# Theory Square_Free_Factorization_Int

```(*
Authors:      Jose Divasón
Sebastiaan Joosten
René Thiemann
*)
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)"
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
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
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"
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
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
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
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"
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"
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"
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
```