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Quellcode-Bibliothek
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Sprache: Isabelle
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(* Title: HOL/Algebra/Indexed_Polynomials.thy
Author: Paulo Emílio de Vilhena
*)
theory Indexed_Polynomials
imports Weak_Morphisms "HOL-Library.Multiset" Polynomial_Divisibility
begin
section \<open>Indexed Polynomials\<close>
text \<open>In this theory, we build a basic framework to the study of polynomials on letters
indexed by a set. The main interest is to then apply these concepts to the construction
of the algebraic closure of a field. \<close>
subsection \<open>Definitions\<close>
text \<open>We formalize indexed monomials as multisets with its support a subset of the index set.
On top of those, we build indexed polynomials which are simply functions mapping a monomial
to its coefficient. \<close>
definition (in ring) indexed_const :: "'a \ ('c multiset \ 'a)"
where "indexed_const k = (\m. if m = {#} then k else \)"
definition (in ring) indexed_pmult :: "('c multiset \ 'a) \ 'c \ ('c multiset \ 'a)" (infixl "\" 65)
where "indexed_pmult P i = (\m. if i \# m then P (m - {# i #}) else \)"
definition (in ring) indexed_padd :: "_ \ _ \ ('c multiset \ 'a)" (infixl "\" 65)
where "indexed_padd P Q = (\m. (P m) \ (Q m))"
definition (in ring) indexed_var :: "'c \ ('c multiset \ 'a)" ("\\")
where "indexed_var i = (indexed_const \) \ i"
definition (in ring) index_free :: "('c multiset \ 'a) \ 'c \ bool"
where "index_free P i \ (\m. i \# m \ P m = \)"
definition (in ring) carrier_coeff :: "('c multiset \ 'a) \ bool"
where "carrier_coeff P \ (\m. P m \ carrier R)"
inductive_set (in ring) indexed_pset :: "'c set \ 'a set \ ('c multiset \ 'a) set" ("_ [\\]" 80)
for I and K where
indexed_const: "k \ K \ indexed_const k \ (K[\\<^bsub>I\<^esub>])"
| indexed_padd: "\ P \ (K[\\<^bsub>I\<^esub>]); Q \ (K[\\<^bsub>I\<^esub>]) \ \ P \ Q \ (K[\\<^bsub>I\<^esub>])"
| indexed_pmult: "\ P \ (K[\\<^bsub>I\<^esub>]); i \ I \ \ P \ i \ (K[\\<^bsub>I\<^esub>])"
fun (in ring) indexed_eval_aux :: "('c multiset \ 'a) list \ 'c \ ('c multiset \ 'a)"
where "indexed_eval_aux Ps i = foldr (\P Q. (Q \ i) \ P) Ps (indexed_const \)"
fun (in ring) indexed_eval :: "('c multiset \ 'a) list \ 'c \ ('c multiset \ 'a)"
where "indexed_eval Ps i = indexed_eval_aux (rev Ps) i"
subsection \<open>Basic Properties\<close>
lemma (in ring) carrier_coeffE:
assumes "carrier_coeff P" shows "P m \ carrier R"
using assms unfolding carrier_coeff_def by simp
lemma (in ring) indexed_zero_def: "indexed_const \ = (\_. \)"
unfolding indexed_const_def by simp
lemma (in ring) indexed_const_index_free: "index_free (indexed_const k) i"
unfolding index_free_def indexed_const_def by auto
lemma (in domain) indexed_var_not_index_free: "\ index_free \\<^bsub>i\<^esub> i"
proof -
have "\\<^bsub>i\<^esub> {# i #} = \"
unfolding indexed_var_def indexed_pmult_def indexed_const_def by simp
thus ?thesis
using one_not_zero unfolding index_free_def by fastforce
qed
lemma (in ring) indexed_pmult_zero [simp]:
shows "indexed_pmult (indexed_const \) i = indexed_const \"
unfolding indexed_zero_def indexed_pmult_def by auto
lemma (in ring) indexed_padd_zero:
assumes "carrier_coeff P" shows "P \ (indexed_const \) = P" and "(indexed_const \) \ P = P"
using assms unfolding carrier_coeff_def indexed_zero_def indexed_padd_def by auto
lemma (in ring) indexed_padd_const:
shows "(indexed_const k1) \ (indexed_const k2) = indexed_const (k1 \ k2)"
unfolding indexed_padd_def indexed_const_def by auto
lemma (in ring) indexed_const_in_carrier:
assumes "K \ carrier R" and "k \ K" shows "\m. (indexed_const k) m \ carrier R"
using assms unfolding indexed_const_def by auto
lemma (in ring) indexed_padd_in_carrier:
assumes "carrier_coeff P" and "carrier_coeff Q" shows "carrier_coeff (indexed_padd P Q)"
using assms unfolding carrier_coeff_def indexed_padd_def by simp
lemma (in ring) indexed_pmult_in_carrier:
assumes "carrier_coeff P" shows "carrier_coeff (P \ i)"
using assms unfolding carrier_coeff_def indexed_pmult_def by simp
lemma (in ring) indexed_eval_aux_in_carrier:
assumes "list_all carrier_coeff Ps" shows "carrier_coeff (indexed_eval_aux Ps i)"
using assms unfolding carrier_coeff_def
by (induct Ps) (auto simp add: indexed_zero_def indexed_padd_def indexed_pmult_def)
lemma (in ring) indexed_eval_in_carrier:
assumes "list_all carrier_coeff Ps" shows "carrier_coeff (indexed_eval Ps i)"
using assms indexed_eval_aux_in_carrier[of "rev Ps"] by auto
lemma (in ring) indexed_pset_in_carrier:
assumes "K \ carrier R" and "P \ (K[\\<^bsub>I\<^esub>])" shows "carrier_coeff P"
using assms(2,1) indexed_const_in_carrier unfolding carrier_coeff_def
by (induction) (auto simp add: indexed_zero_def indexed_padd_def indexed_pmult_def)
subsection \<open>Indexed Eval\<close>
lemma (in ring) exists_indexed_eval_aux_monomial:
assumes "carrier_coeff P" and "list_all carrier_coeff Qs"
and "count n i = k" and "P n \ \" and "list_all (\Q. index_free Q i) Qs"
obtains m where "count m i = length Qs + k" and "(indexed_eval_aux (Qs @ [ P ]) i) m \ \"
proof -
from assms(2,5) have "\m. count m i = length Qs + k \ (indexed_eval_aux (Qs @ [ P ]) i) m \ \"
proof (induct Qs)
case Nil thus ?case
using indexed_padd_zero(2)[OF assms(1)] assms(3-4) by auto
next
case (Cons Q Qs)
then obtain m where m: "count m i = length Qs + k" "(indexed_eval_aux (Qs @ [ P ]) i) m \ \"
by auto
define m' where "m' = m + {# i #}"
hence "Q m' = \"
using Cons(3) unfolding index_free_def by simp
moreover have "(indexed_eval_aux (Qs @ [ P ]) i) m \ carrier R"
using indexed_eval_aux_in_carrier[of "Qs @ [ P ]" i] Cons(2) assms(1) carrier_coeffE by auto
hence "((indexed_eval_aux (Qs @ [ P ]) i) \ i) m' \ carrier R - { \ }"
using m unfolding indexed_pmult_def m'_def by simp
ultimately have "(indexed_eval_aux (Q # (Qs @ [ P ])) i) m' \ \"
by (auto simp add: indexed_padd_def)
moreover from \<open>count m i = length Qs + k\<close> have "count m' i = length (Q # Qs) + k"
unfolding m'_def by simp
ultimately show ?case
by auto
qed
thus thesis
using that by blast
qed
lemma (in ring) indexed_eval_aux_monomial_degree_le:
assumes "list_all carrier_coeff Ps" and "list_all (\P. index_free P i) Ps"
and "(indexed_eval_aux Ps i) m \ \" shows "count m i \ length Ps - 1"
using assms(1-3)
proof (induct Ps arbitrary: m, simp add: indexed_zero_def)
case (Cons P Ps) show ?case
proof (cases "count m i = 0", simp)
assume "count m i \ 0"
hence "P m = \"
using Cons(3) unfolding index_free_def by simp
moreover have "(indexed_eval_aux Ps i) m \ carrier R"
using carrier_coeffE[OF indexed_eval_aux_in_carrier[of Ps i]] Cons(2) by simp
ultimately have "((indexed_eval_aux Ps i) \ i) m \ \"
using Cons(4) by (auto simp add: indexed_padd_def)
with \<open>count m i \<noteq> 0\<close> have "(indexed_eval_aux Ps i) (m - {# i #}) \<noteq> \<zero>"
unfolding indexed_pmult_def by (auto simp del: indexed_eval_aux.simps)
hence "count m i - 1 \ length Ps - 1"
using Cons(1)[of "m - {# i #}"] Cons(2-3) by auto
moreover from \<open>(indexed_eval_aux Ps i) (m - {# i #}) \<noteq> \<zero>\<close> have "length Ps > 0"
by (auto simp add: indexed_zero_def)
moreover from \<open>count m i \<noteq> 0\<close> have "count m i > 0"
by simp
ultimately show ?thesis
by (simp add: Suc_leI le_diff_iff)
qed
qed
lemma (in ring) indexed_eval_aux_is_inj:
assumes "list_all carrier_coeff Ps" and "list_all (\P. index_free P i) Ps"
and "list_all carrier_coeff Qs" and "list_all (\Q. index_free Q i) Qs"
and "indexed_eval_aux Ps i = indexed_eval_aux Qs i" and "length Ps = length Qs"
shows "Ps = Qs"
using assms
proof (induct Ps arbitrary: Qs, simp)
case (Cons P Ps)
from \<open>length (P # Ps) = length Qs\<close> obtain Q' Qs' where Qs: "Qs = Q' # Qs'" and "length Ps = length Qs'"
by (metis Suc_length_conv)
have in_carrier:
"((indexed_eval_aux Ps i) \ i) m \ carrier R" "P m \ carrier R"
"((indexed_eval_aux Qs' i) \ i) m \ carrier R" "Q' m \ carrier R" for m
using indexed_eval_aux_in_carrier[of Ps i]
indexed_eval_aux_in_carrier[of Qs' i] Cons(2,4) carrier_coeffE
unfolding Qs indexed_pmult_def by auto
have "(indexed_eval_aux (P # Ps) i) m = (indexed_eval_aux (Q' # Qs') i) m" for m
using Cons(6) unfolding Qs by simp
hence eq: "((indexed_eval_aux Ps i) \ i) m \ P m = ((indexed_eval_aux Qs' i) \ i) m \ Q' m" for m
by (simp add: indexed_padd_def)
have "P m = Q' m" if "i \# m" for m
using that Cons(3,5) unfolding index_free_def Qs by auto
moreover have "P m = Q' m" if "i \# m" for m
using in_carrier(2,4) eq[of m] that by (auto simp add: indexed_pmult_def)
ultimately have "P = Q'"
by auto
hence "(indexed_eval_aux Ps i) m = (indexed_eval_aux Qs' i) m" for m
using eq[of "m + {# i #}"] in_carrier[of "m + {# i #}"] unfolding indexed_pmult_def by auto
with \<open>length Ps = length Qs'\<close> have "Ps = Qs'"
using Cons(1)[of Qs'] Cons(2-5) unfolding Qs by auto
with \<open>P = Q'\<close> show ?case
unfolding Qs by simp
qed
lemma (in ring) indexed_eval_aux_is_inj':
assumes "list_all carrier_coeff Ps" and "list_all (\P. index_free P i) Ps"
and "list_all carrier_coeff Qs" and "list_all (\Q. index_free Q i) Qs"
and "carrier_coeff P" and "index_free P i" "P \ indexed_const \"
and "carrier_coeff Q" and "index_free Q i" "Q \ indexed_const \"
and "indexed_eval_aux (Ps @ [ P ]) i = indexed_eval_aux (Qs @ [ Q ]) i"
shows "Ps = Qs" and "P = Q"
proof -
obtain m n where "P m \ \" and "Q n \ \"
using assms(7,10) unfolding indexed_zero_def by blast
hence "count m i = 0" and "count n i = 0"
using assms(6,9) unfolding index_free_def by (meson count_inI)+
with \<open>P m \<noteq> \<zero>\<close> and \<open>Q n \<noteq> \<zero>\<close> obtain m' n'
where m': "count m' i = length Ps" "(indexed_eval_aux (Ps @ [ P ]) i) m' \ \"
and n': "count n' i = length Qs" "(indexed_eval_aux (Qs @ [ Q ]) i) n' \ \"
using exists_indexed_eval_aux_monomial[of P Ps m i 0]
exists_indexed_eval_aux_monomial[of Q Qs n i 0] assms(1-5,8)
by (metis (no_types, lifting) add.right_neutral)
have "(indexed_eval_aux (Qs @ [ Q ]) i) m' \ \"
using m'(2) assms(11) by simp
with \<open>count m' i = length Ps\<close> have "length Ps \<le> length Qs"
using indexed_eval_aux_monomial_degree_le[of "Qs @ [ Q ]" i m'] assms(3-4,8-9) by auto
moreover have "(indexed_eval_aux (Ps @ [ P ]) i) n' \ \"
using n'(2) assms(11) by simp
with \<open>count n' i = length Qs\<close> have "length Qs \<le> length Ps"
using indexed_eval_aux_monomial_degree_le[of "Ps @ [ P ]" i n'] assms(1-2,5-6) by auto
ultimately have same_len: "length (Ps @ [ P ]) = length (Qs @ [ Q ])"
by simp
thus "Ps = Qs" and "P = Q"
using indexed_eval_aux_is_inj[of "Ps @ [ P ]" i "Qs @ [ Q ]"] assms(1-6,8-9,11) by auto
qed
lemma (in ring) exists_indexed_eval_monomial:
assumes "carrier_coeff P" and "list_all carrier_coeff Qs"
and "P n \ \" and "list_all (\Q. index_free Q i) Qs"
obtains m where "count m i = length Qs + (count n i)" and "(indexed_eval (P # Qs) i) m \ \"
using exists_indexed_eval_aux_monomial[OF assms(1) _ _ assms(3), of "rev Qs"] assms(2,4) by auto
corollary (in ring) exists_indexed_eval_monomial':
assumes "carrier_coeff P" and "list_all carrier_coeff Qs"
and "P \ indexed_const \" and "list_all (\Q. index_free Q i) Qs"
obtains m where "count m i \ length Qs" and "(indexed_eval (P # Qs) i) m \ \"
proof -
from \<open>P \<noteq> indexed_const \<zero>\<close> obtain n where "P n \<noteq> \<zero>"
unfolding indexed_const_def by auto
then obtain m where "count m i = length Qs + (count n i)" and "(indexed_eval (P # Qs) i) m \ \"
using exists_indexed_eval_monomial[OF assms(1-2) _ assms(4)] by auto
thus thesis
using that by force
qed
lemma (in ring) indexed_eval_monomial_degree_le:
assumes "list_all carrier_coeff Ps" and "list_all (\P. index_free P i) Ps"
and "(indexed_eval Ps i) m \ \" shows "count m i \ length Ps - 1"
using indexed_eval_aux_monomial_degree_le[of "rev Ps"] assms by auto
lemma (in ring) indexed_eval_is_inj:
assumes "list_all carrier_coeff Ps" and "list_all (\P. index_free P i) Ps"
and "list_all carrier_coeff Qs" and "list_all (\Q. index_free Q i) Qs"
and "carrier_coeff P" and "index_free P i" "P \ indexed_const \"
and "carrier_coeff Q" and "index_free Q i" "Q \ indexed_const \"
and "indexed_eval (P # Ps) i = indexed_eval (Q # Qs) i"
shows "Ps = Qs" and "P = Q"
proof -
have rev_cond:
"list_all carrier_coeff (rev Ps)" "list_all (\P. index_free P i) (rev Ps)"
"list_all carrier_coeff (rev Qs)" "list_all (\Q. index_free Q i) (rev Qs)"
using assms(1-4) by auto
show "Ps = Qs" and "P = Q"
using indexed_eval_aux_is_inj'[OF rev_cond assms(5-10)] assms(11) by auto
qed
lemma (in ring) indexed_eval_inj_on_carrier:
assumes "\P. P \ carrier L \ carrier_coeff P" and "\P. P \ carrier L \ index_free P i" and "\\<^bsub>L\<^esub> = indexed_const \"
shows "inj_on (\Ps. indexed_eval Ps i) (carrier (poly_ring L))"
proof -
{ fix Ps
assume "Ps \ carrier (poly_ring L)" and "indexed_eval Ps i = indexed_const \"
have "Ps = []"
proof (rule ccontr)
assume "Ps \ []"
then obtain P' Ps' where Ps: "Ps = P' # Ps'"
using list.exhaust by blast
with \<open>Ps \<in> carrier (poly_ring L)\<close>
have "P' \ indexed_const \" and "list_all carrier_coeff Ps" and "list_all (\P. index_free P i) Ps"
using assms unfolding sym[OF univ_poly_carrier[of L "carrier L"]] polynomial_def
by (simp add: list.pred_set subset_code(1))+
then obtain m where "(indexed_eval Ps i) m \ \"
using exists_indexed_eval_monomial'[of P' Ps'] unfolding Ps by auto
hence "indexed_eval Ps i \ indexed_const \"
unfolding indexed_const_def by auto
with \<open>indexed_eval Ps i = indexed_const \<zero>\<close> show False by simp
qed } note aux_lemma = this
show ?thesis
proof (rule inj_onI)
fix Ps Qs
assume "Ps \ carrier (poly_ring L)" and "Qs \ carrier (poly_ring L)"
show "indexed_eval Ps i = indexed_eval Qs i \ Ps = Qs"
proof (cases)
assume "Qs = []" and "indexed_eval Ps i = indexed_eval Qs i"
with \<open>Ps \<in> carrier (poly_ring L)\<close> show "Ps = Qs"
using aux_lemma by simp
next
assume "Qs \ []" and eq: "indexed_eval Ps i = indexed_eval Qs i"
with \<open>Qs \<in> carrier (poly_ring L)\<close> have "Ps \<noteq> []"
using aux_lemma by auto
from \<open>Ps \<noteq> []\<close> and \<open>Qs \<noteq> []\<close> obtain P' Ps' Q' Qs' where Ps: "Ps = P' # Ps'" and Qs: "Qs = Q' # Qs'"
using list.exhaust by metis
from \<open>Ps \<in> carrier (poly_ring L)\<close> and \<open>Ps = P' # Ps'\<close>
have "carrier_coeff P'" and "index_free P' i" "P' \ indexed_const \"
and "list_all carrier_coeff Ps'" and "list_all (\P. index_free P i) Ps'"
using assms unfolding sym[OF univ_poly_carrier[of L "carrier L"]] polynomial_def
by (simp add: list.pred_set subset_code(1))+
moreover
from \<open>Qs \<in> carrier (poly_ring L)\<close> and \<open>Qs = Q' # Qs'\<close>
have "carrier_coeff Q'" and "index_free Q' i" "Q' \ indexed_const \"
and "list_all carrier_coeff Qs'" and "list_all (\P. index_free P i) Qs'"
using assms unfolding sym[OF univ_poly_carrier[of L "carrier L"]] polynomial_def
by (simp add: list.pred_set subset_code(1))+
ultimately show ?thesis
using indexed_eval_is_inj[of Ps' i Qs' P' Q'] eq unfolding Ps Qs by auto
qed
qed
qed
subsection \<open>Link with Weak Morphisms\<close>
text \<open>We study some elements of the contradiction needed in the algebraic closure existence proof. \<close>
context ring
begin
lemma (in ring) indexed_padd_index_free:
assumes "index_free P i" and "index_free Q i" shows "index_free (P \ Q) i"
using assms unfolding indexed_padd_def index_free_def by auto
lemma (in ring) indexed_pmult_index_free:
assumes "index_free P j" and "i \ j" shows "index_free (P \ i) j"
using assms unfolding index_free_def indexed_pmult_def
by (metis insert_DiffM insert_noteq_member)
lemma (in ring) indexed_eval_index_free:
assumes "list_all (\P. index_free P j) Ps" and "i \ j" shows "index_free (indexed_eval Ps i) j"
proof -
{ fix Ps assume "list_all (\P. index_free P j) Ps" hence "index_free (indexed_eval_aux Ps i) j"
using indexed_padd_index_free[OF indexed_pmult_index_free[OF _ assms(2)]]
by (induct Ps) (auto simp add: indexed_zero_def index_free_def) }
thus ?thesis
using assms(1) by auto
qed
context
fixes L :: "(('c multiset) \ 'a) ring" and i :: 'c
assumes hyps:
\<comment> \<open>i\<close> "field L"
\<comment> \<open>ii\<close> "\<And>P. P \<in> carrier L \<Longrightarrow> carrier_coeff P"
\<comment> \<open>iii\<close> "\<And>P. P \<in> carrier L \<Longrightarrow> index_free P i"
\<comment> \<open>iv\<close> "\<zero>\<^bsub>L\<^esub> = indexed_const \<zero>"
begin
interpretation L: field L
using \<open>field L\<close> .
interpretation UP: principal_domain "poly_ring L"
using L.univ_poly_is_principal[OF L.carrier_is_subfield] .
abbreviation eval_pmod
where "eval_pmod q \ (\p. indexed_eval (L.pmod p q) i)"
abbreviation image_poly
where "image_poly q \ image_ring (eval_pmod q) (poly_ring L)"
lemma indexed_eval_is_weak_ring_morphism:
assumes "q \ carrier (poly_ring L)" shows "weak_ring_morphism (eval_pmod q) (PIdl\<^bsub>poly_ring L\<^esub> q) (poly_ring L)"
proof (rule weak_ring_morphismI)
show "ideal (PIdl\<^bsub>poly_ring L\<^esub> q) (poly_ring L)"
using UP.cgenideal_ideal[OF assms] .
next
fix a b assume in_carrier: "a \ carrier (poly_ring L)" "b \ carrier (poly_ring L)"
note ldiv_closed = in_carrier[THEN L.long_division_closed(2)[OF L.carrier_is_subfield _ assms]]
have "(eval_pmod q) a = (eval_pmod q) b \ L.pmod a q = L.pmod b q"
using inj_onD[OF indexed_eval_inj_on_carrier[OF hyps(2-4)] _ ldiv_closed] by fastforce
also have " ... \ q pdivides\<^bsub>L\<^esub> (a \\<^bsub>poly_ring L\<^esub> b)"
unfolding L.same_pmod_iff_pdivides[OF L.carrier_is_subfield in_carrier assms] ..
also have " ... \ PIdl\<^bsub>poly_ring L\<^esub> (a \\<^bsub>poly_ring L\<^esub> b) \ PIdl\<^bsub>poly_ring L\<^esub> q"
unfolding UP.to_contain_is_to_divide[OF assms UP.minus_closed[OF in_carrier]] pdivides_def ..
also have " ... \ a \\<^bsub>poly_ring L\<^esub> b \ PIdl\<^bsub>poly_ring L\<^esub> q"
unfolding UP.cgenideal_eq_genideal[OF assms] UP.cgenideal_eq_genideal[OF UP.minus_closed[OF in_carrier]]
UP.Idl_subset_ideal'[OF UP.minus_closed[OF in_carrier] assms] ..
finally show "(eval_pmod q) a = (eval_pmod q) b \ a \\<^bsub>poly_ring L\<^esub> b \ PIdl\<^bsub>poly_ring L\<^esub> q" .
qed
lemma eval_norm_eq_id:
assumes "q \ carrier (poly_ring L)" and "degree q > 0" and "a \ carrier L"
shows "((eval_pmod q) \ (ring.poly_of_const L)) a = a"
proof (cases)
assume "a = \\<^bsub>L\<^esub>" thus ?thesis
using L.long_division_zero(2)[OF L.carrier_is_subfield assms(1)] hyps(4)
unfolding ring.poly_of_const_def[OF L.ring_axioms] by auto
next
assume "a \ \\<^bsub>L\<^esub>" then have in_carrier: "[ a ] \ carrier (poly_ring L)"
using assms(3) unfolding sym[OF univ_poly_carrier[of L "carrier L"]] polynomial_def by simp
from \<open>a \<noteq> \<zero>\<^bsub>L\<^esub>\<close> show ?thesis
using L.pmod_const(2)[OF L.carrier_is_subfield in_carrier assms(1)] assms(2)
indexed_padd_zero(2)[OF hyps(2)[OF assms(3)]]
unfolding ring.poly_of_const_def[OF L.ring_axioms] by auto
qed
lemma image_poly_iso_incl:
assumes "q \ carrier (poly_ring L)" and "degree q > 0" shows "id \ ring_hom L (image_poly q)"
proof -
have "((eval_pmod q) \ L.poly_of_const) \ ring_hom L (image_poly q)"
using ring_hom_trans[OF L.canonical_embedding_is_hom[OF L.carrier_is_subring]
UP.weak_ring_morphism_is_hom[OF indexed_eval_is_weak_ring_morphism[OF assms(1)]]]
by simp
thus ?thesis
using eval_norm_eq_id[OF assms(1-2)] L.ring_hom_restrict[of _ "image_poly q" id] by auto
qed
lemma image_poly_is_field:
assumes "q \ carrier (poly_ring L)" and "pirreducible\<^bsub>L\<^esub> (carrier L) q" shows "field (image_poly q)"
using UP.image_ring_is_field[OF indexed_eval_is_weak_ring_morphism[OF assms(1)]] assms(2)
unfolding sym[OF L.rupture_is_field_iff_pirreducible[OF L.carrier_is_subfield assms(1)]] rupture_def
by simp
lemma image_poly_index_free:
assumes "q \ carrier (poly_ring L)" and "P \ carrier (image_poly q)" and "\ index_free P j" "i \ j"
obtains Q where "Q \ carrier L" and "\ index_free Q j"
proof -
from \<open>P \<in> carrier (image_poly q)\<close> obtain p where p: "p \<in> carrier (poly_ring L)" and P: "P = (eval_pmod q) p"
unfolding image_ring_carrier by blast
from \<open>\<not> index_free P j\<close> have "\<not> list_all (\<lambda>P. index_free P j) (L.pmod p q)"
using indexed_eval_index_free[OF _ assms(4), of "L.pmod p q"] unfolding sym[OF P] by auto
then obtain Q where "Q \ set (L.pmod p q)" and "\ index_free Q j"
unfolding list_all_iff by auto
thus ?thesis
using L.long_division_closed(2)[OF L.carrier_is_subfield p assms(1)] that
unfolding sym[OF univ_poly_carrier[of L "carrier L"]] polynomial_def
by auto
qed
lemma eval_pmod_var:
assumes "indexed_const \ ring_hom R L" and "q \ carrier (poly_ring L)" and "degree q > 1"
shows "(eval_pmod q) X\<^bsub>L\<^esub> = \\<^bsub>i\<^esub>" and "\\<^bsub>i\<^esub> \ carrier (image_poly q)"
proof -
have "X\<^bsub>L\<^esub> = [ indexed_const \, indexed_const \ ]" and "X\<^bsub>L\<^esub> \ carrier (poly_ring L)"
using ring_hom_one[OF assms(1)] hyps(4) L.var_closed(1) L.carrier_is_subring unfolding var_def by auto
thus "(eval_pmod q) X\<^bsub>L\<^esub> = \\<^bsub>i\<^esub>"
using L.pmod_const(2)[OF L.carrier_is_subfield _ assms(2), of "X\<^bsub>L\<^esub>"] assms(3)
by (auto simp add: indexed_pmult_def indexed_padd_def indexed_const_def indexed_var_def)
with \<open>X\<^bsub>L\<^esub> \<in> carrier (poly_ring L)\<close> show "\<X>\<^bsub>i\<^esub> \<in> carrier (image_poly q)"
using image_iff unfolding image_ring_carrier by fastforce
qed
lemma image_poly_eval_indexed_var:
assumes "indexed_const \ ring_hom R L"
and "q \ carrier (poly_ring L)" and "degree q > 1" and "pirreducible\<^bsub>L\<^esub> (carrier L) q"
shows "(ring.eval (image_poly q)) q \\<^bsub>i\<^esub> = \\<^bsub>image_poly q\<^esub>"
proof -
let ?surj = "L.rupture_surj (carrier L) q"
let ?Rupt = "Rupt\<^bsub>L\<^esub> (carrier L) q"
let ?f = "eval_pmod q"
interpret UP: ring "poly_ring L"
using L.univ_poly_is_ring[OF L.carrier_is_subring] .
from \<open>pirreducible\<^bsub>L\<^esub> (carrier L) q\<close> interpret Rupt: field ?Rupt
using L.rupture_is_field_iff_pirreducible[OF L.carrier_is_subfield assms(2)] by simp
have weak_morphism: "weak_ring_morphism ?f (PIdl\<^bsub>poly_ring L\<^esub> q) (poly_ring L)"
using indexed_eval_is_weak_ring_morphism[OF assms(2)] .
then interpret I: ideal "PIdl\<^bsub>poly_ring L\<^esub> q" "poly_ring L"
using weak_ring_morphism.axioms(1) by auto
interpret Hom: ring_hom_ring ?Rupt "image_poly q" "\x. the_elem (?f ` x)"
using ring_hom_ring.intro[OF I.quotient_is_ring UP.image_ring_is_ring[OF weak_morphism]]
UP.weak_ring_morphism_is_iso[OF weak_morphism]
unfolding ring_iso_def symmetric[OF ring_hom_ring_axioms_def] rupture_def
by auto
have "set q \ carrier L" and lc: "q \ [] \ lead_coeff q \ carrier L - { \\<^bsub>L\<^esub> }"
using assms(2) unfolding sym[OF univ_poly_carrier] polynomial_def by auto
have map_surj: "set (map (?surj \ L.poly_of_const) q) \ carrier ?Rupt"
proof -
have "L.poly_of_const a \ carrier (poly_ring L)" if "a \ carrier L" for a
using that L.normalize_gives_polynomial[of "[ a ]"]
unfolding univ_poly_carrier ring.poly_of_const_def[OF L.ring_axioms] by simp
hence "(?surj \ L.poly_of_const) a \ carrier ?Rupt" if "a \ carrier L" for a
using ring_hom_memE(1)[OF L.rupture_surj_hom(1)[OF L.carrier_is_subring assms(2)]] that by simp
with \<open>set q \<subseteq> carrier L\<close> show ?thesis
by (induct q) (auto)
qed
have "?surj X\<^bsub>L\<^esub> \ carrier ?Rupt"
using ring_hom_memE(1)[OF L.rupture_surj_hom(1)[OF _ assms(2)] L.var_closed(1)] L.carrier_is_subring by simp
moreover have "map (\x. the_elem (?f ` x)) (map (?surj \ L.poly_of_const) q) = q"
proof -
define g where "g = (?surj \ L.poly_of_const)"
define f where "f = (\x. the_elem (?f ` x))"
have "the_elem (?f ` ((?surj \ L.poly_of_const) a)) = ((eval_pmod q) \ L.poly_of_const) a"
if "a \ carrier L" for a
using that L.normalize_gives_polynomial[of "[ a ]"] UP.weak_ring_morphism_range[OF weak_morphism]
unfolding univ_poly_carrier ring.poly_of_const_def[OF L.ring_axioms] by auto
hence "the_elem (?f ` ((?surj \ L.poly_of_const) a)) = a" if "a \ carrier L" for a
using eval_norm_eq_id[OF assms(2)] that assms(3) by simp
hence "f (g a) = a" if "a \ carrier L" for a
using that unfolding f_def g_def by simp
with \<open>set q \<subseteq> carrier L\<close> have "map f (map g q) = q"
by (induct q) (auto)
thus ?thesis
unfolding f_def g_def by simp
qed
moreover have "(\x. the_elem (?f ` x)) (?surj X\<^bsub>L\<^esub>) = \\<^bsub>i\<^esub>"
using UP.weak_ring_morphism_range[OF weak_morphism L.var_closed(1)[OF L.carrier_is_subring]]
unfolding eval_pmod_var(1)[OF assms(1-3)] by simp
ultimately have "Hom.S.eval q \\<^bsub>i\<^esub> = (\x. the_elem (?f ` x)) (Rupt.eval (map (?surj \ L.poly_of_const) q) (?surj X\<^bsub>L\<^esub>))"
using Hom.eval_hom'[OF _ map_surj] by auto
moreover have "\\<^bsub>?Rupt\<^esub> = ?surj \\<^bsub>poly_ring L\<^esub>"
unfolding rupture_def FactRing_def by (simp add: I.a_rcos_const)
hence "the_elem (?f ` \\<^bsub>?Rupt\<^esub>) = \\<^bsub>image_poly q\<^esub>"
using UP.weak_ring_morphism_range[OF weak_morphism UP.zero_closed]
unfolding image_ring_zero by simp
hence "(\x. the_elem (?f ` x)) (Rupt.eval (map (?surj \ L.poly_of_const) q) (?surj X\<^bsub>L\<^esub>)) = \\<^bsub>image_poly q\<^esub>"
using L.polynomial_rupture[OF L.carrier_is_subring assms(2)] by simp
ultimately show ?thesis
by simp
qed
end (* of fixed L context. *)
end (* of ring context. *)
end
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