header {* 
  \isaheader{wpF computes weakest preconditions} 
*}

theory JBC_wpFcorrect = JBC_SysInv:


section {* Alternative wpF Defintions *}

lemma wpF_Load:
assumes handlesEx: "handlesEx (fst \<Pi>) p' = None"
assumes cmd_p: "cmd \<Pi> p = Some (Load n)"
shows "wpF \<Pi> p p' Q = substE ((map (\<lambda>q. (Pos q, if q = p' then Pos p else FF)) (getPosEx Q))@ 
                               (map (\<lambda>k. (St k,if k=0 then Rg n 
                                      else St (k - 1))) (stkIds Q))) Q"
(*<*) by (simp! add: wpF_def Let_def split_def fst_conv snd_conv) (*>*)


lemma wpF_Store:
assumes handlesEx: "handlesEx (fst \<Pi>) p' = None"
assumes cmd_p: "cmd \<Pi> p = Some (Store n)"
shows "wpF \<Pi> p p' Q = substE ((map (\<lambda>q. (Pos q, if q = p' then Pos p else FF)) (getPosEx Q))@
                      ((Rg n,St 0)# map (\<lambda>k. (St k, St (k+1))) (stkIds Q))) Q"
(*<*) by (simp! add: wpF_def Let_def split_def fst_conv snd_conv) (*>*)


lemma wpF_Push:
assumes handlesEx: "handlesEx (fst \<Pi>) p' = None"
assumes cmd_p: "cmd \<Pi> p = Some (Push v)"
shows "wpF \<Pi> p p' Q = substE ((map (\<lambda>q. (Pos q, if q = p' then Pos p else FF)) (getPosEx Q))@
                       (map (\<lambda>k. (St k,if k=0 then Cn v else St (k - 1))) (stkIds Q))) Q"
(*<*) by (simp! add: wpF_def Let_def split_def fst_conv snd_conv) (*>*)


lemma wpF_New:
assumes handlesEx: "handlesEx (fst \<Pi>) p' = None"
assumes cmd_p: "cmd \<Pi> p = Some (New Cl)"
shows "wpF \<Pi> p p' Q = (let em=((map (\<lambda>q. (Pos q, if q = p' then Pos p else FF)) (getPosEx Q))
                               @(map (\<lambda>k. (St k,if k=0 then NewA 0
                                                 else St (k - 1))) (stkIds Q))@
                     (map (\<lambda>n. (NewA n, NewA (n+1))) (getNewEx Q))); 
                 gfe'=foldl (\<lambda>mp hex. (case hex 
of GF F C ex \<Rightarrow> (let ex'=substE mp ex 
                 in (Gf F C ex,IF ex' \<doteq> NewA 0 
                              THEN Cn (the ((snd (blank (fst \<Pi>) Cl))(F,C))) 
                              ELSE Gf F C ex'))
| TY ex ty   \<Rightarrow> (let ex'=substE mp ex
                 in (Ty ex ty,IF ex' \<doteq> NewA 0 
                             THEN Cn (Bool ((Class Cl) = ty)) 
                             ELSE Ty ex' ty)))#mp)
                      em (remdups' (getHeapEx Q))
                  in substE gfe' Q)"
(*<*) by (simp! add: wpF_def Let_def split_def fst_conv snd_conv) (*>*)


lemma wpF_Getfield:
assumes handlesEx: "handlesEx (fst \<Pi>) p' = None"
assumes cmd_p: "cmd \<Pi> p = Some (Getfield F C)"
shows "wpF \<Pi> p p' Q = substE ((map (\<lambda>q. (Pos q, if q = p' then Pos p else FF)) (getPosEx Q))@[(St 0,Gf F C (St 0))]) Q"
(*<*) by (simp! add: wpF_def Let_def split_def fst_conv snd_conv) (*>*)


lemma wpF_Putfield:
assumes handlesEx: "handlesEx (fst \<Pi>) p' = None"
assumes cmd_p: "cmd \<Pi> p = Some (Putfield F C)"
shows "wpF \<Pi> p p' Q = (let em=(map (\<lambda>q. (Pos q, if q = p' then Pos p else FF)) (getPosEx Q))
                               @(map (\<lambda>k. (St k,St (k+2))) (stkIds Q));
                            gfe'=foldl (\<lambda>mp ex. let ex'=substE mp ex 
                                          in (Gf F C ex,IF (ex' \<doteq> St 1) 
                                             THEN St 0 ELSE Gf F C ex')#mp)
                                 em (remdups' (getGfEx F C Q))
                        in substE gfe' Q)"
(*<*) by (simp! add: wpF_def Let_def split_def fst_conv snd_conv) (*>*)


lemma wpF_Checkcast:
assumes handlesEx: "handlesEx (fst \<Pi>) p' = None"
assumes cmd_p: "cmd \<Pi> p = Some (Checkcast C)"
shows "wpF \<Pi> p p' Q = substE (map (\<lambda>q. (Pos q, if q = p' then Pos p else FF)) (getPosEx Q)) Q"
(*<*) by (simp! add: wpF_def Let_def split_def fst_conv snd_conv) (*>*)


lemma wpF_Invoke:
assumes handlesEx: "handlesEx (fst \<Pi>) p' = None"
assumes cmd_p: "cmd \<Pi> p = Some (Invoke M n)"
shows "wpF \<Pi> p p' Q = substE ((map (\<lambda>q. (Pos q, if q = p' then Pos p else FF)) (getPosEx Q))@(FrNr,FrNr \<oplus> (Cn (Intg 1)))#
                        (map (\<lambda>k. (Rg k,if k \<le> n then St (n-k) 
                                        else (if k \<le> n + fst (snd (snd (snd (snd (method (fst \<Pi>) (fst p') M)))))
                                              then Cn arb
                                              else none))) (rgIds Q))@
                        (map (\<lambda>k. (St k,none)) (stkIds Q))@
                        (map (\<lambda>ex. (Call ex,ex)) (getCallEx Q))@
                        (concat (map (\<lambda>(cn',ex'). 
                        (if catchesEx (fst \<Pi>) cn' p 
                         then [(Catch cn' ex',ex')] 
                         else [(Catch cn' ex',
                               IF (FrNr \<doteq> Cn (Intg 1)) THEN ex' 
                               ELSE Catch cn' ex')])) (getCatchEx Q)))) Q"
(*<*) by (simp! add: wpF_def Let_def split_def fst_conv snd_conv) (*>*)


lemma wpF_Return:
assumes handlesEx: "handlesEx (fst \<Pi>) p' = None"
assumes cmd_p: "cmd \<Pi> p = Some Return"
shows "wpF \<Pi> p p' Q =  (let (C,M,pc)=p; (P,An)=\<Pi>; n = length (fst (snd (method P C M))) 
             in substE ((map (\<lambda>q. (Pos q, if q = p' then Pos p else FF)) (getPosEx Q))
                    @(FrNr,FrNr \<ominus> (Cn (Intg 1)))#
                    (map (\<lambda>k. (St k,if 1 \<le> k then Call (St (n+k)) 
                                     else St 0)) (stkIds Q))@
                   (map (\<lambda>k. (Rg k,Call (Rg k))) (rgIds Q))@
                   (map (\<lambda>ex. (Call ex,Call (Call ex))) (getCallEx Q))@
                   (map (\<lambda>(cn',ex'). (Catch cn' ex',Call (Catch cn' ex'))) 
                        (getCatchEx Q))) Q)"
(*<*) by (simp! add: wpF_def Let_def split_def fst_conv snd_conv) (*>*)


lemma wpF_Pop: 
assumes handlesEx: "handlesEx (fst \<Pi>) p' = None"
assumes cmd_p: "cmd \<Pi> p = Some Pop"
shows "wpF \<Pi> p p' Q = substE (map (\<lambda>q. (Pos q, if q = p' then Pos p else FF)) (getPosEx Q)
                              @(map (\<lambda>k. (St k,St (k+1))) (stkIds Q))) Q"
(*<*) by (simp! add: wpF_def Let_def split_def fst_conv snd_conv) (*>*)


lemma wpF_IBin: 
assumes handlesEx: "handlesEx (fst \<Pi>) p' = None"
assumes cmd_p: "cmd \<Pi> p = Some (IBin no)"
shows "wpF \<Pi> p p' Q = substE (map (\<lambda>q. (Pos q, if q = p' then Pos p else FF)) (getPosEx Q)
                              @ (map (\<lambda>k. (St k,if k=0 then Num (St 1) no (St 0) else (St (k+1)))) (stkIds Q))) Q"
(*<*) by (simp! add: wpF_def Let_def split_def fst_conv snd_conv) (*>*)


lemma wpF_Goto: 
assumes handlesEx: "handlesEx (fst \<Pi>) p' = None"
assumes cmd_p: "cmd \<Pi> p = Some (Goto t)"
shows "wpF \<Pi> p p' Q = substE (map (\<lambda>q. (Pos q, if q = p' then Pos p else FF)) (getPosEx Q)) Q"
(*<*) by (simp! add: wpF_def Let_def split_def fst_conv snd_conv) (*>*)


lemma wpF_CmpEq: 
assumes handlesEx: "handlesEx (fst \<Pi>) p' = None"
assumes cmd_p: "cmd \<Pi> p = Some CmpEq"
shows "wpF \<Pi> p p' Q = substE (map (\<lambda>q. (Pos q, if q = p' then Pos p else FF)) (getPosEx Q)
                              @(map (\<lambda>k. (St k,if k=0 then (St 0)\<doteq>(St 1) 
                                         else (St (k+1)))) (stkIds Q))) Q"
(*<*) by (simp! add: wpF_def Let_def split_def fst_conv snd_conv) (*>*)


lemma wpF_IfIntCmp: 
assumes handlesEx: "handlesEx (fst \<Pi>) p' = None"
assumes cmd_p: "cmd \<Pi> p = Some (IfIntCmp ro t)"
shows "wpF \<Pi> p p' Q = substE (map (\<lambda>q. (Pos q, if q = p' then Pos p else FF)) (getPosEx Q)
                              @(map (\<lambda>k. (St k,St (k+2))) (stkIds Q))) Q"
(*<*) by (simp! add: wpF_def Let_def split_def fst_conv snd_conv) (*>*)


lemma wpF_IfFalse: 
assumes handlesEx: "handlesEx (fst \<Pi>) p' = None"
assumes cmd_p: "cmd \<Pi> p = Some (IfFalse t)"
shows "wpF \<Pi> p p' Q = substE (map (\<lambda>q. (Pos q, if q = p' then Pos p else FF)) (getPosEx Q)
                              @(map (\<lambda>k. (St k,St (k+1))) (stkIds Q))) Q"
(*<*) by (simp! add: wpF_def Let_def split_def fst_conv snd_conv) (*>*)


lemma wpF_Throw_Nrm: 
assumes handlesEx: "handlesEx (fst \<Pi>) p' = None"
assumes cmd_p: "cmd \<Pi> p = Some Throw"
shows "wpF \<Pi> p p' Q = FF"
(*<*) by (simp! add: wpF_def Let_def split_def fst_conv snd_conv) (*>*)

(*
lemma wpF_Except:
assumes handlesEx: "handlesEx (fst \<Pi>) p' = Some cn"
assumes cmd_p: "cmd \<Pi> p = Some i"
shows "wpF \<Pi> p p' Q = (let mp=(map (\<lambda>q. (Pos q, if q = p' then Pos p else FF)) (getPosEx Q)) @
                           (map (\<lambda>k. (St k,if 1\<le>k then none 
                                                   else (if (cmd \<Pi> p = Some Throw) 
                           then (IF St 0 \<doteq> Cn (Null) 
                                 THEN (Cn (Addr (addr_of_sys_xcpt NullPointer))) 
                                 ELSE St 0) 
                           else Cn (Addr (addr_of_sys_xcpt cn))))) (stkIds Q))@
                 (let (C,M,pc)=p; (C',M',pc')=p'; (P,An)=\<Pi> in
                 (if match_ex_table P cn pc (ex_table_of P C M) = Some pc' then [] 
                  else 
                  let rgm=map (\<lambda>k. (Rg k,Catch cn (Rg k))) (rgIds Q);
                      om =map (\<lambda>ex. (Call ex,Catch cn (Call ex))) (getCallEx Q);
                      cm= map (\<lambda>(cn',ex'). (Catch cn' ex', Catch cn (Catch cn' ex'))) 
                              (getCatchEx Q)
                  in (FrNr,Catch cn FrNr)#rgm@om@cm))
              in substE mp Q)"
 by  (simp! add: wpF_def Let_def split_def fst_conv snd_conv) *)

lemma wpF_Except:
assumes handlesEx: "handlesEx (fst \<Pi>) p' = Some cn"
assumes cmd_p: "cmd \<Pi> p = Some i"
shows "wpF \<Pi> p p' Q = (let mp=(map (\<lambda>q. (Pos q, if q = p' then Pos p else FF)) (getPosEx Q)) @
                           (map (\<lambda>k. (St k,if 1\<le>k then none 
                                                   else (if (i = Throw) 
                           then (IF St 0 \<doteq> Cn (Null) 
                                 THEN (Cn (Addr (addr_of_sys_xcpt NullPointer))) 
                                 ELSE St 0) 
                           else Cn (Addr (addr_of_sys_xcpt (sys_xcpt_of i)))))) (stkIds Q))@
                 (let (C,M,pc)=p; (C',M',pc')=p'; (P,An)=\<Pi> in
                 (if match_ex_table P cn pc (ex_table_of P C M) = Some pc' then [] 
                  else 
                  let rgm=map (\<lambda>k. (Rg k,Catch cn (Rg k))) (rgIds Q);
                      om =map (\<lambda>ex. (Call ex,Catch cn (Call ex))) (getCallEx Q);
                      cm= map (\<lambda>(cn',ex'). (Catch cn' ex', Catch cn (Catch cn' ex'))) 
                              (getCatchEx Q)
                  in (FrNr,Catch cn FrNr)#rgm@om@cm))
              in substE mp Q)"
(*<*) 
by (simp! add: wpF_def Let_def split_def fst_conv snd_conv)
(*>*)

section {* Auxiliary Definitions and Lemmas *}

lemma foldl_map_lookup':
"\<And> es. \<forall>hex \<in> set es. heapEx x \<noteq> [hex]  \<Longrightarrow> \<forall>mp'. (foldl (\<lambda>mp hex. (case hex 
 of GF F C ex \<Rightarrow> (let ex'=substE mp ex in (Gf F C ex, 
    IF ex' \<doteq> NewA 0 THEN Cn (the (snd (blank P Cl) (F, C))) ELSE Gf F C ex'))
 | TY ex ty  \<Rightarrow>  (let ex'=substE mp ex in (Ty ex ty, 
    IF ex' \<doteq> NewA 0 THEN Cn (Bool ((Class Cl) = ty)) ELSE Ty ex' ty)))#mp) mp' es) ? x 
 = mp' ? x"
(*<*)
apply (erule rev_mp)
apply (induct_tac "es")
apply simp

apply (rule impI)
apply (rule allI)
apply (simp add: split_def)
apply (erule conjE)+
apply (case_tac "a")
apply simp
apply (case_tac "x = Gf list1 list2 expr")
apply simp

apply (drule_tac t="x" in not_sym)
apply simp

apply simp
apply (case_tac "x = Ty expr ty")
apply simp

apply (drule_tac t="x" in not_sym)
apply simp
done
(*>*)


lemma foldl_map_lookup'':
"\<And> es. \<forall>a\<in> set es. x\<noteq>f a \<Longrightarrow> \<forall>mp'. (foldl (\<lambda>mp a. (f a, g mp a)#mp) mp' es) ? x = mp' ? x"
(*<*)
apply (erule rev_mp)
apply (induct_tac "es")
apply simp

apply (rule impI)
apply (rule allI)
apply simp
apply (erule conjE)+
apply (drule_tac t="x" in not_sym)
apply simp
done
(*>*)


lemma getGfEx_size:
"\<And>ex' . ex' \<in> set (getGfEx F C ex) \<Longrightarrow> size ex' < size ex"
(*<*)
apply (erule rev_mp)
apply (rule_tac expr="ex" in expr_induct')
apply (rule impI | rule conjI | simp add: getGfEx_def split add: split_if split_if_asm | erule disjE)+
done
(*>*)


lemma getGfEx_not_refl:
"\<And> ex'. ex' \<in> set (getGfEx F C ex) \<Longrightarrow> ex' \<noteq> ex"
(*<*)
apply (drule getGfEx_size)
apply (rule classical)
apply simp
done
(*>*)

lemma getHeapEx_GF_size:
"\<And>F C ex . GF F C ex \<in> set (getHeapEx ex') \<Longrightarrow> size ex < size ex'"
(*<*)
apply (erule rev_mp)
apply (rule_tac expr="ex'" in expr_induct')
apply (rule impI | rule conjI | simp add: getHeapEx_def split add: split_if split_if_asm | erule disjE)+
done
(*>*)


lemma getHeapEx_TY_size:
"\<And>ex ty . TY ex ty \<in> set (getHeapEx ex') \<Longrightarrow> size ex < size ex'"
(*<*)
apply (erule rev_mp)
apply (rule_tac expr="ex'" in expr_induct')
apply (rule impI | rule conjI | simp add: getHeapEx_def split add: split_if split_if_asm | erule disjE)+
done
(*>*)


lemma getHeapEx_GF_not_refl:
"\<And> F C ex. GF F C ex \<in> set (getHeapEx ex') \<Longrightarrow> ex \<noteq> ex'"
(*<*)
apply (drule getHeapEx_GF_size)
apply (rule classical)
apply simp
done
(*>*)


lemma getHeapEx_TY_not_refl:
"\<And> ex ty. TY ex ty \<in> set (getHeapEx ex') \<Longrightarrow> ex \<noteq> ex'"
(*<*)
apply (drule getHeapEx_TY_size)
apply (rule classical)
apply simp
done
(*>*)


lemma getGfEx_comp:
"\<And> ex'. \<lbrakk> ex' \<in> set (getGfEx F C ex) \<rbrakk> \<Longrightarrow> \<exists>as bs cs. 
 ((getGfEx F C ex) = as@(getGfEx F C ex')@bs@[ex']@cs \<and> 
  ex' \<notin> set (as@(getGfEx F C ex')@bs))"
(*<*)
apply (erule rev_mp)
apply (rule_tac expr="ex" in expr_induct')
--{* Rg nat *}
apply (simp add: getGfEx_def)
--{* St nat *}
apply (simp add: getGfEx_def)
--{* Lv nat *}
apply (simp add: getGfEx_def)
--{* Cn val *}
apply (simp add: getGfEx_def)
--{* NewA n *}
apply (simp add: getGfEx_def)
--{* Gf list1 list2 expr*}
apply (rule impI)
apply (simp add: getGfEx_def)
apply (rule conjI)
apply (rule impI)
apply (erule disjE)
apply simp
apply (erule exE | erule conjE)+
apply (rule_tac x="as" in exI)
apply (rule_tac x="bs" in exI)
apply simp

apply simp
apply (rule_tac x="[]" in exI)
apply (rule_tac x="[]" in exI)
apply simp

apply (rule impI)
apply (case_tac "F = list1")
apply simp

apply simp
--{* FrNr *}
apply (simp add: getGfEx_def)
--{* Num expr1 numop expr2 *}
apply (rule impI)
apply (simp add: getGfEx_def)
apply (case_tac " ex' \<in> set (foldE (\<lambda>(ex, as). as @ gfEx (F, C, ex)) op @ [] expr1)")
apply simp
apply (erule exE | erule conjE)+
apply (rule_tac x="as" in exI)
apply (rule_tac x="bs" in exI)
apply (rule conjI)
apply (rule_tac x="cs @  foldE (\<lambda>(ex, as). as @ gfEx (F, C, ex)) op @ [] expr2" in exI)
apply simp

apply simp

apply simp
apply (erule exE | erule conjE)+
apply (rule_tac x="foldE (\<lambda>(ex, as). as @ gfEx (F, C, ex)) op @ [] expr1 @ as" in exI)
apply (rule_tac x="bs" in exI)
apply (rule conjI)
apply (rule_tac x="cs" in exI)
apply simp

apply simp
--{* Rel expr1 relop expr2 *}
apply (rule impI)
apply (simp add: getGfEx_def)
apply (case_tac " ex' \<in> set (foldE (\<lambda>(ex, as). as @ gfEx (F, C, ex)) op @ [] expr1)")
apply simp
apply (erule exE | erule conjE)+
apply (rule_tac x="as" in exI)
apply (rule_tac x="bs" in exI)
apply (rule conjI)
apply (rule_tac x="cs @  foldE (\<lambda>(ex, as). as @ gfEx (F, C, ex)) op @ [] expr2" in exI)
apply simp

apply simp

apply simp
apply (erule exE | erule conjE)+
apply (rule_tac x="foldE (\<lambda>(ex, as). as @ gfEx (F, C, ex)) op @ [] expr1 @ as" in exI)
apply (rule_tac x="bs" in exI)
apply (rule conjI)
apply (rule_tac x="cs" in exI)
apply simp

apply simp
--{* IF expr1 THEN expr2 ELSE expr3 *}
apply (rule impI)
apply (simp add: getGfEx_def)
apply (case_tac " ex' \<in> set (foldE (\<lambda>(ex, as). as @ gfEx (F, C, ex)) op @ [] expr1)")
apply (drule mp, assumption)
apply (erule exE | erule conjE)+
apply (rule_tac x="as" in exI)
apply (rule_tac x="bs" in exI)
apply (rule conjI)
apply (rule_tac x="cs @  foldE (\<lambda>(ex, as). as @ gfEx (F, C, ex)) op @ [] expr2 @ foldE (\<lambda>(ex, as). as @ gfEx (F, C, ex)) op @ [] expr3" in exI)
apply simp

apply simp

apply (case_tac " ex' \<in> set (foldE (\<lambda>(ex, as). as @ gfEx (F, C, ex)) op @ [] expr2)")
apply (drule mp, assumption)
apply (erule exE | erule conjE)+
apply (rule_tac x="foldE (\<lambda>(ex, as). as @ gfEx (F, C, ex)) op @ [] expr1 @ as" in exI)
apply (rule_tac x="bs" in exI)
apply (rule conjI)
apply (rule_tac x="cs @ foldE (\<lambda>(ex, as). as @ gfEx (F, C, ex)) op @ [] expr3" in exI)
apply simp

apply simp

apply simp
apply (erule exE | erule conjE)+
apply (rule_tac x="foldE (\<lambda>(ex, as). as @ gfEx (F, C, ex)) op @ [] expr1 @ foldE (\<lambda>(ex, as). as @ gfEx (F, C, ex)) op @ [] expr2 @ as" in exI)
apply (rule_tac x="bs" in exI)
apply (rule conjI)
apply (rule_tac x="cs" in exI)
apply simp

apply simp
--{* Eq expr1 expr2 *}
apply (rule impI)
apply (simp add: getGfEx_def)
apply (case_tac " ex' \<in> set (foldE (\<lambda>(ex, as). as @ gfEx (F, C, ex)) op @ [] expr1)")
apply simp
apply (erule exE | erule conjE)+
apply (rule_tac x="as" in exI)
apply (rule_tac x="bs" in exI)
apply (rule conjI)
apply (rule_tac x="cs @  foldE (\<lambda>(ex, as). as @ gfEx (F, C, ex)) op @ [] expr2" in exI)
apply simp

apply simp

apply simp
apply (erule exE | erule conjE)+
apply (rule_tac x="foldE (\<lambda>(ex, as). as @ gfEx (F, C, ex)) op @ [] expr1 @ as" in exI)
apply (rule_tac x="bs" in exI)
apply (rule conjI)
apply (rule_tac x="cs" in exI)
apply simp

apply simp
--{* Neg expr *}
apply (simp add: getGfEx_def)
--{* Imp expr1 expr2 *}
apply (rule impI)
apply (simp add: getGfEx_def)
apply (case_tac " ex' \<in> set (foldE (\<lambda>(ex, as). as @ gfEx (F, C, ex)) op @ [] expr1)")
apply simp
apply (erule exE | erule conjE)+
apply (rule_tac x="as" in exI)
apply (rule_tac x="bs" in exI)
apply (rule conjI)
apply (rule_tac x="cs @  foldE (\<lambda>(ex, as). as @ gfEx (F, C, ex)) op @ [] expr2" in exI)
apply simp

apply simp

apply simp
apply (erule exE | erule conjE)+
apply (rule_tac x="foldE (\<lambda>(ex, as). as @ gfEx (F, C, ex)) op @ [] expr1 @ as" in exI)
apply (rule_tac x="bs" in exI)
apply (rule conjI)
apply (rule_tac x="cs" in exI)
apply simp

apply simp
--{* Forall nat expr *}
apply (simp add: getGfEx_def)
--{* Ty expr ty *}
apply (simp add: getGfEx_def)
--{* Pos x *}
apply (simp add: getGfEx_def)
--{* Call expr *}
apply (simp add: getGfEx_def)
--{* Catch expr *}
apply (simp add: getGfEx_def)
--{* And (ex  es) *} 
apply (rule impI)
apply (simp add: getGfEx_def)
apply (case_tac "ex' \<in> set (foldE (\<lambda>(ex, as). as @ gfEx (F, C, ex)) op @ [] ex)")
apply (drule mp, assumption)
apply (erule exE | erule conjE)+
apply (rule_tac x="as" in exI)
apply (rule_tac x="bs" in exI)
apply (rule conjI)
apply (rule_tac x="cs @ foldEs (\<lambda>(ex, as). as @ gfEx (F, C, ex)) op @ [] es" in exI)
apply simp

apply simp

apply simp
apply (erule exE | erule conjE)+
apply (rule_tac x=" foldE (\<lambda>(ex, as). as @ gfEx (F, C, ex)) op @ [] ex @ as" in exI)
apply (rule_tac x="bs" in exI)
apply (rule conjI)
apply (rule_tac x="cs" in exI)
apply simp

apply simp
--{* And [] *}
apply (simp add: getGfEx_def)
done
(*>*)


lemma getGFEx_mono:
"\<And> ex'' ex'. \<lbrakk> ex'' \<in> set (getGfEx F C ex'); ex' \<in> set (getGfEx F C ex) \<rbrakk> \<Longrightarrow> \<exists>as bs cs. remdups' (getGfEx F C ex) = as@[ex'']@bs@[ex']@cs"
(*<*)
apply (drule_tac ex'="ex'" in getGfEx_comp)
apply (drule in_set_conv_decomp_fst)
apply (erule exE | erule conjE)+
apply (simp only:)
apply (subgoal_tac "remdups' ((as @ (ys @ ex'' # zs) @ bs) @ [ex'] @ cs) = remdups' (as @ (ys @ ex'' # zs) @ bs) @
[ex'] @ [b \<in> remdups' cs. b \<notin> set ((as @ (ys @ ex'' # zs) @ bs)@[ex'])]") 
 prefer 2
 apply (rule_tac a="ex'" in  remdups'_append_fst)
 apply simp
apply simp
apply (case_tac "ex'' \<in> set (as @ ys)")
apply (drule in_set_conv_decomp_fst)
apply (erule exE)+
apply (subgoal_tac "remdups' (ysa @ [ex''] @ (zsa @ ex'' # zs @ bs)) = remdups' ysa @ [ex''] @ [b \<in> remdups' (zsa @ ex'' # zs @ bs). b \<notin> set (ysa @ [ex''])]")
 prefer 2
 apply (rule remdups'_append_fst)
 apply simp
apply (subgoal_tac "(as @ ys @ ex'' # zs @ bs) = ((as @ ys) @ ex'' # zs @ bs)")
 prefer 2
 apply simp
apply (simp only:)
apply simp
apply (rule_tac x="remdups' ysa" in exI)
apply (rule_tac x="[b\<in>remdups' (zsa @ ex'' # zs @ bs) . b \<noteq> ex'' \<and> b \<notin> set ysa]" in exI) 
apply (rule_tac x="[b\<in>remdups' cs . b \<noteq> ex'' \<and> b \<noteq> ex' \<and> b \<notin> set as \<and> b \<notin> set ys \<and> b \<notin> set zs \<and> b \<notin> set bs]" in exI)
apply (simp only:)

apply (subgoal_tac "remdups' ((as @ ys) @ [ex''] @ (zs @ bs)) = remdups' (as @ ys) @ [ex''] @ [b \<in> remdups' (zs @ bs).
b \<notin> set ((as @ ys) @ [ex''])]")
 prefer 2
 apply (rule remdups'_append_fst)
 apply assumption
apply (subgoal_tac "(as @ ys @ ex'' # zs @ bs) = ((as @ ys) @ [ex''] @ (zs @ bs))")
 prefer 2
 apply simp
apply (simp only:)
apply (rule_tac x="remdups' (as @ ys)" in exI)
apply (rule_tac x="[b\<in>remdups' (zs @ bs) . b \<notin> set ((as @ ys) @ [ex''])]" in exI)
apply (rule_tac x="[b\<in>remdups' cs . b \<noteq> ex'' \<and> b \<noteq> ex' \<and> b \<notin> set as \<and> b \<notin> set ys \<and> b \<notin> set zs \<and> b \<notin> set bs]" in exI)
apply (simp (no_asm))
done
(*>*)


lemma getHeapEx_GF_comp:
"GF F C ex' \<in> set (getHeapEx ex) \<Longrightarrow> \<exists>as bs cs. getHeapEx ex = 
 as @ (getHeapEx ex') @ bs @ [GF F C ex'] @ cs \<and> 
 (GF F C ex') \<notin> set (as @ (getHeapEx ex') @ bs)"
(*<*)
apply (erule rev_mp)
apply (rule_tac expr="ex" in expr_induct')
--{* Rg nat *}
apply (simp add: getHeapEx_def)
--{* St nat *}
apply (simp add: getHeapEx_def)
--{* Lv nat *}
apply (simp add: getHeapEx_def)
--{* Cn val *}
apply (simp add: getHeapEx_def)
--{* NewA n *}
apply (simp add: getHeapEx_def)
--{* Gf list1 list2 expr*}
apply (rule impI)
apply (simp add: getHeapEx_def)
apply (case_tac  "GF F C ex' \<in> set (foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] expr)")
apply (drule mp, assumption)
apply (erule exE | erule conjE)+
apply (rule_tac x="as" in exI)
apply (rule_tac x="bs" in exI)
apply simp

apply simp
apply (rule_tac x="[]" in exI)
apply (rule_tac x="[]" in exI)
apply simp
--{* FrNr *}
apply (simp add: getHeapEx_def)
--{* Num expr1 numop expr2 *}
apply (rule impI)
apply (simp add: getHeapEx_def)
apply (case_tac " GF F C ex' \<in> set (foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] expr1)")
apply simp
apply (erule exE | erule conjE)+
apply (rule_tac x="as" in exI)
apply (rule_tac x="bs" in exI)
apply (rule conjI)
apply (rule_tac x="cs @ foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] expr2 " in exI)
apply simp

apply simp

apply simp
apply (erule exE | erule conjE)+
apply (rule_tac x="foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] expr1 @ as" in exI)
apply (rule_tac x="bs" in exI)
apply (rule conjI)
apply (rule_tac x="cs" in exI)
apply simp

apply simp
--{* Rel expr1 relop expr2 *}
apply (rule impI)
apply (simp add: getHeapEx_def)
apply (case_tac " GF F C ex' \<in> set (foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] expr1)")
apply simp
apply (erule exE | erule conjE)+
apply (rule_tac x="as" in exI)
apply (rule_tac x="bs" in exI)
apply (rule conjI)
apply (rule_tac x="cs @ foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] expr2 " in exI)
apply simp

apply simp

apply simp
apply (erule exE | erule conjE)+
apply (rule_tac x="foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] expr1 @ as" in exI)
apply (rule_tac x="bs" in exI)
apply (rule conjI)
apply (rule_tac x="cs" in exI)
apply simp

apply simp
--{* IF expr1 THEN expr2 ELSE expr3 *}
apply (rule impI)
apply (simp add: getHeapEx_def)
apply (case_tac "GF F C ex' \<in> set (foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] expr1) ")
apply (drule mp, assumption)
apply (erule exE | erule conjE)+
apply (rule_tac x="as" in exI)
apply (rule_tac x="bs" in exI)
apply (rule conjI)
apply (rule_tac x="cs @ foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] expr2 @
                        foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] expr3" in exI)
apply simp

apply simp

apply (case_tac "GF F C ex' \<in> set (foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] expr2) ")
apply (drule mp, assumption)
apply (erule exE | erule conjE)+
apply (rule_tac x="foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] expr1 @ as" in exI)
apply (rule_tac x="bs" in exI)
apply (rule conjI)
apply (rule_tac x="cs @ foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] expr3" in exI)
apply simp

apply simp

apply simp
apply (erule exE | erule conjE)+
apply (rule_tac x=" foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] expr1 @
                        foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] expr2 @ as" in exI)
apply (rule_tac x="bs" in exI)
apply (rule conjI)
apply (rule_tac x="cs" in exI)
apply simp

apply simp
--{* Eq expr1 expr2 *}
apply (rule impI)
apply (simp add: getHeapEx_def)
apply (case_tac " GF F C ex' \<in> set (foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] expr1)")
apply simp
apply (erule exE | erule conjE)+
apply (rule_tac x="as" in exI)
apply (rule_tac x="bs" in exI)
apply (rule conjI)
apply (rule_tac x="cs @ foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] expr2 " in exI)
apply simp

apply simp

apply simp
apply (erule exE | erule conjE)+
apply (rule_tac x="foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] expr1 @ as" in exI)
apply (rule_tac x="bs" in exI)
apply (rule conjI)
apply (rule_tac x="cs" in exI)
apply simp

apply simp
--{* Neg expr *}
apply (simp add: getHeapEx_def)
--{* Imp expr1 expr2 *}
apply (rule impI)
apply (simp add: getHeapEx_def)
apply (case_tac " GF F C ex' \<in> set (foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] expr1)")
apply simp
apply (erule exE | erule conjE)+
apply (rule_tac x="as" in exI)
apply (rule_tac x="bs" in exI)
apply (rule conjI)
apply (rule_tac x="cs @ foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] expr2 " in exI)
apply simp

apply simp

apply simp
apply (erule exE | erule conjE)+
apply (rule_tac x="foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] expr1 @ as" in exI)
apply (rule_tac x="bs" in exI)
apply (rule conjI)
apply (rule_tac x="cs" in exI)
apply simp

apply simp
--{* Forall nat expr *}
apply (simp add: getHeapEx_def)
--{* Ty expr ty *}
apply (simp add: getHeapEx_def)
apply (rule impI)
apply (drule mp, assumption)
apply (erule exE | erule conjE)+
apply (rule_tac x="as" in exI)
apply (rule_tac x="bs" in exI)
apply (rule conjI)
apply (rule_tac x="cs @ [TY expr ty]" in exI)
apply simp

apply simp
--{* Pos x *}
apply (simp add: getHeapEx_def)
--{* Call expr *}
apply (simp add: getHeapEx_def)
--{* Catch expr *}
apply (simp add: getHeapEx_def)
--{* And (ex  es) *} 
apply (rule impI)
apply (simp add: getHeapEx_def)
apply (case_tac " GF F C ex' \<in> set (foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] ex)")
apply (drule mp, assumption)
apply (erule exE | erule conjE)+
apply (rule_tac x="as" in exI)
apply (rule_tac x="bs" in exI)
apply (rule conjI)
apply (rule_tac x="cs @ foldEs (\<lambda>(ex, as). as @ heapEx ex) op @ [] es" in exI)
apply simp

apply simp

apply simp
apply (erule exE | erule conjE)+
apply (rule_tac x="foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] ex @ as" in exI)
apply (rule_tac x="bs" in exI)
apply (rule conjI)
apply (rule_tac x="cs" in exI)
apply simp

apply simp
--{* And [] *}
apply (simp add: getHeapEx_def)
done
(*>*)


lemma getHeapEx_TY_comp:
"TY ex' ty' \<in> set (getHeapEx ex) \<Longrightarrow> \<exists>as bs cs. getHeapEx ex = 
 as @ (getHeapEx ex') @ bs @ [TY ex' ty'] @ cs \<and> 
 (TY ex' ty') \<notin> set (as @ (getHeapEx ex') @ bs)"
(*<*)
apply (erule rev_mp)
apply (rule_tac expr="ex" in expr_induct')
--{* Rg nat *}
apply (simp add: getHeapEx_def)
--{* St nat *}
apply (simp add: getHeapEx_def)
--{* Lv nat *}
apply (simp add: getHeapEx_def)
--{* Cn val *}
apply (simp add: getHeapEx_def)
--{* NewA n *}
apply (simp add: getHeapEx_def)
--{* Gf list1 list2 expr*}
apply (simp add: getHeapEx_def)
apply (rule impI)
apply (drule mp, assumption)
apply (erule exE | erule conjE)+
apply (rule_tac x="as" in exI)
apply (rule_tac x="bs" in exI)
apply (rule conjI)
apply (rule_tac x="cs @ [GF list1 list2 expr]" in exI)
apply simp

apply simp
--{* FrNr *}
apply (simp add: getHeapEx_def)
--{* Num expr1 numop expr2 *}
apply (rule impI)
apply (simp add: getHeapEx_def)
apply (case_tac " TY ex' ty' \<in> set (foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] expr1)")
apply simp
apply (erule exE | erule conjE)+
apply (rule_tac x="as" in exI)
apply (rule_tac x="bs" in exI)
apply (rule conjI)
apply (rule_tac x="cs @ foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] expr2 " in exI)
apply simp

apply simp

apply simp
apply (erule exE | erule conjE)+
apply (rule_tac x="foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] expr1 @ as" in exI)
apply (rule_tac x="bs" in exI)
apply (rule conjI)
apply (rule_tac x="cs" in exI)
apply simp

apply simp
--{* Rel expr1 relop expr2 *}
apply (rule impI)
apply (simp add: getHeapEx_def)
apply (case_tac " TY ex' ty' \<in> set (foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] expr1)")
apply simp
apply (erule exE | erule conjE)+
apply (rule_tac x="as" in exI)
apply (rule_tac x="bs" in exI)
apply (rule conjI)
apply (rule_tac x="cs @ foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] expr2 " in exI)
apply simp

apply simp

apply simp
apply (erule exE | erule conjE)+
apply (rule_tac x="foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] expr1 @ as" in exI)
apply (rule_tac x="bs" in exI)
apply (rule conjI)
apply (rule_tac x="cs" in exI)
apply simp

apply simp
--{* IF expr1 THEN expr2 ELSE expr3 *}
apply (rule impI)
apply (simp add: getHeapEx_def)
apply (case_tac "TY ex' ty' \<in> set (foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] expr1) ")
apply (drule mp, assumption)
apply (erule exE | erule conjE)+
apply (rule_tac x="as" in exI)
apply (rule_tac x="bs" in exI)
apply (rule conjI)
apply (rule_tac x="cs @ foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] expr2 @
                        foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] expr3" in exI)
apply simp

apply simp

apply (case_tac "TY ex' ty' \<in> set (foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] expr2) ")
apply (drule mp, assumption)
apply (erule exE | erule conjE)+
apply (rule_tac x="foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] expr1 @ as" in exI)
apply (rule_tac x="bs" in exI)
apply (rule conjI)
apply (rule_tac x="cs @ foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] expr3" in exI)
apply simp

apply simp

apply simp
apply (erule exE | erule conjE)+
apply (rule_tac x=" foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] expr1 @
                        foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] expr2 @ as" in exI)
apply (rule_tac x="bs" in exI)
apply (rule conjI)
apply (rule_tac x="cs" in exI)
apply simp

apply simp
--{* Eq expr1 expr2 *}
apply (rule impI)
apply (simp add: getHeapEx_def)
apply (case_tac " TY ex' ty' \<in> set (foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] expr1)")
apply simp
apply (erule exE | erule conjE)+
apply (rule_tac x="as" in exI)
apply (rule_tac x="bs" in exI)
apply (rule conjI)
apply (rule_tac x="cs @ foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] expr2 " in exI)
apply simp

apply simp

apply simp
apply (erule exE | erule conjE)+
apply (rule_tac x="foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] expr1 @ as" in exI)
apply (rule_tac x="bs" in exI)
apply (rule conjI)
apply (rule_tac x="cs" in exI)
apply simp

apply simp
--{* Neg expr *}
apply (simp add: getHeapEx_def)
--{* Imp expr1 expr2 *}
apply (rule impI)
apply (simp add: getHeapEx_def)
apply (case_tac " TY ex' ty' \<in> set (foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] expr1)")
apply simp
apply (erule exE | erule conjE)+
apply (rule_tac x="as" in exI)
apply (rule_tac x="bs" in exI)
apply (rule conjI)
apply (rule_tac x="cs @ foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] expr2 " in exI)
apply simp

apply simp

apply simp
apply (erule exE | erule conjE)+
apply (rule_tac x="foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] expr1 @ as" in exI)
apply (rule_tac x="bs" in exI)
apply (rule conjI)
apply (rule_tac x="cs" in exI)
apply simp

apply simp
--{* Forall nat expr *}
apply (simp add: getHeapEx_def)
--{* Ty expr ty *}
apply (rule impI)
apply (simp add: getHeapEx_def)
apply (case_tac  "TY ex' ty' \<in> set (foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] expr)")
apply (drule mp, assumption)
apply (erule exE | erule conjE)+
apply (rule_tac x="as" in exI)
apply (rule_tac x="bs" in exI)
apply simp

apply simp
apply (rule_tac x="[]" in exI)
apply (rule_tac x="[]" in exI)
apply simp
--{* Pos x *}
apply (simp add: getHeapEx_def)
--{* Call expr *}
apply (simp add: getHeapEx_def)
--{* Catch expr *}
apply (simp add: getHeapEx_def)
--{* And (ex  es) *} 
apply (rule impI)
apply (simp add: getHeapEx_def)
apply (case_tac " TY ex' ty' \<in> set (foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] ex)")
apply (drule mp, assumption)
apply (erule exE | erule conjE)+
apply (rule_tac x="as" in exI)
apply (rule_tac x="bs" in exI)
apply (rule conjI)
apply (rule_tac x="cs @ foldEs (\<lambda>(ex, as). as @ heapEx ex) op @ [] es" in exI)
apply simp

apply simp

apply simp
apply (erule exE | erule conjE)+
apply (rule_tac x="foldE (\<lambda>(ex, as). as @ heapEx ex) op @ [] ex @ as" in exI)
apply (rule_tac x="bs" in exI)
apply (rule conjI)
apply (rule_tac x="cs" in exI)
apply simp

apply simp
--{* And [] *}
apply (simp add: getHeapEx_def)
done
(*>*)


lemma getHeapEx_mono_GF_GF:
"\<lbrakk> GF F'' C'' ex'' \<in> set (getHeapEx ex'); GF F' C' ex' \<in> set (getHeapEx ex) \<rbrakk> 
\<Longrightarrow> \<exists>as bs cs. remdups' (getHeapEx ex) = as@[GF F'' C'' ex'']@bs@[GF F' C' ex']@cs"
(*<*)
apply (drule_tac ex'="ex'" in getHeapEx_GF_comp)
apply (drule in_set_conv_decomp_fst)
apply (erule exE | erule conjE)+
apply (simp only:)
apply (subgoal_tac "remdups' ((as @ (ys @ GF F'' C'' ex'' # zs) @ bs) @ [GF F' C' ex'] @ cs) = remdups' (as @ (ys @ GF F'' C'' ex'' # zs) @ bs) @
[GF F' C' ex'] @ [b \<in> remdups' cs. b \<notin> set ((as @ (ys @ GF F'' C'' ex'' # zs) @ bs)@[GF F' C' ex'])]") 
 prefer 2
 apply (rule_tac a="GF F' C' ex'" in  remdups'_append_fst)
 apply simp
apply simp
apply (case_tac " GF F'' C'' ex'' \<in> set (as @ ys)")
apply (drule in_set_conv_decomp_fst)
apply (erule exE)+
apply (subgoal_tac "remdups' (ysa @ [ GF F'' C'' ex''] @ (zsa @  GF F'' C'' ex'' # zs @ bs)) = remdups' ysa @ [ GF F'' C'' ex''] @ [b \<in> remdups' (zsa @ GF F'' C'' ex'' # zs @ bs). b \<notin> set (ysa @ [ GF F'' C'' ex''])]")
 prefer 2
 apply (rule remdups'_append_fst)
 apply simp
apply (subgoal_tac "(as @ ys @  GF F'' C'' ex'' # zs @ bs) = ((as @ ys) @  GF F'' C'' ex'' # zs @ bs)")
 prefer 2
 apply simp
apply (simp only:)
apply simp
apply (rule_tac x="remdups' ysa" in exI)
apply (rule_tac x="[b\<in>remdups' (zsa @ GF F'' C'' ex'' # zs @ bs) . b \<noteq> GF F'' C'' ex'' \<and> b \<notin> set ysa]" in exI) 
apply (rule_tac x="[b\<in>remdups' cs .
              b \<noteq> GF F'' C'' ex'' \<and> b \<noteq> GF F' C' ex' \<and> b \<notin> set as \<and> b \<notin> set ys \<and> b \<notin> set zs \<and> b \<notin> set bs]" in exI)
apply (simp only:)

apply (subgoal_tac "remdups' ((as @ ys) @ [ GF F'' C'' ex''] @ (zs @ bs)) = remdups' (as @ ys) @ [ GF F'' C'' ex''] @ [b \<in> remdups' (zs @ bs).
b \<notin> set ((as @ ys) @ [ GF F'' C'' ex''])]")
 prefer 2
 apply (rule remdups'_append_fst)
 apply assumption
apply (subgoal_tac "(as @ ys @  GF F'' C'' ex'' # zs @ bs) = ((as @ ys) @ [ GF F'' C'' ex''] @ (zs @ bs))")
 prefer 2
 apply simp
apply (simp only:)
apply (rule_tac x="remdups' (as @ ys)" in exI)
apply (rule_tac x="[b\<in>remdups' (zs @ bs) . b \<notin> set ((as @ ys) @ [GF F'' C'' ex''])]" in exI)
apply (rule_tac x="[b\<in>remdups' cs .
              b \<noteq> GF F'' C'' ex'' \<and> b \<noteq> GF F' C' ex' \<and> b \<notin> set as \<and> b \<notin> set ys \<and> b \<notin> set zs \<and> b \<notin> set bs]" in exI)
apply (simp (no_asm))
done
(*>*)


lemma getHeapEx_mono_GF_TY:
"\<lbrakk> GF F'' C'' ex'' \<in> set (getHeapEx ex'); TY ex' ty \<in> set (getHeapEx ex) \<rbrakk> \<Longrightarrow> \<exists>as bs cs. remdups' (getHeapEx ex) = as@[GF F'' C'' ex'']@bs@[TY ex' ty]@cs"
(*<*)
apply (drule_tac ex'="ex'" in getHeapEx_TY_comp)
apply (drule in_set_conv_decomp_fst)
apply (erule exE | erule conjE)+
apply (simp only:)
apply (subgoal_tac "remdups' ((as @ (ys @ GF F'' C'' ex'' # zs) @ bs) @ [TY ex' ty] @ cs) = remdups' (as @ (ys @ GF F'' C'' ex'' # zs) @ bs) @
[TY ex' ty] @ [b \<in> remdups' cs. b \<notin> set ((as @ (ys @ GF F'' C'' ex'' # zs) @ bs)@[TY ex' ty])]") 
 prefer 2
 apply (rule_tac a="TY ex' ty" in  remdups'_append_fst)
 apply simp
apply simp
apply (case_tac " GF F'' C'' ex'' \<in> set (as @ ys)")
apply (drule in_set_conv_decomp_fst)
apply (erule exE)+
apply (subgoal_tac "remdups' (ysa @ [ GF F'' C'' ex''] @ (zsa @  GF F'' C'' ex'' # zs @ bs)) = remdups' ysa @ [ GF F'' C'' ex''] @ [b \<in> remdups' (zsa @ GF F'' C'' ex'' # zs @ bs). b \<notin> set (ysa @ [ GF F'' C'' ex''])]")
 prefer 2
 apply (rule remdups'_append_fst)
 apply simp
apply (subgoal_tac "(as @ ys @  GF F'' C'' ex'' # zs @ bs) = ((as @ ys) @  GF F'' C'' ex'' # zs @ bs)")
 prefer 2
 apply simp
apply (simp only:)
apply simp
apply (rule_tac x="remdups' ysa" in exI)
apply (rule_tac x="[b\<in>remdups' (zsa @ GF F'' C'' ex'' # zs @ bs) . b \<noteq> GF F'' C'' ex'' \<and> b \<notin> set ysa]" in exI) 
apply (rule_tac x="[b\<in>remdups' cs .
              b \<noteq> GF F'' C'' ex'' \<and> b \<noteq> TY ex' ty \<and> b \<notin> set as \<and> b \<notin> set ys \<and> b \<notin> set zs \<and> b \<notin> set bs]" in exI)
apply (simp only:)

apply (subgoal_tac "remdups' ((as @ ys) @ [ GF F'' C'' ex''] @ (zs @ bs)) = remdups' (as @ ys) @ [ GF F'' C'' ex''] @ [b \<in> remdups' (zs @ bs).
b \<notin> set ((as @ ys) @ [ GF F'' C'' ex''])]")
 prefer 2
 apply (rule remdups'_append_fst)
 apply assumption
apply (subgoal_tac "(as @ ys @  GF F'' C'' ex'' # zs @ bs) = ((as @ ys) @ [ GF F'' C'' ex''] @ (zs @ bs))")
 prefer 2
 apply simp
apply (simp only:)
apply (rule_tac x="remdups' (as @ ys)" in exI)
apply (rule_tac x="[b\<in>remdups' (zs @ bs) . b \<notin> set ((as @ ys) @ [GF F'' C'' ex''])]" in exI)
apply (rule_tac x="[b\<in>remdups' cs .
              b \<noteq> GF F'' C'' ex'' \<and> b \<noteq> TY ex' ty \<and> b \<notin> set as \<and> b \<notin> set ys \<and> b \<notin> set zs \<and> b \<notin> set bs]" in exI)
apply (simp (no_asm))
done
(*>*)


lemma getHeapEx_mono_TY_GF:
"\<lbrakk> TY ex'' ty'' \<in> set (getHeapEx ex'); GF F' C' ex' \<in> set (getHeapEx ex) \<rbrakk> 
\<Longrightarrow> \<exists>as bs cs. remdups' (getHeapEx ex) = as@[TY ex'' ty'']@bs@[GF F' C' ex']@cs"
(*<*)
apply (drule_tac ex'="ex'" in getHeapEx_GF_comp)
apply (drule in_set_conv_decomp_fst)
apply (erule exE | erule conjE)+
apply (simp only:)
apply (subgoal_tac "remdups' ((as @ (ys @ TY ex'' ty'' # zs) @ bs) @ [GF F' C' ex'] @ cs) = remdups' (as @ (ys @ TY ex'' ty'' # zs) @ bs) @
[GF F' C' ex'] @ [b \<in> remdups' cs. b \<notin> set ((as @ (ys @ TY ex'' ty'' # zs) @ bs)@[GF F' C' ex'])]") 
 prefer 2
 apply (rule_tac a="GF F' C' ex'" in  remdups'_append_fst)
 apply simp
apply simp
apply (case_tac " TY ex'' ty'' \<in> set (as @ ys)")
apply (drule in_set_conv_decomp_fst)
apply (erule exE)+
apply (subgoal_tac "remdups' (ysa @ [ TY ex'' ty''] @ (zsa @  TY ex'' ty'' # zs @ bs)) = remdups' ysa @ [ TY ex'' ty''] @ [b \<in> remdups' (zsa @ TY ex'' ty'' # zs @ bs). b \<notin> set (ysa @ [ TY ex'' ty''])]")
 prefer 2
 apply (rule remdups'_append_fst)
 apply simp
apply (subgoal_tac "(as @ ys @  TY ex'' ty'' # zs @ bs) = ((as @ ys) @  TY ex'' ty'' # zs @ bs)")
 prefer 2
 apply simp
apply (simp only:)
apply simp
apply (rule_tac x="remdups' ysa" in exI)
apply (rule_tac x="[b\<in>remdups' (zsa @ TY ex'' ty'' # zs @ bs) . b \<noteq> TY ex'' ty'' \<and> b \<notin> set ysa]" in exI) 
apply (rule_tac x="[b\<in>remdups' cs .
              b \<noteq> TY ex'' ty'' \<and> b \<noteq> GF F' C' ex' \<and> b \<notin> set as \<and> b \<notin> set ys \<and> b \<notin> set zs \<and> b \<notin> set bs]" in exI)
apply (simp only:)

apply (subgoal_tac "remdups' ((as @ ys) @ [ TY ex'' ty''] @ (zs @ bs)) = remdups' (as @ ys) @ [ TY ex'' ty''] @ [b \<in> remdups' (zs @ bs).
b \<notin> set ((as @ ys) @ [ TY ex'' ty''])]")
 prefer 2
 apply (rule remdups'_append_fst)
 apply assumption
apply (subgoal_tac "(as @ ys @  TY ex'' ty'' # zs @ bs) = ((as @ ys) @ [ TY ex'' ty''] @ (zs @ bs))")
 prefer 2
 apply simp
apply (simp only:)
apply (rule_tac x="remdups' (as @ ys)" in exI)
apply (rule_tac x="[b\<in>remdups' (zs @ bs) . b \<notin> set ((as @ ys) @ [TY ex'' ty''])]" in exI)
apply (rule_tac x="[b\<in>remdups' cs .
              b \<noteq> TY ex'' ty'' \<and> b \<noteq> GF F' C' ex' \<and> b \<notin> set as \<and> b \<notin> set ys \<and> b \<notin> set zs \<and> b \<notin> set bs]" in exI)
apply (simp (no_asm))
done
(*>*)


lemma getHeapEx_mono_TY_TY:
"\<lbrakk> TY ex'' ty'' \<in> set (getHeapEx ex'); TY ex' ty' \<in> set (getHeapEx ex) \<rbrakk> 
\<Longrightarrow> \<exists>as bs cs. remdups' (getHeapEx ex) = as@[TY ex'' ty'']@bs@[TY ex' ty']@cs"
(*<*)
apply (drule_tac ex'="ex'" in getHeapEx_TY_comp)
apply (drule in_set_conv_decomp_fst)
apply (erule exE | erule conjE)+
apply (simp only:)
apply (subgoal_tac "remdups' ((as @ (ys @ TY ex'' ty'' # zs) @ bs) @ [TY ex' ty'] @ cs) = remdups' (as @ (ys @ TY ex'' ty'' # zs) @ bs) @
[TY ex' ty'] @ [b \<in> remdups' cs. b \<notin> set ((as @ (ys @ TY ex'' ty'' # zs) @ bs)@[TY ex' ty'])]") 
 prefer 2
 apply (rule_tac a="TY ex' ty'" in  remdups'_append_fst)
 apply simp
apply simp
apply (case_tac " TY ex'' ty'' \<in> set (as @ ys)")
apply (drule in_set_conv_decomp_fst)
apply (erule exE)+
apply (subgoal_tac "remdups' (ysa @ [ TY ex'' ty''] @ (zsa @  TY ex'' ty'' # zs @ bs)) = remdups' ysa @ [ TY ex'' ty''] @ [b \<in> remdups' (zsa @ TY ex'' ty'' # zs @ bs). b \<notin> set (ysa @ [ TY ex'' ty''])]")
 prefer 2
 apply (rule remdups'_append_fst)
 apply simp
apply (subgoal_tac "(as @ ys @  TY ex'' ty'' # zs @ bs) = ((as @ ys) @  TY ex'' ty'' # zs @ bs)")
 prefer 2
 apply simp
apply (simp only:)
apply simp
apply (rule_tac x="remdups' ysa" in exI)
apply (rule_tac x="[b\<in>remdups' (zsa @ TY ex'' ty'' # zs @ bs) . b \<noteq> TY ex'' ty'' \<and> b \<notin> set ysa]" in exI) 
apply (rule_tac x="[b\<in>remdups' cs .
              b \<noteq> TY ex'' ty'' \<and> b \<noteq> TY ex' ty' \<and> b \<notin> set as \<and> b \<notin> set ys \<and> b \<notin> set zs \<and> b \<notin> set bs]" in exI)
apply (simp only:)

apply (subgoal_tac "remdups' ((as @ ys) @ [ TY ex'' ty''] @ (zs @ bs)) = remdups' (as @ ys) @ [ TY ex'' ty''] @ [b \<in> remdups' (zs @ bs).
b \<notin> set ((as @ ys) @ [ TY ex'' ty''])]")
 prefer 2
 apply (rule remdups'_append_fst)
 apply assumption
apply (subgoal_tac "(as @ ys @  TY ex'' ty'' # zs @ bs) = ((as @ ys) @ [ TY ex'' ty''] @ (zs @ bs))")
 prefer 2
 apply simp
apply (simp only:)
apply (rule_tac x="remdups' (as @ ys)" in exI)
apply (rule_tac x="[b\<in>remdups' (zs @ bs) . b \<notin> set ((as @ ys) @ [TY ex'' ty''])]" in exI)
apply (rule_tac x="[b\<in>remdups' cs .
              b \<noteq> TY ex'' ty'' \<and> b \<noteq> TY ex' ty' \<and> b \<notin> set as \<and> b \<notin> set ys \<and> b \<notin> set zs \<and> b \<notin> set bs]" in exI)
apply (simp (no_asm))
done
(*>*)

(*<*)
lemmas fms = lookup_lst lookup_map l2o.simps o2l.simps l2o_o2l split_def map_if_notin 
             map_if_notin_pair lookup_concat_map_if del_append 
(*>*)
(*<*)
declare forall_expand[simp]
(*>*)

section {* Simulation between wpF and effS. *}


lemma effS_wpF_Load:
assumes wf_Pi: "wf \<Pi>"
assumes handlesEx: "handlesEx (fst \<Pi>) p' = None"
assumes cmd_p: "cmd \<Pi> p = Some i"
assumes p_domC: "p \<in> set (domC \<Pi>)"
assumes i_def: "i = Load n"
assumes i_instr: "instrs_of P C M ! pc = i"
assumes s_def: "s = (p,\<sigma>,e)"
assumes p_def: "p = (C,M,pc)"
assumes sigma_def: "\<sigma> = (None,h,(stk,loc,p)#frs)"
assumes s'_def: "s' = (p',\<sigma>',e')"
assumes sigma'_def: "\<sigma>' = (None,h,fr'#frs')"
assumes e'_def: "e' = e\<lparr>cs := if \<exists>M n. i = Invoke M n then h # cs e else if i = Return then tl (cs e) else cs e\<rparr>"
assumes p'_def: "p' = snd (snd fr')"
assumes p'_domC: "p' \<in> set (domC \<Pi>)"
assumes check_i: "check_instr' i P h stk loc C M pc frs"  
assumes exec_i : "exec_instr i P h stk loc C M pc frs = \<sigma>'"
assumes Pi_def: "\<Pi> = (P,An)"
shows  "\<forall> I. evalE \<Pi> (p,\<sigma>,e\<lparr>lv:=I\<rparr>) (wpF \<Pi> p p' Q) = evalE \<Pi> (p',\<sigma>',e'\<lparr>lv:=I\<rparr>) Q"
(*<*)
proof -
from handlesEx cmd_p i_def 
show ?thesis
apply -
apply (simp only: wpF_Load)
--{* induction preparation *}
apply (subgoal_tac "\<exists>Q'. Q = Q'")
 prefer 2
 apply fastsimp
apply (erule exE)
apply (subgoal_tac "\<forall>em. substE em Q = substE em Q'")
 prefer 2
 apply simp
apply (subgoal_tac "\<forall> I. evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q = evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q'")
 prefer 2
 apply simp
apply (subgoal_tac "Q' \<in> set (subExpr Q)")
  prefer 2
  apply (simp add: getExpr_refl)
apply (erule_tac V="Q = Q'" in thin_rl)
apply (simp only:)
apply (erule_tac V="\<forall>em.  substE em Q = substE em Q'" in thin_rl)
apply (erule_tac V="\<forall>I. evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q = evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q'" in thin_rl)
apply (erule_tac P="Q' \<in> set (subExpr Q)" in rev_mp)
apply (erule thin_rl)+
apply (cut_tac exec_i[THEN sym] sigma'_def i_def sigma_def check_i p'_def e'_def)
--{* induction on Q' *}
apply (rule_tac expr="Q'" in expr_induct)
   
--{* Rg *}
apply (rule impI, rule allI)
apply (drule subExpr_rgIds)
apply (drule in_set_conv_decomp_fst)
apply (erule exE | erule conjE)+
apply (simp add: fms map_if_notin id_lookup_def)

--{* St *}
apply (rule impI, rule allI)
apply (drule subExpr_stkIds)
apply (drule_tac xs="stkIds Q" in in_set_conv_decomp_fst)
apply (erule exE | erule conjE)+
apply (simp add: fms map_if_notin id_lookup_def)
apply (rule impI)
apply (case_tac "nat")
apply simp

apply simp
--{* Lv *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* Cn *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* NewA *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* Gf list1 list2 expr *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Gf list1 list2 expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def split add: val.split)
--{* FrNr *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* Num expr1 num_op epxr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Num expr1 num_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Num expr1 num_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Rel expr1 rel_op expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Rel expr1 rel_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Rel expr1 rel_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* IF expr1 THEN expr2 ELSE expr3 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr3 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (drule mp,assumption)+

apply (rule allI)
apply (erule_tac x="I" in allE)+
apply (simp add: split_def fms map_if_notin id_lookup_def evalE_evalEs.simps substE_substEs.simps split del: split_if add: bool.split option.split)
apply (case_tac "evalE \<Pi> ((C, M, Suc pc), (None, h, (loc ! n # stk, loc, C, M, Suc pc) # frs), e\<lparr>lv := I\<rparr>) expr1 = \<lfloor>Bool True\<rfloor>")
apply simp

apply simp
--{* Eq expr1 expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Eq expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Eq expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Neg expr *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Neg expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Imp expr1 expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Imp expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Imp expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Forall nat expr *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Forall nat expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def split del: split_if)
--{* Ty expr ty*}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Ty expr ty" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: Let_def split_def fms map_if_notin id_lookup_def split add: val.split)
--{* Pos x *}
apply (rule impI, rule allI)
apply (subgoal_tac "x \<in> set (getPosEx Q)")
 prefer 2
 apply (rule subExpr_getPosEx)
 apply assumption
apply (drule_tac xs="getPosEx Q" in  in_set_conv_decomp_fst)
apply (erule exE |erule conjE)+
apply (simp add: p_def fms map_if_notin id_lookup_def add: callers_sysinv_trans del: callers_sysinv.simps)
apply (case_tac "frs")
apply (cut_tac p'_domC)
apply (cut_tac p_domC)
apply (cut_tac p_def)
apply simp

apply (cut_tac p'_domC)
apply (cut_tac p_domC)
apply (cut_tac p_def)
apply (simp add: callers_simps)
--{* Call *}
apply (rule impI,rule allI)
apply (simp add: Let_def split_def fms map_if_notin id_lookup_def)
apply (rule impI)
apply (simp only: neq_Nil_conv)
apply (erule exE)+
apply (simp (no_asm_simp) add: split_paired_all env_upd_cs)

--{* Catch  *}
apply (rule impI,rule allI)
apply (simp add: Let_def split_def fms map_if_notin id_lookup_def)
apply (rule impI)
apply (simp only: neq_Nil_conv)
apply (erule exE)+
apply (simp (no_asm_simp) add: split_paired_all env_upd_cs)

--{* And es *}
apply (rule impI, rule allI)
apply (simp add: evalE_And fms map_if_notin id_lookup_def del: evalE_evalEs.simps split del: split_if)
apply (rule iffI)
apply (rule ballI)
apply (erule_tac x="ex" in ballE)
apply (erule_tac x="(substE
                  (map (\<lambda>q. (Pos q, if q = (C, M, Suc pc) then Pos p else FF)) (getPosEx Q) @
                   map (\<lambda>k. (St k, if k = 0 then Rg n else St (k - 1))) (stkIds Q))
                  ex)" in ballE)
apply (subgoal_tac "ex \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="And es" in subExpr_Gf)
 apply assumption
 apply (simp add: parts.simps)
apply (simp add: substEs_map) 

apply (simp add: substEs_map)

apply (simp only:)

apply (rule ballI)

apply (simp add: substEs_map image_iff)
apply (erule bexE)
apply (erule_tac x="x" in ballE)
apply (erule_tac x="x" in ballE)
apply (subgoal_tac "x \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="And es" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm_simp) add: parts.simps)
apply simp

apply (simp only:)

apply (simp only:)
done
qed
(*>*)


lemma effS_wpF_Store:
assumes i_def: "i = Store n"
assumes wf_Pi: "wf \<Pi>"
assumes handlesEx: "handlesEx (fst \<Pi>) p' = None"
assumes cmd_p: "cmd \<Pi> p = Some i"
assumes p_domC: "p \<in> set (domC \<Pi>)"
assumes i_instr: "instrs_of P C M ! pc = i"
assumes s_def: "s = (p,\<sigma>,e)"
assumes p_def: "p = (C,M,pc)"
assumes sigma_def: "\<sigma> = (None,h,(stk,loc,p)#frs)"
assumes s'_def: "s' = (p',\<sigma>',e')"
assumes sigma'_def: "\<sigma>' = (None,h,fr'#frs')"
assumes e'_def: "e' = e\<lparr>cs := if \<exists>M n. i = Invoke M n then h # cs e else if i = Return then tl (cs e) else cs e\<rparr>"
assumes p'_def: "p' = snd (snd fr')"
assumes check_i: "check_instr' i P h stk loc C M pc frs"
assumes exec_i : "exec_instr i P h stk loc C M pc frs = \<sigma>'"
assumes Pi_def: "\<Pi> = (P,An)"
shows  "\<forall> I. evalE \<Pi> (p,\<sigma>,e\<lparr>lv:=I\<rparr>) (wpF \<Pi> p p' Q) = evalE \<Pi> (p',\<sigma>',e'\<lparr>lv:=I\<rparr>) Q"
(*<*)
proof -
from handlesEx cmd_p i_def 
show ?thesis
apply -
apply (simp only: wpF_Store)
--{* induction preparation *}
apply (subgoal_tac "\<exists>Q'. Q = Q'")
 prefer 2
 apply fastsimp
apply (erule exE)
apply (subgoal_tac "\<forall>em. substE em Q = substE em Q'")
 prefer 2
 apply simp
apply (subgoal_tac "\<forall> I. evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q = evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q'")
 prefer 2
 apply simp
apply (subgoal_tac "Q' \<in> set (subExpr Q)")
  prefer 2
  apply (simp add: getExpr_refl)
apply (erule_tac V="Q = Q'" in thin_rl)
apply (simp only:)
apply (erule_tac V="\<forall>em.  substE em Q = substE em Q'" in thin_rl)
apply (erule_tac V="\<forall>I. evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q = evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q'" in thin_rl)
apply (erule_tac P="Q' \<in> set (subExpr Q)" in rev_mp)
apply (erule thin_rl)+
apply (cut_tac exec_i[THEN sym] sigma'_def i_def sigma_def check_i p'_def e'_def)
--{* induction on Q' *}
apply (rule_tac expr="Q'" in expr_induct)
   
--{* Rg *}
apply (rule impI, rule allI)
apply (drule subExpr_rgIds)
apply (drule in_set_conv_decomp_fst)
apply (erule exE | erule conjE)+
apply (simp add: fms map_if_notin id_lookup_def)
apply (case_tac "n = nat")
apply (simp add: neq_Nil_conv)
apply (erule exE | erule conjE)+
apply simp
--{* n ~= nat *}
apply (simp add: fms)
--{* St *}
apply (rule impI, rule allI)
apply (drule subExpr_stkIds)
apply (drule_tac xs="stkIds Q" in in_set_conv_decomp_fst)
apply (erule exE | erule conjE)+
apply (simp add: fms map_if_notin id_lookup_def neq_Nil_conv)
apply (erule exE | erule conjE)+
apply (case_tac "nat")
apply simp

apply simp
--{* Lv *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* Cn *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* NewA *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* Gf list1 list2 expr *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Gf list1 list2 expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def split add: val.splits)
--{* FrNr *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* Num expr1 num_op epxr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Num expr1 num_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Num expr1 num_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Rel expr1 rel_op expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Rel expr1 rel_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Rel expr1 rel_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* IF expr1 THEN expr2 ELSE expr3 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr3 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (drule mp,assumption)+

apply (simp add: split_def fms map_if_notin id_lookup_def evalE_evalEs.simps substE_substEs.simps split del: split_if add: bool.split )
apply (rule allI)
apply (erule_tac x="I" in allE)+
apply (case_tac "evalE \<Pi> ((C, M, Suc pc), (None, h, (tl stk, loc[n := hd stk], C, M, Suc pc) # frs), e\<lparr>lv := I\<rparr>) expr1 = \<lfloor>Bool True\<rfloor>")
apply simp

apply simp

--{* Eq expr1 expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Eq expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Eq expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Neg expr *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Neg expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Imp expr1 expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Imp expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Imp expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Forall nat expr *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Forall nat expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def split del: split_if)
--{* Ty expr ty*}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Ty expr ty" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: Let_def split_def fms map_if_notin id_lookup_def split add: val.split)
--{* Pos x *}
apply (rule impI, rule allI)
apply (subgoal_tac "x \<in> set (getPosEx Q)")
 prefer 2
 apply (rule subExpr_getPosEx)
 apply assumption
apply (drule_tac xs="getPosEx Q" in  in_set_conv_decomp_fst)
apply (erule exE |erule conjE)+
apply (simp add: fms map_if_notin id_lookup_def)
apply (rule impI)
apply (subgoal_tac "p' mem domC \<Pi>")
 prefer 2
 apply (cut_tac wf_Pi)
 apply (cut_tac p_domC)
 apply (cut_tac p_def)
 apply (cut_tac cmd_p)
 apply (simp only: in_set_conv_decomp)
 apply (erule exE)+
 apply (simp add: incA_def wf_def checkPos_split succsNormal_def split add: split_if_asm)
apply (cut_tac p_domC)
apply (case_tac "frs")
apply (simp add: p_def mem_iff)

apply (simp add: p_def mem_iff split_def callers_simps)
--{* Call *}
apply (rule impI,rule allI)
apply (simp add: Let_def split_def fms map_if_notin id_lookup_def)
apply (rule impI)
apply (simp only: neq_Nil_conv)
apply (erule exE)+
apply (simp (no_asm_simp) add: split_paired_all env_upd_cs)

--{* Catch  *}
apply (rule impI,rule allI)
apply (simp add: Let_def split_def fms map_if_notin id_lookup_def)
apply (rule impI)
apply (simp only: neq_Nil_conv)
apply (erule exE)+
apply (simp (no_asm_simp) add: split_paired_all env_upd_cs)

--{* And es *}
apply (rule impI, rule allI)
apply (simp add: evalE_And fms map_if_notin id_lookup_def del: evalE_evalEs.simps split del: split_if)
apply (rule iffI)
apply (rule ballI)
apply (erule_tac x="ex" in ballE)
apply (erule_tac x="(substE (map (\<lambda>q. (Pos q, if q = (C, M, Suc pc) then Pos p else FF)) (getPosEx Q) @
                         (Rg n, St 0) # map (\<lambda>k. (St k, St (Suc k))) (stkIds Q))
                  ex)" in ballE)
apply (subgoal_tac "ex \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="And es" in subExpr_Gf)
 apply assumption
 apply (simp add: parts.simps)
apply (simp add: substEs_map) 

apply (simp add: substEs_map)

apply (simp only:)

apply (rule ballI)

apply (simp add: substEs_map image_iff)
apply (erule bexE)
apply (erule_tac x="x" in ballE)
apply (erule_tac x="x" in ballE)
apply (subgoal_tac "x \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="And es" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm_simp) add: parts.simps)
apply simp

apply (simp only:)

apply (simp only:)
done
qed
(*>*)


lemma effS_wpF_Push:
assumes i_def: "i = Push v"
assumes wf_Pi: "wf \<Pi>"
assumes handlesEx: "handlesEx (fst \<Pi>) p' = None"
assumes cmd_p: "cmd \<Pi> p = Some i"
assumes p_domC: "p \<in> set (domC \<Pi>)"
assumes i_instr: "instrs_of P C M ! pc = i"
assumes s_def: "s = (p,\<sigma>,e)"
assumes p_def: "p = (C,M,pc)"
assumes sigma_def: "\<sigma> = (None,h,(stk,loc,p)#frs)"
assumes s'_def: "s' = (p',\<sigma>',e')"
assumes sigma'_def: "\<sigma>' = (None,h,fr'#frs')"
assumes e'_def: "e' = e\<lparr>cs := if \<exists>M n. i = Invoke M n then h # cs e else if i = Return then tl (cs e) else cs e\<rparr>"
assumes p'_def: "p' = snd (snd fr')"
assumes check_i: "check_instr' i P h stk loc C M pc frs"
assumes exec_i : "exec_instr i P h stk loc C M pc frs = \<sigma>'"
assumes Pi_def: "\<Pi> = (P,An)"
shows  "\<forall> I. evalE \<Pi> (p,\<sigma>,e\<lparr>lv:=I\<rparr>) (wpF \<Pi> p p' Q) = evalE \<Pi> (p',\<sigma>',e'\<lparr>lv:=I\<rparr>) Q"
(*<*)
proof -
from handlesEx cmd_p i_def 
show ?thesis
apply -
apply (simp only: wpF_Push)
--{* induction preparation *}
apply (subgoal_tac "\<exists>Q'. Q = Q'")
 prefer 2
 apply fastsimp
apply (erule exE)
apply (subgoal_tac "\<forall>em. substE em Q = substE em Q'")
 prefer 2
 apply simp
apply (subgoal_tac "\<forall> I. evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q = evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q'")
 prefer 2
 apply simp
apply (subgoal_tac "Q' \<in> set (subExpr Q)")
  prefer 2
  apply (simp add: getExpr_refl)
apply (erule_tac V="Q = Q'" in thin_rl)
apply (simp only:)
apply (erule_tac V="\<forall>em.  substE em Q = substE em Q'" in thin_rl)
apply (erule_tac V="\<forall>I. evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q = evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q'" in thin_rl)
apply (erule_tac P="Q' \<in> set (subExpr Q)" in rev_mp)
apply (erule thin_rl)+
apply (cut_tac exec_i[THEN sym] sigma'_def i_def sigma_def check_i p'_def e'_def)
--{* induction on Q' *}
apply (rule_tac expr="Q'" in expr_induct)
   
--{* Rg *}
apply (rule impI, rule allI)
apply (drule subExpr_rgIds)
apply (drule_tac xs="rgIds Q" in in_set_conv_decomp_fst)
apply (erule exE | erule conjE)+
apply (simp add: fms map_if_notin id_lookup_def)
--{* St *}
apply (rule impI, rule allI)
apply (drule subExpr_stkIds)
apply (drule_tac xs="stkIds Q" in in_set_conv_decomp_fst)
apply (erule exE | erule conjE)+
apply (simp add: fms map_if_notin id_lookup_def)
apply (rule impI)
apply (case_tac "nat")
apply simp

apply simp
--{* Lv *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* Cn *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* NewA *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* Gf list1 list2 expr *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Gf list1 list2 expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def split add: val.split)
--{* FrNr *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* Num expr1 num_op epxr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Num expr1 num_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Num expr1 num_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Rel expr1 rel_op expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Rel expr1 rel_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Rel expr1 rel_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* IF expr1 THEN expr2 ELSE expr3 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr3 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (drule mp,assumption)+
apply (rule allI)+
apply (erule_tac x="I" in allE)+
apply (simp add: split_def fms map_if_notin id_lookup_def evalE_evalEs.simps substE_substEs.simps split del: split_if add: bool.split )
apply (case_tac "evalE \<Pi> ((C, M, Suc pc), (None, h, (v # stk, loc, C, M, Suc pc) # frs), e\<lparr>lv := I\<rparr>) expr1 = \<lfloor>Bool True\<rfloor>")
apply simp
apply simp
--{* Eq expr1 expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Eq expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Eq expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Neg expr *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Neg expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Imp expr1 expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Imp expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Imp expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Forall nat expr *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Forall nat expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def split del: split_if)
--{* Ty expr ty*}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Ty expr ty" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: Let_def split_def fms map_if_notin id_lookup_def split add: val.split)
--{* Pos x *}
apply (rule impI, rule allI)
apply (subgoal_tac "x \<in> set (getPosEx Q)")
 prefer 2
 apply (rule subExpr_getPosEx)
 apply assumption
apply (drule_tac xs="getPosEx Q" in  in_set_conv_decomp_fst)
apply (erule exE |erule conjE)+
apply (subgoal_tac "p' mem domC \<Pi>")
 prefer 2
 apply (cut_tac wf_Pi)
 apply (cut_tac p_domC)
 apply (cut_tac p_def)
 apply (cut_tac cmd_p)
 apply (simp only: in_set_conv_decomp)
 apply (erule exE)+
 apply (simp add: incA_def wf_def checkPos_split succsNormal_def split add: split_if_asm)
apply (cut_tac p_domC)
apply (case_tac "frs")
apply (simp add: fms map_if_notin id_lookup_def p_def mem_iff)

apply (simp add: fms map_if_notin id_lookup_def p_def mem_iff callers_simps)
--{* Call *}
apply (rule impI,rule allI)
apply (simp add: Let_def split_def fms map_if_notin id_lookup_def)
apply (rule impI)
apply (simp only: neq_Nil_conv)
apply (erule exE)+
apply (simp (no_asm_simp) add: split_paired_all env_upd_cs)

--{* Catch  *}
apply (rule impI,rule allI)
apply (simp add: Let_def split_def fms map_if_notin id_lookup_def)
apply (rule impI)
apply (simp only: neq_Nil_conv)
apply (erule exE)+
apply (simp (no_asm_simp) add: split_paired_all env_upd_cs)

--{* And es *}
apply (rule impI, rule allI)
apply (simp add: evalE_And fms map_if_notin id_lookup_def del: evalE_evalEs.simps split del: split_if)
apply (rule iffI)
apply (rule ballI)
apply (erule_tac x="ex" in ballE)
apply (erule_tac x="(substE (map (\<lambda>q. (Pos q, if q = (C, M, Suc pc) then Pos p else FF)) (getPosEx Q) @
                         map (\<lambda>k. (St k, if k = 0 then Cn v else St (k - 1))) (stkIds Q))
                  ex)" in ballE)
apply (subgoal_tac "ex \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="And es" in subExpr_Gf)
 apply assumption
 apply (simp add: parts.simps)
apply (simp add: substEs_map) 

apply (simp add: substEs_map)

apply (simp only:)

apply (rule ballI)

apply (simp add: substEs_map image_iff)
apply (erule bexE)
apply (erule_tac x="x" in ballE)
apply (erule_tac x="x" in ballE)
apply (subgoal_tac "x \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="And es" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm_simp) add: parts.simps)
apply simp

apply (simp only:)

apply (simp only:)
done 
qed
(*>*)


lemma effS_wpF_New:
assumes i_def: "i = New Cl"
assumes wf_Pi: "wf \<Pi>"
assumes handlesEx: "handlesEx (fst \<Pi>) p' = None"
assumes cmd_p: "cmd \<Pi> p = Some i"
assumes p_domC: "p \<in> set (domC \<Pi>)"
assumes i_instr: "instrs_of P C M ! pc = i"
assumes s_def: "s = (p,\<sigma>,e)"
assumes p_def: "p = (C,M,pc)"
assumes sigma_def: "\<sigma> = (None,h,(stk,loc,p)#frs)"
assumes s'_def: "s' = (p',\<sigma>',e')"
assumes sigma'_def: "\<sigma>' = (None,h',fr'#frs')"
assumes e'_def: "e' = e\<lparr>cs := if \<exists>M n. i = Invoke M n then h # cs e else if i = Return then tl (cs e) else cs e\<rparr>"
assumes p'_def: "p' = snd (snd fr')"
assumes check_i: "check_instr' i P h stk loc C M pc frs"
assumes exec_i : "exec_instr i P h stk loc C M pc frs = \<sigma>'"
assumes Pi_def: "\<Pi> = (P,An)"
shows  "\<forall> I. evalE \<Pi> (p,\<sigma>,e\<lparr>lv:=I\<rparr>) (wpF \<Pi> p p' Q) = evalE \<Pi> (p',\<sigma>',e'\<lparr>lv:=I\<rparr>) Q"
(*<*)
proof -
from handlesEx cmd_p i_def 
show ?thesis
apply -
apply (simp only: wpF_New)
--{* induction preparation *}
apply (subgoal_tac "\<exists>Q'. Q = Q'")
 prefer 2
 apply fastsimp
apply (erule exE)
apply (subgoal_tac "\<forall>em. substE em Q = substE em Q'")
 prefer 2
 apply simp
apply (subgoal_tac "\<forall> I. evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q = evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q'")
 prefer 2
 apply simp
apply (subgoal_tac "Q' \<in> set (subExpr Q)")
  prefer 2
  apply (simp add: getExpr_refl)
apply (erule_tac V="Q = Q'" in thin_rl)
apply (simp only:)
apply (erule_tac V="\<forall>em.  substE em Q = substE em Q'" in thin_rl)
apply (erule_tac V="\<forall>I. evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q = evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q'" in thin_rl)
apply (erule_tac P="Q' \<in> set (subExpr Q)" in rev_mp)
apply (erule thin_rl)+
apply (cut_tac exec_i[THEN sym] sigma'_def i_def sigma_def check_i p'_def e'_def p_def Pi_def)
--{* induction on Q' *}
apply simp
apply (rule_tac expr="Q'" in expr_induct)
--{* Rg *}
apply (rule impI, rule allI)
apply (subgoal_tac "\<forall>hex \<in> set (remdups' (getHeapEx Q)). heapEx (Rg nat) \<noteq> [hex]")
 prefer 2
 apply simp
apply (drule_tac P="fst \<Pi>" and Cl="Cl" in foldl_map_lookup')
apply (simp add: fms map_if_notin id_lookup_def)
--{* St *} 
apply (rule impI, rule allI)
apply (subgoal_tac "\<forall>hex \<in> set (remdups' (getHeapEx Q)). heapEx (St nat) \<noteq> [hex]")
 prefer 2
 apply simp
apply (drule_tac P="fst \<Pi>" and Cl="Cl" in foldl_map_lookup')
apply (drule subExpr_stkIds)
apply (drule_tac xs="stkIds Q" in  in_set_conv_decomp_fst)
apply (erule exE | erule conjE)+
apply (simp add: fms map_if_notin id_lookup_def neq_Nil_conv)
apply (case_tac "stk")
apply simp

apply (case_tac "nat")
apply simp

apply simp
--{* Lv *}
apply (rule impI, rule allI)
apply (subgoal_tac "\<forall>hex \<in> set (remdups' (getHeapEx Q)). heapEx (Lv nat) \<noteq> [hex]")
 prefer 2
 apply simp
apply (drule_tac P="fst \<Pi>" and Cl="Cl" in foldl_map_lookup')
apply (simp add: fms map_if_notin id_lookup_def)
--{* Cn *}
apply (rule impI, rule allI)
apply (subgoal_tac "\<forall>hex \<in> set (remdups' (getHeapEx Q)). heapEx (Cn val) \<noteq> [hex]")
 prefer 2
 apply simp
apply (drule_tac P="fst \<Pi>" and Cl="Cl" in foldl_map_lookup')
apply (simp add: fms map_if_notin id_lookup_def)

--{* NewA n*}
apply (rule impI, rule allI)
apply (subgoal_tac "\<forall>hex \<in> set (remdups' (getHeapEx Q)). heapEx (NewA n) \<noteq> [hex]")
 prefer 2
 apply simp
apply (drule_tac P="fst \<Pi>" and Cl="Cl" in foldl_map_lookup')
apply (drule subExpr_NewEx)
apply (drule_tac xs="getNewEx Q" in in_set_conv_decomp_fst)
apply (erule exE | erule conjE)+
apply (simp add: fms map_if_notin id_lookup_def)
apply (subgoal_tac "evalNewA (\<lambda>u. if u = a then \<lfloor>blank P Cl\<rfloor> else h u) n = evalNewA (h(a \<mapsto> blank P Cl)) n")
 prefer 2
 apply (induct_tac n)
 apply simp

 apply simp
apply (erule_tac V="h' = h(a \<mapsto> blank P Cl)" in thin_rl)
apply (simp only:)
apply (rule evalNewA_dom)
apply (simp add: dom_def mem_Collect_eq)
--{* Gf list1 list2 expr *}
apply (rule impI, rule allI)
apply (frule subExpr_getHeapEx)
apply (drule_tac x="GF list1 list2 expr" in  in_rd_sp)
apply (erule exE | erule conjE)+
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Gf list1 list2 expr" in subExpr_Gf)
 apply assumption
 apply (simp add: parts.simps)
apply (drule mp, assumption)
apply (subgoal_tac "\<forall>hex \<in> set as. heapEx (Gf list1 list2 expr) \<noteq> [hex]")
 prefer 2
 apply (rule ballI)
 apply (rule classical)
 apply (simp add: split_def)
apply (drule_tac P="fst \<Pi>" and Cl="Cl" in foldl_map_lookup')
apply (subgoal_tac "\<forall>hex \<in> set bs. heapEx (Gf list1 list2 expr) \<noteq> [hex]")
 prefer 2
 apply (rule ballI)
 apply (rule classical)
 apply (simp add: split_def)
apply (drule_tac P="fst \<Pi>" and Cl="Cl" in foldl_map_lookup')
apply (simp (no_asm_simp) only: evalE_evalEs.simps substE_substEs.simps)
apply (subgoal_tac "
               (substE (foldl (\<lambda>mp hex. (case hex of
                                         GF F C ex \<Rightarrow>
                                           let ex' = substE mp ex
                                           in (Gf F C ex, IF ex' \<doteq> NewA 0 THEN Cn (the (snd (blank (fst (P, An)) Cl) (F, C))) ELSE Gf F C ex')
                                         | TY ex ty \<Rightarrow>
                                             let ex' = substE mp ex in (Ty ex ty, IF ex' \<doteq> NewA 0 THEN Cn (Bool (Class Cl = ty)) ELSE Ty ex' ty)) #
                                        mp)
                         ((Gf list1 list2 expr,
                           IF substE (foldl (\<lambda>mp hex. (case hex of
                                                       GF F C ex \<Rightarrow>
                                                         let ex' = substE mp ex
                                                         in (Gf F C ex,
                                                             IF ex' \<doteq> NewA 0 THEN Cn (the (snd (blank (fst (P, An)) Cl) (F, C))) ELSE Gf F C ex')
                                                       | TY ex ty \<Rightarrow>
                                                           let ex' = substE mp ex
                                                           in (Ty ex ty, IF ex' \<doteq> NewA 0 THEN Cn (Bool (Class Cl = ty)) ELSE Ty ex' ty)) #
                                                      mp)
                                       (map (\<lambda>q. (Pos q, if q = (C, M, Suc pc) then Pos p else FF)) (getPosEx Q) @
                                        map (\<lambda>k. (St k, if k = 0 then NewA 0 else St (k - 1))) (stkIds Q) @
                                        map (\<lambda>n. (NewA n, NewA (Suc n))) (getNewEx Q))
                                       as)
                               expr \<doteq>
                              NewA 0 THEN Cn (the (snd (blank P Cl)
                                                    (list1, list2))) ELSE Gf list1 list2
 (substE (foldl (\<lambda>mp hex. (case hex of
                           GF F C ex \<Rightarrow>
                             let ex' = substE mp ex
                             in (Gf F C ex, IF ex' \<doteq> NewA 0 THEN Cn (the (snd (blank (fst (P, An)) Cl) (F, C))) ELSE Gf F C ex')
                           | TY ex ty \<Rightarrow> let ex' = substE mp ex in (Ty ex ty, IF ex' \<doteq> NewA 0 THEN Cn (Bool (Class Cl = ty)) ELSE Ty ex' ty)) #
                          mp)
           (map (\<lambda>q. (Pos q, if q = (C, M, Suc pc) then Pos p else FF)) (getPosEx Q) @
            map (\<lambda>k. (St k, if k = 0 then NewA 0 else St (k - 1))) (stkIds Q) @ map (\<lambda>n. (NewA n, NewA (Suc n))) (getNewEx Q))
           as)
   expr)) #
                          foldl (\<lambda>mp hex. (case hex of
                                           GF F C ex \<Rightarrow>
                                             let ex' = substE mp ex
                                             in (Gf F C ex, IF ex' \<doteq> NewA 0 THEN Cn (the (snd (blank (fst (P, An)) Cl) (F, C))) ELSE Gf F C ex')
                                           | TY ex ty \<Rightarrow>
                                               let ex' = substE mp ex
                                               in (Ty ex ty, IF ex' \<doteq> NewA 0 THEN Cn (Bool (Class Cl = ty)) ELSE Ty ex' ty)) #
                                          mp)
                           (map (\<lambda>q. (Pos q, if q = (C, M, Suc pc) then Pos p else FF)) (getPosEx Q) @
                            map (\<lambda>k. (St k, if k = 0 then NewA 0 else St (k - 1))) (stkIds Q) @ map (\<lambda>n. (NewA n, NewA (Suc n))) (getNewEx Q))
                           as)
                         bs)
                 expr) =  (substE (foldl (\<lambda>mp hex. (case hex of
                                           GF F C ex \<Rightarrow>
                                             let ex' = substE mp ex
                                             in (Gf F C ex, IF ex' \<doteq> NewA 0 THEN Cn (the (snd (blank (fst (P, An)) Cl) (F, C))) ELSE Gf F C ex')
                                           | TY ex ty \<Rightarrow>
                                               let ex' = substE mp ex
                                               in (Ty ex ty, IF ex' \<doteq> NewA 0 THEN Cn (Bool (Class Cl = ty)) ELSE Ty ex' ty)) #
                                          mp)
                           (map (\<lambda>q. (Pos q, if q = (C, M, Suc pc) then Pos p else FF)) (getPosEx Q) @
                            map (\<lambda>k. (St k, if k = 0 then NewA 0 else St (k - 1))) (stkIds Q) @ map (\<lambda>n. (NewA n, NewA (Suc n))) (getNewEx Q))
                           as)
                   expr)" )
 prefer 2
 apply (erule_tac V="\<forall>I. ?P I" in thin_rl)
 apply (erule_tac V="\<forall>mp'. ?P mp'" in thin_rl)
 apply (erule_tac V="\<forall>mp'. ?P mp'" in thin_rl)
 apply (rule substE_eq)
 apply (subgoal_tac "\<exists>expr'. expr = expr'")
  prefer 2
  apply (erule thin_rl)+
  apply fastsimp
 apply (erule exE)+
 apply (subgoal_tac "\<forall> em em'. eqExMps em em' expr = eqExMps em em' expr'")
  prefer 2
  apply simp
 apply (subgoal_tac "expr' \<in> set (subExpr expr)")
  prefer 2
  apply (simp only:)
  apply (rule getExpr_refl)
 apply (erule_tac V="expr = expr'" in thin_rl)
 apply (simp only:)
 apply (erule_tac P="expr' \<in> set (subExpr expr)" in rev_mp)
 apply (rule_tac expr="expr'" in expr_induct)
 --{* Rg nat *}
 apply (subgoal_tac "\<forall> hex\<in>set as. heapEx (Rg nat) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (subgoal_tac "\<forall> hex \<in>set bs. heapEx (Rg nat) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (simp add: fst_conv eqExMps_def lookup.simps fms)
 --{* St nat *}
 apply (subgoal_tac "\<forall>hex\<in>set as. heapEx (St nat) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (subgoal_tac "\<forall> hex\<in>set bs. heapEx (St nat) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (simp add: eqExMps_def lookup.simps fms)
 --{* Lv nat *}
 apply (subgoal_tac "\<forall> hex\<in>set as. heapEx (Lv nat) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (subgoal_tac "\<forall> hex\<in>set bs. heapEx (Lv nat) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (simp add: eqExMps_def lookup.simps fms)
 --{* Cn val *}
 apply (subgoal_tac "\<forall> hex\<in>set as. heapEx (Cn val) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (subgoal_tac "\<forall> hex\<in>set bs. heapEx (Cn val) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (simp add: eqExMps_def lookup.simps fms)
 --{* NewA *}
 apply (subgoal_tac "\<forall> hex\<in>set as. heapEx (NewA n) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (subgoal_tac "\<forall> hex\<in>set bs. heapEx (NewA n) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (simp add: eqExMps_def lookup.simps fms)
 --{* Gf list1a list2a expra *}
 apply (rule impI)
thm subExpr_getHeapEx
 apply (frule_tac ex="expra" in subExpr_getHeapEx)
 apply (subgoal_tac "expra \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="Gf list1a list2a expra" in subExpr_Gf)
  apply assumption
  apply (simp add: parts.simps)
 apply (drule mp, assumption)

 apply (subgoal_tac "\<exists>as' bs' cs'. remdups' (getHeapEx Q) = as'@[GF list1a list2a expra]@bs'@[GF list1 list2 expr]@cs'")
   prefer 2
   apply (rule getHeapEx_mono_GF_GF)
   apply assumption

   apply (rule subExpr_getHeapEx)
   apply assumption
  apply (erule exE)+
 apply (subgoal_tac "distinct (as' @ [GF list1a list2a expra] @ bs' @ [GF list1 list2 expr] @ cs')")
  prefer 2
  apply (drule_tac t="as' @ [GF list1a list2a expra] @ bs' @ [GF list1 list2 expr] @ cs'" in sym)
  apply (erule_tac V="remdups' (getHeapEx Q) = ?c" in thin_rl)
  apply (simp only: distinct_remdups')
 apply (subgoal_tac "distinct (as@[GF list1 list2 expr]@bs)")
  prefer 2
  apply (subgoal_tac "(as @ GF list1 list2 expr # bs) = (as @ [GF list1 list2 expr] @ bs)")
   prefer 2
   apply simp
  apply (drule_tac s="as @ GF list1 list2 expr # bs" in sym)
  apply (simp only: distinct_remdups')
 apply (subgoal_tac "as @ [GF list1 list2 expr] @ bs = (as' @ [GF list1a list2a expra] @ bs') @ [GF list1 list2 expr] @ cs'")
  prefer 2
  apply (drule_tac t="as' @ [GF list1a list2a expra] @ bs' @ [GF list1 list2 expr] @ cs'" in sym)
  apply simp
 apply (drule distinct_list_match)
 apply assumption
 apply simp
 apply (subgoal_tac "\<forall>hex \<in> set bs. heapEx (Gf list1a list2a expra) \<noteq> [hex]")
  prefer 2
   apply (subgoal_tac "GF list1a list2a expra \<notin> set bs")
    prefer 2
    apply simp
  apply (rule ballI)
  apply (rule classical)
  apply (simp add: split_def)

 apply (subgoal_tac "\<forall> hex \<in> set as'. heapEx (Gf list1a list2a expra) \<noteq> [hex]")
  prefer 2
   apply (subgoal_tac "GF list1a list2a expra \<notin> set as'")
    prefer 2
    apply simp
  apply (rule ballI)
  apply (rule classical)
  apply (simp add: split_def)

 apply (subgoal_tac "\<forall> hex \<in> set bs'. heapEx (Gf list1a list2a expra) \<noteq> [hex]")
  prefer 2
   apply (subgoal_tac "GF list1a list2a expra \<notin> set bs'")
    prefer 2
    apply simp
  apply (rule ballI)
  apply (rule classical)
  apply (simp add: split_def)

 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')+

 apply (subgoal_tac "expr \<noteq> expra")
  prefer 2
  apply (drule getHeapEx_GF_not_refl)
  apply (rule not_sym)
  apply assumption

 apply (simp  add: eqExMps_def)
 --{* FrNr *}
 apply (subgoal_tac "\<forall> hex\<in>set as. heapEx FrNr \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (subgoal_tac "\<forall> hex\<in>set bs. heapEx FrNr \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (simp add: eqExMps_def lookup.simps fms)
 --{* Num expr1 numop expr2 *}
 apply (rule impI)
 apply (subgoal_tac "expr1 \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="Num expr1 num_op expr2" in subExpr_Gf)
  apply assumption
  apply simp
 apply (subgoal_tac "expr2 \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="Num expr1 num_op expr2" in subExpr_Gf)
  apply assumption
  apply simp
 apply (subgoal_tac "\<forall> hex\<in>set as. heapEx ( Num expr1 num_op expr2 ) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (subgoal_tac "\<forall> hex\<in>set bs. heapEx (Num expr1 num_op expr2) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (simp add: eqExMps_def  fms)
 --{* Rel expr1 relop expr2 *}
 apply (rule impI)
 apply (subgoal_tac "expr1 \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="Rel expr1 rel_op expr2" in subExpr_Gf)
  apply assumption
  apply simp
 apply (subgoal_tac "expr2 \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="Rel expr1 rel_op expr2" in subExpr_Gf)
  apply assumption
  apply simp
 apply (subgoal_tac "\<forall> hex\<in>set as. heapEx (Rel expr1 rel_op expr2)  \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (subgoal_tac "\<forall> hex\<in>set bs. heapEx (Rel expr1 rel_op expr2) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (simp add: eqExMps_def  fms)

 --{* IF THEN ELSE *}
 apply (rule impI)
 apply (subgoal_tac "expr1 \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
  apply assumption
  apply simp
 apply (subgoal_tac "expr2 \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
  apply assumption
  apply simp 
 apply (subgoal_tac "expr3 \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
  apply assumption
  apply simp
 apply (subgoal_tac "\<forall> hex\<in>set as. heapEx (IF expr1 THEN expr2 ELSE expr3) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (subgoal_tac "\<forall> hex\<in>set bs. heapEx (IF expr1 THEN expr2 ELSE expr3) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (simp add: eqExMps_def  fms)

 --{* Eq expr1 expr2 *}
 apply (rule impI)
 apply (subgoal_tac "expr1 \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="Eq expr1 expr2" in subExpr_Gf)
  apply assumption
  apply simp
 apply (subgoal_tac "expr2 \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="Eq expr1 expr2" in subExpr_Gf)
  apply assumption
  apply simp
 apply (subgoal_tac "\<forall> hex\<in>set as. heapEx (Eq expr1 expr2)  \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (subgoal_tac "\<forall> hex\<in>set bs. heapEx (Eq expr1 expr2) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (simp add: eqExMps_def  fms)

 --{* Neg expra *}
 apply (rule impI)
 apply (subgoal_tac "expra \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="Neg expra" in subExpr_Gf)
  apply assumption
  apply simp
 apply (subgoal_tac "\<forall> hex\<in>set as. heapEx (Neg expra)  \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (subgoal_tac "\<forall> hex\<in>set bs. heapEx (Neg expra) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (simp add: eqExMps_def  fms)

 --{* Imp expr1 expr2 *}
 apply (rule impI)
 apply (subgoal_tac "expr1 \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="Imp expr1 expr2" in subExpr_Gf)
  apply assumption
  apply simp
 apply (subgoal_tac "expr2 \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="Imp expr1 expr2" in subExpr_Gf)
  apply assumption
  apply simp
 apply (subgoal_tac "\<forall> hex\<in>set as. heapEx (Imp expr1 expr2)  \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (subgoal_tac "\<forall> hex\<in>set bs. heapEx (Imp expr1 expr2) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (simp add: eqExMps_def  fms)

 --{* Forall nat expra *}
 apply (rule impI)
 apply (subgoal_tac "expra \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="Forall nat expra" in subExpr_Gf)
  apply assumption
  apply simp
 apply (subgoal_tac "\<forall> hex\<in>set as. heapEx (Forall nat expra)  \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (subgoal_tac "\<forall> hex\<in>set bs. heapEx (Forall nat expra) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (simp add: eqExMps_def  fms)

 --{* Ty expra ty *}
 apply (rule impI)
 apply (subgoal_tac "expra \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="Ty expra ty" in subExpr_Gf)
  apply assumption
  apply simp
 apply (drule mp, assumption)
 apply (subgoal_tac "\<forall> hex\<in>set bs. heapEx (Ty expra ty) \<noteq> [hex]")
  prefer 2
  apply simp
  apply (drule_tac subExpr_getHeapEx_TY)
  apply (subgoal_tac "\<exists> as' bs' cs'. remdups' (getHeapEx Q) = as' @ [TY expra ty] @ bs' @ [ GF list1 list2 expr] @ cs'")
   prefer 2
   apply (rule getHeapEx_mono_TY_GF)
   apply assumption
   
   apply (rule subExpr_getHeapEx)
   apply assumption
  apply (erule exE)+
  apply (subgoal_tac "distinct (as' @ [TY expra ty] @ bs' @ [ GF list1 list2 expr] @ cs')")
   prefer 2
   apply (drule_tac t="as' @ [TY expra ty] @ bs' @ [ GF list1 list2 expr] @ cs'" in sym)
   apply (erule_tac V="remdups' (getHeapEx Q) = ?a" in thin_rl)
   apply (simp only:)
   apply (rule distinct_remdups')
  apply (subgoal_tac "distinct (as @ GF list1 list2 expr # bs)")
   prefer 2
   apply (drule_tac t="as @  GF list1 list2 expr # bs" in sym)
   apply (simp only:)
  apply (subgoal_tac "as = as'@[TY expra ty]@bs' \<and> bs = cs'")
   prefer 2
   apply (rule_tac x="GF list1 list2 expr" in distinct_list_match)
   apply (drule_tac t="as' @ [TY expra ty] @ bs' @ [GF list1 list2 expr] @ cs'" in sym)
   apply simp
   
   apply simp

   apply simp
  apply (rule ballI)
  apply (rule classical)
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (simp add: eqExMps_def  fms)

 --{* Pos x *}
 apply (subgoal_tac "\<forall> hex\<in>set as. heapEx ( Pos x) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (subgoal_tac "\<forall> hex\<in>set bs. heapEx (Pos x) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (simp add: eqExMps_def lookup.simps fms)

 --{* Call expra *}
 apply (subgoal_tac "\<forall> hex\<in>set as. heapEx (Call expra) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (subgoal_tac "\<forall> hex\<in>set bs. heapEx (Call expra) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (simp only: eqExMps_Call)
 apply (simp add: fms)

 --{* Catch list expra *}
 apply (subgoal_tac "\<forall> hex\<in>set as. heapEx (Catch list expra) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (subgoal_tac "\<forall> hex\<in>set bs. heapEx (Catch list expra) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (simp only: eqExMps_Catch)
 apply (simp add: fms)

 --{* And es *}
 apply (rule impI)
 apply (simp only: eqExMps_And')
 apply (rule conjI)
 apply (subgoal_tac "\<forall>hex \<in> set bs. heapEx (And es) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (subgoal_tac "\<forall>hex \<in> set as. heapEx (And es) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')+
 apply (simp add:  fms map_if_notin id_lookup_def del: evalE_evalEs.simps split del: split_if)

 apply (rule ballI)
 apply (erule_tac x="ex" and A="set es" in ballE)
 apply (subgoal_tac "ex \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="And es" in subExpr_Gf)
  apply simp
  apply simp
 apply (simp only:)
 
 apply simp

apply (simp only: foldl_append foldl.simps heapexpr.cases Let_def)
apply simp
apply (subgoal_tac "(\<lambda>u. if u = a then \<lfloor>blank P Cl\<rfloor> else h u) = (h(a \<mapsto> blank P Cl))")
 prefer 2
 apply (rule ext)
 apply simp
apply simp
apply (rule allI | rule impI)+
apply (case_tac "aa = Addr a")
apply (simp add: split_def)

apply (simp add: split_def)
apply (case_tac "aa")
apply simp
apply simp
apply simp
apply simp
apply simp
--{* FrNr *}
apply (rule impI, rule allI)
apply (subgoal_tac "\<forall>hex \<in> set (remdups' (getHeapEx Q)). heapEx FrNr \<noteq> [hex]")
 prefer 2
 apply simp
apply (drule_tac P="fst \<Pi>" and Cl="Cl" in foldl_map_lookup')
apply (simp add: fms map_if_notin id_lookup_def)

--{* Num expr1 numop expr2 *}
apply (rule impI, rule allI)
apply (subgoal_tac "\<forall>hex \<in> set (remdups' (getHeapEx Q)). heapEx (Num expr1 num_op expr2) \<noteq> [hex]")
 prefer 2
 apply simp
apply (drule_tac P="fst \<Pi>" and Cl="Cl" in foldl_map_lookup')
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Num expr1 num_op expr2" in subExpr_Gf)
 apply assumption
 apply simp
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Num expr1 num_op expr2" in subExpr_Gf)
 apply assumption
 apply simp
apply (simp add: fms map_if_notin id_lookup_def)

--{* Rel expr1 numop expr2 *}
apply (rule impI, rule allI)
apply (subgoal_tac "\<forall>hex \<in> set (remdups' (getHeapEx Q)). heapEx (Rel expr1 rel_op expr2) \<noteq> [hex]")
 prefer 2
 apply simp
apply (drule_tac P="fst \<Pi>" and Cl="Cl" in foldl_map_lookup')
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Rel expr1 rel_op expr2" in subExpr_Gf)
 apply assumption
 apply simp
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Rel expr1 rel_op expr2" in subExpr_Gf)
 apply assumption
 apply simp
apply (simp add: fms map_if_notin id_lookup_def)

--{* IF THEN ELSE *}
apply (rule impI, rule allI)
apply (subgoal_tac "\<forall>hex \<in> set (remdups' (getHeapEx Q)). heapEx (IF expr1 THEN expr2 ELSE expr3) \<noteq> [hex]")
 prefer 2
 apply simp
apply (drule_tac P="fst \<Pi>" and Cl="Cl" in foldl_map_lookup')
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply simp
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply simp
apply (subgoal_tac "expr3 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply simp
apply (simp add: fms map_if_notin id_lookup_def split add: bool.split)

--{* Eq expr1 expr2 *}
apply (rule impI, rule allI)
apply (subgoal_tac "\<forall>hex \<in> set (remdups' (getHeapEx Q)). heapEx (Eq expr1 expr2) \<noteq> [hex]")
 prefer 2
 apply simp
apply (drule_tac P="fst \<Pi>" and Cl="Cl" in foldl_map_lookup')
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Eq expr1 expr2" in subExpr_Gf)
 apply assumption
 apply simp
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Eq expr1 expr2" in subExpr_Gf)
 apply assumption
 apply simp
apply (simp add: fms map_if_notin id_lookup_def)

--{* Neg expr *}
apply (rule impI, rule allI)
apply (subgoal_tac "\<forall>hex \<in> set (remdups' (getHeapEx Q)).  heapEx ( Neg expr) \<noteq> [hex]")
 prefer 2
 apply simp
apply (drule_tac P="fst \<Pi>" and Cl="Cl" in foldl_map_lookup')
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Neg expr" in subExpr_Gf)
 apply assumption
 apply simp
apply (simp add: fms map_if_notin id_lookup_def)

--{* Imp expr1 expr2 *}
apply (rule impI, rule allI)
apply (subgoal_tac "\<forall>hex \<in> set (remdups' (getHeapEx Q)).  heapEx (Imp expr1 expr2) \<noteq> [hex]")
 prefer 2
 apply simp
apply (drule_tac P="fst \<Pi>" and Cl="Cl" in foldl_map_lookup')
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Imp expr1 expr2" in subExpr_Gf)
 apply assumption
 apply simp
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Imp expr1 expr2" in subExpr_Gf)
 apply assumption
 apply simp
apply (simp add: fms map_if_notin id_lookup_def)

--{* Foral nat expr *}
apply (rule impI, rule allI)
apply (subgoal_tac "\<forall>hex \<in> set (remdups' (getHeapEx Q)).  heapEx (Forall nat expr) \<noteq> [hex]")
 prefer 2
 apply simp
apply (drule_tac P="fst \<Pi>" and Cl="Cl" in foldl_map_lookup')
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Forall nat expr" in subExpr_Gf)
 apply assumption
 apply simp
apply (simp add: fms map_if_notin id_lookup_def)

--{* Ty expr ty *}
apply (rule impI, rule allI)
apply (frule subExpr_getHeapEx_TY)
apply (drule_tac x="TY expr ty" in  in_rd_sp)
apply (erule exE | erule conjE)+
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Ty expr ty" in subExpr_Gf)
 apply assumption
 apply (simp add: parts.simps)
apply (drule mp, assumption)
apply (subgoal_tac "\<forall>hex \<in> set as. heapEx (Ty expr ty) \<noteq> [hex]")
 prefer 2
 apply (rule ballI)
 apply (rule classical)
 apply (simp add: split_def)
apply (drule_tac P="fst \<Pi>" and Cl="Cl" in foldl_map_lookup')
apply (subgoal_tac "\<forall>hex \<in> set bs. heapEx (Ty expr ty) \<noteq> [hex]")
 prefer 2
 apply (rule ballI)
 apply (rule classical)
 apply (simp add: split_def)
apply (drule_tac P="fst \<Pi>" and Cl="Cl" in foldl_map_lookup')
apply (simp (no_asm_simp))
apply (simp (no_asm_simp) only: substE_substEs.simps)
apply (simp only: evalE_evalEs.simps)
apply simp
apply (subgoal_tac "(substE (foldl (\<lambda>mp hex. (case hex of
                                         GF F C ex \<Rightarrow>
                                           let ex' = substE mp ex
                                           in (Gf F C ex, IF ex' \<doteq> NewA 0 THEN Cn (the (snd (blank (fst (P, An)) Cl) (F, C))) ELSE Gf F C ex')
                                         | TY ex ty \<Rightarrow>
                                             let ex' = substE mp ex in (Ty ex ty, IF ex' \<doteq> NewA 0 THEN Cn (Bool (Class Cl = ty)) ELSE Ty ex' ty)) #
                                        mp)
                         ((Ty expr ty,
                           IF substE (foldl (\<lambda>mp hex. (case hex of
                                                       GF F C ex \<Rightarrow>
                                                         let ex' = substE mp ex
                                                         in (Gf F C ex,
                                                             IF ex' \<doteq> NewA 0 THEN Cn (the (snd (blank (fst (P, An)) Cl) (F, C))) ELSE Gf F C ex')
                                                       | TY ex ty \<Rightarrow>
                                                           let ex' = substE mp ex
                                                           in (Ty ex ty, IF ex' \<doteq> NewA 0 THEN Cn (Bool (Class Cl = ty)) ELSE Ty ex' ty)) #
                                                      mp)
                                       (map (\<lambda>q. (Pos q, if q = (C, M, Suc pc) then Pos (C,M,pc) else FF)) (getPosEx Q) @
                                        map (\<lambda>k. (St k, if k = 0 then NewA 0 else St (k - 1))) (stkIds Q) @ map (\<lambda>n. (NewA n, NewA (Suc n))) (getNewEx Q))
                                       as)
                               expr \<doteq>
                              NewA 0 THEN Cn (Bool (Class Cl =
                                                    ty)) ELSE Ty (substE (foldl (\<lambda>mp hex.
      (case hex of GF F C ex \<Rightarrow> let ex' = substE mp ex in (Gf F C ex, IF ex' \<doteq> NewA 0 THEN Cn (the (snd (blank (fst (P, An)) Cl) (F, C))) ELSE Gf F C ex')
       | TY ex ty \<Rightarrow> let ex' = substE mp ex in (Ty ex ty, IF ex' \<doteq> NewA 0 THEN Cn (Bool (Class Cl = ty)) ELSE Ty ex' ty)) #
      mp)
                                                                           (map (\<lambda>q. (Pos q, if q = (C, M, Suc pc) then Pos (C,M,pc) else FF)) (getPosEx Q) @
                                                                            map (\<lambda>k. (St k, if k = 0 then NewA 0 else St (k - 1))) (stkIds Q) @
                                                                            map (\<lambda>n. (NewA n, NewA (Suc n))) (getNewEx Q))
                                                                           as)
                                                                   expr)
                                                               ty) #
                          foldl (\<lambda>mp hex. (case hex of
                                           GF F C ex \<Rightarrow>
                                             let ex' = substE mp ex
                                             in (Gf F C ex, IF ex' \<doteq> NewA 0 THEN Cn (the (snd (blank (fst (P, An)) Cl) (F, C))) ELSE Gf F C ex')
                                           | TY ex ty \<Rightarrow>
                                               let ex' = substE mp ex in (Ty ex ty, IF ex' \<doteq> NewA 0 THEN Cn (Bool (Class Cl = ty)) ELSE Ty ex' ty)) #
                                          mp)
                           (map (\<lambda>q. (Pos q, if q = (C, M, Suc pc) then Pos (C,M,pc) else FF)) (getPosEx Q) @
                            map (\<lambda>k. (St k, if k = 0 then NewA 0 else St (k - 1))) (stkIds Q) @ map (\<lambda>n. (NewA n, NewA (Suc n))) (getNewEx Q))
                           as)
                         bs)
                 expr) = (substE (foldl (\<lambda>mp hex. (case hex of
                                           GF F C ex \<Rightarrow>
                                             let ex' = substE mp ex
                                             in (Gf F C ex, IF ex' \<doteq> NewA 0 THEN Cn (the (snd (blank (fst (P, An)) Cl) (F, C))) ELSE Gf F C ex')
                                           | TY ex ty \<Rightarrow>
                                               let ex' = substE mp ex in (Ty ex ty, IF ex' \<doteq> NewA 0 THEN Cn (Bool (Class Cl = ty)) ELSE Ty ex' ty)) #
                                          mp)
                           (map (\<lambda>q. (Pos q, if q = (C, M, Suc pc) then Pos (C,M,pc) else FF)) (getPosEx Q) @
                            map (\<lambda>k. (St k, if k = 0 then NewA 0 else St (k - 1))) (stkIds Q) @ map (\<lambda>n. (NewA n, NewA (Suc n))) (getNewEx Q))
                           as)
                   expr)")
 prefer 2
 apply (erule_tac V="\<forall>I. ?P I" in thin_rl)
 apply (erule_tac V="\<forall>mp'. ?P mp'" in thin_rl)
 apply (erule_tac V="\<forall>mp'. ?P mp'" in thin_rl)
 apply (rule substE_eq)
 apply (subgoal_tac "\<exists>expr'. expr = expr'")
  prefer 2
  apply (erule thin_rl)+
  apply fastsimp
 apply (erule exE)+
 apply (subgoal_tac "\<forall> em em'. eqExMps em em' expr = eqExMps em em' expr'")
  prefer 2
  apply simp
 apply (subgoal_tac "expr' \<in> set (subExpr expr)")
  prefer 2
  apply (simp only:)
  apply (rule getExpr_refl)
 apply (erule_tac V="expr = expr'" in thin_rl)
 apply (simp only:)
 apply (erule_tac P="expr' \<in> set (subExpr expr)" in rev_mp)
 apply (rule_tac expr="expr'" in expr_induct)
 --{* Rg nat *}
 apply (subgoal_tac "\<forall> hex\<in>set as. heapEx (Rg nat) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (subgoal_tac "\<forall> hex \<in>set bs. heapEx (Rg nat) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (simp add: fst_conv eqExMps_def lookup.simps fms)
 --{* St nat *}
 apply (subgoal_tac "\<forall>hex\<in>set as. heapEx (St nat) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (subgoal_tac "\<forall> hex\<in>set bs. heapEx (St nat) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (simp add: eqExMps_def lookup.simps fms)
 --{* Lv nat *}
 apply (subgoal_tac "\<forall> hex\<in>set as. heapEx (Lv nat) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (subgoal_tac "\<forall> hex\<in>set bs. heapEx (Lv nat) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (simp add: eqExMps_def lookup.simps fms)
 --{* Cn val *}
 apply (subgoal_tac "\<forall> hex\<in>set as. heapEx (Cn val) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (subgoal_tac "\<forall> hex\<in>set bs. heapEx (Cn val) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (simp add: eqExMps_def lookup.simps fms)
 --{* NewA *}
 apply (subgoal_tac "\<forall> hex\<in>set as. heapEx (NewA n) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (subgoal_tac "\<forall> hex\<in>set bs. heapEx (NewA n) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (simp add: eqExMps_def lookup.simps fms)
 --{* Gf list1 list2 expra *}
 apply (rule impI)
 apply (frule_tac ex="expra" in subExpr_getHeapEx)
 apply (subgoal_tac "expra \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="Gf list1 list2 expra" in subExpr_Gf)
  apply assumption
  apply (simp add: parts.simps)
 apply (drule mp, assumption)

 apply (subgoal_tac "\<exists>as' bs' cs'. remdups' (getHeapEx Q) = as'@[GF list1 list2 expra]@bs'@[TY expr ty]@cs'")
   prefer 2
   apply (rule getHeapEx_mono_GF_TY)
   apply assumption

   apply (rule subExpr_getHeapEx_TY)
   apply assumption
  apply (erule exE)+
 apply (subgoal_tac "distinct (as' @ [GF list1 list2 expra] @ bs' @ [TY expr ty] @ cs')")
  prefer 2
  apply (drule_tac t="as' @ [GF list1 list2 expra] @ bs' @ [TY expr ty] @ cs'" in sym)
  apply (erule_tac V="remdups' (getHeapEx Q) = ?c" in thin_rl)
  apply (simp only: distinct_remdups')
 apply (subgoal_tac "distinct (as@[TY expr ty]@bs)")
  prefer 2
  apply (subgoal_tac "(as @ TY expr ty # bs) = (as @ [TY expr ty] @ bs)")
   prefer 2
   apply simp
  apply (drule_tac s="as @ TY expr ty # bs" in sym)
  apply (simp only: distinct_remdups')
 apply (subgoal_tac "as @ [TY expr ty] @ bs = (as' @ [GF list1 list2 expra] @ bs') @ [TY expr ty] @ cs'")
  prefer 2
  apply (drule_tac t="as' @ [GF list1 list2 expra] @ bs' @ [TY expr ty] @ cs'" in sym)
  apply simp
 apply (drule distinct_list_match)
 apply assumption
 apply simp 
 apply (subgoal_tac "\<forall>hex \<in> set bs. heapEx (Gf list1 list2 expra) \<noteq> [hex]")
  prefer 2
   apply (subgoal_tac "GF list1 list2 expra \<notin> set bs")
    prefer 2
    apply simp
  apply (rule ballI)
  apply (rule classical)
  apply (simp add: split_def)

 apply (subgoal_tac "\<forall> hex \<in> set as'. heapEx (Gf list1 list2 expra) \<noteq> [hex]")
  prefer 2
   apply (subgoal_tac "GF list1 list2 expra \<notin> set as'")
    prefer 2
    apply simp
  apply (rule ballI)
  apply (rule classical)
  apply (simp add: split_def)

 apply (subgoal_tac "\<forall> hex \<in> set bs'. heapEx (Gf list1 list2 expra) \<noteq> [hex]")
  prefer 2
   apply (subgoal_tac "GF list1 list2 expra \<notin> set bs'")
    prefer 2
    apply simp
  apply (rule ballI)
  apply (rule classical)
  apply (simp add: split_def)

 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')+

 apply (subgoal_tac "expr \<noteq> expra")
  prefer 2
  apply (drule getHeapEx_GF_not_refl)
  apply (rule not_sym)
  apply assumption

 apply (simp  add: eqExMps_def)
 --{* FrNr *}
 apply (subgoal_tac "\<forall> hex\<in>set as. heapEx FrNr \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (subgoal_tac "\<forall> hex\<in>set bs. heapEx FrNr \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (simp add: eqExMps_def lookup.simps fms)
 --{* Num expr1 numop expr2 *}
 apply (rule impI)
 apply (subgoal_tac "expr1 \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="Num expr1 num_op expr2" in subExpr_Gf)
  apply assumption
  apply simp
 apply (subgoal_tac "expr2 \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="Num expr1 num_op expr2" in subExpr_Gf)
  apply assumption
  apply simp
 apply (subgoal_tac "\<forall> hex\<in>set as. heapEx ( Num expr1 num_op expr2 ) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (subgoal_tac "\<forall> hex\<in>set bs. heapEx (Num expr1 num_op expr2) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (simp add: eqExMps_def  fms)
 --{* Rel expr1 relop expr2 *}
 apply (rule impI)
 apply (subgoal_tac "expr1 \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="Rel expr1 rel_op expr2" in subExpr_Gf)
  apply assumption
  apply simp
 apply (subgoal_tac "expr2 \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="Rel expr1 rel_op expr2" in subExpr_Gf)
  apply assumption
  apply simp
 apply (subgoal_tac "\<forall> hex\<in>set as. heapEx (Rel expr1 rel_op expr2)  \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (subgoal_tac "\<forall> hex\<in>set bs. heapEx (Rel expr1 rel_op expr2) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (simp add: eqExMps_def  fms)

 --{* IF THEN ELSE *}
 apply (rule impI)
 apply (subgoal_tac "expr1 \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
  apply assumption
  apply simp
 apply (subgoal_tac "expr2 \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
  apply assumption
  apply simp 
 apply (subgoal_tac "expr3 \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
  apply assumption
  apply simp
 apply (subgoal_tac "\<forall> hex\<in>set as. heapEx (IF expr1 THEN expr2 ELSE expr3) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (subgoal_tac "\<forall> hex\<in>set bs. heapEx (IF expr1 THEN expr2 ELSE expr3) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (simp add: eqExMps_def  fms)

 --{* Eq expr1 expr2 *}
 apply (rule impI)
 apply (subgoal_tac "expr1 \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="Eq expr1 expr2" in subExpr_Gf)
  apply assumption
  apply simp
 apply (subgoal_tac "expr2 \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="Eq expr1 expr2" in subExpr_Gf)
  apply assumption
  apply simp
 apply (subgoal_tac "\<forall> hex\<in>set as. heapEx (Eq expr1 expr2)  \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (subgoal_tac "\<forall> hex\<in>set bs. heapEx (Eq expr1 expr2) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (simp add: eqExMps_def  fms)

 --{* Neg expra *}
 apply (rule impI)
 apply (subgoal_tac "expra \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="Neg expra" in subExpr_Gf)
  apply assumption
  apply simp
 apply (subgoal_tac "\<forall> hex\<in>set as. heapEx (Neg expra)  \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (subgoal_tac "\<forall> hex\<in>set bs. heapEx (Neg expra) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (simp add: eqExMps_def  fms)

 --{* Imp expr1 expr2 *}
 apply (rule impI)
 apply (subgoal_tac "expr1 \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="Imp expr1 expr2" in subExpr_Gf)
  apply assumption
  apply simp
 apply (subgoal_tac "expr2 \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="Imp expr1 expr2" in subExpr_Gf)
  apply assumption
  apply simp
 apply (subgoal_tac "\<forall> hex\<in>set as. heapEx (Imp expr1 expr2)  \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (subgoal_tac "\<forall> hex\<in>set bs. heapEx (Imp expr1 expr2) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (simp add: eqExMps_def  fms)

 --{* Forall nat expra *}
 apply (rule impI)
 apply (subgoal_tac "expra \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="Forall nat expra" in subExpr_Gf)
  apply assumption
  apply simp
 apply (subgoal_tac "\<forall> hex\<in>set as. heapEx (Forall nat expra)  \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (subgoal_tac "\<forall> hex\<in>set bs. heapEx (Forall nat expra) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (simp add: eqExMps_def  fms)

 --{* Ty expra tya *}
 apply (rule impI)
 apply (subgoal_tac "expra \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="Ty expra tya" in subExpr_Gf)
  apply assumption
  apply simp
 apply (drule mp, assumption)
 apply (subgoal_tac "\<forall> hex\<in>set bs. heapEx (Ty expra tya) \<noteq> [hex]")
  prefer 2
  apply simp
  apply (drule_tac subExpr_getHeapEx_TY)
  apply (subgoal_tac "\<exists> as' bs' cs'. remdups' (getHeapEx Q) = as' @ [TY expra tya] @ bs' @ [ TY expr ty] @ cs'")
   prefer 2
   apply (rule getHeapEx_mono_TY_TY)
   apply (rule subExpr_getHeapEx_TY)
   apply assumption
   
   apply assumption
  apply (erule exE)+
  apply (subgoal_tac "distinct (as' @ [TY expra tya] @ bs' @ [ TY expr ty] @ cs')")
   prefer 2
   apply (drule_tac t="as' @ [TY expra tya] @ bs' @ [ TY expr ty] @ cs'" in sym)
   apply (erule_tac V="remdups' (getHeapEx Q) = ?a" in thin_rl)
   apply (simp only:)
   apply (rule distinct_remdups')
  apply (subgoal_tac "distinct (as @ TY expr ty # bs)")
   prefer 2
   apply (drule_tac t="as @  TY expr ty # bs" in sym)
   apply (simp only:)
  apply (subgoal_tac "as = as'@[TY expra tya]@bs' \<and> bs = cs'")
   prefer 2
   apply (rule_tac x="TY expr ty" in distinct_list_match)
   apply (drule_tac t="as' @ [TY expra tya] @ bs' @ [TY expr ty] @ cs'" in sym)
   apply simp
   
   apply simp

   apply simp
  apply (rule ballI)
  apply (rule classical)
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (subgoal_tac "expr \<noteq> expra")
  prefer 2
  apply (drule_tac ex="expra" in  subExpr_getHeapEx_TY)
  apply (drule getHeapEx_TY_not_refl)
  apply (rule not_sym)
  apply assumption
 apply (simp add: eqExMps_def  fms split add: bool.split)

 --{* Pos x *}
 apply (subgoal_tac "\<forall> hex\<in>set as. heapEx ( Pos x) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (subgoal_tac "\<forall> hex\<in>set bs. heapEx (Pos x) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (simp add: eqExMps_def lookup.simps fms)



 --{* Call expra *}
 apply (subgoal_tac "\<forall> hex\<in>set as. heapEx (Call expra) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (subgoal_tac "\<forall> hex\<in>set bs. heapEx (Call expra) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (simp only: eqExMps_Call)
 apply (simp add: fms)

 --{* Catch lista expra *}
 apply (subgoal_tac "\<forall> hex\<in>set as. heapEx (Catch list expra) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (subgoal_tac "\<forall> hex\<in>set bs. heapEx (Catch list expra) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')
 apply (simp only: eqExMps_Catch)
 apply (simp add: fms)

 --{* And es *}
 apply (rule impI)
 apply (simp only: eqExMps_And')
 apply (rule conjI)
 apply (subgoal_tac "\<forall>hex \<in> set bs. heapEx (And es) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (subgoal_tac "\<forall>hex \<in> set as. heapEx (And es) \<noteq> [hex]")
  prefer 2
  apply simp
 apply (drule_tac P="fst (P,An)" and Cl="Cl" in foldl_map_lookup')+
 apply (simp add:  fms map_if_notin id_lookup_def del: evalE_evalEs.simps split del: split_if)

 apply (rule ballI)
 apply (erule_tac x="ex" and A="set es" in ballE)
 apply (subgoal_tac "ex \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="And es" in subExpr_Gf)
  apply simp
  apply simp
 apply (simp only:)
 
 apply simp
apply (subgoal_tac "(\<lambda>u. if u = a then \<lfloor>blank P Cl\<rfloor> else h u) =  (h(a \<mapsto> blank P Cl))")
 prefer 2
 apply (rule ext)
 apply simp
apply simp
apply (rule allI | rule impI)+
apply (case_tac "aa = Addr a")
apply (simp add: split_def blank_def  obj_ty_def)
apply blast
apply (case_tac "aa")
apply simp
apply simp
apply simp
apply simp
apply simp

--{* Pos *}
apply (rule impI,rule allI)
apply (drule subExpr_getPosEx)
apply (drule_tac xs="getPosEx Q" in in_set_conv_decomp_fst)
apply (erule exE | erule conjE)+
apply (subgoal_tac "\<forall>hex \<in> set (remdups' (getHeapEx Q)). heapEx (Pos x) \<noteq> [hex]")
 prefer 2
 apply simp
apply (drule_tac P="fst \<Pi>" and Cl="Cl" in foldl_map_lookup')
apply (subgoal_tac "p' mem domC \<Pi>")
 prefer 2
 apply (cut_tac wf_Pi)
 apply (cut_tac p_domC)
 apply (cut_tac p_def)
 apply (cut_tac cmd_p)
 apply (simp only: in_set_conv_decomp)
 apply (erule exE)+
 apply (simp add: incA_def wf_def checkPos_split succsNormal_def split add: split_if_asm)
apply (cut_tac p_domC)
apply (case_tac "frs")
apply (simp add: fms map_if_notin id_lookup_def p_def mem_iff)

apply (simp add: fms map_if_notin id_lookup_def p_def mem_iff callers_simps)

--{* Call expr *} 
apply (rule impI, rule allI)
apply (subgoal_tac "\<forall>hex \<in> set (remdups' (getHeapEx Q)). heapEx (Call expr) \<noteq> [hex]")
 prefer 2
 apply simp
apply (drule_tac P="fst \<Pi>" and Cl="Cl" in foldl_map_lookup')

apply (simp add: fms map_if_notin id_lookup_def)
apply (rule impI)
apply (simp only: neq_Nil_conv)
apply (erule exE)+
apply (simp (no_asm_simp) only: split_paired_all env_upd callstate.simps)

--{* Catch list expr *}
apply (rule impI, rule allI)
apply (subgoal_tac "\<forall>hex \<in> set (remdups' (getHeapEx Q)). heapEx (Catch list expr) \<noteq> [hex]")
 prefer 2
 apply simp
apply (drule_tac P="fst \<Pi>" and Cl="Cl" in foldl_map_lookup')

apply (simp add: fms map_if_notin id_lookup_def)
apply (rule impI)
apply (simp only: neq_Nil_conv)
apply (erule exE)+
apply (simp (no_asm_simp) only: split_paired_all env_upd catchstate.simps)

--{* And ex *}
apply (rule impI, rule allI)
apply (subgoal_tac "\<forall>hex \<in> set (remdups' (getHeapEx Q)). heapEx (And es) \<noteq> [hex]")
 prefer 2
 apply simp
apply (drule_tac P="fst \<Pi>" and Cl="Cl" in foldl_map_lookup')
apply (simp add: substEs_map evalE_And fms map_if_notin id_lookup_def del: evalE_evalEs.simps split del: split_if)
apply (rule iffI)
apply (rule ballI)
apply (erule_tac x="ex" in ballE)+
apply (subgoal_tac "ex \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="And es" in subExpr_Gf)
 apply assumption
 apply (simp add: parts.simps)
apply (drule mp, assumption)
apply simp

apply simp

apply simp

apply (rule ballI)
apply (erule_tac x="ex" in ballE)+
apply (subgoal_tac "ex \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="And es" in subExpr_Gf)
 apply assumption
 apply (simp add: parts.simps)
apply (drule mp, assumption)
apply simp

apply simp

apply simp
done
qed
(*>*)


lemma effS_wpF_Getfield:
assumes i_def: "i = Getfield list1 list2"
assumes wf_Pi: "wf \<Pi>"
assumes handlesEx: "handlesEx (fst \<Pi>) p' = None"
assumes cmd_p: "cmd \<Pi> p = Some i"
assumes p_domC: "p \<in> set (domC \<Pi>)"
assumes i_instr: "instrs_of P C M ! pc = i"
assumes s_def: "s = (p,\<sigma>,e)"
assumes p_def: "p = (C,M,pc)"
assumes sigma_def: "\<sigma> = (None,h,(stk,loc,p)#frs)"
assumes s'_def: "s' = (p',\<sigma>',e')"
assumes sigma'_def: "\<sigma>' = (None,h',fr'#frs')"
assumes e'_def: "e' = e\<lparr>cs := if \<exists>M n. i = Invoke M n then h # cs e else if i = Return then tl (cs e) else cs e\<rparr>"
assumes p'_def: "p' = snd (snd fr')"
assumes check_i: "check_instr' i P h stk loc C M pc frs"
assumes exec_i : "exec_instr i P h stk loc C M pc frs = \<sigma>'"
assumes Pi_def: "\<Pi> = (P,An)"
shows  "\<forall> I. evalE \<Pi> (p,\<sigma>,e\<lparr>lv:=I\<rparr>) (wpF \<Pi> p p' Q) = evalE \<Pi> (p',\<sigma>',e'\<lparr>lv:=I\<rparr>) Q"
(*<*)
proof -
from handlesEx cmd_p i_def 
show ?thesis
apply -
apply (simp only: wpF_Getfield)
--{* induction preparation *}
apply (subgoal_tac "\<exists>Q'. Q = Q'")
 prefer 2
 apply fastsimp
apply (erule exE)
apply (subgoal_tac "\<forall>em. substE em Q = substE em Q'")
 prefer 2
 apply simp
apply (subgoal_tac "\<forall> I. evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q = evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q'")
 prefer 2
 apply simp
apply (subgoal_tac "Q' \<in> set (subExpr Q)")
  prefer 2
  apply (simp add: getExpr_refl)
apply (erule_tac V="Q = Q'" in thin_rl)
apply (simp only:)
apply (erule_tac V="\<forall>em.  substE em Q = substE em Q'" in thin_rl)
apply (erule_tac V="\<forall>I. evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q = evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q'" in thin_rl)
apply (erule_tac P="Q' \<in> set (subExpr Q)" in rev_mp)
apply (erule thin_rl)+
apply (cut_tac exec_i[THEN sym] sigma'_def i_def check_i sigma_def p'_def e'_def p_def Pi_def)
--{* induction on Q' *}
apply simp
apply (case_tac "hd stk = Null")
apply (simp add: split_def)

apply (simp add: split_def)
apply (rule_tac expr="Q'" in expr_induct)
--{* Rg *}
apply (rule impI, rule allI)
apply (drule subExpr_rgIds)
apply (drule_tac xs="rgIds Q" in in_set_conv_decomp_fst)
apply (erule exE | erule conjE)+
apply (simp add: fms map_if_notin id_lookup_def)
--{* St *}
apply (case_tac "stk")
apply simp
apply (rule impI, rule allI)
apply (drule subExpr_stkIds)
apply (drule_tac xs="stkIds Q" in  in_set_conv_decomp_fst)
apply (erule exE | erule conjE)+
apply (simp add: fms map_if_notin id_lookup_def neq_Nil_conv)
apply (case_tac "nat")
apply (simp  split add: val.split )
apply (case_tac "a")
apply (simp add: is_Ref_def is_Ref'_def)
apply (simp add: is_Ref_def is_Ref'_def)
apply (simp add: is_Ref_def is_Ref'_def)
apply (simp add: is_Ref_def is_Ref'_def)
apply (simp add:  is_Ref'_def)
apply (erule exE)+
apply simp

--{* nat = Suc nata *}
apply simp
--{* Lv *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* Cn *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* NewA *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* Gf list1 list2 expr *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Gf list1a list2a expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def split add: val.split)
--{* FrNr *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* Num expr1 num_op epxr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Num expr1 num_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Num expr1 num_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Rel expr1 rel_op expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Rel expr1 rel_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Rel expr1 rel_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* IF expr1 THEN expr2 ELSE expr3 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr3 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (drule mp,assumption)+

apply (simp add: split_def fms map_if_notin id_lookup_def evalE_evalEs.simps substE_substEs.simps split del: split_if add: bool.split )
apply (rule allI)
apply (erule_tac x="I" in allE)
apply (case_tac " evalE (P, An)
               ((C, M, Suc pc), (None, h, (the (snd (the (h (the_Addr (hd stk)))) (list1, list2)) # tl stk, loc, C, M, Suc pc) # frs), e\<lparr>lv := I\<rparr>) expr1 =
              \<lfloor>Bool True\<rfloor>")
apply simp

apply simp
--{* Eq expr1 expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Eq expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Eq expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Neg expr *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Neg expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Imp expr1 expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Imp expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Imp expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Forall nat expr *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Forall nat expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def split del: split_if)
--{* Ty expr ty*}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Ty expr ty" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: Let_def split_def fms map_if_notin id_lookup_def split add: val.split)
--{* Pos x *}
apply (rule impI, rule allI)
apply (subgoal_tac "x \<in> set (getPosEx Q)")
 prefer 2
 apply (rule subExpr_getPosEx)
 apply assumption
apply (drule_tac xs="getPosEx Q" in  in_set_conv_decomp_fst)
apply (erule exE |erule conjE)+
apply (subgoal_tac "p' mem domC \<Pi>")
 prefer 2
 apply (cut_tac wf_Pi)
 apply (cut_tac p_domC)
 apply (cut_tac p_def)
 apply (cut_tac cmd_p)
 apply (simp only: in_set_conv_decomp)
 apply (erule exE)+
 apply (simp add: incA_def wf_def checkPos_split succsNormal_def split add: split_if_asm)
apply (cut_tac p_domC)
apply (case_tac "frs")
apply (simp add: fms map_if_notin id_lookup_def p_def mem_iff)

apply (simp add: fms map_if_notin id_lookup_def p_def mem_iff callers_simps)
--{* Call *}
apply (rule impI,rule allI)
apply (simp add: Let_def split_def fms map_if_notin id_lookup_def)
apply (rule impI)
apply (simp only: neq_Nil_conv)
apply (erule exE)+
apply (simp (no_asm_simp) add: split_paired_all env_upd_cs)

--{* Catch  *}
apply (rule impI,rule allI)
apply (simp add: Let_def split_def fms map_if_notin id_lookup_def)
apply (rule impI)
apply (simp only: neq_Nil_conv)
apply (erule exE)+
apply (simp (no_asm_simp) add: split_paired_all env_upd_cs)

--{* And es *}
apply (rule impI, rule allI)
apply (simp add: evalE_And fms map_if_notin id_lookup_def del: evalE_evalEs.simps split del: split_if)
apply (rule iffI)
apply (rule ballI)
apply (erule_tac x="ex" in ballE)
apply (erule_tac x="(substE (map (\<lambda>q. (Pos q, if q = (C, M, Suc pc) then Pos p else FF)) (getPosEx Q) @ [(St 0, Gf list1 list2 (St 0))]) ex)" in ballE)
apply (subgoal_tac "ex \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="And es" in subExpr_Gf)
 apply assumption
 apply (simp add: parts.simps)
apply (simp add: substEs_map) 

apply (simp add: substEs_map)

apply (simp only:)

apply (rule ballI)

apply (simp add: substEs_map image_iff)
apply (erule bexE)
apply (erule_tac x="x" in ballE)
apply (erule_tac x="x" in ballE)
apply (subgoal_tac "x \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="And es" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm_simp) add: parts.simps)
apply simp

apply (simp only:)

apply (simp only:)

done
qed
(*>*)


lemma effS_wpF_Putfield:
assumes i_def: "i = Putfield list1 list2"
assumes wf_Pi: "wf \<Pi>"
assumes handlesEx: "handlesEx (fst \<Pi>) p' = None"
assumes cmd_p: "cmd \<Pi> p = Some i"
assumes p_domC: "p \<in> set (domC \<Pi>)"
assumes i_instr: "instrs_of P C M ! pc = i"
assumes s_def: "s = (p,\<sigma>,e)"
assumes p_def: "p = (C,M,pc)"
assumes sigma_def: "\<sigma> = (None,h,(stk,loc,p)#frs)"
assumes s'_def: "s' = (p',\<sigma>',e')"
assumes sigma'_def: "\<sigma>' = (None,h',fr'#frs')"
assumes e'_def: "e' = e\<lparr>cs := if \<exists>M n. i = Invoke M n then h # cs e else if i = Return then tl (cs e) else cs e\<rparr>"
assumes p'_def: "p' = snd (snd fr')"
assumes check_i: "check_instr' i P h stk loc C M pc frs"
assumes exec_i : "exec_instr i P h stk loc C M pc frs = \<sigma>'"
assumes Pi_def: "\<Pi> = (P,An)"
shows  "\<forall> I. evalE \<Pi> (p,\<sigma>,e\<lparr>lv:=I\<rparr>) (wpF \<Pi> p p' Q) = evalE \<Pi> (p',\<sigma>',e'\<lparr>lv:=I\<rparr>) Q"
(*<*)
proof -
from handlesEx cmd_p i_def 
show ?thesis
apply -
apply (simp only: wpF_Putfield)
--{* induction preparation *}
apply (subgoal_tac "\<exists>Q'. Q = Q'")
 prefer 2
 apply fastsimp
apply (erule exE)
apply (subgoal_tac "\<forall>em. substE em Q = substE em Q'")
 prefer 2
 apply simp
apply (subgoal_tac "\<forall> I. evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q = evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q'")
 prefer 2
 apply simp
apply (subgoal_tac "Q' \<in> set (subExpr Q)")
  prefer 2
  apply (simp add: getExpr_refl)
apply (erule_tac V="Q = Q'" in thin_rl)
apply (simp only:)
apply (erule_tac V="\<forall>em.  substE em Q = substE em Q'" in thin_rl)
apply (erule_tac V="\<forall>I. evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q = evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q'" in thin_rl)
apply (erule_tac P="Q' \<in> set (subExpr Q)" in rev_mp)
apply (erule thin_rl)+
apply (cut_tac exec_i[THEN sym] sigma'_def i_def sigma_def check_i p'_def e'_def p_def Pi_def)
--{* induction on Q' *}
apply simp
apply (case_tac "hd (tl stk) = Null")
apply (simp add: split_def)

apply (simp add: split_def env_upd)
apply (rule_tac expr="Q'" in expr_induct)
--{* Rg *}
apply (rule impI, rule allI)
apply (subgoal_tac "\<forall>ex \<in> set (remdups' (getGfEx list1 list2 Q)). Rg nat \<noteq> Gf list1 list2 ex")
 prefer 2
 apply simp
apply (drule_tac foldl_map_lookup)
apply (simp add: fms map_if_notin id_lookup_def)
--{* St *}
apply (rule impI, rule allI)
apply (subgoal_tac "\<forall>ex \<in> set (remdups' (getGfEx list1 list2 Q)). St nat \<noteq> Gf list1 list2 ex")
 prefer 2
 apply simp
apply (drule_tac foldl_map_lookup)
apply (drule subExpr_stkIds)
apply (drule_tac xs="stkIds Q" in  in_set_conv_decomp_fst)
apply (erule exE | erule conjE)+
apply (simp add: fms map_if_notin id_lookup_def neq_Nil_conv)
apply (case_tac "stk")
apply simp

apply (case_tac "list")
apply simp

apply (case_tac "lista")
apply simp

apply simp
--{* Lv *}
apply (subgoal_tac "\<forall>ex \<in> set (remdups' (getGfEx list1 list2 Q)). Lv nat \<noteq> Gf list1 list2 ex")
 prefer 2
 apply simp
apply (drule_tac foldl_map_lookup)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Cn *}
apply (subgoal_tac "\<forall>ex \<in> set (remdups' (getGfEx list1 list2 Q)). Cn val \<noteq> Gf list1 list2 ex")
 prefer 2
 apply simp
apply (drule_tac foldl_map_lookup)
apply (simp add: fms map_if_notin id_lookup_def)
--{* NewA *}
apply (subgoal_tac "\<forall>ex \<in> set (remdups' (getGfEx list1 list2 Q)). NewA n \<noteq> Gf list1 list2 ex")
 prefer 2
 apply simp
apply (drule_tac foldl_map_lookup)
apply (simp add: fms map_if_notin id_lookup_def)
apply (rule impI)
apply (erule conjE | erule exE)+
apply (case_tac "hd (tl stk)")
apply (simp add: is_Ref_def is_Ref'_def)+
apply (erule exE)+
apply (rule evalNewA_dom)
apply (rule set_ext)
apply (simp (no_asm_simp) add: mem_Collect_eq Map.dom_def split add: split_if)

--{* Gf list1a list2a expr *}
apply (rule impI)
apply (frule subExpr_getGfEx)
apply (drule_tac x="expr" in  in_rd_sp)
apply (erule exE | erule conjE)+
apply (rule allI)
apply (case_tac "list1a = list1 \<and> list2a = list2")
prefer 2
apply (subgoal_tac "\<forall>ex \<in> set (remdups' (getGfEx list1 list2 Q)). Gf list1a list2a expr \<noteq> Gf list1 list2 ex")
 prefer 2
 apply simp
 apply (rule impI)+
 apply simp
apply (simp (no_asm_simp) only: foldl_map_lookup substE_substEs.simps del: evalE_evalEs.simps)
apply (subgoal_tac "(map (\<lambda>q. (Pos q, if q = (C, M, Suc pc) then Pos p else FF)) (getPosEx Q) @
                  map (\<lambda>k. (St k, St (Suc (Suc k)))) (stkIds Q)) ? Gf list1a list2a expr = None")
 prefer 2 
 apply (simp add: fms)
apply (simp only: option.cases)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Gf list1a list2a expr" in subExpr_Gf)
 apply assumption
 apply (simp add: parts.simps)
apply (drule mp, assumption)
apply (simp add: is_Ref'_def is_Addr'_conv split add: val.split)
apply (rule conjI)
apply (rule impI)
apply (erule conjE)+
apply simp
apply (erule conjE | erule exE)+
apply simp

--{* list1a = list1 and list2a = list2 *}
apply (subgoal_tac "\<forall> ex \<in> set bs. Gf list1 list2 expr \<noteq> Gf list1 list2 ex")
 prefer 2
 apply simp
 apply (rule ballI)
 apply (case_tac "ex = expr")
 apply simp
 apply (rule not_sym)
 apply assumption
apply (subgoal_tac "\<forall> ex \<in> set as. Gf list1 list2 expr \<noteq> Gf list1 list2 ex")
 prefer 2
 apply simp
 apply (rule ballI)
 apply (case_tac "ex = expr")
 apply simp
 apply (rule not_sym)
 apply assumption
apply (simp only: foldl_map_lookup foldl_append foldl.simps substE_substEs.simps lookup.simps fst_conv snd_conv simp_thms bool.cases option.cases)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Gf list1 list2 expr" in subExpr_Gf)
 apply simp
 apply (simp add: parts.simps)
apply (drule mp, assumption)
apply (subgoal_tac "(substE
                 (foldl (\<lambda>mp ex. (Gf list1 list2 ex,
                                  IF substE mp ex \<doteq> St (Suc 0) THEN St 0 ELSE Gf list1 list2 (substE mp ex)) #
                                 mp)
                   ((Gf list1 list2 expr,
                     IF substE
                         (foldl (\<lambda>mp ex. (Gf list1 list2 ex,
                                          IF substE mp ex \<doteq>
                                             St (Suc 0) THEN St 0 ELSE Gf list1 list2 (substE mp ex)) #
                                         mp)
                           (map (\<lambda>q. (Pos q, if q = (C, M, Suc pc) then Pos (C,M,pc) else FF)) (getPosEx Q) @
                            map (\<lambda>k. (St k, St (Suc (Suc k)))) (stkIds Q))
                           as)
                         expr \<doteq>
                        St (Suc 0) THEN St 0 ELSE Gf list1 list2
                                                   (substE
                                                     (foldl (\<lambda>mp ex.
       (Gf list1 list2 ex, IF substE mp ex \<doteq> St (Suc 0) THEN St 0 ELSE Gf list1 list2 (substE mp ex)) # mp)
                                                       (map (\<lambda>q. (Pos q, if q = (C, M, Suc pc) then Pos (C,M,pc) else FF))
(getPosEx Q) @
                                                        map (\<lambda>k. (St k, St (Suc (Suc k)))) (stkIds Q))
                                                       as)
                                                     expr)) #
                    foldl (\<lambda>mp ex. (Gf list1 list2 ex,
                                    IF substE mp ex \<doteq> St (Suc 0) THEN St 0 ELSE Gf list1 list2 (substE mp ex)) #
                                   mp)
                     (map (\<lambda>q. (Pos q, if q = (C, M, Suc pc) then Pos (C,M,pc) else FF)) (getPosEx Q) @
                      map (\<lambda>k. (St k, St (Suc (Suc k)))) (stkIds Q))
                     as)
                   bs)
                 expr)  = substE
                (foldl (\<lambda>mp ex. (Gf list1 list2 ex,
                                 IF substE mp ex \<doteq> St (Suc 0) THEN St 0 ELSE Gf list1 list2 (substE mp ex)) #
                                mp)
                  (map (\<lambda>q. (Pos q, if q = (C, M, Suc pc) then Pos (C,M,pc) else FF)) (getPosEx Q) @
                   map (\<lambda>k. (St k, St (Suc (Suc k)))) (stkIds Q))
                  as)
                expr ")
 prefer 2
 apply (erule_tac V="\<forall>I. ?P I" in thin_rl)
 apply (rule substE_eq)
 apply (subgoal_tac "\<exists>expr'. expr = expr'")
  prefer 2
  apply (erule thin_rl)+
  apply fastsimp
 apply (erule exE)+
 apply (subgoal_tac "\<forall> em em'. eqExMps em em' expr = eqExMps em em' expr'")
  prefer 2
  apply simp
 apply (subgoal_tac "expr' \<in> set (subExpr expr)")
  prefer 2
  apply (simp only:)
  apply (rule getExpr_refl)
 apply (erule_tac V="expr = expr'" in thin_rl)
 apply (simp only:)
 apply (erule_tac P="expr' \<in> set (subExpr expr)" in rev_mp)
 apply (rule_tac expr="expr'" in expr_induct)
 --{* Rg nat *}
 apply (simp add: eqExMps_def lookup.simps fms foldl_map_lookup)
 --{* St nat *}
 apply (simp add: eqExMps_def lookup.simps fms foldl_map_lookup)
 --{* Lv nat *}
 apply (simp add: eqExMps_def lookup.simps fms foldl_map_lookup)
 --{* Cn val *}
 apply (simp add: eqExMps_def lookup.simps fms foldl_map_lookup)
 --{* NewA *}
 apply (simp add: eqExMps_def lookup.simps fms foldl_map_lookup)
 --{* Gf list1aa list2aa expra *}
 apply (rule impI)
 apply (frule_tac ex="expra" in subExpr_getGfEx)
 apply (subgoal_tac "expra \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="Gf list1aa list2aa expra" in subExpr_Gf)
  apply assumption
  apply (simp add: parts.simps)
 apply (drule mp, assumption)
 apply (case_tac "list1aa = list1 \<and> list2aa = list2")
 apply (frule_tac getGfEx_not_refl)
 apply (subgoal_tac "\<exists>as' bs' cs'. remdups' (getGfEx list1 list2 Q) = as'@[expra]@bs'@[expr]@cs'")
   prefer 2
   apply (rule getGFEx_mono)
   apply simp

   apply (rule subExpr_getGfEx)
   apply assumption
  apply (erule exE)+
 apply (subgoal_tac "distinct (as' @ [expra] @ bs' @ [expr] @ cs')")
  prefer 2
  apply (drule_tac t="as' @ [expra] @ bs' @ [expr] @ cs'" in sym)
  apply (erule_tac V="remdups' (getGfEx list1 list2 Q) = ?c" in thin_rl)
  apply (simp only: distinct_remdups')
 apply (subgoal_tac "distinct (as@[expr]@bs)")
  prefer 2
  apply (subgoal_tac "(as @ expr # bs) = (as @ [expr] @ bs)")
   prefer 2
   apply simp
  apply (drule_tac s="as @ expr # bs" in sym)
  apply (simp only: distinct_remdups')
 apply (subgoal_tac "as @ [expr] @ bs = (as' @ [expra] @ bs') @ [expr] @ cs'")
  prefer 2
  apply (drule_tac t="as' @ [expra] @ bs' @ [expr] @ cs'" in sym)
  apply simp
 apply (drule distinct_list_match)
 apply assumption
 apply simp
 apply (subgoal_tac "\<forall> ex \<in> set bs. Gf list1 list2 expra \<noteq> Gf list1 list2 ex")
  prefer 2
   apply (subgoal_tac "expra \<notin> set bs")
    prefer 2
    apply simp
  apply (rule ballI)
  apply (rule classical)
  apply simp
 apply (subgoal_tac "\<forall> ex \<in> set as'. Gf list1 list2 expra \<noteq> Gf list1 list2 ex")
  prefer 2
   apply (subgoal_tac "expra \<notin> set as'")
    prefer 2
    apply simp
  apply (rule ballI)
  apply (rule classical)
  apply simp
 apply (subgoal_tac "\<forall> ex \<in> set bs'. Gf list1 list2 expra \<noteq> Gf list1 list2 ex")
  prefer 2
   apply (subgoal_tac "expra \<notin> set bs'")
    prefer 2
    apply simp
  apply (rule ballI)
  apply (rule classical)
  apply simp

 apply (simp  add: eqExMps_def foldl_map_lookup)
 --{* ~ (list1aa = list1 and list2aa = list2) *}
 apply (frule_tac getGfEx_not_refl)
 apply (subgoal_tac "\<forall>ex \<in> set bs. Gf list1aa list2aa expra \<noteq> Gf list1 list2 ex")
  prefer 2
  apply simp
  apply (rule impI)+
  apply simp
 apply (subgoal_tac "\<forall>ex \<in> set as. Gf list1aa list2aa expra \<noteq> Gf list1 list2 ex")
  prefer 2
  apply simp
  apply (rule impI)+
  apply simp
 apply (simp add: fms eqExMps_def foldl_map_lookup split add: bool.split)
 apply (rule impI)+
 apply (rule not_sym)
 apply assumption

 --{* FrNr *}
 apply (simp add: eqExMps_def lookup.simps fms foldl_map_lookup)
 --{* Num expr1 numop expr2 *}
 apply (rule impI)
 apply (subgoal_tac "expr1 \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="Num expr1 num_op expr2" in subExpr_Gf)
  apply assumption
  apply simp
 apply (subgoal_tac "expr2 \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="Num expr1 num_op expr2" in subExpr_Gf)
  apply assumption
  apply simp
 apply (simp add: eqExMps_def  fms foldl_map_lookup)
 --{* Rel expr1 rop expr2 *}
 apply (rule impI)
 apply (subgoal_tac "expr1 \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="Rel expr1 rel_op expr2" in subExpr_Gf)
  apply assumption
  apply simp
 apply (subgoal_tac "expr2 \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="Rel expr1 rel_op expr2" in subExpr_Gf)
  apply assumption
  apply simp
 apply (simp add: eqExMps_def  fms foldl_map_lookup)
 --{* IF THEN ELSE *}
  apply (rule impI)
 apply (subgoal_tac "expr1 \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
  apply assumption
  apply simp
 apply (subgoal_tac "expr2 \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
  apply assumption
  apply simp
 apply (subgoal_tac "expr3 \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
  apply assumption
  apply simp
 apply (simp add: eqExMps_def  fms foldl_map_lookup)
 --{* Eq expr1 expr2 *}
 apply (rule impI)
 apply (subgoal_tac "expr1 \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="Eq expr1 expr2" in subExpr_Gf)
  apply assumption
  apply simp
 apply (subgoal_tac "expr2 \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="Eq expr1 expr2" in subExpr_Gf)
  apply assumption
  apply simp
 apply (simp add: eqExMps_def  fms foldl_map_lookup)
 --{* Neg *}
 apply (rule impI)
 apply (subgoal_tac "expra \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="Neg expra" in subExpr_Gf)
  apply assumption
  apply simp
 apply (simp add: eqExMps_def  fms foldl_map_lookup)
 --{* Imp expr1 expr2 *}
 apply (rule impI)
 apply (subgoal_tac "expr1 \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="Imp expr1 expr2" in subExpr_Gf)
  apply assumption
  apply simp
 apply (subgoal_tac "expr2 \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="Imp expr1 expr2" in subExpr_Gf)
  apply assumption
  apply simp
 apply (simp add: eqExMps_def  fms foldl_map_lookup)
 --{* Forall nat expra *}
 apply (rule impI)
 apply (subgoal_tac "expra \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="Forall nat expra" in subExpr_Gf)
  apply assumption
  apply simp
 apply (simp add: eqExMps_def  fms foldl_map_lookup)
 --{* Ty expra ty *}
 apply (rule impI)
 apply (subgoal_tac "expra \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="Ty expra ty" in subExpr_Gf)
  apply assumption
  apply simp
 apply (simp add: eqExMps_def  fms foldl_map_lookup)
 --{* Pos *}
 apply (simp add: eqExMps_def  fms foldl_map_lookup)
 --{* Call *}
 apply (rule impI)
 apply (subgoal_tac "\<forall>ex \<in> set bs. Call expra \<noteq> Gf list1 list2 ex")
  prefer 2
  apply simp
 apply (subgoal_tac "\<forall>ex \<in> set as. Call expra \<noteq> Gf list1 list2 ex")
  prefer 2
  apply simp
 apply (simp only: eqExMps_Call)
 apply (simp add: fms foldl_map_lookup)

 --{* Catch  *}
 apply (rule impI)
 apply (subgoal_tac "\<forall>ex \<in> set bs. Call expra \<noteq> Gf list1 list2 ex")
  prefer 2
  apply simp
 apply (subgoal_tac "\<forall>ex \<in> set as. Call expra \<noteq> Gf list1 list2 ex")
  prefer 2
  apply simp
 apply (simp only: eqExMps_Catch)
 apply (simp add: fms foldl_map_lookup)

 --{* And es *}
 apply (rule impI)
 apply (simp only: eqExMps_And')
 apply (rule conjI)
 apply (subgoal_tac "\<forall>ex \<in> set bs. And es \<noteq> Gf list1 list2 ex")
  prefer 2
  apply simp
 apply (subgoal_tac "\<forall>ex \<in> set as. And es \<noteq> Gf list1 list2 ex")
  prefer 2
  apply simp
 apply (simp add:  fms map_if_notin id_lookup_def foldl_map_lookup del: evalE_evalEs.simps split del: split_if)

 apply (rule ballI)
 apply (erule_tac x="ex" and A="set es" in ballE)
  prefer 2
  apply simp
 apply (subgoal_tac "ex \<in> set (subExpr expr)")
  prefer 2
  apply (rule_tac ex'="And es" in subExpr_Gf)
  apply simp
  apply simp
 apply (drule mp, assumption)
 apply (simp only:)

apply (simp only: foldl_append foldl_Cons substE_substEs.simps)
apply (subgoal_tac "\<forall> ex \<in> (set bs). Gf list1 list2 expr \<noteq> Gf list1 list2 ex")
 prefer 2
 apply simp
(* apply (simp only: foldl_map_lookup) *)
apply (simp only: evalE_evalEs.simps the_Bool_def)
apply (case_tac "stk")
apply simp

apply (case_tac "list")
apply simp

apply (simp add: is_Ref'_def is_Addr'_conv)
apply (erule conjE | erule exE)+
apply (rule allI)
apply (rule impI)+
apply (rule conjI)
apply (rule impI)+
apply (case_tac "aa")
apply simp
apply simp
apply simp
apply simp
apply simp
apply (rule impI)+
apply (case_tac "ad")
apply simp
apply simp
apply simp
apply simp
--{* ad = Addr nat *}
apply simp

--{* FrNr *}
apply (rule impI, rule allI)
apply (subgoal_tac "\<forall>ex \<in> set (remdups' (getGfEx list1 list2 Q)). FrNr \<noteq> Gf list1 list2 ex")
 prefer 2
 apply simp
apply (drule_tac foldl_map_lookup)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Num expr1 numop expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Num expr1 num_op expr2" in subExpr_Gf)
 apply assumption
 apply simp
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Num expr1 num_op expr2" in subExpr_Gf)
 apply assumption
 apply simp
apply (simp add: fms foldl_map_lookup)
--{* Rel expr1 rop expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Rel expr1 rel_op expr2" in subExpr_Gf)
 apply assumption
 apply simp
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Rel expr1 rel_op expr2" in subExpr_Gf)
 apply assumption
 apply simp
apply (simp add: fms foldl_map_lookup)
--{* IF THEN ELSE *}
 apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply simp
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply simp
apply (subgoal_tac "expr3 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply simp
apply (simp add: fms foldl_map_lookup split add: bool.split)
--{* Eq expr1 expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Eq expr1 expr2" in subExpr_Gf)
 apply assumption
 apply simp
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Eq expr1 expr2" in subExpr_Gf)
 apply assumption
 apply simp
apply (simp add: fms foldl_map_lookup)
--{* Neg *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Neg expr" in subExpr_Gf)
 apply assumption
 apply simp
apply (simp add: fms foldl_map_lookup)
--{* Imp expr1 expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Imp expr1 expr2" in subExpr_Gf)
 apply assumption
 apply simp
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Imp expr1 expr2" in subExpr_Gf)
 apply assumption
 apply simp
apply (simp add: fms foldl_map_lookup)
--{* Forall nat expra *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Forall nat expr" in subExpr_Gf)
 apply assumption
 apply simp
apply (simp add:  fms foldl_map_lookup)
--{* Ty expr ty *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Ty expr ty" in subExpr_Gf)
 apply assumption
 apply simp
apply (erule conjE | erule exE)+
apply (simp add: is_Ref'_def is_Addr'_conv)
apply (erule conjE | erule exE)+
apply (simp add: fms foldl_map_lookup split add: val.split)
--{* Pos *}
apply (rule impI)
apply (subgoal_tac "x \<in> set (getPosEx Q)")
 prefer 2
 apply (rule subExpr_getPosEx)
 apply assumption
apply (drule_tac xs="(getPosEx Q)" in  in_set_conv_decomp_fst)
apply (erule exE | erule conjE)+
apply (subgoal_tac "\<forall>ex \<in> set (remdups' (getGfEx list1 list2 Q)). Pos x \<noteq> Gf list1 list2 ex")
 prefer 2
 apply simp
apply (subgoal_tac "p' mem domC \<Pi>")
 prefer 2
 apply (cut_tac wf_Pi)
 apply (cut_tac p_domC)
 apply (cut_tac p_def)
 apply (cut_tac cmd_p)
 apply (simp only: in_set_conv_decomp)
 apply (erule exE)+
 apply (simp add: incA_def wf_def checkPos_split succsNormal_def split add: split_if_asm)
apply (cut_tac p_domC)
apply (case_tac "frs")
apply (simp add: fms  foldl_map_lookup map_if_notin id_lookup_def p_def mem_iff)

apply (simp add: fms  foldl_map_lookup map_if_notin id_lookup_def p_def mem_iff callers_simps)

--{* Call *}
apply (rule impI)
apply (subgoal_tac "\<forall>ex \<in> set (remdups' (getGfEx list1 list2 Q)). Call expr \<noteq> Gf list1 list2 ex")
 prefer 2
 apply simp
apply (simp add: fms foldl_map_lookup id_lookup_def)
apply (rule impI)
apply (simp only: neq_Nil_conv)
apply (erule exE)+
apply (simp add: split_paired_all)

 --{* Catch  *}
apply (rule impI)
apply (subgoal_tac "\<forall>ex \<in> set (remdups' (getGfEx list1 list2 Q)). Catch list expr \<noteq> Gf list1 list2 ex")
 prefer 2
 apply simp
apply (simp add: fms foldl_map_lookup id_lookup_def)
apply (rule impI)
apply (simp only: neq_Nil_conv)
apply (erule exE)+
apply (simp add: split_paired_all)
--{* And es*}
apply (rule impI)
apply (simp only: evalE_And substE_substEs.simps substEs_map)
apply simp
apply (rule allI)
apply (rule iffI)
apply (rule ballI)
apply (erule_tac x="ex" in ballE)
 prefer 2
 apply simp
apply (erule_tac x="ex" in ballE)
 prefer 2
 apply simp
apply (subgoal_tac "ex \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="And es" in subExpr_Gf)
 apply assumption
 apply simp
apply (drule mp, assumption)
apply (erule_tac x="I" in allE)
apply (simp only:)

apply (subgoal_tac "(\<lambda>u. if u = the_Addr (hd (tl stk))
                              then \<lfloor>(cname_of h (the_Addr (hd (tl stk))),
                                     \<lambda>u. if u = (list1, list2) then \<lfloor>hd stk\<rfloor> else snd (the (h (the_Addr (hd (tl stk))))) u)\<rfloor>
                              else h u) = h(the_Addr (hd (tl stk)) \<mapsto>
                      (cname_of h (the_Addr (hd (tl stk))), snd (the (h (the_Addr (hd (tl stk)))))((list1, list2) \<mapsto> hd stk)))")
 prefer 2
 apply (rule ext)
 apply simp
 apply (rule impI)
 apply (rule ext)
 apply simp
apply (simp only:)

--{* second direction of iff *}
apply (rule ballI)
apply (erule_tac x="ex" in ballE)
 prefer 2
 apply simp
apply (erule_tac x="ex" in ballE)
 prefer 2
 apply simp
apply (subgoal_tac "ex \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="And es" in subExpr_Gf)
 apply assumption
 apply simp
apply (drule mp, assumption)
apply (erule_tac x="I" in allE)
apply (simp only:)
apply (subgoal_tac "(\<lambda>u. if u = the_Addr (hd (tl stk))
                                then \<lfloor>(cname_of h (the_Addr (hd (tl stk))),
                                       \<lambda>u. if u = (list1, list2) then \<lfloor>hd stk\<rfloor> else snd (the (h (the_Addr (hd (tl stk))))) u)\<rfloor>
                                else h u) = 
                    h(the_Addr (hd (tl stk)) \<mapsto>
                    (cname_of h (the_Addr (hd (tl stk))), snd (the (h (the_Addr (hd (tl stk)))))((list1, list2) \<mapsto> hd stk)))")
 prefer 2
 apply (rule ext)
 apply simp
 apply (rule impI)
 apply (rule ext)
 apply simp
apply (simp only:) 
done
qed
(*>*)


lemma effS_wpF_Checkcast:
assumes i_def: "i = Checkcast Cl"
assumes wf_Pi: "wf \<Pi>"
assumes handlesEx: "handlesEx (fst \<Pi>) p' = None"
assumes cmd_p: "cmd \<Pi> p = Some i"
assumes p_domC: "p \<in> set (domC \<Pi>)"
assumes i_instr: "instrs_of P C M ! pc = i"
assumes s_def: "s = (p,\<sigma>,e)"
assumes p_def: "p = (C,M,pc)"
assumes sigma_def: "\<sigma> = (None,h,(stk,loc,p)#frs)"
assumes s'_def: "s' = (p',\<sigma>',e')"
assumes sigma'_def: "\<sigma>' = (None,h,fr'#frs')"
assumes e'_def: "e' = e\<lparr>cs := if \<exists>M n. i = Invoke M n then h # cs e else if i = Return then tl (cs e) else cs e\<rparr>"
assumes p'_def: "p' = snd (snd fr')"
assumes check_i: "check_instr' i P h stk loc C M pc frs"
assumes exec_i : "exec_instr i P h stk loc C M pc frs = \<sigma>'"
assumes Pi_def: "\<Pi> = (P,An)"
shows  "\<forall> I. evalE \<Pi> (p,\<sigma>,e\<lparr>lv:=I\<rparr>) (wpF \<Pi> p p' Q) = evalE \<Pi> (p',\<sigma>',e'\<lparr>lv:=I\<rparr>) Q"
(*<*)
proof -
from handlesEx cmd_p i_def 
show ?thesis
apply -
apply (simp only: wpF_Checkcast)
--{* induction preparation *}
apply (subgoal_tac "\<exists>Q'. Q = Q'")
 prefer 2
 apply fastsimp
apply (erule exE)
apply (subgoal_tac "\<forall>em. substE em Q = substE em Q'")
 prefer 2
 apply simp
apply (subgoal_tac "\<forall> I. evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q = evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q'")
 prefer 2
 apply simp
apply (subgoal_tac "Q' \<in> set (subExpr Q)")
  prefer 2
  apply (simp add: getExpr_refl)
apply (erule_tac V="Q = Q'" in thin_rl)
apply (simp only:)
apply (erule_tac V="\<forall>em.  substE em Q = substE em Q'" in thin_rl)
apply (erule_tac V="\<forall>I. evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q = evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q'" in thin_rl)
apply (erule_tac P="Q' \<in> set (subExpr Q)" in rev_mp)
apply (erule thin_rl)+
apply (cut_tac exec_i[THEN sym] sigma'_def i_def sigma_def check_i p'_def e'_def p_def)
--{* induction on Q' *}
apply simp
apply (case_tac " \<not> cast_ok P Cl h (hd stk)")
apply (simp add: split_def)

apply (simp add: split_def)
apply (rule_tac expr="Q'" in expr_induct)
--{* Rg *}
apply (rule impI, rule allI)
apply (drule subExpr_rgIds)
apply (drule_tac xs="rgIds Q" in  in_set_conv_decomp_fst)
apply (erule exE | erule conjE)+
apply (simp add: fms map_if_notin id_lookup_def)
--{* St *}
apply (rule impI, rule allI)
apply (drule subExpr_stkIds)
apply (drule_tac xs="stkIds Q" in in_set_conv_decomp_fst)
apply (erule exE | erule conjE)+
apply (simp add: fms map_if_notin id_lookup_def neq_Nil_conv)
--{* Lv *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* Cn *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* NewA *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* Gf list1 list2 expr *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Gf list1 list2 expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def split add: val.split)
--{* FrNr *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* Num expr1 num_op epxr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Num expr1 num_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Num expr1 num_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Rel expr1 rel_op expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Rel expr1 rel_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Rel expr1 rel_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* IF expr1 THEN expr2 ELSE expr3 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr3 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (drule mp,assumption)+

apply (simp add: split_def fms map_if_notin id_lookup_def evalE_evalEs.simps substE_substEs.simps split del: split_if add: bool.split )
apply (rule allI)
apply (case_tac "evalE \<Pi> ((C, M, Suc pc), (None, h, (stk, loc, C, M, Suc pc) # frs), e\<lparr>lv := I\<rparr>) expr1 = \<lfloor>Bool True\<rfloor>")
apply simp

apply simp
--{* Eq expr1 expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Eq expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Eq expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Neg expr *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Neg expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Imp expr1 expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Imp expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Imp expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Forall nat expr *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Forall nat expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def split del: split_if)
--{* Ty expr ty*}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Ty expr ty" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: Let_def split_def fms map_if_notin id_lookup_def split add: val.split)
--{* Pos x *}
apply (rule impI, rule allI)
apply (subgoal_tac "x \<in> set (getPosEx Q)")
 prefer 2
 apply (rule subExpr_getPosEx)
 apply assumption
apply (drule_tac xs="getPosEx Q" in  in_set_conv_decomp_fst)
apply (erule exE |erule conjE)+
apply (subgoal_tac "p' mem domC \<Pi>")
 prefer 2
 apply (cut_tac wf_Pi)
 apply (cut_tac p_domC)
 apply (cut_tac p_def)
 apply (cut_tac cmd_p)
 apply (simp only: in_set_conv_decomp)
 apply (erule exE)+
 apply (simp add: incA_def wf_def checkPos_split succsNormal_def split add: split_if_asm)
apply (cut_tac p_domC)
apply (case_tac "frs")
apply (simp add: fms  map_if_notin id_lookup_def p_def mem_iff)

apply (simp add: fms map_if_notin id_lookup_def p_def mem_iff callers_simps)

--{* Call *}
apply (rule impI,rule allI)
apply (simp add: Let_def split_def fms map_if_notin id_lookup_def)
apply (rule impI)
apply (simp only: neq_Nil_conv)
apply (erule exE)+
apply (simp (no_asm_simp) add: split_paired_all env_upd_cs)

--{* Catch  *}
apply (rule impI,rule allI)
apply (simp add: Let_def split_def fms map_if_notin id_lookup_def)
apply (rule impI)
apply (simp only: neq_Nil_conv)
apply (erule exE)+
apply (simp (no_asm_simp) add: split_paired_all env_upd_cs)

--{* And es *}
apply (rule impI, rule allI)
apply (simp add: evalE_And fms map_if_notin id_lookup_def del: evalE_evalEs.simps split del: split_if)
apply (rule iffI)
apply (rule ballI)
apply (erule_tac x="ex" in ballE)
apply (erule_tac x="(substE (map (\<lambda>q. (Pos q, if q = (C, M, Suc pc) then Pos p else FF)) (getPosEx Q)) ex)" in ballE)
apply (subgoal_tac "ex \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="And es" in subExpr_Gf)
 apply assumption
 apply (simp add: parts.simps)
apply (simp add: substEs_map) 

apply (simp add: substEs_map)

apply (simp only:)

apply (rule ballI)

apply (simp add: substEs_map image_iff)
apply (erule bexE)
apply (erule_tac x="x" in ballE)
apply (erule_tac x="x" in ballE)
apply (subgoal_tac "x \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="And es" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm_simp) add: parts.simps)
apply simp

apply (simp only:)

apply (simp only:)
done
qed
(*>*)


lemma effS_wpF_Invoke:
assumes i_def: "i = Invoke Mn n"
assumes wf_Pi: "wf \<Pi>"
assumes handlesEx: "handlesEx (fst \<Pi>) p' = None"
assumes cmd_p: "cmd \<Pi> p = Some i"
assumes p_domC: "p \<in> set (domC \<Pi>)"
assumes i_instr: "instrs_of P C M ! pc = i"
assumes s_def: "s = (p,\<sigma>,e)"
assumes p_def: "p = (C,M,pc)"
assumes sigma_def: "\<sigma> = (None,h,(stk,loc,p)#frs)"
assumes s'_def: "s' = (p',\<sigma>',e')"
assumes sigma'_def: "\<sigma>' = (None,h,fr'#frs')"
assumes e'_def: "e' = e\<lparr>cs := if \<exists>M n. i = Invoke M n then h # cs e else if i = Return then tl (cs e) else cs e\<rparr>"
assumes p'_def: "p' = snd (snd fr')"
assumes check_i: "check_instr' i P h stk loc C M pc frs"
assumes exec_i : "exec_instr i P h stk loc C M pc frs = \<sigma>'"
assumes Pi_def: "\<Pi> = (P,An)"
(* assumes anF_p: "\<forall> A. anF \<Pi> p = Some A \<longrightarrow> \<Pi>,s \<Turnstile> A" *)
assumes p'_domC: "p' \<in> set (domC \<Pi>)"
shows  "\<forall> I. evalE \<Pi> (p,\<sigma>,e\<lparr>lv:=I\<rparr>) (wpF \<Pi> p p' Q) = evalE \<Pi> (p',\<sigma>',e'\<lparr>lv:=I\<rparr>) Q"
(*<*)
proof -
from handlesEx cmd_p i_def 
show ?thesis
apply -
apply (simp only: wpF_Invoke)
--{* induction preparation *}
apply (subgoal_tac "\<exists>Q'. Q = Q'")
 prefer 2
 apply fastsimp
apply (erule exE)
apply (subgoal_tac "\<forall>em. substE em Q = substE em Q'")
 prefer 2
 apply simp
apply (subgoal_tac "\<forall> I. evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q = evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q'")
 prefer 2
 apply simp
apply (subgoal_tac "Q' \<in> set (subExpr Q)")
  prefer 2
  apply (simp add: getExpr_refl)
apply (erule_tac V="Q = Q'" in thin_rl)
apply (simp only:)
apply (erule_tac V="\<forall>em.  substE em Q = substE em Q'" in thin_rl)
apply (erule_tac V="\<forall>I. evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q = evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q'" in thin_rl)
apply (erule_tac P="Q' \<in> set (subExpr Q)" in rev_mp)
apply (erule thin_rl)+
apply (cut_tac exec_i[THEN sym] sigma'_def i_def sigma_def check_i p'_def e'_def p_def)
--{* induction on Q' *}
apply (case_tac "stk ! n = Null")
apply (simp add: split_def)

apply (rule_tac expr="Q'" in expr_induct)
   
--{* Rg *}
apply (rule impI, rule allI)
apply (drule subExpr_rgIds)
apply (drule_tac xs="rgIds Q" in in_set_conv_decomp_fst)
apply (erule exE | erule conjE)+
apply (simp add:  is_Ref'_def is_Addr'_conv fms map_if_notin id_lookup_def)
apply (erule conjE | erule exE)+ 
apply (drule has_method_has_method)
apply (erule conjE | erule exE)+
apply (rule conjI | rule impI)+
apply (case_tac "nat")
apply simp

apply (subgoal_tac "nata < length stk")
 prefer 2
 apply arith
apply (simp add: nth_append)
apply  (rule rev_take_nth)
apply assumption
apply arith

apply arith

apply (rule impI)+
apply (case_tac "nat")
apply simp

apply (simp add: nth_append)
apply (subgoal_tac "nata - min (length stk) n < fst (snd (snd (snd (snd (method P (cname_of h (the_Addr (stk ! n))) Mn)))))")
 prefer 2
 apply (simp add: min_def linorder_not_le le_simps linorder_not_less Pi_def)
 apply (presburger (abs))

apply (simp add: nth_replicate arb_def)
apply arith

apply arith

apply (case_tac "stk")
apply simp

apply (simp add: none_def linorder_not_less  min_def le_simps linorder_not_le Pi_def)
--{* St *}
apply (rule impI, rule allI)
apply (drule subExpr_stkIds)
apply (drule_tac xs="stkIds Q" in in_set_conv_decomp_fst)
apply (erule exE | erule conjE)+
apply (simp add: fms map_if_notin id_lookup_def neq_Nil_conv none_def)

--{* Lv *}
apply (simp add: fms map_if_notin id_lookup_def cong del: if_weak_cong)
--{* Cn *}
apply (simp add: fms map_if_notin id_lookup_def cong del: if_weak_cong)
--{* NewA *}
apply (simp add: fms map_if_notin id_lookup_def cong del: if_weak_cong)
--{* Gf list1 list2 expr *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Gf list1 list2 expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def cong del: if_weak_cong split add: val.split) 
--{* FrNr *}
apply (simp add: fms map_if_notin id_lookup_def numop_def)
--{* Num expr1 num_op epxr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Num expr1 num_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Num expr1 num_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def cong del: if_weak_cong )
--{* Rel expr1 rel_op expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Rel expr1 rel_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Rel expr1 rel_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def cong del: if_weak_cong)
--{* IF expr1 THEN expr2 ELSE expr3 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr3 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (drule mp,assumption)+

apply (simp add: split_def fms map_if_notin id_lookup_def evalE_evalEs.simps substE_substEs.simps split del: split_if add: bool.split cong del: if_weak_cong )
--{* Eq expr1 expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Eq expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Eq expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def cong del: if_weak_cong)
--{* Neg expr *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Neg expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def cong del: if_weak_cong)
--{* Imp expr1 expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Imp expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Imp expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def cong del: if_weak_cong)
--{* Forall nat expr *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Forall nat expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def split del: split_if cong del: if_weak_cong)
--{* Ty expr ty*}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Ty expr ty" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: Let_def split_def fms map_if_notin id_lookup_def split add: val.split cong del: if_weak_cong)
--{* Pos x *}
apply (rule impI, rule allI)
apply (subgoal_tac "x \<in> set (getPosEx Q)")
 prefer 2
 apply (rule subExpr_getPosEx)
 apply assumption
apply (drule_tac xs="getPosEx Q" in  in_set_conv_decomp_fst)
apply (erule exE |erule conjE)+
apply (subgoal_tac "p' mem domC \<Pi>")
 prefer 2
 apply (cut_tac p'_domC)
 apply (simp only: mem_iff)
 (*
 apply (cut_tac wf_Pi)
 apply (cut_tac p_domC)
 apply (cut_tac p_def)
 apply (cut_tac cmd_p)
 apply (simp only: in_set_conv_decomp)
 apply (erule exE)+
 apply (simp add: is_Ref'_def is_Addr'_conv incA_def wf_def 
        checkPos_split succsNormal_def split add: split_if_asm)
 apply (case_tac "extractTy (a,St n) = []")
 apply simp
 apply (subgoal_tac "\<exists>tp\<in>set (extractTy (a,St n)). \<Pi>,s \<Turnstile>  Ty (St n) tp")
  prefer 2
  apply (rule_tac A="a" in extractTy_sem)
  apply (cut_tac anF_p)
  apply (erule_tac x="a" in allE)
  apply simp
  apply (rule refl)
  apply assumption
 apply (erule bexE)
 apply (case_tac "extractTy (a,St n) = []")
 apply simp
 
 apply (simp add: in_set_conv_decomp not_Nil_case)
 apply (subgoal_tac "\<exists> X. tp = Class X")
   prefer 2
   apply (case_tac "ysc")
   apply simp
   apply (case_tac "tp")
   apply simp
   apply simp
   apply simp
   --{* tp = NT *}
   apply (cut_tac s_def sigma_def)
   apply (case_tac "ac")
   apply simp
   apply simp
   apply simp
   apply simp
   apply simp
   --{* tp = Class list *}
   apply simp
   --{* ysc = ac list *}
   apply (simp add: list_all_iff)
   apply (case_tac "tp")
   apply simp
   apply simp
   apply simp
   --{* tp = NT *}
   apply (cut_tac s_def sigma_def)
   apply (case_tac "ac")
   apply simp
   apply simp
   apply simp
   apply simp
   apply simp
   --{* tp = Class lista *}
   apply simp
  
 apply (erule exE )
 apply simp

 apply (case_tac "ac")
 apply simp
 apply simp
 apply simp
 apply simp
 apply simp
 --{* ac = Addr nat *}
 apply (drule has_method_has_method)
 apply (erule exE | erule conjE)+
 apply (simp add: split_def)
 apply (simp add: obj_ty_def)
 apply (cut_tac s_def)
 apply (simp add: Pi_def)
 *)
apply (cut_tac p_domC)
apply (cut_tac s_def)
apply (case_tac "frs")
apply (simp add: fms  map_if_notin id_lookup_def p_def mem_iff)
apply (rule impI)
apply (case_tac "C = fst (ipc \<Pi>) \<and> M = fst (snd (ipc \<Pi>))")
apply (cut_tac cmd_p)
apply (cut_tac p_def)
apply (simp add: callers_def)

apply simp
apply (case_tac "C = fst (ipc \<Pi>)")
apply simp

apply simp

--{* frs = a list *}
apply (simp add: fms map_if_notin id_lookup_def p_def mem_iff callers_simps)
apply (rule impI)
apply (subgoal_tac "(C, M, pc) \<in> set (callers \<Pi> (fst (method P (cname_of h (the_Addr (stk ! n))) Mn), Mn, 0))")
 prefer 2
 apply (cut_tac cmd_p)
 apply (cut_tac p_def)
 apply (simp add: callers_def)
apply simp

--{* Call *}
apply (rule impI,rule allI)
apply (drule subExpr_getCallEx)
apply (drule_tac xs="getCallEx Q" in in_set_conv_decomp_fst)
apply (erule exE | erule conjE)+

apply (simp add: Let_def split_def fms map_if_notin id_lookup_def env_upd)
--{* Catch  *}
apply (rule impI,rule allI)
apply (drule subExpr_getCatchEx)
apply (drule_tac xs="getCatchEx Q" in  in_set_conv_decomp_fst)
apply (erule exE | erule conjE)+
apply (simp add: Let_def split_def fms map_if_notin id_lookup_def env_upd split del: split_if list.split option.split)
apply (subgoal_tac "(concat (map (\<lambda>e. if catchesEx P (fst e) (C, M, pc)
                                              then FiniteMap.o2l ([(Catch (fst e) (snd e), snd e)] ? Catch list expr)
                                              else FiniteMap.o2l ([(Catch (fst e) (snd e),
                                                       IF FrNr \<doteq> Cn (Intg 1) THEN snd e ELSE Catch (fst e) (snd e))] ? Catch list expr))
                                      ys)) = []")
 prefer 2
 apply (erule_tac P="(list,expr)\<notin> set ys" in rev_mp)
 apply (induct_tac "ys")
 apply simp

 apply (simp (no_asm_simp) split del: split_if)
 apply (rule impI)
 apply (simp (no_asm_simp))
 apply (case_tac "fst a = list \<and> snd a = expr")
 apply (simp add: split_paired_all)

 apply (simp (no_asm_simp))
apply (simp (no_asm_simp) add: Pi_def Let_def split_def fms map_if_notin id_lookup_def env_upd split del: split_if list.split option.split)
apply (case_tac "JVMExceptions.match_ex_table P list pc (ex_table_of P C M)")
apply (subgoal_tac "\<not> catchesEx P list (C,M,pc)")
 prefer 2
 apply (simp add: catchesEx_def)
apply (simp add: env_upd split del: split_if list.split option.split)
apply (case_tac "frs")
apply simp

--{* obsolete record manipulations *}
apply (simp add: Pi_def)



--{* exception caught *}
apply (subgoal_tac "catchesEx P list (C,M,pc)")
 prefer 2
 apply (simp add: catchesEx_def)
apply (simp add: env_upd)


--{* And es *}
apply (rule impI, rule allI)
apply (simp add: evalE_And fms map_if_notin id_lookup_def del: evalE_evalEs.simps split del: split_if)
apply (rule iffI)
apply (rule ballI)
apply (erule_tac x="ex" in ballE)
apply (erule_tac x=" (substE (map (\<lambda>q. (Pos q, if q = (fst (method (fst \<Pi>) (cname_of h (the_Addr (stk ! n))) Mn), Mn, 0) then Pos p else FF))
                          (getPosEx Q) @
                         (FrNr, Num FrNr num_op.Add (Cn (Intg 1))) #
                         map (\<lambda>k. (Rg k, if k \<le> n then St (n - k) else (if k \<le> n + fst (snd (snd (snd (snd (method (fst \<Pi>) (fst p') Mn)))))
                                              then Cn arb
                                              else none))) (rgIds Q) @
                         map (\<lambda>k. (St k, none)) (stkIds Q) @
                         map (\<lambda>ex. (Call ex, ex)) (getCallEx Q) @
                         concat (map (\<lambda>u. if catchesEx P (fst u) (C, M, pc) then [(Catch (fst u) (snd u), snd u)]
                                          else [(Catch (fst u) (snd u), IF FrNr \<doteq> Cn (Intg 1) THEN snd u ELSE Catch (fst u) (snd u))])
                                  (getCatchEx Q)))
                  ex)" in ballE)
apply (subgoal_tac "ex \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="And es" in subExpr_Gf)
 apply assumption
 apply (simp add: parts.simps)
apply (drule mp, assumption)
apply (erule_tac x="I" in allE)
apply (cut_tac Pi_def)
apply (simp add: substEs_map)

apply (simp add: substEs_map Pi_def)

apply (simp only:)

apply (rule ballI)

apply (simp add: substEs_map image_iff)
apply (erule bexE)
apply (erule_tac x="x" in ballE)
apply (erule_tac x="x" in ballE)
apply (subgoal_tac "x \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="And es" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm_simp) add: parts.simps)
apply simp

apply (simp only:)

apply (simp only:)
done
qed
(*>*)


lemma effS_wpF_Return:
assumes i_def: "i = Return"
assumes wf_Pi: "wf \<Pi>"
assumes handlesEx: "handlesEx (fst \<Pi>) p' = None"
assumes cmd_p: "cmd \<Pi> p = Some i"
assumes p_domC: "p \<in> set (domC \<Pi>)"
assumes i_instr: "instrs_of P C M ! pc = i"
assumes s_def: "s = (p,\<sigma>,e)"
assumes p_def: "p = (C,M,pc)"
assumes sigma_def: "\<sigma> = (None,h,(stk,loc,p)#frs)"
assumes s'_def: "s' = (p',\<sigma>',e')"
assumes sigma'_def: "\<sigma>' = (None,h,fr'#frs')"
assumes e'_def: "e' = e\<lparr>cs := if \<exists>M n. i = Invoke M n then h # cs e else if i = Return then tl (cs e) else cs e\<rparr>"
assumes p'_def: "p' = snd (snd fr')"
assumes check_i: "check_instr' i P h stk loc C M pc frs"
assumes exec_i : "exec_instr i P h stk loc C M pc frs = \<sigma>'"
assumes Pi_def: "\<Pi> = (P,An)"
assumes Pos_p: "\<Pi>,(p,\<sigma>,e) \<Turnstile> Pos p"
shows  "\<forall> I. evalE \<Pi> (p,\<sigma>,e\<lparr>lv:=I\<rparr>) (wpF \<Pi> p p' Q) = evalE \<Pi> (p',\<sigma>',e'\<lparr>lv:=I\<rparr>) Q"
(*<*)
proof -
from handlesEx cmd_p i_def 
show ?thesis
apply -
apply (simp only: wpF_Return)
--{* induction preparation *}
apply (subgoal_tac "\<exists>Q'. Q = Q'")
 prefer 2
 apply fastsimp
apply (erule exE)
apply (subgoal_tac "\<forall>em. substE em Q = substE em Q'")
 prefer 2
 apply simp
apply (subgoal_tac "\<forall> I. evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q = evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q'")
 prefer 2
 apply simp
apply (subgoal_tac "Q' \<in> set (subExpr Q)")
  prefer 2
  apply (simp add: getExpr_refl)
apply (erule_tac V="Q = Q'" in thin_rl)
apply (simp only:)
apply (erule_tac V="\<forall>em.  substE em Q = substE em Q'" in thin_rl)
apply (erule_tac V="\<forall>I. evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q = evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q'" in thin_rl)
apply (erule_tac P="Q' \<in> set (subExpr Q)" in rev_mp)
apply (erule thin_rl)+
apply (cut_tac exec_i[THEN sym] sigma'_def i_def sigma_def check_i p'_def e'_def p_def Pi_def)
--{* induction on Q' *}
apply (case_tac "frs")
apply simp

apply (subgoal_tac "\<exists>st_a rg_a p_a. a=(st_a,rg_a,p_a)")
 prefer 2
 apply (erule thin_rl)+
 apply fastsimp
apply (erule exE)+
apply simp
apply (rule_tac expr="Q'" in expr_induct)
--{* Rg *}
apply (rule impI, rule allI)
apply (drule subExpr_rgIds)
apply (drule_tac xs="rgIds Q" in  in_set_conv_decomp_fst)
apply (erule exE | erule conjE)+
apply (simp add: fms map_if_notin id_lookup_def)
--{* St *}
apply (rule impI, rule allI)
apply (drule subExpr_stkIds)
apply (drule_tac xs="stkIds Q" in in_set_conv_decomp_fst)
apply (erule exE | erule conjE)+
apply (simp add: fms map_if_notin id_lookup_def neq_Nil_conv)
apply (erule exE | erule conjE)+
apply (case_tac "nat")
apply simp

apply simp
apply arith
--{* Lv *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* Cn *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* NewA *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* Gf list1 list2 expr *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Gf list1 list2 expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def split add: val.split)
--{* FrNr *}
apply (simp add: fms map_if_notin id_lookup_def  numop_def)
--{* Num expr1 num_op epxr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Num expr1 num_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Num expr1 num_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Rel expr1 rel_op expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Rel expr1 rel_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Rel expr1 rel_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* IF expr1 THEN expr2 ELSE expr3 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr3 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (drule mp,assumption)+

apply (simp add: split_def fms map_if_notin id_lookup_def evalE_evalEs.simps substE_substEs.simps split del: split_if add: bool.split )
apply (rule allI)
apply (erule_tac x="I" in allE)+
apply (case_tac "evalE (P, An)
               ((fst p_a, fst (snd p_a), Suc (snd (snd p_a))),
                (None, h, (hd stk # drop (Suc (length (fst (snd (method P C M))))) st_a, rg_a, fst p_a, fst (snd p_a), Suc (snd (snd p_a))) # list), e
                \<lparr>cs := tl (cs e), lv := I\<rparr>)
               expr1 =
              \<lfloor>Bool True\<rfloor>")
apply simp

apply simp
--{* Eq expr1 expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Eq expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Eq expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Neg expr *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Neg expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Imp expr1 expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Imp expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Imp expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Forall nat expr *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Forall nat expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def split del: split_if)
--{* Ty expr ty*}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Ty expr ty" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: Let_def split_def fms map_if_notin id_lookup_def split add: val.split)
--{* Pos x *}
apply (rule impI, rule allI)
apply (subgoal_tac "x \<in> set (getPosEx Q)")
 prefer 2
 apply (rule subExpr_getPosEx)
 apply assumption
apply (drule_tac xs="getPosEx Q" in  in_set_conv_decomp_fst)
apply (erule exE |erule conjE)+
apply (subgoal_tac "p' mem domC \<Pi>")
 prefer 2
 apply (cut_tac wf_Pi)
 apply (cut_tac p_domC)
 apply (cut_tac p_def)
 apply (cut_tac cmd_p)
 apply (cut_tac Pos_p)
 apply (cut_tac s_def)
 apply (cut_tac sigma_def)
 apply (simp only: in_set_conv_decomp)
 apply (erule exE)+
 apply (simp add: incA_def wf_def checkPos_split succsNormal_def split_def split add: split_if_asm)
 apply (simp add: in_set_conv_decomp)
 apply (erule exE)+
 apply (case_tac "list")
 apply simp

 apply simp

apply (simp add: fms map_if_notin id_lookup_def p_def mem_iff)
apply (rule impI)
apply (cut_tac "p_domC")
apply (cut_tac Pi_def)
apply (cut_tac p_def)
apply (subgoal_tac "p_a \<in> set (callers (P, An) (C, M, pc))")
 prefer 2
 apply (cut_tac Pos_p)
 apply (cut_tac s_def)
 apply (simp add: callers_def)
apply (subgoal_tac "p_a \<in> set (domC (P,An))")
 prefer 2
 apply (cut_tac Pos_p)
 apply (cut_tac s_def)
 apply (simp add: callers_def)

apply (case_tac "list")
apply (simp add: fms map_if_notin id_lookup_def p_def mem_iff callers_simps)

apply (simp add: fms map_if_notin id_lookup_def p_def mem_iff callers_simps)
--{* Call *}
apply (rule impI,rule allI)
apply (drule subExpr_getCallEx)
apply (drule_tac xs="getCallEx Q" in in_set_conv_decomp_fst)
apply (erule exE | erule conjE)+
apply (simp add: Let_def split_def fms map_if_notin id_lookup_def)
apply (rule impI)
apply (simp only: neq_Nil_conv)
apply (erule exE)+
apply (simp (no_asm_simp) add: split_paired_all env_upd_cs)

--{* Catch  *}
apply (rule impI,rule allI)
apply (drule subExpr_getCatchEx)
apply (drule_tac xs="getCatchEx Q" in  in_set_conv_decomp_fst)
apply (erule exE | erule conjE)+
apply (simp add: Let_def split_def fms map_if_notin id_lookup_def)
apply (rule impI)
apply (simp only: neq_Nil_conv)
apply (erule exE)+
apply (simp (no_asm_simp) add: split_paired_all env_upd_cs)

--{* And es *}
apply (rule impI, rule allI)
apply (simp add: evalE_And fms map_if_notin id_lookup_def del: evalE_evalEs.simps split del: split_if)
apply (rule iffI)
apply (rule ballI)
apply (erule_tac x="ex" in ballE)
apply (erule_tac x="(substE (map (\<lambda>q. (Pos q, if q = (fst p_a, fst (snd p_a), Suc (snd (snd p_a))) then Pos p else FF)) (getPosEx Q) @
                         (FrNr, FrNr \<ominus> Cn (Intg 1)) #
                         map (\<lambda>k. (St k, if Suc 0 \<le> k then Call (St (length (fst (snd (method P C M))) + k)) else St 0))
                          (stkIds Q) @
                         map (\<lambda>k. (Rg k, Call (Rg k))) (rgIds Q) @
                         map (\<lambda>ex. (Call ex, Call (Call ex))) (getCallEx Q) @
                         map (\<lambda>u. (Catch (fst u) (snd u), Call (Catch (fst u) (snd u)))) (getCatchEx Q))
                  ex)" in ballE)
apply (subgoal_tac "ex \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="And es" in subExpr_Gf)
 apply assumption
 apply (simp add: parts.simps)
apply (simp add: substEs_map) 

apply (simp add: substEs_map)

apply (simp only:)

apply (rule ballI)

apply (simp add: substEs_map image_iff)
apply (erule bexE)
apply (erule_tac x="x" in ballE)
apply (erule_tac x="x" in ballE)
apply (subgoal_tac "x \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="And es" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm_simp) add: parts.simps)
apply simp

apply (simp only:)

apply (simp only:)
done
qed
(*>*)


lemma effS_wpF_Pop:
assumes i_def: "i = Pop"
assumes wf_Pi: "wf \<Pi>"
assumes handlesEx: "handlesEx (fst \<Pi>) p' = None"
assumes cmd_p: "cmd \<Pi> p = Some i"
assumes p_domC: "p \<in> set (domC \<Pi>)"
assumes i_instr: "instrs_of P C M ! pc = i"
assumes s_def: "s = (p,\<sigma>,e)"
assumes p_def: "p = (C,M,pc)"
assumes sigma_def: "\<sigma> = (None,h,(stk,loc,p)#frs)"
assumes s'_def: "s' = (p',\<sigma>',e')"
assumes sigma'_def: "\<sigma>' = (None,h,fr'#frs')"
assumes e'_def: "e' = e\<lparr>cs := if \<exists>M n. i = Invoke M n then h # cs e else if i = Return then tl (cs e) else cs e\<rparr>"
assumes p'_def: "p' = snd (snd fr')"
assumes check_i: "check_instr' i P h stk loc C M pc frs"
assumes exec_i : "exec_instr i P h stk loc C M pc frs = \<sigma>'"
assumes Pi_def: "\<Pi> = (P,An)"
shows  "\<forall> I. evalE \<Pi> (p,\<sigma>,e\<lparr>lv:=I\<rparr>) (wpF \<Pi> p p' Q) = evalE \<Pi> (p',\<sigma>',e'\<lparr>lv:=I\<rparr>) Q"
(*<*)
proof -
from handlesEx cmd_p i_def 
show ?thesis
apply -
apply (simp only: wpF_Pop)
--{* induction preparation *}
apply (subgoal_tac "\<exists>Q'. Q = Q'")
 prefer 2
 apply fastsimp
apply (erule exE)
apply (subgoal_tac "\<forall>em. substE em Q = substE em Q'")
 prefer 2
 apply simp
apply (subgoal_tac "\<forall> I. evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q = evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q'")
 prefer 2
 apply simp
apply (subgoal_tac "Q' \<in> set (subExpr Q)")
  prefer 2
  apply (simp add: getExpr_refl)
apply (erule_tac V="Q = Q'" in thin_rl)
apply (simp only:)
apply (erule_tac V="\<forall>em.  substE em Q = substE em Q'" in thin_rl)
apply (erule_tac V="\<forall>I. evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q = evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q'" in thin_rl)
apply (erule_tac P="Q' \<in> set (subExpr Q)" in rev_mp)
apply (erule thin_rl)+
apply (cut_tac exec_i[THEN sym] sigma'_def i_def sigma_def check_i p'_def e'_def p_def)
--{* induction on Q' *}
apply simp
apply (rule_tac expr="Q'" in expr_induct)
--{* Rg *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* St *}
apply (rule impI, rule allI)
apply (drule subExpr_stkIds)
apply (drule_tac xs="stkIds Q" in in_set_conv_decomp_fst)
apply (erule exE | erule conjE)+
apply (simp add: fms map_if_notin id_lookup_def neq_Nil_conv)
apply (erule exE)+
apply (case_tac "ysa")
apply simp

apply simp
--{* Lv *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* Cn *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* NewA *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* Gf list1 list2 expr *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Gf list1 list2 expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def split add: val.splits)
--{* FrNr *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* Num expr1 num_op epxr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Num expr1 num_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Num expr1 num_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Rel expr1 rel_op expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Rel expr1 rel_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Rel expr1 rel_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* IF expr1 THEN expr2 ELSE expr3 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr3 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (drule mp,assumption)+
apply (rule allI)
apply (simp add: split_def fms map_if_notin id_lookup_def evalE_evalEs.simps substE_substEs.simps split del: split_if add: bool.split )
apply (case_tac "evalE \<Pi> ((C, M, Suc pc), (None, h, (tl stk, loc, C, M, Suc pc) # frs), e\<lparr>lv := I\<rparr>) expr1 = \<lfloor>Bool True\<rfloor>")
apply simp
apply simp
--{* Eq expr1 expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Eq expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Eq expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Neg expr *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Neg expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Imp expr1 expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Imp expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Imp expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Forall nat expr *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Forall nat expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def split del: split_if)
--{* Ty expr ty*}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Ty expr ty" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: Let_def split_def fms map_if_notin id_lookup_def split add: val.split)
--{* Pos x *}
apply (rule impI, rule allI)
apply (subgoal_tac "x \<in> set (getPosEx Q)")
 prefer 2
 apply (rule subExpr_getPosEx)
 apply assumption
apply (drule_tac xs="getPosEx Q" in  in_set_conv_decomp_fst)
apply (erule exE |erule conjE)+
apply (subgoal_tac "p' mem domC \<Pi>")
 prefer 2
 apply (cut_tac wf_Pi)
 apply (cut_tac p_domC)
 apply (cut_tac p_def)
 apply (cut_tac cmd_p)
 apply (simp only: in_set_conv_decomp)
 apply (erule exE)+
 apply (simp add: incA_def wf_def checkPos_split succsNormal_def split add: split_if_asm)
apply (cut_tac p_domC)
apply (case_tac "frs")
apply (simp add: fms map_if_notin id_lookup_def p_def mem_iff)

apply (simp add: fms map_if_notin id_lookup_def p_def mem_iff callers_simps)
--{* Call *}
apply (rule impI,rule allI)
apply (simp add: Let_def split_def fms map_if_notin id_lookup_def)
apply (rule impI)
apply (simp only: neq_Nil_conv)
apply (erule exE)+
apply (simp (no_asm_simp) add: split_paired_all env_upd_cs)

--{* Catch  *}
apply (rule impI,rule allI)
apply (simp add: Let_def split_def fms map_if_notin id_lookup_def)
apply (rule impI)
apply (simp only: neq_Nil_conv)
apply (erule exE)+
apply (simp (no_asm_simp) add: split_paired_all env_upd_cs)

--{* And es *}
apply (rule impI, rule allI)
apply (simp add: evalE_And fms map_if_notin id_lookup_def del: evalE_evalEs.simps split del: split_if)
apply (rule iffI)
apply (rule ballI)
apply (erule_tac x="ex" in ballE)
apply (erule_tac x="(substE (map (\<lambda>q. (Pos q, if q = (C, M, Suc pc) then Pos p else FF)) (getPosEx Q) @ map (\<lambda>k. (St k, St (Suc k))) (stkIds Q)) ex)" in ballE)
apply (subgoal_tac "ex \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="And es" in subExpr_Gf)
 apply assumption
 apply (simp add: parts.simps)
apply (simp add: substEs_map) 

apply (simp add: substEs_map)

apply (simp only:)

apply (rule ballI)

apply (simp add: substEs_map image_iff)
apply (erule bexE)
apply (erule_tac x="x" in ballE)
apply (erule_tac x="x" in ballE)
apply (subgoal_tac "x \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="And es" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm_simp) add: parts.simps)
apply simp

apply (simp only:)

apply (simp only:)
done
qed
(*>*)


lemma effS_wpF_IBin:
assumes i_def: "i = (IBin no)"
assumes wf_Pi: "wf \<Pi>"
assumes handlesEx: "handlesEx (fst \<Pi>) p' = None"
assumes cmd_p: "cmd \<Pi> p = Some i"
assumes p_domC: "p \<in> set (domC \<Pi>)"
assumes i_instr: "instrs_of P C M ! pc = i"
assumes s_def: "s = (p,\<sigma>,e)"
assumes p_def: "p = (C,M,pc)"
assumes sigma_def: "\<sigma> = (None,h,(stk,loc,p)#frs)"
assumes s'_def: "s' = (p',\<sigma>',e')"
assumes sigma'_def: "\<sigma>' = (None,h,fr'#frs')"
assumes e'_def: "e' = e\<lparr>cs := if \<exists>M n. i = Invoke M n then h # cs e else if i = Return then tl (cs e) else cs e\<rparr>"
assumes p'_def: "p' = snd (snd fr')"
assumes check_i: "check_instr' i P h stk loc C M pc frs"
assumes exec_i : "exec_instr i P h stk loc C M pc frs = \<sigma>'"
assumes Pi_def: "\<Pi> = (P,An)"
shows  "\<forall> I. evalE \<Pi> (p,\<sigma>,e\<lparr>lv:=I\<rparr>) (wpF \<Pi> p p' Q) = evalE \<Pi> (p',\<sigma>',e'\<lparr>lv:=I\<rparr>) Q"
(*<*)
proof -
from handlesEx cmd_p i_def 
show ?thesis
apply -
apply (simp only: wpF_IBin)
--{* induction preparation *}
apply (subgoal_tac "\<exists>Q'. Q = Q'")
 prefer 2
 apply fastsimp
apply (erule exE)
apply (subgoal_tac "\<forall>em. substE em Q = substE em Q'")
 prefer 2
 apply simp
apply (subgoal_tac "\<forall> I. evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q = evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q'")
 prefer 2
 apply simp
apply (subgoal_tac "Q' \<in> set (subExpr Q)")
  prefer 2
  apply (simp add: getExpr_refl)
apply (erule_tac V="Q = Q'" in thin_rl)
apply (simp only:)
apply (erule_tac V="\<forall>em.  substE em Q = substE em Q'" in thin_rl)
apply (erule_tac V="\<forall>I. evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q = evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q'" in thin_rl)
apply (erule_tac P="Q' \<in> set (subExpr Q)" in rev_mp)
apply (erule thin_rl)+
apply (cut_tac exec_i[THEN sym] sigma'_def i_def sigma_def check_i p'_def e'_def p_def)
--{* induction on Q' *}
apply simp
apply (rule_tac expr="Q'" in expr_induct)
--{* Rg *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* St *}
apply (rule impI, rule allI)
apply (drule subExpr_stkIds)
apply (drule_tac xs="stkIds Q" in in_set_conv_decomp_fst)
apply (erule exE | erule conjE)+
apply (case_tac "stk")
apply simp
apply (case_tac "list")
apply simp
apply (simp add: fms map_if_notin id_lookup_def neq_Nil_conv  numop_def isIntg_conv)
apply (erule exE)+
apply (case_tac "nat")
apply simp

apply simp
--{* Lv *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* Cn *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* NewA *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* Gf list1 list2 expr *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Gf list1 list2 expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def split add: val.splits)
--{* FrNr *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* Num expr1 num_op epxr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Num expr1 num_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Num expr1 num_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Rel expr1 rel_op expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Rel expr1 rel_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Rel expr1 rel_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* IF expr1 THEN expr2 ELSE expr3 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr3 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (drule mp,assumption)+
apply (simp add: split_def fms map_if_notin id_lookup_def evalE_evalEs.simps substE_substEs.simps split del: split_if add: bool.split )
apply (case_tac "evalE \<Pi> ((C, M, Suc pc), (None, h, (Intg (numop no (the_Intg (hd stk)) (the_Intg (hd (tl stk)))) # tl (tl stk), loc, C, M, Suc pc) # frs), e\<lparr>lv := I\<rparr>)
                   expr1 =
                  \<lfloor>Bool True\<rfloor>")
apply simp
apply simp
--{* Eq expr1 expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Eq expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Eq expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Neg expr *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Neg expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Imp expr1 expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Imp expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Imp expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Forall nat expr *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Forall nat expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def split del: split_if)
--{* Ty expr ty*}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Ty expr ty" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: Let_def split_def fms map_if_notin id_lookup_def split add: val.split)
--{* Pos x *}
apply (rule impI, rule allI)
apply (subgoal_tac "x \<in> set (getPosEx Q)")
 prefer 2
 apply (rule subExpr_getPosEx)
 apply assumption
apply (drule_tac xs="getPosEx Q" in  in_set_conv_decomp_fst)
apply (erule exE |erule conjE)+
apply (subgoal_tac "p' mem domC \<Pi>")
 prefer 2
 apply (cut_tac wf_Pi)
 apply (cut_tac p_domC)
 apply (cut_tac p_def)
 apply (cut_tac cmd_p)
 apply (simp only: in_set_conv_decomp)
 apply (erule exE)+
 apply (simp add: incA_def wf_def checkPos_split succsNormal_def split add: split_if_asm)
apply (cut_tac p_domC)
apply (case_tac "frs")
apply (simp add: fms map_if_notin id_lookup_def p_def mem_iff)

apply (simp add: fms map_if_notin id_lookup_def p_def mem_iff callers_simps)
--{* Call *}
apply (rule impI,rule allI)
apply (simp add: Let_def split_def fms map_if_notin id_lookup_def)
apply (rule impI)
apply (simp only: neq_Nil_conv)
apply (erule exE)+
apply (simp (no_asm_simp) add: split_paired_all env_upd_cs)

--{* Catch  *}
apply (rule impI,rule allI)
apply (simp add: Let_def split_def fms map_if_notin id_lookup_def)
apply (rule impI)
apply (simp only: neq_Nil_conv)
apply (erule exE)+
apply (simp (no_asm_simp) add: split_paired_all env_upd_cs)

--{* And es *}
apply (rule impI, rule allI)
apply (simp add: evalE_And fms map_if_notin id_lookup_def del: evalE_evalEs.simps split del: split_if)
apply (rule iffI)
apply (rule ballI)
apply (erule_tac x="ex" in ballE)
apply (erule_tac x="(substE (map (\<lambda>q. (Pos q, if q = (C, M, Suc pc) then Pos p else FF)) (getPosEx Q) @
                         map (\<lambda>k. (St k, if k = 0 then Num (St 1) no (St 0) else St (k + 1))) (stkIds Q))
                  ex)" in ballE)
apply (subgoal_tac "ex \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="And es" in subExpr_Gf)
 apply assumption
 apply (simp add: parts.simps)
apply (simp add: substEs_map) 

apply (simp add: substEs_map)

apply (simp only:)

apply (rule ballI)

apply (simp add: substEs_map image_iff)
apply (erule bexE)
apply (erule_tac x="x" in ballE)
apply (erule_tac x="x" in ballE)
apply (subgoal_tac "x \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="And es" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm_simp) add: parts.simps)
apply simp

apply (simp only:)

apply (simp only:)
done
qed
(*>*)


lemma effS_wpF_Goto:
assumes i_def: "i = Goto t"
assumes wf_Pi: "wf \<Pi>"
assumes handlesEx: "handlesEx (fst \<Pi>) p' = None"
assumes cmd_p: "cmd \<Pi> p = Some i"
assumes p_domC: "p \<in> set (domC \<Pi>)"
assumes i_instr: "instrs_of P C M ! pc = i"
assumes s_def: "s = (p,\<sigma>,e)"
assumes p_def: "p = (C,M,pc)"
assumes sigma_def: "\<sigma> = (None,h,(stk,loc,p)#frs)"
assumes s'_def: "s' = (p',\<sigma>',e')"
assumes sigma'_def: "\<sigma>' = (None,h,fr'#frs')"
assumes e'_def: "e' = e\<lparr>cs := if \<exists>M n. i = Invoke M n then h # cs e else if i = Return then tl (cs e) else cs e\<rparr>"
assumes p'_def: "p' = snd (snd fr')"
assumes check_i: "check_instr' i P h stk loc C M pc frs"
assumes exec_i : "exec_instr i P h stk loc C M pc frs = \<sigma>'"
assumes Pi_def: "\<Pi> = (P,An)"
shows  "\<forall> I. evalE \<Pi> (p,\<sigma>,e\<lparr>lv:=I\<rparr>) (wpF \<Pi> p p' Q) = evalE \<Pi> (p',\<sigma>',e'\<lparr>lv:=I\<rparr>) Q"
(*<*)
proof -
from handlesEx cmd_p i_def 
show ?thesis
apply -
apply (simp only: wpF_Goto)
--{* induction preparation *}
apply (subgoal_tac "\<exists>Q'. Q = Q'")
 prefer 2
 apply fastsimp
apply (erule exE)
apply (subgoal_tac "\<forall>em. substE em Q = substE em Q'")
 prefer 2
 apply simp
apply (subgoal_tac "\<forall> I. evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q = evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q'")
 prefer 2
 apply simp
apply (subgoal_tac "Q' \<in> set (subExpr Q)")
  prefer 2
  apply (simp add: getExpr_refl)
apply (erule_tac V="Q = Q'" in thin_rl)
apply (simp only:)
apply (erule_tac V="\<forall>em.  substE em Q = substE em Q'" in thin_rl)
apply (erule_tac V="\<forall>I. evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q = evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q'" in thin_rl)
apply (erule_tac P="Q' \<in> set (subExpr Q)" in rev_mp)
apply (erule thin_rl)+
apply (cut_tac exec_i[THEN sym] sigma'_def i_def sigma_def check_i p'_def e'_def p_def)
--{* induction on Q' *}
apply simp
apply (rule_tac expr="Q'" in expr_induct)
--{* Rg *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* St *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* Lv *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* Cn *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* NewA *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* Gf list1 list2 expr *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Gf list1 list2 expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def split add: val.splits)
--{* FrNr *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* Num expr1 num_op epxr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Num expr1 num_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Num expr1 num_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Rel expr1 rel_op expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Rel expr1 rel_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Rel expr1 rel_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* IF expr1 THEN expr2 ELSE expr3 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr3 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (drule mp,assumption)+
apply (simp add: split_def fms map_if_notin id_lookup_def evalE_evalEs.simps substE_substEs.simps split del: split_if add: bool.split )

apply (case_tac "evalE \<Pi> ((C, M, nat (int pc + t)), (None, h, (stk, loc, C, M, nat (int pc + t)) # frs), e\<lparr>lv := I\<rparr>) expr1 = \<lfloor>Bool True\<rfloor>")
apply simp
apply simp
--{* Eq expr1 expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Eq expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Eq expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Neg expr *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Neg expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Imp expr1 expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Imp expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Imp expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Forall nat expr *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Forall nata expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def split del: split_if)
--{* Ty expr ty*}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Ty expr ty" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: Let_def split_def fms map_if_notin id_lookup_def split add: val.split)
--{* Pos x *}
apply (rule impI, rule allI)
apply (subgoal_tac "x \<in> set (getPosEx Q)")
 prefer 2
 apply (rule subExpr_getPosEx)
 apply assumption
apply (drule_tac xs="getPosEx Q" in  in_set_conv_decomp_fst)
apply (erule exE |erule conjE)+
apply (subgoal_tac "p' mem domC \<Pi>")
 prefer 2
 apply (cut_tac wf_Pi)
 apply (cut_tac p_domC)
 apply (cut_tac p_def)
 apply (cut_tac cmd_p)
 apply (simp only: in_set_conv_decomp)
 apply (erule exE)+
 apply (simp add: incA_def wf_def checkPos_split succsNormal_def split add: split_if_asm)
apply (cut_tac p_domC)
apply (case_tac "frs")
apply (simp add: fms map_if_notin id_lookup_def p_def mem_iff)

apply (simp add: fms map_if_notin id_lookup_def p_def mem_iff callers_simps)

--{* Call *}
apply (rule impI,rule allI)
apply (simp add: Let_def split_def fms map_if_notin id_lookup_def)
apply (rule impI)
apply (simp only: neq_Nil_conv)
apply (erule exE)+
apply (simp (no_asm_simp) add: split_paired_all env_upd_cs)

--{* Catch  *}
apply (rule impI,rule allI)
apply (simp add: Let_def split_def fms map_if_notin id_lookup_def)
apply (rule impI)
apply (simp only: neq_Nil_conv)
apply (erule exE)+
apply (simp (no_asm_simp) add: split_paired_all env_upd_cs)

--{* And es *}
apply (rule impI, rule allI)
apply (simp add: evalE_And fms map_if_notin id_lookup_def del: evalE_evalEs.simps split del: split_if)
apply (rule iffI)
apply (rule ballI)
apply (erule_tac x="ex" in ballE)
apply (erule_tac x="(substE (map (\<lambda>q. (Pos q, if q = (C, M, nat (int pc + t)) then Pos p else FF)) (getPosEx Q)) ex)" in ballE)
apply (subgoal_tac "ex \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="And es" in subExpr_Gf)
 apply assumption
 apply (simp add: parts.simps)
apply (simp add: substEs_map) 

apply (simp add: substEs_map)

apply (simp only:)

apply (rule ballI)

apply (simp add: substEs_map image_iff)
apply (erule bexE)
apply (erule_tac x="x" in ballE)
apply (erule_tac x="x" in ballE)
apply (subgoal_tac "x \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="And es" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm_simp) add: parts.simps)
apply simp

apply (simp only:)

apply (simp only:)

done
qed
(*>*)


lemma effS_wpF_CmpEq:
assumes i_def: "i = CmpEq"
assumes wf_Pi: "wf \<Pi>"
assumes handlesEx: "handlesEx (fst \<Pi>) p' = None"
assumes cmd_p: "cmd \<Pi> p = Some i"
assumes p_domC: "p \<in> set (domC \<Pi>)"
assumes i_instr: "instrs_of P C M ! pc = i"
assumes s_def: "s = (p,\<sigma>,e)"
assumes p_def: "p = (C,M,pc)"
assumes sigma_def: "\<sigma> = (None,h,(stk,loc,p)#frs)"
assumes s'_def: "s' = (p',\<sigma>',e')"
assumes sigma'_def: "\<sigma>' = (None,h,fr'#frs')"
assumes e'_def: "e' = e\<lparr>cs := if \<exists>M n. i = Invoke M n then h # cs e else if i = Return then tl (cs e) else cs e\<rparr>"
assumes p'_def: "p' = snd (snd fr')"
assumes check_i: "check_instr' i P h stk loc C M pc frs"
assumes exec_i : "exec_instr i P h stk loc C M pc frs = \<sigma>'"
assumes Pi_def: "\<Pi> = (P,An)"
shows  "\<forall> I. evalE \<Pi> (p,\<sigma>,e\<lparr>lv:=I\<rparr>) (wpF \<Pi> p p' Q) = evalE \<Pi> (p',\<sigma>',e'\<lparr>lv:=I\<rparr>) Q"
(*<*)
proof -
from handlesEx cmd_p i_def 
show ?thesis
apply -
apply (simp only: wpF_CmpEq)
--{* induction preparation *}
apply (subgoal_tac "\<exists>Q'. Q = Q'")
 prefer 2
 apply fastsimp
apply (erule exE)
apply (subgoal_tac "\<forall>em. substE em Q = substE em Q'")
 prefer 2
 apply simp
apply (subgoal_tac "\<forall> I. evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q = evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q'")
 prefer 2
 apply simp
apply (subgoal_tac "Q' \<in> set (subExpr Q)")
  prefer 2
  apply (simp add: getExpr_refl)
apply (erule_tac V="Q = Q'" in thin_rl)
apply (simp only:)
apply (erule_tac V="\<forall>em.  substE em Q = substE em Q'" in thin_rl)
apply (erule_tac V="\<forall>I. evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q = evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q'" in thin_rl)
apply (erule_tac P="Q' \<in> set (subExpr Q)" in rev_mp)
apply (erule thin_rl)+
apply (cut_tac exec_i[THEN sym] sigma'_def i_def sigma_def check_i p'_def e'_def p_def)
--{* induction on Q' *}
apply simp
apply (rule_tac expr="Q'" in expr_induct)
--{* Rg *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* St *}
apply (rule impI, rule allI)
apply (drule subExpr_stkIds)
apply (drule_tac xs="stkIds Q" in in_set_conv_decomp_fst)
apply (erule exE | erule conjE)+
apply (case_tac "stk")
apply simp
apply (case_tac "list")
apply simp
apply (simp add: fms map_if_notin id_lookup_def neq_Nil_conv liftI_def numop_def)
apply (rule conjI)
apply (rule impI)
apply (rule iffI)
apply (rule sym, assumption)+
apply (rule impI)+
apply (case_tac "nat")
apply simp

apply simp
--{* Lv *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* Cn *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* NewA *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* Gf list1 list2 expr *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Gf list1 list2 expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def split add: val.splits)
--{* FrNr *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* Num expr1 num_op epxr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Num expr1 num_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Num expr1 num_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Rel expr1 rel_op expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Rel expr1 rel_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Rel expr1 rel_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* IF expr1 THEN expr2 ELSE expr3 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr3 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (drule mp,assumption)+

apply (simp add: split_def fms map_if_notin id_lookup_def evalE_evalEs.simps substE_substEs.simps split del: split_if add: bool.split )
apply (case_tac "evalE \<Pi> ((C, M, Suc pc), (None, h, (Bool (hd (tl stk) = hd stk) # tl (tl stk), loc, C, M, Suc pc) # frs), e\<lparr>lv := I\<rparr>) expr1 = \<lfloor>Bool True\<rfloor>")
apply simp
apply simp
--{* Eq expr1 expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Eq expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Eq expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Neg expr *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Neg expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Imp expr1 expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Imp expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Imp expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Forall nat expr *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Forall nat expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def split del: split_if)
--{* Ty expr ty*}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Ty expr ty" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: Let_def split_def fms map_if_notin id_lookup_def split add: val.split)
--{* Pos x *}
apply (rule impI, rule allI)
apply (subgoal_tac "x \<in> set (getPosEx Q)")
 prefer 2
 apply (rule subExpr_getPosEx)
 apply assumption
apply (drule_tac xs="getPosEx Q" in  in_set_conv_decomp_fst)
apply (erule exE |erule conjE)+
apply (subgoal_tac "p' mem domC \<Pi>")
 prefer 2
 apply (cut_tac wf_Pi)
 apply (cut_tac p_domC)
 apply (cut_tac p_def)
 apply (cut_tac cmd_p)
 apply (simp only: in_set_conv_decomp)
 apply (erule exE)+
 apply (simp add: incA_def wf_def checkPos_split succsNormal_def split add: split_if_asm)
apply (cut_tac p_domC)
apply (case_tac "frs")
apply (simp add: fms map_if_notin id_lookup_def p_def mem_iff)

apply (simp add: fms map_if_notin id_lookup_def p_def mem_iff callers_simps)
--{* Call *}
apply (rule impI,rule allI)
apply (simp add: Let_def split_def fms map_if_notin id_lookup_def)
apply (rule impI)
apply (simp only: neq_Nil_conv)
apply (erule exE)+
apply (simp (no_asm_simp) add: split_paired_all env_upd_cs)

--{* Catch  *}
apply (rule impI,rule allI)
apply (simp add: Let_def split_def fms map_if_notin id_lookup_def)
apply (rule impI)
apply (simp only: neq_Nil_conv)
apply (erule exE)+
apply (simp (no_asm_simp) add: split_paired_all env_upd_cs)

--{* And es *}
apply (rule impI, rule allI)
apply (simp add: evalE_And fms map_if_notin id_lookup_def del: evalE_evalEs.simps split del: split_if)
apply (rule iffI)
apply (rule ballI)
apply (erule_tac x="ex" in ballE)
apply (erule_tac x="(substE (map (\<lambda>q. (Pos q, if q = (C, M, Suc pc) then Pos p else FF)) (getPosEx Q) @
                         map (\<lambda>k. (St k, if k = 0 then St 0 \<doteq> St 1 else St (k + 1))) (stkIds Q))
                  ex)" in ballE)
apply (subgoal_tac "ex \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="And es" in subExpr_Gf)
 apply assumption
 apply (simp add: parts.simps)
apply (simp add: substEs_map) 

apply (simp add: substEs_map)

apply (simp only:)

apply (rule ballI)

apply (simp add: substEs_map image_iff)
apply (erule bexE)
apply (erule_tac x="x" in ballE)
apply (erule_tac x="x" in ballE)
apply (subgoal_tac "x \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="And es" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm_simp) add: parts.simps)
apply simp

apply (simp only:)

apply (simp only:)

done
qed
(*>*)


lemma effS_wpF_IfIntCmp:
assumes i_def: "i = IfIntCmp ro t"
assumes wf_Pi: "wf \<Pi>"
assumes handlesEx: "handlesEx (fst \<Pi>) p' = None"
assumes cmd_p: "cmd \<Pi> p = Some i"
assumes p_domC: "p \<in> set (domC \<Pi>)"
assumes i_instr: "instrs_of P C M ! pc = i"
assumes s_def: "s = (p,\<sigma>,e)"
assumes p_def: "p = (C,M,pc)"
assumes sigma_def: "\<sigma> = (None,h,(stk,loc,p)#frs)"
assumes s'_def: "s' = (p',\<sigma>',e')"
assumes sigma'_def: "\<sigma>' = (None,h,fr'#frs')"
assumes e'_def: "e' = e\<lparr>cs := if \<exists>M n. i = Invoke M n then h # cs e else if i = Return then tl (cs e) else cs e\<rparr>"
assumes p'_def: "p' = snd (snd fr')"
assumes check_i: "check_instr' i P h stk loc C M pc frs"
assumes exec_i : "exec_instr i P h stk loc C M pc frs = \<sigma>'"
assumes Pi_def: "\<Pi> = (P,An)"
shows  "\<forall> I. evalE \<Pi> (p,\<sigma>,e\<lparr>lv:=I\<rparr>) (wpF \<Pi> p p' Q) = evalE \<Pi> (p',\<sigma>',e'\<lparr>lv:=I\<rparr>) Q"
(*<*)
proof -
from handlesEx cmd_p i_def 
show ?thesis
apply -
apply (simp only: wpF_IfIntCmp)
--{* induction preparation *}
apply (subgoal_tac "\<exists>Q'. Q = Q'")
 prefer 2
 apply fastsimp
apply (erule exE)
apply (subgoal_tac "\<forall>em. substE em Q = substE em Q'")
 prefer 2
 apply simp
apply (subgoal_tac "\<forall> I. evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q = evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q'")
 prefer 2
 apply simp
apply (subgoal_tac "Q' \<in> set (subExpr Q)")
  prefer 2
  apply (simp add: getExpr_refl)
apply (erule_tac V="Q = Q'" in thin_rl)
apply (simp only:)
apply (erule_tac V="\<forall>em.  substE em Q = substE em Q'" in thin_rl)
apply (erule_tac V="\<forall>I. evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q = evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q'" in thin_rl)
apply (erule_tac P="Q' \<in> set (subExpr Q)" in rev_mp)
apply (erule thin_rl)+
apply (cut_tac exec_i[THEN sym] sigma'_def i_def sigma_def check_i p'_def e'_def p_def)
--{* induction on Q' *}
apply (rule_tac expr="Q'" in expr_induct)
--{* Rg *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* St *}
apply (rule impI)
apply (drule subExpr_stkIds)
apply (drule_tac xs="stkIds Q" in in_set_conv_decomp_fst)
apply (erule exE | erule conjE)+
apply (case_tac "stk")
apply simp

apply (case_tac "list")
apply simp

apply (simp add: fms map_if_notin id_lookup_def)

--{* Lv *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* Cn *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* NewA *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* Gf list1 list2 expr *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Gf list1 list2 expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def split add: val.splits)
--{* FrNr *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* Num expr1 num_op epxr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Num expr1 num_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Num expr1 num_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Rel expr1 rel_op expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Rel expr1 rel_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Rel expr1 rel_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* IF expr1 THEN expr2 ELSE expr3 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr3 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (drule mp,assumption)+

apply (simp add: split_def fms map_if_notin id_lookup_def evalE_evalEs.simps substE_substEs.simps split del: split_if add: bool.split )
apply (rule allI)
apply (case_tac "evalE \<Pi> ((C, M, if relop ro (the_Intg (hd stk)) (the_Intg (hd (tl stk))) then nat (int pc + t) else pc + 1),
                           (None, h, (tl (tl stk), loc, C, M, if relop ro (the_Intg (hd stk)) (the_Intg (hd (tl stk))) then nat (int pc + t) else pc + 1) # frs), e
                           \<lparr>lv := I\<rparr>)
                   expr1 =
                  \<lfloor>Bool True\<rfloor>")
apply simp
apply simp
--{* Eq expr1 expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Eq expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Eq expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Neg expr *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Neg expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Imp expr1 expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Imp expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Imp expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Forall nat expr *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Forall nat expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def split del: split_if)
--{* Ty expr ty*}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Ty expr ty" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: Let_def split_def fms map_if_notin id_lookup_def split add: val.split)
--{* Pos x *}
apply (rule impI, rule allI)
apply (subgoal_tac "x \<in> set (getPosEx Q)")
 prefer 2
 apply (rule subExpr_getPosEx)
 apply assumption
apply (drule_tac xs="getPosEx Q" in  in_set_conv_decomp_fst)
apply (erule exE |erule conjE)+
apply (subgoal_tac "p' mem domC \<Pi>")
 prefer 2
 apply (cut_tac wf_Pi)
 apply (cut_tac p_domC)
 apply (cut_tac p_def)
 apply (cut_tac cmd_p)
 apply (simp only: in_set_conv_decomp)
 apply (erule exE)+
 apply (simp add: incA_def wf_def checkPos_split succsNormal_def split add: split_if_asm)
apply (cut_tac p_domC)
apply (case_tac "frs")
apply (simp add: fms map_if_notin id_lookup_def p_def mem_iff)

apply (simp add: fms map_if_notin id_lookup_def p_def mem_iff callers_simps)
--{* Call *}
apply (rule impI,rule allI)
apply (case_tac "stk")
apply simp
apply (case_tac "frs")
apply (simp add: Let_def split_def fms map_if_notin id_lookup_def env_upd)

apply (simp add: split_paired_all Let_def split_def fms map_if_notin id_lookup_def env_upd)
--{* Catch  *}
apply (rule impI,rule allI)
apply (case_tac "stk")
apply simp
apply (case_tac "frs")
apply (simp add: Let_def split_def fms map_if_notin id_lookup_def env_upd)

apply (simp add: split_paired_all Let_def split_def fms map_if_notin id_lookup_def env_upd)

--{* And es *}
apply (rule impI, rule allI)
apply (simp add: evalE_And fms map_if_notin id_lookup_def del: evalE_evalEs.simps split del: split_if)
apply (rule iffI)
apply (rule ballI)
apply (erule_tac x="ex" in ballE)
apply (erule_tac x="(substE (map (\<lambda>q. (Pos q, if q = (C, M, if relop ro (the_Intg (hd (tl stk))) (the_Intg (hd stk)) then nat (int pc + t) else pc + 1) then Pos p else FF))
                          (getPosEx Q) @
                         map (\<lambda>k. (St k, St (Suc (Suc k)))) (stkIds Q))
                  ex)" in ballE)
apply (subgoal_tac "ex \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="And es" in subExpr_Gf)
 apply assumption
 apply (simp add: parts.simps)
apply (simp add: substEs_map) 

apply (simp add: substEs_map)

apply (simp only:)

apply (rule ballI)

apply (simp add: substEs_map image_iff)
apply (erule bexE)
apply (erule_tac x="x" in ballE)
apply (erule_tac x="x" in ballE)
apply (subgoal_tac "x \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="And es" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm_simp) add: parts.simps)
apply simp

apply (simp only:)

apply (simp only:)
done
qed
(*>*)


lemma effS_wpF_IfFalse:
assumes i_def: "i = IfFalse t"
assumes wf_Pi: "wf \<Pi>"
assumes handlesEx: "handlesEx (fst \<Pi>) p' = None"
assumes cmd_p: "cmd \<Pi> p = Some i"
assumes p_domC: "p \<in> set (domC \<Pi>)"
assumes i_instr: "instrs_of P C M ! pc = i"
assumes s_def: "s = (p,\<sigma>,e)"
assumes p_def: "p = (C,M,pc)"
assumes sigma_def: "\<sigma> = (None,h,(stk,loc,p)#frs)"
assumes s'_def: "s' = (p',\<sigma>',e')"
assumes sigma'_def: "\<sigma>' = (None,h,fr'#frs')"
assumes e'_def: "e' = e\<lparr>cs := if \<exists>M n. i = Invoke M n then h # cs e else if i = Return then tl (cs e) else cs e\<rparr>"
assumes p'_def: "p' = snd (snd fr')"
assumes check_i: "check_instr' i P h stk loc C M pc frs"
assumes exec_i : "exec_instr i P h stk loc C M pc frs = \<sigma>'"
assumes Pi_def: "\<Pi> = (P,An)"
shows  "\<forall> I. evalE \<Pi> (p,\<sigma>,e\<lparr>lv:=I\<rparr>) (wpF \<Pi> p p' Q) = evalE \<Pi> (p',\<sigma>',e'\<lparr>lv:=I\<rparr>) Q"
(*<*)
proof -
from handlesEx cmd_p i_def 
show ?thesis
apply -
apply (simp only: wpF_IfFalse)
--{* induction preparation *}
apply (subgoal_tac "\<exists>Q'. Q = Q'")
 prefer 2
 apply fastsimp
apply (erule exE)
apply (subgoal_tac "\<forall>em. substE em Q = substE em Q'")
 prefer 2
 apply simp
apply (subgoal_tac "\<forall> I. evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q = evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q'")
 prefer 2
 apply simp
apply (subgoal_tac "Q' \<in> set (subExpr Q)")
  prefer 2
  apply (simp add: getExpr_refl)
apply (erule_tac V="Q = Q'" in thin_rl)
apply (simp only:)
apply (erule_tac V="\<forall>em.  substE em Q = substE em Q'" in thin_rl)
apply (erule_tac V="\<forall>I. evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q = evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q'" in thin_rl)
apply (erule_tac P="Q' \<in> set (subExpr Q)" in rev_mp)
apply (erule thin_rl)+
apply (cut_tac exec_i[THEN sym] sigma'_def i_def sigma_def check_i p'_def e'_def p_def)
--{* induction on Q' *}
apply (rule_tac expr="Q'" in expr_induct)
--{* Rg *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* St *}
apply (rule impI)
apply (drule subExpr_stkIds)
apply (drule_tac xs="stkIds Q" in in_set_conv_decomp_fst)
apply (erule exE | erule conjE)+
apply (case_tac "stk")
apply simp

apply (simp add: fms map_if_notin id_lookup_def)
--{* Lv *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* Cn *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* NewA *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* Gf list1 list2 expr *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Gf list1 list2 expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def split add: val.splits)
--{* FrNr *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* Num expr1 num_op epxr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Num expr1 num_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Num expr1 num_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Rel expr1 rel_op expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Rel expr1 rel_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Rel expr1 rel_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* IF expr1 THEN expr2 ELSE expr3 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr3 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (drule mp,assumption)+
apply (rule allI)
apply (simp add: split_def fms map_if_notin id_lookup_def evalE_evalEs.simps substE_substEs.simps split del: split_if add: bool.split )
apply (case_tac "evalE \<Pi> ((C, M, if hd stk = Bool False then nat (int pc + t) else pc + 1),
                       (None, h, (tl stk, loc, C, M, if hd stk = Bool False then nat (int pc + t) else pc + 1) # frs), e\<lparr>lv := I\<rparr>)
               expr1 =
              \<lfloor>Bool True\<rfloor>")
apply simp
apply simp
--{* Eq expr1 expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Eq expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Eq expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Neg expr *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Neg expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Imp expr1 expr2 *}
apply (rule impI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Imp expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Imp expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Forall nat expr *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Forall nat expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def split del: split_if)
--{* Ty expr ty*}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Ty expr ty" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: Let_def split_def fms map_if_notin id_lookup_def split add: val.split)
--{* Pos x *}
apply (rule impI, rule allI)
apply (subgoal_tac "x \<in> set (getPosEx Q)")
 prefer 2
 apply (rule subExpr_getPosEx)
 apply assumption
apply (drule_tac xs="getPosEx Q" in  in_set_conv_decomp_fst)
apply (erule exE |erule conjE)+
apply (subgoal_tac "p' mem domC \<Pi>")
 prefer 2
 apply (cut_tac wf_Pi)
 apply (cut_tac p_domC)
 apply (cut_tac p_def)
 apply (cut_tac cmd_p)
 apply (simp only: in_set_conv_decomp)
 apply (erule exE)+
 apply (simp add: incA_def wf_def checkPos_split succsNormal_def split add: split_if_asm)
apply (cut_tac p_domC)
apply (case_tac "frs")
apply (simp add: fms map_if_notin id_lookup_def p_def mem_iff)

apply (simp add: fms map_if_notin id_lookup_def p_def mem_iff callers_simps)
--{* Call *}
apply (rule impI,rule allI)
apply (case_tac "stk")
apply simp
apply (case_tac "frs")
apply (simp add: Let_def split_def fms map_if_notin id_lookup_def env_upd)

apply (simp add: split_paired_all Let_def split_def fms map_if_notin id_lookup_def env_upd)
--{* Catch  *}
apply (rule impI,rule allI)
apply (case_tac "stk")
apply simp
apply (case_tac "frs")
apply (simp add: Let_def split_def fms map_if_notin id_lookup_def env_upd)

apply (simp add: split_paired_all Let_def split_def fms map_if_notin id_lookup_def env_upd)

--{* And es *}
apply (rule impI, rule allI)
apply (simp add: evalE_And fms map_if_notin id_lookup_def del: evalE_evalEs.simps split del: split_if)
apply (rule iffI)
apply (rule ballI)
apply (erule_tac x="ex" in ballE)
apply (erule_tac x="(substE (map (\<lambda>q. (Pos q, if q = (C, M, if hd stk = Bool False then nat (int pc + t) else pc + 1) then Pos p else FF))
                          (getPosEx Q) @
                         map (\<lambda>k. (St k, St (Suc k))) (stkIds Q))
                  ex)" in ballE)
apply (subgoal_tac "ex \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="And es" in subExpr_Gf)
 apply assumption
 apply (simp add: parts.simps)
apply (simp add: substEs_map) 

apply (simp add: substEs_map)

apply (simp only:)

apply (rule ballI)

apply (simp add: substEs_map image_iff)
apply (erule bexE)
apply (erule_tac x="x" in ballE)
apply (erule_tac x="x" in ballE)
apply (subgoal_tac "x \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="And es" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm_simp) add: parts.simps)
apply simp

apply (simp only:)

apply (simp only:)
done
qed
(*>*)

lemma effS_wpF_Except:
assumes wf_Pi: "wf \<Pi>"
assumes sys_xptn_inv: "\<forall> C \<in> sys_xcpts. (\<exists>ob. (h (addr_of_sys_xcpt C) = \<lfloor>ob\<rfloor> \<and> obj_ty ob = (Class C)))"
assumes handlesEx: "handlesEx (fst \<Pi>) p' = Some cn"
assumes xa_sub_cn: "P \<turnstile> (cname_of h xa)  \<preceq>\<^sup>* cn"
(* assumes cn_noObject: "cn \<noteq> Object" *)
assumes cmd_p: "cmd \<Pi> p = Some i"
assumes p_domC: "p \<in> set (domC \<Pi>)"
assumes i_instr: "instrs_of P C M ! pc = i"
assumes s_def: "s = (p,\<sigma>,e)"
assumes p_def: "p = (C,M,pc)"
assumes sigma_def: "\<sigma> = (None,h,(stk,loc,p)#frs)"
assumes s'_def: "s' = (p',\<sigma>',e')"
assumes check_i: "check_instr' i P h stk loc C M pc frs"
assumes exec_i : "exec_instr i P h stk loc C M pc frs = (\<lfloor>xa\<rfloor>, h', frs'')"
assumes findhandler_s: "find_handler P xa h ((stk, loc, p) # frs) = \<sigma>'"
assumes sigma'_def: "\<sigma>' = (None, h, ([Addr xa], loc', p') # frs')"
assumes e'_def: "e' =  e\<lparr>cs := drop (length frs - length frs') (cs e)\<rparr>"
assumes Pi_def: "\<Pi> = (P,An)"
assumes Pos_p: "\<Pi>,(p,\<sigma>,e) \<Turnstile> Pos p"
shows  "\<forall> I. evalE \<Pi> (p,\<sigma>,e\<lparr>lv:=I\<rparr>) (wpF \<Pi> p p' Q) = evalE \<Pi> (p',\<sigma>',e'\<lparr>lv:=I\<rparr>) Q"
(*<*)
proof -
(*
 from xa_sub_cn cn_noObject wf_Pi Pi_def
 have xa_cn_sysxcpt: "(cname_of h xa) \<in> sys_xcpts \<Longrightarrow> cn = (cname_of h xa)"
   apply -
   apply (rule sysxpct_no_object)
   apply simp+
   done
*)
show ?thesis
proof (cases "match_ex_table P (cname_of h xa) pc (ex_table_of P C M)")
case None
from this xa_sub_cn have JBC_VCG_match_cn_None: "match_ex_table P cn pc (ex_table_of P C M) = None"
  apply (rule_tac cn="cname_of h xa" in  match_ex_table_subtype_None)
  apply assumption+
  done
from this have JVMException_match_cn_None: "JVMExceptions.match_ex_table P cn pc (ex_table_of P C M) = None"
  by (simp add: match_ex_table_def)
from None have find_handler_eq: "find_handler P xa h frs = find_handler P xa h ((stk, loc, C, M, pc)#frs)"
  by simp
obtain C' M' pc' where p'_def: "p' = (C',M',pc')"
 apply atomize
 apply (erule_tac x="fst p'" in allE)
 apply (erule_tac x="fst (snd p')" in allE)
 apply (erule_tac x="snd (snd p')" in allE)
 apply simp
 done
from findhandler_s find_handler_eq sigma'_def p_def 
obtain pfx where frs_pfx_frs': "frs = pfx @ frs' \<and> pfx \<noteq> []"
  apply -
  apply (simp only:)
  apply (drule_tac frs="frs" in find_handler_frs)
  apply fastsimp
  done
from this have frs_not_Nil: "frs \<noteq> []"
  by simp
from wf_Pi Pi_def p'_def findhandler_s s_def s'_def sigma'_def p_def handlesEx frs_pfx_frs'
have catchstate_simp: "\<forall> e. (\<exists>h' stk' pc''.
            catchstate (P, cn, (C, M, pc), (None, h, (stk, loc, C, M, pc) # frs), e) =
            ((C', M', pc''), (None, h', (stk', loc', C', M', pc'') # frs'), e
            \<lparr>cs := drop (length frs - length frs') (cs e)\<rparr>) \<and>
            match_ex_table P (cname_of h xa) pc'' (ex_table_of P C' M') = \<lfloor>pc'\<rfloor>)"
  apply -
  apply (rule allI)
  apply (simp only:)
  apply (drule_tac st'="stk" and stk="stk" and e="ea" in find_handler_catchstate)
  apply assumption+
  apply (simp only: fst_conv)
  apply simp
  done

from None have JVMExceptions_None: "JVMExceptions.match_ex_table P (cname_of h xa) pc (ex_table_of P C M) = None"
  by (simp add: match_ex_table_def)


from cmd_p handlesEx
show ?thesis
apply -
apply (simp only: wpF_Except)
--{* induction preparation *}
apply (subgoal_tac "\<exists>Q'. Q = Q'")
 prefer 2
 apply fastsimp
apply (erule exE)
apply (subgoal_tac "\<forall>em. substE em Q = substE em Q'")
 prefer 2
 apply simp
apply (subgoal_tac "\<forall> I. evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q = evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q'")
 prefer 2
 apply simp
apply (subgoal_tac "Q' \<in> set (subExpr Q)")
  prefer 2
  apply (simp add: getExpr_refl)
apply (erule_tac V="Q = Q'" in thin_rl)
apply (simp only:)
apply (erule_tac V="\<forall>em.  substE em Q = substE em Q'" in thin_rl)
apply (erule_tac V="\<forall>I. evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q = evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q'" in thin_rl)
apply (erule_tac P="Q' \<in> set (subExpr Q)" in rev_mp)
apply (erule thin_rl)+
--{* induction on Q' *}
apply (cut_tac catchstate_simp JBC_VCG_match_cn_None Pi_def p_def sigma_def sigma'_def frs_not_Nil)
apply (rule_tac expr="Q'" in expr_induct)
--{* Rg *}
apply (rule impI, rule allI)
apply (drule subExpr_rgIds)
apply (drule_tac xs="rgIds Q" in in_set_conv_decomp_fst)
apply (erule_tac x="e\<lparr>lv := I\<rparr>" in allE)
apply (erule exE | erule conjE)+
apply (simp add: fms map_if_notin id_lookup_def del: match_ex_table_def)

--{* St nat *}
apply (rule impI, rule allI)
apply (drule subExpr_stkIds)
apply (drule_tac xs="stkIds Q" in in_set_conv_decomp_fst)
apply (erule_tac x="e\<lparr>lv := I\<rparr>" in allE)
apply (erule exE | erule conjE)+
apply (simp add: fms map_if_notin id_lookup_def)
apply (case_tac "i=Throw")
apply (cut_tac "check_i")
apply (simp add: neq_Nil_conv)
apply (erule exE | erule conjE)+
apply (case_tac "hd stk = Null")
apply (cut_tac s_def)
apply (simp only:)
apply (subgoal_tac "\<exists>ob. (h (addr_of_sys_xcpt NullPointer) = \<lfloor>ob\<rfloor> \<and> obj_ty ob = (Class NullPointer))")
 prefer 2
 apply (cut_tac "sys_xptn_inv")
 apply (erule_tac x="NullPointer" in ballE)
  prefer 2
  apply (simp add: NullPointer_def sys_xcpts_def)
 apply assumption
apply (erule exE | erule conjE)+
apply (cut_tac exec_i[THEN sym])
apply (simp add: is_Ref'_def is_Addr'_conv none_def split del: option.split_asm)
--{* hd stk ~= Null *}
apply (cut_tac exec_i[THEN sym])
apply (simp add: is_Ref_def is_Ref'_def is_Addr'_conv split del: split_if split_if_asm option.split option.split_asm)
apply (case_tac "y")
apply simp

apply simp

apply simp

apply simp

apply (simp add: none_def)
--{* i ~= Throw *}
apply simp
apply (cut_tac exec_i[THEN sym] check_i)
apply (case_tac "i")
--{* Load *}
apply simp
--{* Store *}
apply simp
--{* Push *}
apply simp
--{* New *}
apply (cut_tac s_def)
apply simp
apply (subgoal_tac "\<exists>ob. (h (addr_of_sys_xcpt OutOfMemory) = \<lfloor>ob\<rfloor> \<and> obj_ty ob = (Class OutOfMemory))")
 prefer 2
 apply (cut_tac "sys_xptn_inv")
 apply (erule_tac x="OutOfMemory" in ballE)
  prefer 2
  apply (simp add: OutOfMemory_def sys_xcpts_def)
 apply assumption
apply (erule exE)
apply (simp add: sys_xcpt_of_def obj_ty_def none_def)

--{* Getfield *}
apply simp
apply (case_tac "hd stk = Null")
apply (simp add: split_def)
apply (subgoal_tac "\<exists>ob. (h (addr_of_sys_xcpt NullPointer) = \<lfloor>ob\<rfloor> \<and> obj_ty ob = (Class NullPointer))")
 prefer 2
 apply (cut_tac "sys_xptn_inv")
 apply (erule_tac x="NullPointer" in ballE)
  prefer 2
  apply (simp add: NullPointer_def sys_xcpts_def)
 apply assumption
apply (erule exE)
apply (simp add: sys_xcpt_of_def obj_ty_def none_def)

apply (simp add: split_def)

--{* Putfield *}
apply simp
apply (case_tac "hd (tl stk) = Null")
apply (simp add: split_def)
apply (subgoal_tac "\<exists>ob. (h (addr_of_sys_xcpt NullPointer) = \<lfloor>ob\<rfloor> \<and> obj_ty ob = (Class NullPointer))")
 prefer 2
 apply (cut_tac "sys_xptn_inv")
 apply (erule_tac x="NullPointer" in ballE)
  prefer 2
  apply (simp add: NullPointer_def sys_xcpts_def)
 apply assumption
apply (erule exE)
apply (simp add: sys_xcpt_of_def obj_ty_def none_def)

apply (simp add: split_def none_def)
--{* Checkcast *}
apply simp
apply (case_tac "cast_ok P list h (hd stk)")
apply (simp add: split_def)
apply (subgoal_tac "\<exists>ob. (h (addr_of_sys_xcpt ClassCast) = \<lfloor>ob\<rfloor> \<and> obj_ty ob = (Class ClassCast))")
 prefer 2
 apply (cut_tac "sys_xptn_inv")
 apply (erule_tac x="ClassCast" in ballE)
  prefer 2
  apply (simp add: ClassCast_def sys_xcpts_def)
 apply assumption
apply (erule exE)
apply (simp add: sys_xcpt_of_def obj_ty_def none_def)

--{* Invoke *}
apply simp
apply (case_tac " stk ! nata = Null")
apply (simp add: split_def)
apply (subgoal_tac "\<exists>ob. (h (addr_of_sys_xcpt NullPointer) = \<lfloor>ob\<rfloor> \<and> obj_ty ob = (Class NullPointer))")
 prefer 2
 apply (cut_tac "sys_xptn_inv")
 apply (erule_tac x="NullPointer" in ballE)
  prefer 2
  apply (simp add: NullPointer_def sys_xcpts_def)
 apply assumption
apply (erule exE)
apply (simp add: sys_xcpt_of_def obj_ty_def none_def)

apply (simp add: split_def)
--{* Return *}
apply (case_tac "frs")
apply simp
apply (simp add: split_def)
--{* Pop *}
apply simp
--{* IBin *}
apply simp
--{* Goto *}
apply simp
--{* CmpEq *}
apply simp
--{* IfIntLeq *}
apply simp
--{* IfFalse *}
apply simp
--{* Throw *}
apply simp

--{* Lv nat *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* Cn nat *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* NewA *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* Gf list1 list2 expr *}
apply (rule impI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Gf list1 list2 expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def split add: val.split)
--{* FrNr *}
apply (rule impI, rule allI)
apply (erule_tac x="e\<lparr>lv:=I\<rparr>" in allE)
apply (erule exE)+
apply (simp add: fms map_if_notin id_lookup_def)
--{* Num expr1 num_op epxr2 *}
apply (rule impI, rule allI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Num expr1 num_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Num expr1 num_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Rel expr1 rel_op expr2 *}
apply (rule impI, rule allI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Rel expr1 rel_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Rel expr1 rel_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* IF expr1 THEN expr2 ELSE expr3 *}
apply (rule impI, rule allI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr3 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (drule mp,assumption)+

apply (simp add: split_def fms map_if_notin id_lookup_def evalE_evalEs.simps substE_substEs.simps split del: split_if add: bool.split )
apply (case_tac "evalE (P, An) (p', (None, h, ([Addr xa], loc', p') # frs'), e'\<lparr>lv := I\<rparr>) expr1 = \<lfloor>Bool True\<rfloor>")
apply simp
apply simp
--{* Eq expr1 expr2 *}
apply (rule impI, rule allI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Eq expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Eq expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Neg expr *}
apply (rule impI, rule allI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Neg expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Imp expr1 expr2 *}
apply (rule impI, rule allI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Imp expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Imp expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Forall nat expr *}
apply (rule impI, rule allI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Forall nat expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def split del: split_if)
--{* Ty expr ty*}
apply (rule impI, rule allI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Ty expr ty" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: Let_def split_def fms map_if_notin id_lookup_def split add: val.split)
--{* Pos x *}
apply (rule impI, rule allI)
apply (subgoal_tac "x \<in> set (getPosEx Q)")
 prefer 2
 apply (rule subExpr_getPosEx)
 apply assumption
apply (drule_tac xs="getPosEx Q" in  in_set_conv_decomp_fst)
apply (erule exE |erule conjE)+
apply (subgoal_tac "p' mem domC \<Pi>")
 prefer 2
 apply (cut_tac wf_Pi)
 apply (cut_tac p_domC)
 apply (cut_tac p_def)
 apply (cut_tac p'_def)
 apply (subgoal_tac "(C',M') \<in> set (methodnames P)")
  prefer 2
  apply (cut_tac findhandler_s)
  apply (simp only:)
  apply (drule find_handler_frs')
  apply (erule exE | erule conjE)+
  apply (simp add: Let_def fst_conv snd_conv)
  apply (rule_tac xs="pfx" in rev_cases)
  apply simp

  apply (simp add: split_def)
  apply (cut_tac Pos_p)
  apply (simp add: callers_sysinv_trans del: callers_sysinv.simps)
  apply (case_tac "ysa")
  apply simp
  apply (erule conjE)+
  apply (rule_tac pc="snd (snd (snd (snd y)))" and An="An" in domC_methodnames)
  apply simp

  --{* ysa = a list *}
  apply simp

  apply (case_tac "list")
  apply simp
  apply (erule conjE)+
  apply (rule_tac pc="snd (snd (snd (snd y)))" and An="An" in domC_methodnames)
  apply simp
  
  --{* list = aa lista *}
  apply (erule conjE)+
  apply (erule_tac x="length list - 1" in allE)
  apply simp
  apply (erule conjE)+
  apply (rule_tac pc="snd (snd (snd (snd y)))" and An="An" in domC_methodnames)
  apply simp
 apply (erule_tac x="e" in allE)
 apply (erule conjE | erule exE)+
 apply (subgoal_tac "(C',M',pc') \<in> set (domC (P,An))")
  prefer 2
  apply (rule_tac X="cname_of h xa" in wf_ex_table_domC)
  apply (cut_tac "wf_Pi")
  apply (simp add: Pi_def)

  apply assumption

  apply simp
 apply (simp add: Pi_def mem_iff)

apply (cut_tac p_domC)
apply (cut_tac Pos_p)
apply (cut_tac findhandler_s)
apply (simp only:)
apply (drule find_handler_frs')
apply (erule exE | erule conjE)+
apply (simp add: fms map_if_notin id_lookup_def p_def mem_iff callers_sysinv_trans del: callers_sysinv.simps)
apply (case_tac "frs'")
apply (simp add: not_Nil_case)
apply (rule impI)
apply (rule_tac xs="pfx" in rev_cases)
apply simp

apply (rule iffI)
apply (erule_tac x="length pfx - 1" in allE)+
apply (simp add: length_append last_append)

apply (rule allI)
apply (rule conjI)
apply (rule impI)
apply (erule_tac x="i" in allE)
apply (drule_tac s="Suc i" and t="length pfx" in sym)
apply (simp add: nth_append last_append)

apply (rule impI)+
apply (erule_tac x="i" in allE)
apply simp

--{* frs' = a list *}
apply (subgoal_tac "\<exists> Cp Mp pcp. last pfx = (Cp,Mp,pcp)")
 prefer 2
 apply (erule thin_rl)+
 apply (rule_tac x="fst (last pfx)" in exI)
 apply (rule_tac x="fst (snd (last pfx))" in exI)
 apply (rule_tac x="snd (snd (last (pfx)))" in exI)
 apply simp
apply (erule exE)+
apply (cut_tac p'_def)
apply simp
apply (drule_tac t="a # list" and s="frs'" in sym)
apply (simp only:)
apply (subgoal_tac "pfx @ frs' \<noteq> []")
 prefer 2
 apply simp
apply (simp add: not_Nil_case)
apply (rule impI)
apply (erule conjE | erule exE)+
apply (subgoal_tac "snd (snd a) \<in> set (callers (P, An) p')")
 prefer 2
 apply (rule_tac xs="pfx" in rev_cases)
 apply simp
 apply (erule_tac x="length pfx - 2" in allE)
 apply simp
 apply (subgoal_tac "length ysa - Suc 0 < Suc (length ysa + length list)")
  prefer 2
  apply (erule thin_rl)+
  apply arith
 apply simp
 apply (case_tac "length ysa - Suc 0 = length ysa + length list")
 apply simp
 apply (drule_tac t="frs'" and s="a # list" in sym)
 apply (case_tac "ysa")
 apply simp
 apply (case_tac "pfx")
 apply simp
 apply (subgoal_tac "callers \<Pi> (C,M,pc) = callers \<Pi> (C,M,pc')")
  prefer 2
  apply (rule callers_eq)
 apply (erule conjE)+
 apply (drule_tac t="pcp" in sym)
 apply simp

 apply simp

 apply simp
 apply (case_tac "ysa")
 apply simp
 apply (subgoal_tac "callers \<Pi> (C,M,pc) = callers \<Pi> (C,M,pc')")
  prefer 2
  apply (rule callers_eq)
 apply (erule conjE)+
 apply (drule_tac t="pcp" in sym)
 apply (drule_tac t="frs'" in sym)
 apply simp
 --{* ysa = aa lista *}
 apply simp
 apply (drule_tac t="frs'" in sym)
 apply (simp add: nth_append)
 apply (erule conjE)+
 apply (subgoal_tac "callers \<Pi> (fst pcp,fst (snd pcp),pc') = callers \<Pi> (fst pcp,fst (snd pcp),snd (snd pcp))")
  prefer 2
  apply (rule callers_eq)
 apply simp

apply (cut_tac p'_def)
apply simp
apply (rule allI)
apply (rule conjI)
apply (rule impI)

apply (erule_tac x="length pfx + length list - 1" in allE)
apply simp

apply (rule impI)+
apply (erule_tac x="length pfx + i - 1" in allE)
apply (case_tac "pfx")
apply simp
apply (subgoal_tac "length pfx + i - 1 < length pfx + length list")
 prefer 2
 apply simp

apply simp
apply (drule_tac t="frs'" in sym)
apply (simp add: nth_append)
--{* Call *}
apply (rule impI, rule allI)
apply (drule subExpr_getCallEx)
apply (drule_tac xs="getCallEx Q" in in_set_conv_decomp_fst)
apply (erule exE | erule conjE)+
apply (erule_tac x="e\<lparr>lv:=I\<rparr>" in allE)
apply (erule exE)+
apply (simp add: fms id_lookup_def)
apply (rule impI)
apply (simp only: neq_Nil_conv)
apply (erule exE)+
apply (cut_tac e'_def)
apply (simp (no_asm_simp) add: split_paired_all)

--{* Catch  *}
apply (rule impI,rule allI)
apply (drule subExpr_getCatchEx)
apply (drule_tac xs="getCatchEx Q" in in_set_conv_decomp_fst)
apply (erule exE | erule conjE)+
apply (erule_tac x="e\<lparr>lv:=I\<rparr>" in allE)
apply (erule exE)+
apply (simp add: fms id_lookup_def)

apply (rule impI)
apply (simp only: neq_Nil_conv)
apply (erule exE) +
apply (subgoal_tac " (catchstate
             (P, list, (C', M', pc''), (None, h', (stk', loc', C', M', pc'') # frs'), e
              \<lparr>lv:= I, cs := drop (length frs - length frs') (cs e)\<rparr>)) = 
                     (catchstate
             (P, list, (C', M', pc'), (None, h, ([Addr xa], loc', C', M', pc') # frs'), e
              \<lparr>cs := drop (length frs - length frs') (cs e),lv:= I\<rparr>))")
 prefer 2
 apply (simp only:)
 apply (erule thin_rl)+
 apply (cut_tac catchstate_eq)
 apply fastsimp
apply (cut_tac p'_def e'_def)
apply simp
--{* And es *}
apply (rule impI, rule allI)
apply (simp add: evalE_And fms map_if_notin id_lookup_def del: evalE_evalEs.simps split del: split_if)
apply (rule iffI)
apply (rule ballI)
apply (erule_tac x="ex" in ballE)
apply (erule_tac x="(substE (map (\<lambda>q. (Pos q, if q = p' then Pos p else FF)) (getPosEx Q) @
                       map (\<lambda>k. (St k, if Suc 0 \<le> k then none
                                       else if i = Throw
                                            then IF St 0 \<doteq> Cn Null THEN Cn (Addr (addr_of_sys_xcpt NullPointer)) ELSE St 0
                                            else Cn (Addr (addr_of_sys_xcpt  (sys_xcpt_of i)))))
                        (stkIds Q) @
                       (FrNr, Catch cn FrNr) #
                       map (\<lambda>k. (Rg k, Catch cn (Rg k))) (rgIds Q) @
                       map (\<lambda>ex. (Call ex, Catch cn (Call ex))) (getCallEx Q) @
                       map (\<lambda>p. (Catch (fst p) (snd p), Catch cn (Catch (fst p) (snd p)))) (getCatchEx Q))
                ex)" in ballE)
apply (subgoal_tac "ex \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="And es" in subExpr_Gf)
 apply assumption
 apply (simp add: parts.simps)
apply (simp add: substEs_map) 

apply (simp add: substEs_map)

apply (simp only:)

apply (rule ballI)

apply (simp add: substEs_map image_iff)
apply (erule bexE)
apply (erule_tac x="x" in ballE)
apply (erule_tac x="x" in ballE)
apply (subgoal_tac "x \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="And es" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm_simp) add: parts.simps)
apply simp

apply (simp only:)

apply (simp only:)
done

next

case (Some a)
obtain C' M' pc' where p'_def:"p' = (C',M',pc')"
 apply atomize
 apply (erule_tac x="fst p'" in allE)
 apply (erule_tac x="fst (snd p')" in allE)
 apply (erule_tac x="snd (snd p')" in allE)
 apply simp
 done
from Some wf_Pi Pi_def p_def findhandler_s s_def sigma'_def handlesEx p'_def  p'_def obtain d 
where JVMException_match_xa_Some: "JVMExceptions.match_ex_table P (cname_of h xa) pc (ex_table_of P C M) = \<lfloor>(pc',d)\<rfloor>"
  apply atomize
  apply (simp add: JBC_VCG.match_ex_table_def)
  apply fastsimp
  done
from Some wf_Pi Pi_def p_domC p_def findhandler_s s_def sigma'_def handlesEx p'_def
have JBC_VCG_match_cn_Some: "match_ex_table P cn pc (ex_table_of P C M) = \<lfloor>pc'\<rfloor>"
 apply -
 apply (rule_tac cn'="(cname_of h xa)" and An="An" in handlesEx_match_ex_table)
 apply simp
 apply (rule_tac pc="pc" and An="An" in domC_methodnames, simp)
 apply simp
 apply simp
 done
from cmd_p handlesEx
show ?thesis
apply -
apply (simp only: wpF_Except)
--{* induction preparation *}
apply (subgoal_tac "\<exists>Q'. Q = Q'")
 prefer 2
 apply fastsimp
apply (erule exE)
apply (subgoal_tac "\<forall>em. substE em Q = substE em Q'")
 prefer 2
 apply simp
apply (subgoal_tac "\<forall> I. evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q = evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q'")
 prefer 2
 apply simp
apply (subgoal_tac "Q' \<in> set (subExpr Q)")
  prefer 2
  apply (simp add: getExpr_refl)
apply (erule_tac V="Q = Q'" in thin_rl)
apply (simp only:)
apply (erule_tac V="\<forall>em.  substE em Q = substE em Q'" in thin_rl)
apply (erule_tac V="\<forall>I. evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q = evalE \<Pi> (p', \<sigma>', e'\<lparr>lv:=I\<rparr>) Q'" in thin_rl)
apply (erule_tac P="Q' \<in> set (subExpr Q)" in rev_mp)
apply (erule thin_rl)+
--{* induction on Q' *}
apply (cut_tac JVMException_match_xa_Some JBC_VCG_match_cn_Some Pi_def p_def sigma_def sigma'_def p'_def findhandler_s)
apply (rule_tac expr="Q'" in expr_induct)
--{* Rg *}
apply (rule impI, rule allI)
apply (simp only: split_def Let_def fst_conv snd_conv simp_thms if_True)
apply (drule subExpr_rgIds)
apply (drule_tac xs="rgIds Q" in in_set_conv_decomp_fst)
apply (erule exE | erule conjE)+
apply (simp add: fms map_if_notin id_lookup_def del: match_ex_table_def)

--{* St nat *}
apply (rule impI, rule allI)
apply (simp only: split_def Let_def fst_conv snd_conv simp_thms if_True)
apply (drule subExpr_stkIds)
apply (drule_tac xs="stkIds Q" in in_set_conv_decomp_fst)
apply (erule exE | erule conjE)+
apply (simp add: fms map_if_notin id_lookup_def)
apply (case_tac "i=Throw")
apply (cut_tac exec_i[THEN sym] check_i)
apply (simp add: neq_Nil_conv)
apply (erule exE | erule conjE)+
apply (case_tac "hd stk = Null")
apply (subgoal_tac "\<exists>ob. (h (addr_of_sys_xcpt NullPointer) = \<lfloor>ob\<rfloor> \<and> obj_ty ob = (Class NullPointer))")
 prefer 2
 apply (cut_tac "sys_xptn_inv")
 apply (erule_tac x="NullPointer" in ballE)
  prefer 2
  apply (simp add: NullPointer_def sys_xcpts_def)
 apply assumption
apply (erule exE)
apply (simp add: none_def)

--{* hd stk ~= Null *}
apply (simp add: is_Ref_def is_Ref'_def split del: split_if split_if_asm option.split option.split_asm)
apply (case_tac "y")
apply simp

apply simp

apply simp

apply simp

apply (simp add: none_def)
--{* i ~= Throw *}
apply simp
apply (cut_tac exec_i[THEN sym] check_i)
apply (case_tac "i")
--{* Load *}
apply simp
--{* Store *}
apply simp
--{* Push *}
apply simp
--{* New *}
apply simp
apply (subgoal_tac "\<exists>ob. (h (addr_of_sys_xcpt OutOfMemory) = \<lfloor>ob\<rfloor> \<and> obj_ty ob = (Class OutOfMemory))")
 prefer 2
 apply (cut_tac "sys_xptn_inv")
 apply (erule_tac x="OutOfMemory" in ballE)
  prefer 2
  apply (simp add: OutOfMemory_def sys_xcpts_def)
 apply assumption
apply (erule exE)
apply (simp add: sys_xcpt_of_def obj_ty_def none_def)

--{* Getfield *}
apply simp
apply (case_tac "hd stk = Null")
apply (simp add: split_def)
apply (subgoal_tac "\<exists>ob. (h (addr_of_sys_xcpt NullPointer) = \<lfloor>ob\<rfloor> \<and> obj_ty ob = (Class NullPointer))")
 prefer 2
 apply (cut_tac "sys_xptn_inv")
 apply (erule_tac x="NullPointer" in ballE)
  prefer 2
  apply (simp add: NullPointer_def sys_xcpts_def)
 apply assumption
apply (erule exE)
apply (simp add:  sys_xcpt_of_def obj_ty_def none_def)

apply (simp add: split_def)

--{* Putfield *}
apply simp
apply (case_tac "hd (tl stk) = Null")
apply (simp add: split_def)
apply (subgoal_tac "\<exists>ob. (h (addr_of_sys_xcpt NullPointer) = \<lfloor>ob\<rfloor> \<and> obj_ty ob = (Class NullPointer))")
 prefer 2
 apply (cut_tac "sys_xptn_inv")
 apply (erule_tac x="NullPointer" in ballE)
  prefer 2
  apply (simp add: NullPointer_def sys_xcpts_def)
 apply assumption
apply (erule exE)
apply (simp add:  sys_xcpt_of_def obj_ty_def none_def)

apply (simp add: split_def)
--{* Checkcast *}
apply simp
apply (case_tac "cast_ok P list h (hd stk)")
apply (simp add: split_def)
apply (cut_tac  s_def)
apply (simp only:)
apply (subgoal_tac "\<exists>ob. (h (addr_of_sys_xcpt ClassCast) = \<lfloor>ob\<rfloor> \<and> obj_ty ob = (Class ClassCast))")
 prefer 2
 apply (cut_tac "sys_xptn_inv")
 apply (erule_tac x="ClassCast" in ballE)
  prefer 2
  apply (simp add: ClassCast_def sys_xcpts_def)
 apply assumption
apply (erule exE)
apply (simp add:  sys_xcpt_of_def obj_ty_def none_def)

--{* Invoke *}
apply simp
apply (erule conjE)+
apply (case_tac " stk ! nata = Null")
apply (simp add: split_def)
apply (cut_tac s_def)
apply (simp only:)
apply (subgoal_tac "\<exists>ob. (h (addr_of_sys_xcpt NullPointer) = \<lfloor>ob\<rfloor> \<and> obj_ty ob = (Class NullPointer))")
 prefer 2
 apply (cut_tac "sys_xptn_inv")
 apply (erule_tac x="NullPointer" in ballE)
  prefer 2
  apply (simp add: NullPointer_def sys_xcpts_def)
 apply assumption
apply (simp add: sys_xcpt_of_def)



apply (simp add: split_def)

--{* Return *}
apply (case_tac "frs")
apply simp

apply (simp add: split_def)
apply (case_tac "frs'")
apply simp

apply (simp add: split_def)
--{* Pop *}
apply (simp add: none_def)
--{* IBin *}
apply simp
--{* Goto *}
apply simp
--{* CmpEq *}
apply simp
--{* IfIntLeq *}
apply simp
--{* IfFalse *}
apply simp
--{* Throw *}
apply simp

--{* Lv nat *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* Cn nat *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* NewA *}
apply (simp add: fms map_if_notin id_lookup_def)
--{* Gf list1 list2 expr *}
apply (rule impI, rule allI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Gf list1 list2 expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def split add: val.split)
--{* FrNr *}
apply (rule impI, rule allI)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Num expr1 num_op epxr2 *}
apply (rule impI, rule allI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Num expr1 num_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Num expr1 num_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Rel expr1 rel_op expr2 *}
apply (rule impI, rule allI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Rel expr1 rel_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Rel expr1 rel_op expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* IF expr1 THEN expr2 ELSE expr3 *}
apply (rule impI, rule allI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr3 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="IF expr1 THEN expr2 ELSE expr3" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (drule mp,assumption)+

apply (simp add: split_def fms map_if_notin id_lookup_def evalE_evalEs.simps substE_substEs.simps split del: split_if add: bool.split )
apply (case_tac " evalE (P, An) ((C', M', pc'), (None, h, ([Addr xa], loc', C', M', pc') # frs'), e'\<lparr>lv := I\<rparr>) expr1 = \<lfloor>Bool True\<rfloor>")
apply simp
apply simp
--{* Eq expr1 expr2 *}
apply (rule impI, rule allI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Eq expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Eq expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Neg expr *}
apply (rule impI, rule allI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Neg expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Imp expr1 expr2 *}
apply (rule impI, rule allI)
apply (subgoal_tac "expr1 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Imp expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (subgoal_tac "expr2 \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Imp expr1 expr2" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def)
--{* Forall nat expr *}
apply (rule impI, rule allI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Forall nat expr" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: fms map_if_notin id_lookup_def split del: split_if)
--{* Ty expr ty*}
apply (rule impI, rule allI)
apply (subgoal_tac "expr \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="Ty expr ty" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm) add:  parts.simps)
apply (simp add: Let_def split_def fms map_if_notin id_lookup_def split add: val.split)
--{* Pos x *}
apply (rule impI, rule allI)
apply (subgoal_tac "x \<in> set (getPosEx Q)")
 prefer 2
 apply (rule subExpr_getPosEx)
 apply assumption
apply (drule_tac xs="getPosEx Q" in  in_set_conv_decomp_fst)
apply (erule exE |erule conjE)+
apply (subgoal_tac "(C,M,pc') \<in> set (domC (P,An))")
 prefer 2
 apply (rule_tac X="cn" in wf_ex_table_domC)
 apply (cut_tac wf_Pi)
 apply simp
 apply (rule_tac pc="pc" and An="An" in domC_methodnames)
 apply (cut_tac p_domC)
 apply simp
 apply simp
apply (simp add: fms map_if_notin id_lookup_def)
apply (rule impI)
apply (case_tac "frs'")
apply (cut_tac p_domC)
apply (simp add: mem_iff)

apply (cut_tac p_domC)
apply (simp add: mem_iff split_def del: callers_sysinv.simps)
apply (subgoal_tac "callers (P,An) (C',M',pc') = callers (P,An) (C',M',pc)")
 prefer 2
 apply (rule callers_eq)
apply (cut_tac e'_def)
apply simp
--{* Call *}
apply (rule impI,rule allI)
apply (simp add: Let_def split_def fms map_if_notin id_lookup_def)
apply (rule impI)
apply (simp only: neq_Nil_conv)
apply (erule exE)+
apply (cut_tac e'_def)
apply (simp (no_asm_simp) add: split_paired_all)

--{* Catch  *}
apply (rule impI,rule allI)
apply (simp add: Let_def split_def fms map_if_notin id_lookup_def)
apply (rule impI)
apply (simp only: neq_Nil_conv)
apply (erule exE)+
apply (cut_tac e'_def)
apply (simp (no_asm_simp) add: split_paired_all)

--{* And es *}
apply (rule impI, rule allI)
apply (simp add: evalE_And fms map_if_notin id_lookup_def del: evalE_evalEs.simps split del: split_if)
apply (rule iffI)
apply (rule ballI)
apply (erule_tac x="ex" in ballE)
apply (erule_tac x=" (substE (map (\<lambda>q. (Pos q, if q = (C', M', pc') then Pos p else FF)) (getPosEx Q) @
                          map (\<lambda>k. (St k, if Suc 0 \<le> k then none
                                          else if i = Throw
                                               then IF St 0 \<doteq>
                                                       Cn Null THEN Cn (Addr (addr_of_sys_xcpt NullPointer)) ELSE St 0
                                               else Cn (Addr (addr_of_sys_xcpt (sys_xcpt_of i)))))
                           (stkIds Q))
                   ex)" in ballE)
apply (subgoal_tac "ex \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="And es" in subExpr_Gf)
 apply assumption
 apply (simp add: parts.simps)
apply (simp add: substEs_map) 

apply (simp add: substEs_map)

apply (simp only:)

apply (rule ballI)

apply (simp add: substEs_map image_iff)
apply (erule bexE)
apply (erule_tac x="x" in ballE)
apply (erule_tac x="x" in ballE)
apply (subgoal_tac "x \<in> set (subExpr Q)")
 prefer 2
 apply (rule_tac ex'="And es" in subExpr_Gf)
 apply assumption
 apply (simp (no_asm_simp) add: parts.simps)
apply simp

apply (simp only:)

apply (simp only:)
done
qed
qed
(*>*)


lemma effS_wpF:
assumes wf_Pi: "wf \<Pi>"
assumes s_def: "s = (p,\<sigma>,e)"
assumes s'_def: "s' = (p',\<sigma>',e')"
(* assumes anF_p: "(\<exists> Mn n. cmd \<Pi> p = Some (Invoke Mn n)) \<longrightarrow> (\<forall> A. anF \<Pi> p = Some A \<longrightarrow> \<Pi>,s \<Turnstile> A)" *)
assumes s_inv_Pos: "\<Pi>,s \<Turnstile> inv_Pos \<Pi> (fst s)"
assumes s_inv_Ty:"\<Pi>,s \<Turnstile> inv_Ty \<Pi> (fst s)"
assumes s_inv_ExTys:"\<Pi>,s \<Turnstile> inv_ExTys \<Pi> (fst s)"
assumes p'_B_succsF_p: "(p',B) \<in> set (succsF \<Pi> p)"
assumes s_B: "\<Pi>,s \<Turnstile> B"
assumes s_s'_effS: "(s,s') \<in> effS \<Pi>"
shows "\<forall> I. evalE \<Pi> (p,\<sigma>,e\<lparr>lv:=I\<rparr>) (wpF \<Pi> p p' Q) = evalE \<Pi> (p', \<sigma>',e'\<lparr>lv:=I\<rparr>) Q"
(*<*)
proof (rule effS.elims[OF s_s'_effS])
  fix p \<sigma> e p' \<sigma>' e' P C M pc i h stk loc frs
  assume s_s'_def:    "(s, s') = ((p, \<sigma>, e), p', \<sigma>', e')"
  assume P_def:       "P = fst \<Pi>"
  assume p_def:       "p = (C, M, pc)"
  assume i_def:       "i = instrs_of P C M ! pc"
  assume sigma_def:   "\<sigma> = (None, h, (stk, loc, p) # frs)"
  assume has_method_C_M: "JBC_Semantics.has_method P C M"
  { --{* EXCEPTIONAL EXECUTION *}
    fix xa h' frs'' loc' frs' 
    assume exec_i: "exec_instr i P h stk loc C M pc frs = (\<lfloor>xa\<rfloor>, h', frs'')"
    assume find_handler_s: "find_handler P xa h ((stk, loc, p) # frs) = \<sigma>'"
    assume sigma'_def: "\<sigma>' = (None, h, ([Addr xa], loc', p') # frs')"
    assume e'_def: "e' = e\<lparr>cs := drop (length frs - length frs') (cs e)\<rparr>"
    show ?thesis
    proof -
      from s_inv_Pos s_def s_s'_def have Pos_p: "\<Pi>,(p,\<sigma>,e) \<Turnstile> Pos p"
	apply (simp add: inv_Pos_def split_def)
	apply (case_tac "snd (snd \<sigma>)")
	apply simp
	apply simp
	done
      from s_inv_ExTys s_s'_def s_def 
      have sys_xptn_inv: "\<forall>C\<in>sys_xcpts. \<exists>ob. (fst (snd \<sigma>)) (addr_of_sys_xcpt C) = \<lfloor>ob\<rfloor> \<and> obj_ty ob = Class C"
	by (simp add: inv_ExTys_def split_def sys_xcpts_def obj_ty_def split_paired_all)
      from p'_B_succsF_p s_s'_def s_def  have p_domC: "p \<in> set (domC \<Pi>)"
	apply -
	apply (simp add: succsF_def succsNormal_def succsExcept_def addPos_def)
	apply (rule cmd_domC)
	apply simp
	apply (rule cmd_domC)
	apply simp
	done
      from wf_Pi p'_B_succsF_p s_s'_def s_def s'_def have p'_domC: "p' \<in> set (domC \<Pi>)"
	apply -
	apply (drule succsF_domC)
	apply assumption
	apply simp
	done
	
      from sigma_def p_def i_def p_domC wf_Pi s_inv_Ty s_def sigma_def p_def s_s'_def
      have check_i : "check_instr' ((instrs_of P C M)!pc) (fst \<Pi>) h stk loc C M pc frs"
	apply -
	apply (rule wf_invTys_check')
	apply simp
	apply simp
	apply simp
	apply simp
	apply simp
	apply (drule_tac C="C" and M="M" and pc="pc" and \<Pi>="\<Pi>" in domC_cmd_instr_of)
	apply simp
	apply (simp add: P_def)
	done

      from wf_Pi p_domC p_def i_def P_def s_s'_def s_def
      have cmd_p: "cmd \<Pi> p = Some i"
	apply -
	apply simp
	apply (drule domC_cmd_instr_of)
	apply simp
	apply simp
	done

      from P_def obtain An where Pi_def: "\<Pi> = (P,An)"
	apply atomize
	apply (erule_tac x="snd \<Pi>" in allE)
	apply simp
	done

      from wf_Pi p_domC cmd_p p_def exec_i find_handler_s sigma'_def Pi_def s_s'_def s_def s'_def
      obtain cn where handlesEx_p: "handlesEx (fst \<Pi>) p' = Some cn \<and> P \<turnstile> (cname_of h xa)  \<preceq>\<^sup>* cn"
	apply - 
	apply atomize
	apply (subgoal_tac "\<exists>cn. handlesEx P p' = Some cn \<and> P \<turnstile> (cname_of h xa)  \<preceq>\<^sup>* cn")
	 prefer 2
	 apply (subgoal_tac "\<exists> cns cn cns'. handlesEx' P p' = (cns @ cn # cns') \<and> P \<turnstile> (cname_of h xa)  \<preceq>\<^sup>* cn")
          prefer 2
	  apply (subgoal_tac "(\<exists>h' stk' pc''. catchstate (P, cname_of h xa, p, (None, h, (stk, loc, p) # frs), e) =
	  ((fst p', fst (snd p'), pc''), (None, h', (stk', loc', fst p', fst (snd p'), pc'') # frs'), 
           e\<lparr>cs := drop (length frs - length frs') (cs e)\<rparr>) \<and>
	   match_ex_table P (cname_of h xa) pc'' (ex_table_of P (fst p') (fst (snd p'))) = \<lfloor>snd (snd p')\<rfloor>) \<or> frs = frs'")
	   prefer 2
	   apply (rule_tac P="P" and xa="xa" and st'="stk" in find_handler_catchstate')
	   apply simp
          apply (case_tac "frs = frs'")
	  apply simp
	  apply (frule find_handler_frs)
	  apply (erule exE | erule conjE)+
	  apply simp
	  apply (drule_tac C="C" and M="M" and cn="cname_of h xa" and pc="pc" and pc'="fst a" in match_ex_table_handlesEx')
	  apply (rule_tac pc="pc" and An="An" in domC_methodnames)
	   apply simp
	  apply (simp add: JBC_VCG.match_ex_table_def)
	  apply (erule exE | erule conjE)+
	  apply (rule_tac x="cns" in exI)
	  apply (rule_tac x="cn'" in exI)
	  apply simp

	  --{* frs ~= frs' *}
	  apply simp
	  apply (frule_tac C="fst p'" and M="fst (snd p')" and cn="cname_of h xa" and pc="pc''" and pc'="fst a" in match_ex_table_handlesEx')
	  apply (subgoal_tac "(fst p', fst (snd p')) \<in> set (methodnames P)")
	   prefer 2
           apply (cut_tac p'_domC)
	   apply (rule_tac pc="snd (snd p')" and An="An" in domC_methodnames)
	   apply simp
	  apply assumption
	  apply (simp add: JBC_VCG.match_ex_table_def)
	  apply (erule exE | erule conjE)+
	  apply (rule_tac x="cns" in exI)
	  apply (rule_tac x="cn'" in exI)
	  apply simp
	 apply (erule conjE | erule exE)+
	 apply (subgoal_tac "length (handlesEx' P p') \<le> 1")
	  prefer 2
	  apply (rule_tac An="An" in wf_handlesEx'_length)
	  apply (simp only: Pi_def)

	 apply (rule_tac x="cn" in exI)
	 apply (case_tac "cns")
	 apply (simp add: handlesEx_def)

	 apply simp
	apply simp
	done
    
    show ?thesis 
      apply -
      apply (rule_tac s="s" and C="C" and M="M" and pc="pc" and cn="cn" and xa="xa"
                    and h="h" and stk="stk" and loc="loc" and frs="frs" and h'="h'" and frs''="frs''"
		    and s'="s'" and P="P" and An="snd \<Pi>" and loc'="loc'" and frs'="frs'"
                    in effS_wpF_Except)
      apply (rule wf_Pi)

      apply (cut_tac sys_xptn_inv s_def sigma_def s_s'_def)
      apply simp

      apply (cut_tac handlesEx_p s_s'_def s'_def, simp)
 
      apply (cut_tac handlesEx_p s_s'_def s'_def, simp)
 
      apply (cut_tac handlesEx_p s_s'_def s'_def, simp)

      apply (cut_tac cmd_p s_s'_def s_def, simp)

      apply (cut_tac s_s'_def s_def p_domC, simp)
      
      apply (simp only: i_def)
      
      apply (rule s_def)

      apply (cut_tac s_s'_def s_def p_def, simp)

      apply (cut_tac s_s'_def s_def sigma_def, simp)

      apply (rule s'_def)

      apply (cut_tac check_i i_def Pi_def, simp)

      apply (cut_tac s_s'_def s'_def sigma'_def exec_i,simp)

      apply (cut_tac find_handler_s s_s'_def s_def s'_def sigma'_def, simp)

      apply (cut_tac s_s'_def s'_def s_def sigma'_def, simp)

      apply (cut_tac s_s'_def s'_def s_def  e'_def, simp)

      apply (simp add: P_def)

      apply (cut_tac Pos_p s_s'_def s_def)
      apply simp
      done
    qed
 
  next
  
  --{* NORMAL EXECUTION *}
  fix h' fr' frs'
  assume exec_i: "exec_instr i P h stk loc C M pc frs = (None, h', fr' # frs')"
  assume sigma'_def: "\<sigma>' = (None, h', fr' # frs')"
  assume p'_def: "p' = snd (snd fr')"
  assume e'_def: "e' = e\<lparr>cs := if \<exists>M n. i = Invoke M n then h # cs e else if i = Return then tl (cs e) else cs e\<rparr>"
  show ?thesis
    proof -
      from s_inv_Pos s_def s_s'_def have Pos_p: "\<Pi>,(p,\<sigma>,e) \<Turnstile> Pos p"
	apply (simp add: inv_Pos_def split_def)
	apply (case_tac "snd (snd \<sigma>)")
	apply simp
	apply simp
	done

      from s_inv_ExTys s_s'_def s_def 
      have sys_xptn_inv: "\<forall>C\<in>sys_xcpts. \<exists>ob. (fst (snd \<sigma>)) (addr_of_sys_xcpt C) = \<lfloor>ob\<rfloor> \<and> obj_ty ob = Class C"
	by (simp add: inv_ExTys_def split_def sys_xcpts_def obj_ty_def split_paired_all)

      from wf_Pi p'_B_succsF_p s_s'_def s_def s'_def have p_domC' : "p \<in> set (domC \<Pi>)"
      apply -
      apply (drule succsF_domC)
      apply assumption
      apply simp
      done   

    from wf_Pi p'_B_succsF_p s_s'_def s_def s'_def have p'_domC : "p' \<in> set (domC \<Pi>)"
      apply -
      apply (drule succsF_domC)
      apply assumption
      apply simp
      done

    from sigma_def p_def i_def p_domC' wf_Pi s_inv_Ty s_def sigma_def p_def s_s'_def
    have check_i : "check_instr' ((instrs_of P C M)!pc) P h stk loc C M pc frs"
	apply -
	apply (subgoal_tac "check_instr' ((instrs_of P C M)!pc) (fst \<Pi>) h stk loc C M pc frs")
	 prefer 2
	 apply (rule wf_invTys_check')
	 apply simp
	 apply simp
	 apply simp
	 apply simp
	 apply simp
	 apply (drule_tac C="C" and M="M" and pc="pc" and \<Pi>="\<Pi>" in domC_cmd_instr_of)
	 apply simp
	 apply (simp add: P_def)
	apply (simp add: P_def)
	done

    from wf_Pi p_domC' p_def i_def P_def 
    have cmd_p: "cmd \<Pi> p = Some i"
      apply -
      apply simp
      apply (drule domC_cmd_instr_of)
      apply simp
      apply simp
      done
    
    from P_def obtain An where Pi_def: "\<Pi> = (P,An)"
      apply atomize
      apply (erule_tac x="snd \<Pi>" in allE)
      apply simp
      done

    from wf_Pi p_domC' cmd_p p_def p'_def exec_i 
    have handlesEx_p: "handlesEx (fst \<Pi>) p' = None"
      proof -
	from wf_Pi p_domC' cmd_p 
	have throwChk_p: "throwChk (\<Pi>, \<lfloor>i\<rfloor>, anF \<Pi> p, p)"
	  apply (simp only: in_set_conv_decomp)
	  apply (erule exE)+
	  apply (simp add: JBC_VCG.wf_def checkPos_split)
	  apply (erule conjE)+
	  apply (drule split_if_False)
	  apply simp
	  done
	from wf_Pi p_domC' cmd_p 
	have invokeChk_p: "invokeChk (\<Pi>, \<lfloor>i\<rfloor>, anF \<Pi> p, p)"
	  apply (simp only: in_set_conv_decomp)
	  apply (erule exE)+
	  apply (simp add: JBC_VCG.wf_def checkPos_split)
	  apply (erule conjE)+
	  apply (drule split_if_False)
	  apply simp
	  done

	from wf_Pi p_domC' cmd_p
	have p'_succsN_hEx: "p' mem map fst (succsNormal \<Pi> p) \<Longrightarrow> handlesEx (fst \<Pi>) p' = None"
	  apply -
	  apply (simp only: in_set_conv_decomp mem_iff)
	  apply (erule exE)+
	  apply (simp add: JBC_VCG.wf_def checkPos_split mem_iff)
	  apply (erule conjE)+
	  apply (drule split_if_False)
	  apply (erule conjE)+
	  apply simp
	  done

        from p'_B_succsF_p s_s'_def s_def s'_def
	obtain B' where B_def: "B = And [Pos p,B']"
	  apply -
	  apply (simp add: succsF_def addPos_def)
	  apply (erule disjE)
	  apply (erule imageE)
	  apply (simp add: split_def)

	  apply (erule imageE)
	  apply (simp add: split_def)
	  done
	  
	from p'_B_succsF_p s_s'_def s_def s'_def B_def
	have p'_succsNormal_succsExcept: "(p',B') \<in> set (succsNormal \<Pi> p) \<or> (p',B') \<in> set (succsExcept \<Pi> p)"
	  apply -
	  apply (simp add: succsF_def addPos_append addPos_def)
	  apply (erule disjE)
	  apply (erule imageE)
	  apply (simp add: split_def)

	  apply (erule imageE)
	  apply (simp add: split_def)
	  done

	note exec_simp = p_domC' cmd_p exec_i[THEN sym] p_def p'_def
	from exec_simp sigma_def s_def
	have p'_succsN: "p' mem map fst (succsNormal \<Pi> p)"
	  proof (cases i)
	    case (Load n)
	    from this exec_simp show ?thesis
	      by (simp add: succsNormal_def incA_def)
	  next
	    case (Store n)
	    from this exec_simp show ?thesis
	      by (simp add: succsNormal_def incA_def)
	  next
	    case (Push v)
	     from this exec_simp show ?thesis
	      by (simp add: succsNormal_def incA_def)
	  next
	    case (New cn)
	     from this exec_simp show ?thesis
	      by (simp add: succsNormal_def incA_def)
	  next
	    case (Getfield Fn cn)
	    from this exec_simp show ?thesis
	      apply (simp add: succsNormal_def incA_def)
	      apply (case_tac "hd stk = Null")
	      apply (simp add: split_def)+
	      done
	  next
	    case (Putfield Fn Cn) 
	    from this exec_simp show ?thesis
	      apply (simp add: succsNormal_def incA_def)
	      apply (case_tac "hd (tl stk) = Null")
	      apply (simp add: split_def)
	      apply (simp add: split_def)
	      done
	  next
	    case (Checkcast Cn)
	    from this exec_simp show ?thesis 
	      by (simp add: succsNormal_def incA_def)
	  next
	    case (Invoke Mn n)
	    show ?thesis
	      proof (cases "stk ! n = Null")

		case True
		from True Invoke exec_simp
		show ?thesis
		  by (simp add: split_def)

	      next
		case False
		from False have not_Null: "stk ! n \<noteq> Null"
		  by simp
		from p'_succsNormal_succsExcept 
		show ?thesis
		  proof (cases "(p',B') \<in> set (succsExcept \<Pi> p)")
		    case True
		    from True s_B B_def wf_Pi Invoke cmd_p s_s'_def s_def s'_def sigma_def not_Null
		    show ?thesis
		      apply -
		      apply (drule wf_succsXpt_xcpt_cond)
		      apply assumption
		      apply simp
		      apply assumption

		      apply (erule exE)
		      apply (simp add: valid_def sys_xcpt_of_def  xcpt_cond_def evalEs_map)
		      apply (case_tac "n < length stk")
		      apply simp

		      apply simp
		      done

		  next
		    case False
		    from False p'_succsNormal_succsExcept show ?thesis
		      apply -
		      apply (simp add: mem_iff)
		      apply (drule_tac f="fst" in imageI)
		      apply simp
		      done
		  qed
		qed

	  next
	    case Return
	    from this exec_simp p'_B_succsF_p s_s'_def s_def s'_def show ?thesis
	      apply (simp add: succsNormal_def succsF_def incA_def succsExcept_def addPos_def)   
	      apply (erule imageE)
	      apply (erule imageE)
	      apply (simp add: split_def mem_iff map_fst_tuple image_fst_tuple)
	      apply (drule_tac f="(\<lambda>p'. ((fst p', fst (snd p'), Suc (snd (snd p'))), Call (And [aF \<Pi> p', Pos p'])))" and A="set (callers \<Pi> (C,M,pc))" in imageI)
	      apply (drule_tac f="fst" and x="((fst p'a, fst (snd p'a), Suc (snd (snd p'a))), Call (And [aF \<Pi> p'a, Pos p'a]))" in imageI)
	      apply simp
	      done
	  next
	    case Pop
	    from this exec_simp show ?thesis
	      by (simp add: succsNormal_def incA_def)
	  next
	    case IBin
	    from this exec_simp show ?thesis
	      by (simp add: succsNormal_def incA_def)
	  next 	 
	    case (Goto t)
	    from this exec_simp show ?thesis
	      by (simp add: succsNormal_def incA_def)
	  next
	    case CmpEq
	    from this exec_simp show ?thesis
	      by (simp add: succsNormal_def incA_def) 
	  next 
	    case (IfIntCmp ro t)
	    from this exec_simp show ?thesis
	      by (simp add: succsNormal_def incA_def) 
	  next	
	    case (IfFalse t)
	    from this exec_simp show ?thesis
	      by (simp add: succsNormal_def incA_def)
	  next 
	    case Throw
	    from this exec_simp show ?thesis
	      apply simp
	      apply (case_tac "hd stk = Null")
	      apply simp
	      apply simp
	      done
	    qed
	   from p'_succsN_hEx p'_succsN show ?thesis
	      by simp
	  qed --{* handlesEx_p *}

	  show ?thesis 
	  proof (cases i)
	    case (Load n)
	    from this 
	    show ?thesis
	      apply -
	      apply (rule_tac s="s" and C="C" and M="M" and pc="pc" 
                     and h="h" and stk="stk" and loc="loc" and frs="frs" 
		     and s'="s'" and fr'="fr'" and P="P" and An="snd \<Pi>" 
                     in effS_wpF_Load)
	     
	      apply (rule wf_Pi)

	      apply (cut_tac handlesEx_p s_s'_def s'_def, simp)

	      apply (cut_tac cmd_p s_s'_def s_def, simp) 

	      apply (cut_tac s_s'_def s_def p_domC', simp)

	      apply simp

	      apply (simp only: i_def)

	      apply (rule s_def)

	      apply (cut_tac s_s'_def s_def p_def, simp)

	      apply (cut_tac s_s'_def s_def sigma_def, simp)

	      apply (rule s'_def)

	      apply (cut_tac s_s'_def s'_def sigma'_def exec_i,simp)

	      apply (cut_tac s_s'_def s'_def s_def e'_def, simp)

	      apply (cut_tac s_s'_def s'_def p'_def, simp)

	      apply (cut_tac p'_domC s_s'_def s'_def, simp)

	      apply (cut_tac check_i i_def, simp)

	      apply (cut_tac s_s'_def s'_def exec_i sigma'_def, simp)

	      apply (simp add: P_def)
	      done

	    next

	      case (Store n)
	    from this 
	    show ?thesis
	      apply -
	      apply (rule_tac s="s" and C="C" and M="M" and pc="pc" 
                     and h="h" and stk="stk" and loc="loc" and frs="frs" 
		     and s'="s'" and fr'="fr'" and P="P" and An="snd \<Pi>" 
                     in effS_wpF_Store)
	      apply simp
	     
	      apply (rule wf_Pi)

	      apply (cut_tac handlesEx_p s_s'_def s'_def, simp)

	      apply (cut_tac cmd_p s_s'_def s_def, simp)

	      apply (cut_tac s_s'_def s_def p_domC', simp)

	      apply (simp only: i_def)

	      apply (rule s_def)

	      apply (cut_tac s_s'_def s_def p_def, simp)

	      apply (cut_tac s_s'_def s_def sigma_def, simp)

	      apply (rule s'_def)

	      apply (cut_tac s_s'_def s'_def sigma'_def exec_i,simp)

	      apply (cut_tac s_s'_def s'_def s_def e'_def, simp)

	      apply (cut_tac s_s'_def s'_def p'_def, simp)

	      apply (cut_tac check_i i_def, simp)

	      apply (cut_tac s_s'_def s'_def exec_i sigma'_def, simp)

	      apply (simp add: P_def)
	      done

	    next

	      case (Push v)
	    from this 
	    show ?thesis
	      apply -
	      apply (rule_tac s="s" and C="C" and M="M" and pc="pc" 
                     and h="h" and stk="stk" and loc="loc" and frs="frs" 
		     and s'="s'" and fr'="fr'" and P="P" and An="snd \<Pi>" 
                     in effS_wpF_Push)
	      apply simp
	     
	      apply (rule wf_Pi)
	      
	      apply (cut_tac handlesEx_p s_s'_def s'_def, simp)

	      apply (cut_tac cmd_p s_s'_def s_def, simp)

	      apply (cut_tac s_s'_def s_def p_domC', simp)

	      apply (simp only: i_def)

	      apply (rule s_def)

	      apply (cut_tac s_s'_def s_def p_def, simp)

	      apply (cut_tac s_s'_def s_def sigma_def, simp)

	      apply (rule s'_def)

	      apply (cut_tac s_s'_def s'_def sigma'_def exec_i,simp)

	      apply (cut_tac s_s'_def s'_def s_def e'_def, simp)

	      apply (cut_tac s_s'_def s'_def p'_def, simp)

	      apply (cut_tac check_i i_def, simp)

	      apply (cut_tac s_s'_def s'_def exec_i sigma'_def, simp)

	      apply (simp add: P_def)
	      done

	    next

	      case (New Cl)
	    from this 
	    show ?thesis
	      apply -

	      apply (rule_tac s="s" and C="C" and M="M" and pc="pc" and Cl="Cl" 
                     and h="h" and h'="h'" and stk="stk" and loc="loc" and frs="frs" 
		     and s'="s'" and fr'="fr'" and P="P" and An="snd \<Pi>" 
                     in effS_wpF_New)
	      apply simp
	     
	      apply (rule wf_Pi)

	      apply (cut_tac handlesEx_p s_s'_def s'_def, simp)

	      apply (cut_tac cmd_p s_s'_def s_def, simp)

	      apply (cut_tac s_s'_def s_def p_domC', simp)

	      apply (simp only: i_def)

	      apply (rule s_def)

	      apply (cut_tac s_s'_def s_def p_def, simp)

	      apply (cut_tac s_s'_def s_def sigma_def, simp)

	      apply (rule s'_def)

	      apply (cut_tac s_s'_def s'_def sigma'_def exec_i,simp)

	      apply (cut_tac s_s'_def s'_def s_def e'_def, simp)

	      apply (cut_tac s_s'_def s'_def p'_def, simp)

	      apply (cut_tac check_i i_def, simp)

	      apply (cut_tac s_s'_def s'_def exec_i sigma'_def, simp)

	      apply (simp add: P_def)
	      done
	    next
	      case (Getfield F Cl)
	    from this 
	    show ?thesis
	      apply -
	      apply (rule_tac s="s" and C="C" and M="M" and pc="pc" and ?list1.0="F" and ?list2.0="Cl" and h'="h'"
                     and h="h" and stk="stk" and loc="loc" and frs="frs" 
		     and s'="s'" and fr'="fr'" and P="P" and An="snd \<Pi>" 
                     in effS_wpF_Getfield)
	      apply simp
	     
	      apply (rule wf_Pi)

	      apply (cut_tac handlesEx_p s_s'_def s'_def, simp)

	      apply (cut_tac cmd_p s_s'_def s_def, simp)

	      apply (cut_tac s_s'_def s_def p_domC', simp)

	      apply (simp only: i_def)

	      apply (rule s_def)

	      apply (cut_tac s_s'_def s_def p_def, simp)

	      apply (cut_tac s_s'_def s_def sigma_def, simp)

	      apply (rule s'_def)

	      apply (cut_tac s_s'_def s'_def sigma'_def exec_i,simp)

	      apply (cut_tac s_s'_def s'_def s_def e'_def, simp)

	      apply (cut_tac s_s'_def s'_def p'_def, simp)

	      apply (cut_tac check_i i_def, simp)

	      apply (cut_tac s_s'_def s'_def exec_i sigma'_def, simp)

	      apply (simp add: P_def)
	      done
	    
	    next

	      case (Putfield F Cl)
	    from this 
	    show ?thesis
	      apply -
	      apply (rule_tac s="s" and C="C" and M="M" and pc="pc" and ?list1.0="F" and ?list2.0="Cl" and h'="h'"
                     and h="h" and stk="stk" and loc="loc" and frs="frs" 
		     and s'="s'" and fr'="fr'" and P="P" and An="snd \<Pi>" 
                     in effS_wpF_Putfield)
	      apply simp
	     
	      apply (rule wf_Pi)

	      apply (cut_tac handlesEx_p s_s'_def s'_def, simp)

	      apply (cut_tac cmd_p s_s'_def s_def, simp)

	      apply (cut_tac s_s'_def s_def p_domC', simp)

	      apply (simp only: i_def)

	      apply (rule s_def)

	      apply (cut_tac s_s'_def s_def p_def, simp)

	      apply (cut_tac s_s'_def s_def sigma_def, simp)

	      apply (rule s'_def)

	      apply (cut_tac s_s'_def s'_def sigma'_def exec_i,simp)

	      apply (cut_tac s_s'_def s'_def s_def e'_def, simp)

	      apply (cut_tac s_s'_def s'_def p'_def, simp)

	      apply (cut_tac check_i i_def, simp)

	      apply (cut_tac s_s'_def s'_def exec_i sigma'_def, simp)

	      apply (simp add: P_def)
	      done

	    next

	      case (Checkcast Cl)
	    from this 
	    show ?thesis
	      apply -
	      apply (rule_tac s="s" and C="C" and M="M" and pc="pc" and Cl="Cl" 
                     and h="h" and stk="stk" and loc="loc" and frs="frs" 
		     and s'="s'" and fr'="fr'" and P="P" and An="snd \<Pi>" 
                     in effS_wpF_Checkcast)
	      apply simp
	     
	      apply (rule wf_Pi)

	      apply (cut_tac handlesEx_p s_s'_def s'_def, simp)

	      apply (cut_tac cmd_p s_s'_def s_def, simp)

	      apply (cut_tac s_s'_def s_def p_domC', simp)

	      apply (simp only: i_def)

	      apply (rule s_def)

	      apply (cut_tac s_s'_def s_def p_def, simp)

	      apply (cut_tac s_s'_def s_def sigma_def, simp)

	      apply (rule s'_def)

	      apply (cut_tac s_s'_def s'_def sigma'_def exec_i,simp)

	      apply (cut_tac s_s'_def s'_def s_def e'_def, simp)

	      apply (cut_tac s_s'_def s'_def p'_def, simp)

	      apply (cut_tac check_i i_def, simp)

	      apply (cut_tac s_s'_def s'_def exec_i sigma'_def, simp split add: split_if_asm)

	      apply (simp add: P_def)
	      done

	    next 

	      case (Invoke Mn n)

	    from this 
	    show ?thesis
	      apply -
	      apply (rule_tac s="s" and C="C" and M="M" and pc="pc" and Mn="Mn" and n="n"
                     and h="h" and stk="stk" and loc="loc" and frs="frs" 
		     and s'="s'" and fr'="fr'" and P="P" and An="snd \<Pi>" 
                     in effS_wpF_Invoke)
	      apply simp
	     
	      apply (rule wf_Pi)

	      apply (cut_tac handlesEx_p s_s'_def s'_def, simp)

	      apply (cut_tac cmd_p s_s'_def s_def, simp)

	      apply (cut_tac s_s'_def s_def p_domC', simp)

	      apply (simp only: i_def)

	      apply (rule s_def)

	      apply (cut_tac s_s'_def s_def p_def, simp)

	      apply (cut_tac s_s'_def s_def sigma_def, simp)

	      apply (rule s'_def)

	      apply (cut_tac s_s'_def s'_def sigma'_def exec_i,simp add: split_def)

	      apply (cut_tac s_s'_def s'_def s_def e'_def, simp)

	      apply (cut_tac s_s'_def s'_def p'_def, simp)

	      apply (cut_tac check_i i_def, simp)

	      apply (cut_tac s_s'_def s'_def exec_i sigma'_def, simp split add: split_if_asm)

	      apply (simp add: P_def)

	      apply (cut_tac s_s'_def s'_def p'_domC, simp)
	      done

	    next 
	      case Return

	    from this 
	    show ?thesis
	      apply -
	      apply (rule_tac s="s" and C="C" and M="M" and pc="pc" 
                     and h="h" and stk="stk" and loc="loc" and frs="frs" 
		     and s'="s'" and fr'="fr'" and P="P" and An="snd \<Pi>" 
                     in effS_wpF_Return)
	      apply simp
	     
	      apply (rule wf_Pi)

	      apply (cut_tac handlesEx_p s_s'_def s'_def, simp)

	      apply (cut_tac cmd_p s_s'_def s_def, simp)

	      apply (cut_tac s_s'_def s_def p_domC', simp)

	      apply (simp only: i_def)

	      apply (rule s_def)

	      apply (cut_tac s_s'_def s_def p_def, simp)

	      apply (cut_tac s_s'_def s_def sigma_def, simp)

	      apply (rule s'_def)

	      apply (cut_tac s_s'_def s'_def sigma'_def exec_i,simp add: split_def split add: split_if_asm)

	      apply (cut_tac s_s'_def s'_def s_def e'_def, simp)

	      apply (cut_tac s_s'_def s'_def p'_def, simp)

	      apply (cut_tac check_i i_def, simp)

	      apply (cut_tac s_s'_def s'_def exec_i sigma'_def, simp split add: split_if_asm)

	      apply (simp add: P_def)

	      apply (cut_tac Pos_p s_s'_def s_def, simp)
	      done

	    next 
	  
	      case Pop

	    from this 
	    show ?thesis
	      apply -
	      apply (rule_tac s="s" and C="C" and M="M" and pc="pc" 
                     and h="h" and stk="stk" and loc="loc" and frs="frs" 
		     and s'="s'" and fr'="fr'" and P="P" and An="snd \<Pi>" 
                     in effS_wpF_Pop)
	      apply simp
	     
	      apply (rule wf_Pi)

	      apply (cut_tac handlesEx_p s_s'_def s'_def, simp)

	      apply (cut_tac cmd_p s_s'_def s_def, simp)

	      apply (cut_tac s_s'_def s_def p_domC', simp)

	      apply (simp only: i_def)

	      apply (rule s_def)

	      apply (cut_tac s_s'_def s_def p_def, simp)

	      apply (cut_tac s_s'_def s_def sigma_def, simp)

	      apply (rule s'_def)

	      apply (cut_tac s_s'_def s'_def sigma'_def exec_i,simp add: split_def split add: split_if_asm)

	      apply (cut_tac s_s'_def s'_def s_def e'_def, simp)

	      apply (cut_tac s_s'_def s'_def p'_def, simp)

	      apply (cut_tac check_i i_def, simp)

	      apply (cut_tac s_s'_def s'_def exec_i sigma'_def, simp split add: split_if_asm)

	      apply (simp add: P_def)
	      done

	  next 
	  
	    case IBin

	    from this 
	    show ?thesis
	      apply -
	      apply (rule_tac s="s" and C="C" and M="M" and pc="pc" 
                     and h="h" and stk="stk" and loc="loc" and frs="frs" 
		     and s'="s'" and fr'="fr'" and P="P" and An="snd \<Pi>" 
                     in effS_wpF_IBin)
	      apply simp
	     
	      apply (rule wf_Pi)

	      apply (cut_tac handlesEx_p s_s'_def s'_def, simp)

	      apply (cut_tac cmd_p s_s'_def s_def, simp)

	      apply (cut_tac s_s'_def s_def p_domC', simp)

	      apply (simp only: i_def)

	      apply (rule s_def)

	      apply (cut_tac s_s'_def s_def p_def, simp)

	      apply (cut_tac s_s'_def s_def sigma_def, simp)

	      apply (rule s'_def)

	      apply (cut_tac s_s'_def s'_def sigma'_def exec_i,simp add: split_def split add: split_if_asm)

	      apply (cut_tac s_s'_def s'_def s_def e'_def, simp)

	      apply (cut_tac s_s'_def s'_def p'_def, simp)

	      apply (cut_tac check_i i_def, simp)

	      apply (cut_tac s_s'_def s'_def exec_i sigma'_def, simp split add: split_if_asm)

	      apply (simp add: P_def)
	      done

	  next 
	  
	    case (Goto t)

	    from this 
	    show ?thesis
	      apply -
	      apply (rule_tac s="s" and C="C" and M="M" and pc="pc" and t="t"
                     and h="h" and stk="stk" and loc="loc" and frs="frs" 
		     and s'="s'" and fr'="fr'" and P="P" and An="snd \<Pi>" 
                     in effS_wpF_Goto)
	      apply simp
	     
	      apply (rule wf_Pi)

	      apply (cut_tac handlesEx_p s_s'_def s'_def, simp)

	      apply (cut_tac cmd_p s_s'_def s_def, simp)

	      apply (cut_tac s_s'_def s_def p_domC', simp)

	      apply (simp only: i_def)

	      apply (rule s_def)

	      apply (cut_tac s_s'_def s_def p_def, simp)

	      apply (cut_tac s_s'_def s_def sigma_def, simp)

	      apply (rule s'_def)

	      apply (cut_tac s_s'_def s'_def sigma'_def exec_i,simp add: split_def split add: split_if_asm)

	      apply (cut_tac s_s'_def s'_def s_def e'_def, simp)

	      apply (cut_tac s_s'_def s'_def p'_def, simp)

	      apply (cut_tac check_i i_def, simp)

	      apply (cut_tac s_s'_def s'_def exec_i sigma'_def, simp split add: split_if_asm)

	      apply (simp add: P_def)
	      done

	    next 
	  
	      case CmpEq

	    from this 
	    show ?thesis
	      apply -
	      apply (rule_tac s="s" and C="C" and M="M" and pc="pc" 
                     and h="h" and stk="stk" and loc="loc" and frs="frs" 
		     and s'="s'" and fr'="fr'" and P="P" and An="snd \<Pi>" 
                     in effS_wpF_CmpEq)
	      apply simp
	     
	      apply (rule wf_Pi)

	      apply (cut_tac handlesEx_p s_s'_def s'_def, simp)

	      apply (cut_tac cmd_p s_s'_def s_def, simp)

	      apply (cut_tac s_s'_def s_def p_domC', simp)

	      apply (simp only: i_def)

	      apply (rule s_def)

	      apply (cut_tac s_s'_def s_def p_def, simp)

	      apply (cut_tac s_s'_def s_def sigma_def, simp)

	      apply (rule s'_def)

	      apply (cut_tac s_s'_def s'_def sigma'_def exec_i,simp add: split_def split add: split_if_asm)

	      apply (cut_tac s_s'_def s'_def s_def e'_def, simp)

	      apply (cut_tac s_s'_def s'_def p'_def, simp)

	      apply (cut_tac check_i i_def, simp)

	      apply (cut_tac s_s'_def s'_def exec_i sigma'_def, simp split add: split_if_asm)

	      apply (simp add: P_def)
	      done

	  next 
	  
	    case (IfIntCmp ro t)

	    from this 
	    show ?thesis
	      apply -
	      apply (rule_tac s="s" and C="C" and M="M" and pc="pc" and t="t"
                     and h="h" and stk="stk" and loc="loc" and frs="frs" 
		     and s'="s'" and fr'="fr'" and P="P" and An="snd \<Pi>" 
                     in effS_wpF_IfIntCmp)
	      apply simp
	     
	      apply (rule wf_Pi)

	      apply (cut_tac handlesEx_p s_s'_def s'_def, simp)

	      apply (cut_tac cmd_p s_s'_def s_def, simp)

	      apply (cut_tac s_s'_def s_def p_domC', simp)

	      apply (simp only: i_def)

	      apply (rule s_def)

	      apply (cut_tac s_s'_def s_def p_def, simp)

	      apply (cut_tac s_s'_def s_def sigma_def, simp)

	      apply (rule s'_def)

	      apply (cut_tac s_s'_def s'_def sigma'_def exec_i,simp add: split_def split add: split_if_asm)

	      apply (cut_tac s_s'_def s'_def s_def e'_def, simp)

	      apply (cut_tac s_s'_def s'_def p'_def, simp)

	      apply (cut_tac check_i i_def, simp)

	      apply (cut_tac s_s'_def s'_def exec_i sigma'_def, simp split add: split_if_asm)

	      apply (simp add: P_def)
	      done

	  next 
	  
	    case (IfFalse t)

	    from this 
	    show ?thesis
	      apply -
	      apply (rule_tac s="s" and C="C" and M="M" and pc="pc" and t="t"
                     and h="h" and stk="stk" and loc="loc" and frs="frs" 
		     and s'="s'" and fr'="fr'" and P="P" and An="snd \<Pi>" 
                     in effS_wpF_IfFalse)
	      apply simp
	     
	      apply (rule wf_Pi)

	      apply (cut_tac handlesEx_p s_s'_def s'_def, simp)

	      apply (cut_tac cmd_p s_s'_def s_def, simp)

	      apply (cut_tac s_s'_def s_def p_domC', simp)

	      apply (simp only: i_def)

	      apply (rule s_def)

	      apply (cut_tac s_s'_def s_def p_def, simp)

	      apply (cut_tac s_s'_def s_def sigma_def, simp)

	      apply (rule s'_def)

	      apply (cut_tac s_s'_def s'_def sigma'_def exec_i,simp add: split_def split add: split_if_asm)

	      apply (cut_tac s_s'_def s'_def s_def e'_def, simp)

	      apply (cut_tac s_s'_def s'_def p'_def, simp)

	      apply (cut_tac check_i i_def, simp)

	      apply (cut_tac s_s'_def s'_def exec_i sigma'_def, simp split add: split_if_asm)

	      apply (simp add: P_def)
	      done

	  next 

	    case Throw

	    from this exec_i s_s'_def s_def s'_def sigma_def sigma'_def show ?thesis
	      apply -
	      apply (simp split add: split_if_asm)
	      done
	  qed
	qed
  }
  qed
(*>*)

end