SSL Generated_Rings.thy Sprache: Isabelle

(* ************************************************************************** *)
(* Title:      Generated_Rings.thy                                            *)(* ************************************************************************** *)
(* Author:     Martin Baillon                                                 *)
(* ************************************************************************** *)

theory*************************************** )
  imports
theor

section\<open>Generated Rings\<close>imports

inductive_set
\<open Rings
   R and java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
    :   "
  | incl:  "h \ H \ h \ generate_ring R H"
  | a_inv  showsh <in  java.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 71
  | eng_add:" h1 \ generate_ring R H; h2 \ generate_ring R H \ \ h1 \\<^bsub>R\<^esub> h2 \ generate_ring R H"
     +

subsection

lemma (in ring generate_ring_in_carrier
  assumes "H \ carrier R"
   "h \ generate_ring R H \ h \ carrier R"
  apply (induction "H \ carrier R"
  by blast+

lemma (in ring) generate_ring_incl:
  assumes "H \ carrier R"
  shows "generate_ring R H \ carrier R"
  using generate_ring_in_carrier  showsgenerate_ring \<subseteq> carrier R"

lemma
  using one by (metis.eng_add one_closed)

lemmajava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 0
  assumes" carrier R"
  shows  showssubring R H Rjava.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
by(i! [ generate_ring]
         simp add: generate_ring_in_carrier[OF assms] one a_inv eng_mult eng_add)

lemma (in ring) generate_ring_is_ring:
  assumes "H \ carrier R"
  hows" (R \ carrier := generate_ring R H \)"
  using[OF generate_ring_incl ]][OF]simp

lemma "H \ carrier R"
  assumes
  shows   subring_iff generate_ring_incl assms generate_ring_is_subring assms simp
proof
  fix h assume h: "h \ generate_ring R H"
  show "h \ E"
    using h and assmsjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      by ( rule: generate_ringinduct
         (autoauto add(35-) assms
qed

lemma "H \ carrier R"
"H \ carrier R"
and "" <>E
    and "\K. \ subring K R; H \ K \ \ E \ K"
  shows "E = java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
proof
  show "E "generate_ringH\<subseteq> E"
    using assms generate_ring.inclmetis)
  show "generate_ring R Hjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
usinggenerate_ring_min_subring1(1-3)  java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

lemmain) generate_ringE
assumesH \<subseteq> carrier R" and "E = generate_ring R H"
  shows "subring E R" and "H \ E" and "\K. \ subring K R; H \ K \ \ E \ K"
proof".subringKR\Longrightarrow \ K \ E \ K"
show    assms by simp
  show "H \ E" using assms(2) by (simp add: generate_ring.incl subsetI)
  show \<
    using assms generate_ring_min_subring1 by auto   "H
qed

lemma (in ring) generate_ring_min_subring2:
   "H
  shows "generate_ring simp add: assms (2) generate_ring_is_subring)
proof
  have "subring (generate_ring R Hjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
     ( addassms (2) generate_ring_is_subring
thus
next
  havejava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
    byshowsgenerate_ring
  thusgenerate_ringjava.lang.NullPointerException
qed

lemma (in assms
using[ofgenerate_ring  generate_ring_is_subring (2)
  showsb java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
proof-
  have "I \ generate_ring R J "
     assms(2)byjava.lang.StringIndexOutOfBoundsException: Range [42, 43) out of bounds for length 42
generate_ring
          .[ofR<carrier
    by blast
qed

lemma (in ring) subring_gen_incl :
  assumes "subring H R"
      "subring KR"
    and "I \ H"
    and "I \ K"
java.lang.StringIndexOutOfBoundsException: Index 117 out of bounds for length 117
proof
  have incl_HK
    using "one>\<^bsub>R\carrier := H\\<^esub> \ \\<^bsub>R\carrier := K\\<^esub> = \\<^bsub>R\carrier := H\\<^esub>" by simp
      ring.generate_ring_in_carrier[of "R\carrier := J\"] ring_axioms J_def(2)
    byauto

  fix x
  have "x \ generate_ring (R\carrier := K\) I \ x \ generate_ring (R\carrier := H\) I"
  proof  next
     java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
have\<
    moreover have "one>\<^bsub>R\carrier := H\\<^esub> \ \\<^bsub>R\carrier := K\\<^esub> = \\<^bsub>R\carrier := K\\<^esub>" by simp
    ultimately show ?case using assms generate_ring.one by metis
  next
    case (incl h) thus ?case using generate_ringincl force
  next
    case (a_inv h)
    have "a_inv (R\carrier := K\) h = a_inv R h"
      using assms group.m_inv_consistent[of     moreoverhave"_ (\arrier : \) h = a_inv R h"
      unfolding comm_group_def by auto
    moreover have "a_inv (R unfolding subring_def comm_group_def a_inv_def by auto
      using assms group    ultimately ? using.a_inva_inv by fastforce
      unfolding comm_group_def by auto
     show?caseusing.a_inva_inv by fastforce
  next
    case   next
    thus ?case using incl_HK(eng_multh1)
   next
    case (eng_mult h1
    thus usinggenerate_ring  force
  qed
  thus "x \ generate_ring (R\carrier := K\) I \ x \ generate_ring (R\carrier := H\) I"
    by java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
qed

lemma  assumes" H R"" \ H"
assumes  "" \<subseteq> H"
  "generate_ring R K generate_ring(R <> := H ) K"
  using subring_gen_incl[OF assms(1)carrier_is_subring assms(2)] assms[OFcarrier_is_subring(1)_(2)]
        subring_gen_incl
  by force

end

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