiation poly :: (zero) zero begin lift_definition zero_poly :: "'a poly" is "λ_. 0" by (rule MOST_I) simp instance .. end lemma coeff_0 [simp]: "coeff 0 n = 0" by transfer rule lemma degree_0 [simp]: "degree 0 = 0" by (rule order_antisym [OF degree_le le0]) simp 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 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_0 [simp]: "coeff (pCons a p) 0 = a" by transfer simp lemma coeff_pCons_Suc [simp]: "coeff (pCons a p) (Suc n) = coeff p n" by (simp add: coeff_pCons) lemma degree_pCons_le: "degree (pCons a p) ≤ Suc (degree p)" by (rule degree_le) (simp add: coeff_eq_0 coeff_pCons split: nat.split) lemma degree_pCons_eq: "p ≠ 0 ⟹ degree (pCons a p) = Suc (degree p)" apply (rule order_antisym [OF degree_pCons_le]) apply (rule le_degree, simp) done lemma degree_pCons_0: "degree (pCons a 0) = 0" apply (rule order_antisym [OF _ le0]) apply (rule degree_le, simp add: coeff_pCons split: nat.split) done lemma degree_pCons_eq_if [simp]: "degree (pCons a p) = (if p = 0 then 0 else Suc (degree p))" apply (cases "p = 0", simp_all) apply (rule order_antisym [OF _ le0]) apply (rule degree_le, simp add: coeff_pCons split: nat.split) apply (rule order_antisym [OF degree_pCons_le]) apply (rule le_degree, simp) done lemma pCons_0_0 [simp]: "pCons 0 0 = 0" by (rule poly_eqI) (simp add: coeff_pCons split: nat.split) lemma pCons_eq_iff [simp]: "pCons a p = pCons b q ⟷ a = b ∧ p = q" proof safe assume "pCons a p = pCons b q" then have "coeff (pCons a p) 0 = coeff (pCons b q) 0" by simp then show "a = b" by simp next assume "pCons a p = pCons b q" then have "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) subsubsection ‹Reconstructing the polynomial from the list› ― ‹contributed by Sebastiaan J.C. Joosten and RenÃ© Thiemann› definition poly_of_list :: "'a::comm_monoid_add list ⇒ 'a poly" where [simp]: "poly_of_list = Poly" lemma poly_of_list_impl [code abstract]: "coeffs (poly_of_list as) = strip_while (HOL.eq 0) as" by simp subsection ‹Fold combinator for polynomials› definition fold_coeffs :: "('a::zero ⇒ 'b ⇒ 'b) ⇒ 'a poly ⇒ 'b ⇒ 'b" where "fold_coeffs f p = foldr f (coeffs p)" lemma fold_coeffs_0_eq [simp]: "fold_coeffs f 0 = id" 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 = fold_coeffs (λa f x. a + x * f x) p (λx. 0)" ― ‹The Horner Schema› lemma poly_0 [simp]: "poly 0 x = 0" by (simp add: poly_def) lemma poly_pCons [simp]: "poly (pCons a p) x = a + x * poly p x" by (cases "p = 0 ∧ a = 0") (auto simp add: poly_def) 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 = pCons a 0" by (rule poly_eqI) (simp add: coeff_pCons split: nat.split) lemma monom_Suc: "monom a (Suc n) = pCons 0 (monom a n)" by (rule poly_eqI) (simp add: coeff_pCons split: nat.split) lemma monom_eq_0 [simp]: "monom 0 n = 0" by (rule poly_eqI) simp lemma monom_eq_0_iff [simp]: "monom a n = 0 ⟷ a = 0" by (simp add: poly_eq_iff) lemma monom_eq_iff [simp]: "monom a n = monom b n ⟷ a = b" by (simp add: poly_eq_iff) lemma degree_monom_le: "degree (monom a n) ≤ n" by (rule degree_le, simp) lemma degree_monom_eq: "a ≠ 0 ⟹ degree (monom a n) = n" apply (rule order_antisym [OF degree_monom_le]) apply (rule le_degree) apply simp done lemma coeffs_monom [code abstract]: "coeffs (monom a n) = (if a = 0 then [] else replicate n 0 @ [a])" by (induct n) (simp_all add: monom_0 monom_Suc) lemma fold_coeffs_monom [simp]: "a ≠ 0 ⟹ fold_coeffs f (monom a n) = f 0 ^^ n ∘ f a" by (simp add: fold_coeffs_def coeffs_monom fun_eq_iff) lemma poly_monom: "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_def) lemma monom_eq_iff': "monom c n = monom d m ⟷ c = d ∧ (c = 0 ∨ n = m)" by (auto simp: poly_eq_iff) lemma monom_eq_const_iff: "monom c n = [:d:] ⟷ c = d ∧ (c = 0 ∨ n = 0)" using monom_eq_iff'[of c n d 0] by (simp add: monom_0) 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: "degree p < degree q ⟹ degree (p + q) = degree q" apply (cases "q = 0") apply simp apply (rule order_antisym) apply (simp add: degree_add_le) apply (rule le_degree) apply (simp add: coeff_eq_0) done lemma degree_add_eq_left: "degree q < degree p ⟹ degree (p + q) = degree p" using degree_add_eq_right [of q p] by (simp add: add.commute) lemma degree_minus [simp]: "degree (- p) = degree p" 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" apply (induct p arbitrary: q) apply simp apply (case_tac q, simp, simp add: algebra_simps) done 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 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 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 degree_mult_le: "degree (p * q) ≤ degree p + degree q" apply (rule degree_le) apply (induct p) apply simp apply (simp add: coeff_eq_0 coeff_pCons split: nat.split) done lemma mult_monom: "monom a m * monom b n = monom (a * b) (m + n)" by (induct m) (simp add: monom_0 smult_monom, simp add: monom_Suc) instantiation poly :: (comm_semiring_1) comm_semiring_1 begin 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 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 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) 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 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) 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 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_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 elim!: dvdE 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_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) ⟷ pos_poly p ∨ (p = 0 ∧ 0 < a)" by (simp add: pos_poly_def) lemma not_pos_poly_0 [simp]: "¬ pos_poly 0" by (simp add: pos_poly_def) lemma pos_poly_add: "pos_poly p ⟹ pos_poly q ⟹ pos_poly (p + q)" apply (induct p arbitrary: q) apply simp apply (case_tac q) apply (force simp add: pos_poly_pCons add_pos_pos) done lemma pos_poly_mult: "pos_poly p ⟹ pos_poly q ⟹ pos_poly (p * q)" unfolding pos_poly_def apply (subgoal_tac "p ≠ 0 ∧ q ≠ 0") apply (simp add: degree_mult_eq coeff_mult_degree_sum) apply auto done lemma pos_poly_total: "p = 0 ∨ pos_poly p ∨ pos_poly (- p)" for p :: "'a::linordered_idom poly" by (induct p) (auto simp: pos_poly_pCons) lemma pos_poly_coeffs [code]: "pos_poly p ⟷ (let as = coeffs p in as ≠ [] ∧ last as > 0)" (is "?lhs ⟷ ?rhs") proof assume ?rhs then show ?lhs by (auto simp add: pos_poly_def last_coeffs_eq_coeff_degree) next assume ?lhs then have *: "0 < coeff p (degree p)" by (simp add: pos_poly_def) then have "p ≠ 0" by auto with * show ?rhs by (simp add: last_coeffs_eq_coeff_degree) qed instantiation poly :: (linordered_idom) linordered_idom begin definition "x < y ⟷ pos_poly (y - x)" definition "x ≤ y ⟷ x = y ∨ pos_poly (y - x)" definition "¦x::'a poly¦ = (if x < 0 then - x else x)" definition "sgn (x::'a poly) = (if x = 0 then 0 else if 0 < x then 1 else - 1)" instance proof fix x y z :: "'a poly" show "x < y ⟷ x ≤ y ∧ ¬ y ≤ x" unfolding less_eq_poly_def less_poly_def apply safe apply simp apply (drule (1) pos_poly_add) apply simp done show "x ≤ x" by (simp add: less_eq_poly_def) show "x ≤ y ⟹ y ≤ z ⟹ x ≤ z" unfolding less_eq_poly_def apply safe apply (drule (1) pos_poly_add) apply (simp add: algebra_simps) done show "x ≤ y ⟹ y ≤ x ⟹ x = y" unfolding less_eq_poly_def apply safe apply (drule (1) pos_poly_add) apply simp done show "x ≤ y ⟹ z + x ≤ z + y" unfolding less_eq_poly_def apply safe apply (simp add: algebra_simps) done show "x ≤ y ∨ y ≤ x" unfolding less_eq_poly_def using pos_poly_total [of "x - y"] by auto show "x < y ⟹ 0 < z ⟹ z * x < z * y" by (simp add: less_poly_def right_diff_distrib [symmetric] pos_poly_mult) show "¦x¦ = (if x < 0 then - x else x)" by (rule abs_poly_def) show "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)" by (rule sgn_poly_def) qed end text ‹TODO: Simplification rules for comparisons› subsection ‹Synthetic division and polynomial roots› subsubsection ‹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 = fold_coeffs (λa (q, r). (pCons r q, a + c * r)) p (0, 0)" definition synthetic_div :: "'a::comm_semiring_0 poly ⇒ 'a ⇒ 'a poly" where "synthetic_div p c = fst (synthetic_divmod p c)" lemma synthetic_divmod_0 [simp]: "synthetic_divmod 0 c = (0, 0)" by (simp add: synthetic_divmod_def) lemma synthetic_divmod_pCons [simp]: "synthetic_divmod (pCons a p) c = (λ(q, r). (pCons r q, a + c * r)) (synthetic_divmod p c)" by (cases "p = 0 ∧ a = 0") (auto simp add: synthetic_divmod_def) lemma synthetic_div_0 [simp]: "synthetic_div 0 c = 0" by (simp add: synthetic_div_def) lemma synthetic_div_unique_lemma: "smult c p = pCons a p ⟹ p = 0" by (induct p arbitrary: a) simp_all lemma snd_synthetic_divmod: "snd (synthetic_divmod p c) = poly p c" by (induct p) (simp_all add: split_def) lemma synthetic_div_pCons [simp]: "synthetic_div (pCons a p) c = pCons (poly p c) (synthetic_div p c)" by (simp add: synthetic_div_def split_def snd_synthetic_divmod) lemma synthetic_div_eq_0_iff: "synthetic_div p c = 0 ⟷ degree p = 0" proof (induct p) case 0 then show ?case by simp next case (pCons a p) then show ?case by (cases p) simp qed lemma degree_synthetic_div: "degree (synthetic_div p c) = degree p - 1" by (induct p) (simp_all add: synthetic_div_eq_0_iff) lemma synthetic_div_correct: "p + smult c (synthetic_div p c) = pCons (poly p c) (synthetic_div p c)" by (induct p) simp_all lemma synthetic_div_unique: "p + smult c q = pCons r q ⟹ r = poly p c ∧ q = synthetic_div p c" apply (induct p arbitrary: q r) apply simp apply (frule synthetic_div_unique_lemma) apply simp apply (case_tac q, force) done lemma synthetic_div_correct': "[:-c, 1:] * synthetic_div p c + [:poly p c:] = p" for c :: "'a::comm_ring_1" using synthetic_div_correct [of p c] by (simp add: algebra_simps) subsubsection ‹Polynomial roots› lemma poly_eq_0_iff_dvd: "poly p c = 0 ⟷ [:- c, 1:] dvd p" (is "?lhs ⟷ ?rhs") for c :: "'a::comm_ring_1" proof assume ?lhs with synthetic_div_correct' [of c p] have "p = [:-c, 1:] * synthetic_div p c" by simp then show ?rhs .. next assume ?rhs then obtain k where "p = [:-c, 1:] * k" by (rule dvdE) then show ?lhs by simp qed lemma dvd_iff_poly_eq_0: "[:c, 1:] dvd p ⟷ poly p (- c) = 0" for c :: "'a::comm_ring_1" by (simp add: poly_eq_0_iff_dvd) lemma poly_roots_finite: "p ≠ 0 ⟹ finite {x. poly p x = 0}" for p :: "'a::{comm_ring_1,ring_no_zero_divisors} poly" proof (induct n ≡ "degree p" arbitrary: p) case 0 then obtain a where "a ≠ 0" and "p = [:a:]" by (cases p) (simp split: if_splits) then show "finite {x. poly p x = 0}" by simp next case (Suc n) 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 from this ‹k ≠ 0› have "finite {x. poly k x = 0}" by (rule Suc.hyps) then have "finite (insert a {x. poly k x = 0})" by simp then show "finite {x. poly p x = 0}" by (simp add: k Collect_disj_eq del: mult_pCons_left) qed qed lemma poly_eq_poly_eq_iff: "poly p = poly q ⟷ p = q" (is "?lhs ⟷ ?rhs") for p q :: "'a::{comm_ring_1,ring_no_zero_divisors,ring_char_0} poly" proof assume ?rhs then show ?lhs by simp next assume ?lhs have "poly p = poly 0 ⟷ p = 0" for p :: "'a poly" apply (cases "p = 0") apply simp_all apply (drule poly_roots_finite) apply (auto simp add: infinite_UNIV_char_0) done from ‹?lhs› and this [of "p - q"] show ?rhs by auto qed lemma poly_all_0_iff_0: "(∀x. poly p x = 0) ⟷ p = 0" for p :: "'a::{ring_char_0,comm_ring_1,ring_no_zero_divisors} poly" by (auto simp add: poly_eq_poly_eq_iff [symmetric]) subsubsection ‹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: "coeff ([:a, 1:] ^ n) n = 1" for a :: "'a::comm_semiring_1" apply (induct n) apply simp_all apply (subst coeff_eq_0) apply (auto intro: le_less_trans degree_power_le) done lemma degree_linear_power: "degree ([:a, 1:] ^ n) = n" for a :: "'a::comm_semiring_1" apply (rule order_antisym) apply (rule ord_le_eq_trans [OF degree_power_le]) apply simp apply (rule le_degree) apply (simp add: coeff_linear_power) done lemma order_1: "[:-a, 1:] ^ order a p dvd p" apply (cases "p = 0") apply simp apply (cases "order a p") apply simp apply (subgoal_tac "nat < (LEAST n. ¬ [:-a, 1:] ^ Suc n dvd p)") apply (drule not_less_Least) apply simp apply (fold order_def) apply simp done lemma order_2: "p ≠ 0 ⟹ ¬ [:-a, 1:] ^ Suc (order a p) dvd p" unfolding order_def apply (rule LeastI_ex) apply (rule_tac x="degree p" in exI) apply (rule notI) apply (drule (1) dvd_imp_degree_le) apply (simp only: degree_linear_power) done lemma order: "p ≠ 0 ⟹ [:-a, 1:] ^ order a p dvd p ∧ ¬ [:-a, 1:] ^ Suc (order a p) dvd p" by (rule conjI [OF order_1 order_2]) lemma order_degree: assumes p: "p ≠ 0" shows "order a p ≤ degree p" proof - have "order a p = degree ([:-a, 1:] ^ order a p)" by (simp only: degree_linear_power) also from order_1 p have "… ≤ degree p" by (rule dvd_imp_degree_le) finally show ?thesis . qed lemma order_root: "poly p a = 0 ⟷ p = 0 ∨ order a p ≠ 0" apply (cases "p = 0") apply simp_all apply (rule iffI) apply (metis order_2 not_gr0 poly_eq_0_iff_dvd power_0 power_Suc_0 power_one_right) unfolding poly_eq_0_iff_dvd apply (metis dvd_power dvd_trans order_1) done lemma order_0I: "poly p a ≠ 0 ⟹ order a p = 0" by (subst (asm) order_root) auto lemma order_unique_lemma: fixes p :: "'a::idom poly" assumes "[:-a, 1:] ^ n dvd p" "¬ [:-a, 1:] ^ Suc n dvd p" shows "n = order a p" unfolding Polynomial.order_def apply (rule Least_equality [symmetric]) apply (fact assms) apply (rule classical) apply (erule notE) unfolding not_less_eq_eq using assms(1) apply (rule power_le_dvd) apply assumption done lemma order_mult: "p * q ≠ 0 ⟹ order a (p * q) = order a p + order a q" proof - define i where "i = order a p" define j where "j = order a q" define t where "t = [:-a, 1:]" have t_dvd_iff: "⋀u. t dvd u ⟷ poly u a = 0" by (simp add: t_def dvd_iff_poly_eq_0) assume "p * q ≠ 0" then show "order a (p * q) = i + j" apply clarsimp apply (drule order [where a=a and p=p, folded i_def t_def]) apply (drule order [where a=a and p=q, folded j_def t_def]) apply clarify apply (erule dvdE)+ apply (rule order_unique_lemma [symmetric], fold t_def) apply (simp_all add: power_add t_dvd_iff) done qed lemma order_smult: assumes "c ≠ 0" shows "order x (smult c p) = order x p" proof (cases "p = 0") case True then show ?thesis by simp next case False have "smult c p = [:c:] * p" by simp also from assms False have "order x … = order x [:c:] + order x p" by (subst order_mult) simp_all also have "order x [:c:] = 0" by (rule order_0I) (use assms in auto) finally show ?thesis by simp qed (* Next three lemmas contributed by Wenda Li *) lemma order_1_eq_0 [simp]:"order x 1 = 0" by (metis order_root poly_1 zero_neq_one) lemma order_uminus[simp]: "order x (-p) = order x p" by (metis neg_equal_0_iff_equal order_smult smult_1_left smult_minus_left) lemma order_power_n_n: "order a ([:-a,1:]^n)=n" proof (induct n) (*might be proved more concisely using nat_less_induct*) case 0 then show ?case by (metis order_root poly_1 power_0 zero_neq_one) next case (Suc n) have "order a ([:- a, 1:] ^ Suc n) = order a ([:- a, 1:] ^ n) + order a [:-a,1:]" by (metis (no_types, hide_lams) One_nat_def add_Suc_right monoid_add_class.add.right_neutral one_neq_zero order_mult pCons_eq_0_iff power_add power_eq_0_iff power_one_right) moreover have "order a [:-a,1:] = 1" unfolding order_def proof (rule Least_equality, rule notI) assume "[:- a, 1:] ^ Suc 1 dvd [:- a, 1:]" then have "degree ([:- a, 1:] ^ Suc 1) ≤ degree ([:- a, 1:])" by (rule dvd_imp_degree_le) auto then show False by auto next fix y assume *: "¬ [:- a, 1:] ^ Suc y dvd [:- a, 1:]" show "1 ≤ y" proof (rule ccontr) assume "¬ 1 ≤ y" then have "y = 0" by auto then have "[:- a, 1:] ^ Suc y dvd [:- a, 1:]" by auto with * show False by auto qed qed ultimately show ?case using Suc by auto qed lemma order_0_monom [simp]: "c ≠ 0 ⟹ order 0 (monom c n) = n" using order_power_n_n[of 0 n] by (simp add: monom_altdef order_smult) lemma dvd_imp_order_le: "q ≠ 0 ⟹ p dvd q ⟹ Polynomial.order a p ≤ Polynomial.order a q" by (auto simp: order_mult elim: dvdE) text ‹Now justify the standard squarefree decomposition, i.e. ‹f / gcd f f'›.› lemma order_divides: "[:-a, 1:] ^ n dvd p ⟷ p = 0 ∨ n ≤ order a p" apply (cases "p = 0") apply auto apply (drule order_2 [where a=a and p=p]) apply (metis not_less_eq_eq power_le_dvd) apply (erule power_le_dvd [OF order_1]) done lemma order_decomp: assumes "p ≠ 0" shows "∃q. p = [:- a, 1:] ^ order a p * q ∧ ¬ [:- a, 1:] dvd q" proof - from assms have *: "[:- a, 1:] ^ order a p dvd p" and **: "¬ [:- a, 1:] ^ Suc (order a p) dvd p" by (auto dest: order) from * obtain q where q: "p = [:- a, 1:] ^ order a p * q" .. with ** have "¬ [:- a, 1:] ^ Suc (order a p) dvd [:- a, 1:] ^ order a p * q" by simp then have "¬ [:- a, 1:] ^ order a p * [:- a, 1:] dvd [:- a, 1:] ^ order a p * q" by simp with idom_class.dvd_mult_cancel_left [of "[:- a, 1:] ^ order a p" "[:- a, 1:]" q] have "¬ [:- a, 1:] dvd q" by auto with q show ?thesis by blast qed lemma monom_1_dvd_iff: "p ≠ 0 ⟹ monom 1 n dvd p ⟷ n ≤ order 0 p" using order_divides[of 0 n p] by (simp add: monom_altdef) subsection ‹Additional induction rules on polynomials› text ‹ An induction rule for induction over the roots of a polynomial with a certain property. (e.g. all positive roots) › lemma poly_root_induct [case_names 0 no_roots root]: fixes p :: "'a :: idom poly" assumes "Q 0" and "⋀p. (⋀a. P a ⟹ poly p a ≠ 0) ⟹ Q p" and "⋀a p. P a ⟹ Q p ⟹ Q ([:a, -1:] * p)" shows "Q p" proof (induction "degree p" arbitrary: p rule: less_induct) case (less p) show ?case proof (cases "p = 0") case True with assms(1) show ?thesis by simp next case False show ?thesis proof (cases "∃a. P a ∧ poly p a = 0") case False then show ?thesis by (intro assms(2)) blast next case True then obtain a where a: "P a" "poly p a = 0" by blast then have "-[:-a, 1:] dvd p" by (subst minus_dvd_iff) (simp add: poly_eq_0_iff_dvd) then obtain q where q: "p = [:a, -1:] * q" by (elim dvdE) simp with False have "q ≠ 0" by auto have "degree p = Suc (degree q)" by (subst q, subst degree_mult_eq) (simp_all add: ‹q ≠ 0›) then have "Q q" by (intro less) simp with a(1) have "Q ([:a, -1:] * q)" by (rule assms(3)) with q show ?thesis by simp qed qed qed lemma dropWhile_replicate_append: "dropWhile ((=) a) (replicate n a @ ys) = dropWhile ((=) a) ys" by (induct n) simp_all lemma Poly_append_replicate_0: "Poly (xs @ replicate n 0) = Poly xs" by (subst coeffs_eq_iff) (simp_all add: strip_while_def dropWhile_replicate_append) text ‹ An induction rule for simultaneous induction over two polynomials, prepending one coefficient in each step. › lemma poly_induct2 [case_names 0 pCons]: assumes "P 0 0" "⋀a p b q. P p q ⟹ P (pCons a p) (pCons b q)" shows "P p q" proof - define n where "n = max (length (coeffs p)) (length (coeffs q))" define xs where "xs = coeffs p @ (replicate (n - length (coeffs p)) 0)" define ys where "ys = coeffs q @ (replicate (n - length (coeffs q)) 0)" have "length xs = length ys" by (simp add: xs_def ys_def n_def) then have "P (Poly xs) (Poly ys)" by (induct rule: list_induct2) (simp_all add: assms) also have "Poly xs = p" by (simp add: xs_def Poly_append_replicate_0) also have "Poly ys = q" by (simp add: ys_def Poly_append_replicate_0) finally show ?thesis . qed subsection ‹Composition of polynomials› (* Several lemmas contributed by RenÃ© Thiemann and Akihisa Yamada *) definition pcompose :: "'a::comm_semiring_0 poly ⇒ 'a poly ⇒ 'a poly" where "pcompose p q = fold_coeffs (λa c. [:a:] + q * c) p 0" notation pcompose (infixl "∘⇩_{p}" 71) lemma pcompose_0 [simp]: "pcompose 0 q = 0" by (simp add: pcompose_def) lemma pcompose_pCons: "pcompose (pCons a p) q = [:a:] + q * pcompose p q" by (cases "p = 0 ∧ a = 0") (auto simp add: pcompose_def) lemma pcompose_1: "pcompose 1 p = 1" for p :: "'a::comm_semiring_1 poly" by (auto simp: one_pCons pcompose_pCons) lemma poly_pcompose: "poly (pcompose p q) x = poly p (poly q x)" by (induct p) (simp_all add: pcompose_pCons) lemma degree_pcompose_le: "degree (pcompose p q) ≤ degree p * degree q" apply (induct p) apply simp apply (simp add: pcompose_pCons) apply clarify apply (rule degree_add_le) apply simp apply (rule order_trans [OF degree_mult_le]) apply simp done lemma pcompose_add: "pcompose (p + q) r = pcompose p r + pcompose q r" for p q r :: "'a::{comm_semiring_0, ab_semigroup_add} poly" proof (induction p q rule: poly_induct2) case 0 then show ?case by simp next case (pCons a p b q) have "pcompose (pCons a p + pCons b q) r = [:a + b:] + r * pcompose p r + r * pcompose q r" by (simp_all add: pcompose_pCons pCons.IH algebra_simps) also have "[:a + b:] = [:a:] + [:b:]" by simp also have "… + r * pcompose p r + r * pcompose q r = pcompose (pCons a p) r + pcompose (pCons b q) r" by (simp only: pcompose_pCons add_ac) finally show ?case . qed lemma pcompose_uminus: "pcompose (-p) r = -pcompose p r" for p r :: "'a::comm_ring poly" by (induct p) (simp_all add: pcompose_pCons) lemma pcompose_diff: "pcompose (p - q) r = pcompose p r - pcompose q r" for p q r :: "'a::comm_ring poly" using pcompose_add[of p "-q"] by (simp add: pcompose_uminus) lemma pcompose_smult: "pcompose (smult a p) r = smult a (pcompose p r)" for p r :: "'a::comm_semiring_0 poly" by (induct p) (simp_all add: pcompose_pCons pcompose_add smult_add_right) lemma pcompose_mult: "pcompose (p * q) r = pcompose p r * pcompose q r" for p q r :: "'a::comm_semiring_0 poly" by (induct p arbitrary: q) (simp_all add: pcompose_add pcompose_smult pcompose_pCons algebra_simps) lemma pcompose_assoc: "pcompose p (pcompose q r) = pcompose (pcompose p q) r" for p q r :: "'a::comm_semiring_0 poly" by (induct p arbitrary: q) (simp_all add: pcompose_pCons pcompose_add pcompose_mult) lemma pcompose_idR[simp]: "pcompose p [: 0, 1 :] = p" for p :: "'a::comm_semiring_1 poly" by (induct p) (simp_all add: pcompose_pCons) lemma pcompose_sum: "pcompose (sum f A) p = sum (λi. pcompose (f i) p) A" by (induct A rule: infinite_finite_induct) (simp_all add: pcompose_1 pcompose_add) lemma pcompose_prod: "pcompose (prod f A) p = prod (λi. pcompose (f i) p) A" by (induct A rule: infinite_finite_induct) (simp_all add: pcompose_1 pcompose_mult) lemma pcompose_const [simp]: "pcompose [:a:] q = [:a:]" by (subst pcompose_pCons) simp lemma pcompose_0': "pcompose p 0 = [:coeff p 0:]" by (induct p) (auto simp add: pcompose_pCons) lemma degree_pcompose: "degree (pcompose p q) = degree p * degree q" for p q :: "'a::{comm_semiring_0,semiring_no_zero_divisors} poly" proof (induct p) case 0 then show ?case by auto next case (pCons a p) consider "degree (q * pcompose p q) = 0" | "degree (q * pcompose p q) > 0" by blast then show ?case proof cases case prems: 1 show ?thesis proof (cases "p = 0") case True then show ?thesis by auto next case False from prems have "degree q = 0 ∨ pcompose p q = 0" by (auto simp add: degree_mult_eq_0) moreover have False if "pcompose p q = 0" "degree q ≠ 0" proof - from pCons.hyps(2) that have "degree p = 0" by auto then obtain a1 where "p = [:a1:]" by (metis degree_pCons_eq_if old.nat.distinct(2) pCons_cases) with ‹pcompose p q = 0› ‹p ≠ 0› show False by auto qed ultimately have "degree (pCons a p) * degree q = 0" by auto moreover have "degree (pcompose (pCons a p) q) = 0" proof - from prems have "0 = max (degree [:a:]) (degree (q * pcompose p q))" by simp also have "… ≥ degree ([:a:] + q * pcompose p q)" by (rule degree_add_le_max) finally show ?thesis by (auto simp add: pcompose_pCons) qed ultimately show ?thesis by simp qed next case prems: 2 then have "p ≠ 0" "q ≠ 0" "pcompose p q ≠ 0" by auto from prems degree_add_eq_right [of "[:a:]"] have "degree (pcompose (pCons a p) q) = degree (q * pcompose p q)" by (auto simp: pcompose_pCons) with pCons.hyps(2) degree_mult_eq[OF ‹q≠0› ‹pcompose p q≠0›] show ?thesis by auto qed qed lemma pcompose_eq_0: fixes p q :: "'a::{comm_semiring_0,semiring_no_zero_divisors} poly" assumes "pcompose p q = 0" "degree q > 0" shows "p = 0" proof - from assms degree_pcompose [of p q] have "degree p = 0" by auto then obtain a where "p = [:a:]" by (metis degree_pCons_eq_if gr0_conv_Suc neq0_conv pCons_cases) with assms(1) have "a = 0" by auto with ‹p = [:a:]› show ?thesis by simp qed lemma lead_coeff_comp: fixes p q :: "'a::{comm_semiring_1,semiring_no_zero_divisors} poly" assumes "degree q > 0" shows "lead_coeff (pcompose p q) = lead_coeff p * lead_coeff q ^ (degree p)" proof (induct p) case 0 then show ?case by auto next case (pCons a p) consider "degree (q * pcompose p q) = 0" | "degree (q * pcompose p q) > 0" by blast then show ?case proof cases case prems: 1 then have "pcompose p q = 0" by (metis assms degree_0 degree_mult_eq_0 neq0_conv) with pcompose_eq_0[OF _ ‹degree q > 0›] have "p = 0" by simp then show ?thesis by auto next case prems: 2 then have "degree [:a:] < degree (q * pcompose p q)" by simp then have "lead_coeff ([:a:] + q * p ∘⇩_{p}q) = lead_coeff (q * p ∘⇩_{p}q)" by (rule lead_coeff_add_le) then have "lead_coeff (pcompose (pCons a p) q) = lead_coeff (q * pcompose p q)" by (simp add: pcompose_pCons) also have "… = lead_coeff q * (lead_coeff p * lead_coeff q ^ degree p)" using pCons.hyps(2) lead_coeff_mult[of q "pcompose p q"] by simp also have "… = lead_coeff p * lead_coeff q ^ (degree p + 1)" by (auto simp: mult_ac) finally show ?thesis by auto qed qed subsection ‹Shifting polynomials› definition poly_shift :: "nat ⇒ 'a::zero poly ⇒ 'a poly" where "poly_shift n p = Abs_poly (λi. coeff p (i + n))" lemma nth_default_drop: "nth_default x (drop n xs) m = nth_default x xs (m + n)" by (auto simp add: nth_default_def add_ac) lemma nth_default_take: "nth_default x (take n xs) m = (if m < n then nth_default x xs m else x)" by (auto simp add: nth_default_def add_ac) lemma coeff_poly_shift: "coeff (poly_shift n p) i = coeff p (i + n)" proof - from MOST_coeff_eq_0[of p] obtain m where "∀k>m. coeff p k = 0" by (auto simp: MOST_nat) then have "∀k>m. coeff p (k + n) = 0" by auto then have "∀⇩_{∞}k. coeff p (k + n) = 0" by (auto simp: MOST_nat) then show ?thesis by (simp add: poly_shift_def poly.Abs_poly_inverse) qed lemma poly_shift_id [simp]: "poly_shift 0 = (λx. x)" by (simp add: poly_eq_iff fun_eq_iff coeff_poly_shift) lemma poly_shift_0 [simp]: "poly_shift n 0 = 0" by (simp add: poly_eq_iff coeff_poly_shift) lemma poly_shift_1: "poly_shift n 1 = (if n = 0 then 1 else 0)" by (simp add: poly_eq_iff coeff_poly_shift) lemma poly_shift_monom: "poly_shift n (monom c m) = (if m ≥ n then monom c (m - n) else 0)" by (auto simp add: poly_eq_iff coeff_poly_shift) lemma coeffs_shift_poly [code abstract]: "coeffs (poly_shift n p) = drop n (coeffs p)" proof (cases "p = 0") case True then show ?thesis by simp next case False then show ?thesis by (intro coeffs_eqI) (simp_all add: coeff_poly_shift nth_default_drop nth_default_coeffs_eq) qed subsection ‹Truncating polynomials› definition poly_cutoff where "poly_cutoff n p = Abs_poly (λk. if k < n then coeff p k else 0)" lemma coeff_poly_cutoff: "coeff (poly_cutoff n p) k = (if k < n then coeff p k else 0)" unfolding poly_cutoff_def by (subst poly.Abs_poly_inverse) (auto simp: MOST_nat intro: exI[of _ n]) lemma poly_cutoff_0 [simp]: "poly_cutoff n 0 = 0" by (simp add: poly_eq_iff coeff_poly_cutoff) lemma poly_cutoff_1 [simp]: "poly_cutoff n 1 = (if n = 0 then 0 else 1)" by (simp add: poly_eq_iff coeff_poly_cutoff) lemma coeffs_poly_cutoff [code abstract]: "coeffs (poly_cutoff n p) = strip_while ((=) 0) (take n (coeffs p))" proof (cases "strip_while ((=) 0) (take n (coeffs p)) = []") case True then have "coeff (poly_cutoff n p) k = 0" for k unfolding coeff_poly_cutoff by (auto simp: nth_default_coeffs_eq [symmetric] nth_default_def set_conv_nth) then have "poly_cutoff n p = 0" by (simp add: poly_eq_iff) then show ?thesis by (subst True) simp_all next case False have "no_trailing ((=) 0) (strip_while ((=) 0) (take n (coeffs p)))" by simp with False have "last (strip_while ((=) 0) (take n (coeffs p))) ≠ 0" unfolding no_trailing_unfold by auto then show ?thesis by (intro coeffs_eqI) (simp_all add: coeff_poly_cutoff nth_default_take nth_default_coeffs_eq) qed subsection ‹Reflecting polynomials› definition reflect_poly :: "'a::zero poly ⇒ 'a poly" where "reflect_poly p = Poly (rev (coeffs p))" lemma coeffs_reflect_poly [code abstract]: "coeffs (reflect_poly p) = rev (dropWhile ((=) 0) (coeffs p))" by (simp add: reflect_poly_def) lemma reflect_poly_0 [simp]: "reflect_poly 0 = 0" by (simp add: reflect_poly_def) lemma reflect_poly_1 [simp]: "reflect_poly 1 = 1" by (simp add: reflect_poly_def one_pCons) lemma coeff_reflect_poly: "coeff (reflect_poly p) n = (if n > degree p then 0 else coeff p (degree p - n))" by (cases "p = 0") (auto simp add: reflect_poly_def nth_default_def rev_nth degree_eq_length_coeffs coeffs_nth not_less dest: le_imp_less_Suc) lemma coeff_0_reflect_poly_0_iff [simp]: "coeff (reflect_poly p) 0 = 0 ⟷ p = 0" by (simp add: coeff_reflect_poly) lemma reflect_poly_at_0_eq_0_iff [simp]: "poly (reflect_poly p) 0 = 0 ⟷ p = 0" by (simp add: coeff_reflect_poly poly_0_coeff_0) lemma reflect_poly_pCons': "p ≠ 0 ⟹ reflect_poly (pCons c p) = reflect_poly p + monom c (Suc (degree p))" by (intro poly_eqI) (auto simp: coeff_reflect_poly coeff_pCons not_less Suc_diff_le split: nat.split) lemma reflect_poly_const [simp]: "reflect_poly [:a:] = [:a:]" by (cases "a = 0") (simp_all add: reflect_poly_def) lemma poly_reflect_poly_nz: "x ≠ 0 ⟹ poly (reflect_poly p) x = x ^ degree p * poly p (inverse x)" for x :: "'a::field" by (induct rule: pCons_induct) (simp_all add: field_simps reflect_poly_pCons' poly_monom) lemma coeff_0_reflect_poly [simp]: "coeff (reflect_poly p) 0 = lead_coeff p" by (simp add: coeff_reflect_poly) lemma poly_reflect_poly_0 [simp]: "poly (reflect_poly p) 0 = lead_coeff p" by (simp add: poly_0_coeff_0) lemma reflect_poly_reflect_poly [simp]: "coeff p 0 ≠ 0 ⟹ reflect_poly (reflect_poly p) = p" by (cases p rule: pCons_cases) (simp add: reflect_poly_def ) lemma degree_reflect_poly_le: "degree (reflect_poly p) ≤ degree p" by (simp add: degree_eq_length_coeffs coeffs_reflect_poly length_dropWhile_le diff_le_mono) lemma reflect_poly_pCons: "a ≠ 0 ⟹ reflect_poly (pCons a p) = Poly (rev (a # coeffs p))" by (subst coeffs_eq_iff) (simp add: coeffs_reflect_poly) lemma degree_reflect_poly_eq [simp]: "coeff p 0 ≠ 0 ⟹ degree (reflect_poly p) = degree p" by (cases p rule: pCons_cases) (simp add: reflect_poly_pCons degree_eq_length_coeffs) (* TODO: does this work with zero divisors as well? Probably not. *) lemma reflect_poly_mult: "reflect_poly (p * q) = reflect_poly p * reflect_poly q" for p q :: "'a::{comm_semiring_0,semiring_no_zero_divisors} poly" proof (cases "p = 0 ∨ q = 0") case False then have [simp]: "p ≠ 0" "q ≠ 0" by auto show ?thesis proof (rule poly_eqI) show "coeff (reflect_poly (p * q)) i = coeff (reflect_poly p * reflect_poly q) i" for i proof (cases "i ≤ degree (p * q)") case True define A where "A = {..i} ∩ {i - degree q..degree p}" define B where "B = {..degree p} ∩ {degree p - i..degree (p*q) - i}" let ?f = "λj. degree p - j" from True have "coeff (reflect_poly (p * q)) i = coeff (p * q) (degree (p * q) - i)" by (simp add: coeff_reflect_poly) also have "… = (∑j≤degree (p * q) - i. coeff p j * coeff q (degree (p * q) - i - j))" by (simp add: coeff_mult) also have "… = (∑j∈B. coeff p j * coeff q (degree (p * q) - i - j))" by (intro sum.mono_neutral_right) (auto simp: B_def degree_mult_eq not_le coeff_eq_0) also from True have "… = (∑j∈A. coeff p (degree p - j) * coeff q (degree q - (i - j)))" by (intro sum.reindex_bij_witness[of _ ?f ?f]) (auto simp: A_def B_def degree_mult_eq add_ac) also have "… = (∑j≤i. if j ∈ {i - degree q..degree p} then coeff p (degree p - j) * coeff q (degree q - (i - j)) else 0)" by (subst sum.inter_restrict [symmetric]) (simp_all add: A_def) also have "… = coeff (reflect_poly p * reflect_poly q) i" by (fastforce simp: coeff_mult coeff_reflect_poly intro!: sum.cong) finally show ?thesis . qed (auto simp: coeff_mult coeff_reflect_poly coeff_eq_0 degree_mult_eq intro!: sum.neutral) qed qed auto lemma reflect_poly_smult: "reflect_poly (smult c p) = smult c (reflect_poly p)" for p :: "'a::{comm_semiring_0,semiring_no_zero_divisors} poly" using reflect_poly_mult[of "[:c:]" p] by simp lemma reflect_poly_power: "reflect_poly (p ^ n) = reflect_poly p ^ n" for p :: "'a::{comm_semiring_1,semiring_no_zero_divisors} poly" by (induct n) (simp_all add: reflect_poly_mult) lemma reflect_poly_prod: "reflect_poly (prod f A) = prod (λx. reflect_poly (f x)) A" for f :: "_ ⇒ _::{comm_semiring_0,semiring_no_zero_divisors} poly" by (induct A rule: infinite_finite_induct) (simp_all add: reflect_poly_mult) lemma reflect_poly_prod_list: "reflect_poly (prod_list xs) = prod_list (map reflect_poly xs)" for xs :: "_::{comm_semiring_0,semiring_no_zero_divisors} poly list" by (induct xs) (simp_all add: reflect_poly_mult) lemma reflect_poly_Poly_nz: "no_trailing (HOL.eq 0) xs ⟹ reflect_poly (Poly xs) = Poly (rev xs)" by (simp add: reflect_poly_def) lemmas reflect_poly_simps = reflect_poly_0 reflect_poly_1 reflect_poly_const reflect_poly_smult reflect_poly_mult reflect_poly_power reflect_poly_prod reflect_poly_prod_list subsection ‹Derivatives› function pderiv :: "('a :: {comm_semiring_1,semiring_no_zero_divisors}) poly ⇒ 'a poly" where "pderiv (pCons a p) = (if p = 0 then 0 else p + pCons 0 (pderiv p))" by (auto intro: pCons_cases) termination pderiv by (relation "measure degree") simp_all declare pderiv.simps[simp del] lemma pderiv_0 [simp]: "pderiv 0 = 0" using pderiv.simps [of 0 0] by simp lemma pderiv_pCons: "pderiv (pCons a p) = p + pCons 0 (pderiv p)" by (simp add: pderiv.simps) lemma pderiv_1 [simp]: "pderiv 1 = 0" by (simp add: one_pCons pderiv_pCons) lemma pderiv_of_nat [simp]: "pderiv (of_nat n) = 0" and pderiv_numeral [simp]: "pderiv (numeral m) = 0" by (simp_all add: of_nat_poly numeral_poly pderiv_pCons) lemma coeff_pderiv: "coeff (pderiv p) n = of_nat (Suc n) * coeff p (Suc n)" by (induct p arbitrary: n) (auto simp add: pderiv_pCons coeff_pCons algebra_simps split: nat.split) fun pderiv_coeffs_code :: "'a::{comm_semiring_1,semiring_no_zero_divisors} ⇒ 'a list ⇒ 'a list" where "pderiv_coeffs_code f (x # xs) = cCons (f * x) (pderiv_coeffs_code (f+1) xs)" | "pderiv_coeffs_code f [] = []" definition pderiv_coeffs :: "'a::{comm_semiring_1,semiring_no_zero_divisors} list ⇒ 'a list" where "pderiv_coeffs xs = pderiv_coeffs_code 1 (tl xs)" (* Efficient code for pderiv contributed by RenÃ© Thiemann and Akihisa Yamada *) lemma pderiv_coeffs_code: "nth_default 0 (pderiv_coeffs_code f xs) n = (f + of_nat n) * nth_default 0 xs n" proof (induct xs arbitrary: f n) case Nil then show ?case by simp next case (Cons x xs) show ?case proof (cases n) case 0 then show ?thesis by (cases "pderiv_coeffs_code (f + 1) xs = [] ∧ f * x = 0") (auto simp: cCons_def) next case n: (Suc m) show ?thesis proof (cases "pderiv_coeffs_code (f + 1) xs = [] ∧ f * x = 0") case False then have "nth_default 0 (pderiv_coeffs_code f (x # xs)) n = nth_default 0 (pderiv_coeffs_code (f + 1) xs) m" by (auto simp: cCons_def n) also have "… = (f + of_nat n) * nth_default 0 xs m" by (simp add: Cons n add_ac) finally show ?thesis by (simp add: n) next case True have empty: "pderiv_coeffs_code g xs = [] ⟹ g + of_nat m = 0 ∨ nth_default 0 xs m = 0" for g proof (induct xs arbitrary: g m) case Nil then show ?case by simp next case (Cons x xs) from Cons(2) have empty: "pderiv_coeffs_code (g + 1) xs = []" and g: "g = 0 ∨ x = 0" by (auto simp: cCons_def split: if_splits) note IH = Cons(1)[OF empty] from IH[of m] IH[of "m - 1"] g show ?case by (cases m) (auto simp: field_simps) qed from True have "nth_default 0 (pderiv_coeffs_code f (x # xs)) n = 0" by (auto simp: cCons_def n) moreover from True have "(f + of_nat n) * nth_default 0 (x # xs) n = 0" by (simp add: n) (use empty[of "f+1"] in ‹auto simp: field_simps›) ultimately show ?thesis by simp qed qed qed lemma coeffs_pderiv_code [code abstract]: "coeffs (pderiv p) = pderiv_coeffs (coeffs p)" unfolding pderiv_coeffs_def proof (rule coeffs_eqI, unfold pderiv_coeffs_code coeff_pderiv, goal_cases) case (1 n) have id: "coeff p (Suc n) = nth_default 0 (map (λi. coeff p (Suc i)) [0..<degree p]) n" by (cases "n < degree p") (auto simp: nth_default_def coeff_eq_0) show ?case unfolding coeffs_def map_upt_Suc by (auto simp: id) next case 2 obtain n :: 'a and xs where defs: "tl (coeffs p) = xs" "1 = n" by simp from 2 show ?case unfolding defs by (induct xs arbitrary: n) (auto simp: cCons_def) qed lemma pderiv_eq_0_iff: "pderiv p = 0 ⟷ degree p = 0" for p :: "'a::{comm_semiring_1,semiring_no_zero_divisors,semiring_char_0} poly" apply (rule iffI) apply (cases p) apply simp apply (simp add: poly_eq_iff coeff_pderiv del: of_nat_Suc) apply (simp add: poly_eq_iff coeff_pderiv coeff_eq_0) done lemma degree_pderiv: "degree (pderiv p) = degree p - 1" for p :: "'a::{comm_semiring_1,semiring_no_zero_divisors,semiring_char_0} poly" apply (rule order_antisym [OF degree_le]) apply (simp add: coeff_pderiv coeff_eq_0) apply (cases "degree p") apply simp apply (rule le_degree) apply (simp add: coeff_pderiv del: of_nat_Suc) apply (metis degree_0 leading_coeff_0_iff nat.distinct(1)) done lemma not_dvd_pderiv: fixes p :: "'a::{comm_semiring_1,semiring_no_zero_divisors,semiring_char_0} poly" assumes "degree p ≠ 0" shows "¬ p dvd pderiv p" proof assume dvd: "p dvd pderiv p" then obtain q where p: "pderiv p = p * q" unfolding dvd_def by auto from dvd have le: "degree p ≤ degree (pderiv p)" by (simp add: assms dvd_imp_degree_le pderiv_eq_0_iff) from assms and this [unfolded degree_pderiv] show False by auto qed lemma dvd_pderiv_iff [simp]: "p dvd pderiv p ⟷ degree p = 0" for p :: "'a::{comm_semiring_1,semiring_no_zero_divisors,semiring_char_0} poly" using not_dvd_pderiv[of p] by (auto simp: pderiv_eq_0_iff [symmetric]) lemma pderiv_singleton [simp]: "pderiv [:a:] = 0" by (simp add: pderiv_pCons) lemma pderiv_add: "pderiv (p + q) = pderiv p + pderiv q" by (rule poly_eqI) (simp add: coeff_pderiv algebra_simps) lemma pderiv_minus: "pderiv (- p :: 'a :: idom poly) = - pderiv p" by (rule poly_eqI) (simp add: coeff_pderiv algebra_simps) lemma pderiv_diff: "pderiv ((p :: _ :: idom poly) - q) = pderiv p - pderiv q" by (rule poly_eqI) (simp add: coeff_pderiv algebra_simps) lemma pderiv_smult: "pderiv (smult a p) = smult a (pderiv p)" by (rule poly_eqI) (simp add: coeff_pderiv algebra_simps) lemma pderiv_mult: "pderiv (p * q) = p * pderiv q + q * pderiv p" by (induct p) (auto simp: pderiv_add pderiv_smult pderiv_pCons algebra_simps) lemma pderiv_power_Suc: "pderiv (p ^ Suc n) = smult (of_nat (Suc n)) (p ^ n) * pderiv p" apply (induct n) apply simp apply (subst power_Suc) apply (subst pderiv_mult) apply (erule ssubst) apply (simp only: of_nat_Suc smult_add_left smult_1_left) apply (simp add: algebra_simps) done lemma pderiv_pcompose: "pderiv (pcompose p q) = pcompose (pderiv p) q * pderiv q" by (induction p rule: pCons_induct) (auto simp: pcompose_pCons pderiv_add pderiv_mult pderiv_pCons pcompose_add algebra_simps) lemma pderiv_prod: "pderiv (prod f (as)) = (∑a∈as. prod f (as - {a}) * pderiv (f a))" proof (induct as rule: infinite_finite_induct) case (insert a as) then have id: "prod f (insert a as) = f a * prod f as" "⋀g. sum g (insert a as) = g a + sum g as" "insert a as - {a} = as" by auto have "prod f (insert a as - {b}) = f a * prod f (as - {b})" if "b ∈ as" for b proof - from ‹a ∉ as› that have *: "insert a as - {b} = insert a (as - {b})" by auto show ?thesis unfolding * by (subst prod.insert) (use insert in auto) qed then show ?case unfolding id pderiv_mult insert(3) sum_distrib_left by (auto simp add: ac_simps intro!: sum.cong) qed auto lemma DERIV_pow2: "DERIV (λx. x ^ Suc n) x :> real (Suc n) * (x ^ n)" by (rule DERIV_cong, rule DERIV_pow) simp declare DERIV_pow2 [simp] DERIV_pow [simp] lemma DERIV_add_const: "DERIV f x :> D ⟹ DERIV (λx. a + f x :: 'a::real_normed_field) x :> D" by (rule DERIV_cong, rule DERIV_add) auto lemma poly_DERIV [simp]: "DERIV (λx. poly p x) x :> poly (pderiv p) x" by (induct p) (auto intro!: derivative_eq_intros simp add: pderiv_pCons) lemma poly_isCont[simp]: fixes x::"'a::real_normed_field" shows "isCont (λx. poly p x) x" by (rule poly_DERIV [THEN DERIV_isCont]) lemma tendsto_poly [tendsto_intros]: "(f ⤏ a) F ⟹ ((λx. poly p (f x)) ⤏ poly p a) F" for f :: "_ ⇒ 'a::real_normed_field" by (rule isCont_tendsto_compose [OF poly_isCont]) lemma continuous_within_poly: "continuous (at z within s) (poly p)" for z :: "'a::{real_normed_field}" by (simp add: continuous_within tendsto_poly) lemma continuous_poly [continuous_intros]: "continuous F f ⟹ continuous F (λx. poly p (f x))" for f :: "_ ⇒ 'a::real_normed_field" unfolding continuous_def by (rule tendsto_poly) lemma continuous_on_poly [continuous_intros]: fixes p :: "'a :: {real_normed_field} poly" assumes "continuous_on A f" shows "continuous_on A (λx. poly p (f x))" by (metis DERIV_continuous_on assms continuous_on_compose2 poly_DERIV subset_UNIV) text ‹Consequences of the derivative theorem above.› lemma poly_differentiable[simp]: "(λx. poly p x) differentiable (at x)" for x :: real by (simp add: real_differentiable_def) (blast intro: poly_DERIV) lemma poly_IVT_pos: "a < b ⟹ poly p a < 0 ⟹ 0 < poly p b ⟹ ∃x. a < x ∧ x < b ∧ poly p x = 0" for a b :: real using IVT_objl [of "poly p" a 0 b] by (auto simp add: order_le_less) lemma poly_IVT_neg: "a < b ⟹ 0 < poly p a ⟹ poly p b < 0 ⟹ ∃x. a < x ∧ x < b ∧ poly p x = 0" for a b :: real using poly_IVT_pos [where p = "- p"] by simp lemma poly_IVT: "a < b ⟹ poly p a * poly p b < 0 ⟹ ∃x>a. x < b ∧ poly p x = 0" for p :: "real poly" by (metis less_not_sym mult_less_0_iff poly_IVT_neg poly_IVT_pos) lemma poly_MVT: "a < b ⟹ ∃x. a < x ∧ x < b ∧ poly p b - poly p a = (b - a) * poly (pderiv p) x" for a b :: real using MVT [of a b "poly p"] apply auto apply (rule_tac x = z in exI) apply (auto simp add: mult_left_cancel poly_DERIV [THEN DERIV_unique]) done lemma poly_MVT': fixes a b :: real assumes "{min a b..max a b} ⊆ A" shows "∃x∈A. poly p b - poly p a = (b - a) * poly (pderiv p) x" proof (cases a b rule: linorder_cases) case less from poly_MVT[OF less, of p] guess x by (elim exE conjE) then show ?thesis by (intro bexI[of _ x]) (auto intro!: subsetD[OF assms]) next case greater from poly_MVT[OF greater, of p] guess x by (elim exE conjE) then show ?thesis by (intro bexI[of _ x]) (auto simp: algebra_simps intro!: subsetD[OF assms]) qed (use assms in auto) lemma poly_pinfty_gt_lc: fixes p :: "real poly" assumes "lead_coeff p > 0" shows "∃n. ∀ x ≥ n. poly p x ≥ lead_coeff p" using assms proof (induct p) case 0 then show ?case by auto next case (pCons a p) from this(1) consider "a ≠ 0" "p = 0" | "p ≠ 0" by auto then show ?case proof cases case 1 then show ?thesis by auto next case 2 with pCons obtain n1 where gte_lcoeff: "∀x≥n1. lead_coeff p ≤ poly p x" by auto from pCons(3) ‹p ≠ 0› have gt_0: "lead_coeff p > 0" by auto define n where "n = max n1 (1 + ¦a¦ / lead_coeff p)" have "lead_coeff (pCons a p) ≤ poly (pCons a p) x" if "n ≤ x" for x proof - from gte_lcoeff that have "lead_coeff p ≤ poly p x" by (auto simp: n_def) with gt_0 have "¦a¦ / lead_coeff p ≥ ¦a¦ / poly p x" and "poly p x > 0" by (auto intro: frac_le) with ‹n ≤ x›[unfolded n_def] have "x ≥ 1 + ¦a¦ / poly p x" by auto with ‹lead_coeff p ≤ poly p x› ‹poly p x > 0› ‹p ≠ 0› show "lead_coeff (pCons a p) ≤ poly (pCons a p) x" by (auto simp: field_simps) qed then show ?thesis by blast qed qed lemma lemma_order_pderiv1: "pderiv ([:- a, 1:] ^ Suc n * q) = [:- a, 1:] ^ Suc n * pderiv q + smult (of_nat (Suc n)) (q * [:- a, 1:] ^ n)" by (simp only: pderiv_mult pderiv_power_Suc) (simp del: power_Suc of_nat_Suc add: pderiv_pCons) lemma lemma_order_pderiv: fixes p :: "'a :: field_char_0 poly" assumes n: "0 < n" and pd: "pderiv p ≠ 0" and pe: "p = [:- a, 1:] ^ n * q" and nd: "¬ [:- a, 1:] dvd q" shows "n = Suc (order a (pderiv p))" proof - from assms have "pderiv ([:- a, 1:] ^ n * q) ≠ 0" by auto from assms obtain n' where "n = Suc n'" "0 < Suc n'" "pderiv ([:- a, 1:] ^ Suc n' * q) ≠ 0" by (cases n) auto have *: "k dvd k * pderiv q + smult (of_nat (Suc n')) l ⟹ k dvd l" for k l by (auto simp del: of_nat_Suc simp: dvd_add_right_iff dvd_smult_iff) have "n' = order a (pderiv ([:- a, 1:] ^ Suc n' * q))" proof (rule order_unique_lemma) show "[:- a, 1:] ^ n' dvd pderiv ([:- a, 1:] ^ Suc n' * q)" apply (subst lemma_order_pderiv1) apply (rule dvd_add) apply (metis dvdI dvd_mult2 power_Suc2) apply (metis dvd_smult dvd_triv_right) done show "¬ [:- a, 1:] ^ Suc n' dvd pderiv ([:- a, 1:] ^ Suc n' * q)" apply (subst lemma_order_pderiv1) apply (metis * nd dvd_mult_cancel_right power_not_zero pCons_eq_0_iff power_Suc zero_neq_one) done qed then show ?thesis by (metis ‹n = Suc n'› pe) qed lemma order_pderiv: "pderiv p ≠ 0 ⟹ order a p ≠ 0 ⟹ order a p = Suc (order a (pderiv p))" for p :: "'a::field_char_0 poly" apply (cases "p = 0") apply simp apply (drule_tac a = a and p = p in order_decomp) using neq0_conv apply (blast intro: lemma_order_pderiv) done lemma poly_squarefree_decomp_order: fixes p :: "'a::field_char_0 poly" assumes "pderiv p ≠ 0" and p: "p = q * d" and p': "pderiv p = e * d" and d: "d = r * p + s * pderiv p" shows "order a q = (if order a p = 0 then 0 else 1)" proof (rule classical) assume 1: "¬ ?thesis" from ‹pderiv p ≠ 0› have "p ≠ 0" by auto with p have "order a p = order a q + order a d" by (simp add: order_mult) with 1 have "order a p ≠ 0" by (auto split: if_splits) from ‹pderiv p ≠ 0› ‹pderiv p = e * d› have "order a (pderiv p) = order a e + order a d" by (simp add: order_mult) from ‹pderiv p ≠ 0› ‹order a p ≠ 0› have "order a p = Suc (order a (pderiv p))" by (rule order_pderiv) from ‹p ≠ 0› ‹p = q * d› have "d ≠ 0" by simp have "([:-a, 1:] ^ (order a (pderiv p))) dvd d" apply (simp add: d) apply (rule dvd_add) apply (rule dvd_mult) apply (simp add: order_divides ‹p ≠ 0› ‹order a p = Suc (order a (pderiv p))›) apply (rule dvd_mult) apply (simp add: order_divides) done with ‹d ≠ 0› have "order a (pderiv p) ≤ order a d" by (simp add: order_divides) show ?thesis using ‹order a p = order a q + order a d› and ‹order a (pderiv p) = order a e + order a d› and ‹order a p = Suc (order a (pderiv p))› and ‹order a (pderiv p) ≤ order a d› by auto qed lemma poly_squarefree_decomp_order2: "pderiv p ≠ 0 ⟹ p = q * d ⟹ pderiv p = e * d ⟹ d = r * p + s * pderiv p ⟹ ∀a. order a q = (if order a p = 0 then 0 else 1)" for p :: "'a::field_char_0 poly" by (blast intro: poly_squarefree_decomp_order) lemma order_pderiv2: "pderiv p ≠ 0 ⟹ order a p ≠ 0 ⟹ order a (pderiv p) = n ⟷ order a p = Suc n" for p :: "'a::field_char_0 poly" by (auto dest: order_pderiv) definition rsquarefree :: "'a::idom poly ⇒ bool" where "rsquarefree p ⟷ p ≠ 0 ∧ (∀a. order a p = 0 ∨ order a p = 1)" lemma pderiv_iszero: "pderiv p = 0 ⟹ ∃h. p = [:h:]" for p :: "'a::{semidom,semiring_char_0} poly" by (cases p) (auto simp: pderiv_eq_0_iff split: if_splits) lemma rsquarefree_roots: "rsquarefree p ⟷ (∀a. ¬ (poly p a = 0 ∧ poly (pderiv p) a = 0))" for p :: "'a::field_char_0 poly" apply (simp add: rsquarefree_def) apply (case_tac "p = 0") apply simp apply simp apply (case_tac "pderiv p = 0") apply simp apply (drule pderiv_iszero, clarsimp) apply (metis coeff_0 coeff_pCons_0 degree_pCons_0 le0 le_antisym order_degree) apply (force simp add: order_root order_pderiv2) done lemma poly_squarefree_decomp: fixes p :: "'a::field_char_0 poly" assumes "pderiv p ≠ 0" and "p = q * d" and "pderiv p = e * d" and "d = r * p + s * pderiv p" shows "rsquarefree q ∧ (∀a. poly q a = 0 ⟷ poly p a = 0)" proof - from ‹pderiv p ≠ 0› have "p ≠ 0" by auto with ‹p = q * d› have "q ≠ 0" by simp from assms have "∀a. order a q = (if order a p = 0 then 0 else 1)" by (rule poly_squarefree_decomp_order2) with ‹p ≠ 0› ‹q ≠ 0› show ?thesis by (simp add: rsquarefree_def order_root) qed subsection ‹Algebraic numbers› text ‹ Algebraic numbers can be defined in two equivalent ways: all real numbers that are roots of rational polynomials or of integer polynomials. The Algebraic-Numbers AFP entry uses the rational definition, but we need the integer definition. The equivalence is obvious since any rational polynomial can be multiplied with the LCM of its coefficients, yielding an integer polynomial with the same roots. › definition algebraic :: "'a :: field_char_0 ⇒ bool" where "algebraic x ⟷ (∃p. (∀i. coeff p i ∈ ℤ) ∧ p ≠ 0 ∧ poly p x = 0)" lemma algebraicI: "(⋀i. coeff p i ∈ ℤ) ⟹ p ≠ 0 ⟹ poly p x = 0 ⟹ algebraic x" unfolding algebraic_def by blast lemma algebraicE: assumes "algebraic x" obtains p where "⋀i. coeff p i ∈ ℤ" "p ≠ 0" "poly p x = 0" using assms unfolding algebraic_def by blast lemma algebraic_altdef: "algebraic x ⟷ (∃p. (∀i. coeff p i ∈ ℚ) ∧ p ≠ 0 ∧ poly p x = 0)" for p :: "'a::field_char_0 poly" proof safe fix p assume rat: "∀i. coeff p i ∈ ℚ" and root: "poly p x = 0" and nz: "p ≠ 0" define cs where "cs = coeffs p" from rat have "∀c∈range (coeff p). ∃c'. c = of_rat c'" unfolding Rats_def by blast then obtain f where f: "coeff p i = of_rat (f (coeff p i))" for i by (subst (asm) bchoice_iff) blast define cs' where "cs' = map (quotient_of ∘ f) (coeffs p)" define d where "d = Lcm (set (map snd cs'))" define p' where "p' = smult (of_int d) p" have "coeff p' n ∈ ℤ" for n proof (cases "n ≤ degree p") case True define c where "c = coeff p n" define a where "a = fst (quotient_of (f (coeff p n)))" define b where "b = snd (quotient_of (f (coeff p n)))" have b_pos: "b > 0" unfolding b_def using quotient_of_denom_pos' by simp have "coeff p' n = of_int d * coeff p n" by (simp add: p'_def) also have "coeff p n = of_rat (of_int a / of_int b)" unfolding a_def b_def by (subst quotient_of_div [of "f (coeff p n)", symmetric]) (simp_all add: f [symmetric]) also have "of_int d * … = of_rat (of_int (a*d) / of_int b)" by (simp add: of_rat_mult of_rat_divide) also from nz True have "b ∈ snd ` set cs'" by (force simp: cs'_def o_def b_def coeffs_def simp del: upt_Suc) then have "b dvd (a * d)" by (simp add: d_def) then have "of_int (a * d) / of_int b ∈ (ℤ :: rat set)" by (rule of_int_divide_in_Ints) then have "of_rat (of_int (a * d) / of_int b) ∈ ℤ" by (elim Ints_cases) auto finally show ?thesis . next case False then show ?thesis by (auto simp: p'_def not_le coeff_eq_0) qed moreover have "set (map snd cs') ⊆ {0<..}" unfolding cs'_def using quotient_of_denom_pos' by (auto simp: coeffs_def simp del: upt_Suc) then have "d ≠ 0" unfolding d_def by (induct cs') simp_all with nz have "p' ≠ 0" by (simp add: p'_def) moreover from root have "poly p' x = 0" by (simp add: p'_def) ultimately show "algebraic x" unfolding algebraic_def by blast next assume "algebraic x" then obtain p where p: "coeff p i ∈ ℤ" "poly p x = 0" "p ≠ 0" for i by (force simp: algebraic_def) moreover have "coeff p i ∈ ℤ ⟹ coeff p i ∈ ℚ" for i by (elim Ints_cases) simp ultimately show "∃p. (∀i. coeff p i ∈ ℚ) ∧ p ≠ 0 ∧ poly p x = 0" by auto qed subsection ‹Division of polynomials› subsubsection ‹Division in general› instantiation poly :: (idom_divide) idom_divide begin fun divide_poly_main :: "'a ⇒ 'a poly ⇒ 'a poly ⇒ 'a poly ⇒ nat ⇒ nat ⇒ 'a poly" where "divide_poly_main lc q r d dr (Suc n) = (let cr = coeff r dr; a = cr div lc; mon = monom a n in if False ∨ a * lc = cr then ― ‹‹False ∨› is only because of problem in function-package› divide_poly_main lc (q + mon) (r - mon * d) d (dr - 1) n else 0)" | "divide_poly_main lc q r d dr 0 = q" definition divide_poly :: "'a poly ⇒ 'a poly ⇒ 'a poly" where "divide_poly f g = (if g = 0 then 0 else divide_poly_main (coeff g (degree g)) 0 f g (degree f) (1 + length (coeffs f) - length (coeffs g)))" lemma divide_poly_main: assumes d: "d ≠ 0" "lc = coeff d (degree d)" and "degree (d * r) ≤ dr" "divide_poly_main lc q (d * r) d dr n = q'" and "n = 1 + dr - degree d ∨ dr = 0 ∧ n = 0 ∧ d * r = 0" shows "q' = q + r" using assms(3-) proof (induct n arbitrary: q r dr) case (Suc n) let ?rr = "d * r" let ?a = "coeff ?rr dr" let ?qq = "?a div lc" define b where [simp]: "b = monom ?qq n" let ?rrr = "d * (r - b)" let ?qqq = "q + b" note res = Suc(3) from Suc(4) have dr: "dr = n + degree d" by auto from d have lc: "lc ≠ 0" by auto have "coeff (b * d) dr = coeff b n * coeff d (degree d)" proof (cases "?qq = 0") case True then show ?thesis by simp next case False then have n: "n = degree b" by (simp add: degree_monom_eq) show ?thesis unfolding n dr by (simp add: coeff_mult_degree_sum) qed also have "… = lc * coeff b n" by (simp add: d) finally have c2: "coeff (b * d) dr = lc * coeff b n" . have rrr: "?rrr = ?rr - b * d" by (simp add: field_simps) have c1: "coeff (d * r) dr = lc * coeff r n" proof (cases "degree r = n") case True with Suc(2) show ?thesis unfolding dr using coeff_mult_degree_sum[of d r] d by (auto simp: ac_simps) next case False from dr Suc(2) have "degree r ≤ n" by auto (metis add.commute add_le_cancel_left d(1) degree_0 degree_mult_eq diff_is_0_eq diff_zero le_cases) with False have r_n: "degree r < n" by auto then have right: "lc * coeff r n = 0" by (simp add: coeff_eq_0) have "coeff (d * r) dr = coeff (d * r) (degree d + n)" by (simp add: dr ac_simps) also from r_n have "… = 0" by (metis False Suc.prems(1) add.commute add_left_imp_eq coeff_degree_mult coeff_eq_0 coeff_mult_degree_sum degree_mult_le dr le_eq_less_or_eq) finally show ?thesis by (simp only: right) qed have c0: "coeff ?rrr dr = 0" and id: "lc * (coeff (d * r) dr div lc) = coeff (d * r) dr" unfolding rrr coeff_diff c2 unfolding b_def coeff_monom coeff_smult c1 using lc by auto from res[unfolded divide_poly_main.simps[of lc q] Let_def] id have res: "divide_poly_main lc ?qqq ?rrr d (dr - 1) n = q'" by (simp del: divide_poly_main.simps add: field_simps) note IH = Suc(1)[OF _ res] from Suc(4) have dr: "dr = n + degree d" by auto from Suc(2) have deg_rr: "degree ?rr ≤ dr" by auto have deg_bd: "degree (b * d) ≤ dr" unfolding dr b_def by (rule order.trans[OF degree_mult_le]) (auto simp: degree_monom_le) have "degree ?rrr ≤ dr" unfolding rrr by (rule degree_diff_le[OF deg_rr deg_bd]) with c0 have deg_rrr: "degree ?rrr ≤ (dr - 1)" by (rule coeff_0_degree_minus_1) have "n = 1 + (dr - 1) - degree d ∨ dr - 1 = 0 ∧ n = 0 ∧ ?rrr = 0" proof (cases dr) case 0 with Suc(4) have 0: "dr = 0" "n = 0" "degree d = 0" by auto with deg_rrr have "degree ?rrr = 0" by simp from degree_eq_zeroE[OF this] obtain a where rrr: "?rrr = [:a:]" by metis show ?thesis unfolding 0 using c0 unfolding rrr 0 by simp next case _: Suc with Suc(4) show ?thesis by auto qed from IH[OF deg_rrr this] show ?case by simp next case 0 show ?case proof (cases "r = 0") case True with 0 show ?thesis by auto next case False from d False have "degree (d * r) = degree d + degree r" by (subst degree_mult_eq) auto with 0 d show ?thesis by auto qed qed lemma divide_poly_main_0: "divide_poly_main 0 0 r d dr n = 0" proof (induct n arbitrary: r d dr) case 0 then show ?case by simp next case Suc show ?case unfolding divide_poly_main.simps[of _ _ r] Let_def by (simp add: Suc del: divide_poly_main.simps) qed lemma divide_poly: assumes g: "g ≠ 0" shows "(f * g) div g = (f :: 'a poly)" proof - have len: "length (coeffs f) = Suc (degree f)" if "f ≠ 0" for f :: "'a poly" using that unfolding degree_eq_length_coeffs by auto have "divide_poly_main (coeff g (degree g)) 0 (g * f) g (degree (g * f)) (1 + length (coeffs (g * f)) - length (coeffs g)) = (f * g) div g" by (simp add: divide_poly_def Let_def ac_simps) note main = divide_poly_main[OF g refl le_refl this] have "(f * g) div g = 0 + f" proof (rule main, goal_cases) case 1 show ?case proof (cases "f = 0") case True with g show ?thesis by (auto simp: degree_eq_length_coeffs) next case False with g have fg: "g * f ≠ 0" by auto show ?thesis unfolding len[OF fg] len[OF g] by auto qed qed then show ?thesis by simp qed lemma divide_poly_0: "f div 0 = 0" for f :: "'a poly" by (simp add: divide_poly_def Let_def divide_poly_main_0) instance by standard (auto simp: divide_poly divide_poly_0) end instance poly :: (idom_divide) algebraic_semidom .. lemma div_const_poly_conv_map_poly: assumes "[:c:] dvd p" shows "p div [:c:] = map_poly (λx. x div c) p" proof (cases "c = 0") case True then show ?thesis by (auto intro!: poly_eqI simp: coeff_map_poly) next case False from assms obtain q where p: "p = [:c:] * q" by (rule dvdE) moreover { have "smult c q = [:c:] * q" by simp also have "… div [:c:] = q" by (rule nonzero_mult_div_cancel_left) (use False in auto) finally have "smult c q div [:c:] = q" . } ultimately show ?thesis by (intro poly_eqI) (auto simp: coeff_map_poly False) qed lemma is_unit_monom_0: fixes a :: "'a::field" assumes "a ≠ 0" shows "is_unit (monom a 0)" proof from assms show "1 = monom a 0 * monom (inverse a) 0" by (simp add: mult_monom) qed lemma is_unit_triv: "a ≠ 0 ⟹ is_unit [:a:]" for a :: "'a::field" by (simp add: is_unit_monom_0 monom_0 [symmetric]) lemma is_unit_iff_degree: fixes p :: "'a::field poly" assumes "p ≠ 0" shows "is_unit p ⟷ degree p = 0" (is "?lhs ⟷ ?rhs") proof assume ?rhs then obtain a where "p = [:a:]" by (rule degree_eq_zeroE) with assms show ?lhs by (simp add: is_unit_triv) next assume ?lhs then obtain q where "q ≠ 0" "p * q = 1" .. then have "degree (p * q) = degree 1" by simp with ‹p ≠ 0› ‹q ≠ 0› have "degree p + degree q = 0" by (simp add: degree_mult_eq) then show ?rhs by simp qed lemma is_unit_pCons_iff: "is_unit (pCons a p) ⟷ p = 0 ∧ a ≠ 0" for p :: "'a::field poly" by (cases "p = 0") (auto simp: is_unit_triv is_unit_iff_degree) lemma is_unit_monom_trival: "is_unit p ⟹ monom (coeff p (degree p)) 0 = p" for p :: "'a::field poly" by (cases p) (simp_all add: monom_0 is_unit_pCons_iff) lemma is_unit_const_poly_iff: "[:c:] dvd 1 ⟷ c dvd 1" for c :: "'a::{comm_semiring_1,semiring_no_zero_divisors}" by (auto simp: one_pCons) lemma is_unit_polyE: fixes p :: "'a :: {comm_semiring_1,semiring_no_zero_divisors} poly" assumes "p dvd 1" obtains c where "p = [:c:]" "c dvd 1" proof - from assms obtain q where "1 = p * q" by (rule dvdE) then have "p ≠ 0" and "q ≠ 0" by auto from ‹1 = p * q› have "degree 1 = degree (p * q)" by simp also from ‹p ≠ 0› and ‹q ≠ 0› have "… = degree p + degree q" by (simp add: degree_mult_eq) finally have "degree p = 0" by simp with degree_eq_zeroE obtain c where c: "p = [:c:]" . with ‹p dvd 1› have "c dvd 1" by (simp add: is_unit_const_poly_iff) with c show thesis .. qed lemma is_unit_polyE': fixes p :: "'a::field poly" assumes "is_unit p" obtains a where "p = monom a 0" and "a ≠ 0" proof - obtain a q where "p = pCons a q" by (cases p) with assms have "p = [:a:]" and "a ≠ 0" by (simp_all add: is_unit_pCons_iff) with that show thesis by (simp add: monom_0) qed lemma is_unit_poly_iff: "p dvd 1 ⟷ (∃c. p = [:c:] ∧ c dvd 1)" for p :: "'a::{comm_semiring_1,semiring_no_zero_divisors} poly" by (auto elim: is_unit_polyE simp add: is_unit_const_poly_iff) subsubsection ‹Pseudo-Division› text ‹This part is by RenÃ© Thiemann and Akihisa Yamada.› fun pseudo_divmod_main :: "'a :: comm_ring_1 ⇒ 'a poly ⇒ 'a poly ⇒ 'a poly ⇒ nat ⇒ nat ⇒ 'a poly × 'a poly" where "pseudo_divmod_main lc q r d dr (Suc n) = (let rr = smult lc r; qq = coeff r dr; rrr = rr - monom qq n * d; qqq = smult lc q + monom qq n in pseudo_divmod_main lc qqq rrr d (dr - 1) n)" | "pseudo_divmod_main lc q r d dr 0 = (q,r)" definition pseudo_divmod :: "'a :: comm_ring_1 poly ⇒ 'a poly ⇒ 'a poly × 'a poly" where "pseudo_divmod p q ≡ if q = 0 then (0, p) else pseudo_divmod_main (coeff q (degree q)) 0 p q (degree p) (1 + length (coeffs p) - length (coeffs q))" lemma pseudo_divmod_main: assumes d: "d ≠ 0" "lc = coeff d (degree d)" and "degree r ≤ dr" "pseudo_divmod_main lc q r d dr n = (q',r')" and "n = 1 + dr - degree d ∨ dr = 0 ∧ n = 0 ∧ r = 0" shows "(r' = 0 ∨ degree r' < degree d) ∧ smult (lc^n) (d * q + r) = d * q' + r'" using assms(3-) proof (induct n arbitrary: q r dr) case 0 then show ?case by auto next case (Suc n) let ?rr = "smult lc r" let ?qq = "coeff r dr" define b where [simp]: "b = monom ?qq n" let ?rrr = "?rr - b * d" let ?qqq = "smult lc q + b" note res = Suc(3) from res[unfolded pseudo_divmod_main.simps[of lc q] Let_def] have res: "pseudo_divmod_main lc ?qqq ?rrr d (dr - 1) n = (q',r')" by (simp del: pseudo_divmod_main.simps) from Suc(4) have dr: "dr = n + degree d" by auto have "coeff (b * d) dr = coeff b n * coeff d (degree d)" proof (cases "?qq = 0") case True then show ?thesis by auto next case False then have n: "n = degree b" by (simp add: degree_monom_eq) show ?thesis unfolding n dr by (simp add: coeff_mult_degree_sum) qed also have "… = lc * coeff b n" by (simp add: d) finally have "coeff (b * d) dr = lc * coeff b n" . moreover have "coeff ?rr dr = lc * coeff r dr" by simp ultimately have c0: "coeff ?rrr dr = 0" by auto from Suc(4) have dr: "dr = n + degree d" by auto have deg_rr: "degree ?rr ≤ dr" using Suc(2) degree_smult_le dual_order.trans by blast have deg_bd: "degree (b * d) ≤ dr" unfolding dr by (rule order.trans[OF degree_mult_le]) (auto simp: degree_monom_le) have "degree ?rrr ≤ dr" using degree_diff_le[OF deg_rr deg_bd] by auto with c0 have deg_rrr: "degree ?rrr ≤ (dr - 1)" by (rule coeff_0_degree_minus_1) have "n = 1 + (dr - 1) - degree d ∨ dr - 1 = 0 ∧ n = 0 ∧ ?rrr = 0" proof (cases dr) case 0 with Suc(4) have 0: "dr = 0" "n = 0" "degree d = 0" by auto with deg_rrr have "degree ?rrr = 0" by simp then have "∃a. ?rrr = [:a:]" by (metis degree_pCons_eq_if old.nat.distinct(2) pCons_cases) from this obtain a where rrr: "?rrr = [:a:]" by auto show ?thesis unfolding 0 using c0 unfolding rrr 0 by simp next case _: Suc with Suc(4) show ?thesis by auto qed note IH = Suc(1)[OF deg_rrr res this] show ?case proof (intro conjI) from IH show "r' = 0 ∨ degree r' < degree d" by blast show "smult (lc ^ Suc n) (d * q + r) = d * q' + r'" unfolding IH[THEN conjunct2,symmetric] by (simp add: field_simps smult_add_right) qed qed lemma pseudo_divmod: assumes g: "g ≠ 0" and *: "pseudo_divmod f g = (q,r)" shows "smult (coeff g (degree g) ^ (Suc (degree f) - degree g)) f = g * q + r" (is ?A) and "r = 0 ∨ degree r < degree g" (is ?B) proof - from *[unfolded pseudo_divmod_def Let_def] have "pseudo_divmod_main (coeff g (degree g)) 0 f g (degree f) (1 + length (coeffs f) - length (coeffs g)) = (q, r)" by (auto simp: g) note main = pseudo_divmod_main[OF _ _ _ this, OF g refl le_refl] from g have "1 + length (coeffs f) - length (coeffs g) = 1 + degree f - degree g ∨ degree f = 0 ∧ 1 + length (coeffs f) - length (coeffs g) = 0 ∧ f = 0" by (cases "f = 0"; cases "coeffs g") (auto simp: degree_eq_length_coeffs) note main' = main[OF this] then show "r = 0 ∨ degree r < degree g" by auto show "smult (coeff g (degree g) ^ (Suc (degree f) - degree g)) f = g * q + r" by (subst main'[THEN conjunct2, symmetric], simp add: degree_eq_length_coeffs, cases "f = 0"; cases "coeffs g", use g in auto) qed definition "pseudo_mod_main lc r d dr n = snd (pseudo_divmod_main lc 0 r d dr n)" lemma snd_pseudo_divmod_main: "snd (pseudo_divmod_main lc q r d dr n) = snd (pseudo_divmod_main lc q' r d dr n)" by (induct n arbitrary: q q' lc r d dr) (simp_all add: Let_def) definition pseudo_mod :: "'a::{comm_ring_1,semiring_1_no_zero_divisors} poly ⇒ 'a poly ⇒ 'a poly" where "pseudo_mod f g = snd (pseudo_divmod f g)" lemma pseudo_mod: fixes f g :: "'a::{comm_ring_1,semiring_1_no_zero_divisors} poly" defines "r ≡ pseudo_mod f g" assumes g: "g ≠ 0" shows "∃a q. a ≠ 0 ∧ smult a f = g * q + r" "r = 0 ∨ degree r < degree g" proof - let ?cg = "coeff g (degree g)" let ?cge = "?cg ^ (Suc (degree f) - degree g)" define a where "a = ?cge" from r_def[unfolded pseudo_mod_def] obtain q where pdm: "pseudo_divmod f g = (q, r)" by (cases "pseudo_divmod f g") auto from pseudo_divmod[OF g pdm] have id: "smult a f = g * q + r" and "r = 0 ∨ degree r < degree g" by (auto simp: a_def) show "r = 0 ∨ degree r < degree g" by fact from g have "a ≠ 0" by (auto simp: a_def) with id show "∃a q. a ≠ 0 ∧ smult a f = g * q + r" by auto qed lemma fst_pseudo_divmod_main_as_divide_poly_main: assumes d: "d ≠ 0" defines lc: "lc ≡ coeff d (degree d)" shows "fst (pseudo_divmod_main lc q r d dr n) = divide_poly_main lc (smult (lc^n) q) (smult (lc^n) r) d dr n" proof (induct n arbitrary: q r dr) case 0 then show ?case by simp next case (Suc n) note lc0 = leading_coeff_neq_0[OF d, folded lc] then have "pseudo_divmod_main lc q r d dr (Suc n) = pseudo_divmod_main lc (smult lc q + monom (coeff r dr) n) (smult lc r - monom (coeff r dr) n * d) d (dr - 1) n" by (simp add: Let_def ac_simps) also have "fst … = divide_poly_main lc (smult (lc^n) (smult lc q + monom (coeff r dr) n)) (smult (lc^n) (smult lc r - monom (coeff r dr) n * d)) d (dr - 1) n" by (simp only: Suc[unfolded divide_poly_main.simps Let_def]) also have "… = divide_poly_main lc (smult (lc ^ Suc n) q) (smult (lc ^ Suc n) r) d dr (Suc n)" unfolding smult_monom smult_distribs mult_smult_left[symmetric] using lc0 by (simp add: Let_def ac_simps) finally show ?case . qed subsubsection ‹Division in polynomials over fields› lemma pseudo_divmod_field: fixes g :: "'a::field poly" assumes g: "g ≠ 0" and *: "pseudo_divmod f g = (q,r)" defines "c ≡ coeff g (degree g) ^ (Suc (degree f) - degree g)" shows "f = g * smult (1/c) q + smult (1/c) r" proof - from leading_coeff_neq_0[OF g] have c0: "c ≠ 0" by (auto simp: c_def) from pseudo_divmod(1)[OF g *, folded c_def] have "smult c f = g * q + r" by auto also have "smult (1 / c) … = g * smult (1 / c) q + smult (1 / c) r" by (simp add: smult_add_right) finally show ?thesis using c0 by auto qed lemma divide_poly_main_field: fixes d :: "'a::field poly" assumes d: "d ≠ 0" defines lc: "lc ≡ coeff d (degree d)" shows "divide_poly_main lc q r d dr n = fst (pseudo_divmod_main lc (smult ((1 / lc)^n) q) (smult ((1 / lc)^n) r) d dr n)" unfolding lc by (subst fst_pseudo_divmod_main_as_divide_poly_main) (auto simp: d power_one_over) lemma divide_poly_field: fixes f g :: "'a::field poly" defines "f' ≡ smult ((1 / coeff g (degree g)) ^ (Suc (degree f) - degree g)) f" shows "f div g = fst (pseudo_divmod f' g)" proof (cases "g = 0") case True show ?thesis unfolding divide_poly_def pseudo_divmod_def Let_def f'_def True by (simp add: divide_poly_main_0) next case False from leading_coeff_neq_0[OF False] have "degree f' = degree f" by (auto simp: f'_def) then show ?thesis using length_coeffs_degree[of f'] length_coeffs_degree[of f] unfolding divide_poly_def pseudo_divmod_def Let_def divide_poly_main_field[OF False] length_coeffs_degree[OF False] f'_def by force qed instantiation poly :: ("{semidom_divide_unit_factor,idom_divide}") normalization_semidom begin definition unit_factor_poly :: "'a poly ⇒ 'a poly" where "unit_factor_poly p = [:unit_factor (lead_coeff p):]" definition normalize_poly :: "'a poly ⇒ 'a poly" where "normalize p = p div [:unit_factor (lead_coeff p):]" instance proof fix p :: "'a poly" show "unit_factor p * normalize p = p" proof (cases "p = 0") case True then show ?thesis by (simp add: unit_factor_poly_def normalize_poly_def) next case False then have "lead_coeff p ≠ 0" by simp then have *: "unit_factor (lead_coeff p) ≠ 0" using unit_factor_is_unit [of "lead_coeff p"] by auto then have "unit_factor (lead_coeff p) dvd 1" by (auto intro: unit_factor_is_unit) then have **: "unit_factor (lead_coeff p) dvd c" for c by (rule dvd_trans) simp have ***: "unit_factor (lead_coeff p) * (c div unit_factor (lead_coeff p)) = c" for c proof - from ** obtain b where "c = unit_factor (lead_coeff p) * b" .. with False * show ?thesis by simp qed have "p div [:unit_factor (lead_coeff p):] = map_poly (λc. c div unit_factor (lead_coeff p)) p" by (simp add: const_poly_dvd_iff div_const_poly_conv_map_poly **) then show ?thesis by (simp add: normalize_poly_def unit_factor_poly_def smult_conv_map_poly map_poly_map_poly o_def ***) qed next fix p :: "'a poly" assume "is_unit p" then obtain c where p: "p = [:c:]" "c dvd 1" by (auto simp: is_unit_poly_iff) then show "unit_factor p = p" by (simp add: unit_factor_poly_def monom_0 is_unit_unit_factor) next fix p :: "'a poly" assume "p ≠ 0" then show "is_unit (unit_factor p)" by (simp add: unit_factor_poly_def monom_0 is_unit_poly_iff unit_factor_is_unit) qed (simp_all add: normalize_poly_def unit_factor_poly_def monom_0 lead_coeff_mult unit_factor_mult) end lemma normalize_poly_eq_map_poly: "normalize p = map_poly (λx. x div unit_factor (lead_coeff p)) p" proof - have "[:unit_factor (lead_coeff p):] dvd p" by (metis unit_factor_poly_def unit_factor_self) then show ?thesis by (simp add: normalize_poly_def div_const_poly_conv_map_poly) qed lemma coeff_normalize [simp]: "coeff (normalize p) n = coeff p n div unit_factor (lead_coeff p)" by (simp add: normalize_poly_eq_map_poly coeff_map_poly) class field_unit_factor = field + unit_factor + assumes unit_factor_field [simp]: "unit_factor = id" begin subclass semidom_divide_unit_factor proof fix a assume "a ≠ 0" then have "1 = a * inverse a" by simp then have "a dvd 1" .. then show "unit_factor a dvd 1" by simp qed simp_all end lemma unit_factor_pCons: "unit_factor (pCons a p) = (if p = 0 then [:unit_factor a:] else unit_factor p)" by (simp add: unit_factor_poly_def) lemma normalize_monom [simp]: "normalize (monom a n) = monom (normalize a) n" by (cases "a = 0") (simp_all add: map_poly_monom normalize_poly_eq_map_poly degree_monom_eq) lemma unit_factor_monom [simp]: "unit_factor (monom a n) = [:unit_factor a:]" by (cases "a = 0") (simp_all add: unit_factor_poly_def degree_monom_eq) lemma normalize_const_poly: "normalize [:c:] = [:normalize c:]" by (simp add: normalize_poly_eq_map_poly map_poly_pCons) lemma normalize_smult: "normalize (smult c p) = smult (normalize c) (normalize p)" proof - have "smult c p = [:c:] * p" by simp also have "normalize … = smult (normalize c) (normalize p)" by (subst normalize_mult) (simp add: normalize_const_poly) finally show ?thesis . qed inductive eucl_rel_poly :: "'a::field poly ⇒ 'a poly ⇒ 'a poly × 'a poly ⇒ bool" where eucl_rel_poly_by0: "eucl_rel_poly x 0 (0, x)" | eucl_rel_poly_dividesI: "y ≠ 0 ⟹ x = q * y ⟹ eucl_rel_poly x y (q, 0)" | eucl_rel_poly_remainderI: "y ≠ 0 ⟹ degree r < degree y ⟹ x = q * y + r ⟹ eucl_rel_poly x y (q, r)" lemma eucl_rel_poly_iff: "eucl_rel_poly x y (q, r) ⟷ x = q * y + r ∧ (if y = 0 then q = 0 else r = 0 ∨ degree r < degree y)" by (auto elim: eucl_rel_poly.cases intro: eucl_rel_poly_by0 eucl_rel_poly_dividesI eucl_rel_poly_remainderI) lemma eucl_rel_poly_0: "eucl_rel_poly 0 y (0, 0)" by (simp add: eucl_rel_poly_iff) lemma eucl_rel_poly_by_0: "eucl_rel_poly x 0 (0, x)" by (simp add: eucl_rel_poly_iff) lemma eucl_rel_poly_pCons: assumes rel: "eucl_rel_poly x y (q, r)" assumes y: "y ≠ 0" assumes b: "b = coeff (pCons a r) (degree y) / coeff y (degree y)" shows "eucl_rel_poly (pCons a x) y (pCons b q, pCons a r - smult b y)" (is "eucl_rel_poly ?x y (?q, ?r)") proof - from assms have x: "x = q * y + r" and r: "r = 0 ∨ degree r < degree y" by (simp_all add: eucl_rel_poly_iff) from b x have "?x = ?q * y + ?r" by simp moreover have "?r = 0 ∨ degree ?r < degree y" proof (rule eq_zero_or_degree_less) show "degree ?r ≤ degree y" proof (rule degree_diff_le) from r show "degree (pCons a r) ≤ degree y" by auto show "degree (smult b y) ≤ degree y" by (rule degree_smult_le) qed from ‹y ≠ 0› show "coeff ?r (degree y) = 0" by (simp add: b) qed ultimately show ?thesis unfolding eucl_rel_poly_iff using ‹y ≠ 0› by simp qed lemma eucl_rel_poly_exists: "∃q r. eucl_rel_poly x y (q, r)" apply (cases "y = 0") apply (fast intro!: eucl_rel_poly_by_0) apply (induct x) apply (fast intro!: eucl_rel_poly_0) apply (fast intro!: eucl_rel_poly_pCons) done lemma eucl_rel_poly_unique: assumes 1: "eucl_rel_poly x y (q1, r1)" assumes 2: "eucl_rel_poly x y (q2, r2)" shows "q1 = q2 ∧ r1 = r2" proof (cases "y = 0") assume "y = 0" with assms show ?thesis by (simp add: eucl_rel_poly_iff) next assume [simp]: "y ≠ 0" from 1 have q1: "x = q1 * y + r1" and r1: "r1 = 0 ∨ degree r1 < degree y" unfolding eucl_rel_poly_iff by simp_all from 2 have q2: "x = q2 * y + r2" and r2: "r2 = 0 ∨ degree r2 < degree y" unfolding eucl_rel_poly_iff 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 classical) assume "¬ ?thesis" 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 from ‹q1 ≠ q2› have "… = degree ((q1 - q2) * y)" by (simp add: degree_mult_eq) also from q3 have "… = degree (r2 - r1)" by simp finally have "degree (r2 - r1) < degree (r2 - r1)" . then show ?thesis by simp qed qed lemma eucl_rel_poly_0_iff: "eucl_rel_poly 0 y (q, r) ⟷ q = 0 ∧ r = 0" by (auto dest: eucl_rel_poly_unique intro: eucl_rel_poly_0) lemma eucl_rel_poly_by_0_iff: "eucl_rel_poly x 0 (q, r) ⟷ q = 0 ∧ r = x" by (auto dest: eucl_rel_poly_unique intro: eucl_rel_poly_by_0) lemmas eucl_rel_poly_unique_div = eucl_rel_poly_unique [THEN conjunct1] lemmas eucl_rel_poly_unique_mod = eucl_rel_poly_unique [THEN conjunct2] instantiation poly :: (field) semidom_modulo begin definition modulo_poly :: "'a poly ⇒ 'a poly ⇒ 'a poly" where mod_poly_def: "f mod g = (if g = 0 then f else pseudo_mod (smult ((1 / lead_coeff g) ^ (Suc (degree f) - degree g)) f) g)" instance proof fix x y :: "'a poly" show "x div y * y + x mod y = x" proof (cases "y = 0") case True then show ?thesis by (simp add: divide_poly_0 mod_poly_def) next case False then have "pseudo_divmod (smult ((1 / lead_coeff y) ^ (Suc (degree x) - degree y)) x) y = (x div y, x mod y)" by (simp add: divide_poly_field mod_poly_def pseudo_mod_def) with False pseudo_divmod [OF False this] show ?thesis by (simp add: power_mult_distrib [symmetric] ac_simps) qed qed end lemma eucl_rel_poly: "eucl_rel_poly x y (x div y, x mod y)" unfolding eucl_rel_poly_iff proof show "x = x div y * y + x mod y" by (simp add: div_mult_mod_eq) show "if y = 0 then x div y = 0 else x mod y = 0 ∨ degree (x mod y) < degree y" proof (cases "y = 0") case True then show ?thesis by auto next case False with pseudo_mod[OF this] show ?thesis by (simp add: mod_poly_def) qed qed lemma div_poly_eq: "eucl_rel_poly x y (q, r) ⟹ x div y = q" for x :: "'a::field poly" by (rule eucl_rel_poly_unique_div [OF eucl_rel_poly]) lemma mod_poly_eq: "eucl_rel_poly x y (q, r) ⟹ x mod y = r" for x :: "'a::field poly" by (rule eucl_rel_poly_unique_mod [OF eucl_rel_poly]) instance poly :: (field) idom_modulo .. lemma div_pCons_eq: "pCons a p div q = (if q = 0 then 0 else pCons (coeff (pCons a (p mod q)) (degree q) / lead_coeff q) (p div q))" using eucl_rel_poly_pCons [OF eucl_rel_poly _ refl, of q a p] by (auto intro: div_poly_eq) lemma mod_pCons_eq: "pCons a p mod q = (if q = 0 then pCons a p else pCons a (p mod q) - smult (coeff (pCons a (p mod q)) (degree q) / lead_coeff q) q)" using eucl_rel_poly_pCons [OF eucl_rel_poly _ refl, of q a p] by (auto intro: mod_poly_eq) lemma div_mod_fold_coeffs: "(p div q, p mod q) = (if q = 0 then (0, p) else fold_coeffs (λa (s, r). let b = coeff (pCons a r) (degree q) / coeff q (degree q) in (pCons b s, pCons a r - smult b q)) p (0, 0))" by (rule sym, induct p) (auto simp: div_pCons_eq mod_pCons_eq Let_def) lemma degree_mod_less: "y ≠ 0 ⟹ x mod y = 0 ∨ degree (x mod y) < degree y" using eucl_rel_poly [of x y] unfolding eucl_rel_poly_iff by simp lemma degree_mod_less': "b ≠ 0 ⟹ a mod b ≠ 0 ⟹ degree (a mod b) < degree b" using degree_mod_less[of b a] by auto lemma div_poly_less: fixes x :: "'a::field poly" assumes "degree x < degree y" shows "x div y = 0" proof - from assms have "eucl_rel_poly x y (0, x)" by (simp add: eucl_rel_poly_iff) then show "x div y = 0" by (rule div_poly_eq) qed lemma mod_poly_less: assumes "degree x < degree y" shows "x mod y = x" proof - from assms have "eucl_rel_poly x y (0, x)" by (simp add: eucl_rel_poly_iff) then show "x mod y = x" by (rule mod_poly_eq) qed lemma eucl_rel_poly_smult_left: "eucl_rel_poly x y (q, r) ⟹ eucl_rel_poly (smult a x) y (smult a q, smult a r)" by (simp add: eucl_rel_poly_iff smult_add_right) lemma div_smult_left: "(smult a x) div y = smult a (x div y)" for x y :: "'a::field poly" by (rule div_poly_eq, rule eucl_rel_poly_smult_left, rule eucl_rel_poly) lemma mod_smult_left: "(smult a x) mod y = smult a (x mod y)" by (rule mod_poly_eq, rule eucl_rel_poly_smult_left, rule eucl_rel_poly) lemma poly_div_minus_left [simp]: "(- x) div y = - (x div y)" for x y :: "'a::field poly" using div_smult_left [of "- 1::'a"] by simp lemma poly_mod_minus_left [simp]: "(- x) mod y = - (x mod y)" for x y :: "'a::field poly" using mod_smult_left [of "- 1::'a"] by simp lemma eucl_rel_poly_add_left: assumes "eucl_rel_poly x y (q, r)" assumes "eucl_rel_poly x' y (q', r')" shows "eucl_rel_poly (x + x') y (q + q', r + r')" using assms unfolding eucl_rel_poly_iff by (auto simp: algebra_simps degree_add_less) lemma poly_div_add_left: "(x + y) div z = x div z + y div z" for x y z :: "'a::field poly" using eucl_rel_poly_add_left [OF eucl_rel_poly eucl_rel_poly] by (rule div_poly_eq) lemma poly_mod_add_left: "(x + y) mod z = x mod z + y mod z" for x y z :: "'a::field poly" using eucl_rel_poly_add_left [OF eucl_rel_poly eucl_rel_poly] by (rule mod_poly_eq) lemma poly_div_diff_left: "(x - y) div z = x div z - y div z" for x y z :: "'a::field poly" by (simp only: diff_conv_add_uminus poly_div_add_left poly_div_minus_left) lemma poly_mod_diff_left: "(x - y) mod z = x mod z - y mod z" for x y z :: "'a::field poly" by (simp only: diff_conv_add_uminus poly_mod_add_left poly_mod_minus_left) lemma eucl_rel_poly_smult_right: "a ≠ 0 ⟹ eucl_rel_poly x y (q, r) ⟹ eucl_rel_poly x (smult a y) (smult (inverse a) q, r)" by (simp add: eucl_rel_poly_iff) lemma div_smult_right: "a ≠ 0 ⟹ x div (smult a y) = smult (inverse a) (x div y)" for x y :: "'a::field poly" by (rule div_poly_eq, erule eucl_rel_poly_smult_right, rule eucl_rel_poly) lemma mod_smult_right: "a ≠ 0 ⟹ x mod (smult a y) = x mod y" by (rule mod_poly_eq, erule eucl_rel_poly_smult_right, rule eucl_rel_poly) lemma poly_div_minus_right [simp]: "x div (- y) = - (x div y)" for x y :: "'a::field poly" using div_smult_right [of "- 1::'a"] by (simp add: nonzero_inverse_minus_eq) lemma poly_mod_minus_right [simp]: "x mod (- y) = x mod y" for x y :: "'a::field poly" using mod_smult_right [of "- 1::'a"] by simp lemma eucl_rel_poly_mult: "eucl_rel_poly x y (q, r) ⟹ eucl_rel_poly q z (q', r') ⟹ eucl_rel_poly x (y * z) (q', y * r' + r)" apply (cases "z = 0", simp add: eucl_rel_poly_iff) apply (cases "y = 0", simp add: eucl_rel_poly_by_0_iff eucl_rel_poly_0_iff) apply (cases "r = 0") apply (cases "r' = 0") apply (simp add: eucl_rel_poly_iff) apply (simp add: eucl_rel_poly_iff field_simps degree_mult_eq) apply (cases "r' = 0") apply (simp add: eucl_rel_poly_iff degree_mult_eq) apply (simp add: eucl_rel_poly_iff field_simps) apply (simp add: degree_mult_eq degree_add_less) done lemma poly_div_mult_right: "x div (y * z) = (x div y) div z" for x y z :: "'a::field poly" by (rule div_poly_eq, rule eucl_rel_poly_mult, (rule eucl_rel_poly)+) lemma poly_mod_mult_right: "x mod (y * z) = y * (x div y mod z) + x mod y" for x y z :: "'a::field poly" by (rule mod_poly_eq, rule eucl_rel_poly_mult, (rule eucl_rel_poly)+) lemma mod_pCons: fixes a :: "'a::field" and x y :: "'a::field poly" assumes y: "y ≠ 0" defines "b ≡ coeff (pCons a (x mod y)) (degree y) / coeff y (degree y)" shows "(pCons a x) mod y = pCons a (x mod y) - smult b y" unfolding b_def by (rule mod_poly_eq, rule eucl_rel_poly_pCons [OF eucl_rel_poly y refl]) subsubsection ‹List-based versions for fast implementation› (* Subsection by: Sebastiaan Joosten RenÃ© Thiemann Akihisa Yamada *) fun minus_poly_rev_list :: "'a :: group_add list ⇒ 'a list ⇒ 'a list" where "minus_poly_rev_list (x # xs) (y # ys) = (x - y) # (minus_poly_rev_list xs ys)" | "minus_poly_rev_list xs [] = xs" | "minus_poly_rev_list [] (y # ys) = []" fun pseudo_divmod_main_list :: "'a::comm_ring_1 ⇒ 'a list ⇒ 'a list ⇒ 'a list ⇒ nat ⇒ 'a list × 'a list" where "pseudo_divmod_main_list lc q r d (Suc n) = (let rr = map (( * ) lc) r; a = hd r; qqq = cCons a (map (( * ) lc) q); rrr = tl (if a = 0 then rr else minus_poly_rev_list rr (map (( * ) a) d)) in pseudo_divmod_main_list lc qqq rrr d n)" | "pseudo_divmod_main_list lc q r d 0 = (q, r)" fun pseudo_mod_main_list :: "'a::comm_ring_1 ⇒ 'a list ⇒ 'a list ⇒ nat ⇒ 'a list" where "pseudo_mod_main_list lc r d (Suc n) = (let rr = map (( * ) lc) r; a = hd r; rrr = tl (if a = 0 then rr else minus_poly_rev_list rr (map (( * ) a) d)) in pseudo_mod_main_list lc rrr d n)" | "pseudo_mod_main_list lc r d 0 = r" fun divmod_poly_one_main_list :: "'a::comm_ring_1 list ⇒ 'a list ⇒ 'a list ⇒ nat ⇒ 'a list × 'a list" where "divmod_poly_one_main_list q r d (Suc n) = (let a = hd r; qqq = cCons a q; rr = tl (if a = 0 then r else minus_poly_rev_list r (map (( * ) a) d)) in divmod_poly_one_main_list qqq rr d n)" | "divmod_poly_one_main_list q r d 0 = (q, r)" fun mod_poly_one_main_list :: "'a::comm_ring_1 list ⇒ 'a list ⇒ nat ⇒ 'a list" where "mod_poly_one_main_list r d (Suc n) = (let a = hd r; rr = tl (if a = 0 then r else minus_poly_rev_list r (map (( * ) a) d)) in mod_poly_one_main_list rr d n)" | "mod_poly_one_main_list r d 0 = r" definition pseudo_divmod_list :: "'a::comm_ring_1 list ⇒ 'a list ⇒ 'a list × 'a list" where "pseudo_divmod_list p q = (if q = [] then ([], p) else (let rq = rev q; (qu,re) = pseudo_divmod_main_list (hd rq) [] (rev p) rq (1 + length p - length q) in (qu, rev re)))" definition pseudo_mod_list :: "'a::comm_ring_1 list ⇒ 'a list ⇒ 'a list" where "pseudo_mod_list p q = (if q = [] then p else (let rq = rev q; re = pseudo_mod_main_list (hd rq) (rev p) rq (1 + length p - length q) in rev re))" lemma minus_zero_does_nothing: "minus_poly_rev_list x (map (( * ) 0) y) = x" for x :: "'a::ring list" by (induct x y rule: minus_poly_rev_list.induct) auto lemma length_minus_poly_rev_list [simp]: "length (minus_poly_rev_list xs ys) = length xs" by (induct xs ys rule: minus_poly_rev_list.induct) auto lemma if_0_minus_poly_rev_list: "(if a = 0 then x else minus_poly_rev_list x (map (( * ) a) y)) = minus_poly_rev_list x (map (( * ) a) y)" for a :: "'a::ring" by(cases "a = 0") (simp_all add: minus_zero_does_nothing) lemma Poly_append: "Poly (a @ b) = Poly a + monom 1 (length a) * Poly b" for a :: "'a::comm_semiring_1 list" by (induct a) (auto simp: monom_0 monom_Suc) lemma minus_poly_rev_list: "length p ≥ length q ⟹ Poly (rev (minus_poly_rev_list (rev p) (rev q))) = Poly p - monom 1 (length p - length q) * Poly q" for p q :: "'a :: comm_ring_1 list" proof (induct "rev p" "rev q" arbitrary: p q rule: minus_poly_rev_list.induct) case (1 x xs y ys) then have "length (rev q) ≤ length (rev p)" by simp from this[folded 1(2,3)] have ys_xs: "length ys ≤ length xs" by simp then have *: "Poly (rev (minus_poly_rev_list xs ys)) = Poly (rev xs) - monom 1 (length xs - length ys) * Poly (rev ys)" by (subst "1.hyps"(1)[of "rev xs" "rev ys", unfolded rev_rev_ident length_rev]) auto have "Poly p - monom 1 (length p - length q) * Poly q = Poly (rev (rev p)) - monom 1 (length (rev (rev p)) - length (rev (rev q))) * Poly (rev (rev q))" by simp also have "… = Poly (rev (x # xs)) - monom 1 (length (x # xs) - length (y # ys)) * Poly (rev (y # ys))" unfolding 1(2,3) by simp also from ys_xs have "… = Poly (rev xs) + monom x (length xs) - (monom 1 (length xs - length ys) * Poly (rev ys) + monom y (length xs))" by (simp add: Poly_append distrib_left mult_monom smult_monom) also have "… = Poly (rev (minus_poly_rev_list xs ys)) + monom (x - y) (length xs)" unfolding * diff_monom[symmetric] by simp finally show ?case by (simp add: 1(2,3)[symmetric] smult_monom Poly_append) qed auto lemma smult_monom_mult: "smult a (monom b n * f) = monom (a * b) n * f" using smult_monom [of a _ n] by (metis mult_smult_left) lemma head_minus_poly_rev_list: "length d ≤ length r ⟹ d ≠ [] ⟹ hd (minus_poly_rev_list (map (( * ) (last d)) r) (map (( * ) (hd r)) (rev d))) = 0" for d r :: "'a::comm_ring list" proof (induct r) case Nil then show ?case by simp next case (Cons a rs) then show ?case by (cases "rev d") (simp_all add: ac_simps) qed lemma Poly_map: "Poly (map (( * ) a) p) = smult a (Poly p)" proof (induct p) case Nil then show ?case by simp next case (Cons x xs) then show ?case by (cases "Poly xs = 0") auto qed lemma last_coeff_is_hd: "xs ≠ [] ⟹ coeff (Poly xs) (length xs - 1) = hd (rev xs)" by (simp_all add: hd_conv_nth rev_nth nth_default_nth nth_append) lemma pseudo_divmod_main_list_invar: assumes leading_nonzero: "last d ≠ 0" and lc: "last d = lc" and "d ≠ []" and "pseudo_divmod_main_list lc q (rev r) (rev d) n = (q', rev r')" and "n = 1 + length r - length d" shows "pseudo_divmod_main lc (monom 1 n * Poly q) (Poly r) (Poly d) (length r - 1) n = (Poly q', Poly r')" using assms(4-) proof (induct n arbitrary: r q) case (Suc n) from Suc.prems have *: "¬ Suc (length r) ≤ length d" by simp with ‹d ≠ []› have "r ≠ []" using Suc_leI length_greater_0_conv list.size(3) by fastforce let ?a = "(hd (rev r))" let ?rr = "map (( * ) lc) (rev r)" let ?rrr = "rev (tl (minus_poly_rev_list ?rr (map (( * ) ?a) (rev d))))" let ?qq = "cCons ?a (map (( * ) lc) q)" from * Suc(3) have n: "n = (1 + length r - length d - 1)" by simp from * have rr_val:"(length ?rrr) = (length r - 1)" by auto with ‹r ≠ []› * have rr_smaller: "(1 + length r - length d - 1) = (1 + length ?rrr - length d)" by auto from * have id: "Suc (length r) - length d = Suc (length r - length d)" by auto from Suc.prems * have "pseudo_divmod_main_list lc ?qq (rev ?rrr) (rev d) (1 + length r - length d - 1) = (q', rev r')" by (simp add: Let_def if_0_minus_poly_rev_list id) with n have v: "pseudo_divmod_main_list lc ?qq (rev ?rrr) (rev d) n = (q', rev r')" by auto from * have sucrr:"Suc (length r) - length d = Suc (length r - length d)" using Suc_diff_le not_less_eq_eq by blast from Suc(3) ‹r ≠ []› have n_ok : "n = 1 + (length ?rrr) - length d" by simp have cong: "⋀x1 x2 x3 x4 y1 y2 y3 y4. x1 = y1 ⟹ x2 = y2 ⟹ x3 = y3 ⟹ x4 = y4 ⟹ pseudo_divmod_main lc x1 x2 x3 x4 n = pseudo_divmod_main lc y1 y2 y3 y4 n" by simp have hd_rev: "coeff (Poly r) (length r - Suc 0) = hd (rev r)" using last_coeff_is_hd[OF ‹r ≠ []›] by simp show ?case unfolding Suc.hyps(1)[OF v n_ok, symmetric] pseudo_divmod_main.simps Let_def proof (rule cong[OF _ _ refl], goal_cases) case 1 show ?case by (simp add: monom_Suc hd_rev[symmetric] smult_monom Poly_map) next case 2 show ?case proof (subst Poly_on_rev_starting_with_0, goal_cases) show "hd (minus_poly_rev_list (map (( * ) lc) (rev r)) (map (( * ) (hd (rev r))) (rev d))) = 0" by (fold lc, subst head_minus_poly_rev_list, insert * ‹d ≠ []›, auto) from * have "length d ≤ length r" by simp then show "smult lc (Poly r) - monom (coeff (Poly r) (length r - 1)) n * Poly d = Poly (rev (minus_poly_rev_list (map (( * ) lc) (rev r)) (map (( * ) (hd (rev r))) (rev d))))" by (fold rev_map) (auto simp add: n smult_monom_mult Poly_map hd_rev [symmetric] minus_poly_rev_list) qed qed simp qed simp lemma pseudo_divmod_impl [code]: "pseudo_divmod f g = map_prod poly_of_list poly_of_list (pseudo_divmod_list (coeffs f) (coeffs g))" for f g :: "'a::comm_ring_1 poly" proof (cases "g = 0") case False then have "last (coeffs g) ≠ 0" and "last (coeffs g) = lead_coeff g" and "coeffs g ≠ []" by (simp_all add: last_coeffs_eq_coeff_degree) moreover obtain q r where qr: "pseudo_divmod_main_list (last (coeffs g)) (rev []) (rev (coeffs f)) (rev (coeffs g)) (1 + length (coeffs f) - length (coeffs g)) = (q, rev (rev r))" by force ultimately have "(Poly q, Poly (rev r)) = pseudo_divmod_main (lead_coeff g) 0 f g (length (coeffs f) - Suc 0) (Suc (length (coeffs f)) - length (coeffs g))" by (subst pseudo_divmod_main_list_invar [symmetric]) auto moreover have "pseudo_divmod_main_list (hd (rev (coeffs g))) [] (rev (coeffs f)) (rev (coeffs g)) (1 + length (coeffs f) - length (coeffs g)) = (q, r)" using qr hd_rev [OF ‹coeffs g ≠ []›] by simp ultimately show ?thesis by (auto simp: degree_eq_length_coeffs pseudo_divmod_def pseudo_divmod_list_def Let_def) next case True then show ?thesis by (auto simp add: pseudo_divmod_def pseudo_divmod_list_def) qed lemma pseudo_mod_main_list: "snd (pseudo_divmod_main_list l q xs ys n) = pseudo_mod_main_list l xs ys n" by (induct n arbitrary: l q xs ys) (auto simp: Let_def) lemma pseudo_mod_impl[code]: "pseudo_mod f g = poly_of_list (pseudo_mod_list (coeffs f) (coeffs g))" proof - have snd_case: "⋀f g p. snd ((λ(x,y). (f x, g y)) p) = g (snd p)" by auto show ?thesis unfolding pseudo_mod_def pseudo_divmod_impl pseudo_divmod_list_def pseudo_mod_list_def Let_def by (simp add: snd_case pseudo_mod_main_list) qed subsubsection ‹Improved Code-Equations for Polynomial (Pseudo) Division› lemma pdivmod_pdivmodrel: "eucl_rel_poly p q (r, s) ⟷ (p div q, p mod q) = (r, s)" by (metis eucl_rel_poly eucl_rel_poly_unique) lemma pdivmod_via_pseudo_divmod: "(f div g, f mod g) = (if g = 0 then (0, f) else let ilc = inverse (coeff g (degree g)); h = smult ilc g; (q,r) = pseudo_divmod f h in (smult ilc q, r))" (is "?l = ?r") proof (cases "g = 0") case True then show ?thesis by simp next case False define lc where "lc = inverse (coeff g (degree g))" define h where "h = smult lc g" from False have h1: "coeff h (degree h) = 1" and lc: "lc ≠ 0" by (auto simp: h_def lc_def) then have h0: "h ≠ 0" by auto obtain q r where p: "pseudo_divmod f h = (q, r)" by force from False have id: "?r = (smult lc q, r)" by (auto simp: Let_def h_def[symmetric] lc_def[symmetric] p) from pseudo_divmod[OF h0 p, unfolded h1] have f: "f = h * q + r" and r: "r = 0 ∨ degree r < degree h" by auto from f r h0 have "eucl_rel_poly f h (q, r)" by (auto simp: eucl_rel_poly_iff) then have "(f div h, f mod h) = (q, r)" by (simp add: pdivmod_pdivmodrel) with lc have "(f div g, f mod g) = (smult lc q, r)" by (auto simp: h_def div_smult_right[OF lc] mod_smult_right[OF lc]) with id show ?thesis by auto qed lemma pdivmod_via_pseudo_divmod_list: "(f div g, f mod g) = (let cg = coeffs g in if cg = [] then (0, f) else let cf = coeffs f; ilc = inverse (last cg); ch = map (( * ) ilc) cg; (q, r) = pseudo_divmod_main_list 1 [] (rev cf) (rev ch) (1 + length cf - length cg) in (poly_of_list (map (( * ) ilc) q), poly_of_list (rev r)))" proof - note d = pdivmod_via_pseudo_divmod pseudo_divmod_impl pseudo_divmod_list_def show ?thesis proof (cases "g = 0") case True with d show ?thesis by auto next case False define ilc where "ilc = inverse (coeff g (degree g))" from False have ilc: "ilc ≠ 0" by (auto simp: ilc_def) with False have id: "g = 0 ⟷ False" "coeffs g = [] ⟷ False" "last (coeffs g) = coeff g (degree g)" "coeffs (smult ilc g) = [] ⟷ False" by (auto simp: last_coeffs_eq_coeff_degree) have id2: "hd (rev (coeffs (smult ilc g))) = 1" by (subst hd_rev, insert id ilc, auto simp: coeffs_smult, subst last_map, auto simp: id ilc_def) have id3: "length (coeffs (smult ilc g)) = length (coeffs g)" "rev (coeffs (smult ilc g)) = rev (map (( * ) ilc) (coeffs g))" unfolding coeffs_smult using ilc by auto obtain q r where pair: "pseudo_divmod_main_list 1 [] (rev (coeffs f)) (rev (map (( * ) ilc) (coeffs g))) (1 + length (coeffs f) - length (coeffs g)) = (q, r)" by force show ?thesis unfolding d Let_def id if_False ilc_def[symmetric] map_prod_def[symmetric] id2 unfolding id3 pair map_prod_def split by (auto simp: Poly_map) qed qed lemma pseudo_divmod_main_list_1: "pseudo_divmod_main_list 1 = divmod_poly_one_main_list" proof (intro ext, goal_cases) case (1 q r d n) have *: "map (( * ) 1) xs = xs" for xs :: "'a list" by (induct xs) auto show ?case by (induct n arbitrary: q r d) (auto simp: * Let_def) qed fun divide_poly_main_list :: "'a::idom_divide ⇒ 'a list ⇒ 'a list ⇒ 'a list ⇒ nat ⇒ 'a list" where "divide_poly_main_list lc q r d (Suc n) = (let cr = hd r in if cr = 0 then divide_poly_main_list lc (cCons cr q) (tl r) d n else let a = cr div lc; qq = cCons a q; rr = minus_poly_rev_list r (map (( * ) a) d) in if hd rr = 0 then divide_poly_main_list lc qq (tl rr) d n else [])" | "divide_poly_main_list lc q r d 0 = q" lemma divide_poly_main_list_simp [simp]: "divide_poly_main_list lc q r d (Suc n) = (let cr = hd r; a = cr div lc; qq = cCons a q; rr = minus_poly_rev_list r (map (( * ) a) d) in if hd rr = 0 then divide_poly_main_list lc qq (tl rr) d n else [])" by (simp add: Let_def minus_zero_does_nothing) declare divide_poly_main_list.simps(1)[simp del] definition divide_poly_list :: "'a::idom_divide poly ⇒ 'a poly ⇒ 'a poly" where "divide_poly_list f g = (let cg = coeffs g in if cg = [] then g else let cf = coeffs f; cgr = rev cg in poly_of_list (divide_poly_main_list (hd cgr) [] (rev cf) cgr (1 + length cf - length cg)))" lemmas pdivmod_via_divmod_list = pdivmod_via_pseudo_divmod_list[unfolded pseudo_divmod_main_list_1] lemma mod_poly_one_main_list: "snd (divmod_poly_one_main_list q r d n) = mod_poly_one_main_list r d n" by (induct n arbitrary: q r d) (auto simp: Let_def) lemma mod_poly_code [code]: "f mod g = (let cg = coeffs g in if cg = [] then f else let cf = coeffs f; ilc = inverse (last cg); ch = map (( * ) ilc) cg; r = mod_poly_one_main_list (rev cf) (rev ch) (1 + length cf - length cg) in poly_of_list (rev r))" (is "_ = ?rhs") proof - have "snd (f div g, f mod g) = ?rhs" unfolding pdivmod_via_divmod_list Let_def mod_poly_one_main_list [symmetric, of _ _ _ Nil] by (auto split: prod.splits) then show ?thesis by simp qed definition div_field_poly_impl :: "'a :: field poly ⇒ 'a poly ⇒ 'a poly" where "div_field_poly_impl f g = (let cg = coeffs g in if cg = [] then 0 else let cf = coeffs f; ilc = inverse (last cg); ch = map (( * ) ilc) cg; q = fst (divmod_poly_one_main_list [] (rev cf) (rev ch) (1 + length cf - length cg)) in poly_of_list ((map (( * ) ilc) q)))" text ‹We do not declare the following lemma as code equation, since then polynomial division on non-fields will no longer be executable. However, a code-unfold is possible, since ‹div_field_poly_impl› is a bit more efficient than the generic polynomial division.› lemma div_field_poly_impl[code_unfold]: "(div) = div_field_poly_impl" proof (intro ext) fix f g :: "'a poly" have "fst (f div g, f mod g) = div_field_poly_impl f g" unfolding div_field_poly_impl_def pdivmod_via_divmod_list Let_def by (auto split: prod.splits) then show "f div g = div_field_poly_impl f g" by simp qed lemma divide_poly_main_list: assumes lc0: "lc ≠ 0" and lc: "last d = lc" and d: "d ≠ []" and "n = (1 + length r - length d)" shows "Poly (divide_poly_main_list lc q (rev r) (rev d) n) = divide_poly_main lc (monom 1 n * Poly q) (Poly r) (Poly d) (length r - 1) n" using assms(4-) proof (induct "n" arbitrary: r q) case (Suc n) from Suc.prems have ifCond: "¬ Suc (length r) ≤ length d" by simp with d have r: "r ≠ []" using Suc_leI length_greater_0_conv list.size(3) by fastforce then obtain rr lcr where r: "r = rr @ [lcr]" by (cases r rule: rev_cases) auto from d lc obtain dd where d: "d = dd @ [lc]" by (cases d rule: rev_cases) auto from Suc(2) ifCond have n: "n = 1 + length rr - length d" by (auto simp: r) from ifCond have len: "length dd ≤ length rr" by (simp add: r d) show ?case proof (cases "lcr div lc * lc = lcr") case False with r d show ?thesis unfolding Suc(2)[symmetric] by (auto simp add: Let_def nth_default_append) next case True with r d have id: "?thesis ⟷ Poly (divide_poly_main_list lc (cCons (lcr div lc) q) (rev (rev (minus_poly_rev_list (rev rr) (rev (map (( * ) (lcr div lc)) dd))))) (rev d) n) = divide_poly_main lc (monom 1 (Suc n) * Poly q + monom (lcr div lc) n) (Poly r - monom (lcr div lc) n * Poly d) (Poly d) (length rr - 1) n" by (cases r rule: rev_cases; cases "d" rule: rev_cases) (auto simp add: Let_def rev_map nth_default_append) have cong: "⋀x1 x2 x3 x4 y1 y2 y3 y4. x1 = y1 ⟹ x2 = y2 ⟹ x3 = y3 ⟹ x4 = y4 ⟹ divide_poly_main lc x1 x2 x3 x4 n = divide_poly_main lc y1 y2 y3 y4 n" by simp show ?thesis unfolding id proof (subst Suc(1), simp add: n, subst minus_poly_rev_list, force simp: len, rule cong[OF _ _ refl], goal_cases) case 2 have "monom lcr (length rr) = monom (lcr div lc) (length rr - length dd) * monom lc (length dd)" by (simp add: mult_monom len True) then show ?case unfolding r d Poly_append n ring_distribs by (auto simp: Poly_map smult_monom smult_monom_mult) qed (auto simp: len monom_Suc smult_monom) qed qed simp lemma divide_poly_list[code]: "f div g = divide_poly_list f g" proof - note d = divide_poly_def divide_poly_list_def show ?thesis proof (cases "g = 0") case True show ?thesis by (auto simp: d True) next case False then obtain cg lcg where cg: "coeffs g = cg @ [lcg]" by (cases "coeffs g" rule: rev_cases) auto with False have id: "(g = 0) = False" "(cg @ [lcg] = []) = False" by auto from cg False have lcg: "coeff g (degree g) = lcg" using last_coeffs_eq_coeff_degree last_snoc by force with False have "lcg ≠ 0" by auto from cg Poly_coeffs [of g] have ltp: "Poly (cg @ [lcg]) = g" by auto show ?thesis unfolding d cg Let_def id if_False poly_of_list_def by (subst divide_poly_main_list, insert False cg ‹lcg ≠ 0›) (auto simp: lcg ltp, simp add: degree_eq_length_coeffs) qed qed subsection ‹Primality and irreducibility in polynomial rings› lemma prod_mset_const_poly: "(∏x∈#A. [:f x:]) = [:prod_mset (image_mset f A):]" by (induct A) (simp_all add: ac_simps) lemma irreducible_const_poly_iff: fixes c :: "'a :: {comm_semiring_1,semiring_no_zero_divisors}" shows "irreducible [:c:] ⟷ irreducible c" proof assume A: "irreducible c" show "irreducible [:c:]" proof (rule irreducibleI) fix a b assume ab: "[:c:] = a * b" hence "degree [:c:] = degree (a * b)" by (simp only: ) also from A ab have "a ≠ 0" "b ≠ 0" by auto hence "degree (a * b) = degree a + degree b" by (simp add: degree_mult_eq) finally have "degree a = 0" "degree b = 0" by auto then obtain a' b' where ab': "a = [:a':]" "b = [:b':]" by (auto elim!: degree_eq_zeroE) from ab have "coeff [:c:] 0 = coeff (a * b) 0" by (simp only: ) hence "c = a' * b'" by (simp add: ab' mult_ac) from A and this have "a' dvd 1 ∨ b' dvd 1" by (rule irreducibleD) with ab' show "a dvd 1 ∨ b dvd 1" by (auto simp add: is_unit_const_poly_iff) qed (insert A, auto simp: irreducible_def is_unit_poly_iff) next assume A: "irreducible [:c:]" then have "c ≠ 0" and "¬ c dvd 1" by (auto simp add: irreducible_def is_unit_const_poly_iff) then show "irreducible c" proof (rule irreducibleI) fix a b assume ab: "c = a * b" hence "[:c:] = [:a:] * [:b:]" by (simp add: mult_ac) from A and this have "[:a:] dvd 1 ∨ [:b:] dvd 1" by (rule irreducibleD) then show "a dvd 1 ∨ b dvd 1" by (auto simp add: is_unit_const_poly_iff) qed qed lemma lift_prime_elem_poly: assumes "prime_elem (c :: 'a :: semidom)" shows "prime_elem [:c:]" proof (rule prime_elemI) fix a b assume *: "[:c:] dvd a * b" from * have dvd: "c dvd coeff (a * b) n" for n by (subst (asm) const_poly_dvd_iff) blast { define m where "m = (GREATEST m. ¬c dvd coeff b m)" assume "¬[:c:] dvd b" hence A: "∃i. ¬c dvd coeff b i" by (subst (asm) const_poly_dvd_iff) blast have B: "∀i. ¬c dvd coeff b i ⟶ i ≤ degree b" by (auto intro: le_degree) have coeff_m: "¬c dvd coeff b m" unfolding m_def by (rule GreatestI_ex_nat[OF A B]) have "i ≤ m" if "¬c dvd coeff b i" for i unfolding m_def by (rule Greatest_le_nat[OF that B]) hence dvd_b: "c dvd coeff b i" if "i > m" for i using that by force have "c dvd coeff a i" for i proof (induction i rule: nat_descend_induct[of "degree a"]) case (base i) thus ?case by (simp add: coeff_eq_0) next case (descend i) let ?A = "{..i+m} - {i}" have "c dvd coeff (a * b) (i + m)" by (rule dvd) also have "coeff (a * b) (i + m) = (∑k≤i + m. coeff a k * coeff b (i + m - k))" by (simp add: coeff_mult) also have "{..i+m} = insert i ?A" by auto also have "(∑k∈…. coeff a k * coeff b (i + m - k)) = coeff a i * coeff b m + (∑k∈?A. coeff a k * coeff b (i + m - k))" (is "_ = _ + ?S") by (subst sum.insert) simp_all finally have eq: "c dvd coeff a i * coeff b m + ?S" . moreover have "c dvd ?S" proof (rule dvd_sum) fix k assume k: "k ∈ {..i+m} - {i}" show "c dvd coeff a k * coeff b (i + m - k)" proof (cases "k < i") case False with k have "c dvd coeff a k" by (intro descend.IH) simp thus ?thesis by simp next case True hence "c dvd coeff b (i + m - k)" by (intro dvd_b) simp thus ?thesis by simp qed qed ultimately have "c dvd coeff a i * coeff b m" by (simp add: dvd_add_left_iff) with assms coeff_m show "c dvd coeff a i" by (simp add: prime_elem_dvd_mult_iff) qed hence "[:c:] dvd a" by (subst const_poly_dvd_iff) blast } then show "[:c:] dvd a ∨ [:c:] dvd b" by blast next from assms show "[:c:] ≠ 0" and "¬ [:c:] dvd 1" by (simp_all add: prime_elem_def is_unit_const_poly_iff) qed lemma prime_elem_const_poly_iff: fixes c :: "'a :: semidom" shows "prime_elem [:c:] ⟷ prime_elem c" proof assume A: "prime_elem [:c:]" show "prime_elem c" proof (rule prime_elemI) fix a b assume "c dvd a * b" hence "[:c:] dvd [:a:] * [:b:]" by (simp add: mult_ac) from A and this have "[:c:] dvd [:a:] ∨ [:c:] dvd [:b:]" by (rule prime_elem_dvd_multD) thus "c dvd a ∨ c dvd b" by simp qed (insert A, auto simp: prime_elem_def is_unit_poly_iff) qed (auto intro: lift_prime_elem_poly) subsection ‹Content and primitive part of a polynomial› definition content :: "'a::semiring_gcd poly ⇒ 'a" where "content p = gcd_list (coeffs p)" lemma content_eq_fold_coeffs [code]: "content p = fold_coeffs gcd p 0" by (simp add: content_def Gcd_fin.set_eq_fold fold_coeffs_def foldr_fold fun_eq_iff ac_simps) lemma content_0 [simp]: "content 0 = 0" by (simp add: content_def) lemma content_1 [simp]: "content 1 = 1" by (simp add: content_def) lemma content_const [simp]: "content [:c:] = normalize c" by (simp add: content_def cCons_def) lemma const_poly_dvd_iff_dvd_content: "[:c:] dvd p ⟷ c dvd content p" for c :: "'a::semiring_gcd" proof (cases "p = 0") case True then show ?thesis by simp next case False have "[:c:] dvd p ⟷ (∀n. c dvd coeff p n)" by (rule const_poly_dvd_iff) also have "… ⟷ (∀a∈set (coeffs p). c dvd a)" proof safe fix n :: nat assume "∀a∈set (coeffs p). c dvd a" then show "c dvd coeff p n" by (cases "n ≤ degree p") (auto simp: coeff_eq_0 coeffs_def split: if_splits) qed (<