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Mainboard.pas.~985~
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(* Title: HOL/Algebra/Polynomial_Divisibility.thy
Author: Paulo Emílio de Vilhena
*)
theory Polynomial_Divisibility
imports Polynomials Embedded_Algebras "HOL-Library.Multiset"
begin
section \<open>Divisibility of Polynomials\<close>
subsection \<open>Definitions\<close>
abbreviation poly_ring :: "_ \ ('a list) ring"
where "poly_ring R \ univ_poly R (carrier R)"
abbreviation pirreducible :: "_ \ 'a set \ 'a list \ bool" ("pirreducible\")
where "pirreducible\<^bsub>R\<^esub> K p \ ring_irreducible\<^bsub>(univ_poly R K)\<^esub> p"
abbreviation pprime :: "_ \ 'a set \ 'a list \ bool" ("pprime\")
where "pprime\<^bsub>R\<^esub> K p \ ring_prime\<^bsub>(univ_poly R K)\<^esub> p"
definition pdivides :: "_ \ 'a list \ 'a list \ bool" (infix "pdivides\" 65)
where "p pdivides\<^bsub>R\<^esub> q = p divides\<^bsub>(univ_poly R (carrier R))\<^esub> q"
definition rupture :: "_ \ 'a set \ 'a list \ (('a list) set) ring" ("Rupt\")
where "Rupt\<^bsub>R\<^esub> K p = (K[X]\<^bsub>R\<^esub>) Quot (PIdl\<^bsub>K[X]\<^bsub>R\<^esub>\<^esub> p)"
abbreviation (in ring) rupture_surj :: "'a set \ 'a list \ 'a list \ ('a list) set"
where "rupture_surj K p \ (\q. (PIdl\<^bsub>K[X]\<^esub> p) +>\<^bsub>K[X]\<^esub> q)"
subsection \<open>Basic Properties\<close>
lemma (in ring) carrier_polynomial_shell [intro]:
assumes "subring K R" and "p \ carrier (K[X])" shows "p \ carrier (poly_ring R)"
using carrier_polynomial[OF assms(1), of p] assms(2) unfolding sym[OF univ_poly_carrier] by simp
lemma (in domain) pdivides_zero:
assumes "subring K R" and "p \ carrier (K[X])" shows "p pdivides []"
using ring.divides_zero[OF univ_poly_is_ring[OF carrier_is_subring]
carrier_polynomial_shell[OF assms]]
unfolding univ_poly_zero pdivides_def .
lemma (in domain) zero_pdivides_zero: "[] pdivides []"
using pdivides_zero[OF carrier_is_subring] univ_poly_carrier by blast
lemma (in domain) zero_pdivides:
shows "[] pdivides p \ p = []"
using ring.zero_divides[OF univ_poly_is_ring[OF carrier_is_subring]]
unfolding univ_poly_zero pdivides_def .
lemma (in domain) pprime_iff_pirreducible:
assumes "subfield K R" and "p \ carrier (K[X])"
shows "pprime K p \ pirreducible K p"
using principal_domain.primeness_condition[OF univ_poly_is_principal] assms by simp
lemma (in domain) pirreducibleE:
assumes "subring K R" "p \ carrier (K[X])" "pirreducible K p"
shows "p \ []" "p \ Units (K[X])"
and "\q r. \ q \ carrier (K[X]); r \ carrier (K[X])\ \
p = q \<otimes>\<^bsub>K[X]\<^esub> r \<Longrightarrow> q \<in> Units (K[X]) \<or> r \<in> Units (K[X])"
using domain.ring_irreducibleE[OF univ_poly_is_domain[OF assms(1)] _ assms(3)] assms(2)
by (auto simp add: univ_poly_zero)
lemma (in domain) pirreducibleI:
assumes "subring K R" "p \ carrier (K[X])" "p \ []" "p \ Units (K[X])"
and "\q r. \ q \ carrier (K[X]); r \ carrier (K[X])\ \
p = q \<otimes>\<^bsub>K[X]\<^esub> r \<Longrightarrow> q \<in> Units (K[X]) \<or> r \<in> Units (K[X])"
shows "pirreducible K p"
using domain.ring_irreducibleI[OF univ_poly_is_domain[OF assms(1)] _ assms(4)] assms(2-3,5)
by (auto simp add: univ_poly_zero)
lemma (in domain) univ_poly_carrier_units_incl:
shows "Units ((carrier R) [X]) \ { [ k ] | k. k \ carrier R - { \ } }"
proof
fix p assume "p \ Units ((carrier R) [X])"
then obtain q
where p: "polynomial (carrier R) p" and q: "polynomial (carrier R) q" and pq: "poly_mult p q = [ \ ]"
unfolding Units_def univ_poly_def by auto
hence not_nil: "p \ []" and "q \ []"
using poly_mult_integral[OF carrier_is_subring p q] poly_mult_zero[OF polynomial_incl[OF p]] by auto
hence "degree p = 0"
using poly_mult_degree_eq[OF carrier_is_subring p q] unfolding pq by simp
hence "length p = 1"
using not_nil by (metis One_nat_def Suc_pred length_greater_0_conv)
then obtain k where k: "p = [ k ]"
by (metis One_nat_def length_0_conv length_Suc_conv)
hence "k \ carrier R - { \ }"
using p unfolding polynomial_def by auto
thus "p \ { [ k ] | k. k \ carrier R - { \ } }"
unfolding k by blast
qed
lemma (in field) univ_poly_carrier_units:
"Units ((carrier R) [X]) = { [ k ] | k. k \ carrier R - { \ } }"
proof
show "Units ((carrier R) [X]) \ { [ k ] | k. k \ carrier R - { \ } }"
using univ_poly_carrier_units_incl by simp
next
show "{ [ k ] | k. k \ carrier R - { \ } } \ Units ((carrier R) [X])"
proof (auto)
fix k assume k: "k \ carrier R" "k \ \"
hence inv_k: "inv k \ carrier R" "inv k \ \" and "k \ inv k = \" "inv k \ k = \"
using subfield_m_inv[OF carrier_is_subfield, of k] by auto
hence "poly_mult [ k ] [ inv k ] = [ \ ]" and "poly_mult [ inv k ] [ k ] = [ \ ]"
by (auto simp add: k)
moreover have "polynomial (carrier R) [ k ]" and "polynomial (carrier R) [ inv k ]"
using const_is_polynomial k inv_k by auto
ultimately show "[ k ] \ Units ((carrier R) [X])"
unfolding Units_def univ_poly_def by (auto simp del: poly_mult.simps)
qed
qed
lemma (in domain) univ_poly_units_incl:
assumes "subring K R" shows "Units (K[X]) \ { [ k ] | k. k \ K - { \ } }"
using domain.univ_poly_carrier_units_incl[OF subring_is_domain[OF assms]]
univ_poly_consistent[OF assms] by auto
lemma (in ring) univ_poly_units:
assumes "subfield K R" shows "Units (K[X]) = { [ k ] | k. k \ K - { \ } }"
using field.univ_poly_carrier_units[OF subfield_iff(2)[OF assms]]
univ_poly_consistent[OF subfieldE(1)[OF assms]] by auto
lemma (in domain) univ_poly_units':
assumes "subfield K R" shows "p \ Units (K[X]) \ p \ carrier (K[X]) \ p \ [] \ degree p = 0"
unfolding univ_poly_units[OF assms] sym[OF univ_poly_carrier] polynomial_def
by (auto, metis hd_in_set le_0_eq le_Suc_eq length_0_conv length_Suc_conv list.sel(1) subsetD)
corollary (in domain) rupture_one_not_zero:
assumes "subfield K R" and "p \ carrier (K[X])" and "degree p > 0"
shows "\\<^bsub>Rupt K p\<^esub> \ \\<^bsub>Rupt K p\<^esub>"
proof (rule ccontr)
interpret UP: principal_domain "K[X]"
using univ_poly_is_principal[OF assms(1)] .
assume "\ \\<^bsub>Rupt K p\<^esub> \ \\<^bsub>Rupt K p\<^esub>"
then have "PIdl\<^bsub>K[X]\<^esub> p +>\<^bsub>K[X]\<^esub> \\<^bsub>K[X]\<^esub> = PIdl\<^bsub>K[X]\<^esub> p"
unfolding rupture_def FactRing_def by simp
hence "\\<^bsub>K[X]\<^esub> \ PIdl\<^bsub>K[X]\<^esub> p"
using ideal.rcos_const_imp_mem[OF UP.cgenideal_ideal[OF assms(2)]] by auto
then obtain q where "q \ carrier (K[X])" and "\\<^bsub>K[X]\<^esub> = q \\<^bsub>K[X]\<^esub> p"
using assms(2) unfolding cgenideal_def by auto
hence "p \ Units (K[X])"
unfolding Units_def using assms(2) UP.m_comm by auto
hence "degree p = 0"
unfolding univ_poly_units[OF assms(1)] by auto
with \<open>degree p > 0\<close> show False
by simp
qed
corollary (in ring) pirreducible_degree:
assumes "subfield K R" "p \ carrier (K[X])" "pirreducible K p"
shows "degree p \ 1"
proof (rule ccontr)
assume "\ degree p \ 1" then have "length p \ 1"
by simp
moreover have "p \ []" and "p \ Units (K[X])"
using assms(3) by (auto simp add: ring_irreducible_def irreducible_def univ_poly_zero)
ultimately obtain k where k: "p = [ k ]"
by (metis append_butlast_last_id butlast_take diff_is_0_eq le_refl self_append_conv2 take0 take_all)
hence "k \ K" and "k \ \"
using assms(2) by (auto simp add: polynomial_def univ_poly_def)
hence "p \ Units (K[X])"
using univ_poly_units[OF assms(1)] unfolding k by auto
from \<open>p \<in> Units (K[X])\<close> and \<open>p \<notin> Units (K[X])\<close> show False by simp
qed
corollary (in domain) univ_poly_not_field:
assumes "subring K R" shows "\ field (K[X])"
proof -
have "X \ carrier (K[X]) - { \\<^bsub>(K[X])\<^esub> }" and "X \ { [ k ] | k. k \ K - { \ } }"
using var_closed(1)[OF assms] unfolding univ_poly_zero var_def by auto
thus ?thesis
using field.field_Units[of "K[X]"] univ_poly_units_incl[OF assms] by blast
qed
lemma (in domain) rupture_is_field_iff_pirreducible:
assumes "subfield K R" and "p \ carrier (K[X])"
shows "field (Rupt K p) \ pirreducible K p"
proof
assume "pirreducible K p" thus "field (Rupt K p)"
using principal_domain.field_iff_prime[OF univ_poly_is_principal[OF assms(1)]] assms(2)
pprime_iff_pirreducible[OF assms] pirreducibleE(1)[OF subfieldE(1)[OF assms(1)]]
by (simp add: univ_poly_zero rupture_def)
next
interpret UP: principal_domain "K[X]"
using univ_poly_is_principal[OF assms(1)] .
assume field: "field (Rupt K p)"
have "p \ []"
proof (rule ccontr)
assume "\ p \ []" then have p: "p = []"
by simp
hence "Rupt K p \ (K[X])"
using UP.FactRing_zeroideal(1) UP.genideal_zero
UP.cgenideal_eq_genideal[OF UP.zero_closed]
by (simp add: rupture_def univ_poly_zero)
then obtain h where h: "h \ ring_iso (Rupt K p) (K[X])"
unfolding is_ring_iso_def by blast
moreover have "ring (Rupt K p)"
using field by (simp add: cring_def domain_def field_def)
ultimately interpret R: ring_hom_ring "Rupt K p" "K[X]" h
unfolding ring_hom_ring_def ring_hom_ring_axioms_def ring_iso_def
using UP.ring_axioms by simp
have "field (K[X])"
using field.ring_iso_imp_img_field[OF field h] by simp
thus False
using univ_poly_not_field[OF subfieldE(1)[OF assms(1)]] by simp
qed
thus "pirreducible K p"
using UP.field_iff_prime pprime_iff_pirreducible[OF assms] assms(2) field
by (simp add: univ_poly_zero rupture_def)
qed
lemma (in domain) rupture_surj_hom:
assumes "subring K R" and "p \ carrier (K[X])"
shows "(rupture_surj K p) \ ring_hom (K[X]) (Rupt K p)"
and "ring_hom_ring (K[X]) (Rupt K p) (rupture_surj K p)"
proof -
interpret UP: domain "K[X]"
using univ_poly_is_domain[OF assms(1)] .
interpret I: ideal "PIdl\<^bsub>K[X]\<^esub> p" "K[X]"
using UP.cgenideal_ideal[OF assms(2)] .
show "(rupture_surj K p) \ ring_hom (K[X]) (Rupt K p)"
and "ring_hom_ring (K[X]) (Rupt K p) (rupture_surj K p)"
using ring_hom_ring.intro[OF UP.ring_axioms I.quotient_is_ring] I.rcos_ring_hom
unfolding symmetric[OF ring_hom_ring_axioms_def] rupture_def by auto
qed
corollary (in domain) rupture_surj_norm_is_hom:
assumes "subring K R" and "p \ carrier (K[X])"
shows "((rupture_surj K p) \ poly_of_const) \ ring_hom (R \ carrier := K \) (Rupt K p)"
using ring_hom_trans[OF canonical_embedding_is_hom[OF assms(1)] rupture_surj_hom(1)[OF assms]] .
lemma (in domain) norm_map_in_poly_ring_carrier:
assumes "p \ carrier (poly_ring R)" and "\a. a \ carrier R \ f a \ carrier (poly_ring R)"
shows "ring.normalize (poly_ring R) (map f p) \ carrier (poly_ring (poly_ring R))"
proof -
have "set p \ carrier R"
using assms(1) unfolding sym[OF univ_poly_carrier] polynomial_def by auto
hence "set (map f p) \ carrier (poly_ring R)"
using assms(2) by auto
thus ?thesis
using ring.normalize_gives_polynomial[OF univ_poly_is_ring[OF carrier_is_subring]]
unfolding univ_poly_carrier by simp
qed
lemma (in domain) map_in_poly_ring_carrier:
assumes "p \ carrier (poly_ring R)" and "\a. a \ carrier R \ f a \ carrier (poly_ring R)"
and "\a. a \ \ \ f a \ []"
shows "map f p \ carrier (poly_ring (poly_ring R))"
proof -
interpret UP: ring "poly_ring R"
using univ_poly_is_ring[OF carrier_is_subring] .
have "lead_coeff p \ \" if "p \ []"
using that assms(1) unfolding sym[OF univ_poly_carrier] polynomial_def by auto
hence "ring.normalize (poly_ring R) (map f p) = map f p"
by (cases p) (simp_all add: assms(3) univ_poly_zero)
thus ?thesis
using norm_map_in_poly_ring_carrier[of p f] assms(1-2) by simp
qed
lemma (in domain) map_norm_in_poly_ring_carrier:
assumes "subring K R" and "p \ carrier (K[X])"
shows "map poly_of_const p \ carrier (poly_ring (K[X]))"
using domain.map_in_poly_ring_carrier[OF subring_is_domain[OF assms(1)]]
proof -
have "\a. a \ K \ poly_of_const a \ carrier (K[X])"
and "\a. a \ \ \ poly_of_const a \ []"
using ring_hom_memE(1)[OF canonical_embedding_is_hom[OF assms(1)]]
by (auto simp: poly_of_const_def)
thus ?thesis
using domain.map_in_poly_ring_carrier[OF subring_is_domain[OF assms(1)]] assms(2)
unfolding univ_poly_consistent[OF assms(1)] by simp
qed
lemma (in domain) polynomial_rupture:
assumes "subring K R" and "p \ carrier (K[X])"
shows "(ring.eval (Rupt K p)) (map ((rupture_surj K p) \ poly_of_const) p) (rupture_surj K p X) = \\<^bsub>Rupt K p\<^esub>"
proof -
let ?surj = "rupture_surj K p"
interpret UP: domain "K[X]"
using univ_poly_is_domain[OF assms(1)] .
interpret Hom: ring_hom_ring "K[X]" "Rupt K p" ?surj
using rupture_surj_hom(2)[OF assms] .
have "(Hom.S.eval) (map (?surj \ poly_of_const) p) (?surj X) = ?surj ((UP.eval) (map poly_of_const p) X)"
using Hom.eval_hom[OF UP.carrier_is_subring var_closed(1)[OF assms(1)]
map_norm_in_poly_ring_carrier[OF assms]] by simp
also have " ... = ?surj p"
unfolding sym[OF eval_rewrite[OF assms]] ..
also have " ... = \\<^bsub>Rupt K p\<^esub>"
using UP.a_rcos_zero[OF UP.cgenideal_ideal[OF assms(2)] UP.cgenideal_self[OF assms(2)]]
unfolding rupture_def FactRing_def by simp
finally show ?thesis .
qed
subsection \<open>Division\<close>
definition (in ring) long_divides :: "'a list \ 'a list \ ('a list \ 'a list) \ bool"
where "long_divides p q t \
\<comment> \<open>i\<close> (t \<in> carrier (poly_ring R) \<times> carrier (poly_ring R)) \<and>
\<comment> \<open>ii\<close> (p = (q \<otimes>\<^bsub>poly_ring R\<^esub> (fst t)) \<oplus>\<^bsub>poly_ring R\<^esub> (snd t)) \<and>
\<comment> \<open>iii\<close> (snd t = [] \<or> degree (snd t) < degree q)"
definition (in ring) long_division :: "'a list \ 'a list \ ('a list \ 'a list)"
where "long_division p q = (THE t. long_divides p q t)"
definition (in ring) pdiv :: "'a list \ 'a list \ 'a list" (infixl "pdiv" 65)
where "p pdiv q = (if q = [] then [] else fst (long_division p q))"
definition (in ring) pmod :: "'a list \ 'a list \ 'a list" (infixl "pmod" 65)
where "p pmod q = (if q = [] then p else snd (long_division p q))"
lemma (in ring) long_dividesI:
assumes "b \ carrier (poly_ring R)" and "r \ carrier (poly_ring R)"
and "p = (q \\<^bsub>poly_ring R\<^esub> b) \\<^bsub>poly_ring R\<^esub> r" and "r = [] \ degree r < degree q"
shows "long_divides p q (b, r)"
using assms unfolding long_divides_def by auto
lemma (in domain) exists_long_division:
assumes "subfield K R" and "p \ carrier (K[X])" and "q \ carrier (K[X])" "q \ []"
obtains b r where "b \ carrier (K[X])" and "r \ carrier (K[X])" and "long_divides p q (b, r)"
using subfield_long_division_theorem_shell[OF assms(1-3)] assms(4)
carrier_polynomial_shell[OF subfieldE(1)[OF assms(1)]]
unfolding long_divides_def univ_poly_zero univ_poly_add univ_poly_mult by auto
lemma (in domain) exists_unique_long_division:
assumes "subfield K R" and "p \ carrier (K[X])" and "q \ carrier (K[X])" "q \ []"
shows "\!t. long_divides p q t"
proof -
let ?padd = "\a b. a \\<^bsub>poly_ring R\<^esub> b"
let ?pmult = "\a b. a \\<^bsub>poly_ring R\<^esub> b"
let ?pminus = "\a b. a \\<^bsub>poly_ring R\<^esub> b"
interpret UP: domain "poly_ring R"
using univ_poly_is_domain[OF carrier_is_subring] .
obtain b r where ldiv: "long_divides p q (b, r)"
using exists_long_division[OF assms] by metis
moreover have "(b, r) = (b', r')" if "long_divides p q (b', r')" for b' r'
proof -
have q: "q \ carrier (poly_ring R)" "q \ []"
using assms(3-4) carrier_polynomial[OF subfieldE(1)[OF assms(1)]]
unfolding univ_poly_carrier by auto
hence in_carrier: "q \ carrier (poly_ring R)"
"b \ carrier (poly_ring R)" "r \ carrier (poly_ring R)"
"b' \ carrier (poly_ring R)" "r' \ carrier (poly_ring R)"
using assms(3) that ldiv unfolding long_divides_def by auto
have "?pminus (?padd (?pmult q b) r) r' = ?pminus (?padd (?pmult q b') r') r'"
using ldiv and that unfolding long_divides_def by auto
hence eq: "?padd (?pmult q (?pminus b b')) (?pminus r r') = \\<^bsub>poly_ring R\<^esub>"
using in_carrier by algebra
have "b = b'"
proof (rule ccontr)
assume "b \ b'"
hence pminus: "?pminus b b' \ \\<^bsub>poly_ring R\<^esub>" "?pminus b b' \ carrier (poly_ring R)"
using in_carrier(2,4) by (metis UP.add.inv_closed UP.l_neg UP.minus_eq UP.minus_unique, algebra)
hence degree_ge: "degree (?pmult q (?pminus b b')) \ degree q"
using poly_mult_degree_eq[OF carrier_is_subring, of q "?pminus b b'"] q
unfolding univ_poly_zero univ_poly_carrier univ_poly_mult by simp
have "?pminus b b' = \\<^bsub>poly_ring R\<^esub>" if "?pminus r r' = \\<^bsub>poly_ring R\<^esub>"
using eq pminus(2) q UP.integral univ_poly_zero unfolding that by auto
hence "?pminus r r' \ []"
using pminus(1) unfolding univ_poly_zero by blast
moreover have "?pminus r r' = []" if "r = []" and "r' = []"
using univ_poly_a_inv_def'[OF carrier_is_subring UP.zero_closed] that
unfolding a_minus_def univ_poly_add univ_poly_zero by auto
ultimately have "r \ [] \ r' \ []"
by blast
hence "max (degree r) (degree r') < degree q"
using ldiv and that unfolding long_divides_def by auto
moreover have "degree (?pminus r r') \ max (degree r) (degree r')"
using poly_add_degree[of r "map (a_inv R) r'"]
unfolding a_minus_def univ_poly_add univ_poly_a_inv_def'[OF carrier_is_subring in_carrier(5)]
by auto
ultimately have degree_lt: "degree (?pminus r r') < degree q"
by linarith
have is_poly: "polynomial (carrier R) (?pmult q (?pminus b b'))" "polynomial (carrier R) (?pminus r r')"
using in_carrier pminus(2) unfolding univ_poly_carrier by algebra+
have "degree (?padd (?pmult q (?pminus b b')) (?pminus r r')) = degree (?pmult q (?pminus b b'))"
using poly_add_degree_eq[OF carrier_is_subring is_poly] degree_ge degree_lt
unfolding univ_poly_carrier sym[OF univ_poly_add[of R "carrier R"]] max_def by simp
hence "degree (?padd (?pmult q (?pminus b b')) (?pminus r r')) > 0"
using degree_ge degree_lt by simp
moreover have "degree (?padd (?pmult q (?pminus b b')) (?pminus r r')) = 0"
using eq unfolding univ_poly_zero by simp
ultimately show False by simp
qed
hence "?pminus r r' = \\<^bsub>poly_ring R\<^esub>"
using in_carrier eq by algebra
hence "r = r'"
using in_carrier by (metis UP.add.inv_closed UP.add.right_cancel UP.minus_eq UP.r_neg)
with \<open>b = b'\<close> show ?thesis
by simp
qed
ultimately show ?thesis
by auto
qed
lemma (in domain) long_divisionE:
assumes "subfield K R" and "p \ carrier (K[X])" and "q \ carrier (K[X])" "q \ []"
shows "long_divides p q (p pdiv q, p pmod q)"
using theI'[OF exists_unique_long_division[OF assms]] assms(4)
unfolding pmod_def pdiv_def long_division_def by auto
lemma (in domain) long_divisionI:
assumes "subfield K R" and "p \ carrier (K[X])" and "q \ carrier (K[X])" "q \ []"
shows "long_divides p q (b, r) \ (b, r) = (p pdiv q, p pmod q)"
using exists_unique_long_division[OF assms] long_divisionE[OF assms] by metis
lemma (in domain) long_division_closed:
assumes "subfield K R" and "p \ carrier (K[X])" "q \ carrier (K[X])"
shows "p pdiv q \ carrier (K[X])" and "p pmod q \ carrier (K[X])"
proof -
have "p pdiv q \ carrier (K[X]) \ p pmod q \ carrier (K[X])"
using assms univ_poly_zero_closed[of R] long_divisionI[of K] exists_long_division[OF assms]
by (cases "q = []") (simp add: pdiv_def pmod_def, metis Pair_inject)+
thus "p pdiv q \ carrier (K[X])" and "p pmod q \ carrier (K[X])"
by auto
qed
lemma (in domain) pdiv_pmod:
assumes "subfield K R" and "p \ carrier (K[X])" "q \ carrier (K[X])"
shows "p = (q \\<^bsub>K[X]\<^esub> (p pdiv q)) \\<^bsub>K[X]\<^esub> (p pmod q)"
proof (cases)
interpret UP: ring "K[X]"
using univ_poly_is_ring[OF subfieldE(1)[OF assms(1)]] .
assume "q = []" thus ?thesis
using assms(2) unfolding pdiv_def pmod_def sym[OF univ_poly_zero[of R K]] by simp
next
assume "q \ []" thus ?thesis
using long_divisionE[OF assms] unfolding long_divides_def univ_poly_mult univ_poly_add by simp
qed
lemma (in domain) pmod_degree:
assumes "subfield K R" and "p \ carrier (K[X])" and "q \ carrier (K[X])" "q \ []"
shows "p pmod q = [] \ degree (p pmod q) < degree q"
using long_divisionE[OF assms] unfolding long_divides_def by auto
lemma (in domain) pmod_const:
assumes "subfield K R" and "p \ carrier (K[X])" "q \ carrier (K[X])" and "degree q > degree p"
shows "p pdiv q = []" and "p pmod q = p"
proof -
have "p pdiv q = [] \ p pmod q = p"
proof (cases)
interpret UP: ring "K[X]"
using univ_poly_is_ring[OF subfieldE(1)[OF assms(1)]] .
assume "q \ []"
have "p = (q \\<^bsub>K[X]\<^esub> []) \\<^bsub>K[X]\<^esub> p"
using assms(2-3) unfolding sym[OF univ_poly_zero[of R K]] by simp
moreover have "([], p) \ carrier (poly_ring R) \ carrier (poly_ring R)"
using carrier_polynomial_shell[OF subfieldE(1)[OF assms(1)] assms(2)] by auto
ultimately have "long_divides p q ([], p)"
using assms(4) unfolding long_divides_def univ_poly_mult univ_poly_add by auto
with \<open>q \<noteq> []\<close> show ?thesis
using long_divisionI[OF assms(1-3)] by auto
qed (simp add: pmod_def pdiv_def)
thus "p pdiv q = []" and "p pmod q = p"
by auto
qed
lemma (in domain) long_division_zero:
assumes "subfield K R" and "q \ carrier (K[X])" shows "[] pdiv q = []" and "[] pmod q = []"
proof -
interpret UP: ring "poly_ring R"
using univ_poly_is_ring[OF carrier_is_subring] .
have "[] pdiv q = [] \ [] pmod q = []"
proof (cases)
assume "q \ []"
have "q \ carrier (poly_ring R)"
using carrier_polynomial_shell[OF subfieldE(1)[OF assms(1)] assms(2)] .
hence "long_divides [] q ([], [])"
unfolding long_divides_def sym[OF univ_poly_zero[of R "carrier R"]] by auto
with \<open>q \<noteq> []\<close> show ?thesis
using long_divisionI[OF assms(1) univ_poly_zero_closed assms(2)] by simp
qed (simp add: pmod_def pdiv_def)
thus "[] pdiv q = []" and "[] pmod q = []"
by auto
qed
lemma (in domain) long_division_a_inv:
assumes "subfield K R" and "p \ carrier (K[X])" "q \ carrier (K[X])"
shows "((\\<^bsub>K[X]\<^esub> p) pdiv q) = \\<^bsub>K[X]\<^esub> (p pdiv q)" (is "?pdiv")
and "((\\<^bsub>K[X]\<^esub> p) pmod q) = \\<^bsub>K[X]\<^esub> (p pmod q)" (is "?pmod")
proof -
interpret UP: ring "K[X]"
using univ_poly_is_ring[OF subfieldE(1)[OF assms(1)]] .
have "?pdiv \ ?pmod"
proof (cases)
assume "q = []" thus ?thesis
unfolding pmod_def pdiv_def sym[OF univ_poly_zero[of R K]] by simp
next
assume not_nil: "q \ []"
have "\\<^bsub>K[X]\<^esub> p = \\<^bsub>K[X]\<^esub> ((q \\<^bsub>K[X]\<^esub> (p pdiv q)) \\<^bsub>K[X]\<^esub> (p pmod q))"
using pdiv_pmod[OF assms] by simp
hence "\\<^bsub>K[X]\<^esub> p = (q \\<^bsub>K[X]\<^esub> (\\<^bsub>K[X]\<^esub> (p pdiv q))) \\<^bsub>K[X]\<^esub> (\\<^bsub>K[X]\<^esub> (p pmod q))"
using assms(2-3) long_division_closed[OF assms] by algebra
moreover have "\\<^bsub>K[X]\<^esub> (p pdiv q) \ carrier (K[X])" "\\<^bsub>K[X]\<^esub> (p pmod q) \ carrier (K[X])"
using long_division_closed[OF assms] by algebra+
hence "(\\<^bsub>K[X]\<^esub> (p pdiv q), \\<^bsub>K[X]\<^esub> (p pmod q)) \ carrier (poly_ring R) \ carrier (poly_ring R)"
using carrier_polynomial_shell[OF subfieldE(1)[OF assms(1)]] by auto
moreover have "\\<^bsub>K[X]\<^esub> (p pmod q) = [] \ degree (\\<^bsub>K[X]\<^esub> (p pmod q)) < degree q"
using univ_poly_a_inv_length[OF subfieldE(1)[OF assms(1)]
long_division_closed(2)[OF assms]] pmod_degree[OF assms not_nil]
by auto
ultimately have "long_divides (\\<^bsub>K[X]\<^esub> p) q (\\<^bsub>K[X]\<^esub> (p pdiv q), \\<^bsub>K[X]\<^esub> (p pmod q))"
unfolding long_divides_def univ_poly_mult univ_poly_add by simp
thus ?thesis
using long_divisionI[OF assms(1) UP.a_inv_closed[OF assms(2)] assms(3) not_nil] by simp
qed
thus ?pdiv and ?pmod
by auto
qed
lemma (in domain) long_division_add:
assumes "subfield K R" and "a \ carrier (K[X])" "b \ carrier (K[X])" "q \ carrier (K[X])"
shows "(a \\<^bsub>K[X]\<^esub> b) pdiv q = (a pdiv q) \\<^bsub>K[X]\<^esub> (b pdiv q)" (is "?pdiv")
and "(a \\<^bsub>K[X]\<^esub> b) pmod q = (a pmod q) \\<^bsub>K[X]\<^esub> (b pmod q)" (is "?pmod")
proof -
let ?pdiv_add = "(a pdiv q) \\<^bsub>K[X]\<^esub> (b pdiv q)"
let ?pmod_add = "(a pmod q) \\<^bsub>K[X]\<^esub> (b pmod q)"
interpret UP: ring "K[X]"
using univ_poly_is_ring[OF subfieldE(1)[OF assms(1)]] .
have "?pdiv \ ?pmod"
proof (cases)
assume "q = []" thus ?thesis
using assms(2-3) unfolding pmod_def pdiv_def sym[OF univ_poly_zero[of R K]] by simp
next
note in_carrier = long_division_closed[OF assms(1,2,4)]
long_division_closed[OF assms(1,3,4)]
assume "q \ []"
have "a \\<^bsub>K[X]\<^esub> b = ((q \\<^bsub>K[X]\<^esub> (a pdiv q)) \\<^bsub>K[X]\<^esub> (a pmod q)) \\<^bsub>K[X]\<^esub>
((q \<otimes>\<^bsub>K[X]\<^esub> (b pdiv q)) \<oplus>\<^bsub>K[X]\<^esub> (b pmod q))"
using assms(2-3)[THEN pdiv_pmod[OF assms(1) _ assms(4)]] by simp
hence "a \\<^bsub>K[X]\<^esub> b = (q \\<^bsub>K[X]\<^esub> ?pdiv_add) \\<^bsub>K[X]\<^esub> ?pmod_add"
using assms(4) in_carrier by algebra
moreover have "(?pdiv_add, ?pmod_add) \ carrier (poly_ring R) \ carrier (poly_ring R)"
using in_carrier carrier_polynomial_shell[OF subfieldE(1)[OF assms(1)]] by auto
moreover have "?pmod_add = [] \ degree ?pmod_add < degree q"
proof (cases)
assume "?pmod_add \ []"
hence "a pmod q \ [] \ b pmod q \ []"
using in_carrier(2,4) unfolding sym[OF univ_poly_zero[of R K]] by auto
moreover from \<open>q \<noteq> []\<close>
have "a pmod q = [] \ degree (a pmod q) < degree q" and "b pmod q = [] \ degree (b pmod q) < degree q"
using assms(2-3)[THEN pmod_degree[OF assms(1) _ assms(4)]] by auto
ultimately have "max (degree (a pmod q)) (degree (b pmod q)) < degree q"
by auto
thus ?thesis
using poly_add_degree le_less_trans unfolding univ_poly_add by blast
qed simp
ultimately have "long_divides (a \\<^bsub>K[X]\<^esub> b) q (?pdiv_add, ?pmod_add)"
unfolding long_divides_def univ_poly_mult univ_poly_add by simp
with \<open>q \<noteq> []\<close> show ?thesis
using long_divisionI[OF assms(1) UP.a_closed[OF assms(2-3)] assms(4)] by simp
qed
thus ?pdiv and ?pmod
by auto
qed
lemma (in domain) long_division_add_iff:
assumes "subfield K R"
and "a \ carrier (K[X])" "b \ carrier (K[X])" "c \ carrier (K[X])" "q \ carrier (K[X])"
shows "a pmod q = b pmod q \ (a \\<^bsub>K[X]\<^esub> c) pmod q = (b \\<^bsub>K[X]\<^esub> c) pmod q"
proof -
interpret UP: ring "K[X]"
using univ_poly_is_ring[OF subfieldE(1)[OF assms(1)]] .
show ?thesis
using assms(2-4)[THEN long_division_closed(2)[OF assms(1) _ assms(5)]]
unfolding assms(2-3)[THEN long_division_add(2)[OF assms(1) _ assms(4-5)]] by auto
qed
lemma (in domain) pdivides_iff:
assumes "subfield K R" and "polynomial K p" "polynomial K q"
shows "p pdivides q \ p divides\<^bsub>K[X]\<^esub> q"
proof
show "p divides\<^bsub>K [X]\<^esub> q \ p pdivides q"
using carrier_polynomial[OF subfieldE(1)[OF assms(1)]]
unfolding pdivides_def factor_def univ_poly_mult univ_poly_carrier by auto
next
interpret UP: ring "poly_ring R"
using univ_poly_is_ring[OF carrier_is_subring] .
have in_carrier: "p \ carrier (poly_ring R)" "q \ carrier (poly_ring R)"
using carrier_polynomial[OF subfieldE(1)[OF assms(1)]] assms
unfolding univ_poly_carrier by auto
assume "p pdivides q"
then obtain b where "b \ carrier (poly_ring R)" and "q = p \\<^bsub>poly_ring R\<^esub> b"
unfolding pdivides_def factor_def by blast
show "p divides\<^bsub>K[X]\<^esub> q"
proof (cases)
assume "p = []"
with \<open>b \<in> carrier (poly_ring R)\<close> and \<open>q = p \<otimes>\<^bsub>poly_ring R\<^esub> b\<close> have "q = []"
unfolding univ_poly_mult sym[OF univ_poly_carrier]
using poly_mult_zero(1)[OF polynomial_incl] by simp
with \<open>p = []\<close> show ?thesis
using poly_mult_zero(2)[of "[]"]
unfolding factor_def univ_poly_mult by auto
next
interpret UP: ring "poly_ring R"
using univ_poly_is_ring[OF carrier_is_subring] .
assume "p \ []"
from \<open>p pdivides q\<close> obtain b where "b \<in> carrier (poly_ring R)" and "q = p \<otimes>\<^bsub>poly_ring R\<^esub> b"
unfolding pdivides_def factor_def by blast
moreover have "p \ carrier (poly_ring R)" and "q \ carrier (poly_ring R)"
using assms carrier_polynomial[OF subfieldE(1)[OF assms(1)]] unfolding univ_poly_carrier by auto
ultimately have "q = (p \\<^bsub>poly_ring R\<^esub> b) \\<^bsub>poly_ring R\<^esub> \\<^bsub>poly_ring R\<^esub>"
by algebra
with \<open>b \<in> carrier (poly_ring R)\<close> have "long_divides q p (b, [])"
unfolding long_divides_def univ_poly_zero by auto
with \<open>p \<noteq> []\<close> have "b \<in> carrier (K[X])"
using long_divisionI[of K q p b] long_division_closed[of K q p] assms
unfolding univ_poly_carrier by auto
with \<open>q = p \<otimes>\<^bsub>poly_ring R\<^esub> b\<close> show ?thesis
unfolding factor_def univ_poly_mult by blast
qed
qed
lemma (in domain) pdivides_iff_shell:
assumes "subfield K R" and "p \ carrier (K[X])" "q \ carrier (K[X])"
shows "p pdivides q \ p divides\<^bsub>K[X]\<^esub> q"
using pdivides_iff assms by (simp add: univ_poly_carrier)
lemma (in domain) pmod_zero_iff_pdivides:
assumes "subfield K R" and "p \ carrier (K[X])" "q \ carrier (K[X])"
shows "p pmod q = [] \ q pdivides p"
proof -
interpret UP: domain "K[X]"
using univ_poly_is_domain[OF subfieldE(1)[OF assms(1)]] .
show ?thesis
proof
assume pmod: "p pmod q = []"
have "p pdiv q \ carrier (K[X])" and "p pmod q \ carrier (K[X])"
using long_division_closed[OF assms] by auto
hence "p = q \\<^bsub>K[X]\<^esub> (p pdiv q)"
using pdiv_pmod[OF assms] assms(3) unfolding pmod sym[OF univ_poly_zero[of R K]] by algebra
with \<open>p pdiv q \<in> carrier (K[X])\<close> show "q pdivides p"
unfolding pdivides_iff_shell[OF assms(1,3,2)] factor_def by blast
next
assume "q pdivides p" show "p pmod q = []"
proof (cases)
assume "q = []" with \<open>q pdivides p\<close> show ?thesis
using zero_pdivides unfolding pmod_def by simp
next
assume "q \ []"
from \<open>q pdivides p\<close> obtain r where "r \<in> carrier (K[X])" and "p = q \<otimes>\<^bsub>K[X]\<^esub> r"
unfolding pdivides_iff_shell[OF assms(1,3,2)] factor_def by blast
hence "p = (q \\<^bsub>K[X]\<^esub> r) \\<^bsub>K[X]\<^esub> []"
using assms(2) unfolding sym[OF univ_poly_zero[of R K]] by simp
moreover from \<open>r \<in> carrier (K[X])\<close> have "r \<in> carrier (poly_ring R)"
using carrier_polynomial_shell[OF subfieldE(1)[OF assms(1)]] by auto
ultimately have "long_divides p q (r, [])"
unfolding long_divides_def univ_poly_mult univ_poly_add by auto
with \<open>q \<noteq> []\<close> show ?thesis
using long_divisionI[OF assms] by simp
qed
qed
qed
lemma (in domain) same_pmod_iff_pdivides:
assumes "subfield K R" and "a \ carrier (K[X])" "b \ carrier (K[X])" "q \ carrier (K[X])"
shows "a pmod q = b pmod q \ q pdivides (a \\<^bsub>K[X]\<^esub> b)"
proof -
interpret UP: domain "K[X]"
using univ_poly_is_domain[OF subfieldE(1)[OF assms(1)]] .
have "a pmod q = b pmod q \ (a \\<^bsub>K[X]\<^esub> (\\<^bsub>K[X]\<^esub> b)) pmod q = (b \\<^bsub>K[X]\<^esub> (\\<^bsub>K[X]\<^esub> b)) pmod q"
using long_division_add_iff[OF assms(1-3) UP.a_inv_closed[OF assms(3)] assms(4)] .
also have " ... \ (a \\<^bsub>K[X]\<^esub> b) pmod q = \\<^bsub>K[X]\<^esub> pmod q"
using assms(2-3) by algebra
also have " ... \ q pdivides (a \\<^bsub>K[X]\<^esub> b)"
using pmod_zero_iff_pdivides[OF assms(1) UP.minus_closed[OF assms(2-3)] assms(4)]
unfolding univ_poly_zero long_division_zero(2)[OF assms(1,4)] .
finally show ?thesis .
qed
lemma (in domain) pdivides_imp_degree_le:
assumes "subring K R" and "p \ carrier (K[X])" "q \ carrier (K[X])" "q \ []"
shows "p pdivides q \ degree p \ degree q"
proof -
assume "p pdivides q"
then obtain r where r: "polynomial (carrier R) r" "q = poly_mult p r"
unfolding pdivides_def factor_def univ_poly_mult univ_poly_carrier by blast
moreover have p: "polynomial (carrier R) p"
using assms(2) carrier_polynomial[OF assms(1)] unfolding univ_poly_carrier by auto
moreover have "p \ []" and "r \ []"
using poly_mult_zero(2)[OF polynomial_incl[OF p]] r(2) assms(4) by auto
ultimately show "degree p \ degree q"
using poly_mult_degree_eq[OF carrier_is_subring, of p r] by auto
qed
lemma (in domain) pprimeE:
assumes "subfield K R" "p \ carrier (K[X])" "pprime K p"
shows "p \ []" "p \ Units (K[X])"
and "\q r. \ q \ carrier (K[X]); r \ carrier (K[X])\ \
p pdivides (q \<otimes>\<^bsub>K[X]\<^esub> r) \<Longrightarrow> p pdivides q \<or> p pdivides r"
using assms(2-3) poly_mult_closed[OF subfieldE(1)[OF assms(1)]] pdivides_iff[OF assms(1)]
unfolding ring_prime_def prime_def
by (auto simp add: univ_poly_mult univ_poly_carrier univ_poly_zero)
lemma (in domain) pprimeI:
assumes "subfield K R" "p \ carrier (K[X])" "p \ []" "p \ Units (K[X])"
and "\q r. \ q \ carrier (K[X]); r \ carrier (K[X])\ \
p pdivides (q \<otimes>\<^bsub>K[X]\<^esub> r) \<Longrightarrow> p pdivides q \<or> p pdivides r"
shows "pprime K p"
using assms(2-5) poly_mult_closed[OF subfieldE(1)[OF assms(1)]] pdivides_iff[OF assms(1)]
unfolding ring_prime_def prime_def
by (auto simp add: univ_poly_mult univ_poly_carrier univ_poly_zero)
lemma (in domain) associated_polynomials_iff:
assumes "subfield K R" and "p \ carrier (K[X])" "q \ carrier (K[X])"
shows "p \\<^bsub>K[X]\<^esub> q \ (\k \ K - { \ }. p = [ k ] \\<^bsub>K[X]\<^esub> q)"
using domain.ring_associated_iff[OF univ_poly_is_domain[OF subfieldE(1)[OF assms(1)]] assms(2-3)]
unfolding univ_poly_units[OF assms(1)] by auto
corollary (in domain) associated_polynomials_imp_same_length: (* stronger than "imp_same_degree" *)
assumes "subring K R" and "p \ carrier (K[X])" and "q \ carrier (K[X])"
shows "p \\<^bsub>K[X]\<^esub> q \ length p = length q"
proof -
{ fix p q
assume p: "p \ carrier (K[X])" and q: "q \ carrier (K[X])" and "p \\<^bsub>K[X]\<^esub> q"
have "length p \ length q"
proof (cases "q = []")
case True with \<open>p \<sim>\<^bsub>K[X]\<^esub> q\<close> have "p = []"
unfolding associated_def True factor_def univ_poly_def by auto
thus ?thesis
using True by simp
next
case False
from \<open>p \<sim>\<^bsub>K[X]\<^esub> q\<close> have "p divides\<^bsub>K [X]\<^esub> q"
unfolding associated_def by simp
hence "p divides\<^bsub>poly_ring R\<^esub> q"
using carrier_polynomial[OF assms(1)]
unfolding factor_def univ_poly_carrier univ_poly_mult by auto
with \<open>q \<noteq> []\<close> have "degree p \<le> degree q"
using pdivides_imp_degree_le[OF assms(1) p q] unfolding pdivides_def by simp
with \<open>q \<noteq> []\<close> show ?thesis
by (cases "p = []", auto simp add: Suc_leI le_diff_iff)
qed
} note aux_lemma = this
interpret UP: domain "K[X]"
using univ_poly_is_domain[OF assms(1)] .
assume "p \\<^bsub>K[X]\<^esub> q" thus ?thesis
using aux_lemma[OF assms(2-3)] aux_lemma[OF assms(3,2) UP.associated_sym] by simp
qed
lemma (in ring) divides_pirreducible_condition:
assumes "pirreducible K q" and "p \ carrier (K[X])"
shows "p divides\<^bsub>K[X]\<^esub> q \ p \ Units (K[X]) \ p \\<^bsub>K[X]\<^esub> q"
using divides_irreducible_condition[of "K[X]" q p] assms
unfolding ring_irreducible_def by auto
subsection \<open>Polynomial Power\<close>
lemma (in domain) polynomial_pow_not_zero:
assumes "p \ carrier (poly_ring R)" and "p \ []"
shows "p [^]\<^bsub>poly_ring R\<^esub> (n::nat) \ []"
proof -
interpret UP: domain "poly_ring R"
using univ_poly_is_domain[OF carrier_is_subring] .
from assms UP.integral show ?thesis
unfolding sym[OF univ_poly_zero[of R "carrier R"]]
by (induction n, auto)
qed
lemma (in domain) subring_polynomial_pow_not_zero:
assumes "subring K R" and "p \ carrier (K[X])" and "p \ []"
shows "p [^]\<^bsub>K[X]\<^esub> (n::nat) \ []"
using domain.polynomial_pow_not_zero[OF subring_is_domain, of K p n] assms
unfolding univ_poly_consistent[OF assms(1)] by simp
lemma (in domain) polynomial_pow_degree:
assumes "p \ carrier (poly_ring R)"
shows "degree (p [^]\<^bsub>poly_ring R\<^esub> n) = n * degree p"
proof -
interpret UP: domain "poly_ring R"
using univ_poly_is_domain[OF carrier_is_subring] .
show ?thesis
proof (induction n)
case 0 thus ?case
using UP.nat_pow_0 unfolding univ_poly_one by auto
next
let ?ppow = "\n. p [^]\<^bsub>poly_ring R\<^esub> n"
case (Suc n) thus ?case
proof (cases "p = []")
case True thus ?thesis
using univ_poly_zero[of R "carrier R"] UP.r_null assms by auto
next
case False
hence "?ppow n \ carrier (poly_ring R)" and "?ppow n \ []" and "p \ []"
using polynomial_pow_not_zero[of p n] assms by (auto simp add: univ_poly_one)
thus ?thesis
using poly_mult_degree_eq[OF carrier_is_subring, of "?ppow n" p] Suc assms
unfolding univ_poly_carrier univ_poly_zero
by (auto simp add: add.commute univ_poly_mult)
qed
qed
qed
lemma (in domain) subring_polynomial_pow_degree:
assumes "subring K R" and "p \ carrier (K[X])"
shows "degree (p [^]\<^bsub>K[X]\<^esub> n) = n * degree p"
using domain.polynomial_pow_degree[OF subring_is_domain, of K p n] assms
unfolding univ_poly_consistent[OF assms(1)] by simp
lemma (in domain) polynomial_pow_division:
assumes "p \ carrier (poly_ring R)" and "(n::nat) \ m"
shows "(p [^]\<^bsub>poly_ring R\<^esub> n) pdivides (p [^]\<^bsub>poly_ring R\<^esub> m)"
proof -
interpret UP: domain "poly_ring R"
using univ_poly_is_domain[OF carrier_is_subring] .
let ?ppow = "\n. p [^]\<^bsub>poly_ring R\<^esub> n"
have "?ppow n \\<^bsub>poly_ring R\<^esub> ?ppow k = ?ppow (n + k)" for k
using assms(1) by (simp add: UP.nat_pow_mult)
thus ?thesis
using dividesI[of "?ppow (m - n)" "poly_ring R" "?ppow m" "?ppow n"] assms
unfolding pdivides_def by auto
qed
lemma (in domain) subring_polynomial_pow_division:
assumes "subring K R" and "p \ carrier (K[X])" and "(n::nat) \ m"
shows "(p [^]\<^bsub>K[X]\<^esub> n) divides\<^bsub>K[X]\<^esub> (p [^]\<^bsub>K[X]\<^esub> m)"
using domain.polynomial_pow_division[OF subring_is_domain, of K p n m] assms
unfolding univ_poly_consistent[OF assms(1)] pdivides_def by simp
lemma (in domain) pirreducible_pow_pdivides_iff:
assumes "subfield K R" "p \ carrier (K[X])" "q \ carrier (K[X])" "r \ carrier (K[X])"
and "pirreducible K p" and "\ (p pdivides q)"
shows "(p [^]\<^bsub>K[X]\<^esub> (n :: nat)) pdivides (q \\<^bsub>K[X]\<^esub> r) \ (p [^]\<^bsub>K[X]\<^esub> n) pdivides r"
proof -
interpret UP: principal_domain "K[X]"
using univ_poly_is_principal[OF assms(1)] .
show ?thesis
proof (cases "r = []")
case True with \<open>q \<in> carrier (K[X])\<close> have "q \<otimes>\<^bsub>K[X]\<^esub> r = []" and "r = []"
unfolding sym[OF univ_poly_zero[of R K]] by auto
thus ?thesis
using pdivides_zero[OF subfieldE(1),of K] assms by auto
next
case False then have not_zero: "p \ []" "q \ []" "r \ []" "q \\<^bsub>K[X]\<^esub> r \ []"
using subfieldE(1) pdivides_zero[OF _ assms(2)] assms(1-2,5-6) pirreducibleE(1)
UP.integral_iff[OF assms(3-4)] univ_poly_zero[of R K] by auto
from \<open>p \<noteq> []\<close>
have ppow: "p [^]\<^bsub>K[X]\<^esub> (n :: nat) \ []" "p [^]\<^bsub>K[X]\<^esub> (n :: nat) \ carrier (K[X])"
using subring_polynomial_pow_not_zero[OF subfieldE(1)] assms(1-2) by auto
have not_pdiv: "\ (p divides\<^bsub>mult_of (K[X])\<^esub> q)"
using assms(6) pdivides_iff_shell[OF assms(1-3)] unfolding pdivides_def by auto
have prime: "prime (mult_of (K[X])) p"
using assms(5) pprime_iff_pirreducible[OF assms(1-2)]
unfolding sym[OF UP.prime_eq_prime_mult[OF assms(2)]] ring_prime_def by simp
have "a pdivides b \ a divides\<^bsub>mult_of (K[X])\<^esub> b"
if "a \ carrier (K[X])" "a \ \\<^bsub>K[X]\<^esub>" "b \ carrier (K[X])" "b \ \\<^bsub>K[X]\<^esub>" for a b
using that UP.divides_imp_divides_mult[of a b] divides_mult_imp_divides[of "K[X]" a b]
unfolding pdivides_iff_shell[OF assms(1) that(1,3)] by blast
thus ?thesis
using UP.mult_of.prime_pow_divides_iff[OF _ _ _ prime not_pdiv, of r] ppow not_zero assms(2-4)
unfolding nat_pow_mult_of carrier_mult_of mult_mult_of sym[OF univ_poly_zero[of R K]]
by (metis DiffI UP.m_closed singletonD)
qed
qed
lemma (in domain) subring_degree_one_imp_pirreducible:
assumes "subring K R" and "a \ Units (R \ carrier := K \)" and "b \ K"
shows "pirreducible K [ a, b ]"
proof (rule pirreducibleI[OF assms(1)])
have "a \ K" and "a \ \"
using assms(2) subringE(1)[OF assms(1)] unfolding Units_def by auto
thus "[ a, b ] \ carrier (K[X])" and "[ a, b ] \ []" and "[ a, b ] \ Units (K [X])"
using univ_poly_units_incl[OF assms(1)] assms(2-3)
unfolding sym[OF univ_poly_carrier] polynomial_def by auto
next
interpret UP: domain "K[X]"
using univ_poly_is_domain[OF assms(1)] .
{ fix q r
assume q: "q \ carrier (K[X])" and r: "r \ carrier (K[X])" and "[ a, b ] = q \\<^bsub>K[X]\<^esub> r"
hence not_zero: "q \ []" "r \ []"
by (metis UP.integral_iff list.distinct(1) univ_poly_zero)+
have "degree (q \\<^bsub>K[X]\<^esub> r) = degree q + degree r"
using not_zero poly_mult_degree_eq[OF assms(1)] q r
by (simp add: univ_poly_carrier univ_poly_mult)
with sym[OF \<open>[ a, b ] = q \<otimes>\<^bsub>K[X]\<^esub> r\<close>] have "degree q + degree r = 1" and "q \<noteq> []" "r \<noteq> []"
using not_zero by auto
} note aux_lemma1 = this
{ fix q r
assume q: "q \ carrier (K[X])" "q \ []" and r: "r \ carrier (K[X])" "r \ []"
and "[ a, b ] = q \\<^bsub>K[X]\<^esub> r" and "degree q = 1" and "degree r = 0"
hence "length q = Suc (Suc 0)" and "length r = Suc 0"
by (linarith, metis add.right_neutral add_eq_if length_0_conv)
from \<open>length q = Suc (Suc 0)\<close> obtain c d where q_def: "q = [ c, d ]"
by (metis length_0_conv length_Cons list.exhaust nat.inject)
from \<open>length r = Suc 0\<close> obtain e where r_def: "r = [ e ]"
by (metis length_0_conv length_Suc_conv)
from \<open>r = [ e ]\<close> and \<open>q = [ c, d ]\<close>
have c: "c \ K" "c \ \" and d: "d \ K" and e: "e \ K" "e \ \"
using r q subringE(1)[OF assms(1)] unfolding sym[OF univ_poly_carrier] polynomial_def by auto
with sym[OF \<open>[ a, b ] = q \<otimes>\<^bsub>K[X]\<^esub> r\<close>] have "a = c \<otimes> e"
using poly_mult_lead_coeff[OF assms(1), of q r]
unfolding polynomial_def sym[OF univ_poly_mult[of R K]] r_def q_def by auto
obtain inv_a where a: "a \ K" and inv_a: "inv_a \ K" "a \ inv_a = \" "inv_a \ a = \"
using assms(2) unfolding Units_def by auto
hence "a \ \" and "inv_a \ \"
using subringE(1)[OF assms(1)] integral_iff by auto
with \<open>c \<in> K\<close> and \<open>c \<noteq> \<zero>\<close> have in_carrier: "[ c \<otimes> inv_a ] \<in> carrier (K[X])"
using subringE(1,6)[OF assms(1)] inv_a integral
unfolding sym[OF univ_poly_carrier] polynomial_def
by (auto, meson subsetD)
moreover have "[ c \ inv_a ] \\<^bsub>K[X]\<^esub> r = [ \ ]"
using \<open>a = c \<otimes> e\<close> a inv_a c e subsetD[OF subringE(1)[OF assms(1)]]
unfolding r_def univ_poly_mult by (auto) (simp add: m_assoc m_lcomm integral_iff)+
ultimately have "r \ Units (K[X])"
using r(1) UP.m_comm[OF in_carrier r(1)] unfolding sym[OF univ_poly_one[of R K]] Units_def by auto
} note aux_lemma2 = this
fix q r
assume q: "q \ carrier (K[X])" and r: "r \ carrier (K[X])" and qr: "[ a, b ] = q \\<^bsub>K[X]\<^esub> r"
thus "q \ Units (K[X]) \ r \ Units (K[X])"
using aux_lemma1[OF q r qr] aux_lemma2[of q r] aux_lemma2[of r q] UP.m_comm add_is_1 by auto
qed
lemma (in domain) degree_one_imp_pirreducible:
assumes "subfield K R" and "p \ carrier (K[X])" and "degree p = 1"
shows "pirreducible K p"
proof -
from \<open>degree p = 1\<close> have "length p = Suc (Suc 0)"
by simp
then obtain a b where p: "p = [ a, b ]"
by (metis length_0_conv length_Cons nat.inject neq_Nil_conv)
with \<open>p \<in> carrier (K[X])\<close> show ?thesis
using subring_degree_one_imp_pirreducible[OF subfieldE(1)[OF assms(1)], of a b]
subfield.subfield_Units[OF assms(1)]
unfolding sym[OF univ_poly_carrier] polynomial_def by auto
qed
lemma (in ring) degree_oneE[elim]:
assumes "p \ carrier (K[X])" and "degree p = 1"
and "\a b. \ a \ K; a \ \; b \ K; p = [ a, b ] \ \ P"
shows P
proof -
from \<open>degree p = 1\<close> have "length p = Suc (Suc 0)"
by simp
then obtain a b where "p = [ a, b ]"
by (metis length_0_conv length_Cons nat.inject neq_Nil_conv)
with \<open>p \<in> carrier (K[X])\<close> have "a \<in> K" and "a \<noteq> \<zero>" and "b \<in> K"
unfolding sym[OF univ_poly_carrier] polynomial_def by auto
with \<open>p = [ a, b ]\<close> show ?thesis
using assms(3) by simp
qed
lemma (in domain) subring_degree_one_associatedI:
assumes "subring K R" and "a \ K" "a' \ K" and "b \ K" and "a \ a' = \"
shows "[ a , b ] \\<^bsub>K[X]\<^esub> [ \, a' \ b ]"
proof -
from \<open>a \<otimes> a' = \<one>\<close> have not_zero: "a \<noteq> \<zero>" "a' \<noteq> \<zero>"
using subringE(1)[OF assms(1)] assms(2-3) by auto
hence "[ a, b ] = [ a ] \\<^bsub>K[X]\<^esub> [ \, a' \ b ]"
using assms(2-4)[THEN subsetD[OF subringE(1)[OF assms(1)]]] assms(5) m_assoc
unfolding univ_poly_mult by fastforce
moreover have "[ a, b ] \ carrier (K[X])" and "[ \, a' \ b ] \ carrier (K[X])"
using subringE(1,3,6)[OF assms(1)] not_zero one_not_zero assms
unfolding sym[OF univ_poly_carrier] polynomial_def by auto
moreover have "[ a ] \ Units (K[X])"
proof -
from \<open>a \<noteq> \<zero>\<close> and \<open>a' \<noteq> \<zero>\<close> have "[ a ] \<in> carrier (K[X])" and "[ a' ] \<in> carrier (K[X])"
using assms(2-3) unfolding sym[OF univ_poly_carrier] polynomial_def by auto
moreover have "a' \ a = \"
using subsetD[OF subringE(1)[OF assms(1)]] assms m_comm by simp
hence "[ a ] \\<^bsub>K[X]\<^esub> [ a' ] = [ \ ]" and "[ a' ] \\<^bsub>K[X]\<^esub> [ a ] = [ \ ]"
using assms unfolding univ_poly_mult by auto
ultimately show ?thesis
unfolding sym[OF univ_poly_one[of R K]] Units_def by blast
qed
ultimately show ?thesis
using domain.ring_associated_iff[OF univ_poly_is_domain[OF assms(1)]] by blast
qed
lemma (in domain) degree_one_associatedI:
assumes "subfield K R" and "p \ carrier (K[X])" and "degree p = 1"
shows "p \\<^bsub>K[X]\<^esub> [ \, inv (lead_coeff p) \ (const_term p) ]"
proof -
from \<open>p \<in> carrier (K[X])\<close> and \<open>degree p = 1\<close>
obtain a b where "p = [ a, b ]" and "a \ K" "a \ \" and "b \ K"
by auto
thus ?thesis
using subring_degree_one_associatedI[OF subfieldE(1)[OF assms(1)]]
subfield_m_inv[OF assms(1)] subsetD[OF subfieldE(3)[OF assms(1)]]
unfolding const_term_def
by auto
qed
subsection \<open>Ideals\<close>
lemma (in domain) exists_unique_gen:
assumes "subfield K R" "ideal I (K[X])" "I \ { [] }"
shows "\!p \ carrier (K[X]). lead_coeff p = \ \ I = PIdl\<^bsub>K[X]\<^esub> p"
(is "\!p. ?generator p")
proof -
interpret UP: principal_domain "K[X]"
using univ_poly_is_principal[OF assms(1)] .
obtain q where q: "q \ carrier (K[X])" "I = PIdl\<^bsub>K[X]\<^esub> q"
using UP.exists_gen[OF assms(2)] by blast
hence not_nil: "q \ []"
using UP.genideal_zero UP.cgenideal_eq_genideal[OF UP.zero_closed] assms(3)
by (auto simp add: univ_poly_zero)
hence "lead_coeff q \ K - { \ }"
using q(1) unfolding univ_poly_def polynomial_def by auto
hence inv_lc_q: "inv (lead_coeff q) \ K - { \ }" "inv (lead_coeff q) \ lead_coeff q = \"
using subfield_m_inv[OF assms(1)] by auto
define p where "p = [ inv (lead_coeff q) ] \\<^bsub>K[X]\<^esub> q"
have is_poly: "polynomial K [ inv (lead_coeff q) ]" "polynomial K q"
using inv_lc_q(1) q(1) unfolding univ_poly_def polynomial_def by auto
hence in_carrier: "p \ carrier (K[X])"
using UP.m_closed unfolding univ_poly_carrier p_def by simp
have lc_p: "lead_coeff p = \"
using poly_mult_lead_coeff[OF subfieldE(1)[OF assms(1)] is_poly _ not_nil] inv_lc_q(2)
unfolding p_def univ_poly_mult[of R K] by simp
moreover have PIdl_p: "I = PIdl\<^bsub>K[X]\<^esub> p"
using UP.associated_iff_same_ideal[OF in_carrier q(1)] q(2) inv_lc_q(1) p_def
associated_polynomials_iff[OF assms(1) in_carrier q(1)]
by auto
ultimately have "?generator p"
using in_carrier by simp
moreover
have "\r. \ r \ carrier (K[X]); lead_coeff r = \; I = PIdl\<^bsub>K[X]\<^esub> r \ \ r = p"
proof -
fix r assume r: "r \ carrier (K[X])" "lead_coeff r = \" "I = PIdl\<^bsub>K[X]\<^esub> r"
have "subring K R"
by (simp add: \<open>subfield K R\<close> subfieldE(1))
obtain k where k: "k \ K - { \ }" "r = [ k ] \\<^bsub>K[X]\<^esub> p"
using UP.associated_iff_same_ideal[OF r(1) in_carrier] PIdl_p r(3)
associated_polynomials_iff[OF assms(1) r(1) in_carrier]
by auto
hence "polynomial K [ k ]"
unfolding polynomial_def by simp
moreover have "p \ []"
using not_nil UP.associated_iff_same_ideal[OF in_carrier q(1)] q(2) PIdl_p
associated_polynomials_imp_same_length[OF \<open>subring K R\<close> in_carrier q(1)] by auto
ultimately have "lead_coeff r = k \ (lead_coeff p)"
using poly_mult_lead_coeff[OF subfieldE(1)[OF assms(1)]] in_carrier k(2)
unfolding univ_poly_def by (auto simp del: poly_mult.simps)
hence "k = \"
using lc_p r(2) k(1) subfieldE(3)[OF assms(1)] by auto
hence "r = map ((\) \) p"
using poly_mult_const(1)[OF subfieldE(1)[OF assms(1)] _ k(1), of p] in_carrier
unfolding k(2) univ_poly_carrier[of R K] univ_poly_mult[of R K] by auto
moreover have "set p \ carrier R"
using polynomial_in_carrier[OF subfieldE(1)[OF assms(1)]]
in_carrier univ_poly_carrier[of R K] by auto
hence "map ((\) \) p = p"
by (induct p) (auto)
ultimately show "r = p" by simp
qed
ultimately show ?thesis by blast
qed
proposition (in domain) exists_unique_pirreducible_gen:
assumes "subfield K R" "ring_hom_ring (K[X]) R h"
and "a_kernel (K[X]) R h \ { [] }" "a_kernel (K[X]) R h \ carrier (K[X])"
shows "\!p \ carrier (K[X]). pirreducible K p \ lead_coeff p = \ \ a_kernel (K[X]) R h = PIdl\<^bsub>K[X]\<^esub> p"
(is "\!p. ?generator p")
proof -
interpret UP: principal_domain "K[X]"
using univ_poly_is_principal[OF assms(1)] .
have "ideal (a_kernel (K[X]) R h) (K[X])"
using ring_hom_ring.kernel_is_ideal[OF assms(2)] .
then obtain p
where p: "p \ carrier (K[X])" "lead_coeff p = \" "a_kernel (K[X]) R h = PIdl\<^bsub>K[X]\<^esub> p"
and unique:
"\q. \ q \ carrier (K[X]); lead_coeff q = \; a_kernel (K[X]) R h = PIdl\<^bsub>K[X]\<^esub> q \ \ q = p"
using exists_unique_gen[OF assms(1) _ assms(3)] by metis
have "p \ carrier (K[X]) - { [] }"
using UP.genideal_zero UP.cgenideal_eq_genideal[OF UP.zero_closed] assms(3) p(1,3)
by (auto simp add: univ_poly_zero)
hence "pprime K p"
using ring_hom_ring.primeideal_vimage[OF assms(2) UP.is_cring zeroprimeideal]
UP.primeideal_iff_prime[of p]
unfolding univ_poly_zero sym[OF p(3)] a_kernel_def' by simp
hence "pirreducible K p"
using pprime_iff_pirreducible[OF assms(1) p(1)] by simp
thus ?thesis
using p unique by metis
qed
lemma (in domain) cgenideal_pirreducible:
assumes "subfield K R" and "p \ carrier (K[X])" "pirreducible K p"
shows "\ pirreducible K q; q \ PIdl\<^bsub>K[X]\<^esub> p \ \ p \\<^bsub>K[X]\<^esub> q"
proof -
interpret UP: principal_domain "K[X]"
using univ_poly_is_principal[OF assms(1)] .
assume q: "pirreducible K q" "q \ PIdl\<^bsub>K[X]\<^esub> p"
hence in_carrier: "q \ carrier (K[X])"
using additive_subgroup.a_subset[OF ideal.axioms(1)[OF UP.cgenideal_ideal[OF assms(2)]]] by auto
hence "p divides\<^bsub>K[X]\<^esub> q"
by (meson q assms(2) UP.cgenideal_ideal UP.cgenideal_minimal UP.to_contain_is_to_divide)
then obtain r where r: "r \ carrier (K[X])" "q = p \\<^bsub>K[X]\<^esub> r"
by auto
hence "r \ Units (K[X])"
using pirreducibleE(3)[OF _ in_carrier q(1) assms(2) r(1)] subfieldE(1)[OF assms(1)]
pirreducibleE(2)[OF _ assms(2-3)] by auto
thus "p \\<^bsub>K[X]\<^esub> q"
using UP.ring_associated_iff[OF in_carrier assms(2)] r(2) UP.associated_sym
unfolding UP.m_comm[OF assms(2) r(1)] by auto
qed
subsection \<open>Roots and Multiplicity\<close>
definition (in ring) is_root :: "'a list \ 'a \ bool"
where "is_root p x \ (x \ carrier R \ eval p x = \ \ p \ [])"
definition (in ring) alg_mult :: "'a list \ 'a \ nat"
where "alg_mult p x =
(if p = [] then 0 else
(if x \<in> carrier R then Greatest (\<lambda> n. ([ \<one>, \<ominus> x ] [^]\<^bsub>poly_ring R\<^esub> n) pdivides p) else 0))"
definition (in ring) roots :: "'a list \ 'a multiset"
where "roots p = Abs_multiset (alg_mult p)"
definition (in ring) roots_on :: "'a set \ 'a list \ 'a multiset"
where "roots_on K p = roots p \# mset_set K"
definition (in ring) splitted :: "'a list \ bool"
where "splitted p \ size (roots p) = degree p"
definition (in ring) splitted_on :: "'a set \ 'a list \ bool"
where "splitted_on K p \ size (roots_on K p) = degree p"
lemma (in domain) pdivides_imp_root_sharing:
assumes "p \ carrier (poly_ring R)" "p pdivides q" and "a \ carrier R"
shows "eval p a = \ \ eval q a = \"
proof -
from \<open>p pdivides q\<close> obtain r where r: "q = p \<otimes>\<^bsub>poly_ring R\<^esub> r" "r \<in> carrier (poly_ring R)"
unfolding pdivides_def factor_def by auto
hence "eval q a = (eval p a) \ (eval r a)"
using ring_hom_memE(2)[OF eval_is_hom[OF carrier_is_subring assms(3)] assms(1) r(2)] by simp
thus "eval p a = \ \ eval q a = \"
using ring_hom_memE(1)[OF eval_is_hom[OF carrier_is_subring assms(3)] r(2)] by auto
qed
lemma (in domain) degree_one_root:
assumes "subfield K R" and "p \ carrier (K[X])" and "degree p = 1"
shows "eval p (\ (inv (lead_coeff p) \ (const_term p))) = \"
and "inv (lead_coeff p) \ (const_term p) \ K"
proof -
from \<open>degree p = 1\<close> have "length p = Suc (Suc 0)"
by simp
then obtain a b where p: "p = [ a, b ]"
by (metis (no_types, hide_lams) Suc_length_conv length_0_conv)
hence "a \ K - { \ }" "b \ K" and in_carrier: "a \ carrier R" "b \ carrier R"
using assms(2) subfieldE(3)[OF assms(1)] unfolding sym[OF univ_poly_carrier] polynomial_def by auto
hence inv_a: "inv a \ carrier R" "a \ inv a = \" and "inv a \ K"
using subfield_m_inv(1-2)[OF assms(1), of a] subfieldE(3)[OF assms(1)] by auto
hence "eval p (\ (inv a \ b)) = a \ (\ (inv a \ b)) \ b"
using in_carrier unfolding p by simp
also have " ... = \ (a \ (inv a \ b)) \ b"
using inv_a in_carrier by (simp add: r_minus)
also have " ... = \"
using in_carrier(2) unfolding sym[OF m_assoc[OF in_carrier(1) inv_a(1) in_carrier(2)]] inv_a(2) by algebra
finally have "eval p (\ (inv a \ b)) = \" .
moreover have ct: "const_term p = b"
using in_carrier unfolding p const_term_def by auto
ultimately show "eval p (\ (inv (lead_coeff p) \ (const_term p))) = \"
unfolding p by simp
from \<open>inv a \<in> K\<close> and \<open>b \<in> K\<close>
show "inv (lead_coeff p) \ (const_term p) \ K"
using p subringE(6)[OF subfieldE(1)[OF assms(1)]] unfolding ct by auto
qed
lemma (in domain) is_root_imp_pdivides:
assumes "p \ carrier (poly_ring R)"
shows "is_root p x \ [ \, \ x ] pdivides p"
proof -
let ?b = "[ \ , \ x ]"
interpret UP: domain "poly_ring R"
using univ_poly_is_domain[OF carrier_is_subring] .
assume "is_root p x" hence x: "x \ carrier R" and is_root: "eval p x = \"
unfolding is_root_def by auto
hence b: "?b \ carrier (poly_ring R)"
unfolding sym[OF univ_poly_carrier] polynomial_def by auto
then obtain q r where q: "q \ carrier (poly_ring R)" and r: "r \ carrier (poly_ring R)"
and long_divides: "p = (?b \\<^bsub>poly_ring R\<^esub> q) \\<^bsub>poly_ring R\<^esub> r" "r = [] \ degree r < degree ?b"
using long_division_theorem[OF carrier_is_subring, of p ?b] assms by (auto simp add: univ_poly_carrier)
show ?thesis
proof (cases "r = []")
case True then have "r = \\<^bsub>poly_ring R\<^esub>"
unfolding univ_poly_zero[of R "carrier R"] .
thus ?thesis
using long_divides(1) q r b dividesI[OF q, of p ?b] by (simp add: pdivides_def)
next
case False then have "length r = Suc 0"
using long_divides(2) le_SucE by fastforce
then obtain a where "r = [ a ]" and a: "a \ carrier R" and "a \ \"
using r unfolding sym[OF univ_poly_carrier] polynomial_def
by (metis False length_0_conv length_Suc_conv list.sel(1) list.set_sel(1) subset_code(1))
have "eval p x = ((eval ?b x) \ (eval q x)) \ (eval r x)"
using long_divides(1) ring_hom_memE[OF eval_is_hom[OF carrier_is_subring x]] by (simp add: b q r)
also have " ... = eval r x"
using ring_hom_memE[OF eval_is_hom[OF carrier_is_subring x]] x b q r by (auto, algebra)
finally have "a = \"
using a unfolding \<open>r = [ a ]\<close> is_root by simp
with \<open>a \<noteq> \<zero>\<close> have False .. thus ?thesis ..
qed
qed
lemma (in domain) pdivides_imp_is_root:
assumes "p \ []" and "x \ carrier R"
shows "[ \, \ x ] pdivides p \ is_root p x"
proof -
assume "[ \, \ x ] pdivides p"
then obtain q where q: "q \ carrier (poly_ring R)" and pdiv: "p = [ \, \ x ] \\<^bsub>poly_ring R\<^esub> q"
unfolding pdivides_def by auto
moreover have "[ \, \ x ] \ carrier (poly_ring R)"
using assms(2) unfolding sym[OF univ_poly_carrier] polynomial_def by simp
ultimately have "eval p x = \"
using ring_hom_memE[OF eval_is_hom[OF carrier_is_subring, of x]] assms(2) by (auto, algebra)
with \<open>p \<noteq> []\<close> and \<open>x \<in> carrier R\<close> show "is_root p x"
unfolding is_root_def by simp
qed
lemma (in domain) associated_polynomials_imp_same_is_root:
assumes "p \ carrier (poly_ring R)" and "q \ carrier (poly_ring R)" and "p \\<^bsub>poly_ring R\<^esub> q"
shows "is_root p x \ is_root q x"
proof (cases "p = []")
case True with \<open>p \<sim>\<^bsub>poly_ring R\<^esub> q\<close> have "q = []"
unfolding associated_def True factor_def univ_poly_def by auto
thus ?thesis
using True unfolding is_root_def by simp
next
case False
interpret UP: domain "poly_ring R"
using univ_poly_is_domain[OF carrier_is_subring] .
{ fix p q
assume p: "p \ carrier (poly_ring R)" and q: "q \ carrier (poly_ring R)" and pq: "p \\<^bsub>poly_ring R\<^esub> q"
have "is_root p x \ is_root q x"
proof -
assume is_root: "is_root p x"
then have "[ \, \ x ] pdivides p" and "p \ []" and "x \ carrier R"
using is_root_imp_pdivides[OF p] unfolding is_root_def by auto
moreover have "[ \, \ x ] \ carrier (poly_ring R)"
using is_root unfolding is_root_def sym[OF univ_poly_carrier] polynomial_def by simp
ultimately have "[ \, \ x ] pdivides q"
using UP.divides_cong_r[OF _ pq ] unfolding pdivides_def by simp
--> --------------------
--> maximum size reached
--> --------------------
[ Dauer der Verarbeitung: 0.83 Sekunden
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