theory flash7Rev imports flashPub
begin
section{*Main defintions*}

lemma lemmaOnPtrAtGetX[intro]:
  assumes  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "
    and a5:"formEval ( eqn ( IVar ( Para ''UniMsg_Cmd'' iInv1) )  ( Const UNI_GetX )) s" and
    a6:"formEval   ( eqn ( IVar ( Para ''UniMsg_proc'' iInv1) )   (Const iInv2)) s " and
    a7:"\<forall>f. f \<in> (invariants   N) \<longrightarrow> formEval f s "
 shows  "formEval ( eqn ( IVar ( Global ''Dir_HeadPtr'') )   (Const iInv2) )  s"
 proof(rule ccontr)
  assume b0:"\<not>formEval ( eqn ( IVar ( Global ''Dir_HeadPtr'') )   (Const iInv2) )  s"
  have b1:"formEval (inv2 iInv1 iInv2) s"
    apply(cut_tac a7)

    apply(drule_tac x="inv2  iInv1 iInv2" in spec)
    apply(cut_tac a7 a2 a3 a4,simp)
    done
 with b0 a5 a6 show False
  by auto
qed

lemma lemmaOnPtrAtGet[intro]:
  assumes  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "
    and a5:"formEval ( eqn ( IVar ( Para ''UniMsg_Cmd'' iInv1) )  ( Const UNI_Get )) s" and
    a6:"formEval   ( eqn ( IVar ( Para ''UniMsg_proc'' iInv1) )   (Const iInv2)) s " and
    a7:"\<forall>f. f \<in> (invariants   N) \<longrightarrow> formEval f s "
 shows  "formEval ( eqn ( IVar ( Global ''Dir_HeadPtr'') )   (Const iInv2) )  s"
 proof(rule ccontr)
  assume b0:"\<not>formEval ( eqn ( IVar ( Global ''Dir_HeadPtr'') )   (Const iInv2) )  s"
  have b1:"formEval (inv3 iInv1 iInv2) s"
    apply(cut_tac a7)

    apply(drule_tac x="inv3  iInv1 iInv2" in spec)
    apply(cut_tac a7 a2 a3 a4,simp)
    done
 with b0 a5 a6 show False
  by auto
qed

lemma lemmaOnPtr1[intro]:
  assumes  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "
    and a5:"formEval ( eqn ( IVar ( Para ''UniMsg_Cmd'' iInv1) )  ( Const UNI_Get )) s" and
    a6:"formEval   ( eqn ( IVar ( Para ''UniMsg_proc'' iInv1) )   (Const iInv2)) s " and
    a7:"\<forall>f. f \<in> (invariants   N) \<longrightarrow> formEval f s " and
    b2:"iInv1' \<le> N" and  b3:"iInv2' \<le> N" and  b4:"iInv1'~=iInv2'  "
    and b5:"formEval ( eqn ( IVar ( Para ''UniMsg_Cmd'' iInv1') )  ( Const UNI_Get )) s" and
    b6:"formEval   ( eqn ( IVar ( Para ''UniMsg_proc'' iInv1') )   (Const iInv2')) s " and
    b7:"iInv2\<noteq>iInv2'"
 shows False
 proof -
  have c1:"formEval ( eqn ( IVar ( Global ''Dir_HeadPtr'') )   (Const iInv2) )  s"
    by(metis a2 a3 a4 a5 a6 a7 lemmaOnPtrAtGet)
  have c2:"formEval ( eqn ( IVar ( Global ''Dir_HeadPtr'') )   (Const iInv2') )  s"
    by(metis b2 b3 b4 b5 b6 a7 lemmaOnPtrAtGet)
 with c1 b7  show False
  by auto
qed

lemma lemmaOnPtr2[intro]:
  assumes  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "
    and a5:"formEval ( eqn ( IVar ( Para ''UniMsg_Cmd'' iInv1) )  ( Const UNI_GetX )) s" and
    a6:"formEval   ( eqn ( IVar ( Para ''UniMsg_proc'' iInv1) )   (Const iInv2)) s " and
    a7:"\<forall>f. f \<in> (invariants   N) \<longrightarrow> formEval f s " and
    b2:"iInv1' \<le> N" and  b3:"iInv2' \<le> N" and  b4:"iInv1'~=iInv2'  "
    and b5:"formEval ( eqn ( IVar ( Para ''UniMsg_Cmd'' iInv1') )  ( Const UNI_GetX )) s" and
    b6:"formEval   ( eqn ( IVar ( Para ''UniMsg_proc'' iInv1') )   (Const iInv2')) s " and
    b7:"iInv2\<noteq>iInv2'"
 shows False
 proof -
  have c1:"formEval ( eqn ( IVar ( Global ''Dir_HeadPtr'') )   (Const iInv2) )  s"
    by(metis a2 a3 a4 a5 a6 a7 lemmaOnPtrAtGetX)
  have c2:"formEval ( eqn ( IVar ( Global ''Dir_HeadPtr'') )   (Const iInv2') )  s"
    by(metis b2 b3 b4 b5 b6 a7 lemmaOnPtrAtGetX)
 with c1 b7  show False
  by auto
qed

lemma lemmaOnPtr3[intro]:
  assumes  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "
    and a5:"formEval ( eqn ( IVar ( Para ''UniMsg_Cmd'' iInv1) )  ( Const UNI_Get )) s" and
    a6:"formEval   ( eqn ( IVar ( Para ''UniMsg_proc'' iInv1) )   (Const iInv2)) s " and
    a7:"\<forall>f. f \<in> (invariants   N) \<longrightarrow> formEval f s " and
    b2:"iInv1' \<le> N" and  b3:"iInv2' \<le> N" and  b4:"iInv1'~=iInv2'  "
    and b5:"formEval ( eqn ( IVar ( Para ''UniMsg_Cmd'' iInv1') )  ( Const UNI_GetX )) s" and
    b6:"formEval   ( eqn ( IVar ( Para ''UniMsg_proc'' iInv1') )   (Const iInv2')) s " and
    b7:"iInv2\<noteq>iInv2'"
 shows False
 proof -
  have c1:"formEval ( eqn ( IVar ( Global ''Dir_HeadPtr'') )   (Const iInv2) )  s"
    by(metis a2 a3 a4 a5 a6 a7 lemmaOnPtrAtGet)
  have c2:"formEval ( eqn ( IVar ( Global ''Dir_HeadPtr'') )   (Const iInv2') )  s"
    by(metis b2 b3 b4 b5 b6 a7 lemmaOnPtrAtGetX)
 with c1 b7  show False
  by auto
qed

abbreviation N::"nat"   where   "N\<equiv> M"
lemma NI_FAckVsInv7:
  (*Rule0VsPInv2*)
  assumes   a1:"iInv1 \<le> N" and  a2:"iInv2 \<le> N" and  a3:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_FAck ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -
  fix s

      have "?P1 s"

        apply(cut_tac  a1  a2  a3 , auto)

        done

        then  show "?P1 s\<or> ?P2 s\<or> ?P3 s"

       by blast

 qed
lemma NI_InvVsInv7:
    (*Rule1VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_Inv  iRule1 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

    by (cut_tac  a1  a2  a3  a4 , auto)
 lemma NI_InvAck_1VsInv7:
    (*Rule2VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iRule2 \<le> N" and  a3:"iInv1 \<le> N" and  a4:"iInv2 \<le> N" and  a5:"iInv1~=iInv2  " and  a6:"iRule1~=iRule2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_InvAck_1  iRule1  iRule2 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have "?P2 s"

     by (cut_tac a1 a2 a3 a4 a5 a6,auto  )
   then show "?P1 s\<or>?P2 s\<or>?P3 s"
      by auto
 qed
  lemma NI_InvAck_1_HomeVsInv7:
    (*Rule1VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_InvAck_1_Home  iRule1 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

    by (cut_tac  a1  a2  a3  a4 , auto)
 lemma NI_InvAck_2VsInv7:
    (*Rule1VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_InvAck_2 N  iRule1 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

    by (cut_tac  a1  a2  a3  a4 , auto)
 lemma NI_Local_GetX_GetXVsInv7:
    (*Rule1VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_Local_GetX_GetX  iRule1 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have allCases:"(iRule1=iInv1)   \<or>(iRule1=iInv2)   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 ))   "
	                      by( cut_tac  a1  a2  a3  a4  , auto)
moreover
                {assume b1:"(iRule1=iInv1)"

      have "?P1 s"

        apply(cut_tac  a1  a2  a3  a4  b1 , auto)

        done

        then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

       by blast

	                }
moreover
                {assume b1:"(iRule1=iInv2)"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 ))"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
   ultimately show "?P1 s\<or> ?P2 s\<or> ?P3 s"

	                         by metis

 qed
lemma NI_Local_GetX_Nak1VsInv7:
    (*Rule1VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_Local_GetX_Nak1  iRule1 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have allCases:"(iRule1=iInv1)   \<or>(iRule1=iInv2)   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 ))   "
	                      by( cut_tac  a1  a2  a3  a4  , auto)
moreover
                {assume b1:"(iRule1=iInv1)"

      have "?P1 s"

        apply(cut_tac  a1  a2  a3  a4  b1 , auto)

        done

        then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

       by blast

	                }
moreover
                {assume b1:"(iRule1=iInv2)"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 ))"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
   ultimately show "?P1 s\<or> ?P2 s\<or> ?P3 s"

	                         by metis

 qed
lemma NI_Local_GetX_Nak2VsInv7:
    (*Rule1VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_Local_GetX_Nak2  iRule1 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have allCases:"(iRule1=iInv1)   \<or>(iRule1=iInv2)   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 ))   "
	                      by( cut_tac  a1  a2  a3  a4  , auto)
moreover
                {assume b1:"(iRule1=iInv1)"

      have "?P1 s"

        apply(cut_tac  a1  a2  a3  a4  b1 , auto)

        done

        then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

       by blast

	                }
moreover
                {assume b1:"(iRule1=iInv2)"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 ))"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
   ultimately show "?P1 s\<or> ?P2 s\<or> ?P3 s"

	                         by metis

 qed
lemma NI_Local_GetX_Nak3VsInv7:
    (*Rule1VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_Local_GetX_Nak3  iRule1 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have allCases:"(iRule1=iInv1)   \<or>(iRule1=iInv2)   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 ))   "
	                      by( cut_tac  a1  a2  a3  a4  , auto)
moreover
                {assume b1:"(iRule1=iInv1)"

      have "?P1 s"

        apply(cut_tac  a1  a2  a3  a4  b1 , auto)

        done

        then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

       by blast

	                }
moreover
                {assume b1:"(iRule1=iInv2)"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 ))"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
   ultimately show "?P1 s\<or> ?P2 s\<or> ?P3 s"

	                         by metis

 qed
lemma NI_Local_GetX_PutX1VsInv7:
    (*Rule1VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_Local_GetX_PutX1 N  iRule1 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have allCases:"(iRule1=iInv1)   \<or>(iRule1=iInv2)   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 ))   "
	                      by( cut_tac  a1  a2  a3  a4  , auto)
moreover
                {assume b1:"(iRule1=iInv1)"

      have "?P1 s"

        apply(cut_tac  a1  a2  a3  a4  b1 , auto)

        done

        then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

       by blast

	                }
moreover
                {assume b1:"(iRule1=iInv2)"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 ))"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
   ultimately show "?P1 s\<or> ?P2 s\<or> ?P3 s"

	                         by metis

 qed
lemma NI_Local_GetX_PutX2VsInv7:
    (*Rule1VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_Local_GetX_PutX2 N  iRule1 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have allCases:"(iRule1=iInv1)   \<or>(iRule1=iInv2)   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 ))   "
	                      by( cut_tac  a1  a2  a3  a4  , auto)
moreover
                {assume b1:"(iRule1=iInv1)"

      have "?P1 s"

        apply(cut_tac  a1  a2  a3  a4  b1 , auto)

        done

        then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

       by blast

	                }
moreover
                {assume b1:"(iRule1=iInv2)"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 ))"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
   ultimately show "?P1 s\<or> ?P2 s\<or> ?P3 s"

	                         by metis

 qed
lemma NI_Local_GetX_PutX3VsInv7:
    (*Rule1VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_Local_GetX_PutX3 N  iRule1 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have allCases:"(iRule1=iInv1)   \<or>(iRule1=iInv2)   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 ))   "
	                      by( cut_tac  a1  a2  a3  a4  , auto)
moreover
                {assume b1:"(iRule1=iInv1)"

      have "?P1 s"

        apply(cut_tac  a1  a2  a3  a4  b1 , auto)

        done

        then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

       by blast

	                }
moreover
                {assume b1:"(iRule1=iInv2)"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 ))"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
   ultimately show "?P1 s\<or> ?P2 s\<or> ?P3 s"

	                         by metis

 qed
lemma NI_Local_GetX_PutX4VsInv7:
    (*Rule1VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_Local_GetX_PutX4 N  iRule1 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have allCases:"(iRule1=iInv1)   \<or>(iRule1=iInv2)   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 ))   "
	                      by( cut_tac  a1  a2  a3  a4  , auto)
moreover
                {assume b1:"(iRule1=iInv1)"

      have "?P1 s"

        apply(cut_tac  a1  a2  a3  a4  b1 , auto)

        done

        then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

       by blast

	                }
moreover
                {assume b1:"(iRule1=iInv2)"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 ))"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
   ultimately show "?P1 s\<or> ?P2 s\<or> ?P3 s"

	                         by metis

 qed
lemma NI_Local_GetX_PutX5VsInv7:
    (*Rule1VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_Local_GetX_PutX5 N  iRule1 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have allCases:"(iRule1=iInv1)   \<or>(iRule1=iInv2)   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 ))   "
	                      by( cut_tac  a1  a2  a3  a4  , auto)
moreover
                {assume b1:"(iRule1=iInv1)"

      have "?P1 s"

        apply(cut_tac  a1  a2  a3  a4  b1 , auto)

        done

        then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

       by blast

	                }
moreover
                {assume b1:"(iRule1=iInv2)"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 ))"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
   ultimately show "?P1 s\<or> ?P2 s\<or> ?P3 s"

	                         by metis

 qed
lemma NI_Local_GetX_PutX6VsInv7:
    (*Rule1VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_Local_GetX_PutX6 N  iRule1 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have allCases:"(iRule1=iInv1)   \<or>(iRule1=iInv2)   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 ))   "
	                      by( cut_tac  a1  a2  a3  a4  , auto)
moreover
                {assume b1:"(iRule1=iInv1)"

      have "?P1 s"

        apply(cut_tac  a1  a2  a3  a4  b1 , auto)

        done

        then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

       by blast

	                }
moreover
                {assume b1:"(iRule1=iInv2)"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 ))"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
   ultimately show "?P1 s\<or> ?P2 s\<or> ?P3 s"

	                         by metis

 qed
lemma NI_Local_GetX_PutX7VsInv7:
    (*Rule1VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_Local_GetX_PutX7 N  iRule1 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have allCases:"(iRule1=iInv1)   \<or>(iRule1=iInv2)   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 ))   "
	                      by( cut_tac  a1  a2  a3  a4  , auto)
moreover
                {assume b1:"(iRule1=iInv1)"

      have "?P1 s"

        apply(cut_tac  a1  a2  a3  a4  b1 , auto)

        done

        then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

       by blast

	                }
moreover
                {assume b1:"(iRule1=iInv2)"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 ))"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
   ultimately show "?P1 s\<or> ?P2 s\<or> ?P3 s"

	                         by metis

 qed
lemma NI_Local_GetX_PutX8VsInv7:
    (*Rule2VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iRule2 \<le> N" and  a3:"iInv1 \<le> N" and  a4:"iInv2 \<le> N" and  a5:"iInv1~=iInv2  " and  a6:"iRule1~=iRule2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_Local_GetX_PutX8 N  iRule1  iRule2 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have allCases:"(iRule1=iInv1\<and>iRule2=iInv2)   \<or>(iRule1=iInv1\<and>(iRule2~=iInv1 \<and>iRule2~=iInv2 ))   \<or>(iRule1=iInv2\<and>iRule2=iInv1)   \<or>(iRule1=iInv2\<and>(iRule2~=iInv1 \<and>iRule2~=iInv2 ))   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 )\<and>iRule2=iInv1)   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 )\<and>iRule2=iInv2)   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 )\<and>(iRule2~=iInv1 \<and>iRule2~=iInv2 ))   "
	                      by( cut_tac  a1  a2  a3  a4  a5  a6  , auto)
moreover
                {assume b1:"(iRule1=iInv1\<and>iRule2=iInv2)"

      have "?P1 s"

        apply(cut_tac  a1  a2  a3  a4  a5  a6  b1 , auto)

        done

        then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

       by blast

	                }
moreover
                {assume b1:"(iRule1=iInv1\<and>(iRule2~=iInv1 \<and>iRule2~=iInv2 ))"

      have "?P1 s"

        apply(cut_tac  a1  a2  a3  a4  a5  a6  b1 , auto)

        done

        then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

       by blast

	                }
moreover
                {assume b1:"(iRule1=iInv2\<and>iRule2=iInv1)"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  a5  a6  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"(iRule1=iInv2\<and>(iRule2~=iInv1 \<and>iRule2~=iInv2 ))"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  a5  a6  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 )\<and>iRule2=iInv1)"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  a5  a6  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 )\<and>iRule2=iInv2)"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  a5  a6  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 )\<and>(iRule2~=iInv1 \<and>iRule2~=iInv2 ))"

                  have "?P1 s \<or> ?P2 s"

  apply(cut_tac  a1  a2  a3  a4  a5  a6  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
   ultimately show "?P1 s\<or> ?P2 s\<or> ?P3 s"

	                         by metis

 qed
lemma NI_Local_GetX_PutX8_homeVsInv7:
    (*Rule1VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_Local_GetX_PutX8_home N  iRule1 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have allCases:"(iRule1=iInv1)   \<or>(iRule1=iInv2)   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 ))   "
	                      by( cut_tac  a1  a2  a3  a4  , auto)
moreover
                {assume b1:"(iRule1=iInv1)"

      have "?P1 s"

        apply(cut_tac  a1  a2  a3  a4  b1 , auto)

        done

        then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

       by blast

	                }
moreover
                {assume b1:"(iRule1=iInv2)"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 ))"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
   ultimately show "?P1 s\<or> ?P2 s\<or> ?P3 s"

	                         by metis

 qed
lemma NI_Local_GetX_PutX9VsInv7:
    (*Rule1VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_Local_GetX_PutX9 N  iRule1 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have allCases:"(iRule1=iInv1)   \<or>(iRule1=iInv2)   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 ))   "
	                      by( cut_tac  a1  a2  a3  a4  , auto)
moreover
                {assume b1:"(iRule1=iInv1)"

      have "?P1 s"

        apply(cut_tac  a1  a2  a3  a4  b1 , auto)

        done

        then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

       by blast

	                }
moreover
                {assume b1:"(iRule1=iInv2)"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 ))"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
   ultimately show "?P1 s\<or> ?P2 s\<or> ?P3 s"

	                         by metis

 qed
lemma NI_Local_GetX_PutX10VsInv7:
    (*Rule2VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iRule2 \<le> N" and  a3:"iInv1 \<le> N" and  a4:"iInv2 \<le> N" and  a5:"iInv1~=iInv2  " and  a6:"iRule1~=iRule2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_Local_GetX_PutX10 N  iRule1  iRule2 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have allCases:"(iRule1=iInv1\<and>iRule2=iInv2)   \<or>(iRule1=iInv1\<and>(iRule2~=iInv1 \<and>iRule2~=iInv2 ))   \<or>(iRule1=iInv2\<and>iRule2=iInv1)   \<or>(iRule1=iInv2\<and>(iRule2~=iInv1 \<and>iRule2~=iInv2 ))   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 )\<and>iRule2=iInv1)   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 )\<and>iRule2=iInv2)   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 )\<and>(iRule2~=iInv1 \<and>iRule2~=iInv2 ))   "
	                      by( cut_tac  a1  a2  a3  a4  a5  a6  , auto)
moreover
                {assume b1:"(iRule1=iInv1\<and>iRule2=iInv2)"

      have "?P1 s"

        apply(cut_tac  a1  a2  a3  a4  a5  a6  b1 , auto)

        done

        then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

       by blast

	                }
moreover
                {assume b1:"(iRule1=iInv1\<and>(iRule2~=iInv1 \<and>iRule2~=iInv2 ))"

      have "?P1 s"

        apply(cut_tac  a1  a2  a3  a4  a5  a6  b1 , auto)

        done

        then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

       by blast

	                }
moreover
                {assume b1:"(iRule1=iInv2\<and>iRule2=iInv1)"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  a5  a6  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"(iRule1=iInv2\<and>(iRule2~=iInv1 \<and>iRule2~=iInv2 ))"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  a5  a6  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 )\<and>iRule2=iInv1)"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  a5  a6  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 )\<and>iRule2=iInv2)"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  a5  a6  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 )\<and>(iRule2~=iInv1 \<and>iRule2~=iInv2 ))"

                  have "?P1 s \<or> ?P2 s"

  apply(cut_tac  a1  a2  a3  a4  a5  a6  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
   ultimately show "?P1 s\<or> ?P2 s\<or> ?P3 s"

	                         by metis

 qed
lemma NI_Local_GetX_PutX10_homeVsInv7:
    (*Rule1VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_Local_GetX_PutX10_home N  iRule1 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have allCases:"(iRule1=iInv1)   \<or>(iRule1=iInv2)   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 ))   "
	                      by( cut_tac  a1  a2  a3  a4  , auto)
moreover
                {assume b1:"(iRule1=iInv1)"

      have "?P1 s"

        apply(cut_tac  a1  a2  a3  a4  b1 , auto)

        done

        then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

       by blast

	                }
moreover
                {assume b1:"(iRule1=iInv2)"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 ))"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
   ultimately show "?P1 s\<or> ?P2 s\<or> ?P3 s"

	                         by metis

 qed
lemma NI_Local_GetX_PutX11VsInv7:
    (*Rule1VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_Local_GetX_PutX11 N  iRule1 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have allCases:"(iRule1=iInv1)   \<or>(iRule1=iInv2)   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 ))   "
	                      by( cut_tac  a1  a2  a3  a4  , auto)
moreover
                {assume b1:"(iRule1=iInv1)"

      have "?P1 s"

        apply(cut_tac  a1  a2  a3  a4  b1 , auto)

        done

        then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

       by blast

	                }
moreover
                {assume b1:"(iRule1=iInv2)"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 ))"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
   ultimately show "?P1 s\<or> ?P2 s\<or> ?P3 s"

	                         by metis

 qed
lemma NI_Local_Get_GetVsInv7:
    (*Rule1VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_Local_Get_Get  iRule1 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have allCases:"(iRule1=iInv1)   \<or>(iRule1=iInv2)   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 ))   "
	                      by( cut_tac  a1  a2  a3  a4  , auto)
moreover
                {assume b1:"(iRule1=iInv1)"

      have "?P3 s"

        apply(   cut_tac  a1  a2  a3  a4  b1 , simp)

        apply(rule_tac x=" (neg ( andForm ( eqn ( IVar ( Global ''ShWbMsg_Cmd'') )  ( Const SHWB_FAck ))    ( eqn ( IVar ( Global ''Dir_Pending'') )  ( Const false ))  ) ) " in exI,auto)

        done

       then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"(iRule1=iInv2)"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 ))"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
   ultimately show "?P1 s\<or> ?P2 s\<or> ?P3 s"

	                         by metis

 qed
lemma NI_Local_Get_Nak1VsInv7:
    (*Rule1VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_Local_Get_Nak1  iRule1 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have allCases:"(iRule1=iInv1)   \<or>(iRule1=iInv2)   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 ))   "
	                      by( cut_tac  a1  a2  a3  a4  , auto)
moreover
                {assume b1:"(iRule1=iInv1)"

      have "?P1 s"

        apply(cut_tac  a1  a2  a3  a4  b1 , auto)

        done

        then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

       by blast

	                }
moreover
                {assume b1:"(iRule1=iInv2)"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 ))"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
   ultimately show "?P1 s\<or> ?P2 s\<or> ?P3 s"

	                         by metis

 qed
lemma NI_Local_Get_Nak2VsInv7:
    (*Rule1VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_Local_Get_Nak2  iRule1 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have allCases:"(iRule1=iInv1)   \<or>(iRule1=iInv2)   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 ))   "
	                      by( cut_tac  a1  a2  a3  a4  , auto)
moreover
                {assume b1:"(iRule1=iInv1)"

      have "?P1 s"

        apply(cut_tac  a1  a2  a3  a4  b1 , auto)

        done

        then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

       by blast

	                }
moreover
                {assume b1:"(iRule1=iInv2)"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 ))"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
   ultimately show "?P1 s\<or> ?P2 s\<or> ?P3 s"

	                         by metis

 qed
lemma NI_Local_Get_Nak3VsInv7:
    (*Rule1VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_Local_Get_Nak3  iRule1 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have allCases:"(iRule1=iInv1)   \<or>(iRule1=iInv2)   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 ))   "
	                      by( cut_tac  a1  a2  a3  a4  , auto)
moreover
                {assume b1:"(iRule1=iInv1)"

      have "?P1 s"

        apply(cut_tac  a1  a2  a3  a4  b1 , auto)

        done

        then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

       by blast

	                }
moreover
                {assume b1:"(iRule1=iInv2)"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 ))"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
   ultimately show "?P1 s\<or> ?P2 s\<or> ?P3 s"

	                         by metis

 qed
lemma NI_Local_Get_Put1VsInv7:
    (*Rule1VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_Local_Get_Put1 N  iRule1 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have allCases:"(iRule1=iInv1)   \<or>(iRule1=iInv2)   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 ))   "
	                      by( cut_tac  a1  a2  a3  a4  , auto)
moreover
                {assume b1:"(iRule1=iInv1)"

      have "?P1 s"

        apply(cut_tac  a1  a2  a3  a4  b1 , auto)

        done

        then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

       by blast

	                }
moreover
                {assume b1:"(iRule1=iInv2)"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 ))"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
   ultimately show "?P1 s\<or> ?P2 s\<or> ?P3 s"

	                         by metis

 qed
lemma NI_Local_Get_Put2VsInv7:
    (*Rule1VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_Local_Get_Put2  iRule1 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have allCases:"(iRule1=iInv1)   \<or>(iRule1=iInv2)   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 ))   "
	                      by( cut_tac  a1  a2  a3  a4  , auto)
moreover
                {assume b1:"(iRule1=iInv1)"

      have "?P1 s"

        apply(cut_tac  a1  a2  a3  a4  b1 , auto)

        done

        then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

       by blast

	                }
moreover
                {assume b1:"(iRule1=iInv2)"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 ))"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
   ultimately show "?P1 s\<or> ?P2 s\<or> ?P3 s"

	                         by metis

 qed
lemma NI_Local_Get_Put3VsInv7:
    (*Rule1VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_Local_Get_Put3  iRule1 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have allCases:"(iRule1=iInv1)   \<or>(iRule1=iInv2)   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 ))   "
	                      by( cut_tac  a1  a2  a3  a4  , auto)
moreover
                {assume b1:"(iRule1=iInv1)"

      have "?P1 s"

        apply(cut_tac  a1  a2  a3  a4  b1 , auto)

        done

        then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

       by blast

	                }
moreover
                {assume b1:"(iRule1=iInv2)"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 ))"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
   ultimately show "?P1 s\<or> ?P2 s\<or> ?P3 s"

	                         by metis

 qed
lemma NI_Local_PutVsInv7:
    (*Rule0VsPInv2*)
  assumes   a1:"iInv1 \<le> N" and  a2:"iInv2 \<le> N" and  a3:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_Local_Put ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have "?P2 s"

     by (cut_tac a1 a2 a3, auto  )
   then show "?P1 s\<or>?P2 s\<or>?P3 s"
      by auto
 qed
  lemma NI_Local_PutXAcksDoneVsInv7:
    (*Rule0VsPInv2*)
  assumes   a1:"iInv1 \<le> N" and  a2:"iInv2 \<le> N" and  a3:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_Local_PutXAcksDone ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have "?P2 s"

     by (cut_tac a1 a2 a3, auto  )
   then show "?P1 s\<or>?P2 s\<or>?P3 s"
      by auto
 qed
  lemma NI_NakVsInv7:
    (*Rule1VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_Nak  iRule1 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have allCases:"(iRule1=iInv1)   \<or>(iRule1=iInv2)   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 ))   "
	                      by( cut_tac  a1  a2  a3  a4  , auto)
moreover
                {assume b1:"(iRule1=iInv1)"

      have "?P1 s"

        apply(cut_tac  a1  a2  a3  a4  b1 , auto)

        done

        then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

       by blast

	                }
moreover
                {assume b1:"(iRule1=iInv2)"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 ))"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
   ultimately show "?P1 s\<or> ?P2 s\<or> ?P3 s"

	                         by metis

 qed
lemma NI_Nak_ClearVsInv7:
    (*Rule0VsPInv2*)
  assumes   a1:"iInv1 \<le> N" and  a2:"iInv2 \<le> N" and  a3:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_Nak_Clear ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have "?P2 s"

     by (cut_tac a1 a2 a3, auto  )
   then show "?P1 s\<or>?P2 s\<or>?P3 s"
      by auto
 qed
  lemma NI_Nak_HomeVsInv7:
    (*Rule0VsPInv2*)
  assumes   a1:"iInv1 \<le> N" and  a2:"iInv2 \<le> N" and  a3:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_Nak_Home ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have "?P2 s"

     by (cut_tac a1 a2 a3, auto  )
   then show "?P1 s\<or>?P2 s\<or>?P3 s"
      by auto
 qed
  lemma NI_Remote_GetX_NakVsInv7:
    (*Rule2VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iRule2 \<le> N" and  a3:"iInv1 \<le> N" and  a4:"iInv2 \<le> N" and  a5:"iInv1~=iInv2  " and  a6:"iRule1~=iRule2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_Remote_GetX_Nak  iRule1  iRule2 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have allCases:"(iRule1=iInv1\<and>iRule2=iInv2)   \<or>(iRule1=iInv1\<and>(iRule2~=iInv1 \<and>iRule2~=iInv2 ))   \<or>(iRule1=iInv2\<and>iRule2=iInv1)   \<or>(iRule1=iInv2\<and>(iRule2~=iInv1 \<and>iRule2~=iInv2 ))   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 )\<and>iRule2=iInv1)   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 )\<and>iRule2=iInv2)   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 )\<and>(iRule2~=iInv1 \<and>iRule2~=iInv2 ))   "
	                      by( cut_tac  a1  a2  a3  a4  a5  a6  , auto)
moreover
                {assume b1:"(iRule1=iInv1\<and>iRule2=iInv2)"

      have "?P1 s"

        apply(cut_tac  a1  a2  a3  a4  a5  a6  b1 , auto)

        done

        then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

       by blast

	                }
moreover
                {assume b1:"(iRule1=iInv1\<and>(iRule2~=iInv1 \<and>iRule2~=iInv2 ))"

      have "?P1 s"

        apply(cut_tac  a1  a2  a3  a4  a5  a6  b1 , auto)

        done

        then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

       by blast

	                }
moreover
                {assume b1:"(iRule1=iInv2\<and>iRule2=iInv1)"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  a5  a6  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"(iRule1=iInv2\<and>(iRule2~=iInv1 \<and>iRule2~=iInv2 ))"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  a5  a6  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 )\<and>iRule2=iInv1)"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  a5  a6  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 )\<and>iRule2=iInv2)"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  a5  a6  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 )\<and>(iRule2~=iInv1 \<and>iRule2~=iInv2 ))"

                  have "?P1 s \<or> ?P2 s"

  apply(cut_tac  a1  a2  a3  a4  a5  a6  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
   ultimately show "?P1 s\<or> ?P2 s\<or> ?P3 s"

	                         by metis

 qed
lemma NI_Remote_GetX_Nak_HomeVsInv7:
    (*Rule1VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_Remote_GetX_Nak_Home  iRule1 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

    by (cut_tac  a1  a2  a3  a4 , auto)
 lemma NI_Remote_GetX_PutXVsInv7Aux:
    (*Rule2VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iRule2 \<le> N" and  a3:"iInv1 \<le> N" and  a4:"iInv2 \<le> N" and  a5:"iInv1~=iInv2  " and  a6:"iRule1~=iRule2  "

  shows  "invHoldForRule'' s (inv7  iInv1  iInv2 ) (NI_Remote_GetX_PutX  iRule1  iRule2 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have allCases:"(iRule1=iInv1\<and>iRule2=iInv2)   \<or>(iRule1=iInv1\<and>(iRule2~=iInv1 \<and>iRule2~=iInv2 ))   \<or>(iRule1=iInv2\<and>iRule2=iInv1)   \<or>(iRule1=iInv2\<and>(iRule2~=iInv1 \<and>iRule2~=iInv2 ))   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 )\<and>iRule2=iInv1)   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 )\<and>iRule2=iInv2)   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 )\<and>(iRule2~=iInv1 \<and>iRule2~=iInv2 ))   "
	                      by( cut_tac  a1  a2  a3  a4  a5  a6  , auto)
moreover
                {assume b1:"(iRule1=iInv1\<and>iRule2=iInv2)"

      have "?P1 s"

        apply(cut_tac  a1  a2  a3  a4  a5  a6  b1 , auto)

        done

        then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

       by blast

	                }
moreover
                {assume b1:"(iRule1=iInv1\<and>(iRule2~=iInv1 \<and>iRule2~=iInv2 ))"

      have "?P1 s"

        apply(cut_tac  a1  a2  a3  a4  a5  a6  b1 , auto)

        done

        then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

       by blast

	                }
moreover
                {assume b1:"(iRule1=iInv2\<and>iRule2=iInv1)"

      have "invHoldForRule3' s (inv7  iInv1  iInv2 ) (NI_Remote_GetX_PutX  iRule1  iRule2 ) (invariants   N)"

        apply(   cut_tac  a1  a2  a3  a4  a5  a6  b1 , simp)

        apply(rule_tac x=" (neg ( andForm ( eqn ( IVar ( Para ''UniMsg_Cmd'' iInv1) )  ( Const UNI_Get ))    ( eqn ( IVar ( Para ''CacheState'' iInv1) )  ( Const CACHE_E ))  ) ) " in exI,auto)

        done
     then have "?P3 s"
       by(rule weak3)

       then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"(iRule1=iInv2\<and>(iRule2~=iInv1 \<and>iRule2~=iInv2 ))"

       have "invHoldForRule3' s (inv7  iInv1  iInv2 ) (NI_Remote_GetX_PutX  iRule1  iRule2 ) (invariants   N)"

        apply(   cut_tac  a1  a2  a3  a4  a5  a6  b1 , simp)

        apply(rule_tac x=" (neg ( andForm ( andForm ( eqn ( IVar ( Para ''UniMsg_Cmd'' iInv1) )  ( Const UNI_Get ))    ( eqn ( IVar ( Para ''CacheState'' iRule2) )  ( Const CACHE_E ))  )    ( eqn ( IVar ( Para ''UniMsg_proc'' iInv1) )   (Const iInv2))  ) ) " in exI,auto)

        done

       then have "?P3 s"
        by(rule weak3)

       then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 )\<and>iRule2=iInv1)"

      have "invHoldForRule3' s (inv7  iInv1  iInv2 ) (NI_Remote_GetX_PutX  iRule1  iRule2 ) (invariants   N)"

        apply(   cut_tac  a1  a2  a3  a4  a5  a6  b1 , simp)

        apply(rule_tac x=" (neg ( andForm ( eqn ( IVar ( Para ''UniMsg_Cmd'' iInv1) )  ( Const UNI_Get ))    ( eqn ( IVar ( Para ''CacheState'' iInv1) )  ( Const CACHE_E ))  ) ) " in exI,auto)

        done
      then have "?P3 s"
        by(rule weak3)

       then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 )\<and>iRule2=iInv2)"

        have "invHoldForRule3' s (inv7  iInv1  iInv2 ) (NI_Remote_GetX_PutX  iRule1  iRule2 ) (invariants   N)"

        apply(   cut_tac  a1  a2  a3  a4  a5  a6  b1 , simp)

        apply(rule_tac x=" (neg ( andForm ( andForm ( andForm ( eqn ( IVar ( Para ''UniMsg_Cmd'' iInv1) )  ( Const UNI_Get ))    ( eqn ( IVar ( Para ''UniMsg_proc'' iRule1) )   (Const iInv2))  )    ( eqn ( IVar ( Para ''UniMsg_Cmd'' iRule1) )  ( Const UNI_GetX ))  )    ( eqn ( IVar ( Para ''UniMsg_proc'' iInv1) )   (Const iInv2))  ) ) " in exI,auto)

        done
      then have "?P3 s"
        by(rule weak3)

       then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 )\<and>(iRule2~=iInv1 \<and>iRule2~=iInv2 ))"

                  have "?P3 s"
                  proof( unfold invHoldForRule3''_def,simp only:Let_def,( rule impI)+,                   rule ccontr)
                   assume c1:"\<forall>f'. f' \<in> invariants N \<longrightarrow> formEval f' s" and c2:"formEval (pre (NI_Remote_GetX_PutX iRule1 iRule2)) s "
                    and c3:"\<not> formEval (substFormByStatement (inv7 iInv1 iInv2) (act (NI_Remote_GetX_PutX iRule1 iRule2))) s"
                   have d1:" formEval (eqn (IVar (Para ''UniMsg_Cmd'' iInv1)) (Const UNI_Get)) s" by(cut_tac b1 c2 c3, auto)
                   have d2:"formEval (eqn (IVar (Para ''UniMsg_Cmd'' iRule1)) (Const UNI_GetX)) s" by(cut_tac b1 c2 c3, auto)
                   have d3:"formEval (eqn (IVar (Para ''UniMsg_proc'' iInv1)) (Const iInv2)) s"  by(cut_tac b1 c2 c3, auto)
                   have d4:"formEval (eqn (IVar (Para ''UniMsg_proc'' iRule1)) (Const iRule2)) s"  by(cut_tac b1 c2 c3, auto)
                    show False thm lemmaOnPtr2
                    apply(cut_tac a1 a2 a3 a4 a5 a6 b1 c1 c2 c3 d1 d2 d3 d4,blast )
                    done
                   qed

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
   ultimately show "?P1 s\<or> ?P2 s\<or> ?P3 s"

	                         by metis

 qed
lemma NI_Remote_GetX_PutXVsInv7:
    (*Rule2VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iRule2 \<le> N" and  a3:"iInv1 \<le> N" and  a4:"iInv2 \<le> N" and  a5:"iInv1~=iInv2  " and  a6:"iRule1~=iRule2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_Remote_GetX_PutX  iRule1  iRule2 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")
by (metis NI_Remote_GetX_PutXVsInv7Aux a2 a3 a4 a5 a6 local.a1 strengthEn)

lemma NI_Remote_GetX_PutX_HomeVsInv7:
    (*Rule1VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_Remote_GetX_PutX_Home  iRule1 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

    by (cut_tac  a1  a2  a3  a4 , auto)
 lemma NI_Remote_Get_Nak1VsInv7:
    (*Rule1VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_Remote_Get_Nak1  iRule1 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

    by (cut_tac  a1  a2  a3  a4 , auto)
 lemma NI_Remote_Get_Nak2VsInv7:
    (*Rule2VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iRule2 \<le> N" and  a3:"iInv1 \<le> N" and  a4:"iInv2 \<le> N" and  a5:"iInv1~=iInv2  " and  a6:"iRule1~=iRule2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_Remote_Get_Nak2  iRule1  iRule2 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have allCases:"(iRule1=iInv1\<and>iRule2=iInv2)   \<or>(iRule1=iInv1\<and>(iRule2~=iInv1 \<and>iRule2~=iInv2 ))   \<or>(iRule1=iInv2\<and>iRule2=iInv1)   \<or>(iRule1=iInv2\<and>(iRule2~=iInv1 \<and>iRule2~=iInv2 ))   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 )\<and>iRule2=iInv1)   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 )\<and>iRule2=iInv2)   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 )\<and>(iRule2~=iInv1 \<and>iRule2~=iInv2 ))   "
	                      by( cut_tac  a1  a2  a3  a4  a5  a6  , auto)
moreover
                {assume b1:"(iRule1=iInv1\<and>iRule2=iInv2)"

      have "?P1 s"

        apply(cut_tac  a1  a2  a3  a4  a5  a6  b1 , auto)

        done

        then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

       by blast

	                }
moreover
                {assume b1:"(iRule1=iInv1\<and>(iRule2~=iInv1 \<and>iRule2~=iInv2 ))"

      have "?P1 s"

        apply(cut_tac  a1  a2  a3  a4  a5  a6  b1 , auto)

        done

        then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

       by blast

	                }
moreover
                {assume b1:"(iRule1=iInv2\<and>iRule2=iInv1)"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  a5  a6  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"(iRule1=iInv2\<and>(iRule2~=iInv1 \<and>iRule2~=iInv2 ))"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  a5  a6  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 )\<and>iRule2=iInv1)"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  a5  a6  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 )\<and>iRule2=iInv2)"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  a5  a6  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 )\<and>(iRule2~=iInv1 \<and>iRule2~=iInv2 ))"

                  have "?P1 s \<or> ?P2 s"

  apply(cut_tac  a1  a2  a3  a4  a5  a6  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
   ultimately show "?P1 s\<or> ?P2 s\<or> ?P3 s"

	                         by metis

 qed
lemma NI_Remote_Get_Put1VsInv7:
    (*Rule1VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_Remote_Get_Put1  iRule1 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

    by (cut_tac  a1  a2  a3  a4 , auto)
 lemma NI_Remote_Get_Put2VsInv7:
    (*Rule2VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iRule2 \<le> N" and  a3:"iInv1 \<le> N" and  a4:"iInv2 \<le> N" and  a5:"iInv1~=iInv2  " and  a6:"iRule1~=iRule2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_Remote_Get_Put2  iRule1  iRule2 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have allCases:"(iRule1=iInv1\<and>iRule2=iInv2)   \<or>(iRule1=iInv1\<and>(iRule2~=iInv1 \<and>iRule2~=iInv2 ))   \<or>(iRule1=iInv2\<and>iRule2=iInv1)   \<or>(iRule1=iInv2\<and>(iRule2~=iInv1 \<and>iRule2~=iInv2 ))   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 )\<and>iRule2=iInv1)   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 )\<and>iRule2=iInv2)   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 )\<and>(iRule2~=iInv1 \<and>iRule2~=iInv2 ))   "
	                      by( cut_tac  a1  a2  a3  a4  a5  a6  , auto)
moreover
                {assume b1:"(iRule1=iInv1\<and>iRule2=iInv2)"

      have "?P1 s"

        apply(cut_tac  a1  a2  a3  a4  a5  a6  b1 , auto)

        done

        then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

       by blast

	                }
moreover
                {assume b1:"(iRule1=iInv1\<and>(iRule2~=iInv1 \<and>iRule2~=iInv2 ))"

      have "?P1 s"

        apply(cut_tac  a1  a2  a3  a4  a5  a6  b1 , auto)

        done

        then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

       by blast

	                }
moreover
                {assume b1:"(iRule1=iInv2\<and>iRule2=iInv1)"

      have "?P1 s"

        apply(cut_tac  a1  a2  a3  a4  a5  a6  b1 , auto)

        done

        then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

       by blast

	                }
moreover
                {assume b1:"(iRule1=iInv2\<and>(iRule2~=iInv1 \<and>iRule2~=iInv2 ))"

      have "?P1 s"

        apply(cut_tac  a1  a2  a3  a4  a5  a6  b1 , auto)

        done

        then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

       by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 )\<and>iRule2=iInv1)"

      have "?P1 s"

        apply(cut_tac  a1  a2  a3  a4  a5  a6  b1 , auto)

        done

        then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

       by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 )\<and>iRule2=iInv2)"

      have "?P1 s"

        apply(cut_tac  a1  a2  a3  a4  a5  a6  b1 , auto)

        done

        then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

       by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 )\<and>(iRule2~=iInv1 \<and>iRule2~=iInv2 ))"

                  have "?P1 s \<or> ?P2 s"

  apply(cut_tac  a1  a2  a3  a4  a5  a6  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
   ultimately show "?P1 s\<or> ?P2 s\<or> ?P3 s"

	                         by metis

 qed
lemma NI_Remote_PutVsInv7:
    (*Rule1VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_Remote_Put  iRule1 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have allCases:"(iRule1=iInv1)   \<or>(iRule1=iInv2)   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 ))   "
	                      by( cut_tac  a1  a2  a3  a4  , auto)
moreover
                {assume b1:"(iRule1=iInv1)"

      have "?P1 s"

        apply(cut_tac  a1  a2  a3  a4  b1 , auto)

        done

        then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

       by blast

	                }
moreover
                {assume b1:"(iRule1=iInv2)"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 ))"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
   ultimately show "?P1 s\<or> ?P2 s\<or> ?P3 s"

	                         by metis

 qed
lemma NI_Remote_PutXVsInv7:
    (*Rule1VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_Remote_PutX  iRule1 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have allCases:"(iRule1=iInv1)   \<or>(iRule1=iInv2)   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 ))   "
	                      by( cut_tac  a1  a2  a3  a4  , auto)
moreover
                {assume b1:"(iRule1=iInv1)"

      have "?P1 s"

        apply(cut_tac  a1  a2  a3  a4  b1 , auto)

        done

        then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

       by blast

	                }
moreover
                {assume b1:"(iRule1=iInv2)"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 ))"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
   ultimately show "?P1 s\<or> ?P2 s\<or> ?P3 s"

	                         by metis

 qed
lemma NI_ReplaceVsInv7:
    (*Rule1VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_Replace  iRule1 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

    by (cut_tac  a1  a2  a3  a4 , auto)
 lemma NI_ReplaceHomeVsInv7:
    (*Rule0VsPInv2*)
  assumes   a1:"iInv1 \<le> N" and  a2:"iInv2 \<le> N" and  a3:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_ReplaceHome ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have "?P2 s"

     by (cut_tac a1 a2 a3, auto  )
   then show "?P1 s\<or>?P2 s\<or>?P3 s"
      by auto
 qed
  lemma NI_ReplaceHomeShrVldVsInv7:
    (*Rule0VsPInv2*)
  assumes   a1:"iInv1 \<le> N" and  a2:"iInv2 \<le> N" and  a3:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_ReplaceHomeShrVld ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have "?P2 s"

     by (cut_tac a1 a2 a3, auto  )
   then show "?P1 s\<or>?P2 s\<or>?P3 s"
      by auto
 qed
  lemma NI_ReplaceShrVldVsInv7:
    (*Rule1VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_ReplaceShrVld  iRule1 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

    by (cut_tac  a1  a2  a3  a4 , auto)
 lemma NI_ShWbVsInv7:
  (*Rule0VsPInv2*)
  assumes   a1:"iInv1 \<le> N" and  a2:"iInv2 \<le> N" and  a3:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_ShWb N ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -
  fix s

      have "?P1 s"

        apply(cut_tac  a1  a2  a3 , auto)

        done

        then  show "?P1 s\<or> ?P2 s\<or> ?P3 s"

       by blast

 qed
lemma NI_WbVsInv7:
    (*Rule0VsPInv2*)
  assumes   a1:"iInv1 \<le> N" and  a2:"iInv2 \<le> N" and  a3:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (NI_Wb ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have "?P2 s"

     by (cut_tac a1 a2 a3, auto  )
   then show "?P1 s\<or>?P2 s\<or>?P3 s"
      by auto
 qed
  lemma PI_Local_GetX_GetX1VsInv7:
    (*Rule0VsPInv2*)
  assumes   a1:"iInv1 \<le> N" and  a2:"iInv2 \<le> N" and  a3:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (PI_Local_GetX_GetX1 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have "?P2 s"

     by (cut_tac a1 a2 a3, auto  )
   then show "?P1 s\<or>?P2 s\<or>?P3 s"
      by auto
 qed
  lemma PI_Local_GetX_GetX2VsInv7:
    (*Rule0VsPInv2*)
  assumes   a1:"iInv1 \<le> N" and  a2:"iInv2 \<le> N" and  a3:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (PI_Local_GetX_GetX2 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have "?P2 s"

     by (cut_tac a1 a2 a3, auto  )
   then show "?P1 s\<or>?P2 s\<or>?P3 s"
      by auto
 qed
  lemma PI_Local_GetX_PutX1VsInv7:
    (*Rule0VsPInv2*)
  assumes   a1:"iInv1 \<le> N" and  a2:"iInv2 \<le> N" and  a3:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (PI_Local_GetX_PutX1 N ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have "?P2 s"

     by (cut_tac a1 a2 a3, auto  )
   then show "?P1 s\<or>?P2 s\<or>?P3 s"
      by auto
 qed
  lemma PI_Local_GetX_PutX2VsInv7:
    (*Rule0VsPInv2*)
  assumes   a1:"iInv1 \<le> N" and  a2:"iInv2 \<le> N" and  a3:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (PI_Local_GetX_PutX2 N ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have "?P2 s"

     by (cut_tac a1 a2 a3, auto  )
   then show "?P1 s\<or>?P2 s\<or>?P3 s"
      by auto
 qed
  lemma PI_Local_GetX_PutX3VsInv7:
    (*Rule0VsPInv2*)
  assumes   a1:"iInv1 \<le> N" and  a2:"iInv2 \<le> N" and  a3:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (PI_Local_GetX_PutX3 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have "?P2 s"

     by (cut_tac a1 a2 a3, auto  )
   then show "?P1 s\<or>?P2 s\<or>?P3 s"
      by auto
 qed
  lemma PI_Local_GetX_PutX4VsInv7:
    (*Rule0VsPInv2*)
  assumes   a1:"iInv1 \<le> N" and  a2:"iInv2 \<le> N" and  a3:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (PI_Local_GetX_PutX4 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have "?P2 s"

     by (cut_tac a1 a2 a3, auto  )
   then show "?P1 s\<or>?P2 s\<or>?P3 s"
      by auto
 qed
  lemma PI_Local_Get_GetVsInv7:
    (*Rule0VsPInv2*)
  assumes   a1:"iInv1 \<le> N" and  a2:"iInv2 \<le> N" and  a3:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (PI_Local_Get_Get ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have "?P2 s"

     by (cut_tac a1 a2 a3, auto  )
   then show "?P1 s\<or>?P2 s\<or>?P3 s"
      by auto
 qed
  lemma PI_Local_Get_PutVsInv7:
    (*Rule0VsPInv2*)
  assumes   a1:"iInv1 \<le> N" and  a2:"iInv2 \<le> N" and  a3:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (PI_Local_Get_Put ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have "?P2 s"

     by (cut_tac a1 a2 a3, auto  )
   then show "?P1 s\<or>?P2 s\<or>?P3 s"
      by auto
 qed
  lemma PI_Local_PutXVsInv7:
    (*Rule0VsPInv2*)
  assumes   a1:"iInv1 \<le> N" and  a2:"iInv2 \<le> N" and  a3:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (PI_Local_PutX ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have "?P2 s"

     by (cut_tac a1 a2 a3, auto  )
   then show "?P1 s\<or>?P2 s\<or>?P3 s"
      by auto
 qed
  lemma PI_Local_ReplaceVsInv7:
    (*Rule0VsPInv2*)
  assumes   a1:"iInv1 \<le> N" and  a2:"iInv2 \<le> N" and  a3:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (PI_Local_Replace ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have "?P2 s"

     by (cut_tac a1 a2 a3, auto  )
   then show "?P1 s\<or>?P2 s\<or>?P3 s"
      by auto
 qed
  lemma PI_Remote_GetVsInv7:
    (*Rule1VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (PI_Remote_Get  iRule1 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have allCases:"(iRule1=iInv1)   \<or>(iRule1=iInv2)   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 ))   "
	                      by( cut_tac  a1  a2  a3  a4  , auto)
moreover
                {assume b1:"(iRule1=iInv1)"

      have "?P1 s"

        apply(cut_tac  a1  a2  a3  a4  b1 , auto)

        done

        then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

       by blast

	                }
moreover
                {assume b1:"(iRule1=iInv2)"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 ))"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
   ultimately show "?P1 s\<or> ?P2 s\<or> ?P3 s"

	                         by metis

 qed
lemma PI_Remote_GetXVsInv7:
    (*Rule1VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (PI_Remote_GetX  iRule1 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have allCases:"(iRule1=iInv1)   \<or>(iRule1=iInv2)   \<or>((iRule1~=iInv1 \<and>iRule1~=iInv2 ))   "
	                      by( cut_tac  a1  a2  a3  a4  , auto)
moreover
                {assume b1:"(iRule1=iInv1)"

      have "?P1 s"

        apply(cut_tac  a1  a2  a3  a4  b1 , auto)

        done

        then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

       by blast

	                }
moreover
                {assume b1:"(iRule1=iInv2)"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
moreover
                {assume b1:"((iRule1~=iInv1 \<and>iRule1~=iInv2 ))"

                  have "?P2 s"

  apply(cut_tac  a1  a2  a3  a4  b1 , auto intro!:forallVars1 simp  add :invHoldForRule2'_def varsOfVar_def)

  done

  then  have "?P1 s\<or> ?P2 s \<or> ?P3 s"

        by blast

	                }
   ultimately show "?P1 s\<or> ?P2 s\<or> ?P3 s"

	                         by metis

 qed
lemma PI_Remote_PutXVsInv7:
    (*Rule1VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (PI_Remote_PutX  iRule1 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

    by (cut_tac  a1  a2  a3  a4 , auto)
 lemma PI_Remote_ReplaceVsInv7:
    (*Rule1VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (PI_Remote_Replace  iRule1 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

    by (cut_tac  a1  a2  a3  a4 , auto)
 lemma StoreVsInv7:
    (*Rule1VsPInv2*)
  assumes   a1:"iRule1 \<le> N" and  a2:"iInv1 \<le> N" and  a3:"iInv2 \<le> N" and  a4:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (Store  iRule1 ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

    by (cut_tac  a1  a2  a3  a4 , auto)
 lemma StoreHomeVsInv7:
    (*Rule0VsPInv2*)
  assumes   a1:"iInv1 \<le> N" and  a2:"iInv2 \<le> N" and  a3:"iInv1~=iInv2  "

  shows  "invHoldForRule' s (inv7  iInv1  iInv2 ) (StoreHome ) (invariants   N)" (is " ?P1 s\<or>?P2 s\<or>?P3 s")

  proof -

     have "?P2 s"

     by (cut_tac a1 a2 a3, auto  )
   then show "?P1 s\<or>?P2 s\<or>?P3 s"
      by auto
 qed
  end


section \<open> Circus State Machine Semantics \<close>

theory Circus_SM_Semantics
  imports "RoboChart.RoboChart" Actions
begin recall_syntax

subsection \<open> Alphabet \<close>

alphabet robochart_ctrl =
  rc_ctrl :: ID

type_synonym 's rcst = "'s robochart_ctrl_scheme"
type_synonym ('s, 'e) RoboAction = "('s robochart_ctrl_scheme, 'e) Actions.Action"
type_synonym 's RoboPred = "'s robochart_ctrl_scheme upred"

translations
  (type) "'s rcst" <= (type) "'s robochart_ctrl_scheme"
  (type) "('s, 'e) RoboAction" <= (type) "('s rcst, 'e) Action"
  (type) "'s RoboPred" <= (type) "'s rcst upred"

abbreviation "rc_state \<equiv> robochart_ctrl.more\<^sub>L"

notation rc_state ("\<^bold>r")

syntax
  "_svid_rc_state"  :: "svid" ("\<^bold>r")

translations
  "_svid_rc_state" == "CONST rc_state"
  "_svid_dot \<^bold>r x" <= "x ;\<^sub>L \<^bold>r"

type_synonym ('s, 'e) RTransition = "('s upred, ('s, 'e) Action, unit) STransition"
type_synonym ('s, 'e) RNode = "('s upred, ('s, 'e) Action, unit) SNode"
type_synonym ('s, 'e) RStateMachine = "('s upred, ('s, 'e) Action, unit) SStateMachine"

subsection \<open> State Machine Semantics \<close>

named_theorems sm_sem

abbreviation "trigger_semantics t \<equiv>
  (case tn_trigger t of
    Some e \<Rightarrow> e |
    None \<Rightarrow> skip)"

definition tn_condition :: "('s, 'e) RTransition \<Rightarrow> 's upred" where
[sm_sem]: "tn_condition t = case_option true_upred id (tn_cond t)"

definition tn_action :: "('s, 'e) RTransition \<Rightarrow> ('s, 'e) Action" where
[sm_sem]: "tn_action t = case_option skip id (tn_act t)"

no_utp_lift tn_condition

definition tr_semantics :: "('s, 'e) RTransition \<Rightarrow> (unit, 'e) chan \<Rightarrow> ('s, 'e) RoboAction" ("\<lbrakk>_\<rbrakk>\<^sub>T") where
[sm_sem]: "tr_semantics t \<epsilon> \<equiv>
  let tsem = trigger_semantics t ; tn_action t
  in
  tn_condition t \<oplus>\<^sub>p rc_state \<^bold>&
  rc_state:[if productive tsem then tsem else tsem ; sync \<epsilon>]\<^sub>A\<^sup>+ ; rc_ctrl := \<guillemotleft>tn_target t\<guillemotright>"

definition [sm_sem]: "n_entry_sem n = case_option skip id (n_entry n)"

definition [sm_sem]: "n_exit_sem n = case_option skip id (n_exit n)"

definition node_semantics ::
  "('s, 'e) RStateMachine \<Rightarrow> (unit, 'e) chan \<Rightarrow> ('s, 'e) RNode \<Rightarrow> ('s, 'e) RoboAction" ("_;_ \<turnstile> \<lbrakk>_\<rbrakk>\<^sub>N" [10,0,0] 10) where
[sm_sem]: "node_semantics M \<epsilon> node  =
  (rc_state:[n_entry_sem node]\<^sub>A\<^sup>+ ;
   (foldr (\<box>) (map (\<lambda> t. \<lbrakk>t\<rbrakk>\<^sub>T \<epsilon>) (the (Tmap\<^bsub>M\<^esub> (n_name node)))) stop) ;
   rc_state:[n_exit_sem node]\<^sub>A\<^sup>+)"

dataspace stm_context =
  channels null_event :: unit

context stm_context
begin

notation null_event ("\<epsilon>")

end

definition sm_semantics :: "('st, 'ch) RStateMachine \<Rightarrow> _ \<Rightarrow> ('st, 'ch) RoboAction" ("\<lbrakk>_\<rbrakk>\<^sub>M") where
[sm_sem]:
"sm_semantics M null_event =
    (rc_ctrl := \<guillemotleft>sm_initial M\<guillemotright> ;
    iteration (map (\<lambda> n. (&rc_ctrl =\<^sub>u \<guillemotleft>n_name n\<guillemotright>, M;null_event \<turnstile> \<lbrakk>n\<rbrakk>\<^sub>N)) (sm_inters M)))"

lemma tr_semantics_subst_ctrl: "[&rc_ctrl \<mapsto>\<^sub>s \<guillemotleft>k\<guillemotright>] \<dagger> (\<lbrakk>a\<rbrakk>\<^sub>T null_event) = \<lbrakk>a\<rbrakk>\<^sub>T null_event"
  by (simp add: tr_semantics_def action_simp action_subst usubst unrest frame_asubst Let_unfold)

lemma tr_choice_subst_ctrl:
  "[&rc_ctrl \<mapsto>\<^sub>s \<guillemotleft>k\<guillemotright>] \<dagger> foldr (\<box>) (map (\<lambda>t. \<lbrakk>t\<rbrakk>\<^sub>T null_event) ts) stop = foldr (\<box>) (map (\<lambda>t. \<lbrakk>t\<rbrakk>\<^sub>T null_event) ts) stop"
  by (induct ts, simp_all add: action_simp action_subst usubst tr_semantics_subst_ctrl)

lemma sm_semantics_subst_ctrl:
  "[&rc_ctrl \<mapsto>\<^sub>s \<guillemotleft>k\<guillemotright>] \<dagger> node_semantics M null_event node = node_semantics M null_event node"
  by (simp add: node_semantics_def action_simp action_subst frame_asubst tr_choice_subst_ctrl unrest)

(* Tests *)

definition "circus_predT stT = Type @{type_name upred} [stT]"
definition "circus_actionT stT evT = Type @{type_name Action} [stT, evT]"
definition "circus_probT stT = @{typ unit}"
definition "actionN = STR ''action''"

definition "action_eq =
  mk_equals (free actionN) (const @{const_name sm_semantics} $ free machineN $ free STR ''null_event'')"

definition "add_stm_context smd =
  smd\<lparr> uses := STR ''stm_context'' # uses smd \<rparr>"

code_reflect RC_Circus_Semantics
  functions circus_predT circus_actionT circus_probT action_eq add_stm_context

setup \<open>
  let open RC_Compiler; open RC_Circus_Semantics; open Specification
    val cont = snd o definition NONE [] [] ((Binding.empty, []), action_eq)
    val circus_types = RCTypes_ext (circus_predT, @{nonterminal utp_expr}, circus_actionT, @{nonterminal "raction"}, circus_probT, @{nonterminal utp_expr}, ())
    val circus_semantics =
      stm_sem_update (K (context_Stm_Semantics cont o add_stm_context)) null_RCSem_Proc
      |> rctypes_update (K circus_types)
  in
    Stm_Sem.put circus_semantics
  end
\<close>

lemma entry_basic_Node [simp]: "n_entry_sem (basic_Node n) = skip"
  by (simp add: n_entry_sem_def basic_Node_def)

lemma exit_basic_Node [simp]: "n_exit_sem (basic_Node n) = skip"
  by (simp add: n_exit_sem_def basic_Node_def)

lemma basic_Node_sem:
  "(M;\<epsilon> \<turnstile> \<lbrakk>basic_Node n\<rbrakk>\<^sub>N) = foldr (\<box>) (map (\<lambda>t. \<lbrakk>t\<rbrakk>\<^sub>T \<epsilon>) (the (Tmap\<^bsub>M\<^esub> n))) stop"
  by (simp add: node_semantics_def action_simp)

end


           (*-------------------------------------------*
            |        CSP-Prover on Isabelle2005         |
            |               December 2005  (modified)   |
            |                  April 2006  (modified)   |
            |                  March 2007  (modified)   |
            |                                           |
            |        Yoshinao Isobe (AIST JAPAN)        |
            *-------------------------------------------*)

theory CSP_T_law_step_ext
imports CSP_T_law_basic
begin

(*****************************************************************

         1. step laws
         2.
         3.
         4.

 *****************************************************************)

(*********************************************************
             Parallel expansion & distribution
 *********************************************************)

lemma cspT_Parallel_Timeout_split:
  "((? :Y -> Pf) [> P) |[X]| ((? :Z -> Qf) [> Q) =T[M,M]
     (? x:((X Int Y Int Z) Un (Y - X) Un (Z - X))
         -> IF (x : X) THEN (Pf x |[X]| Qf x)
               ELSE IF (x : Y & x : Z) THEN ((Pf x |[X]| ((? x:Z -> Qf x) [> Q))
                                        |~| (((? x:Y -> Pf x) [> P) |[X]| Qf x))
               ELSE IF (x : Y) THEN (Pf x |[X]| ((? x:Z -> Qf x) [> Q))
               ELSE (((? x:Y -> Pf x) [> P) |[X]| Qf x))
     [> (((P |[X]| ((? :Z -> Qf) [> Q)) |~|
         (((? :Y -> Pf) [> P) |[X]| Q)))"
apply (fold Timeout_def)
apply (simp add: cspT_semantics)
apply (rule order_antisym)

(* => *)
 apply (rule)
 apply (simp add: in_traces_Timeout in_traces)
 apply (elim conjE exE disjE)
 apply (simp_all)
  apply (force)
  apply (force)
  apply (force)
  apply (force)

  apply (simp add: par_tr_head_Ev_Ev)
  apply (elim conjE exE disjE)

   apply (simp)
   apply (rule disjI1)
   apply (rule_tac x="c" in exI)
   apply (rule_tac x="v" in exI)
   apply (force)

   apply (simp)
   apply (rule disjI1)
   apply (rule_tac x="c" in exI)
   apply (rule_tac x="v" in exI)
   apply (force)

   apply (simp)
   apply (rule disjI1)
   apply (rule_tac x="c" in exI)
   apply (rule_tac x="v" in exI)
   apply (force)

  apply (force)
  apply (force)
  apply (force)

(* <= *)
 apply (rule)
 apply (simp add: in_traces_Timeout in_traces)
 apply (elim conjE exE disjE)
 apply (simp_all)

  apply (simp add: in_traces)
  apply (elim conjE exE)
  apply (rule_tac x="<Ev a> ^^^ sa" in exI)
  apply (rule_tac x="<Ev a> ^^^ ta" in exI)
  apply (simp add: par_tr_head)

  apply (simp add: in_traces_IF)
  apply (case_tac "a : Z")

  (* a : Z *)
   apply (simp add: in_traces_Timeout in_traces)
   apply (elim conjE exE disjE)
   apply (simp_all)

    apply (simp add: par_tr_nil_right)
    apply (rule_tac x="<Ev a> ^^^ sa" in exI)
    apply (rule_tac x="<>" in exI)
    apply (simp add: par_tr_nil_right)
    apply (force)

    apply (rule_tac x="<Ev a> ^^^ sa" in exI)
    apply (rule_tac x="ta" in exI)
    apply (simp add: par_tr_head)

    apply (rule_tac x="<Ev a> ^^^ sa" in exI)
    apply (rule_tac x="ta" in exI)
    apply (simp add: par_tr_head)

    apply (rule_tac x="sa" in exI)
    apply (rule_tac x="<Ev a> ^^^ ta" in exI)
    apply (simp add: par_tr_head)

    apply (rule_tac x="sa" in exI)
    apply (rule_tac x="<Ev a> ^^^ ta" in exI)
    apply (simp add: par_tr_head)

    apply (rule_tac x="sa" in exI)
    apply (rule_tac x="<Ev a> ^^^ ta" in exI)
    apply (simp add: par_tr_head)

  (* a ~: Z *)
   apply (simp add: in_traces_Timeout in_traces)
   apply (elim conjE exE disjE)

    apply (simp add: par_tr_nil_right)
    apply (rule_tac x="<Ev a> ^^^ sa" in exI)
    apply (rule_tac x="<>" in exI)
    apply (simp add: par_tr_nil_right)
    apply (force)

    apply (rule_tac x="<Ev a> ^^^ sa" in exI)
    apply (rule_tac x="ta" in exI)
    apply (simp add: par_tr_head)

    apply (rule_tac x="<Ev a> ^^^ sa" in exI)
    apply (rule_tac x="ta" in exI)
    apply (simp add: par_tr_head)

  apply (simp add: in_traces_IF)
  apply (case_tac "a : Y")

  (* a : Y *)
   apply (simp add: in_traces_Timeout in_traces)
   apply (elim conjE exE disjE)
   apply (simp_all)

    apply (simp add: par_tr_nil_right)
    apply (rule_tac x="<Ev a> ^^^ sa" in exI)
    apply (rule_tac x="<>" in exI)
    apply (simp add: par_tr_nil_right)
    apply (force)

    apply (rule_tac x="<Ev a> ^^^ sa" in exI)
    apply (rule_tac x="ta" in exI)
    apply (simp add: par_tr_head)

    apply (rule_tac x="<Ev a> ^^^ sa" in exI)
    apply (rule_tac x="ta" in exI)
    apply (simp add: par_tr_head)

    apply (rule_tac x="sa" in exI)
    apply (rule_tac x="<Ev a> ^^^ ta" in exI)
    apply (simp add: par_tr_head)

    apply (rule_tac x="sa" in exI)
    apply (rule_tac x="<Ev a> ^^^ ta" in exI)
    apply (simp add: par_tr_head)

    apply (rule_tac x="sa" in exI)
    apply (rule_tac x="<Ev a> ^^^ ta" in exI)
    apply (simp add: par_tr_head)

  (* a ~:Y *)
   apply (simp add: in_traces_Timeout in_traces)
   apply (elim conjE exE disjE)

    apply (simp add: par_tr_nil_left)
    apply (rule_tac x="<>" in exI)
    apply (rule_tac x="<Ev a> ^^^ ta" in exI)
    apply (simp add: par_tr_nil_left)
    apply (force)

    apply (rule_tac x="sa" in exI)
    apply (rule_tac x="<Ev a> ^^^ ta" in exI)
    apply (simp add: par_tr_head)

    apply (rule_tac x="sa" in exI)
    apply (rule_tac x="<Ev a> ^^^ ta" in exI)
    apply (simp add: par_tr_head)

  (* *)
  apply (force)
  apply (force)
  apply (force)
  apply (force)
  apply (force)
  apply (force)
done

(*********************************************************
            Parallel expansion & distribution 2
 *********************************************************)

lemma cspT_Parallel_Timeout_input_l:
  "((? :Y -> Pf) [> P) |[X]| (? :Z -> Qf) =T[M,M]
     (? x:((X Int Y Int Z) Un (Y - X) Un (Z - X))
         -> IF (x : X) THEN (Pf x |[X]| Qf x)
               ELSE IF (x : Y & x : Z) THEN ((Pf x |[X]| (? x:Z -> Qf x))
                                        |~| (((? x:Y -> Pf x) [> P) |[X]| Qf x))
               ELSE IF (x : Y) THEN (Pf x |[X]| (? x:Z -> Qf x))
               ELSE (((? x:Y -> Pf x) [> P) |[X]| Qf x))
     [> (((P |[X]| (? :Z -> Qf))))"
apply (fold Timeout_def)
apply (simp add: cspT_semantics)
apply (rule order_antisym)

(* => *)
 apply (rule)
 apply (simp add: in_traces_Timeout in_traces)
 apply (elim conjE exE disjE)
 apply (simp_all)

  apply (simp add: par_tr_nil_left)
  apply (rule disjI1)
  apply (rule_tac x="a" in exI)
  apply (rule_tac x="sa" in exI)
  apply (elim conjE)
  apply (simp add: image_iff)
  apply (case_tac "a:Y")
   apply (simp)
   apply (rule disjI2)
   apply (rule_tac x="<>" in exI)
   apply (rule_tac x="sa" in exI)
   apply (simp add: par_tr_nil_left)

   apply (simp)
   apply (rule_tac x="<>" in exI)
   apply (rule_tac x="sa" in exI)
   apply (simp add: par_tr_nil_left)

  apply (simp add: par_tr_nil_right)
  apply (rule disjI1)
  apply (rule_tac x="a" in exI)
  apply (rule_tac x="sa" in exI)
  apply (elim conjE)
  apply (simp add: image_iff)
  apply (case_tac "a:Z")
   apply (simp)
   apply (rule disjI1)
   apply (rule_tac x="sa" in exI)
   apply (rule_tac x="<>" in exI)
   apply (simp add: par_tr_nil_right)

   apply (simp)
   apply (rule_tac x="sa" in exI)
   apply (rule_tac x="<>" in exI)
   apply (simp add: par_tr_nil_right)

  apply (force)

  apply (simp add: par_tr_head_Ev_Ev)
  apply (elim conjE exE disjE)

   apply (simp)
   apply (rule disjI1)
   apply (rule_tac x="c" in exI)
   apply (rule_tac x="v" in exI)
   apply (force)

   apply (simp)
   apply (rule disjI1)
   apply (rule_tac x="c" in exI)
   apply (rule_tac x="v" in exI)
   apply (force)

   apply (simp)
   apply (rule disjI1)
   apply (rule_tac x="c" in exI)
   apply (rule_tac x="v" in exI)
   apply (force)

  apply (force)

(* <= *)
 apply (rule)
 apply (simp add: in_traces_Timeout in_traces)
 apply (elim conjE exE disjE)
 apply (simp_all)

  apply (simp add: in_traces)
  apply (elim conjE exE)
  apply (rule_tac x="<Ev a> ^^^ sa" in exI)
  apply (rule_tac x="<Ev a> ^^^ ta" in exI)
  apply (simp add: par_tr_head)

  apply (simp add: in_traces_IF)
  apply (case_tac "a : Z")

  (* a : Z *)
   apply (simp add: in_traces_Timeout in_traces)
   apply (elim conjE exE disjE)
   apply (simp_all)

    apply (simp add: par_tr_nil_right)
    apply (rule_tac x="<Ev a> ^^^ sa" in exI)
    apply (rule_tac x="<>" in exI)
    apply (simp add: par_tr_nil_right)
    apply (force)

    apply (rule_tac x="<Ev a> ^^^ sa" in exI)
    apply (rule_tac x="ta" in exI)
    apply (simp add: par_tr_head)

    apply (rule_tac x="<>" in exI)
    apply (rule_tac x="<Ev a> ^^^ ta" in exI)
    apply (simp add: par_tr_head)

    apply (rule_tac x="<Ev aa> ^^^ sb" in exI)
    apply (rule_tac x="<Ev a> ^^^ ta" in exI)
    apply (simp add: par_tr_head)

    apply (rule_tac x="sa" in exI)
    apply (rule_tac x="<Ev a> ^^^ ta" in exI)
    apply (simp add: par_tr_head)

  (* a ~: Z *)
   apply (simp add: in_traces_Timeout in_traces)
   apply (elim conjE exE disjE)

    apply (simp add: par_tr_nil_right)
    apply (rule_tac x="<Ev a> ^^^ sa" in exI)
    apply (rule_tac x="<>" in exI)
    apply (simp add: par_tr_nil_right)
    apply (force)

    apply (rule_tac x="<Ev a> ^^^ sa" in exI)
    apply (rule_tac x="ta" in exI)
    apply (simp add: par_tr_head)

  apply (simp add: in_traces_IF)
  apply (case_tac "a : Y")

  (* a : Y *)
   apply (simp add: in_traces_Timeout in_traces)
   apply (elim conjE exE disjE)
   apply (simp_all)

    apply (simp add: par_tr_nil_right)
    apply (rule_tac x="<Ev a> ^^^ sa" in exI)
    apply (rule_tac x="<>" in exI)
    apply (simp add: par_tr_nil_right)
    apply (force)

    apply (rule_tac x="<Ev a> ^^^ sa" in exI)
    apply (rule_tac x="ta" in exI)
    apply (simp add: par_tr_head)

    apply (simp add: par_tr_nil_left)
    apply (rule_tac x="<>" in exI)
    apply (rule_tac x="<Ev a> ^^^ ta" in exI)
    apply (simp add: par_tr_nil_left)
    apply (force)

    apply (rule_tac x="<Ev aa> ^^^ sb" in exI)
    apply (rule_tac x="<Ev a> ^^^ ta" in exI)
    apply (simp add: par_tr_head)

    apply (rule_tac x="sa" in exI)
    apply (rule_tac x="<Ev a> ^^^ ta" in exI)
    apply (simp add: par_tr_head)

  (* a ~:Y *)
   apply (simp add: in_traces_Timeout in_traces)
   apply (elim conjE exE disjE)

    apply (simp add: par_tr_nil_left)
    apply (rule_tac x="<>" in exI)
    apply (rule_tac x="<Ev a> ^^^ ta" in exI)
    apply (simp add: par_tr_nil_left)
    apply (force)

    apply (rule_tac x="sa" in exI)
    apply (rule_tac x="<Ev a> ^^^ ta" in exI)
    apply (simp add: par_tr_head)

    apply (rule_tac x="sa" in exI)
    apply (rule_tac x="<Ev a> ^^^ ta" in exI)
    apply (simp add: par_tr_head)

  (* *)
  apply (force)
  apply (force)
done

lemma cspT_Parallel_Timeout_input_r:
  "(? :Y -> Pf) |[X]| ((? :Z -> Qf) [> Q) =T[M,M]
     (? x:((X Int Y Int Z) Un (Y - X) Un (Z - X))
         -> IF (x : X) THEN (Pf x |[X]| Qf x)
               ELSE IF (x : Y & x : Z) THEN ((Pf x |[X]| ((? x:Z -> Qf x) [> Q))
                                        |~| ((? x:Y -> Pf x) |[X]| Qf x))
               ELSE IF (x : Y) THEN (Pf x |[X]| ((? x:Z -> Qf x) [> Q))
               ELSE ((? x:Y -> Pf x) |[X]| Qf x))
     [> ((((? :Y -> Pf) |[X]| Q)))"
apply (rule cspT_rw_left)
apply (rule cspT_commut)

apply (rule cspT_rw_left)
apply (rule cspT_Parallel_Timeout_input_l)

apply (rule cspT_decompo)
apply (rule cspT_decompo)
apply (rule cspT_decompo)
apply (force)

apply (case_tac "a : X")
apply (simp)
apply (rule cspT_rw_left)
apply (rule cspT_IF)
apply (rule cspT_rw_right)
apply (rule cspT_IF)
apply (rule cspT_commut)

apply (simp)
apply (rule cspT_rw_left)
apply (rule cspT_IF)
apply (rule cspT_rw_right)
apply (rule cspT_IF)

apply (case_tac "(a : Z & a : Y)")
apply (simp)
apply (rule cspT_rw_left)
apply (rule cspT_IF)
apply (rule cspT_rw_right)
apply (rule cspT_IF)

apply (rule cspT_rw_left)
apply (rule cspT_commut)
apply (rule cspT_decompo)
apply (rule cspT_rw_left)
apply (rule cspT_commut)
apply (simp)
apply (rule cspT_rw_left)
apply (rule cspT_commut)
apply (simp)

apply (simp (no_asm_simp))
apply (rule cspT_rw_left)
apply (rule cspT_IF)
apply (subgoal_tac "~(a : Y & a : Z)")
apply (simp (no_asm_simp))
apply (rule cspT_rw_right)
apply (rule cspT_IF)

apply (case_tac "a : Z")
apply (simp)
apply (rule cspT_rw_left)
apply (rule cspT_IF)
apply (rule cspT_rw_right)
apply (rule cspT_IF)
apply (rule cspT_commut)
apply (simp)
apply (rule cspT_rw_left)
apply (rule cspT_IF)
apply (rule cspT_rw_right)
apply (rule cspT_IF)
apply (rule cspT_commut)

apply (fast)
apply (simp)
apply (rule cspT_commut)
done

lemmas cspT_Parallel_Timeout_input =
       cspT_Parallel_Timeout_input_l
       cspT_Parallel_Timeout_input_r

(*** cspT_step_ext ***)

lemmas cspT_step_ext = cspT_Parallel_Timeout_split
                       cspT_Parallel_Timeout_input

end


(*  Title:      HOL/Bali/Type.thy
    Author:     David von Oheimb
*)

subsection \<open>Java types\<close>

theory Type imports Name begin

text \<open>
simplifications:
\begin{itemize}
\item only the most important primitive types
\item the null type is regarded as reference type
\end{itemize}
\<close>

datatype prim_ty        \<comment> \<open>primitive type, cf. 4.2\<close>
        = Void          \<comment> \<open>result type of void methods\<close>
        | Boolean
        | Integer

datatype ref_ty         \<comment> \<open>reference type, cf. 4.3\<close>
        = NullT         \<comment> \<open>null type, cf. 4.1\<close>
        | IfaceT qtname \<comment> \<open>interface type\<close>
        | ClassT qtname \<comment> \<open>class type\<close>
        | ArrayT ty     \<comment> \<open>array type\<close>

and ty                  \<comment> \<open>any type, cf. 4.1\<close>
        = PrimT prim_ty \<comment> \<open>primitive type\<close>
        | RefT  ref_ty  \<comment> \<open>reference type\<close>

abbreviation "NT == RefT NullT"
abbreviation "Iface I == RefT (IfaceT I)"
abbreviation "Class C == RefT (ClassT C)"
abbreviation Array :: "ty \<Rightarrow> ty"  ("_.[]" [90] 90)
  where "T.[] == RefT (ArrayT T)"

definition
  the_Class :: "ty \<Rightarrow> qtname"
  where "the_Class T = (SOME C. T = Class C)" (** primrec not possible here **)

lemma the_Class_eq [simp]: "the_Class (Class C)= C"
by (auto simp add: the_Class_def)

end