section ‹More on Polynomials› text ‹This theory contains several results on content, gcd, primitive part, etc.. Moreover, there is a slightly improved code-equation for computing the gcd.› theory Missing_Polynomial_Factorial imports "HOL-Computational_Algebra.Polynomial_Factorial" Polynomial_Interpolation.Missing_Polynomial begin text ‹Improved code equation for @{const gcd_poly_code} which avoids computing the content twice.› lemma gcd_poly_code_code[code]: "gcd_poly_code p q = (if p = 0 then normalize q else if q = 0 then normalize p else let c1 = content p; c2 = content q; p' = map_poly (λ x. x div c1) p; q' = map_poly (λ x. x div c2) q in smult (gcd c1 c2) (gcd_poly_code_aux p' q'))" unfolding gcd_poly_code_def Let_def primitive_part_def by simp lemma gcd_smult: fixes f g :: "'a :: {factorial_ring_gcd,semiring_gcd_mult_normalize} poly" defines cf: "cf ≡ content f" and cg: "cg ≡ content g" shows "gcd (smult a f) g = (if a = 0 ∨ f = 0 then normalize g else smult (gcd a (cg div (gcd cf cg))) (gcd f g))" proof (cases "a = 0 ∨ f = 0") case False let ?c = "content" let ?pp = primitive_part let ?ua = "unit_factor a" let ?na = "normalize a" define H where "H = gcd (?c f) (?c g)" have "H dvd ?c f" unfolding H_def by auto then obtain F where fh: "?c f = H * F" unfolding dvd_def by blast from False have cf0: "?c f ≠ 0" by auto hence H: "H ≠ 0" unfolding H_def by auto from arg_cong[OF fh, of "λ f. f div H"] H have F: "F = ?c f div H" by auto have "H dvd ?c g" unfolding H_def by auto then obtain G where gh: "?c g = H * G" unfolding dvd_def by blast from arg_cong[OF gh, of "λ f. f div H"] H have G: "G = ?c g div H" by auto have "coprime F G" using H unfolding F G H_def using cf0 div_gcd_coprime by blast have "is_unit ?ua" using False by simp then have ua: "is_unit [: ?ua :]" by (simp add: is_unit_const_poly_iff) have "gcd (smult a f) g = smult (gcd (?na * ?c f) (?c g)) (gcd (smult ?ua (?pp f)) (?pp g))" unfolding gcd_poly_decompose[of "smult a f"] content_smult primitive_part_smult by simp also have "smult ?ua (?pp f) = ?pp f * [: ?ua :]" by simp also have "gcd … (?pp g) = gcd (?pp f) (?pp g)" unfolding gcd_mult_unit1[OF ua] .. also have "gcd (?na * ?c f) (?c g) = gcd ((?na * F) * H) (G * H)" unfolding fh gh by (simp add: ac_simps) also have "… = gcd (?na * F) G * normalize H" unfolding gcd_mult_right gcd.commute[of G] by (simp add: normalize_mult) also have "normalize H = H" by (metis H_def normalize_gcd) finally have "gcd (smult a f) g = smult (gcd (?na * F) G) (smult H (gcd (?pp f) (?pp g)))" by simp also have "smult H (gcd (?pp f) (?pp g)) = gcd f g" unfolding H_def by (rule gcd_poly_decompose[symmetric]) also have "gcd (?na * F) G = gcd (F * ?na) G" by (simp add: ac_simps) also have "… = gcd ?na G" using ‹coprime F G› by (simp add: gcd_mult_right_left_cancel ac_simps) finally show ?thesis unfolding G H_def cg cf using False by simp next case True hence "gcd (smult a f) g = normalize g" by (cases "a = 0", auto) thus ?thesis using True by simp qed lemma gcd_smult_ex: assumes "a ≠ 0" shows "∃ b. gcd (smult a f) g = smult b (gcd f g) ∧ b ≠ 0" proof (cases "f = 0") case True thus ?thesis by (intro exI[of _ 1], auto) next case False hence id: "(a = 0 ∨ f = 0) = False" using assms by auto show ?thesis unfolding gcd_smult id if_False by (intro exI conjI, rule refl, insert assms, auto) qed lemma primitive_part_idemp[simp]: fixes f :: "'a :: {semiring_gcd,normalization_semidom_multiplicative} poly" shows "primitive_part (primitive_part f) = primitive_part f" by (metis content_primitive_part[of f] primitive_part_eq_0_iff primitive_part_prim) lemma content_gcd_primitive: "f ≠ 0 ⟹ content (gcd (primitive_part f) g) = 1" "f ≠ 0 ⟹ content (gcd (primitive_part f) (primitive_part g)) = 1" by (metis (no_types, lifting) content_dvd_contentI content_primitive_part gcd_dvd1 is_unit_content_iff)+ lemma content_gcd_content: "content (gcd f g) = gcd (content f) (content g)" (is "?l = ?r") proof - let ?c = "content" have "?l = normalize (gcd (?c f) (?c g)) * ?c (gcd (primitive_part f) (primitive_part g))" unfolding gcd_poly_decompose[of f g] content_smult .. also have "… = gcd (?c f) (?c g) * ?c (gcd (primitive_part f) (primitive_part g))" by simp also have "… = ?r" using content_gcd_primitive[of f g] by (metis (no_types, lifting) content_dvd_contentI content_eq_zero_iff content_primitive_part gcd_dvd2 gcd_eq_0_iff is_unit_content_iff mult_cancel_left1) finally show ?thesis . qed lemma gcd_primitive_part: "gcd (primitive_part f) (primitive_part g) = normalize (primitive_part (gcd f g))" proof(cases "f = 0") case True show ?thesis unfolding gcd_poly_decompose[of f g] gcd_0_left primitive_part_0 True by (simp add: associatedI primitive_part_dvd_primitive_partI) next case False have "normalize 1 = normalize (unit_factor (gcd (content f) (content g)))" by (simp add: False) then show ?thesis unfolding gcd_poly_decompose[of f g] by (metis (no_types) Polynomial.normalize_smult content_gcd_primitive(1)[OF False] content_times_primitive_part normalize_gcd primitive_part_smult) qed lemma primitive_part_gcd: "primitive_part (gcd f g) = unit_factor (gcd f g) * gcd (primitive_part f) (primitive_part g)" unfolding gcd_primitive_part by (metis (no_types, lifting) content_times_primitive_part gcd.normalize_idem mult_cancel_left2 mult_smult_left normalize_eq_0_iff normalize_mult_unit_factor primitive_part_eq_0_iff smult_content_normalize_primitive_part unit_factor_mult_normalize) lemma primitive_part_normalize: fixes f :: "'a :: {semiring_gcd,idom_divide,normalization_semidom_multiplicative} poly" shows "primitive_part (normalize f) = normalize (primitive_part f)" proof (cases "f = 0") case True thus ?thesis by simp next case False have "normalize (content (normalize (primitive_part f))) = 1" using content_primitive_part[OF False] content_dvd content_const content_dvd_contentI dvd_normalize_iff is_unit_content_iff by (metis (no_types)) then have "content (normalize (primitive_part f)) = 1" by fastforce then have "content (normalize f) = 1 * content f" by (metis (no_types) content_smult mult.commute normalize_content smult_content_normalize_primitive_part) then have "content f = content (normalize f)" by simp then show ?thesis unfolding smult_content_normalize_primitive_part[of f,symmetric] by (metis (no_types) False content_times_primitive_part mult.commute mult_cancel_left mult_smult_right smult_content_normalize_primitive_part) qed lemma length_coeffs_primitive_part[simp]: "length (coeffs (primitive_part f)) = length (coeffs f)" proof (cases "f = 0") case False hence "length (coeffs f) ≠ 0" "length (coeffs (primitive_part f)) ≠ 0" by auto thus ?thesis using degree_primitive_part[of f, unfolded degree_eq_length_coeffs] by linarith qed simp lemma degree_unit_factor[simp]: "degree (unit_factor f) = 0" by (simp add: monom_0 unit_factor_poly_def) lemma degree_normalize[simp]: "degree (normalize f) = degree f" proof (cases "f = 0") case False have "degree f = degree (unit_factor f * normalize f)" by simp also have "… = degree (unit_factor f) + degree (normalize f)" by (rule degree_mult_eq, insert False, auto) finally show ?thesis by simp qed simp lemma content_iff: "x dvd content p ⟷ (∀ c ∈ set (coeffs p). x dvd c)" by (simp add: content_def dvd_gcd_list_iff) lemma is_unit_field_poly[simp]: "(p::'a::field poly) dvd 1 ⟷ p ≠ 0 ∧ degree p = 0" proof(intro iffI conjI, unfold conj_imp_eq_imp_imp) assume "is_unit p" then obtain q where *: "p * q = 1" by (elim dvdE, auto) from * show p0: "p ≠ 0" by auto from * have q0: "q ≠ 0" by auto from * degree_mult_eq[OF p0 q0] show "degree p = 0" by auto next assume "degree p = 0" from degree0_coeffs[OF this] obtain c where c: "p = [:c:]" by auto assume "p ≠ 0" with c have "c ≠ 0" by auto with c have "1 = p * [:1/c:]" by auto from dvdI[OF this] show "is_unit p". qed definition primitive where "primitive f ⟷ (∀x. (∀y ∈ set (coeffs f). x dvd y) ⟶ x dvd 1)" lemma primitiveI: assumes "(⋀x. (⋀y. y ∈ set (coeffs f) ⟹ x dvd y) ⟹ x dvd 1)" shows "primitive f" by (insert assms, auto simp: primitive_def) lemma primitiveD: assumes "primitive f" shows "(⋀y. y ∈ set (coeffs f) ⟹ x dvd y) ⟹ x dvd 1" by (insert assms, auto simp: primitive_def) lemma not_primitiveE: assumes "¬ primitive f" and "⋀x. (⋀y. y ∈ set (coeffs f) ⟹ x dvd y) ⟹ ¬ x dvd 1 ⟹ thesis" shows thesis by (insert assms, auto simp: primitive_def) lemma primitive_iff_content_eq_1[simp]: fixes f :: "'a :: semiring_gcd poly" shows "primitive f ⟷ content f = 1" proof(intro iffI primitiveI) fix x assume "(⋀y. y ∈ set (coeffs f) ⟹ x dvd y)" from gcd_list_greatest[of "coeffs f", OF this] have "x dvd content f" by (simp add: content_def) also assume "content f = 1" finally show "x dvd 1". next assume "primitive f" from primitiveD[OF this list_gcd[of _ "coeffs f"], folded content_def] show "content f = 1" by simp qed lemma primitive_prod_list: fixes fs :: "'a :: {factorial_semiring,semiring_Gcd,normalization_semidom_multiplicative} poly list" assumes "primitive (prod_list fs)" and "f ∈ set fs" shows "primitive f" proof (insert assms, induct fs arbitrary: f) case (Cons f' fs) from Cons.prems have "is_unit (content f' * content (prod_list fs))" by (auto simp: content_mult) from this[unfolded is_unit_mult_iff] have "content f' = 1" and "content (prod_list fs) = 1" by auto moreover from Cons.prems have "f = f' ∨ f ∈ set fs" by auto ultimately show ?case using Cons.hyps[of f] by auto qed auto lemma irreducible_imp_primitive: fixes f :: "'a :: {idom,semiring_gcd} poly" assumes irr: "irreducible f" and deg: "degree f ≠ 0" shows "primitive f" proof (rule ccontr) assume not: "¬ ?thesis" then have "¬ [:content f:] dvd 1" by simp moreover have "f = [:content f:] * primitive_part f" by simp note Factorial_Ring.irreducibleD[OF irr this] ultimately have "primitive_part f dvd 1" by auto from this[unfolded poly_dvd_1] have "degree f = 0" by auto with deg show False by auto qed lemma irreducible_primitive_connect: fixes f :: "'a :: {idom,semiring_gcd} poly" assumes cf: "primitive f" shows "irreducible⇩_{d}f ⟷ irreducible f" (is "?l ⟷ ?r") proof assume l: ?l show ?r proof(rule ccontr, elim not_irreducibleE) from l have deg: "degree f > 0" by (auto dest: irreducible⇩_{d}D) from cf have f0: "f ≠ 0" by auto then show "f = 0 ⟹ False" by auto show "f dvd 1 ⟹ False" using deg by (auto simp:poly_dvd_1) fix a b assume fab: "f = a * b" and a1: "¬ a dvd 1" and b1: "¬ b dvd 1" then have af: "a dvd f" and bf: "b dvd f" by auto with f0 have a0: "a ≠ 0" and b0: "b ≠ 0" by auto from irreducible⇩_{d}D(2)[OF l, of a] af dvd_imp_degree_le[OF af f0] have "degree a = 0 ∨ degree a = degree f" by (metis degree_smult_le irreducible⇩_{d}_dvd_smult l le_antisym Nat.neq0_conv) then show False proof(elim disjE) assume "degree a = 0" then obtain c where ac: "a = [:c:]" by (auto dest: degree0_coeffs) from fab[unfolded ac] have "c dvd content f" by (simp add: content_iff coeffs_smult) with cf have "c dvd 1" by simp then have "a dvd 1" by (auto simp: ac) with a1 show False by auto next assume dega: "degree a = degree f" with f0 degree_mult_eq[OF a0 b0] fab have "degree b = 0" by (auto simp: ac_simps) then obtain c where bc: "b = [:c:]" by (auto dest: degree0_coeffs) from fab[unfolded bc] have "c dvd content f" by (simp add: content_iff coeffs_smult) with cf have "c dvd 1" by simp then have "b dvd 1" by (auto simp: bc) with b1 show False by auto qed qed next assume r: ?r show ?l proof(intro irreducible⇩_{d}I) show "degree f > 0" proof (rule ccontr) assume "¬degree f > 0" then obtain f0 where f: "f = [:f0:]" by (auto dest: degree0_coeffs) from cf[unfolded this] have "normalize f0 = 1" by auto then have "f0 dvd 1" by (unfold normalize_1_iff) with r[unfolded f irreducible_const_poly_iff] show False by auto qed next fix g h assume deg_g: "degree g > 0" and deg_gf: "degree g < degree f" and fgh: "f = g * h" with r have "g dvd 1 ∨ h dvd 1" by auto with deg_g have "degree h = 0" by (auto simp: poly_dvd_1) with deg_gf[unfolded fgh] degree_mult_eq[of g h] show False by (cases "g = 0 ∨ h = 0", auto) qed qed lemma deg_not_zero_imp_not_unit: fixes f:: "'a::{idom_divide,semidom_divide_unit_factor} poly" assumes deg_f: "degree f > 0" shows "¬ is_unit f" proof - have "degree (normalize f) > 0" using deg_f degree_normalize by auto hence "normalize f ≠ 1" by fastforce thus "¬ is_unit f" using normalize_1_iff by auto qed lemma content_pCons[simp]: "content (pCons a p) = gcd a (content p)" proof(induct p arbitrary: a) case 0 show ?case by simp next case (pCons c p) then show ?case by (cases "p = 0", auto simp: content_def cCons_def) qed lemma content_field_poly: fixes f :: "'a :: {field,semiring_gcd} poly" shows "content f = (if f = 0 then 0 else 1)" by(induct f, auto simp: dvd_field_iff is_unit_normalize) end