Theory Missing_Dvd_Int_Poly

theory Missing_Dvd_Int_Poly
imports Hensel_Lifting Subresultant Cancel_Card_Constraint
```(*
Authors:    Jose Divasón
Sebastiaan Joosten
René Thiemann
*)

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)"
have j2: "degree (q* u + smult m r) = degree (q*u)"
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"
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"
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
have [transfer_rule]: "MP_Rel g G" unfolding MP_Rel_def
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
have g': "Mp g = to_int_poly G" unfolding g
have f'': "of_int_poly (Mp f) = F"
by (metis (no_types, lifting) Dp_Mp_eq Mp_f_representative
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
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"
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)"
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]

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
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]:
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)))))
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
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"
have b[transfer_rule]: "MP_Rel g ?G"
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
```