(* Title: HOL/Library/Polynomial.thy Author: Brian Huffman Author: Clemens Ballarin Author: Florian Haftmann *) section ‹Polynomials as type over a ring structure› theory Polynomial imports Main "~~/src/HOL/GCD" "~~/src/HOL/Library/More_List" "~~/src/HOL/Library/Infinite_Set" begin subsection ‹Auxiliary: operations for lists (later) representing coefficients› definition cCons :: "'a::zero ⇒ 'a list ⇒ 'a list" (infixr "##" 65) where "x ## xs = (if xs = [] ∧ x = 0 then [] else x # xs)" lemma cCons_0_Nil_eq [simp]: "0 ## [] = []" 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› have "∃n. coeff p n ≠ 0" by (simp add: poly_eq_iff) then obtain n where "coeff p n ≠ 0" .. hence "n ≤ degree p" by (rule le_degree) with ‹coeff p n ≠ 0› and ‹degree p = 0› show "coeff p (degree p) ≠ 0" by simp next case (Suc n) from ‹degree p = Suc n› have "n < degree p" by simp hence "∃i>n. coeff p i ≠ 0" by (rule less_degree_imp) then obtain i where "n < i" and "coeff p i ≠ 0" by fast from ‹degree p = Suc n› and ‹n < i› have "degree p ≤ i" by simp also from ‹coeff p i ≠ 0› have "i ≤ degree p" by (rule le_degree) finally have "degree p = i" . with ‹coeff p i ≠ 0› show "coeff p (degree p) ≠ 0" by simp qed lemma leading_coeff_0_iff [simp]: "coeff p (degree p) = 0 ⟷ p = 0" by (cases "p = 0", simp, simp add: leading_coeff_neq_0) subsection ‹List-style constructor for polynomials› 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 "∀n. coeff (pCons a p) (Suc n) = coeff (pCons b q) (Suc n)" by simp then show "p = q" by (simp add: poly_eq_iff) qed lemma pCons_eq_0_iff [simp]: "pCons a p = 0 ⟷ a = 0 ∧ p = 0" using pCons_eq_iff [of a p 0 0] by simp lemma pCons_cases [cases type: poly]: obtains (pCons) a q where "p = pCons a q" proof show "p = pCons (coeff p 0) (Abs_poly (λn. coeff p (Suc n)))" by transfer (simp_all add: MOST_inj[where f=Suc and P="λn. p n = 0" for p] fun_eq_iff Abs_poly_inverse split: nat.split) qed lemma pCons_induct [case_names 0 pCons, induct type: poly]: assumes zero: "P 0" assumes pCons: "⋀a p. a ≠ 0 ∨ p ≠ 0 ⟹ P p ⟹ P (pCons a p)" shows "P p" proof (induct p rule: measure_induct_rule [where f=degree]) case (less p) obtain a q where "p = pCons a q" by (rule pCons_cases) have "P q" proof (cases "q = 0") case True then show "P q" by (simp add: zero) next case False then have "degree (pCons a q) = Suc (degree q)" by (rule degree_pCons_eq) then have "degree q < degree p" using ‹p = pCons a q› by simp then show "P q" by (rule less.hyps) qed have "P (pCons a q)" proof (cases "a ≠ 0 ∨ q ≠ 0") case True with ‹P q› show ?thesis by (auto intro: pCons) next case False with zero show ?thesis by simp qed then show ?case using ‹p = pCons a q› by simp qed lemma degree_eq_zeroE: fixes p :: "'a::zero poly" assumes "degree p = 0" obtains a where "p = pCons a 0" proof - obtain a q where p: "p = pCons a q" by (cases p) with assms have "q = 0" by (cases "q = 0") simp_all with p have "p = pCons a 0" by simp with that show thesis . qed subsection ‹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 degree_Poly: "degree (Poly xs) ≤ length xs" by (induction xs) simp_all 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 - { fix ms :: "nat list" and f :: "nat ⇒ 'a" and x :: "'a" assume "∀m∈set ms. m > 0" then have "map (case_nat x f) ms = map f (map (λn. n - 1) ms)" by (induct ms) (auto split: nat.split) } note * = this show ?thesis by (simp add: coeffs_def * upt_conv_Cons coeff_pCons map_decr_upt del: upt_Suc) qed lemma length_coeffs: "p ≠ 0 ⟹ length (coeffs p) = degree p + 1" by (simp add: coeffs_def) lemma coeffs_nth: assumes "p ≠ 0" "n ≤ degree p" shows "coeffs p ! n = coeff p n" using assms unfolding coeffs_def by (auto simp del: upt_Suc) lemma not_0_cCons_eq [simp]: "p ≠ 0 ⟹ a ## coeffs p = a # coeffs p" 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) have "(∀n. as ≠ replicate n 0) ⟷ (∃a∈set as. a ≠ 0)" using replicate_length_same [of as 0] by (auto dest: sym [of _ as]) with Cons show ?case by auto qed lemma last_coeffs_not_0: "p ≠ 0 ⟹ last (coeffs p) ≠ 0" by (induct p) (auto simp add: cCons_def) lemma strip_while_coeffs [simp]: "strip_while (HOL.eq 0) (coeffs p) = coeffs p" by (cases "p = 0") (auto dest: last_coeffs_not_0 intro: strip_while_not_last) lemma coeffs_eq_iff: "p = q ⟷ coeffs p = coeffs q" (is "?P ⟷ ?Q") proof assume ?P then show ?Q by simp next assume ?Q then have "Poly (coeffs p) = Poly (coeffs q)" by simp then show ?P by simp qed lemma coeff_Poly_eq: "coeff (Poly xs) n = nth_default 0 xs n" apply (induct xs arbitrary: n) apply simp_all by (metis nat.case not0_implies_Suc nth_default_Cons_0 nth_default_Cons_Suc pCons.rep_eq) lemma nth_default_coeffs_eq: "nth_default 0 (coeffs p) = coeff p" by (simp add: fun_eq_iff coeff_Poly_eq [symmetric]) lemma [code]: "coeff p = nth_default 0 (coeffs p)" by (simp add: nth_default_coeffs_eq) lemma coeffs_eqI: assumes coeff: "⋀n. coeff p n = nth_default 0 xs n" assumes zero: "xs ≠ [] ⟹ last xs ≠ 0" shows "coeffs p = xs" proof - from coeff have "p = Poly xs" by (simp add: poly_eq_iff coeff_Poly_eq) with zero show ?thesis by simp (cases xs, simp_all) qed lemma degree_eq_length_coeffs [code]: "degree p = length (coeffs p) - 1" by (simp add: coeffs_def) lemma length_coeffs_degree: "p ≠ 0 ⟹ length (coeffs p) = Suc (degree p)" by (induct p) (auto simp add: cCons_def) lemma [code abstract]: "coeffs 0 = []" by (fact coeffs_0_eq_Nil) lemma [code abstract]: "coeffs (pCons a p) = a ## coeffs p" by (fact coeffs_pCons_eq_cCons) instantiation poly :: ("{zero, equal}") equal begin definition [code]: "HOL.equal (p::'a poly) q ⟷ HOL.equal (coeffs p) (coeffs q)" instance by standard (simp add: equal equal_poly_def coeffs_eq_iff) end lemma [code nbe]: "HOL.equal (p :: _ poly) p ⟷ True" by (fact equal_refl) definition is_zero :: "'a::zero poly ⇒ bool" where [code]: "is_zero p ⟷ List.null (coeffs p)" lemma is_zero_null [code_abbrev]: "is_zero p ⟷ p = 0" by (simp add: is_zero_def null_def) 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 :: 'a :: {comm_semiring_0, semiring_1}) = (∑i≤degree p. coeff p i * x ^ i)" proof (induction p rule: pCons_induct) case (pCons a p) show ?case proof (cases "p = 0") case False let ?p' = "pCons a p" note poly_pCons[of a p x] also note pCons.IH also have "a + x * (∑i≤degree p. coeff p i * x ^ i) = coeff ?p' 0 * x^0 + (∑i≤degree p. coeff ?p' (Suc i) * x^Suc i)" by (simp add: field_simps setsum_right_distrib coeff_pCons) also note setsum_atMost_Suc_shift[symmetric] also note degree_pCons_eq[OF ‹p ≠ 0›, of a, symmetric] finally show ?thesis . qed simp qed simp lemma poly_0_coeff_0: "poly p 0 = coeff p 0" by (cases p) (auto simp: poly_altdef) subsection ‹Monomials› lift_definition monom :: "'a ⇒ nat ⇒ 'a::zero poly" is "λa m n. if m = n then a else 0" 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, simp) done lemma coeffs_monom [code abstract]: "coeffs (monom a n) = (if a = 0 then [] else replicate n 0 @ [a])" by (induct n) (simp_all add: monom_0 monom_Suc) lemma fold_coeffs_monom [simp]: "a ≠ 0 ⟹ fold_coeffs f (monom a n) = f 0 ^^ n ∘ f a" by (simp add: fold_coeffs_def coeffs_monom fun_eq_iff) lemma poly_monom: fixes a x :: "'a::{comm_semiring_1}" shows "poly (monom a n) x = a * x ^ n" by (cases "a = 0", simp_all) (induct n, simp_all add: mult.left_commute poly_def) 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", 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" unfolding degree_def by simp lemma degree_diff_le_max: fixes p q :: "'a :: ab_group_add poly" shows "degree (p - q) ≤ max (degree p) (degree q)" using degree_add_le [where p=p and q="-q"] by simp lemma degree_diff_le: fixes p q :: "'a :: ab_group_add poly" assumes "degree p ≤ n" and "degree q ≤ n" shows "degree (p - q) ≤ n" using assms degree_add_le [of p n "- q"] by simp lemma degree_diff_less: fixes p q :: "'a :: ab_group_add poly" assumes "degree p < n" and "degree q < n" shows "degree (p - q) < n" using assms degree_add_less [of p n "- q"] by simp lemma add_monom: "monom a n + monom b n = monom (a + b) n" by (rule poly_eqI) simp lemma diff_monom: "monom a n - monom b n = monom (a - b) n" by (rule poly_eqI) simp lemma minus_monom: "- monom a n = monom (-a) n" by (rule poly_eqI) simp lemma coeff_setsum: "coeff (∑x∈A. p x) i = (∑x∈A. coeff (p x) i)" by (cases "finite A", induct set: finite, simp_all) lemma monom_setsum: "monom (∑x∈A. a x) n = (∑x∈A. monom (a x) n)" by (rule poly_eqI) (simp add: coeff_setsum) fun plus_coeffs :: "'a::comm_monoid_add list ⇒ 'a list ⇒ 'a list" where "plus_coeffs xs [] = xs" | "plus_coeffs [] ys = ys" | "plus_coeffs (x # xs) (y # ys) = (x + y) ## plus_coeffs xs ys" lemma coeffs_plus_eq_plus_coeffs [code abstract]: "coeffs (p + q) = plus_coeffs (coeffs p) (coeffs q)" proof - { fix xs ys :: "'a list" and n have "nth_default 0 (plus_coeffs xs ys) n = nth_default 0 xs n + nth_default 0 ys n" proof (induct xs ys arbitrary: n rule: plus_coeffs.induct) case (3 x xs y ys n) then show ?case by (cases n) (auto simp add: cCons_def) qed simp_all } note * = this { fix xs ys :: "'a list" assume "xs ≠ [] ⟹ last xs ≠ 0" and "ys ≠ [] ⟹ last ys ≠ 0" moreover assume "plus_coeffs xs ys ≠ []" ultimately have "last (plus_coeffs xs ys) ≠ 0" proof (induct xs ys rule: plus_coeffs.induct) case (3 x xs y ys) then show ?case by (auto simp add: cCons_def) metis qed simp_all } note ** = this show ?thesis apply (rule coeffs_eqI) apply (simp add: * nth_default_coeffs_eq) apply (rule **) apply (auto dest: last_coeffs_not_0) done qed lemma coeffs_uminus [code abstract]: "coeffs (- p) = map (λa. - a) (coeffs p)" by (rule coeffs_eqI) (simp_all add: not_0_coeffs_not_Nil last_map last_coeffs_not_0 nth_default_map_eq nth_default_coeffs_eq) lemma [code]: fixes p q :: "'a::ab_group_add poly" shows "p - q = p + - q" 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, simp) apply (case_tac q, simp, simp add: algebra_simps) done lemma poly_minus [simp]: fixes x :: "'a::comm_ring" shows "poly (- p) x = - poly p x" by (induct p) simp_all lemma poly_diff [simp]: fixes x :: "'a::comm_ring" shows "poly (p - q) x = poly p x - poly q x" using poly_add [of p "- q" x] by simp lemma poly_setsum: "poly (∑k∈A. p k) x = (∑k∈A. poly (p k) x)" by (induct A rule: infinite_finite_induct) simp_all lemma degree_setsum_le: "finite S ⟹ (⋀ p . p ∈ S ⟹ degree (f p) ≤ n) ⟹ degree (setsum f S) ≤ n" proof (induct S rule: finite_induct) case (insert p S) hence "degree (setsum f S) ≤ n" "degree (f p) ≤ n" by auto thus ?case unfolding setsum.insert[OF insert(1-2)] by (metis degree_add_le) qed simp lemma poly_as_sum_of_monoms': assumes n: "degree p ≤ n" shows "(∑i≤n. monom (coeff p i) i) = p" proof - have eq: "⋀i. {..n} ∩ {i} = (if i ≤ n then {i} else {})" by auto show ?thesis using n by (simp add: poly_eq_iff coeff_setsum coeff_eq_0 setsum.If_cases eq if_distrib[where f="λx. x * a" for a]) qed lemma poly_as_sum_of_monoms: "(∑i≤degree p. monom (coeff p i) i) = p" by (intro poly_as_sum_of_monoms' order_refl) lemma Poly_snoc: "Poly (xs @ [x]) = Poly xs + monom x (length xs)" by (induction xs) (simp_all add: monom_0 monom_Suc) subsection ‹Multiplication by a constant, polynomial multiplication and the unit polynomial› lift_definition smult :: "'a::comm_semiring_0 ⇒ 'a poly ⇒ 'a poly" is "λa p n. a * coeff p n" proof - fix a :: 'a and p :: "'a poly" show "∀⇩_{∞}i. a * coeff p i = 0" using MOST_coeff_eq_0[of p] by eventually_elim simp qed lemma coeff_smult [simp]: "coeff (smult a p) n = a * coeff p n" 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::'a::comm_ring) (- p) = - smult a p" by (rule poly_eqI, simp) lemma smult_minus_left [simp]: "smult (- a::'a::comm_ring) p = - smult a p" by (rule poly_eqI, simp) lemma smult_diff_right: "smult (a::'a::comm_ring) (p - q) = smult a p - smult a q" by (rule poly_eqI, simp add: algebra_simps) lemma smult_diff_left: "smult (a - b::'a::comm_ring) p = smult a p - smult b p" by (rule poly_eqI, simp add: algebra_simps) lemmas smult_distribs = smult_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 add: monom_0, simp add: monom_Suc) lemma degree_smult_eq [simp]: fixes a :: "'a::idom" shows "degree (smult a p) = (if a = 0 then 0 else degree p)" by (cases "a = 0", simp, simp add: degree_def) lemma smult_eq_0_iff [simp]: fixes a :: "'a::idom" shows "smult a p = 0 ⟷ a = 0 ∨ p = 0" by (simp add: poly_eq_iff) lemma coeffs_smult [code abstract]: fixes p :: "'a::idom poly" shows "coeffs (smult a p) = (if a = 0 then [] else map (Groups.times a) (coeffs p))" by (rule coeffs_eqI) (auto simp add: not_0_coeffs_not_Nil last_map last_coeffs_not_0 nth_default_map_eq nth_default_coeffs_eq) instantiation poly :: (comm_semiring_0) comm_semiring_0 begin definition "p * q = fold_coeffs (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 add: mult_poly_0_left, simp) lemma mult_pCons_right [simp]: "p * pCons a q = smult a p + pCons 0 (p * q)" by (induct p) (simp add: mult_poly_0_left, simp add: algebra_simps) lemmas mult_poly_0 = mult_poly_0_left mult_poly_0_right lemma mult_smult_left [simp]: "smult a p * q = smult a (p * q)" by (induct p) (simp add: mult_poly_0, simp add: smult_add_right) lemma mult_smult_right [simp]: "p * smult a q = smult a (p * q)" by (induct q) (simp add: mult_poly_0, simp add: smult_add_right) lemma mult_poly_add_left: fixes p q r :: "'a poly" shows "(p + q) * r = p * r + q * r" by (induct r) (simp add: mult_poly_0, simp add: smult_distribs algebra_simps) instance proof fix p q r :: "'a poly" show 0: "0 * p = 0" by (rule mult_poly_0_left) show "p * 0 = 0" by (rule mult_poly_0_right) show "(p + q) * r = p * r + q * r" by (rule mult_poly_add_left) show "(p * q) * r = p * (q * r)" by (induct p, simp add: mult_poly_0, simp add: mult_poly_add_left) show "p * q = q * p" by (induct p, simp add: mult_poly_0, simp) qed end instance poly :: (comm_semiring_0_cancel) comm_semiring_0_cancel .. lemma coeff_mult: "coeff (p * q) n = (∑i≤n. coeff p i * coeff q (n-i))" proof (induct p arbitrary: n) case 0 show ?case by simp next case (pCons a p n) thus ?case by (cases n, simp, simp add: setsum_atMost_Suc_shift del: setsum_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 definition one_poly_def: "1 = pCons 1 0" instance proof show "1 * p = p" for p :: "'a poly" unfolding one_poly_def by simp show "0 ≠ (1::'a poly)" unfolding one_poly_def by simp qed end instance poly :: (comm_ring) comm_ring .. instance poly :: (comm_ring_1) comm_ring_1 .. lemma coeff_1 [simp]: "coeff 1 n = (if n = 0 then 1 else 0)" unfolding one_poly_def by (simp add: coeff_pCons split: nat.split) lemma monom_eq_1 [simp]: "monom 1 0 = 1" by (simp add: monom_0 one_poly_def) lemma degree_1 [simp]: "degree 1 = 0" unfolding one_poly_def by (rule degree_pCons_0) lemma coeffs_1_eq [simp, code abstract]: "coeffs 1 = [1]" by (simp add: one_poly_def) lemma degree_power_le: "degree (p ^ n) ≤ degree p * n" by (induct n) (auto intro: order_trans degree_mult_le) lemma poly_smult [simp]: "poly (smult a p) x = a * poly p x" by (induct p, simp, simp add: algebra_simps) lemma poly_mult [simp]: "poly (p * q) x = poly p x * poly q x" by (induct p, simp_all, simp add: algebra_simps) lemma poly_1 [simp]: "poly 1 x = 1" by (simp add: one_poly_def) lemma poly_power [simp]: fixes p :: "'a::{comm_semiring_1} poly" shows "poly (p ^ n) x = poly p x ^ n" by (induct n) simp_all lemma poly_setprod: "poly (∏k∈A. p k) x = (∏k∈A. poly (p k) x)" by (induct A rule: infinite_finite_induct) simp_all lemma degree_setprod_setsum_le: "finite S ⟹ degree (setprod f S) ≤ setsum (degree o f) S" proof (induct S rule: finite_induct) case (insert a S) show ?case unfolding setprod.insert[OF insert(1-2)] setsum.insert[OF insert(1-2)] by (rule le_trans[OF degree_mult_le], insert insert, auto) qed simp subsection ‹Conversions from natural numbers› lemma of_nat_poly: "of_nat n = [:of_nat n :: 'a :: comm_semiring_1:]" proof (induction n) case (Suc n) hence "of_nat (Suc n) = 1 + (of_nat n :: 'a poly)" by simp also have "(of_nat n :: 'a poly) = [: of_nat n :]" by (subst Suc) (rule refl) also have "1 = [:1:]" by (simp add: one_poly_def) finally show ?case by (subst (asm) add_pCons) simp qed simp lemma degree_of_nat [simp]: "degree (of_nat n) = 0" by (simp add: of_nat_poly) lemma degree_numeral [simp]: "degree (numeral n) = 0" by (subst of_nat_numeral [symmetric], subst of_nat_poly) simp lemma numeral_poly: "numeral n = [:numeral n:]" by (subst of_nat_numeral [symmetric], subst of_nat_poly) simp subsection ‹Lemmas about divisibility› lemma dvd_smult: "p dvd q ⟹ p dvd smult a q" proof - assume "p dvd q" then obtain k where "q = p * k" .. then have "smult a q = p * smult a k" by simp then show "p dvd smult a q" .. qed lemma dvd_smult_cancel: fixes a :: "'a :: field" shows "p dvd smult a q ⟹ a ≠ 0 ⟹ p dvd q" by (drule dvd_smult [where a="inverse a"]) simp lemma dvd_smult_iff: fixes a :: "'a::field" shows "a ≠ 0 ⟹ p dvd smult a q ⟷ p dvd q" by (safe elim!: dvd_smult dvd_smult_cancel) lemma smult_dvd_cancel: "smult a p dvd q ⟹ p dvd q" proof - assume "smult a p dvd q" then obtain k where "q = smult a p * k" .. then have "q = p * smult a k" by simp then show "p dvd q" .. qed lemma smult_dvd: fixes a :: "'a::field" shows "p dvd q ⟹ a ≠ 0 ⟹ smult a p dvd q" by (rule smult_dvd_cancel [where a="inverse a"]) simp lemma smult_dvd_iff: fixes a :: "'a::field" shows "smult a p dvd q ⟷ (if a = 0 then q = 0 else p dvd q)" by (auto elim: smult_dvd smult_dvd_cancel) subsection ‹Polynomials form an integral domain› lemma coeff_mult_degree_sum: "coeff (p * q) (degree p + degree q) = coeff p (degree p) * coeff q (degree q)" by (induct p, simp, simp add: coeff_eq_0) instance poly :: (idom) idom proof fix p q :: "'a poly" assume "p ≠ 0" and "q ≠ 0" have "coeff (p * q) (degree p + degree q) = coeff p (degree p) * coeff q (degree q)" by (rule coeff_mult_degree_sum) also have "coeff p (degree p) * coeff q (degree q) ≠ 0" using ‹p ≠ 0› and ‹q ≠ 0› by simp finally have "∃n. coeff (p * q) n ≠ 0" .. thus "p * q ≠ 0" by (simp add: poly_eq_iff) qed lemma degree_mult_eq: fixes p q :: "'a::semidom poly" shows "⟦p ≠ 0; q ≠ 0⟧ ⟹ degree (p * q) = degree p + degree q" apply (rule order_antisym [OF degree_mult_le le_degree]) apply (simp add: coeff_mult_degree_sum) done lemma degree_mult_right_le: fixes p q :: "'a::semidom poly" assumes "q ≠ 0" shows "degree p ≤ degree (p * q)" using assms by (cases "p = 0") (simp_all add: degree_mult_eq) lemma coeff_degree_mult: fixes p q :: "'a::semidom poly" shows "coeff (p * q) (degree (p * q)) = coeff q (degree q) * coeff p (degree p)" by (cases "p = 0 ∨ q = 0") (auto simp add: degree_mult_eq coeff_mult_degree_sum mult_ac) lemma dvd_imp_degree_le: fixes p q :: "'a::semidom poly" shows "⟦p dvd q; q ≠ 0⟧ ⟹ degree p ≤ degree q" by (erule dvdE, hypsubst, subst degree_mult_eq) auto lemma divides_degree: assumes pq: "p dvd (q :: 'a :: semidom poly)" shows "degree p ≤ degree q ∨ q = 0" by (metis dvd_imp_degree_le pq) subsection ‹Polynomials form an ordered integral domain› definition pos_poly :: "'a::linordered_idom poly ⇒ bool" where "pos_poly p ⟷ 0 < coeff p (degree p)" lemma pos_poly_pCons: "pos_poly (pCons a p) ⟷ pos_poly p ∨ (p = 0 ∧ 0 < a)" unfolding pos_poly_def by simp lemma not_pos_poly_0 [simp]: "¬ pos_poly 0" unfolding pos_poly_def by simp lemma pos_poly_add: "⟦pos_poly p; pos_poly q⟧ ⟹ pos_poly (p + q)" apply (induct p arbitrary: q, simp) apply (case_tac q, 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)" by (induct p) (auto simp add: pos_poly_pCons) lemma last_coeffs_eq_coeff_degree: "p ≠ 0 ⟹ last (coeffs p) = coeff p (degree p)" by (simp add: coeffs_def) lemma pos_poly_coeffs [code]: "pos_poly p ⟷ (let as = coeffs p in as ≠ [] ∧ last as > 0)" (is "?P ⟷ ?Q") proof assume ?Q then show ?P by (auto simp add: pos_poly_def last_coeffs_eq_coeff_degree) next assume ?P then have *: "0 < coeff p (degree p)" by (simp add: pos_poly_def) then have "p ≠ 0" by auto with * show ?Q by (simp add: last_coeffs_eq_coeff_degree) qed instantiation poly :: (linordered_idom) linordered_idom begin definition "x < y ⟷ pos_poly (y - x)" definition "x ≤ y ⟷ x = y ∨ pos_poly (y - x)" definition "¦x::'a poly¦ = (if x < 0 then - x else x)" definition "sgn (x::'a poly) = (if x = 0 then 0 else if 0 < x then 1 else - 1)" instance proof fix x y z :: "'a poly" show "x < y ⟷ x ≤ y ∧ ¬ y ≤ x" unfolding less_eq_poly_def less_poly_def apply safe apply simp apply (drule (1) pos_poly_add) apply simp done show "x ≤ x" unfolding less_eq_poly_def by simp show "x ≤ y ⟹ y ≤ z ⟹ x ≤ z" unfolding less_eq_poly_def apply safe 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" unfolding less_poly_def by (simp add: right_diff_distrib [symmetric] pos_poly_mult) show "¦x¦ = (if x < 0 then - x else x)" by (rule abs_poly_def) show "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)" by (rule sgn_poly_def) qed end text ‹TODO: Simplification rules for comparisons› subsection ‹Synthetic division and polynomial roots› text ‹ Synthetic division is simply division by the linear polynomial @{term "x - c"}. › definition synthetic_divmod :: "'a::comm_semiring_0 poly ⇒ 'a ⇒ 'a poly × 'a" where "synthetic_divmod p c = fold_coeffs (λa (q, r). (pCons r q, a + c * r)) p (0, 0)" definition synthetic_div :: "'a::comm_semiring_0 poly ⇒ 'a ⇒ 'a poly" where "synthetic_div p c = fst (synthetic_divmod p c)" lemma synthetic_divmod_0 [simp]: "synthetic_divmod 0 c = (0, 0)" 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" unfolding synthetic_div_def by simp lemma synthetic_div_unique_lemma: "smult c p = pCons a p ⟹ p = 0" by (induct p arbitrary: a) simp_all lemma snd_synthetic_divmod: "snd (synthetic_divmod p c) = poly p c" by (induct p, simp, simp add: split_def) lemma synthetic_div_pCons [simp]: "synthetic_div (pCons a p) c = pCons (poly p c) (synthetic_div p c)" unfolding synthetic_div_def by (simp add: split_def snd_synthetic_divmod) lemma synthetic_div_eq_0_iff: "synthetic_div p c = 0 ⟷ degree p = 0" by (induct p, simp, case_tac p, simp) lemma degree_synthetic_div: "degree (synthetic_div p c) = degree p - 1" by (induct p, simp, simp add: synthetic_div_eq_0_iff) lemma synthetic_div_correct: "p + smult c (synthetic_div p c) = pCons (poly p c) (synthetic_div p c)" by (induct p) simp_all lemma synthetic_div_unique: "p + smult c q = pCons r q ⟹ r = poly p c ∧ q = synthetic_div p c" apply (induct p arbitrary: q r) apply (simp, frule synthetic_div_unique_lemma, simp) apply (case_tac q, force) done lemma synthetic_div_correct': fixes c :: "'a::comm_ring_1" shows "[:-c, 1:] * synthetic_div p c + [:poly p c:] = p" using synthetic_div_correct [of p c] by (simp add: algebra_simps) lemma poly_eq_0_iff_dvd: fixes c :: "'a::idom" shows "poly p c = 0 ⟷ [:-c, 1:] dvd p" proof assume "poly p c = 0" with synthetic_div_correct' [of c p] have "p = [:-c, 1:] * synthetic_div p c" by simp then show "[:-c, 1:] dvd p" .. next assume "[:-c, 1:] dvd p" then obtain k where "p = [:-c, 1:] * k" by (rule dvdE) then show "poly p c = 0" by simp qed lemma dvd_iff_poly_eq_0: fixes c :: "'a::idom" shows "[:c, 1:] dvd p ⟷ poly p (-c) = 0" by (simp add: poly_eq_0_iff_dvd) lemma poly_roots_finite: fixes p :: "'a::idom poly" shows "p ≠ 0 ⟹ finite {x. poly p x = 0}" proof (induct n ≡ "degree p" arbitrary: p) case (0 p) then obtain a where "a ≠ 0" and "p = [:a:]" by (cases p, simp split: if_splits) then show "finite {x. poly p x = 0}" by simp next case (Suc n p) show "finite {x. poly p x = 0}" proof (cases "∃x. poly p x = 0") case False then show "finite {x. poly p x = 0}" by simp next case True then obtain a where "poly p a = 0" .. then have "[:-a, 1:] dvd p" by (simp only: poly_eq_0_iff_dvd) then obtain k where k: "p = [:-a, 1:] * k" .. with ‹p ≠ 0› have "k ≠ 0" by auto with k have "degree p = Suc (degree k)" by (simp add: degree_mult_eq del: mult_pCons_left) with ‹Suc n = degree p› have "n = degree k" by simp then have "finite {x. poly k x = 0}" using ‹k ≠ 0› by (rule Suc.hyps) then have "finite (insert a {x. poly k x = 0})" by simp then show "finite {x. poly p x = 0}" by (simp add: k Collect_disj_eq del: mult_pCons_left) qed qed lemma poly_eq_poly_eq_iff: fixes p q :: "'a::{idom,ring_char_0} poly" shows "poly p = poly q ⟷ p = q" (is "?P ⟷ ?Q") proof assume ?Q then show ?P by simp next { fix p :: "'a::{idom,ring_char_0} poly" have "poly p = poly 0 ⟷ p = 0" apply (cases "p = 0", simp_all) apply (drule poly_roots_finite) apply (auto simp add: infinite_UNIV_char_0) done } note this [of "p - q"] moreover assume ?P ultimately show ?Q by auto qed lemma poly_all_0_iff_0: fixes p :: "'a::{ring_char_0, idom} poly" shows "(∀x. poly p x = 0) ⟷ p = 0" by (auto simp add: poly_eq_poly_eq_iff [symmetric]) subsection ‹Long division of polynomials› definition pdivmod_rel :: "'a::field poly ⇒ 'a poly ⇒ 'a poly ⇒ 'a poly ⇒ bool" where "pdivmod_rel x y q r ⟷ x = q * y + r ∧ (if y = 0 then q = 0 else r = 0 ∨ degree r < degree y)" lemma pdivmod_rel_0: "pdivmod_rel 0 y 0 0" unfolding pdivmod_rel_def by simp lemma pdivmod_rel_by_0: "pdivmod_rel x 0 0 x" unfolding pdivmod_rel_def by simp lemma eq_zero_or_degree_less: assumes "degree p ≤ n" and "coeff p n = 0" shows "p = 0 ∨ degree p < n" proof (cases n) case 0 with ‹degree p ≤ n› and ‹coeff p n = 0› have "coeff p (degree p) = 0" by simp then have "p = 0" by simp then show ?thesis .. next case (Suc m) have "∀i>n. coeff p i = 0" using ‹degree p ≤ n› by (simp add: coeff_eq_0) then have "∀i≥n. coeff p i = 0" using ‹coeff p n = 0› by (simp add: le_less) then have "∀i>m. coeff p i = 0" using ‹n = Suc m› by (simp add: less_eq_Suc_le) then have "degree p ≤ m" by (rule degree_le) then have "degree p < n" using ‹n = Suc m› by (simp add: less_Suc_eq_le) then show ?thesis .. qed lemma pdivmod_rel_pCons: assumes rel: "pdivmod_rel x y q r" assumes y: "y ≠ 0" assumes b: "b = coeff (pCons a r) (degree y) / coeff y (degree y)" shows "pdivmod_rel (pCons a x) y (pCons b q) (pCons a r - smult b y)" (is "pdivmod_rel ?x y ?q ?r") proof - have x: "x = q * y + r" and r: "r = 0 ∨ degree r < degree y" using assms unfolding pdivmod_rel_def by simp_all have 1: "?x = ?q * y + ?r" using b x by simp have 2: "?r = 0 ∨ degree ?r < degree y" proof (rule eq_zero_or_degree_less) show "degree ?r ≤ degree y" proof (rule degree_diff_le) show "degree (pCons a r) ≤ degree y" using r by auto show "degree (smult b y) ≤ degree y" by (rule degree_smult_le) qed next show "coeff ?r (degree y) = 0" using ‹y ≠ 0› unfolding b by simp qed from 1 2 show ?thesis unfolding pdivmod_rel_def using ‹y ≠ 0› by simp qed lemma pdivmod_rel_exists: "∃q r. pdivmod_rel x y q r" apply (cases "y = 0") apply (fast intro!: pdivmod_rel_by_0) apply (induct x) apply (fast intro!: pdivmod_rel_0) apply (fast intro!: pdivmod_rel_pCons) done lemma pdivmod_rel_unique: assumes 1: "pdivmod_rel x y q1 r1" assumes 2: "pdivmod_rel x y q2 r2" shows "q1 = q2 ∧ r1 = r2" proof (cases "y = 0") assume "y = 0" with assms show ?thesis by (simp add: pdivmod_rel_def) next assume [simp]: "y ≠ 0" from 1 have q1: "x = q1 * y + r1" and r1: "r1 = 0 ∨ degree r1 < degree y" unfolding pdivmod_rel_def by simp_all from 2 have q2: "x = q2 * y + r2" and r2: "r2 = 0 ∨ degree r2 < degree y" unfolding pdivmod_rel_def by simp_all from q1 q2 have q3: "(q1 - q2) * y = r2 - r1" by (simp add: algebra_simps) from r1 r2 have r3: "(r2 - r1) = 0 ∨ degree (r2 - r1) < degree y" by (auto intro: degree_diff_less) show "q1 = q2 ∧ r1 = r2" proof (rule ccontr) assume "¬ (q1 = q2 ∧ r1 = r2)" with q3 have "q1 ≠ q2" and "r1 ≠ r2" by auto with r3 have "degree (r2 - r1) < degree y" by simp also have "degree y ≤ degree (q1 - q2) + degree y" by simp also have "… = degree ((q1 - q2) * y)" using ‹q1 ≠ q2› by (simp add: degree_mult_eq) also have "… = degree (r2 - r1)" using q3 by simp finally have "degree (r2 - r1) < degree (r2 - r1)" . then show "False" by simp qed qed lemma pdivmod_rel_0_iff: "pdivmod_rel 0 y q r ⟷ q = 0 ∧ r = 0" by (auto dest: pdivmod_rel_unique intro: pdivmod_rel_0) lemma pdivmod_rel_by_0_iff: "pdivmod_rel x 0 q r ⟷ q = 0 ∧ r = x" by (auto dest: pdivmod_rel_unique intro: pdivmod_rel_by_0) lemmas pdivmod_rel_unique_div = pdivmod_rel_unique [THEN conjunct1] lemmas pdivmod_rel_unique_mod = pdivmod_rel_unique [THEN conjunct2] instantiation poly :: (field) ring_div begin definition divide_poly where div_poly_def: "x div y = (THE q. ∃r. pdivmod_rel x y q r)" definition mod_poly where "x mod y = (THE r. ∃q. pdivmod_rel x y q r)" lemma div_poly_eq: "pdivmod_rel x y q r ⟹ x div y = q" unfolding div_poly_def by (fast elim: pdivmod_rel_unique_div) lemma mod_poly_eq: "pdivmod_rel x y q r ⟹ x mod y = r" unfolding mod_poly_def by (fast elim: pdivmod_rel_unique_mod) lemma pdivmod_rel: "pdivmod_rel x y (x div y) (x mod y)" proof - from pdivmod_rel_exists obtain q r where "pdivmod_rel x y q r" by fast thus ?thesis by (simp add: div_poly_eq mod_poly_eq) qed instance proof fix x y :: "'a poly" show "x div y * y + x mod y = x" using pdivmod_rel [of x y] by (simp add: pdivmod_rel_def) next fix x :: "'a poly" have "pdivmod_rel x 0 0 x" by (rule pdivmod_rel_by_0) thus "x div 0 = 0" by (rule div_poly_eq) next fix y :: "'a poly" have "pdivmod_rel 0 y 0 0" by (rule pdivmod_rel_0) thus "0 div y = 0" by (rule div_poly_eq) next fix x y z :: "'a poly" assume "y ≠ 0" hence "pdivmod_rel (x + z * y) y (z + x div y) (x mod y)" using pdivmod_rel [of x y] by (simp add: pdivmod_rel_def distrib_right) thus "(x + z * y) div y = z + x div y" by (rule div_poly_eq) next fix x y z :: "'a poly" assume "x ≠ 0" show "(x * y) div (x * z) = y div z" proof (cases "y ≠ 0 ∧ z ≠ 0") have "⋀x::'a poly. pdivmod_rel x 0 0 x" by (rule pdivmod_rel_by_0) then have [simp]: "⋀x::'a poly. x div 0 = 0" by (rule div_poly_eq) have "⋀x::'a poly. pdivmod_rel 0 x 0 0" by (rule pdivmod_rel_0) then have [simp]: "⋀x::'a poly. 0 div x = 0" by (rule div_poly_eq) case False then show ?thesis by auto next case True then have "y ≠ 0" and "z ≠ 0" by auto with ‹x ≠ 0› have "⋀q r. pdivmod_rel y z q r ⟹ pdivmod_rel (x * y) (x * z) q (x * r)" by (auto simp add: pdivmod_rel_def algebra_simps) (rule classical, simp add: degree_mult_eq) moreover from pdivmod_rel have "pdivmod_rel y z (y div z) (y mod z)" . ultimately have "pdivmod_rel (x * y) (x * z) (y div z) (x * (y mod z))" . then show ?thesis by (simp add: div_poly_eq) qed qed end lemma is_unit_monom_0: fixes a :: "'a::field" assumes "a ≠ 0" shows "is_unit (monom a 0)" proof from assms show "1 = monom a 0 * monom (1 / a) 0" by (simp add: mult_monom) qed lemma is_unit_triv: fixes a :: "'a::field" assumes "a ≠ 0" shows "is_unit [:a:]" using assms by (simp add: is_unit_monom_0 monom_0 [symmetric]) lemma is_unit_iff_degree: assumes "p ≠ 0" shows "is_unit p ⟷ degree p = 0" (is "?P ⟷ ?Q") proof assume ?Q then obtain a where "p = [:a:]" by (rule degree_eq_zeroE) with assms show ?P by (simp add: is_unit_triv) next assume ?P then obtain q where "q ≠ 0" "p * q = 1" .. then have "degree (p * q) = degree 1" by simp with ‹p ≠ 0› ‹q ≠ 0› have "degree p + degree q = 0" by (simp add: degree_mult_eq) then show ?Q by simp qed lemma is_unit_pCons_iff: "is_unit (pCons a p) ⟷ p = 0 ∧ a ≠ 0" (is "?P ⟷ ?Q") by (cases "p = 0") (auto simp add: is_unit_triv is_unit_iff_degree) lemma is_unit_monom_trival: fixes p :: "'a::field poly" assumes "is_unit p" shows "monom (coeff p (degree p)) 0 = p" using assms by (cases p) (simp_all add: monom_0 is_unit_pCons_iff) lemma is_unit_polyE: assumes "is_unit p" obtains a where "p = monom a 0" and "a ≠ 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 instantiation poly :: (field) normalization_semidom begin definition normalize_poly :: "'a poly ⇒ 'a poly" where "normalize_poly p = smult (1 / coeff p (degree p)) p" definition unit_factor_poly :: "'a poly ⇒ 'a poly" where "unit_factor_poly p = monom (coeff p (degree p)) 0" instance proof fix p :: "'a poly" show "unit_factor p * normalize p = p" by (simp add: normalize_poly_def unit_factor_poly_def) (simp only: mult_smult_left [symmetric] smult_monom, simp) next show "normalize 0 = (0::'a poly)" by (simp add: normalize_poly_def) next show "unit_factor 0 = (0::'a poly)" by (simp add: unit_factor_poly_def) next fix p :: "'a poly" assume "is_unit p" then obtain a where "p = monom a 0" and "a ≠ 0" by (rule is_unit_polyE) then show "normalize p = 1" by (auto simp add: normalize_poly_def smult_monom degree_monom_eq) next fix p q :: "'a poly" assume "q ≠ 0" from ‹q ≠ 0› have "is_unit (monom (coeff q (degree q)) 0)" by (auto intro: is_unit_monom_0) then show "is_unit (unit_factor q)" by (simp add: unit_factor_poly_def) next fix p q :: "'a poly" have "monom (coeff (p * q) (degree (p * q))) 0 = monom (coeff p (degree p)) 0 * monom (coeff q (degree q)) 0" by (simp add: monom_0 coeff_degree_mult) then show "unit_factor (p * q) = unit_factor p * unit_factor q" by (simp add: unit_factor_poly_def) qed end lemma degree_mod_less: "y ≠ 0 ⟹ x mod y = 0 ∨ degree (x mod y) < degree y" using pdivmod_rel [of x y] unfolding pdivmod_rel_def by simp lemma div_poly_less: "degree x < degree y ⟹ x div y = 0" proof - assume "degree x < degree y" hence "pdivmod_rel x y 0 x" by (simp add: pdivmod_rel_def) thus "x div y = 0" by (rule div_poly_eq) qed lemma mod_poly_less: "degree x < degree y ⟹ x mod y = x" proof - assume "degree x < degree y" hence "pdivmod_rel x y 0 x" by (simp add: pdivmod_rel_def) thus "x mod y = x" by (rule mod_poly_eq) qed lemma pdivmod_rel_smult_left: "pdivmod_rel x y q r ⟹ pdivmod_rel (smult a x) y (smult a q) (smult a r)" unfolding pdivmod_rel_def by (simp add: smult_add_right) lemma div_smult_left: "(smult a x) div y = smult a (x div y)" by (rule div_poly_eq, rule pdivmod_rel_smult_left, rule pdivmod_rel) lemma mod_smult_left: "(smult a x) mod y = smult a (x mod y)" by (rule mod_poly_eq, rule pdivmod_rel_smult_left, rule pdivmod_rel) lemma poly_div_minus_left [simp]: fixes x y :: "'a::field poly" shows "(- x) div y = - (x div y)" using div_smult_left [of "- 1::'a"] by simp lemma poly_mod_minus_left [simp]: fixes x y :: "'a::field poly" shows "(- x) mod y = - (x mod y)" using mod_smult_left [of "- 1::'a"] by simp lemma pdivmod_rel_add_left: assumes "pdivmod_rel x y q r" assumes "pdivmod_rel x' y q' r'" shows "pdivmod_rel (x + x') y (q + q') (r + r')" using assms unfolding pdivmod_rel_def by (auto simp add: algebra_simps degree_add_less) lemma poly_div_add_left: fixes x y z :: "'a::field poly" shows "(x + y) div z = x div z + y div z" using pdivmod_rel_add_left [OF pdivmod_rel pdivmod_rel] by (rule div_poly_eq) lemma poly_mod_add_left: fixes x y z :: "'a::field poly" shows "(x + y) mod z = x mod z + y mod z" using pdivmod_rel_add_left [OF pdivmod_rel pdivmod_rel] by (rule mod_poly_eq) lemma poly_div_diff_left: fixes x y z :: "'a::field poly" shows "(x - y) div z = x div z - y div z" by (simp only: diff_conv_add_uminus poly_div_add_left poly_div_minus_left) lemma poly_mod_diff_left: fixes x y z :: "'a::field poly" shows "(x - y) mod z = x mod z - y mod z" by (simp only: diff_conv_add_uminus poly_mod_add_left poly_mod_minus_left) lemma pdivmod_rel_smult_right: "⟦a ≠ 0; pdivmod_rel x y q r⟧ ⟹ pdivmod_rel x (smult a y) (smult (inverse a) q) r" unfolding pdivmod_rel_def by simp lemma div_smult_right: "a ≠ 0 ⟹ x div (smult a y) = smult (inverse a) (x div y)" by (rule div_poly_eq, erule pdivmod_rel_smult_right, rule pdivmod_rel) lemma mod_smult_right: "a ≠ 0 ⟹ x mod (smult a y) = x mod y" by (rule mod_poly_eq, erule pdivmod_rel_smult_right, rule pdivmod_rel) lemma poly_div_minus_right [simp]: fixes x y :: "'a::field poly" shows "x div (- y) = - (x div y)" using div_smult_right [of "- 1::'a"] by (simp add: nonzero_inverse_minus_eq) lemma poly_mod_minus_right [simp]: fixes x y :: "'a::field poly" shows "x mod (- y) = x mod y" using mod_smult_right [of "- 1::'a"] by simp lemma pdivmod_rel_mult: "⟦pdivmod_rel x y q r; pdivmod_rel q z q' r'⟧ ⟹ pdivmod_rel x (y * z) q' (y * r' + r)" apply (cases "z = 0", simp add: pdivmod_rel_def) apply (cases "y = 0", simp add: pdivmod_rel_by_0_iff pdivmod_rel_0_iff) apply (cases "r = 0") apply (cases "r' = 0") apply (simp add: pdivmod_rel_def) apply (simp add: pdivmod_rel_def field_simps degree_mult_eq) apply (cases "r' = 0") apply (simp add: pdivmod_rel_def degree_mult_eq) apply (simp add: pdivmod_rel_def field_simps) apply (simp add: degree_mult_eq degree_add_less) done lemma poly_div_mult_right: fixes x y z :: "'a::field poly" shows "x div (y * z) = (x div y) div z" by (rule div_poly_eq, rule pdivmod_rel_mult, (rule pdivmod_rel)+) lemma poly_mod_mult_right: fixes x y z :: "'a::field poly" shows "x mod (y * z) = y * (x div y mod z) + x mod y" by (rule mod_poly_eq, rule pdivmod_rel_mult, (rule pdivmod_rel)+) lemma mod_pCons: fixes a and x assumes y: "y ≠ 0" defines b: "b ≡ coeff (pCons a (x mod y)) (degree y) / coeff y (degree y)" shows "(pCons a x) mod y = (pCons a (x mod y) - smult b y)" unfolding b apply (rule mod_poly_eq) apply (rule pdivmod_rel_pCons [OF pdivmod_rel y refl]) done definition pdivmod :: "'a::field poly ⇒ 'a poly ⇒ 'a poly × 'a poly" where "pdivmod p q = (p div q, p mod q)" lemma div_poly_code [code]: "p div q = fst (pdivmod p q)" by (simp add: pdivmod_def) lemma mod_poly_code [code]: "p mod q = snd (pdivmod p q)" by (simp add: pdivmod_def) lemma pdivmod_0: "pdivmod 0 q = (0, 0)" by (simp add: pdivmod_def) lemma pdivmod_pCons: "pdivmod (pCons a p) q = (if q = 0 then (0, pCons a p) else (let (s, r) = pdivmod p q; b = coeff (pCons a r) (degree q) / coeff q (degree q) in (pCons b s, pCons a r - smult b q)))" apply (simp add: pdivmod_def Let_def, safe) apply (rule div_poly_eq) apply (erule pdivmod_rel_pCons [OF pdivmod_rel _ refl]) apply (rule mod_poly_eq) apply (erule pdivmod_rel_pCons [OF pdivmod_rel _ refl]) done lemma pdivmod_fold_coeffs [code]: "pdivmod p q = (if q = 0 then (0, p) else fold_coeffs (λa (s, r). let b = coeff (pCons a r) (degree q) / coeff q (degree q) in (pCons b s, pCons a r - smult b q) ) p (0, 0))" apply (cases "q = 0") apply (simp add: pdivmod_def) apply (rule sym) apply (induct p) apply (simp_all add: pdivmod_0 pdivmod_pCons) apply (case_tac "a = 0 ∧ p = 0") apply (auto simp add: pdivmod_def) done subsection ‹Order of polynomial roots› definition order :: "'a::idom ⇒ 'a poly ⇒ nat" where "order a p = (LEAST n. ¬ [:-a, 1:] ^ Suc n dvd p)" lemma coeff_linear_power: fixes a :: "'a::comm_semiring_1" shows "coeff ([:a, 1:] ^ n) n = 1" apply (induct n, simp_all) apply (subst coeff_eq_0) apply (auto intro: le_less_trans degree_power_le) done lemma degree_linear_power: fixes a :: "'a::comm_semiring_1" shows "degree ([:a, 1:] ^ n) = n" apply (rule order_antisym) apply (rule ord_le_eq_trans [OF degree_power_le], simp) apply (rule le_degree, simp add: coeff_linear_power) done lemma order_1: "[:-a, 1:] ^ order a p dvd p" apply (cases "p = 0", simp) apply (cases "order a p", simp) apply (subgoal_tac "nat < (LEAST n. ¬ [:-a, 1:] ^ Suc n dvd p)") apply (drule not_less_Least, simp) apply (fold order_def, simp) done lemma order_2: "p ≠ 0 ⟹ ¬ [:-a, 1:] ^ Suc (order a p) dvd p" unfolding order_def apply (rule LeastI_ex) apply (rule_tac x="degree p" in exI) apply (rule notI) apply (drule (1) dvd_imp_degree_le) apply (simp only: degree_linear_power) done lemma order: "p ≠ 0 ⟹ [:-a, 1:] ^ order a p dvd p ∧ ¬ [:-a, 1:] ^ Suc (order a p) dvd p" by (rule conjI [OF order_1 order_2]) lemma order_degree: assumes p: "p ≠ 0" shows "order a p ≤ degree p" proof - have "order a p = degree ([:-a, 1:] ^ order a p)" by (simp only: degree_linear_power) also have "… ≤ degree p" using order_1 p by (rule dvd_imp_degree_le) finally show ?thesis . qed lemma order_root: "poly p a = 0 ⟷ p = 0 ∨ order a p ≠ 0" apply (cases "p = 0", simp_all) apply (rule iffI) apply (metis order_2 not_gr0 poly_eq_0_iff_dvd power_0 power_Suc_0 power_one_right) unfolding poly_eq_0_iff_dvd apply (metis dvd_power dvd_trans order_1) done lemma order_0I: "poly p a ≠ 0 ⟹ order a p = 0" by (subst (asm) order_root) auto subsection ‹GCD of polynomials› instantiation poly :: (field) gcd begin function gcd_poly :: "'a::field poly ⇒ 'a poly ⇒ 'a poly" where "gcd (x::'a poly) 0 = smult (inverse (coeff x (degree x))) x" | "y ≠ 0 ⟹ gcd (x::'a poly) y = gcd y (x mod y)" by auto termination "gcd :: _ poly ⇒ _" by (relation "measure (λ(x, y). if y = 0 then 0 else Suc (degree y))") (auto dest: degree_mod_less) declare gcd_poly.simps [simp del] definition lcm_poly :: "'a::field poly ⇒ 'a poly ⇒ 'a poly" where "lcm_poly a b = a * b div smult (coeff a (degree a) * coeff b (degree b)) (gcd a b)" instance .. end lemma fixes x y :: "_ poly" shows poly_gcd_dvd1 [iff]: "gcd x y dvd x" and poly_gcd_dvd2 [iff]: "gcd x y dvd y" apply (induct x y rule: gcd_poly.induct) apply (simp_all add: gcd_poly.simps) apply (fastforce simp add: smult_dvd_iff dest: inverse_zero_imp_zero) apply (blast dest: dvd_mod_imp_dvd) done lemma poly_gcd_greatest: fixes k x y :: "_ poly" shows "k dvd x ⟹ k dvd y ⟹ k dvd gcd x y" by (induct x y rule: gcd_poly.induct) (simp_all add: gcd_poly.simps dvd_mod dvd_smult) lemma dvd_poly_gcd_iff [iff]: fixes k x y :: "_ poly" shows "k dvd gcd x y ⟷ k dvd x ∧ k dvd y" by (auto intro!: poly_gcd_greatest intro: dvd_trans [of _ "gcd x y"]) lemma poly_gcd_monic: fixes x y :: "_ poly" shows "coeff (gcd x y) (degree (gcd x y)) = (if x = 0 ∧ y = 0 then 0 else 1)" by (induct x y rule: gcd_poly.induct) (simp_all add: gcd_poly.simps nonzero_imp_inverse_nonzero) lemma poly_gcd_zero_iff [simp]: fixes x y :: "_ poly" shows "gcd x y = 0 ⟷ x = 0 ∧ y = 0" by (simp only: dvd_0_left_iff [symmetric] dvd_poly_gcd_iff) lemma poly_gcd_0_0 [simp]: "gcd (0::_ poly) 0 = 0" by simp lemma poly_dvd_antisym: fixes p q :: "'a::idom poly" assumes coeff: "coeff p (degree p) = coeff q (degree q)" assumes dvd1: "p dvd q" and dvd2: "q dvd p" shows "p = q" proof (cases "p = 0") case True with coeff show "p = q" by simp next case False with coeff have "q ≠ 0" by auto have degree: "degree p = degree q" using ‹p dvd q› ‹q dvd p› ‹p ≠ 0› ‹q ≠ 0› by (intro order_antisym dvd_imp_degree_le) from ‹p dvd q› obtain a where a: "q = p * a" .. with ‹q ≠ 0› have "a ≠ 0" by auto with degree a ‹p ≠ 0› have "degree a = 0" by (simp add: degree_mult_eq) with coeff a show "p = q" by (cases a, auto split: if_splits) qed lemma poly_gcd_unique: fixes d x y :: "_ poly" assumes dvd1: "d dvd x" and dvd2: "d dvd y" and greatest: "⋀k. k dvd x ⟹ k dvd y ⟹ k dvd d" and monic: "coeff d (degree d) = (if x = 0 ∧ y = 0 then 0 else 1)" shows "gcd x y = d" proof - have "coeff (gcd x y) (degree (gcd x y)) = coeff d (degree d)" by (simp_all add: poly_gcd_monic monic) moreover have "gcd x y dvd d" using poly_gcd_dvd1 poly_gcd_dvd2 by (rule greatest) moreover have "d dvd gcd x y" using dvd1 dvd2 by (rule poly_gcd_greatest) ultimately show ?thesis by (rule poly_dvd_antisym) qed interpretation gcd_poly: abel_semigroup "gcd :: _ poly ⇒ _" proof fix x y z :: "'a poly" show "gcd (gcd x y) z = gcd x (gcd y z)" by (rule poly_gcd_unique) (auto intro: dvd_trans simp add: poly_gcd_monic) show "gcd x y = gcd y x" by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic) qed lemmas poly_gcd_assoc = gcd_poly.assoc lemmas poly_gcd_commute = gcd_poly.commute lemmas poly_gcd_left_commute = gcd_poly.left_commute lemmas poly_gcd_ac = poly_gcd_assoc poly_gcd_commute poly_gcd_left_commute lemma poly_gcd_1_left [simp]: "gcd 1 y = (1 :: _ poly)" by (rule poly_gcd_unique) simp_all lemma poly_gcd_1_right [simp]: "gcd x 1 = (1 :: _ poly)" by (rule poly_gcd_unique) simp_all lemma poly_gcd_minus_left [simp]: "gcd (- x) y = gcd x (y :: _ poly)" by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic) lemma poly_gcd_minus_right [simp]: "gcd x (- y) = gcd x (y :: _ poly)" by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic) lemma poly_gcd_code [code]: "gcd x y = (if y = 0 then smult (inverse (coeff x (degree x))) x else gcd y (x mod (y :: _ poly)))" by (simp add: gcd_poly.simps) subsection ‹Additional induction rules on polynomials› text ‹ An induction rule for induction over the roots of a polynomial with a certain property. (e.g. all positive roots) › lemma poly_root_induct [case_names 0 no_roots root]: fixes p :: "'a :: idom poly" assumes "Q 0" assumes "⋀p. (⋀a. P a ⟹ poly p a ≠ 0) ⟹ Q p" assumes "⋀a p. P a ⟹ Q p ⟹ Q ([:a, -1:] * p)" shows "Q p" proof (induction "degree p" arbitrary: p rule: less_induct) case (less p) show ?case proof (cases "p = 0") assume nz: "p ≠ 0" show ?case proof (cases "∃a. P a ∧ poly p a = 0") case False thus ?thesis by (intro assms(2)) blast next case True then obtain a where a: "P a" "poly p a = 0" by blast hence "-[:-a, 1:] dvd p" by (subst minus_dvd_iff) (simp add: poly_eq_0_iff_dvd) then obtain q where q: "p = [:a, -1:] * q" by (elim dvdE) simp with nz have q_nz: "q ≠ 0" by auto have "degree p = Suc (degree q)" by (subst q, subst degree_mult_eq) (simp_all add: q_nz) hence "Q q" by (intro less) simp from a(1) and this have "Q ([:a, -1:] * q)" by (rule assms(3)) with q show ?thesis by simp qed qed (simp add: assms(1)) qed lemma dropWhile_replicate_append: "dropWhile (op= a) (replicate n a @ ys) = dropWhile (op= a) ys" by (induction n) simp_all lemma Poly_append_replicate_0: "Poly (xs @ replicate n 0) = Poly xs" by (subst coeffs_eq_iff) (simp_all add: strip_while_def dropWhile_replicate_append) text ‹ An induction rule for simultaneous induction over two polynomials, prepending one coefficient in each step. › lemma poly_induct2 [case_names 0 pCons]: assumes "P 0 0" "⋀a p b q. P p q ⟹ P (pCons a p) (pCons b q)" shows "P p q" proof - def n ≡ "max (length (coeffs p)) (length (coeffs q))" def xs ≡ "coeffs p @ (replicate (n - length (coeffs p)) 0)" def ys ≡ "coeffs q @ (replicate (n - length (coeffs q)) 0)" have "length xs = length ys" by (simp add: xs_def ys_def n_def) hence "P (Poly xs) (Poly ys)" by (induction rule: list_induct2) (simp_all add: assms) also have "Poly xs = p" 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 RenĂ© 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: fixes p :: "'a :: comm_semiring_1 poly" shows "pcompose 1 p = 1" unfolding one_poly_def by (auto simp: pcompose_pCons) lemma poly_pcompose: "poly (pcompose p q) x = poly p (poly q x)" by (induct p) (simp_all add: pcompose_pCons) lemma degree_pcompose_le: "degree (pcompose p q) ≤ degree p * degree q" apply (induct p, simp) apply (simp add: pcompose_pCons, clarify) apply (rule degree_add_le, simp) apply (rule order_trans [OF degree_mult_le], simp) done lemma pcompose_add: fixes p q r :: "'a :: {comm_semiring_0, ab_semigroup_add} poly" shows "pcompose (p + q) r = pcompose p r + pcompose q r" proof (induction p q rule: poly_induct2) case (pCons a p b q) have "pcompose (pCons a p + pCons b q) r = [:a + b:] + r * pcompose p r + r * pcompose q r" by (simp_all add: pcompose_pCons pCons.IH algebra_simps) also have "[:a + b:] = [:a:] + [:b:]" by simp also have "… + r * pcompose p r + r * pcompose q r = pcompose (pCons a p) r + pcompose (pCons b q) r" by (simp only: pcompose_pCons add_ac) finally show ?case . qed simp lemma pcompose_uminus: fixes p r :: "'a :: comm_ring poly" shows "pcompose (-p) r = -pcompose p r" by (induction p) (simp_all add: pcompose_pCons) lemma pcompose_diff: fixes p q r :: "'a :: comm_ring poly" shows "pcompose (p - q) r = pcompose p r - pcompose q r" using pcompose_add[of p "-q"] by (simp add: pcompose_uminus) lemma pcompose_smult: fixes p r :: "'a :: comm_semiring_0 poly" shows "pcompose (smult a p) r = smult a (pcompose p r)" by (induction p) (simp_all add: pcompose_pCons pcompose_add smult_add_right) lemma pcompose_mult: fixes p q r :: "'a :: comm_semiring_0 poly" shows "pcompose (p * q) r = pcompose p r * pcompose q r" by (induction p arbitrary: q) (simp_all add: pcompose_add pcompose_smult pcompose_pCons algebra_simps) lemma pcompose_assoc: "pcompose p (pcompose q r :: 'a :: comm_semiring_0 poly ) = pcompose (pcompose p q) r" by (induction p arbitrary: q) (simp_all add: pcompose_pCons pcompose_add pcompose_mult) lemma pcompose_idR[simp]: fixes p :: "'a :: comm_semiring_1 poly" shows "pcompose p [: 0, 1 :] = p" by (induct p; simp add: pcompose_pCons) (* The remainder of this section and the next were contributed by Wenda Li *) lemma degree_mult_eq_0: fixes p q:: "'a :: semidom poly" shows "degree (p*q) = 0 ⟷ p=0 ∨ q=0 ∨ (p≠0 ∧ q≠0 ∧ degree p =0 ∧ degree q =0)" by (auto simp add:degree_mult_eq) lemma pcompose_const[simp]:"pcompose [:a:] q = [:a:]" by (subst pcompose_pCons,simp) lemma pcompose_0': "pcompose p 0 = [:coeff p 0:]" by (induct p) (auto simp add:pcompose_pCons) lemma degree_pcompose: fixes p q:: "'a::semidom poly" shows "degree (pcompose p q) = degree p * degree q" proof (induct p) case 0 thus ?case by auto next case (pCons a p) have "degree (q * pcompose p q) = 0 ⟹ ?case" proof (cases "p=0") case True thus ?thesis by auto next case False assume "degree (q * pcompose p q) = 0" hence "degree q=0 ∨ pcompose p q=0" by (auto simp add: degree_mult_eq_0) moreover have "⟦pcompose p q=0;degree q≠0⟧ ⟹ False" using pCons.hyps(2) ‹p≠0› proof - assume "pcompose p q=0" "degree q≠0" hence "degree p=0" using pCons.hyps(2) by auto then obtain a1 where "p=[:a1:]" by (metis degree_pCons_eq_if old.nat.distinct(2) pCons_cases) thus False using ‹pcompose p q=0› ‹p≠0› by auto qed ultimately have "degree (pCons a p) * degree q=0" by auto moreover have "degree (pcompose (pCons a p) q) = 0" proof - have" 0 = max (degree [:a:]) (degree (q*pcompose p q))" using ‹degree (q * pcompose p q) = 0› by simp also have "... ≥ degree ([:a:] + q * pcompose p q)" by (rule degree_add_le_max) finally show ?thesis by (auto simp add:pcompose_pCons) qed ultimately show ?thesis by simp qed moreover have "degree (q * pcompose p q)>0 ⟹ ?case" proof - assume asm:"0 < degree (q * pcompose p q)" hence "p≠0" "q≠0" "pcompose p q≠0" by auto have "degree (pcompose (pCons a p) q) = degree ( q * pcompose p q)" unfolding pcompose_pCons using degree_add_eq_right[of "[:a:]" ] asm by auto thus ?thesis using pCons.hyps(2) degree_mult_eq[OF ‹q≠0› ‹pcompose p q≠0›] by auto qed ultimately show ?case by blast qed lemma pcompose_eq_0: fixes p q:: "'a :: semidom poly" assumes "pcompose p q = 0" "degree q > 0" shows "p = 0" proof - have "degree p=0" using assms degree_pcompose[of p q] by auto then obtain a where "p=[:a:]" by (metis degree_pCons_eq_if gr0_conv_Suc neq0_conv pCons_cases) hence "a=0" using assms(1) by auto thus ?thesis using ‹p=[:a:]› by simp qed subsection ‹Leading coefficient› definition 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" unfolding lead_coeff_def by auto lemma lead_coeff_0[simp]:"lead_coeff 0 =0" unfolding lead_coeff_def by auto lemma lead_coeff_mult: fixes p q::"'a ::idom poly" shows "lead_coeff (p * q) = lead_coeff p * lead_coeff q" by (unfold lead_coeff_def,cases "p=0 ∨ q=0",auto simp add:coeff_mult_degree_sum degree_mult_eq) lemma lead_coeff_add_le: assumes "degree p < degree q" shows "lead_coeff (p+q) = lead_coeff q" using assms unfolding lead_coeff_def 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 lead_coeff_def) lemma lead_coeff_comp: fixes p q:: "'a::idom poly" assumes "degree q > 0" shows "lead_coeff (pcompose p q) = lead_coeff p * lead_coeff q ^ (degree p)" proof (induct p) case 0 thus ?case unfolding lead_coeff_def by auto next case (pCons a p) have "degree ( q * pcompose p q) = 0 ⟹ ?case" proof - assume "degree ( q * pcompose p q) = 0" hence "pcompose p q = 0" by (metis assms degree_0 degree_mult_eq_0 neq0_conv) hence "p=0" using pcompose_eq_0[OF _ ‹degree q > 0›] by simp thus ?thesis by auto qed moreover have "degree ( q * pcompose p q) > 0 ⟹ ?case" proof - assume "degree ( q * pcompose p q) > 0" hence "lead_coeff (pcompose (pCons a p) q) =lead_coeff ( q * pcompose p q)" by (auto simp add:pcompose_pCons lead_coeff_add_le) 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 finally show ?thesis by auto qed ultimately show ?case by blast qed lemma lead_coeff_smult: "lead_coeff (smult c p :: 'a :: idom poly) = c * lead_coeff p" 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 add: lead_coeff_def) lemma lead_coeff_of_nat [simp]: "lead_coeff (of_nat n) = (of_nat n :: 'a :: {comm_semiring_1,semiring_char_0})" by (induction n) (simp_all add: lead_coeff_def of_nat_poly) lemma lead_coeff_numeral [simp]: "lead_coeff (numeral n) = numeral n" unfolding lead_coeff_def by (subst of_nat_numeral [symmetric], subst of_nat_poly) simp lemma lead_coeff_power: "lead_coeff (p ^ n :: 'a :: idom poly) = lead_coeff p ^ n" by (induction n) (simp_all add: lead_coeff_mult) lemma lead_coeff_nonzero: "p ≠ 0 ⟹ lead_coeff p ≠ 0" by (simp add: lead_coeff_def) no_notation cCons (infixr "##" 65) end