(* Authors: Jose Divasón Sebastiaan Joosten René Thiemann Akihisa Yamada *) theory Square_Free_Factorization_Int imports Square_Free_Int_To_Square_Free_GFp Suitable_Prime Code_Abort_Gcd Gcd_Finite_Field_Impl begin definition yun_wrel :: "int poly ⇒ rat ⇒ rat poly ⇒ bool" where "yun_wrel F c f = (map_poly rat_of_int F = smult c f)" definition yun_rel :: "int poly ⇒ rat ⇒ rat poly ⇒ bool" where "yun_rel F c f = (yun_wrel F c f ∧ content F = 1 ∧ lead_coeff F > 0 ∧ monic f)" definition yun_erel :: "int poly ⇒ rat poly ⇒ bool" where "yun_erel F f = (∃ c. yun_rel F c f)" lemma yun_wrelD: assumes "yun_wrel F c f" shows "map_poly rat_of_int F = smult c f" using assms unfolding yun_wrel_def by auto lemma yun_relD: assumes "yun_rel F c f" shows "yun_wrel F c f" "map_poly rat_of_int F = smult c f" "degree F = degree f" "F ≠ 0" "lead_coeff F > 0" "monic f" "f = 1 ⟷ F = 1" "content F = 1" proof - note * = assms[unfolded yun_rel_def yun_wrel_def, simplified] then have "degree (map_poly rat_of_int F) = degree f" by auto then show deg: "degree F = degree f" by simp show "F ≠ 0" "lead_coeff F > 0" "monic f" "content F = 1" "map_poly rat_of_int F = smult c f" "yun_wrel F c f" using * by (auto simp: yun_wrel_def) { assume "f = 1" with deg have "degree F = 0" by auto from degree0_coeffs[OF this] obtain c where F: "F = [:c:]" and c: "c = lead_coeff F" by auto from c * have c0: "c > 0" by auto hence cF: "content F = c" unfolding F content_def by auto with * have "c = 1" by auto with F have "F = 1" by simp } moreover { assume "F = 1" with deg have "degree f = 0" by auto with ‹monic f› have "f = 1" using monic_degree_0 by blast } ultimately show "(f = 1) ⟷ (F = 1)" by auto qed lemma yun_erel_1_eq: assumes "yun_erel F f" shows "(F = 1) ⟷ (f = 1)" proof - from assms[unfolded yun_erel_def] obtain c where "yun_rel F c f" by auto from yun_relD[OF this] show ?thesis by simp qed lemma yun_rel_1[simp]: "yun_rel 1 1 1" by (auto simp: yun_rel_def yun_wrel_def content_def) lemma yun_erel_1[simp]: "yun_erel 1 1" unfolding yun_erel_def using yun_rel_1 by blast lemma yun_rel_mult: "yun_rel F c f ⟹ yun_rel G d g ⟹ yun_rel (F * G) (c * d) (f * g)" unfolding yun_rel_def yun_wrel_def content_mult lead_coeff_mult by (auto simp: monic_mult hom_distribs) lemma yun_erel_mult: "yun_erel F f ⟹ yun_erel G g ⟹ yun_erel (F * G) (f * g)" unfolding yun_erel_def using yun_rel_mult[of F _ f G _ g] by blast lemma yun_rel_pow: assumes "yun_rel F c f" shows "yun_rel (F^n) (c^n) (f^n)" by (induct n, insert assms yun_rel_mult, auto) lemma yun_erel_pow: "yun_erel F f ⟹ yun_erel (F^n) (f^n)" using yun_rel_pow unfolding yun_erel_def by blast lemma yun_wrel_pderiv: assumes "yun_wrel F c f" shows "yun_wrel (pderiv F) c (pderiv f)" by (unfold yun_wrel_def, simp add: yun_wrelD[OF assms] pderiv_smult hom_distribs) lemma yun_wrel_minus: assumes "yun_wrel F c f" "yun_wrel G c g" shows "yun_wrel (F - G) c (f - g)" using assms unfolding yun_wrel_def by (auto simp: smult_diff_right hom_distribs) lemma yun_wrel_div: assumes f: "yun_wrel F c f" and g: "yun_wrel G d g" and dvd: "G dvd F" "g dvd f" and G0: "G ≠ 0" shows "yun_wrel (F div G) (c / d) (f div g)" proof - let ?r = "rat_of_int" let ?rp = "map_poly ?r" from dvd obtain H h where fgh: "F = G * H" "f = g * h" unfolding dvd_def by auto from G0 yun_wrelD[OF g] have g0: "g ≠ 0" and d0: "d ≠ 0" by auto from arg_cong[OF fgh(1), of "λ x. x div G"] have H: "H = F div G" using G0 by simp from arg_cong[OF fgh(1), of ?rp] have "?rp F = ?rp G * ?rp H" by (auto simp: hom_distribs) from arg_cong[OF this, of "λ x. x div ?rp G"] G0 have id: "?rp H = ?rp F div ?rp G" by auto have "?rp (F div G) = ?rp F div ?rp G" unfolding H[symmetric] id by simp also have "… = smult c f div smult d g" using f g unfolding yun_wrel_def by auto also have "… = smult (c / d) (f div g)" unfolding div_smult_right div_smult_left by (simp add: field_simps) finally show ?thesis unfolding yun_wrel_def by simp qed lemma yun_rel_div: assumes f: "yun_rel F c f" and g: "yun_rel G d g" and dvd: "G dvd F" "g dvd f" shows "yun_rel (F div G) (c / d) (f div g)" proof - note ff = yun_relD[OF f] note gg = yun_relD[OF g] show ?thesis unfolding yun_rel_def proof (intro conjI) from yun_wrel_div[OF ff(1) gg(1) dvd gg(4)] show "yun_wrel (F div G) (c / d) (f div g)" by auto from dvd have fg: "f = g * (f div g)" by auto from arg_cong[OF fg, of monic] ff(6) gg(6) show "monic (f div g)" using monic_factor by blast from dvd have FG: "F = G * (F div G)" by auto from arg_cong[OF FG, of content, unfolded content_mult] ff(8) gg(8) show "content (F div G) = 1" by simp from arg_cong[OF FG, of lead_coeff, unfolded lead_coeff_mult] ff(5) gg(5) show "lead_coeff (F div G) > 0" by (simp add: zero_less_mult_iff) qed qed lemma yun_wrel_gcd: assumes "yun_wrel F c' f" "yun_wrel G c g" and c: "c' ≠ 0" "c ≠ 0" and d: "d = rat_of_int (lead_coeff (gcd F G))" "d ≠ 0" shows "yun_wrel (gcd F G) d (gcd f g)" proof - let ?r = "rat_of_int" let ?rp = "map_poly ?r" have "smult d (gcd f g) = smult d (gcd (smult c' f) (smult c g))" by (simp add: c gcd_smult_left gcd_smult_right) also have "… = smult d (gcd (?rp F) (?rp G))" using assms(1-2)[unfolded yun_wrel_def] by simp also have "… = smult (d * inverse d) (?rp (gcd F G))" unfolding gcd_rat_to_gcd_int d by simp also have "d * inverse d = 1" using d by auto finally show ?thesis unfolding yun_wrel_def by simp qed lemma yun_rel_gcd: assumes f: "yun_rel F c f" and g: "yun_wrel G c' g" and c': "c' ≠ 0" and d: "d = rat_of_int (lead_coeff (gcd F G))" shows "yun_rel (gcd F G) d (gcd f g)" unfolding yun_rel_def proof (intro conjI) note ff = yun_relD[OF f] from ff have c0: "c ≠ 0" by auto from ff d have d0: "d ≠ 0" by auto from yun_wrel_gcd[OF ff(1) g c0 c' d d0] show "yun_wrel (gcd F G) d (gcd f g)" by auto from ff have "gcd f g ≠ 0" by auto thus "monic (gcd f g)" by (simp add: poly_gcd_monic) obtain H where H: "gcd F G = H" by auto obtain lc where lc: "coeff H (degree H) = lc" by auto from ff have "gcd F G ≠ 0" by auto hence "H ≠ 0" "lc ≠ 0" unfolding H[symmetric] lc[symmetric] by auto thus "0 < lead_coeff (gcd F G)" unfolding arg_cong[OF normalize_gcd[of F G], of lead_coeff, symmetric] unfolding normalize_poly_eq_map_poly H by (auto, subst Polynomial.coeff_map_poly, auto, subst Polynomial.degree_map_poly, auto simp: sgn_if) have "H dvd F" unfolding H[symmetric] by auto then obtain K where F: "F = H * K" unfolding dvd_def by auto from arg_cong[OF this, of content, unfolded content_mult ff(8)] content_ge_0_int[of H] have "content H = 1" by (auto simp add: zmult_eq_1_iff) thus "content (gcd F G) = 1" unfolding H . qed lemma yun_factorization_main_int: assumes f: "f = p div gcd p (pderiv p)" and "g = pderiv p div gcd p (pderiv p)" "monic p" and "yun_gcd.yun_factorization_main gcd f g i hs = res" and "yun_gcd.yun_factorization_main gcd F G i Hs = Res" and "yun_rel F c f" "yun_wrel G c g" "list_all2 (rel_prod yun_erel (=)) Hs hs" shows "list_all2 (rel_prod yun_erel (=)) Res res" proof - let ?P = "λ f g. ∀ i hs res F G Hs Res c. yun_gcd.yun_factorization_main gcd f g i hs = res ⟶ yun_gcd.yun_factorization_main gcd F G i Hs = Res ⟶ yun_rel F c f ⟶ yun_wrel G c g ⟶ list_all2 (rel_prod yun_erel (=)) Hs hs ⟶ list_all2 (rel_prod yun_erel (=)) Res res" note simps = yun_gcd.yun_factorization_main.simps note rel = yun_relD let ?rel = "λ F f. map_poly rat_of_int F = smult (rat_of_int (lead_coeff F)) f" show ?thesis proof (induct rule: yun_factorization_induct[of ?P, rule_format, OF _ _ assms]) case (1 f g i hs res F G Hs Res c) from rel[OF 1(4)] 1(1) have "f = 1" "F = 1" by auto from 1(2-3)[unfolded simps[of _ 1] this] have "res = hs" "Res = Hs" by auto with 1(6) show ?case by simp next case (2 f g i hs res F G Hs Res c) define d where "d = g - pderiv f" define a where "a = gcd f d" define D where "D = G - pderiv F" define A where "A = gcd F D" note f = 2(5) note g = 2(6) note hs = 2(7) note f1 = 2(1) from f1 rel[OF f] have *: "(f = 1) = False" "(F = 1) = False" and c: "c ≠ 0" by auto note res = 2(3)[unfolded simps[of _ f] * if_False Let_def, folded d_def a_def] note Res = 2(4)[unfolded simps[of _ F] * if_False Let_def, folded D_def A_def] note IH = 2(2)[folded d_def a_def, OF res Res] obtain c' where c': "c' = rat_of_int (lead_coeff (gcd F D))" by auto show ?case proof (rule IH) from yun_wrel_minus[OF g yun_wrel_pderiv[OF rel(1)[OF f]]] have d: "yun_wrel D c d" unfolding D_def d_def . have a: "yun_rel A c' a" unfolding A_def a_def by (rule yun_rel_gcd[OF f d c c']) hence "yun_erel A a" unfolding yun_erel_def by auto thus "list_all2 (rel_prod yun_erel (=)) ((A, i) # Hs) ((a, i) # hs)" using hs by auto have A0: "A ≠ 0" by (rule rel(4)[OF a]) have "A dvd D" "a dvd d" unfolding A_def a_def by auto from yun_wrel_div[OF d rel(1)[OF a] this A0] show "yun_wrel (D div A) (c / c') (d div a)" . have "A dvd F" "a dvd f" unfolding A_def a_def by auto from yun_rel_div[OF f a this] show "yun_rel (F div A) (c / c') (f div a)" . qed qed qed lemma yun_monic_factorization_int_yun_rel: assumes res: "yun_gcd.yun_monic_factorization gcd f = res" and Res: "yun_gcd.yun_monic_factorization gcd F = Res" and f: "yun_rel F c f" shows "list_all2 (rel_prod yun_erel (=)) Res res" proof - note ff = yun_relD[OF f] let ?g = "gcd f (pderiv f)" let ?yf = "yun_gcd.yun_factorization_main gcd (f div ?g) (pderiv f div ?g) 0 []" let ?G = "gcd F (pderiv F)" let ?yF = "yun_gcd.yun_factorization_main gcd (F div ?G) (pderiv F div ?G) 0 []" obtain r R where r: "?yf = r" and R: "?yF = R" by blast from res[unfolded yun_gcd.yun_monic_factorization_def Let_def r] have res: "res = [(a, i)←r . a ≠ 1]" by simp from Res[unfolded yun_gcd.yun_monic_factorization_def Let_def R] have Res: "Res = [(A, i)←R . A ≠ 1]" by simp from yun_wrel_pderiv[OF ff(1)] have f': "yun_wrel (pderiv F) c (pderiv f)" . from ff have c: "c ≠ 0" by auto from yun_rel_gcd[OF f f' c refl] obtain d where g: "yun_rel ?G d ?g" .. from yun_rel_div[OF f g] have 1: "yun_rel (F div ?G) (c / d) (f div ?g)" by auto from yun_wrel_div[OF f' yun_relD(1)[OF g] _ _ yun_relD(4)[OF g]] have 2: "yun_wrel (pderiv F div ?G) (c / d) (pderiv f div ?g)" by auto from yun_factorization_main_int[OF refl refl ff(6) r R 1 2] have "list_all2 (rel_prod yun_erel (=)) R r" by simp thus ?thesis unfolding res Res by (induct R r rule: list_all2_induct, auto dest: yun_erel_1_eq) qed lemma yun_rel_same_right: assumes "yun_rel f c G" "yun_rel g d G" shows "f = g" proof - note f = yun_relD[OF assms(1)] note g = yun_relD[OF assms(2)] let ?r = "rat_of_int" let ?rp = "map_poly ?r" from g have d: "d ≠ 0" by auto obtain a b where quot: "quotient_of (c / d) = (a,b)" by force from quotient_of_nonzero[of "c/d", unfolded quot] have b: "b ≠ 0" by simp note f(2) also have "smult c G = smult (c / d) (smult d G)" using d by (auto simp: field_simps) also have "smult d G = ?rp g" using g(2) by simp also have cd: "c / d = (?r a / ?r b)" using quotient_of_div[OF quot] . finally have fg: "?rp f = smult (?r a / ?r b) (?rp g)" by simp from f have "c ≠ 0" by auto with cd d have a: "a ≠ 0" by auto from arg_cong[OF fg, of "λ x. smult (?r b) x"] have "smult (?r b) (?rp f) = smult (?r a) (?rp g)" using b by auto hence "?rp (smult b f) = ?rp (smult a g)" by (auto simp: hom_distribs) then have fg: "[:b:] * f = [:a:] * g" by auto from arg_cong[OF this, of content, unfolded content_mult f(8) g(8)] have "content [: b :] = content [: a :]" by simp hence abs: "abs a = abs b" unfolding content_def using b a by auto from arg_cong[OF fg, of "λ x. lead_coeff x > 0", unfolded lead_coeff_mult] f(5) g(5) a b have "(a > 0) = (b > 0)" by (simp add: zero_less_mult_iff) with a b abs have "a = b" by auto with arg_cong[OF fg, of "λ x. x div [:b:]"] b show ?thesis by (metis nonzero_mult_div_cancel_left pCons_eq_0_iff) qed definition square_free_factorization_int_main :: "int poly ⇒ (int poly × nat) list" where "square_free_factorization_int_main f = (case square_free_heuristic f of None ⇒ yun_gcd.yun_monic_factorization gcd f | Some p ⇒ [(f,0)])" lemma square_free_factorization_int_main: assumes res: "square_free_factorization_int_main f = fs" and ct: "content f = 1" and lc: "lead_coeff f > 0" and deg: "degree f ≠ 0" shows "square_free_factorization f (1,fs) ∧ (∀ fi i. (fi, i) ∈ set fs ⟶ content fi = 1 ∧ lead_coeff fi > 0) ∧ distinct (map snd fs)" proof (cases "square_free_heuristic f") case None from lc have f0: "f ≠ 0" by auto from res None have fs: "yun_gcd.yun_monic_factorization gcd f = fs" unfolding square_free_factorization_int_main_def by auto let ?r = "rat_of_int" let ?rp = "map_poly ?r" define G where "G = smult (inverse (lead_coeff (?rp f))) (?rp f)" have "?rp f ≠ 0" using f0 by auto hence mon: "monic G" unfolding G_def coeff_smult by simp obtain Fs where Fs: "yun_gcd.yun_monic_factorization gcd G = Fs" by blast from lc have lg: "lead_coeff (?rp f) ≠ 0" by auto let ?c = "lead_coeff (?rp f)" define c where "c = ?c" have rp: "?rp f = smult c G" unfolding G_def c_def by (simp add: field_simps) have in_rel: "yun_rel f c G" unfolding yun_rel_def yun_wrel_def using rp mon lc ct by auto from yun_monic_factorization_int_yun_rel[OF Fs fs in_rel] have out_rel: "list_all2 (rel_prod yun_erel (=)) fs Fs" by auto from yun_monic_factorization[OF Fs mon] have "square_free_factorization G (1, Fs)" and dist: "distinct (map snd Fs)" by auto note sff = square_free_factorizationD[OF this(1)] from out_rel have "map snd fs = map snd Fs" by (induct fs Fs rule: list_all2_induct, auto) with dist have dist': "distinct (map snd fs)" by auto have main: "square_free_factorization f (1, fs) ∧ (∀ fi i. (fi, i) ∈ set fs ⟶ content fi = 1 ∧ lead_coeff fi > 0)" unfolding square_free_factorization_def split proof (intro conjI allI impI) from ct have "f ≠ 0" by auto thus "f = 0 ⟹ 1 = 0" "f = 0 ⟹ fs = []" by auto from dist' show "distinct fs" by (simp add: distinct_map) { fix a i assume a: "(a,i) ∈ set fs" with out_rel obtain bj where "bj ∈ set Fs" and "rel_prod yun_erel (=) (a,i) bj" unfolding list_all2_conv_all_nth set_conv_nth by fastforce then obtain b where b: "(b,i) ∈ set Fs" and ab: "yun_erel a b" by (cases bj, auto simp: rel_prod.simps) from sff(2)[OF b] have b': "square_free b" "degree b ≠ 0" by auto from ab obtain c where rel: "yun_rel a c b" unfolding yun_erel_def by auto note aa = yun_relD[OF this] from aa have c0: "c ≠ 0" by auto from b' aa(3) show "degree a > 0" by simp from square_free_smult[OF c0 b'(1), folded aa(2)] show "square_free a" unfolding square_free_def by (force simp: dvd_def hom_distribs) show cnt: "content a = 1" and lc: "lead_coeff a > 0" using aa by auto fix A I assume A: "(A,I) ∈ set fs" and diff: "(a,i) ≠ (A,I)" from a[unfolded set_conv_nth] obtain k where k: "fs ! k = (a,i)" "k < length fs" by auto from A[unfolded set_conv_nth] obtain K where K: "fs ! K = (A,I)" "K < length fs" by auto from diff k K have kK: "k ≠ K" by auto from dist'[unfolded distinct_conv_nth length_map, rule_format, OF k(2) K(2) kK] have iI: "i ≠ I" using k K by simp from A out_rel obtain Bj where "Bj ∈ set Fs" and "rel_prod yun_erel (=) (A,I) Bj" unfolding list_all2_conv_all_nth set_conv_nth by fastforce then obtain B where B: "(B,I) ∈ set Fs" and AB: "yun_erel A B" by (cases Bj, auto simp: rel_prod.simps) then obtain C where Rel: "yun_rel A C B" unfolding yun_erel_def by auto note AA = yun_relD[OF this] from iI have "(b,i) ≠ (B,I)" by auto from sff(3)[OF b B this] have cop: "coprime b B" by simp from AA have C: "C ≠ 0" by auto from yun_rel_gcd[OF rel AA(1) C refl] obtain c where "yun_rel (gcd a A) c (gcd b B)" by auto note rel = yun_relD[OF this] from rel(2) cop have "?rp (gcd a A) = [: c :]" by simp from arg_cong[OF this, of degree] have "degree (gcd a A) = 0" by simp from degree0_coeffs[OF this] obtain c where gcd: "gcd a A = [: c :]" by auto from rel(8) rel(5) show "Rings.coprime a A" by (auto intro!: gcd_eq_1_imp_coprime simp add: gcd) } let ?prod = "λ fs. (∏(a, i)∈set fs. a ^ Suc i)" let ?pr = "λ fs. (∏(a, i)←fs. a ^ Suc i)" define pr where "pr = ?prod fs" from ‹distinct fs› have pfs: "?prod fs = ?pr fs" by (rule prod.distinct_set_conv_list) from ‹distinct Fs› have pFs: "?prod Fs = ?pr Fs" by (rule prod.distinct_set_conv_list) from out_rel have "yun_erel (?prod fs) (?prod Fs)" unfolding pfs pFs proof (induct fs Fs rule: list_all2_induct) case (Cons ai fs Ai Fs) obtain a i where ai: "ai = (a,i)" by force from Cons(1) ai obtain A where Ai: "Ai = (A,i)" and rel: "yun_erel a A" by (cases Ai, auto simp: rel_prod.simps) show ?case unfolding ai Ai using yun_erel_mult[OF yun_erel_pow[OF rel, of "Suc i"] Cons(3)] by auto qed simp also have "?prod Fs = G" using sff(1) by simp finally obtain d where rel: "yun_rel pr d G" unfolding yun_erel_def pr_def by auto with in_rel have "f = pr" by (rule yun_rel_same_right) thus "f = smult 1 (?prod fs)" unfolding pr_def by simp qed from main dist' show ?thesis by auto next case (Some p) from res[unfolded square_free_factorization_int_main_def Some] have fs: "fs = [(f,0)]" by auto from lc have f0: "f ≠ 0" by auto from square_free_heuristic[OF Some] poly_mod_prime.separable_impl(1)[of p f] square_free_mod_imp_square_free[of p f] deg show ?thesis unfolding fs by (auto simp: ct lc square_free_factorization_def f0 poly_mod_prime_def) qed definition square_free_factorization_int' :: "int poly ⇒ int × (int poly × nat)list" where "square_free_factorization_int' f = (if degree f = 0 then (lead_coeff f,[]) else (let ― ‹content factorization› c = content f; d = (sgn (lead_coeff f) * c); g = sdiv_poly f d ― ‹and ‹square_free› factorization› in (d, square_free_factorization_int_main g)))" lemma square_free_factorization_int': assumes res: "square_free_factorization_int' f = (d, fs)" shows "square_free_factorization f (d,fs)" "(fi, i) ∈ set fs ⟹ content fi = 1 ∧ lead_coeff fi > 0" "distinct (map snd fs)" proof - note res = res[unfolded square_free_factorization_int'_def Let_def] have "square_free_factorization f (d,fs) ∧ ((fi, i) ∈ set fs ⟶ content fi = 1 ∧ lead_coeff fi > 0) ∧ distinct (map snd fs)" proof (cases "degree f = 0") case True from degree0_coeffs[OF True] obtain c where f: "f = [: c :]" by auto thus ?thesis using res by (simp add: square_free_factorization_def) next case False let ?s = "sgn (lead_coeff f)" have s: "?s ∈ {-1,1}" using False unfolding sgn_if by auto define g where "g = smult ?s f" let ?d = "?s * content f" have "content g = content ([:?s:] * f)" unfolding g_def by simp also have "… = content [:?s:] * content f" unfolding content_mult by simp also have "content [:?s:] = 1" using s by (auto simp: content_def) finally have cg: "content g = content f" by simp from False res have d: "d = ?d" and fs: "fs = square_free_factorization_int_main (sdiv_poly f ?d)" by auto let ?g = "primitive_part g" define ng where "ng = primitive_part g" note fs also have "sdiv_poly f ?d = sdiv_poly g (content g)" unfolding cg unfolding g_def by (rule poly_eqI, unfold coeff_sdiv_poly coeff_smult, insert s, auto simp: div_minus_right) finally have fs: "square_free_factorization_int_main ng = fs" unfolding primitive_part_alt_def ng_def by simp have "lead_coeff f ≠ 0" using False by auto hence lg: "lead_coeff g > 0" unfolding g_def lead_coeff_smult by (meson linorder_neqE_linordered_idom sgn_greater sgn_less zero_less_mult_iff) hence g0: "g ≠ 0" by auto from g0 have "content g ≠ 0" by simp from arg_cong[OF content_times_primitive_part[of g], of lead_coeff, unfolded lead_coeff_smult] lg content_ge_0_int[of g] have lg': "lead_coeff ng > 0" unfolding ng_def by (metis ‹content g ≠ 0› dual_order.antisym dual_order.strict_implies_order zero_less_mult_iff) from content_primitive_part[OF g0] have c_ng: "content ng = 1" unfolding ng_def . have "degree ng = degree f" using ‹content [:sgn (lead_coeff f):] = 1› g_def ng_def by (auto simp add: sgn_eq_0_iff) with False have "degree ng ≠ 0" by auto note main = square_free_factorization_int_main[OF fs c_ng lg' this] show ?thesis proof (intro conjI impI) { assume "(fi, i) ∈ set fs" with main show "content fi = 1" "0 < lead_coeff fi" by auto } have d0: "d ≠ 0" using ‹content [:?s:] = 1› d by (auto simp:sgn_eq_0_iff) have "smult d ng = smult ?s (smult (content g) (primitive_part g))" unfolding ng_def d cg by simp also have "smult (content g) (primitive_part g) = g" using content_times_primitive_part . also have "smult ?s g = f" unfolding g_def using s by auto finally have id: "smult d ng = f" . from main have "square_free_factorization ng (1, fs)" by auto from square_free_factorization_smult[OF d0 this] show "square_free_factorization f (d,fs)" unfolding id by simp show "distinct (map snd fs)" using main by auto qed qed thus "square_free_factorization f (d,fs)" "(fi, i) ∈ set fs ⟹ content fi = 1 ∧ lead_coeff fi > 0" "distinct (map snd fs)" by auto qed definition x_split :: "'a :: semiring_0 poly ⇒ nat × 'a poly" where "x_split f = (let fs = coeffs f; zs = takeWhile ((=) 0) fs in case zs of [] ⇒ (0,f) | _ ⇒ (length zs, poly_of_list (dropWhile ((=) 0) fs)))" lemma x_split: assumes "x_split f = (n, g)" shows "f = monom 1 n * g" "n ≠ 0 ∨ f ≠ 0 ⟹ ¬ monom 1 1 dvd g" proof - define zs where "zs = takeWhile ((=) 0) (coeffs f)" note res = assms[unfolded zs_def[symmetric] x_split_def Let_def] have "f = monom 1 n * g ∧ ((n ≠ 0 ∨ f ≠ 0) ⟶ ¬ (monom 1 1 dvd g))" (is "_ ∧ (_ ⟶ ¬ (?x dvd _))") proof (cases "f = 0") case True with res have "n = 0" "g = 0" unfolding zs_def by auto thus ?thesis using True by auto next case False note f = this show ?thesis proof (cases "zs = []") case True hence choice: "coeff f 0 ≠ 0" using f unfolding zs_def coeff_f_0_code poly_compare_0_code by (cases "coeffs f", auto) have dvd: "?x dvd h ⟷ coeff h 0 = 0" for h by (simp add: monom_1_dvd_iff') from True choice res f show ?thesis unfolding dvd by auto next case False define ys where "ys = dropWhile ((=) 0) (coeffs f)" have dvd: "?x dvd h ⟷ coeff h 0 = 0" for h by (simp add: monom_1_dvd_iff') from res False have n: "n = length zs" and g: "g = poly_of_list ys" unfolding ys_def by (cases zs, auto)+ obtain xx where xx: "coeffs f = xx" by auto have "coeffs f = zs @ ys" unfolding zs_def ys_def by auto also have "zs = replicate n 0" unfolding zs_def n xx by (induct xx, auto) finally have ff: "coeffs f = replicate n 0 @ ys" by auto from f have "lead_coeff f ≠ 0" by auto then have nz: "coeffs f ≠ []" "last (coeffs f) ≠ 0" by (simp_all add: last_coeffs_eq_coeff_degree) have ys: "ys ≠ []" using nz[unfolded ff] by auto with ys_def have hd: "hd ys ≠ 0" by (metis (full_types) hd_dropWhile) hence "coeff (poly_of_list ys) 0 ≠ 0" unfolding poly_of_list_def coeff_Poly using ys by (cases ys, auto) moreover have "coeffs (Poly ys) = ys" by (simp add: ys_def strip_while_dropWhile_commute) then have "coeffs (monom_mult n (Poly ys)) = replicate n 0 @ ys" by (simp add: coeffs_eq_iff monom_mult_def [symmetric] ff ys monom_mult_code) ultimately show ?thesis unfolding dvd g by (auto simp add: coeffs_eq_iff monom_mult_def [symmetric] ff) qed qed thus "f = monom 1 n * g" "n ≠ 0 ∨ f ≠ 0 ⟹ ¬ monom 1 1 dvd g" by auto qed definition square_free_factorization_int :: "int poly ⇒ int × (int poly × nat)list" where "square_free_factorization_int f = (case x_split f of (n,g) ― ‹extract ‹x^n›› ⇒ case square_free_factorization_int' g of (d,fs) ⇒ if n = 0 then (d,fs) else (d, (monom 1 1, n - 1) # fs))" lemma square_free_factorization_int: assumes res: "square_free_factorization_int f = (d, fs)" shows "square_free_factorization f (d,fs)" "(fi, i) ∈ set fs ⟹ primitive fi ∧ lead_coeff fi > 0" proof - obtain n g where xs: "x_split f = (n,g)" by force obtain c hs where sf: "square_free_factorization_int' g = (c,hs)" by force from res[unfolded square_free_factorization_int_def xs sf split] have d: "d = c" and fs: "fs = (if n = 0 then hs else (monom 1 1, n - 1) # hs)" by (cases n, auto) note sff = square_free_factorization_int'(1-2)[OF sf] note xs = x_split[OF xs] let ?x = "monom 1 1 :: int poly" have x: "primitive ?x ∧ lead_coeff ?x = 1 ∧ degree ?x = 1" by (auto simp add: degree_monom_eq content_def monom_Suc) thus "(fi, i) ∈ set fs ⟹ primitive fi ∧ lead_coeff fi > 0" using sff(2) unfolding fs by (cases n, auto) show "square_free_factorization f (d,fs)" proof (cases n) case 0 with d fs sff xs show ?thesis by auto next case (Suc m) with xs have fg: "f = monom 1 (Suc m) * g" and dvd: "¬ ?x dvd g" by auto from Suc have fs: "fs = (?x,m) # hs" unfolding fs by auto have degx: "degree ?x = 1" by code_simp from irreducible⇩_{d}_square_free[OF linear_irreducible⇩_{d}[OF this]] have sfx: "square_free ?x" by auto have fg: "f = ?x ^ n * g" unfolding fg Suc by (metis x_pow_n) have eq0: "?x ^ n * g = 0 ⟷ g = 0" by simp note sf = square_free_factorizationD[OF sff(1)] { fix a i assume ai: "(a,i) ∈ set hs" with sf(4) have g0: "g ≠ 0" by auto from split_list[OF ai] obtain ys zs where hs: "hs = ys @ (a,i) # zs" by auto have "a dvd g" unfolding square_free_factorization_prod_list[OF sff(1)] hs by (rule dvd_smult, simp add: ac_simps) moreover have "¬ ?x dvd g" using xs[unfolded Suc] by auto ultimately have dvd: "¬ ?x dvd a" using dvd_trans by blast from sf(2)[OF ai] have "a ≠ 0" by auto have "1 = gcd ?x a" proof (rule gcdI) fix d assume d: "d dvd ?x" "d dvd a" from content_dvd_contentI[OF d(1)] x have cnt: "is_unit (content d)" by auto show "is_unit d" proof (cases "degree d = 1") case False with divides_degree[OF d(1), unfolded degx] have "degree d = 0" by auto from degree0_coeffs[OF this] obtain c where dc: "d = [:c:]" by auto from cnt[unfolded dc] have "is_unit c" by (auto simp: content_def, cases "c = 0", auto) hence "d * d = 1" unfolding dc by (cases "c = -1"; cases "c = 1", auto) thus "is_unit d" by (metis dvd_triv_right) next case True from d(1) obtain e where xde: "?x = d * e" unfolding dvd_def by auto from arg_cong[OF this, of degree] degx have "degree d + degree e = 1" by (metis True add.right_neutral degree_0 degree_mult_eq one_neq_zero) with True have "degree e = 0" by auto from degree0_coeffs[OF this] xde obtain e where xde: "?x = [:e:] * d" by auto from arg_cong[OF this, of content, unfolded content_mult] x have "content [:e:] * content d = 1" by auto also have "content [:e :] = abs e" by (auto simp: content_def, cases "e = 0", auto) finally have "¦e¦ * content d = 1" . from pos_zmult_eq_1_iff_lemma[OF this] have "e * e = 1" by (cases "e = 1"; cases "e = -1", auto) with arg_cong[OF xde, of "smult e"] have "d = ?x * [:e:]" by auto hence "?x dvd d" unfolding dvd_def by blast with d(2) have "?x dvd a" by (metis dvd_trans) with dvd show ?thesis by auto qed qed auto hence "coprime ?x a" by (simp add: gcd_eq_1_imp_coprime) note this dvd } note hs_dvd_x = this from hs_dvd_x[of ?x m] have nmem: "(?x,m) ∉ set hs" by auto hence eq: "?x ^ n * g = smult c (∏(a, i)∈set fs. a ^ Suc i)" unfolding sf(1) unfolding fs Suc by simp show ?thesis unfolding fg d unfolding square_free_factorization_def split eq0 unfolding eq proof (intro conjI allI impI, rule refl) fix a i assume ai: "(a,i) ∈ set fs" thus "square_free a" "degree a > 0" using sf(2) sfx degx unfolding fs by auto fix b j assume bj: "(b,j) ∈ set fs" and diff: "(a,i) ≠ (b,j)" consider (hs_hs) "(a,i) ∈ set hs" "(b,j) ∈ set hs" | (hs_x) "(a,i) ∈ set hs" "b = ?x" | (x_hs) "(b,j) ∈ set hs" "a = ?x" using ai bj diff unfolding fs by auto then show "Rings.coprime a b" proof cases case hs_hs from sf(3)[OF this diff] show ?thesis . next case hs_x from hs_dvd_x(1)[OF hs_x(1)] show ?thesis unfolding hs_x(2) by (simp add: ac_simps) next case x_hs from hs_dvd_x(1)[OF x_hs(1)] show ?thesis unfolding x_hs(2) by simp qed next show "g = 0 ⟹ c = 0" using sf(4) by auto show "g = 0 ⟹ fs = []" using sf(4) xs Suc by auto show "distinct fs" using sf(5) nmem unfolding fs by auto qed qed qed end