(* Title: HOL/Algebra/Generated_Fields.thy
Author: Martin Baillon
*)
theory Generated_Fields
imports Generated_Rings Subrings Multiplicative_Group
begin
inductive_set
generate_field :: "('a, 'b) ring_scheme \ 'a set \ 'a set"
for R and H where
one : "\\<^bsub>R\<^esub> \ generate_field R H"
| incl : "h \ H \ h \ generate_field R H"
| a_inv: "h \ generate_field R H \ \\<^bsub>R\<^esub> h \ generate_field R H"
| m_inv: "\ h \ generate_field R H; h \ \\<^bsub>R\<^esub> \ \ inv\<^bsub>R\<^esub> h \ generate_field R H"
| eng_add : "\ h1 \ generate_field R H; h2 \ generate_field R H \ \ h1 \\<^bsub>R\<^esub> h2 \ generate_field R H"
| eng_mult: "\ h1 \ generate_field R H; h2 \ generate_field R H \ \ h1 \\<^bsub>R\<^esub> h2 \ generate_field R H"
subsection\<open>Basic Properties of Generated Rings - First Part\<close>
lemma (in field) generate_field_in_carrier:
assumes "H \ carrier R"
shows "h \ generate_field R H \ h \ carrier R"
apply (induction rule: generate_field.induct)
using assms field_Units
by blast+
lemma (in field) generate_field_incl:
assumes "H \ carrier R"
shows "generate_field R H \ carrier R"
using generate_field_in_carrier[OF assms] by auto
lemma (in field) zero_in_generate: "\\<^bsub>R\<^esub> \ generate_field R H"
using one a_inv generate_field.eng_add one_closed r_neg
by metis
lemma (in field) generate_field_is_subfield:
assumes "H \ carrier R"
shows "subfield (generate_field R H) R"
proof (intro subfieldI', simp_all add: m_inv)
show "subring (generate_field R H) R"
by (auto intro: subringI[of "generate_field R H"]
simp add: eng_add a_inv eng_mult one generate_field_in_carrier[OF assms])
qed
lemma (in field) generate_field_is_add_subgroup:
assumes "H \ carrier R"
shows "subgroup (generate_field R H) (add_monoid R)"
using subring.axioms(1)[OF subfieldE(1)[OF generate_field_is_subfield[OF assms]]] .
lemma (in field) generate_field_is_field :
assumes "H \ carrier R"
shows "field (R \ carrier := generate_field R H \)"
using subfield_iff generate_field_is_subfield assms by simp
lemma (in field) generate_field_min_subfield1:
assumes "H \ carrier R"
and "subfield E R" "H \ E"
shows "generate_field R H \ E"
proof
fix h
assume h: "h \ generate_field R H"
show "h \ E"
using h and assms(3) and subfield_m_inv[OF assms(2)]
by (induct rule: generate_field.induct)
(auto simp add: subringE(3,5-7)[OF subfieldE(1)[OF assms(2)]])
qed
lemma (in field) generate_fieldI:
assumes "H \ carrier R"
and "subfield E R" "H \ E"
and "\K. \ subfield K R; H \ K \ \ E \ K"
shows "E = generate_field R H"
proof
show "E \ generate_field R H"
using assms generate_field_is_subfield generate_field.incl by (metis subset_iff)
show "generate_field R H \ E"
using generate_field_min_subfield1[OF assms(1-3)] by simp
qed
lemma (in field) generate_fieldE:
assumes "H \ carrier R" and "E = generate_field R H"
shows "subfield E R" and "H \ E" and "\K. \ subfield K R; H \ K \ \ E \ K"
proof -
show "subfield E R" using assms generate_field_is_subfield by simp
show "H \ E" using assms(2) by (simp add: generate_field.incl subsetI)
show "\K. subfield K R \ H \ K \ E \ K"
using assms generate_field_min_subfield1 by auto
qed
lemma (in field) generate_field_min_subfield2:
assumes "H \ carrier R"
shows "generate_field R H = \{K. subfield K R \ H \ K}"
proof
have "subfield (generate_field R H) R \ H \ generate_field R H"
by (simp add: assms generate_fieldE(2) generate_field_is_subfield)
thus "\{K. subfield K R \ H \ K} \ generate_field R H" by blast
next
have "\K. subfield K R \ H \ K \ generate_field R H \ K"
by (simp add: assms generate_field_min_subfield1)
thus "generate_field R H \ \{K. subfield K R \ H \ K}" by blast
qed
lemma (in field) mono_generate_field:
assumes "I \ J" and "J \ carrier R"
shows "generate_field R I \ generate_field R J"
proof-
have "I \ generate_field R J "
using assms generate_fieldE(2) by blast
thus "generate_field R I \ generate_field R J"
using generate_field_min_subfield1[of I "generate_field R J"] assms generate_field_is_subfield[OF assms(2)]
by blast
qed
lemma (in field) subfield_gen_incl :
assumes "subfield H R"
and "subfield K R"
and "I \ H"
and "I \ K"
shows "generate_field (R\carrier := K\) I \ generate_field (R\carrier := H\) I"
proof
{fix J assume J_def : "subfield J R" "I \ J"
have "generate_field (R \ carrier := J \) I \ J"
using field.mono_generate_field[of "(R\carrier := J\)" I J] subfield_iff(2)[OF J_def(1)]
field.generate_field_in_carrier[of "R\carrier := J\"] field_axioms J_def
by auto}
note incl_HK = this
{fix x have "x \ generate_field (R\carrier := K\) I \ x \ generate_field (R\carrier := H\) I"
proof (induction rule : generate_field.induct)
case one
have "\\<^bsub>R\carrier := H\\<^esub> \ \\<^bsub>R\carrier := K\\<^esub> = \\<^bsub>R\carrier := H\\<^esub>" by simp
moreover have "\\<^bsub>R\carrier := H\\<^esub> \ \\<^bsub>R\carrier := K\\<^esub> = \\<^bsub>R\carrier := K\\<^esub>" by simp
ultimately show ?case using assms generate_field.one by metis
next
case (incl h) thus ?case using generate_field.incl by force
next
case (a_inv h)
note hyp = this
have "a_inv (R\carrier := K\) h = a_inv R h"
using assms group.m_inv_consistent[of "add_monoid R" K] a_comm_group incl_HK[of K] hyp
subring.axioms(1)[OF subfieldE(1)[OF assms(2)]]
unfolding comm_group_def a_inv_def by auto
moreover have "a_inv (R\carrier := H\) h = a_inv R h"
using assms group.m_inv_consistent[of "add_monoid R" H] a_comm_group incl_HK[of H] hyp
subring.axioms(1)[OF subfieldE(1)[OF assms(1)]]
unfolding comm_group_def a_inv_def by auto
ultimately show ?case using generate_field.a_inv a_inv.IH by fastforce
next
case (m_inv h)
note hyp = this
have h_K : "h \ (K - {\})" using incl_HK[OF assms(2) assms(4)] hyp by auto
hence "m_inv (R\carrier := K\) h = m_inv R h"
using field.m_inv_mult_of[OF subfield_iff(2)[OF assms(2)]]
group.m_inv_consistent[of "mult_of R" "K - {\}"] field_mult_group units_of_inv
subgroup_mult_of subfieldE[OF assms(2)] unfolding mult_of_def apply simp
by (metis h_K mult_of_def mult_of_is_Units subgroup.mem_carrier units_of_carrier assms(2))
moreover have h_H : "h \ (H - {\})" using incl_HK[OF assms(1) assms(3)] hyp by auto
hence "m_inv (R\carrier := H\) h = m_inv R h"
using field.m_inv_mult_of[OF subfield_iff(2)[OF assms(1)]]
group.m_inv_consistent[of "mult_of R" "H - {\}"] field_mult_group
subgroup_mult_of[OF assms(1)] unfolding mult_of_def apply simp
by (metis h_H field_Units m_inv_mult_of mult_of_is_Units subgroup.mem_carrier units_of_def)
ultimately show ?case using generate_field.m_inv m_inv.IH h_H by fastforce
next
case (eng_add h1 h2)
thus ?case using incl_HK assms generate_field.eng_add by force
next
case (eng_mult h1 h2)
thus ?case using generate_field.eng_mult by force
qed}
thus "\x. x \ generate_field (R\carrier := K\) I \ x \ generate_field (R\carrier := H\) I"
by auto
qed
lemma (in field) subfield_gen_equality:
assumes "subfield H R" "K \ H"
shows "generate_field R K = generate_field (R \ carrier := H \) K"
using subfield_gen_incl[OF assms(1) carrier_is_subfield assms(2)] assms subringE(1)
subfield_gen_incl[OF carrier_is_subfield assms(1) _ assms(2)] subfieldE(1)[OF assms(1)]
by force
end
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