Theory Polynomial_Factorization.Gauss_Lemma
section ‹Gauss Lemma›
text ‹We formalized Gauss Lemma, that the content of a product of two polynomials $p$ and $q$
is the product of the contents of $p$ and $q$. As a corollary we provide an algorithm
to convert a rational factor of an integer polynomial into an integer factor.
In contrast to the theory on unique factorization domains -- where Gauss Lemma is also proven
in a more generic setting --
we are here in an executable setting and do not use the unspecified $some-gcd$ function.
Moreover, there is a slight difference in the definition of content: in this theory it is only
defined for integer-polynomials, whereas in the UFD theory, the content is defined for
polynomials in the fraction field.›
theory Gauss_Lemma
imports
"HOL-Computational_Algebra.Primes"
"HOL-Computational_Algebra.Field_as_Ring"
Polynomial_Interpolation.Ring_Hom_Poly
Missing_Polynomial_Factorial
begin
lemma primitive_part_alt_def:
"primitive_part p = sdiv_poly p (content p)"
by (simp add: primitive_part_def sdiv_poly_def)
definition common_denom :: "rat list ⇒ int × int list" where
"common_denom xs ≡ let
nds = map quotient_of xs;
denom = list_lcm (map snd nds);
ints = map (λ (n,d). n * denom div d) nds
in (denom, ints)"
definition rat_to_int_poly :: "rat poly ⇒ int × int poly" where
"rat_to_int_poly p ≡ let
ais = coeffs p;
d = fst (common_denom ais)
in (d, map_poly (λ x. case quotient_of x of (p,q) ⇒ p * d div q) p)"
definition rat_to_normalized_int_poly :: "rat poly ⇒ rat × int poly" where
"rat_to_normalized_int_poly p ≡ if p = 0 then (1,0) else case rat_to_int_poly p of (s,q)
⇒ (of_int (content q) / of_int s, primitive_part q)"
lemma rat_to_normalized_int_poly_code[code]:
"rat_to_normalized_int_poly p = (if p = 0 then (1,0) else case rat_to_int_poly p of (s,q)
⇒ let c = content q in (of_int c / of_int s, sdiv_poly q c))"
unfolding Let_def rat_to_normalized_int_poly_def primitive_part_alt_def ..
lemma common_denom: assumes cd: "common_denom xs = (dd,ys)"
shows "xs = map (λ i. of_int i / of_int dd) ys" "dd > 0"
"⋀x. x ∈ set xs ⟹ rat_of_int (case quotient_of x of (n, x) ⇒ n * dd div x) / rat_of_int dd = x"
proof -
let ?nds = "map quotient_of xs"
define nds where "nds = ?nds"
let ?denom = "list_lcm (map snd nds)"
let ?ints = "map (λ (n,d). n * dd div d) nds"
from cd[unfolded common_denom_def Let_def]
have dd: "dd = ?denom" and ys: "ys = ?ints" unfolding nds_def by auto
show dd0: "dd > 0" unfolding dd
by (intro list_lcm_pos(3), auto simp: nds_def quotient_of_nonzero)
{
fix x
assume x: "x ∈ set xs"
obtain p q where quot: "quotient_of x = (p,q)" by force
from x have "(p,q) ∈ set nds" unfolding nds_def using quot by force
hence "q ∈ set (map snd nds)" by force
from list_lcm[OF this] have q: "q dvd dd" unfolding dd .
show "rat_of_int (case quotient_of x of (n, x) ⇒ n * dd div x) / rat_of_int dd = x"
unfolding quot split unfolding quotient_of_div[OF quot]
proof -
have f1: "q * (dd div q) = dd"
using dvd_mult_div_cancel q by blast
have "rat_of_int (dd div q) ≠ 0"
using dd0 dvd_mult_div_cancel q by fastforce
thus "rat_of_int (p * dd div q) / rat_of_int dd = rat_of_int p / rat_of_int q"
using f1 by (metis (no_types) div_mult_swap mult_divide_mult_cancel_right of_int_mult q)
qed
} note main = this
show "xs = map (λ i. of_int i / of_int dd) ys" unfolding ys map_map o_def nds_def
by (rule sym, rule map_idI, rule main)
qed
lemma rat_to_int_poly: assumes "rat_to_int_poly p = (d,q)"
shows "p = smult (inverse (of_int d)) (map_poly of_int q)" "d > 0"
proof -
let ?f = "λ x. case quotient_of x of (pa, x) ⇒ pa * d div x"
define f where "f = ?f"
from assms[unfolded rat_to_int_poly_def Let_def]
obtain xs where cd: "common_denom (coeffs p) = (d,xs)"
and q: "q = map_poly f p" unfolding f_def by (cases "common_denom (coeffs p)", auto)
from common_denom[OF cd] have d: "d > 0" and
id: "⋀ x. x ∈ set (coeffs p) ⟹ rat_of_int (f x) / rat_of_int d = x"
unfolding f_def by auto
have f0: "f 0 = 0" unfolding f_def by auto
have id: "rat_of_int (f (coeff p n)) / rat_of_int d = coeff p n" for n
using id[of "coeff p n"] f0 range_coeff by (cases "coeff p n = 0", auto)
show "d > 0" by fact
show "p = smult (inverse (of_int d)) (map_poly of_int q)"
unfolding q smult_as_map_poly using id f0
by (intro poly_eqI, auto simp: field_simps coeff_map_poly)
qed
lemma content_ge_0_int: "content p ≥ (0 :: int)"
unfolding content_def
by (cases "coeffs p", auto)
lemma abs_content_int[simp]: fixes p :: "int poly"
shows "abs (content p) = content p" using content_ge_0_int[of p] by auto
lemma content_smult_int: fixes p :: "int poly"
shows "content (smult a p) = abs a * content p" by simp
lemma normalize_non_0_smult: "∃ a. (a :: 'a :: semiring_gcd) ≠ 0 ∧ smult a (primitive_part p) = p"
by (cases "p = 0", rule exI[of _ 1], simp, rule exI[of _ "content p"], auto)
lemma rat_to_normalized_int_poly: assumes "rat_to_normalized_int_poly p = (d,q)"
shows "p = smult d (map_poly of_int q)" "d > 0" "p ≠ 0 ⟹ content q = 1" "degree q = degree p"
proof -
have "p = smult d (map_poly of_int q) ∧ d > 0 ∧ (p ≠ 0 ⟶ content q = 1)"
proof (cases "p = 0")
case True
thus ?thesis using assms unfolding rat_to_normalized_int_poly_def
by (auto simp: eval_poly_def)
next
case False
hence p0: "p ≠ 0" by auto
obtain s r where id: "rat_to_int_poly p = (s,r)" by force
let ?cr = "rat_of_int (content r)"
let ?s = "rat_of_int s"
let ?q = "map_poly rat_of_int q"
from rat_to_int_poly[OF id] have p: "p = smult (inverse ?s) (map_poly of_int r)"
and s: "s > 0" by auto
let ?q = "map_poly rat_of_int q"
from p0 assms[unfolded rat_to_normalized_int_poly_def id split]
have d: "d = ?cr / ?s" and q: "q = primitive_part r" by auto
from content_times_primitive_part[of r, folded q] have qr: "smult (content r) q = r" .
have "smult d ?q = smult (?cr / ?s) ?q"
unfolding d by simp
also have "?cr / ?s = ?cr * inverse ?s" by (rule divide_inverse)
also have "… = inverse ?s * ?cr" by simp
also have "smult (inverse ?s * ?cr) ?q = smult (inverse ?s) (smult ?cr ?q)" by simp
also have "smult ?cr ?q = map_poly of_int (smult (content r) q)" by (simp add: hom_distribs)
also have "… = map_poly of_int r" unfolding qr ..
finally have pq: "p = smult d ?q" unfolding p by simp
from p p0 have r0: "r ≠ 0" by auto
from content_eq_zero_iff[of r] content_ge_0_int[of r] r0 have cr: "?cr > 0" by linarith
with s have d0: "d > 0" unfolding d by auto
from content_primitive_part[OF r0] have cq: "content q = 1" unfolding q .
from pq d0 cq show ?thesis by auto
qed
thus p: "p = smult d (map_poly of_int q)" and d: "d > 0" and "p ≠ 0 ⟹ content q = 1" by auto
show "degree q = degree p" unfolding p smult_as_map_poly
by (rule sym, subst map_poly_map_poly, force+, rule degree_map_poly, insert d, auto)
qed
lemma content_dvd_1:
"content g = 1" if "content f = (1 :: 'a :: semiring_gcd)" "g dvd f"
proof -
from ‹g dvd f› have "content g dvd content f"
by (rule content_dvd_contentI)
with ‹content f = 1› show ?thesis
by simp
qed
lemma dvd_smult_int: fixes c :: int assumes c: "c ≠ 0"
and dvd: "q dvd (smult c p)"
shows "primitive_part q dvd p"
proof (cases "p = 0")
case True thus ?thesis by auto
next
case False note p0 = this
let ?cp = "smult c p"
from p0 c have cp0: "?cp ≠ 0" by auto
from dvd obtain r where prod: "?cp = q * r" unfolding dvd_def by auto
from prod cp0 have q0: "q ≠ 0" and r0: "r ≠ 0" by auto
let ?c = "content :: int poly ⇒ int"
let ?n = "primitive_part :: int poly ⇒ int poly"
let ?pn = "λ p. smult (?c p) (?n p)"
have cq: "(?c q = 0) = False" using content_eq_zero_iff q0 by auto
from prod have id1: "?cp = ?pn q * ?pn r" unfolding content_times_primitive_part by simp
from arg_cong[OF this, of content, unfolded content_smult_int content_mult
content_primitive_part[OF r0] content_primitive_part[OF q0], symmetric]
p0[folded content_eq_zero_iff] c
have "abs c dvd ?c q * ?c r" unfolding dvd_def by auto
hence "c dvd ?c q * ?c r" by auto
then obtain d where id: "?c q * ?c r = c * d" unfolding dvd_def by auto
have "?cp = ?pn q * ?pn r" by fact
also have "… = smult (c * d) (?n q * ?n r)" unfolding id [symmetric]
by (metis content_mult content_times_primitive_part primitive_part_mult)
finally have id: "?cp = smult c (?n q * smult d (?n r))" by (simp add: mult.commute)
interpret map_poly_inj_zero_hom "(*) c" using c by (unfold_locales, auto)
have "p = ?n q * smult d (?n r)" using id[unfolded smult_as_map_poly[of c]] by auto
thus dvd: "?n q dvd p" unfolding dvd_def by blast
qed
lemma irreducible⇩d_primitive_part:
fixes p :: "int poly"
shows "irreducible⇩d (primitive_part p) ⟷ irreducible⇩d p" (is "?l ⟷ ?r")
proof (rule iffI, rule irreducible⇩dI)
assume l: ?l
show "degree p > 0" using l by auto
have dpp: "degree (primitive_part p) = degree p" by simp
fix q r
assume deg: "degree q < degree p" "degree r < degree p" and "p = q * r"
then have pp: "primitive_part p = primitive_part q * primitive_part r" by (simp add: primitive_part_mult)
have "¬ irreducible⇩d (primitive_part p)"
apply (intro reducible⇩dI, rule exI[of _ "primitive_part q"], rule exI[of _ "primitive_part r"], unfold dpp)
using deg pp by auto
with l show False by auto
next
show "?r ⟹ ?l" by (metis irreducible⇩d_smultI normalize_non_0_smult)
qed
lemma irreducible⇩d_smult_int:
fixes c :: int assumes c: "c ≠ 0"
shows "irreducible⇩d (smult c p) = irreducible⇩d p" (is "?l = ?r")
using irreducible⇩d_primitive_part[of "smult c p", unfolded primitive_part_smult] c
apply (cases "c < 0", simp)
apply (metis add.inverse_inverse add.inverse_neutral c irreducible⇩d_smultI normalize_non_0_smult smult_1_left smult_minus_left)
apply (simp add: irreducible⇩d_primitive_part)
done
lemma irreducible⇩d_as_irreducible:
fixes p :: "int poly"
shows "irreducible⇩d p ⟷ irreducible (primitive_part p)"
using irreducible_primitive_connect[of "primitive_part p"]
by (cases "p = 0", auto simp: irreducible⇩d_primitive_part)
lemma rat_to_int_factor_content_1: fixes p :: "int poly"
assumes cp: "content p = 1"
and pgh: "map_poly rat_of_int p = g * h"
and g: "rat_to_normalized_int_poly g = (r,rg)"
and h: "rat_to_normalized_int_poly h = (s,sh)"
and p: "p ≠ 0"
shows "p = rg * sh"
proof -
let ?r = "rat_of_int"
let ?rp = "map_poly ?r"
from p have rp0: "?rp p ≠ 0" by simp
with pgh have g0: "g ≠ 0" and h0: "h ≠ 0" by auto
from rat_to_normalized_int_poly[OF g] g0
have r: "r > 0" "r ≠ 0" and g: "g = smult r (?rp rg)" and crg: "content rg = 1" by auto
from rat_to_normalized_int_poly[OF h] h0
have s: "s > 0" "s ≠ 0" and h: "h = smult s (?rp sh)" and csh: "content sh = 1" by auto
let ?irs = "inverse (r * s)"
from r s have irs0: "?irs ≠ 0" by (auto simp: field_simps)
have "?rp (rg * sh) = ?rp rg * ?rp sh" by (simp add: hom_distribs)
also have "… = smult ?irs (?rp p)" unfolding pgh g h using r s
by (simp add: field_simps)
finally have id: "?rp (rg * sh) = smult ?irs (?rp p)" by auto
have rsZ: "?irs ∈ ℤ"
proof (rule ccontr)
assume not: "¬ ?irs ∈ ℤ"
obtain n d where irs': "quotient_of ?irs = (n,d)" by force
from quotient_of_denom_pos[OF irs'] have "d > 0" .
from not quotient_of_div[OF irs'] have "d ≠ 1" "d ≠ 0" and irs: "?irs = ?r n / ?r d" by auto
with irs0 have n0: "n ≠ 0" by auto
from ‹d > 0› ‹d ≠ 1› have "d ≥ 2" and "¬ d dvd 1" by auto
with content_iff[of d p, unfolded cp] obtain c where
c: "c ∈ set (coeffs p)" and dc: "¬ d dvd c"
by auto
from c range_coeff[of p] obtain i where "c = coeff p i" by auto
from arg_cong[OF id, of "λ p. coeff p i",
unfolded coeff_smult of_int_hom.coeff_map_poly_hom this[symmetric] irs]
have "?r n / ?r d * ?r c ∈ ℤ" by (metis Ints_of_int)
also have "?r n / ?r d * ?r c = ?r (n * c) / ?r d" by simp
finally have inZ: "?r (n * c) / ?r d ∈ ℤ" .
have cop: "coprime n d" by (rule quotient_of_coprime[OF irs'])
define prod where "prod = ?r (n * c) / ?r d"
obtain n' d' where quot: "quotient_of prod = (n',d')" by force
have qr: "⋀ x. quotient_of (?r x) = (x, 1)"
using Rat.of_int_def quotient_of_int by auto
from quotient_of_denom_pos[OF quot] have "d' > 0" .
with quotient_of_div[OF quot] inZ[folded prod_def] have "d' = 1"
by (metis Ints_cases Rat.of_int_def old.prod.inject quot quotient_of_int)
with quotient_of_div[OF quot] have "prod = ?r n'" by auto
from arg_cong[OF this, of quotient_of, unfolded prod_def rat_divide_code qr Let_def split]
have "Rat.normalize (n * c, d) = (n',1)" by simp
from normalize_crossproduct[OF ‹d ≠ 0›, of 1 "n * c" n', unfolded this]
have id: "n * c = n' * d" by auto
from quotient_of_coprime[OF irs'] have "coprime n d" .
with id have "d dvd c"
by (metis coprime_commute coprime_dvd_mult_right_iff dvd_triv_right)
with dc show False ..
qed
then obtain irs where irs: "?irs = ?r irs" unfolding Ints_def by blast
from id[unfolded irs, folded hom_distribs, unfolded of_int_poly_hom.eq_iff]
have p: "rg * sh = smult irs p" by auto
have "content (rg * sh) = 1" unfolding content_mult crg csh by auto
from this[unfolded p content_smult_int cp] have "abs irs = 1" by simp
hence "abs ?irs = 1" using irs by auto
with r s have "?irs = 1" by auto
with irs have "irs = 1" by auto
with p show p: "p = rg * sh" by auto
qed
lemma rat_to_int_factor_explicit: fixes p :: "int poly"
assumes pgh: "map_poly rat_of_int p = g * h"
and g: "rat_to_normalized_int_poly g = (r,rg)"
shows "∃ r. p = rg * smult (content p) r"
proof -
show ?thesis
proof (cases "p = 0")
case True
show ?thesis unfolding True
by (rule exI[of _ 0], auto simp: degree_monom_eq)
next
case False
hence p: "p ≠ 0" by auto
let ?r = "rat_of_int"
let ?rp = "map_poly ?r"
define q where "q = primitive_part p"
from content_times_primitive_part[of p, folded q_def] content_eq_zero_iff[of p] p
obtain a where a: "a ≠ 0" and pq: "p = smult a q" and acp: "content p = a" by metis
from a pq p have ra: "?r a ≠ 0" and q0: "q ≠ 0" by auto
from content_primitive_part[OF p, folded q_def] have cq: "content q = 1" by auto
obtain s sh where h: "rat_to_normalized_int_poly (smult (inverse (?r a)) h) = (s,sh)" by force
from arg_cong[OF pgh[unfolded pq], of "smult (inverse (?r a))"] ra
have "?rp q = g * smult (inverse (?r a)) h" by (auto simp: hom_distribs)
from rat_to_int_factor_content_1[OF cq this g h q0]
have qrs: "q = rg * sh" .
show ?thesis unfolding acp unfolding pq qrs
by (rule exI[of _ sh], auto)
qed
qed
lemma rat_to_int_factor: fixes p :: "int poly"
assumes pgh: "map_poly rat_of_int p = g * h"
shows "∃ g' h'. p = g' * h' ∧ degree g' = degree g ∧ degree h' = degree h"
proof(cases "p = 0")
case True
with pgh have "g = 0 ∨ h = 0" by auto
then show ?thesis
by (metis True degree_0 mult_hom.hom_zero mult_zero_left rat_to_normalized_int_poly(4) surj_pair)
next
case False
obtain r rg where ri: "rat_to_normalized_int_poly (smult (1 / of_int (content p)) g) = (r,rg)" by force
obtain q qh where ri2: "rat_to_normalized_int_poly h = (q,qh)" by force
show ?thesis
proof (intro exI conjI)
have "of_int_poly (primitive_part p) = smult (1 / of_int (content p)) (g * h)"
apply (auto simp: primitive_part_def pgh[symmetric] smult_map_poly map_poly_map_poly o_def intro!: map_poly_cong)
by (metis (no_types, lifting) content_dvd_coeffs div_by_0 dvd_mult_div_cancel floor_of_int nonzero_mult_div_cancel_left of_int_hom.hom_zero of_int_mult)
also have "… = smult (1 / of_int (content p)) g * h" by simp
finally have "of_int_poly (primitive_part p) = …".
note main = rat_to_int_factor_content_1[OF _ this ri ri2, simplified, OF False]
show "p = smult (content p) rg * qh" by (simp add: main[symmetric])
from ri2 show "degree qh = degree h" by (fact rat_to_normalized_int_poly)
from rat_to_normalized_int_poly(4)[OF ri] False
show "degree (smult (content p) rg) = degree g" by auto
qed
qed
lemma rat_to_int_factor_normalized_int_poly: fixes p :: "rat poly"
assumes pgh: "p = g * h"
and p: "rat_to_normalized_int_poly p = (i,ip)"
shows "∃ g' h'. ip = g' * h' ∧ degree g' = degree g"
proof -
from rat_to_normalized_int_poly[OF p]
have p: "p = smult i (map_poly rat_of_int ip)" and i: "i ≠ 0" by auto
from arg_cong[OF p, of "smult (inverse i)", unfolded pgh] i
have "map_poly rat_of_int ip = g * smult (inverse i) h" by auto
from rat_to_int_factor[OF this] show ?thesis by auto
qed
lemma irreducible_smult [simp]:
fixes c :: "'a :: field"
shows "irreducible (smult c p) ⟷ irreducible p ∧ c ≠ 0"
using irreducible_mult_unit_left[of "[:c:]", simplified] by force
text ‹A polynomial with integer coefficients is
irreducible over the rationals, if it is irreducible over the integers.›
theorem irreducible⇩d_int_rat: fixes p :: "int poly"
assumes p: "irreducible⇩d p"
shows "irreducible⇩d (map_poly rat_of_int p)"
proof (rule irreducible⇩dI)
from irreducible⇩dD[OF p]
have p: "degree p ≠ 0" and irr: "⋀ q r. degree q < degree p ⟹ degree r < degree p ⟹ p ≠ q * r" by auto
let ?r = "rat_of_int"
let ?rp = "map_poly ?r"
from p show rp: "degree (?rp p) > 0" by auto
from p have p0: "p ≠ 0" by auto
fix g h :: "rat poly"
assume deg: "degree g > 0" "degree g < degree (?rp p)" "degree h > 0" "degree h < degree (?rp p)" and pgh: "?rp p = g * h"
from rat_to_int_factor[OF pgh] obtain g' h' where p: "p = g' * h'" and dg: "degree g' = degree g" "degree h' = degree h"
by auto
from irr[of g' h'] deg[unfolded dg]
show False using degree_mult_eq[of g' h'] by (auto simp: p dg)
qed
corollary irreducible⇩d_rat_to_normalized_int_poly:
assumes rp: "rat_to_normalized_int_poly rp = (a, ip)"
and ip: "irreducible⇩d ip"
shows "irreducible⇩d rp"
proof -
from rat_to_normalized_int_poly[OF rp]
have rp: "rp = smult a (map_poly rat_of_int ip)" and a: "a ≠ 0" by auto
with irreducible⇩d_int_rat[OF ip] show ?thesis by auto
qed
lemma dvd_content_dvd: assumes dvd: "content f dvd content g" "primitive_part f dvd primitive_part g"
shows "f dvd g"
proof -
let ?cf = "content f" let ?nf = "primitive_part f"
let ?cg = "content g" let ?ng = "primitive_part g"
have "f dvd g = (smult ?cf ?nf dvd smult ?cg ?ng)"
unfolding content_times_primitive_part by auto
from dvd(1) obtain ch where cg: "?cg = ?cf * ch" unfolding dvd_def by auto
from dvd(2) obtain nh where ng: "?ng = ?nf * nh" unfolding dvd_def by auto
have "f dvd g = (smult ?cf ?nf dvd smult ?cg ?ng)"
unfolding content_times_primitive_part[of f] content_times_primitive_part[of g] by auto
also have "… = (smult ?cf ?nf dvd smult ?cf ?nf * smult ch nh)" unfolding cg ng
by (metis mult.commute mult_smult_right smult_smult)
also have "…" by (rule dvd_triv_left)
finally show ?thesis .
qed
lemma sdiv_poly_smult: "c ≠ 0 ⟹ sdiv_poly (smult c f) c = f"
by (intro poly_eqI, unfold coeff_sdiv_poly coeff_smult, auto)
lemma primitive_part_smult_int: fixes f :: "int poly" shows
"primitive_part (smult d f) = smult (sgn d) (primitive_part f)"
proof (cases "d = 0 ∨ f = 0")
case False
obtain cf where cf: "content f = cf" by auto
with False have 0: "d ≠ 0" "f ≠ 0" "cf ≠ 0" by auto
show ?thesis
proof (rule poly_eqI, unfold primitive_part_alt_def coeff_sdiv_poly content_smult_int coeff_smult cf)
fix n
consider (pos) "d > 0" | (neg) "d < 0" using 0(1) by linarith
thus "d * coeff f n div (¦d¦ * cf) = sgn d * (coeff f n div cf)"
proof cases
case neg
hence "?thesis = (d * coeff f n div - (d * cf) = - (coeff f n div cf))" by auto
also have "d * coeff f n div - (d * cf) = - (d * coeff f n div (d * cf))"
by (subst dvd_div_neg, insert 0(1), auto simp: cf[symmetric])
also have "d * coeff f n div (d * cf) = coeff f n div cf" using 0(1) by auto
finally show ?thesis by simp
qed auto
qed
qed auto
lemma gcd_smult_left: assumes "c ≠ 0"
shows "gcd (smult c f) g = gcd f (g :: 'b :: {field_gcd} poly)"
proof -
from assms have "normalize c = 1"
by (meson dvd_field_iff is_unit_normalize)
then show ?thesis
by (metis (no_types) Polynomial.normalize_smult gcd.commute gcd.left_commute gcd_left_idem gcd_self smult_1_left)
qed
lemma gcd_smult_right: "c ≠ 0 ⟹ gcd f (smult c g) = gcd f (g :: 'b :: {field_gcd} poly)"
using gcd_smult_left[of c g f] by (simp add: gcd.commute)
lemma gcd_rat_to_gcd_int: "gcd (of_int_poly f :: rat poly) (of_int_poly g) =
smult (inverse (of_int (lead_coeff (gcd f g)))) (of_int_poly (gcd f g))"
proof (cases "f = 0 ∧ g = 0")
case True
thus ?thesis by simp
next
case False
let ?r = rat_of_int
let ?rp = "map_poly ?r"
from False have gcd0: "gcd f g ≠ 0" by auto
hence lc0: "lead_coeff (gcd f g) ≠ 0" by auto
hence inv: "inverse (?r (lead_coeff (gcd f g))) ≠ 0" by auto
show ?thesis
proof (rule sym, rule gcdI, goal_cases)
case 1
have "gcd f g dvd f" by auto
then obtain h where f: "f = gcd f g * h" unfolding dvd_def by auto
show ?case by (rule smult_dvd[OF _ inv], insert arg_cong[OF f, of ?rp], simp add: hom_distribs)
next
case 2
have "gcd f g dvd g" by auto
then obtain h where g: "g = gcd f g * h" unfolding dvd_def by auto
show ?case by (rule smult_dvd[OF _ inv], insert arg_cong[OF g, of ?rp], simp add: hom_distribs)
next
case (3 h)
show ?case
proof (rule dvd_smult)
obtain ch ph where h: "rat_to_normalized_int_poly h = (ch, ph)" by force
from 3 obtain ff where f: "?rp f = h * ff" unfolding dvd_def by auto
from 3 obtain gg where g: "?rp g = h * gg" unfolding dvd_def by auto
from rat_to_int_factor_explicit[OF f h] obtain f' where f: "f = ph * f'" by blast
from rat_to_int_factor_explicit[OF g h] obtain g' where g: "g = ph * g'" by blast
from f g have "ph dvd gcd f g" by auto
then obtain gg where gcd: "gcd f g = ph * gg" unfolding dvd_def by auto
note * = rat_to_normalized_int_poly[OF h]
show "h dvd ?rp (gcd f g)" unfolding gcd *(1)
by (rule smult_dvd, insert *(2), auto)
qed
next
case 4
have [simp]: "[:1:] = 1" by simp
show ?case unfolding normalize_poly_def
by (rule poly_eqI, simp)
qed
qed
end