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Datei: Complete_Lattice.thy   Sprache: Unknown

(*  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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