imports Hensel_Lifting Subresultant Cancel_Card_Constraint

(* Authors: Jose Divasón Sebastiaan Joosten René Thiemann Akihisa Yamada License: BSD *) section ‹Executable dvdm operation› text ‹This theory contains some results about division of integer polynomials which are not part of Polynomial\_Factorization.Dvd\_Int\_Poly.thy. Essentially, we give an executable implementation of division modulo m. › theory Missing_Dvd_Int_Poly imports Berlekamp_Zassenhaus.Poly_Mod_Finite_Field Berlekamp_Zassenhaus.Polynomial_Record_Based Berlekamp_Zassenhaus.Hensel_Lifting (* for thm pdivmod_monic, should be moved *) Subresultants.Subresultant (* for abbreviation pdivmod *) Perron_Frobenius.Cancel_Card_Constraint begin (*In mathematics, this lemma also works if q = 0 since degree 0 = -∞, but in Isabelle degree 0 = 0.*) lemma degree_div_mod_smult: fixes g::"int poly" assumes g: "degree g < j" and r: "degree r < d" and u: "degree u = d" and g1: "g = q * u + smult m r" and q: "q ≠ 0" and m_not0: "m ≠ 0" shows "degree q < j - d" proof - have u_not0: "u≠0" using u r by auto have d_uq: "d ≤ degree (u*q)" using u degree_mult_right_le[OF q] by auto have j: "j > degree (q* u + smult m r)" using g1 g by auto have "degree (smult m r) < d" using degree_smult_eq m_not0 r by auto also have "... ≤ degree (u*q)" using d_uq by auto finally have deg_mr_uq: "degree (smult m r) < degree (q*u)" by (simp add: mult.commute) have j2: "degree (q* u + smult m r) = degree (q*u)" by (rule degree_add_eq_left[OF deg_mr_uq]) also have "... = degree q + degree u" by (rule degree_mult_eq[OF q u_not0]) finally have "degree q = degree g - degree u" using g1 by auto thus ?thesis using j j2 ‹degree (q * u) = degree q + degree u› u by linarith qed subsection ‹Uniqueness of division algorithm for polynomials› lemma uniqueness_algorithm_division_poly: fixes f::"'a::{comm_ring,semiring_1_no_zero_divisors} poly" assumes f1: "f = g * q1 + r1" and f2: "f = g * q2 + r2" and g: "g ≠ 0" and r1: "r1 = 0 ∨ degree r1 < degree g" and r2: "r2 = 0 ∨ degree r2 < degree g" shows "q1 = q2 ∧ r1 = r2" proof - have "0 = g * q1 + r1 - (g * q2 + r2)" using f1 f2 by auto also have "... = g * (q1 - q2) + r1 - r2" by (simp add: right_diff_distrib) finally have eq: "g * (q1 - q2) = r2 - r1" by auto have q_eq: "q1 = q2" proof (rule ccontr) assume q1_not_q2: "q1 ≠ q2" hence nz: "g * (q1 - q2) ≠ 0" using g by auto hence "degree (g * (q1 - q2)) ≥ degree g" by (simp add: degree_mult_right_le) moreover have "degree (r2 - r1) < degree g" using eq nz degree_diff_less r1 r2 by auto ultimately show False using eq by auto qed moreover have "r1 = r2" using eq q_eq by auto ultimately show ?thesis by simp qed lemma pdivmod_eq_pdivmod_monic: assumes g: "monic g" shows "pdivmod f g = pdivmod_monic f g" proof - obtain q r where qr: "pdivmod f g = (q,r)" by simp obtain Q R where QR: "pdivmod_monic f g = (Q,R)" by (meson surj_pair) have g0: "g ≠ 0" using g by auto have f1: "f = g * q + r" by (metis Pair_inject mult_div_mod_eq qr) have r: "r=0 ∨ degree r < degree g" by (metis Pair_inject assms degree_mod_less leading_coeff_0_iff qr zero_neq_one) have f2: "f = g * Q + R" by (simp add: QR assms pdivmod_monic(1)) have R: "R=0 ∨ degree R < degree g" by (rule pdivmod_monic[OF g QR]) have "q=Q ∧ r=R" by (rule uniqueness_algorithm_division_poly[OF f1 f2 g0 r R]) thus ?thesis using qr QR by auto qed context poly_mod begin definition "pdivmod2 f g = (if Mp g = 0 then (0, f) else let ilc = inverse_p m ((lead_coeff (Mp g))); h = Polynomial.smult ilc (Mp g); (q, r) = pseudo_divmod (Mp f) (Mp h) in (Polynomial.smult ilc q, r))" end context poly_mod_prime_type begin lemma dvdm_iff_pdivmod0: assumes f: "(F :: 'a mod_ring poly) = of_int_poly f" and g: "(G :: 'a mod_ring poly) = of_int_poly g" shows "g dvdm f = (snd (pdivmod F G) = 0)" proof - have [transfer_rule]: "MP_Rel f F" unfolding MP_Rel_def by (simp add: Mp_f_representative f) have [transfer_rule]: "MP_Rel g G" unfolding MP_Rel_def by (simp add: Mp_f_representative g) have "(snd (pdivmod F G) = 0) = (G dvd F)" unfolding dvd_eq_mod_eq_0 by auto from this [untransferred] show ?thesis by simp qed lemma of_int_poly_Mp_0[simp]: "(of_int_poly (Mp a) = (0:: 'a mod_ring poly)) = (Mp a = 0)" by (auto, metis Mp_f_representative map_poly_0 poly_mod.Mp_Mp) lemma uniqueness_algorithm_division_of_int_poly: assumes g0: "Mp g ≠ 0" and f: "(F :: 'a mod_ring poly) = of_int_poly f" and g: "(G :: 'a mod_ring poly) = of_int_poly g" and F: "F = G * Q + R" and R: "R = 0 ∨ degree R < degree G" and Mp_f: "Mp f = Mp g * q + r" and r: "r = 0 ∨ degree r < degree (Mp g)" shows "Q = of_int_poly q ∧ R = of_int_poly r" proof (rule uniqueness_algorithm_division_poly[OF F _ _ R]) have f': "Mp f = to_int_poly F" unfolding f by (simp add: Mp_f_representative) have g': "Mp g = to_int_poly G" unfolding g by (simp add: Mp_f_representative) have f'': "of_int_poly (Mp f) = F" by (metis (no_types, lifting) Dp_Mp_eq Mp_f_representative Mp_smult_m_0 add_cancel_left_right f map_poly_zero of_int_hom.map_poly_hom_add to_int_mod_ring_hom.hom_zero to_int_mod_ring_hom.injectivity) have g'': "of_int_poly (Mp g) = G" by (metis (no_types, lifting) Dp_Mp_eq Mp_f_representative Mp_smult_m_0 add_cancel_left_right g map_poly_zero of_int_hom.map_poly_hom_add to_int_mod_ring_hom.hom_zero to_int_mod_ring_hom.injectivity) have "F = of_int_poly (Mp g * q + r)" using Mp_f f'' by auto also have "... = G * of_int_poly q + of_int_poly r" by (simp add: g'' of_int_poly_hom.hom_add of_int_poly_hom.hom_mult) finally show "F = G * of_int_poly q + of_int_poly r" . show "of_int_poly r = 0 ∨ degree (of_int_poly r::'a mod_ring poly) < degree G" proof (cases "r = 0") case True hence "of_int_poly r = 0" by auto then show ?thesis by auto next case False have "degree (of_int_poly r::'a mod_ring poly) ≤ degree (r)" by (simp add: degree_map_poly_le) also have "... < degree (Mp g)" using r False by auto also have "... = degree G" by (simp add: g') finally show ?thesis by auto qed show "G ≠ 0" using g0 unfolding g''[symmetric] by simp qed corollary uniqueness_algorithm_division_to_int_poly: assumes g0: "Mp g ≠ 0" and f: "(F :: 'a mod_ring poly) = of_int_poly f" and g: "(G :: 'a mod_ring poly) = of_int_poly g" and F: "F = G * Q + R" and R: "R = 0 ∨ degree R < degree G" and Mp_f: "Mp f = Mp g * q + r" and r: "r = 0 ∨ degree r < degree (Mp g)" shows "Mp q = to_int_poly Q ∧ Mp r = to_int_poly R" using uniqueness_algorithm_division_of_int_poly[OF assms] by (auto simp add: Mp_f_representative) lemma uniqueness_algorithm_division_Mp_Rel: assumes monic_Mpg: "monic (Mp g)" and f: "(F :: 'a mod_ring poly) = of_int_poly f" and g: "(G :: 'a mod_ring poly) = of_int_poly g" and qr: "pseudo_divmod (Mp f) (Mp g) = (q,r)" and QR: "pseudo_divmod F G = (Q,R)" shows "MP_Rel q Q ∧ MP_Rel r R " proof (unfold MP_Rel_def, rule uniqueness_algorithm_division_to_int_poly[OF _ f g]) show f_gq_r: "Mp f = Mp g * q + r" by (rule pdivmod_monic(1)[OF monic_Mpg], simp add: pdivmod_monic_pseudo_divmod qr monic_Mpg) have monic_G: "monic G" using monic_Mpg using Mp_f_representative g by auto show "F = G * Q + R" by (rule pdivmod_monic(1)[OF monic_G], simp add: pdivmod_monic_pseudo_divmod QR monic_G) show "Mp g ≠ 0" using monic_Mpg by auto show "R = 0 ∨ degree R < degree G" by (rule pdivmod_monic(2)[OF monic_G], auto simp add: pdivmod_monic_pseudo_divmod monic_G intro: QR) show "r = 0 ∨ degree r < degree (Mp g)" by (rule pdivmod_monic(2)[OF monic_Mpg], auto simp add: pdivmod_monic_pseudo_divmod monic_Mpg intro: qr) qed definition "MP_Rel_Pair A B ≡ (let (a,b) = A; (c,d) = B in MP_Rel a c ∧ MP_Rel b d)" lemma pdivmod2_rel[transfer_rule]: "(MP_Rel ===> MP_Rel ===> MP_Rel_Pair) (pdivmod2) (pdivmod)" proof (auto simp add: rel_fun_def MP_Rel_Pair_def) interpret pm: prime_field m using m unfolding prime_field_def mod_ring_locale_def by auto have p: "prime_field TYPE('a) m" using m unfolding prime_field_def mod_ring_locale_def by auto fix f F g G a b assume 1[transfer_rule]: "MP_Rel f F" and 2[transfer_rule]: "MP_Rel g G" and 3: "pdivmod2 f g = (a, b)" have "MP_Rel a (F div G) ∧ MP_Rel b (F mod G)" proof (cases "Mp g ≠ 0") case True note Mp_g = True have G: "G ≠ 0" using Mp_g 2 unfolding MP_Rel_def by auto have gG[transfer_rule]: "pm.mod_ring_rel (lead_coeff (Mp g)) (lead_coeff G)" using 2 unfolding pm.mod_ring_rel_def MP_Rel_def by auto have [transfer_rule]: "(pm.mod_ring_rel ===> pm.mod_ring_rel) (inverse_p m) inverse" by (rule prime_field.mod_ring_inverse[OF p]) hence rel_inverse_p[transfer_rule]: "pm.mod_ring_rel (inverse_p m ((lead_coeff (Mp g)))) (inverse (lead_coeff G))" using gG unfolding rel_fun_def by auto let ?h= "(Polynomial.smult (inverse_p m (lead_coeff (Mp g))) g)" define h where h: "h = Polynomial.smult (inverse_p m (lead_coeff (Mp g))) (Mp g)" define H where H: "H = Polynomial.smult (inverse (lead_coeff G)) G" have hH': "MP_Rel ?h H" unfolding MP_Rel_def unfolding H by (metis (mono_tags, hide_lams) "2" MP_Rel_def M_to_int_mod_ring Mp_f_representative rel_inverse_p functional_relation left_total_MP_Rel of_int_hom.map_poly_hom_smult pm.mod_ring_rel_def right_unique_MP_Rel to_int_mod_ring_hom.injectivity to_int_mod_ring_of_int_M) have "Mp (Polynomial.smult (inverse_p m (lead_coeff (Mp g))) g) = Mp (Polynomial.smult (inverse_p m (lead_coeff (Mp g))) (Mp g))" by simp hence hH: "MP_Rel h H" using hH' h unfolding MP_Rel_def by auto obtain q x where pseudo_fh: "pseudo_divmod (Mp f) (Mp h) = (q, x)" by (meson surj_pair) hence lc_G: "(lead_coeff G) ≠ 0" using G by auto have a: "a = Polynomial.smult (inverse_p m ((lead_coeff (Mp g)))) q" using 3 pseudo_fh Mp_g unfolding pdivmod2_def Let_def h by auto have b: "b = x" using 3 pseudo_fh Mp_g unfolding pdivmod2_def Let_def h by auto have Mp_Rel_FH: "MP_Rel q (F div H) ∧ MP_Rel x (F mod H)" proof (rule uniqueness_algorithm_division_Mp_Rel) show "monic (Mp h)" proof - have aux: "(inverse_p m (lead_coeff (Mp g))) = to_int_mod_ring (inverse (lead_coeff G))" using rel_inverse_p unfolding pm.mod_ring_rel_def by auto hence "M (inverse_p m (M (poly.coeff g (degree (Mp g))))) = to_int_mod_ring (inverse (lead_coeff G))" by (simp add: M_to_int_mod_ring Mp_coeff) thus ?thesis unfolding h unfolding Mp_coeff by auto (metis (no_types, lifting) "2" H MP_Rel_def Mp_coeff aux degree_smult_eq gG hH' inverse_zero_imp_zero lc_G left_inverse pm.mod_ring_rel_def to_int_mod_ring_hom.degree_map_poly_hom to_int_mod_ring_hom.hom_one to_int_mod_ring_times) qed hence monic_H: "monic H" using hH H lc_G by auto show f: "F = of_int_poly f" using 1 unfolding MP_Rel_def by (simp add: Mp_f_representative poly_eq_iff) have "pdivmod F H = pdivmod_monic F H" by (rule pdivmod_eq_pdivmod_monic[OF monic_H]) also have "... = pseudo_divmod F H" by (rule pdivmod_monic_pseudo_divmod[OF monic_H]) finally show "pseudo_divmod F H = (F div H, F mod H)" by simp show "H = of_int_poly h" by (meson MP_Rel_def Mp_f_representative hH right_unique_MP_Rel right_unique_def) show "pseudo_divmod (Mp f) (Mp h) = (q, x)" by (rule pseudo_fh) qed hence Mp_Rel_F_div_H: "MP_Rel q (F div H)" and Mp_Rel_F_mod_H: "MP_Rel x (F mod H)" by auto have "F div H = Polynomial.smult (lead_coeff G) (F div G)" unfolding H using div_smult_right[OF lc_G] inverse_inverse_eq by (metis div_smult_right inverse_zero) hence F_div_G: "(F div G) = Polynomial.smult (inverse (lead_coeff G)) (F div H)" using lc_G by auto have "MP_Rel a (F div G)" proof - have "of_int_poly (Polynomial.smult (inverse_p m ((lead_coeff (Mp g)))) q) = smult (inverse (lead_coeff G)) (F div H)" by (metis (mono_tags) MP_Rel_def M_to_int_mod_ring Mp_Rel_F_div_H Mp_f_representative of_int_hom.map_poly_hom_smult pm.mod_ring_rel_def rel_inverse_p right_unique_MP_Rel right_unique_def to_int_mod_ring_hom.injectivity to_int_mod_ring_of_int_M) thus ?thesis using Mp_Rel_F_div_H unfolding MP_Rel_def a F_div_G Mp_f_representative by auto qed moreover have "MP_Rel b (F mod G)" using Mp_Rel_F_mod_H b H inverse_zero_imp_zero lc_G by (metis mod_smult_right) ultimately show ?thesis by auto next assume Mp_g_0: "¬ Mp g ≠ 0" hence "pdivmod2 f g = (0, f)" unfolding pdivmod2_def by auto hence a: "a = 0" and b: "b = f" using 3 by auto have G0: "G = 0" using Mp_g_0 2 unfolding MP_Rel_def by auto have "MP_Rel a (F div G)" unfolding MP_Rel_def G0 a by auto moreover have "MP_Rel b (F mod G)" using 1 unfolding MP_Rel_def G0 a b by auto ultimately show ?thesis by simp qed thus "MP_Rel a (F div G)" and "MP_Rel b (F mod G)" by auto qed subsection ‹Executable division operation modulo $m$ for polynomials› lemma dvdm_iff_Mp_pdivmod2: shows "g dvdm f = (Mp (snd (pdivmod2 f g)) = 0)" proof - let ?F="(of_int_poly f)::'a mod_ring poly" let ?G="(of_int_poly g)::'a mod_ring poly" have a[transfer_rule]: "MP_Rel f ?F" by (simp add: MP_Rel_def Mp_f_representative) have b[transfer_rule]: "MP_Rel g ?G" by (simp add: MP_Rel_def Mp_f_representative) have "MP_Rel_Pair (pdivmod2 f g) (pdivmod ?F ?G)" using pdivmod2_rel unfolding rel_fun_def using a b by auto hence "MP_Rel (snd (pdivmod2 f g)) (snd (pdivmod ?F ?G))" unfolding MP_Rel_Pair_def by auto hence "(Mp (snd (pdivmod2 f g)) = 0) = (snd (pdivmod ?F ?G) = 0)" unfolding MP_Rel_def by auto thus ?thesis using dvdm_iff_pdivmod0 by auto qed end lemmas (in poly_mod_prime) dvdm_pdivmod = poly_mod_prime_type.dvdm_iff_Mp_pdivmod2 [unfolded poly_mod_type_simps, internalize_sort "'a :: prime_card", OF type_to_set, unfolded remove_duplicate_premise, cancel_type_definition, OF non_empty] lemma (in poly_mod) dvdm_code: "g dvdm f = (if prime m then Mp (snd (pdivmod2 f g)) = 0 else Code.abort (STR ''dvdm error: m is not a prime number'') (λ _. g dvdm f))" using poly_mod_prime.dvdm_pdivmod[unfolded poly_mod_prime_def] by auto declare poly_mod.pdivmod2_def[code] declare poly_mod.dvdm_code[code] end