# HG changeset patch
# User kleing
# Date 987075615 -7200
# Node ID 71c00cb091d2dbf40e0200c974d5a279cd08935c
# Parent  a6816d47f41db00a5b5836dd52d2a0ed2f192609
cleanup, tuned

diff -r a6816d47f41d -r 71c00cb091d2 src/HOL/MicroJava/BV/BVSpecTypeSafe.thy
--- a/src/HOL/MicroJava/BV/BVSpecTypeSafe.thy	Thu Apr 12 12:45:05 2001 +0200
+++ b/src/HOL/MicroJava/BV/BVSpecTypeSafe.thy	Thu Apr 12 13:40:15 2001 +0200
@@ -2,7 +2,6 @@
     ID:         $Id$
     Author:     Cornelia Pusch
     Copyright   1999 Technische Universitaet Muenchen
-
 Proof that the specification of the bytecode verifier only admits type safe
 programs.
 *)
@@ -14,6 +13,8 @@
 lemmas defs1 = sup_state_conv correct_state_def correct_frame_def 
                wt_instr_def step_def
 
+lemmas widen_rules[intro] = approx_val_widen approx_loc_widen approx_stk_widen
+
 lemmas [simp del] = split_paired_All
 
 lemma wt_jvm_prog_impl_wt_instr_cor:
@@ -38,11 +39,7 @@
     G,phi \<turnstile>JVM (None, hp, (stk,loc,C,sig,pc)#frs)\<surd> |] 
 ==> G,phi \<turnstile>JVM state'\<surd>"
 apply (clarsimp simp add: defs1 map_eq_Cons)
-apply (rule conjI)
- apply (rule approx_loc_imp_approx_val_sup)
- apply simp+
-apply (blast intro: approx_stk_imp_approx_stk_sup 
-                    approx_loc_imp_approx_loc_sup)
+apply (blast intro: approx_loc_imp_approx_val_sup)
 done
 
 lemma Store_correct:
@@ -54,10 +51,7 @@
   G,phi \<turnstile>JVM (None, hp, (stk,loc,C,sig,pc)#frs)\<surd> |]
 ==> G,phi \<turnstile>JVM state'\<surd>"
 apply (clarsimp simp add: defs1 map_eq_Cons)
-apply (rule conjI)
- apply (blast intro: approx_stk_imp_approx_stk_sup)
-apply (blast intro: approx_loc_imp_approx_loc_subst 
-                    approx_loc_imp_approx_loc_sup)
+apply (blast intro: approx_loc_subst)
 done
 
 
@@ -68,10 +62,9 @@
     wt_instr (ins!pc) G rT (phi C sig) maxs (length ins) pc; 
     Some state' = exec (G, None, hp, (stk,loc,C,sig,pc)#frs);
     G,phi \<turnstile>JVM (None, hp, (stk,loc,C,sig,pc)#frs)\<surd> |]
-==> G,phi \<turnstile>JVM state'\<surd>"
-apply (clarsimp simp add: defs1 approx_val_def sup_PTS_eq map_eq_Cons)
-apply (blast dest: conf_litval intro: conf_widen approx_stk_imp_approx_stk_sup 
-                    approx_loc_imp_approx_loc_sup)
+==> G,phi \<turnstile>JVM state'\<surd>" 
+apply (clarsimp simp add: defs1 sup_PTS_eq map_eq_Cons)
+apply (blast dest: conf_litval intro: conf_widen)
 done
 
 
@@ -81,7 +74,7 @@
   ==> G,h\<turnstile>v::\<preceq>T"
 apply (unfold cast_ok_def)
 apply (frule widen_Class)
-apply (elim exE disjE)
+apply (elim exE disjE) 
  apply (simp add: null)
 apply (clarsimp simp add: conf_def obj_ty_def)
 apply (cases v)
@@ -97,9 +90,8 @@
     Some state' = exec (G, None, hp, (stk,loc,C,sig,pc)#frs) ; 
     G,phi \<turnstile>JVM (None, hp, (stk,loc,C,sig,pc)#frs)\<surd> |] 
 ==> G,phi \<turnstile>JVM state'\<surd>"
-apply (clarsimp simp add: defs1 map_eq_Cons raise_xcpt_def approx_val_def)
-apply (blast intro: approx_stk_imp_approx_stk_sup 
-                    approx_loc_imp_approx_loc_sup Cast_conf2)
+apply (clarsimp simp add: defs1 map_eq_Cons raise_xcpt_def)
+apply (blast intro: Cast_conf2)
 done
 
 
@@ -111,18 +103,17 @@
   Some state' = exec (G, None, hp, (stk,loc,C,sig,pc)#frs) ; 
   G,phi \<turnstile>JVM (None, hp, (stk,loc,C,sig,pc)#frs)\<surd> |] 
 ==> G,phi \<turnstile>JVM state'\<surd>"
-apply (clarsimp simp add: defs1 raise_xcpt_def map_eq_Cons approx_val_def
+apply (clarsimp simp add: defs1 raise_xcpt_def map_eq_Cons
                 split: option.split)
 apply (frule conf_widen)
 apply assumption+
 apply (drule conf_RefTD)
-apply (clarsimp simp add: defs1 approx_val_def)
+apply (clarsimp simp add: defs1)
 apply (rule conjI)
  apply (drule widen_cfs_fields)
  apply assumption+
  apply (force intro: conf_widen simp add: hconf_def oconf_def lconf_def)
-apply (blast intro: approx_stk_imp_approx_stk_sup 
-                    approx_loc_imp_approx_loc_sup)
+apply blast
 done
 
 lemma Putfield_correct:
@@ -134,19 +125,18 @@
   G,phi \<turnstile>JVM (None, hp, (stk,loc,C,sig,pc)#frs)\<surd> |] 
 ==> G,phi \<turnstile>JVM state'\<surd>"
 apply (clarsimp simp add: defs1 raise_xcpt_def split: option.split List.split)
-apply (clarsimp simp add: approx_val_def)
 apply (frule conf_widen)
 prefer 2
-apply assumption
-apply assumption
+  apply assumption
+ apply assumption
 apply (drule conf_RefTD)
 apply clarsimp
 apply (blast 
        intro: 
-         hext_upd_obj approx_stk_imp_approx_stk_sup_heap
-         approx_stk_imp_approx_stk_sup approx_loc_imp_approx_loc_sup_heap 
-         approx_loc_imp_approx_loc_sup hconf_imp_hconf_field_update
-         correct_frames_imp_correct_frames_field_update conf_widen 
+         hext_upd_obj approx_stk_sup_heap
+         approx_loc_sup_heap 
+         hconf_field_update
+         correct_frames_field_update conf_widen 
        dest: 
          widen_cfs_fields)
 done
@@ -175,8 +165,8 @@
     frame:      "correct_frame G hp (ST,LT) maxl ins (stk, loc, C, sig, pc)" and
     frames:     "correct_frames G hp phi rT sig frs"
     by (auto simp add: correct_state_def iff del: not_None_eq)
+
   from phi_pc ins wt
-
   obtain ST' LT' where
     is_class_X: "is_class G X" and
     maxs:       "maxs < length ST" and
@@ -194,10 +184,8 @@
   moreover
   { assume "xp' = Some OutOfMemory"
     with exec ins meth new_Addr [symmetric]
-    have "state' = (Some OutOfMemory, hp, (stk, loc, C, sig, Suc pc) # frs)"
-      by simp
-    hence ?thesis
-      by (simp add: correct_state_def)
+    have "state' = (Some OutOfMemory, hp, (stk,loc,C,sig,Suc pc)#frs)" by simp
+    hence ?thesis by (simp add: correct_state_def)
   }
   moreover
   { assume hp: "hp oref = None" and "xp' = None"
@@ -210,7 +198,7 @@
     moreover
     from wf hp heap_ok is_class_X
     have hp': "G \<turnstile>h ?hp' \<surd>"
-      by - (rule hconf_imp_hconf_newref, 
+      by - (rule hconf_newref, 
             auto simp add: oconf_def dest: fields_is_type)
     moreover
     from hp
@@ -219,15 +207,12 @@
     have "correct_frame G ?hp' (ST', LT') maxl ins ?f"
       apply (unfold correct_frame_def sup_state_conv)
       apply (clarsimp simp add: map_eq_Cons conf_def fun_upd_apply approx_val_def)
-      apply (blast intro: approx_stk_imp_approx_stk_sup_heap 
-                          approx_stk_imp_approx_stk_sup
-                          approx_loc_imp_approx_loc_sup_heap 
-                          approx_loc_imp_approx_loc_sup sup)
+      apply (blast intro: approx_stk_sup_heap approx_loc_sup_heap sup)
       done      
     moreover
     from hp frames wf heap_ok is_class_X
     have "correct_frames G ?hp' phi rT sig frs"
-      by - (rule correct_frames_imp_correct_frames_newref,
+      by - (rule correct_frames_newref,
             auto simp add: oconf_def dest: fields_is_type)
     ultimately
     have ?thesis
@@ -237,7 +222,6 @@
   show ?thesis by auto
 qed
 
-
 lemmas [simp del] = split_paired_Ex
 
 lemma Invoke_correct: 
@@ -272,13 +256,13 @@
 
   from ins wti phi_pc
   obtain apTs X ST LT D' rT body where 
-    s: "s = (rev apTs @ X # ST, LT)" and
-    l: "length apTs = length pTs" and
     is_class: "is_class G C'" and
-    X: "G\<turnstile> X \<preceq> Class C'" and
-    w: "\<forall>x\<in>set (zip apTs pTs). x \<in> widen G" and
-    mC': "method (G, C') (mn, pTs) = Some (D', rT, body)" and
-    pc:  "Suc pc < length ins" and
+    s:  "s = (rev apTs @ X # ST, LT)" and
+    l:  "length apTs = length pTs" and
+    X:  "G\<turnstile> X \<preceq> Class C'" and
+    w:  "\<forall>x\<in>set (zip apTs pTs). x \<in> widen G" and
+    mC':"method (G, C') (mn, pTs) = Some (D', rT, body)" and
+    pc: "Suc pc < length ins" and
     step: "G \<turnstile> step (Invoke C' mn pTs) G (Some s) <=' phi C sig!Suc pc"
     by (simp add: wt_instr_def del: not_None_eq) (elim exE conjE, rule that)
 
@@ -307,7 +291,7 @@
     stk':   "stk = opTs @ oX # stk'" and
     l_o:    "length opTs = length apTs" 
             "length stk' = length ST"  
-    by - (drule approx_stk_append_lemma, simp, elim, simp)
+    by - (drule approx_stk_append, simp, elim, simp)
 
   from oX 
   have "\<exists>T'. typeof (option_map obj_ty \<circ> hp) oX = Some T' \<and> G \<turnstile> T' \<preceq> X"
@@ -322,15 +306,13 @@
     by (elim, simp)
 
   from stk' l_o l
-  have oX_pos: "last (take (Suc (length pTs)) stk) = oX" 
-    by simp
+  have oX_pos: "last (take (Suc (length pTs)) stk) = oX" by simp
 
   with state' method ins 
   have Null_ok: "oX = Null ==> ?thesis"
     by (simp add: correct_state_def raise_xcpt_def)
   
-  { fix ref
-    assume oX_Addr: "oX = Addr ref"
+  { fix ref assume oX_Addr: "oX = Addr ref"
     
     with oX_Ref
     obtain obj where
@@ -424,29 +406,20 @@
       have a: "approx_val G hp (Addr ref) (OK (Class D''))"
         by (simp add: approx_val_def conf_def)
 
-      from w l
-      have "\<forall>x\<in>set (zip (rev apTs) (rev pTs)). x \<in> widen G"
-        by (auto simp add: zip_rev)
-      with wfprog l l_o opTs
-      have "approx_loc G hp opTs (map OK (rev pTs))"
-        by (auto intro: assConv_approx_stk_imp_approx_loc)
-      hence "approx_stk G hp opTs (rev pTs)"
-        by (simp add: approx_stk_def)
-      hence "approx_stk G hp (rev opTs) pTs"
-        by (simp add: approx_stk_rev)
+      from opTs w l l_o wfprog 
+      have "approx_stk G hp opTs (rev pTs)" 
+	by (auto elim!: approx_stk_all_widen simp add: zip_rev)
+      hence "approx_stk G hp (rev opTs) pTs" by (subst approx_stk_rev)
 
       with r a l_o l
       have "approx_loc G hp (Addr ref # rev opTs @ replicate mxl' arbitrary) ?LT"
         (is "approx_loc G hp ?lt ?LT")
         by (auto simp add: approx_loc_append approx_stk_def)
 
-      from wfprog this sup_loc
-      have "approx_loc G hp ?lt LT0"
-        by (rule approx_loc_imp_approx_loc_sup)
-
+      from this sup_loc wfprog
+      have "approx_loc G hp ?lt LT0" by (rule approx_loc_widen)
       with start l_o l
-      show ?thesis
-        by (simp add: correct_frame_def)
+      show ?thesis by (simp add: correct_frame_def)
     qed
 
     have c_f': "correct_frame G hp (tl ST', LT') maxl ins ?f'"
@@ -456,28 +429,26 @@
         by (simp add: step_def sup_state_conv)
       with wfprog a_loc
       have a: "approx_loc G hp loc LT'"
-        by (rule approx_loc_imp_approx_loc_sup)
+        by - (rule approx_loc_widen)
       moreover
       from s s' mC' step l
       have  "G \<turnstile> map OK ST <=l map OK (tl ST')"
         by (auto simp add: step_def sup_state_conv map_eq_Cons)
       with wfprog a_stk'
       have "approx_stk G hp stk' (tl ST')"
-        by (rule approx_stk_imp_approx_stk_sup)
+        by - (rule approx_stk_widen)
       ultimately
-      show ?thesis
-        by (simp add: correct_frame_def suc_l pc)
+      show ?thesis by (simp add: correct_frame_def suc_l pc)
     qed
 
     with state'_val heap_ok mD'' ins method phi_pc s X' l 
          frames s' LT0 c_f c_f' is_class_C
-    have ?thesis
-      by (simp add: correct_state_def) (intro exI conjI, force+)
+    have ?thesis 
+      by (simp add: correct_state_def) (intro exI conjI, blast, assumption+)
   }
   
   with Null_ok oX_Ref
-  show "G,phi \<turnstile>JVM state'\<surd>"
-    by - (cases oX, auto)
+  show "G,phi \<turnstile>JVM state'\<surd>" by - (cases oX, auto)
 qed
 
 lemmas [simp del] = map_append
@@ -493,12 +464,14 @@
 apply (clarsimp simp add: neq_Nil_conv defs1 split: split_if_asm iff del: not_None_eq)
 apply (frule wt_jvm_prog_impl_wt_instr)
 apply (assumption, assumption, erule Suc_lessD)
+apply (clarsimp simp add: map_eq_Cons append_eq_conv_conj defs1)
 apply (unfold wt_jvm_prog_def)
-apply (fastsimp
-  dest: subcls_widen_methd
-  elim: widen_trans [COMP swap_prems_rl]
-  intro: conf_widen
-  simp: approx_val_def append_eq_conv_conj map_eq_Cons defs1)
+apply (frule subcls_widen_methd, assumption+)
+apply clarify
+apply simp
+apply (erule conf_widen, assumption)
+apply (erule widen_trans)+
+apply blast
 done
 
 lemmas [simp] = map_append
@@ -512,8 +485,7 @@
   G,phi \<turnstile>JVM (None, hp, (stk,loc,C,sig,pc)#frs)\<surd> |] 
 ==> G,phi \<turnstile>JVM state'\<surd>"
 apply (clarsimp simp add: defs1)
-apply (fast intro: approx_loc_imp_approx_loc_sup 
-                   approx_stk_imp_approx_stk_sup)
+apply fast
 done
 
 
@@ -526,8 +498,7 @@
   G,phi \<turnstile>JVM (None, hp, (stk,loc,C,sig,pc)#frs)\<surd> |] 
 ==> G,phi \<turnstile>JVM state'\<surd>"
 apply (clarsimp simp add: defs1)
-apply (fast intro: approx_loc_imp_approx_loc_sup 
-                   approx_stk_imp_approx_stk_sup)
+apply fast
 done
 
 lemma Pop_correct:
@@ -539,8 +510,7 @@
   G,phi \<turnstile>JVM (None, hp, (stk,loc,C,sig,pc)#frs)\<surd> |] 
 ==> G,phi \<turnstile>JVM state'\<surd>"
 apply (clarsimp simp add: defs1)
-apply (fast intro: approx_loc_imp_approx_loc_sup 
-                   approx_stk_imp_approx_stk_sup)
+apply fast
 done
 
 lemma Dup_correct:
@@ -552,10 +522,7 @@
   G,phi \<turnstile>JVM (None, hp, (stk,loc,C,sig,pc)#frs)\<surd> |] 
 ==> G,phi \<turnstile>JVM state'\<surd>"
 apply (clarsimp simp add: defs1 map_eq_Cons)
-apply (force intro: approx_stk_imp_approx_stk_sup 
-                    approx_val_imp_approx_val_sup 
-                    approx_loc_imp_approx_loc_sup
-             simp add: defs1 map_eq_Cons)
+apply (blast intro: conf_widen)
 done
 
 lemma Dup_x1_correct:
@@ -567,10 +534,7 @@
   G,phi \<turnstile>JVM (None, hp, (stk,loc,C,sig,pc)#frs)\<surd> |] 
 ==> G,phi \<turnstile>JVM state'\<surd>"
 apply (clarsimp simp add: defs1 map_eq_Cons)
-apply (force intro: approx_stk_imp_approx_stk_sup 
-                    approx_val_imp_approx_val_sup 
-                    approx_loc_imp_approx_loc_sup
-             simp add: defs1 map_eq_Cons)
+apply (blast intro: conf_widen)
 done
 
 lemma Dup_x2_correct:
@@ -582,10 +546,7 @@
   G,phi \<turnstile>JVM (None, hp, (stk,loc,C,sig,pc)#frs)\<surd> |] 
 ==> G,phi \<turnstile>JVM state'\<surd>"
 apply (clarsimp simp add: defs1 map_eq_Cons)
-apply (force intro: approx_stk_imp_approx_stk_sup 
-                    approx_val_imp_approx_val_sup 
-                    approx_loc_imp_approx_loc_sup
-             simp add: defs1 map_eq_Cons)
+apply (blast intro: conf_widen)
 done
 
 lemma Swap_correct:
@@ -597,10 +558,7 @@
   G,phi \<turnstile>JVM (None, hp, (stk,loc,C,sig,pc)#frs)\<surd> |] 
 ==> G,phi \<turnstile>JVM state'\<surd>"
 apply (clarsimp simp add: defs1 map_eq_Cons)
-apply (force intro: approx_stk_imp_approx_stk_sup 
-                    approx_val_imp_approx_val_sup 
-                    approx_loc_imp_approx_loc_sup
-             simp add: defs1 map_eq_Cons)
+apply (blast intro: conf_widen)
 done
 
 lemma IAdd_correct:
@@ -612,9 +570,7 @@
   G,phi \<turnstile>JVM (None, hp, (stk,loc,C,sig,pc)#frs)\<surd> |] 
 ==> G,phi \<turnstile>JVM state'\<surd>"
 apply (clarsimp simp add: defs1 map_eq_Cons approx_val_def conf_def)
-apply (blast intro: approx_stk_imp_approx_stk_sup 
-                    approx_val_imp_approx_val_sup 
-                    approx_loc_imp_approx_loc_sup)
+apply blast
 done
 
 
@@ -650,7 +606,6 @@
 apply (rule Ifcmpeq_correct, assumption+)
 done
 
-
 section {* Main *}
 
 lemma correct_state_impl_Some_method:
diff -r a6816d47f41d -r 71c00cb091d2 src/HOL/MicroJava/BV/Correct.thy
--- a/src/HOL/MicroJava/BV/Correct.thy	Thu Apr 12 12:45:05 2001 +0200
+++ b/src/HOL/MicroJava/BV/Correct.thy	Thu Apr 12 13:40:15 2001 +0200
@@ -10,6 +10,12 @@
 
 theory Correct = BVSpec:
 
+lemma list_all2_rev [simp]:
+  "list_all2 P (rev l) (rev l') = list_all2 P l l'"
+  apply (unfold list_all2_def)
+  apply (simp add: zip_rev cong: conj_cong)
+  done
+
 constdefs
   approx_val :: "[jvm_prog,aheap,val,ty err] => bool"
   "approx_val G h v any == case any of Err => True | OK T => G,h\<turnstile>v::\<preceq>T"
@@ -32,7 +38,7 @@
 "correct_frames G hp phi rT0 sig0 [] = True"
 
 "correct_frames G hp phi rT0 sig0 (f#frs) =
-	(let (stk,loc,C,sig,pc) = f in
+  (let (stk,loc,C,sig,pc) = f in
   (\<exists>ST LT rT maxs maxl ins.
     phi C sig ! pc = Some (ST,LT) \<and> is_class G C \<and> 
     method (G,C) sig = Some(C,rT,(maxs,maxl,ins)) \<and>
@@ -71,254 +77,248 @@
  correct_state :: "[jvm_prog,prog_type,jvm_state] => bool"
                   ("_,_ |-JVM _ [ok]"  [51,51] 50)
 
+
+lemma sup_ty_opt_OK:
+  "(G \<turnstile> X <=o (OK T')) = (\<exists>T. X = OK T \<and> G \<turnstile> T \<preceq> T')"
+  apply (cases X)
+  apply auto
+  done
+
+
 section {* approx-val *}
 
-lemma approx_val_Err:
+lemma approx_val_Err [simp,intro!]:
   "approx_val G hp x Err"
-by (simp add: approx_val_def)
-
-lemma approx_val_Null:
-  "approx_val G hp Null (OK (RefT x))"
-by (auto intro: null simp add: approx_val_def)
+  by (simp add: approx_val_def)
 
-lemma approx_val_imp_approx_val_assConvertible [rule_format]: 
-  "wf_prog wt G ==> approx_val G hp v (OK T) --> G\<turnstile> T\<preceq>T' 
-  --> approx_val G hp v (OK T')"
-by (cases T) (auto intro: conf_widen simp add: approx_val_def)
+lemma approx_val_OK [iff]: 
+  "approx_val G hp x (OK T) = (G,hp \<turnstile> x ::\<preceq> T)"
+  by (simp add: approx_val_def)
 
-lemma approx_val_imp_approx_val_sup_heap [rule_format]:
-  "approx_val G hp v at --> hp \<le>| hp' --> approx_val G hp' v at"
-apply (simp add: approx_val_def split: err.split)
-apply (blast intro: conf_hext)
-done
+lemma approx_val_Null [simp,intro!]:
+  "approx_val G hp Null (OK (RefT x))"
+  by (auto simp add: approx_val_def)
 
-lemma approx_val_imp_approx_val_heap_update:
-  "[|hp a = Some obj'; G,hp\<turnstile> v::\<preceq>T; obj_ty obj = obj_ty obj'|] 
-  ==> G,hp(a\<mapsto>obj)\<turnstile> v::\<preceq>T"
-by (cases v, auto simp add: obj_ty_def conf_def)
+lemma approx_val_sup_heap:
+  "\<lbrakk> approx_val G hp v T; hp \<le>| hp' \<rbrakk> \<Longrightarrow> approx_val G hp' v T"
+  by (cases T) (blast intro: conf_hext)+
 
-lemma approx_val_imp_approx_val_sup [rule_format]:
-  "wf_prog wt G ==> (approx_val G h v us) --> (G \<turnstile> us <=o us') 
-  --> (approx_val G h v us')"
-apply (simp add: sup_PTS_eq approx_val_def split: err.split)
-apply (blast intro: conf_widen)
-done
+lemma approx_val_heap_update:
+  "[| hp a = Some obj'; G,hp\<turnstile> v::\<preceq>T; obj_ty obj = obj_ty obj'|] 
+  ==> G,hp(a\<mapsto>obj)\<turnstile> v::\<preceq>T"
+  by (cases v, auto simp add: obj_ty_def conf_def)
 
-lemma approx_loc_imp_approx_val_sup:
-  "[| wf_prog wt G; approx_loc G hp loc LT; idx < length LT; 
-      v = loc!idx; G \<turnstile> LT!idx <=o at |]
-  ==> approx_val G hp v at"
-apply (unfold approx_loc_def)
-apply (unfold list_all2_def)
-apply (auto intro: approx_val_imp_approx_val_sup 
-            simp add: split_def all_set_conv_all_nth)
-done
-
+lemma approx_val_widen:
+  "\<lbrakk> approx_val G hp v T; G \<turnstile> T <=o T'; wf_prog wt G \<rbrakk>
+  \<Longrightarrow> approx_val G hp v T'"
+  by (cases T', auto simp add: sup_ty_opt_OK intro: conf_widen)
 
 section {* approx-loc *}
 
+lemma approx_loc_Nil [simp,intro!]:
+  "approx_loc G hp [] []"
+  by (simp add: approx_loc_def)
+
 lemma approx_loc_Cons [iff]:
-  "approx_loc G hp (s#xs) (l#ls) = 
-  (approx_val G hp s l \<and> approx_loc G hp xs ls)"
+  "approx_loc G hp (l#ls) (L#LT) = 
+  (approx_val G hp l L \<and> approx_loc G hp ls LT)"
 by (simp add: approx_loc_def)
 
-lemma assConv_approx_stk_imp_approx_loc [rule_format]:
-  "wf_prog wt G ==> (\<forall>tt'\<in>set (zip tys_n ts). tt' \<in> widen G) 
-  --> length tys_n = length ts --> approx_stk G hp s tys_n --> 
-  approx_loc G hp s (map OK ts)"
-apply (unfold approx_stk_def approx_loc_def list_all2_def)
-apply (clarsimp simp add: all_set_conv_all_nth)
-apply (rule approx_val_imp_approx_val_assConvertible)
-apply auto
+lemma approx_loc_nth:
+  "\<lbrakk> approx_loc G hp loc LT; n < length LT \<rbrakk>
+  \<Longrightarrow> approx_val G hp (loc!n) (LT!n)"
+  by (simp add: approx_loc_def list_all2_conv_all_nth)
+
+lemma approx_loc_imp_approx_val_sup:
+  "\<lbrakk>approx_loc G hp loc LT; n < length LT; LT ! n = OK T; G \<turnstile> T \<preceq> T'; wf_prog wt G\<rbrakk> 
+  \<Longrightarrow> G,hp \<turnstile> (loc!n) ::\<preceq> T'"
+  apply (drule approx_loc_nth, assumption) 
+  apply simp
+  apply (erule conf_widen, assumption+)
+  done
+
+lemma approx_loc_conv_all_nth:
+  "approx_loc G hp loc LT = 
+  (length loc = length LT \<and> (\<forall>n < length loc. approx_val G hp (loc!n) (LT!n)))"
+  by (simp add: approx_loc_def list_all2_conv_all_nth)
+
+lemma approx_loc_sup_heap:
+  "\<lbrakk> approx_loc G hp loc LT; hp \<le>| hp' \<rbrakk>
+  \<Longrightarrow> approx_loc G hp' loc LT"
+  apply (clarsimp simp add: approx_loc_conv_all_nth)
+  apply (blast intro: approx_val_sup_heap)
+  done
+
+lemma approx_loc_widen:
+  "\<lbrakk> approx_loc G hp loc LT; G \<turnstile> LT <=l LT'; wf_prog wt G \<rbrakk>
+  \<Longrightarrow> approx_loc G hp loc LT'"
+apply (unfold Listn.le_def lesub_def sup_loc_def)
+apply (simp (no_asm_use) only: list_all2_conv_all_nth approx_loc_conv_all_nth)
+apply (simp (no_asm_simp))
+apply clarify
+apply (erule allE, erule impE) 
+ apply simp
+apply (erule approx_val_widen)
+ apply simp
+apply assumption
 done
 
-
-lemma approx_loc_imp_approx_loc_sup_heap [rule_format]:
-  "\<forall>lvars. approx_loc G hp lvars lt --> hp \<le>| hp' 
-  --> approx_loc G hp' lvars lt"
-apply (unfold approx_loc_def list_all2_def)
-apply (cases lt)
- apply simp
-apply clarsimp
-apply (rule approx_val_imp_approx_val_sup_heap)
-apply auto
-done
-
-lemma approx_loc_imp_approx_loc_sup [rule_format]:
-  "wf_prog wt G ==> approx_loc G hp lvars lt --> G \<turnstile> lt <=l lt' 
-  --> approx_loc G hp lvars lt'"
-apply (unfold Listn.le_def lesub_def sup_loc_def approx_loc_def list_all2_def)
-apply (auto simp add: all_set_conv_all_nth)
-apply (auto elim: approx_val_imp_approx_val_sup)
-done
-
-
-lemma approx_loc_imp_approx_loc_subst [rule_format]:
-  "\<forall>loc idx x X. (approx_loc G hp loc LT) --> (approx_val G hp x X) 
-  --> (approx_loc G hp (loc[idx:=x]) (LT[idx:=X]))"
+lemma approx_loc_subst:
+  "\<lbrakk> approx_loc G hp loc LT; approx_val G hp x X \<rbrakk>
+  \<Longrightarrow> approx_loc G hp (loc[idx:=x]) (LT[idx:=X])"
 apply (unfold approx_loc_def list_all2_def)
 apply (auto dest: subsetD [OF set_update_subset_insert] simp add: zip_update)
 done
 
-
-lemmas [cong] = conj_cong 
-
-lemma approx_loc_append [rule_format]:
-  "\<forall>L1 l2 L2. length l1=length L1 --> 
+lemma approx_loc_append:
+  "length l1=length L1 \<Longrightarrow>
   approx_loc G hp (l1@l2) (L1@L2) = 
   (approx_loc G hp l1 L1 \<and> approx_loc G hp l2 L2)"
-apply (unfold approx_loc_def list_all2_def)
-apply simp
-apply blast
-done
-
-lemmas [cong del] = conj_cong
-
+  apply (unfold approx_loc_def list_all2_def)
+  apply (simp cong: conj_cong)
+  apply blast
+  done
 
 section {* approx-stk *}
 
 lemma approx_stk_rev_lem:
   "approx_stk G hp (rev s) (rev t) = approx_stk G hp s t"
-apply (unfold approx_stk_def approx_loc_def list_all2_def)
-apply (auto simp add: zip_rev sym [OF rev_map])
-done
+  apply (unfold approx_stk_def approx_loc_def)
+  apply (simp add: rev_map [THEN sym])
+  done
 
 lemma approx_stk_rev:
   "approx_stk G hp (rev s) t = approx_stk G hp s (rev t)"
-by (auto intro: subst [OF approx_stk_rev_lem])
-
+  by (auto intro: subst [OF approx_stk_rev_lem])
 
-lemma approx_stk_imp_approx_stk_sup_heap [rule_format]:
-  "\<forall>lvars. approx_stk G hp lvars lt --> hp \<le>| hp' 
-  --> approx_stk G hp' lvars lt"
-by (auto intro: approx_loc_imp_approx_loc_sup_heap simp add: approx_stk_def)
+lemma approx_stk_sup_heap:
+  "\<lbrakk> approx_stk G hp stk ST; hp \<le>| hp' \<rbrakk> \<Longrightarrow> approx_stk G hp' stk ST"
+  by (auto intro: approx_loc_sup_heap simp add: approx_stk_def)
 
-lemma approx_stk_imp_approx_stk_sup [rule_format]:
-  "wf_prog wt G ==> approx_stk G hp lvars st --> (G \<turnstile> map OK st <=l (map OK st')) 
-  --> approx_stk G hp lvars st'" 
-by (auto intro: approx_loc_imp_approx_loc_sup simp add: approx_stk_def)
+lemma approx_stk_widen:
+  "\<lbrakk> approx_stk G hp stk ST; G \<turnstile> map OK ST <=l map OK ST'; wf_prog wt G \<rbrakk>
+  \<Longrightarrow> approx_stk G hp stk ST'" 
+  by (auto elim: approx_loc_widen simp add: approx_stk_def)
 
 lemma approx_stk_Nil [iff]:
   "approx_stk G hp [] []"
-by (simp add: approx_stk_def approx_loc_def)
+  by (simp add: approx_stk_def)
 
 lemma approx_stk_Cons [iff]:
-  "approx_stk G hp (x # stk) (S#ST) = 
-   (approx_val G hp x (OK S) \<and> approx_stk G hp stk ST)"
-by (simp add: approx_stk_def approx_loc_def)
+  "approx_stk G hp (x#stk) (S#ST) = 
+  (approx_val G hp x (OK S) \<and> approx_stk G hp stk ST)"
+  by (simp add: approx_stk_def)
 
 lemma approx_stk_Cons_lemma [iff]:
   "approx_stk G hp stk (S#ST') = 
   (\<exists>s stk'. stk = s#stk' \<and> approx_val G hp s (OK S) \<and> approx_stk G hp stk' ST')"
-by (simp add: list_all2_Cons2 approx_stk_def approx_loc_def)
+  by (simp add: list_all2_Cons2 approx_stk_def approx_loc_def)
+
+lemma approx_stk_append:
+  "approx_stk G hp stk (S@S') \<Longrightarrow>
+  (\<exists>s stk'. stk = s@stk' \<and> length s = length S \<and> length stk' = length S' \<and> 
+            approx_stk G hp s S \<and> approx_stk G hp stk' S')"
+  by (simp add: list_all2_append2 approx_stk_def approx_loc_def)
 
-lemma approx_stk_append_lemma:
-  "approx_stk G hp stk (S@ST') ==>
-   (\<exists>s stk'. stk = s@stk' \<and> length s = length S \<and> length stk' = length ST' \<and> 
-             approx_stk G hp s S \<and> approx_stk G hp stk' ST')"
-by (simp add: list_all2_append2 approx_stk_def approx_loc_def)
-
+lemma approx_stk_all_widen:
+  "\<lbrakk> approx_stk G hp stk ST; \<forall>x \<in> set (zip ST ST'). x \<in> widen G; length ST = length ST'; wf_prog wt G \<rbrakk> 
+  \<Longrightarrow> approx_stk G hp stk ST'"
+apply (unfold approx_stk_def)
+apply (clarsimp simp add: approx_loc_conv_all_nth all_set_conv_all_nth)
+apply (erule allE, erule impE, assumption)
+apply (erule allE, erule impE, assumption)
+apply (erule conf_widen, assumption+)
+done
 
 section {* oconf *}
 
-lemma correct_init_obj:
-  "[|is_class G C; wf_prog wt G|] ==> 
-  G,h \<turnstile> (C, map_of (map (\<lambda>(f,fT).(f,default_val fT)) (fields(G,C)))) \<surd>"
-apply (unfold oconf_def lconf_def)
-apply (simp add: map_of_map)
-apply (force intro: defval_conf dest: map_of_SomeD fields_is_type)
-done
-
-
-lemma oconf_imp_oconf_field_update [rule_format]:
+lemma oconf_field_update:
   "[|map_of (fields (G, oT)) FD = Some T; G,hp\<turnstile>v::\<preceq>T; G,hp\<turnstile>(oT,fs)\<surd> |]
   ==> G,hp\<turnstile>(oT, fs(FD\<mapsto>v))\<surd>"
-by (simp add: oconf_def lconf_def)
-
-lemma oconf_imp_oconf_heap_newref [rule_format]:
-"hp oref = None --> G,hp\<turnstile>obj\<surd> --> G,hp\<turnstile>obj'\<surd> --> G,(hp(oref\<mapsto>obj'))\<turnstile>obj\<surd>"
-apply (unfold oconf_def lconf_def)
-apply simp
-apply (fast intro: conf_hext hext_new)
-done
+  by (simp add: oconf_def lconf_def)
 
-lemma oconf_imp_oconf_heap_update [rule_format]:
-  "hp a = Some obj' --> obj_ty obj' = obj_ty obj'' --> G,hp\<turnstile>obj\<surd> 
-  --> G,hp(a\<mapsto>obj'')\<turnstile>obj\<surd>"
-apply (unfold oconf_def lconf_def)
-apply simp
-apply (force intro: approx_val_imp_approx_val_heap_update)
-done
+lemma oconf_newref:
+  "\<lbrakk>hp oref = None; G,hp \<turnstile> obj \<surd>; G,hp \<turnstile> obj' \<surd>\<rbrakk> \<Longrightarrow> G,hp(oref\<mapsto>obj') \<turnstile> obj \<surd>"
+  apply (unfold oconf_def lconf_def)
+  apply simp
+  apply (blast intro: conf_hext hext_new)
+  done
 
+lemma oconf_heap_update:
+  "\<lbrakk> hp a = Some obj'; obj_ty obj' = obj_ty obj''; G,hp\<turnstile>obj\<surd> \<rbrakk>
+  \<Longrightarrow> G,hp(a\<mapsto>obj'')\<turnstile>obj\<surd>"
+  apply (unfold oconf_def lconf_def)
+  apply (fastsimp intro: approx_val_heap_update)
+  done
 
 section {* hconf *}
 
-lemma hconf_imp_hconf_newref [rule_format]:
-  "hp oref = None --> G\<turnstile>h hp\<surd> --> G,hp\<turnstile>obj\<surd> --> G\<turnstile>h hp(oref\<mapsto>obj)\<surd>"
-apply (simp add: hconf_def)
-apply (fast intro: oconf_imp_oconf_heap_newref)
-done
+lemma hconf_newref:
+  "\<lbrakk> hp oref = None; G\<turnstile>h hp\<surd>; G,hp\<turnstile>obj\<surd> \<rbrakk> \<Longrightarrow> G\<turnstile>h hp(oref\<mapsto>obj)\<surd>"
+  apply (simp add: hconf_def)
+  apply (fast intro: oconf_newref)
+  done
 
-lemma hconf_imp_hconf_field_update [rule_format]:
-  "map_of (fields (G, oT)) (F, D) = Some T \<and> hp oloc = Some(oT,fs) \<and> 
-  G,hp\<turnstile>v::\<preceq>T \<and> G\<turnstile>h hp\<surd> --> G\<turnstile>h hp(oloc \<mapsto> (oT, fs((F,D)\<mapsto>v)))\<surd>"
-apply (simp add: hconf_def)
-apply (force intro: oconf_imp_oconf_heap_update oconf_imp_oconf_field_update 
-             simp add: obj_ty_def)
-done
+lemma hconf_field_update:
+  "\<lbrakk> map_of (fields (G, oT)) X = Some T; hp a = Some(oT,fs); 
+     G,hp\<turnstile>v::\<preceq>T; G\<turnstile>h hp\<surd> \<rbrakk> 
+  \<Longrightarrow> G\<turnstile>h hp(a \<mapsto> (oT, fs(X\<mapsto>v)))\<surd>"
+  apply (simp add: hconf_def)
+  apply (fastsimp intro: oconf_heap_update oconf_field_update 
+                  simp add: obj_ty_def)
+  done
 
 section {* correct-frames *}
 
 lemmas [simp del] = fun_upd_apply
 
-lemma correct_frames_imp_correct_frames_field_update [rule_format]:
-  "\<forall>rT C sig. correct_frames G hp phi rT sig frs --> 
-  hp a = Some (C,od) --> map_of (fields (G, C)) fl = Some fd --> 
+lemma correct_frames_field_update [rule_format]:
+  "\<forall>rT C sig. 
+  correct_frames G hp phi rT sig frs --> 
+  hp a = Some (C,fs) --> 
+  map_of (fields (G, C)) fl = Some fd --> 
   G,hp\<turnstile>v::\<preceq>fd 
-  --> correct_frames G (hp(a \<mapsto> (C, od(fl\<mapsto>v)))) phi rT sig frs";
+  --> correct_frames G (hp(a \<mapsto> (C, fs(fl\<mapsto>v)))) phi rT sig frs";
 apply (induct frs)
  apply simp
 apply clarify
-apply simp
+apply (simp (no_asm_use))
 apply clarify
 apply (unfold correct_frame_def)
 apply (simp (no_asm_use))
-apply clarsimp
+apply clarify
 apply (intro exI conjI)
-     apply simp
-    apply simp
-   apply force
-  apply simp
- apply (rule approx_stk_imp_approx_stk_sup_heap)
-  apply simp
- apply (rule hext_upd_obj)
- apply simp
-apply (rule approx_loc_imp_approx_loc_sup_heap)
- apply simp
-apply (rule hext_upd_obj)
-apply simp
+    apply assumption+
+   apply (erule approx_stk_sup_heap)
+   apply (erule hext_upd_obj)
+  apply (erule approx_loc_sup_heap)
+  apply (erule hext_upd_obj)
+ apply assumption+
+apply blast
 done
 
-lemma correct_frames_imp_correct_frames_newref [rule_format]:
-  "\<forall>rT C sig. hp x = None --> correct_frames G hp phi rT sig frs \<and> oconf G hp obj 
-  --> correct_frames G (hp(x \<mapsto> obj)) phi rT sig frs"
+lemma correct_frames_newref [rule_format]:
+  "\<forall>rT C sig. 
+  hp x = None \<longrightarrow> 
+  correct_frames G hp phi rT sig frs \<longrightarrow>
+  G,hp \<turnstile> obj \<surd>
+  \<longrightarrow> correct_frames G (hp(x \<mapsto> obj)) phi rT sig frs"
 apply (induct frs)
  apply simp
-apply (clarsimp simp add: correct_frame_def)
+apply clarify
+apply (simp (no_asm_use))
+apply clarify
+apply (unfold correct_frame_def)
+apply (simp (no_asm_use))
+apply clarify
 apply (intro exI conjI)
-     apply simp
-    apply simp
-   apply force
-  apply simp
- apply (rule approx_stk_imp_approx_stk_sup_heap)
-  apply simp
- apply (rule hext_new)
- apply simp
-apply (rule approx_loc_imp_approx_loc_sup_heap)
- apply simp
-apply (rule hext_new)
-apply simp
+    apply assumption+
+   apply (erule approx_stk_sup_heap)
+   apply (erule hext_new)
+  apply (erule approx_loc_sup_heap)
+  apply (erule hext_new)
+ apply assumption+
+apply blast
 done
 
 end
-