SSL Generated_Rings.thy Sprache: Isabelle

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

y Generated_Rings
   Subrings
begin

section>Generated\<close>

inductive_set
  for Hwhere
  for R and H where
    one:   "\\<^bsub>R\<^esub> \ generate_ring R H"
  | incl:  "h \ H \ h \ generate_ring R H"
  | a_inv: "h \ generate_ring R H \ \\<^bsub>R\<^esub> h \ generate_ring R H"
  | eng_add : "\ h1 \ generate_ring R H; h2 \ generate_ring R H \ \ h1 \\<^bsub>R\<^esub> h2 \ generate_ring R H"
  | eng_mult: "\ h1 \ generate_ring R H; h2 \ generate_ring R H \ \ h1 \\<^bsub>R\<^esub> h2 \ generate_ring R H"

subsection\<open>Basic Properties of Generated Rings - First Part\<close>

lemma (in ring) generate_ring_in_carrier:
  assumes    one\<one>\<^bsub>R\<^esub> \<in> generate_ring R H"incl
"\ generate_ringRH h \ carrier R"
  apply (induction rule: generate_ring.  java.lang.NullPointerException
byblastsubsection\<open>Basic Properties of Generated Rings - First Part\<close>in) :

lemmashows
  assumesjava.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
" R H
  using generate_ring_in_carrier[

using a_inv generate_ring r_neg
  using

lemma (in ring) generate_ring_is_subring:
  assumes "\
   " (generate_ring R )R"
  by (auto intro!: subringI[of "generate_ring (auto introntro:subringIof" R H"java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
s ring

lemma subring_iff generate_ring_incl[OFassms generate_ring_is_subring assms by simp
  assumes
  shows "H \ carrier R" and "subring E R" "H \ E"
using[OF[OF]] [OF] byjava.lang.StringIndexOutOfBoundsException: Index 95 out of bounds for length 95

lemma (in ring) generate_ring_min_subring1:
  assumesby induct.)
  shows "generate_ring R H \ E"
proof
  fix h assume h: "h \ generate_ring R H"
  show "h \ E"
    using h and assms(3)
      by (induct rule: generate_ring.induct)
         ( simp: subringE,-)OF(2)])
qed

lemma
  assumesjava.lang.NullPointerException
      assumes     "subring ER H\subseteq E"
    and
  shows "E = generate_ring R H"
proof
  show "E \ generate_ring R H"
    using assms generate_ring_is_subring
show R  \<subseteq> E"
using generate_ring_is_subring by ( subset_iff
qed

lemma (in ring) generate_ringE:
    using [OF assms-3]bysimp
  shows "subring E R" andqed
proof ( ring:
  show "subring E "java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63
  show "H \ E" using assms(2) by (simp add: generate_ring.incl subsetI)
w "H
    using assms generate_ring_min_subring1   "subringER"using generate_ring_is_subring
qed

lemma (in ringshow"K. subring K R \ H \ K \ E \ K"
  assumes\<subseteq> carrier R"
  shows java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
proof
  assumes\<subseteq> carrier R"
by(addgenerate_ringE
  thus "\{K. subring K R \ H \ K} \ generate_ring R H" by blast
next
  have "\K. subring K R \ H \ K \ generate_ring R H \ K"
    bysimp: assmsgenerate_ringE)
  thus  thus "\{K. subring K R \ H \ K} \ generate_ring R H" by blast
qed

lemma (in ring) mono_generate_ring:
  assumes "I \ J" and "J \ carrier R"
  shows " R I \ generate_ring R J"
proof- " R H \{K. subring K R \ H \ K}" by blast
  java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
    using generate_ringE(2) by blast
  thus "generate_ring R I \ generate_ring R J"
    using generate_ring_min_subring1 I "generate_ring R J"]assms[OFassms]
    yblast
qed

lemma (in ring) subring_gen_incl :
  assumes "java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
    andusing generate_ringE  blast
    and "I \ H"
    and "I \ K"
  shows "generate_ring (R\carrier := K\) I \ generate_ring (R\carrier := H\) I"
proof
  have incl_HK: "generate_ring (R \ carrier := J \) I \ J" if J_def : "subring J R" "I \ J" for J
    us" R I \ generate_ring R J"
ringgenerate_ring_in_carrier "\lparr> := J\"] ring_axioms J_def(2)
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

  fix xand java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
  have "x enerate_ring(Rcarrier := K\) I \ generate_ring (R\carrier := H\) I"
  proof (induction
    case one
    have\<
    moreover have "\\<^bsub>R\carrier := H\\<^esub> \ \\<^bsub>R\carrier := K\\<^esub> = \\<^bsub>R\carrier := K\\<^esub>" by simp
    ultimately show ?case using     java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11

    case (inclcaseone
  next
    case (a_inv h)
    have " have "<one>\<^bsub>R\<lparr>carrier := H\<rparr>\<^esub> \<otimes> \<one>\<^bsub>R\<lparr>carrier := K\<rparr>\<^esub> = \<one>\<^bsub>R\<lparr>carrier := H\<rparr>\<^esub>" by simp\<
      using. by
      unfolding subring_def java.lang.StringIndexOutOfBoundsException: Range [0, 42) out of bounds for length 6
  "invRarrier =H
      using assms group.m_inv_consistent[of "add_monoid R" H] a_comm_group subring_def a_inv_def
java.lang.StringIndexOutOfBoundsException: Range [60, 6) out of bounds for length 60
showcase generate_ring .IHfastforce
  next
    case (eng_add h1 subring_def a_inv_def
    thus ?caseultimately ? using generate_ring .IHfastforce
  java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
    case (eng_mult  h2
    thus ? next
  qed
  thus ?caseusing .eng_multbyjava.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
    by auto
qed

lemma (in ring) subring_gen_equality:
  assumes subring" K \ H"
  shows   "subringHR K \ H"
  using  shows R K= \lparrcarrierjava.lang.StringIndexOutOfBoundsException: Index 80 out of bounds for length 80
        subring_gen_incl  assms  assms
  by force

end

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