--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Bali/TypeSafe.thy Mon Jan 28 17:00:19 2002 +0100
@@ -0,0 +1,1211 @@
+(* Title: isabelle/Bali/TypeSafe.thy
+ ID: $Id$
+ Author: David von Oheimb
+ Copyright 1997 Technische Universitaet Muenchen
+*)
+header {* The type soundness proof for Java *}
+
+
+theory TypeSafe = Eval + WellForm + Conform:
+
+section "result conformance"
+
+constdefs
+ assign_conforms :: "st \<Rightarrow> (val \<Rightarrow> state \<Rightarrow> state) \<Rightarrow> ty \<Rightarrow> env_ \<Rightarrow> bool"
+ ("_\<le>|_\<preceq>_\<Colon>\<preceq>_" [71,71,71,71] 70)
+ "s\<le>|f\<preceq>T\<Colon>\<preceq>E \<equiv>
+ \<forall>s' w. Norm s'\<Colon>\<preceq>E \<longrightarrow> fst E,s'\<turnstile>w\<Colon>\<preceq>T \<longrightarrow> s\<le>|s' \<longrightarrow> assign f w (Norm s')\<Colon>\<preceq>E"
+
+ rconf :: "prog \<Rightarrow> lenv \<Rightarrow> st \<Rightarrow> term \<Rightarrow> vals \<Rightarrow> tys \<Rightarrow> bool"
+ ("_,_,_\<turnstile>_\<succ>_\<Colon>\<preceq>_" [71,71,71,71,71,71] 70)
+ "G,L,s\<turnstile>t\<succ>v\<Colon>\<preceq>T
+ \<equiv> case T of
+ Inl T \<Rightarrow> if (\<exists>vf. t=In2 vf)
+ then G,s\<turnstile>fst (the_In2 v)\<Colon>\<preceq>T \<and> s\<le>|snd (the_In2 v)\<preceq>T\<Colon>\<preceq>(G,L)
+ else G,s\<turnstile>the_In1 v\<Colon>\<preceq>T
+ | Inr Ts \<Rightarrow> list_all2 (conf G s) (the_In3 v) Ts"
+
+lemma rconf_In1 [simp]:
+ "G,L,s\<turnstile>In1 ec\<succ>In1 v \<Colon>\<preceq>Inl T = G,s\<turnstile>v\<Colon>\<preceq>T"
+apply (unfold rconf_def)
+apply (simp (no_asm))
+done
+
+lemma rconf_In2 [simp]:
+ "G,L,s\<turnstile>In2 va\<succ>In2 vf\<Colon>\<preceq>Inl T = (G,s\<turnstile>fst vf\<Colon>\<preceq>T \<and> s\<le>|snd vf\<preceq>T\<Colon>\<preceq>(G,L))"
+apply (unfold rconf_def)
+apply (simp (no_asm))
+done
+
+lemma rconf_In3 [simp]:
+ "G,L,s\<turnstile>In3 es\<succ>In3 vs\<Colon>\<preceq>Inr Ts = list_all2 (\<lambda>v T. G,s\<turnstile>v\<Colon>\<preceq>T) vs Ts"
+apply (unfold rconf_def)
+apply (simp (no_asm))
+done
+
+section "fits and conf"
+
+(* unused *)
+lemma conf_fits: "G,s\<turnstile>v\<Colon>\<preceq>T \<Longrightarrow> G,s\<turnstile>v fits T"
+apply (unfold fits_def)
+apply clarify
+apply (erule swap, simp (no_asm_use))
+apply (drule conf_RefTD)
+apply auto
+done
+
+lemma fits_conf:
+ "\<lbrakk>G,s\<turnstile>v\<Colon>\<preceq>T; G\<turnstile>T\<preceq>? T'; G,s\<turnstile>v fits T'; ws_prog G\<rbrakk> \<Longrightarrow> G,s\<turnstile>v\<Colon>\<preceq>T'"
+apply (auto dest!: fitsD cast_PrimT2 cast_RefT2)
+apply (force dest: conf_RefTD intro: conf_AddrI)
+done
+
+lemma fits_Array:
+ "\<lbrakk>G,s\<turnstile>v\<Colon>\<preceq>T; G\<turnstile>T'.[]\<preceq>T.[]; G,s\<turnstile>v fits T'; ws_prog G\<rbrakk> \<Longrightarrow> G,s\<turnstile>v\<Colon>\<preceq>T'"
+apply (auto dest!: fitsD widen_ArrayPrimT widen_ArrayRefT)
+apply (force dest: conf_RefTD intro: conf_AddrI)
+done
+
+
+
+section "gext"
+
+lemma halloc_gext: "\<And>s1 s2. G\<turnstile>s1 \<midarrow>halloc oi\<succ>a\<rightarrow> s2 \<Longrightarrow> snd s1\<le>|snd s2"
+apply (simp (no_asm_simp) only: split_tupled_all)
+apply (erule halloc.induct)
+apply (auto dest!: new_AddrD)
+done
+
+lemma sxalloc_gext: "\<And>s1 s2. G\<turnstile>s1 \<midarrow>sxalloc\<rightarrow> s2 \<Longrightarrow> snd s1\<le>|snd s2"
+apply (simp (no_asm_simp) only: split_tupled_all)
+apply (erule sxalloc.induct)
+apply (auto dest!: halloc_gext)
+done
+
+lemma eval_gext_lemma [rule_format (no_asm)]:
+ "G\<turnstile>s \<midarrow>t\<succ>\<rightarrow> (w,s') \<Longrightarrow> snd s\<le>|snd s' \<and> (case w of
+ In1 v \<Rightarrow> True
+ | In2 vf \<Rightarrow> normal s \<longrightarrow> (\<forall>v x s. s\<le>|snd (assign (snd vf) v (x,s)))
+ | In3 vs \<Rightarrow> True)"
+apply (erule eval_induct)
+prefer 24
+ apply (case_tac "inited C (globs s0)", clarsimp, erule thin_rl) (* Init *)
+apply (auto del: conjI dest!: not_initedD gext_new sxalloc_gext halloc_gext
+ simp add: lvar_def fvar_def2 avar_def2 init_lvars_def2
+ split del: split_if_asm split add: sum3.split)
+(* 6 subgoals *)
+apply force+
+done
+
+lemma evar_gext_f:
+ "G\<turnstile>Norm s1 \<midarrow>e=\<succ>vf \<rightarrow> s2 \<Longrightarrow> s\<le>|snd (assign (snd vf) v (x,s))"
+apply (drule eval_gext_lemma [THEN conjunct2])
+apply auto
+done
+
+lemmas eval_gext = eval_gext_lemma [THEN conjunct1]
+
+lemma eval_gext': "G\<turnstile>(x1,s1) \<midarrow>t\<succ>\<rightarrow> (w,x2,s2) \<Longrightarrow> s1\<le>|s2"
+apply (drule eval_gext)
+apply auto
+done
+
+lemma init_yields_initd: "G\<turnstile>Norm s1 \<midarrow>Init C\<rightarrow> s2 \<Longrightarrow> initd C s2"
+apply (erule eval_cases , auto split del: split_if_asm)
+apply (case_tac "inited C (globs s1)")
+apply (clarsimp split del: split_if_asm)+
+apply (drule eval_gext')+
+apply (drule init_class_obj_inited)
+apply (erule inited_gext)
+apply (simp (no_asm_use))
+done
+
+
+section "Lemmas"
+
+lemma obj_ty_obj_class1:
+ "\<lbrakk>wf_prog G; is_type G (obj_ty obj)\<rbrakk> \<Longrightarrow> is_class G (obj_class obj)"
+apply (case_tac "tag obj")
+apply (auto simp add: obj_ty_def obj_class_def)
+done
+
+lemma oconf_init_obj:
+ "\<lbrakk>wf_prog G;
+ (case r of Heap a \<Rightarrow> is_type G (obj_ty obj) | Stat C \<Rightarrow> is_class G C)
+\<rbrakk> \<Longrightarrow> G,s\<turnstile>obj \<lparr>values:=init_vals (var_tys G (tag obj) r)\<rparr>\<Colon>\<preceq>\<surd>r"
+apply (auto intro!: oconf_init_obj_lemma unique_fields)
+done
+
+(*
+lemma obj_split: "P obj = (\<forall> oi vs. obj = \<lparr>tag=oi,values=vs\<rparr> \<longrightarrow> ?P \<lparr>tag=oi,values=vs\<rparr>)"
+apply auto
+apply (case_tac "obj")
+apply auto
+*)
+
+lemma conforms_newG: "\<lbrakk>globs s oref = None; (x, s)\<Colon>\<preceq>(G,L);
+ wf_prog G; case oref of Heap a \<Rightarrow> is_type G (obj_ty \<lparr>tag=oi,values=vs\<rparr>)
+ | Stat C \<Rightarrow> is_class G C\<rbrakk> \<Longrightarrow>
+ (x, init_obj G oi oref s)\<Colon>\<preceq>(G, L)"
+apply (unfold init_obj_def)
+apply (auto elim!: conforms_gupd dest!: oconf_init_obj
+ simp add: obj.update_defs)
+done
+
+lemma conforms_init_class_obj:
+ "\<lbrakk>(x,s)\<Colon>\<preceq>(G, L); wf_prog G; class G C=Some y; \<not> inited C (globs s)\<rbrakk> \<Longrightarrow>
+ (x,init_class_obj G C s)\<Colon>\<preceq>(G, L)"
+apply (rule not_initedD [THEN conforms_newG])
+apply (auto)
+done
+
+
+lemma fst_init_lvars[simp]:
+ "fst (init_lvars G C sig (invmode m e) a' pvs (x,s)) =
+ (if static m then x else (np a') x)"
+apply (simp (no_asm) add: init_lvars_def2)
+done
+
+
+lemma halloc_conforms: "\<And>s1. \<lbrakk>G\<turnstile>s1 \<midarrow>halloc oi\<succ>a\<rightarrow> s2; wf_prog G; s1\<Colon>\<preceq>(G, L);
+ is_type G (obj_ty \<lparr>tag=oi,values=fs\<rparr>)\<rbrakk> \<Longrightarrow> s2\<Colon>\<preceq>(G, L)"
+apply (simp (no_asm_simp) only: split_tupled_all)
+apply (case_tac "aa")
+apply (auto elim!: halloc_elim_cases dest!: new_AddrD
+ intro!: conforms_newG [THEN conforms_xconf] conf_AddrI)
+done
+
+lemma halloc_type_sound: "\<And>s1. \<lbrakk>G\<turnstile>s1 \<midarrow>halloc oi\<succ>a\<rightarrow> (x,s); wf_prog G; s1\<Colon>\<preceq>(G, L);
+ T = obj_ty \<lparr>tag=oi,values=fs\<rparr>; is_type G T\<rbrakk> \<Longrightarrow>
+ (x,s)\<Colon>\<preceq>(G, L) \<and> (x = None \<longrightarrow> G,s\<turnstile>Addr a\<Colon>\<preceq>T)"
+apply (auto elim!: halloc_conforms)
+apply (case_tac "aa")
+apply (subst obj_ty_eq)
+apply (auto elim!: halloc_elim_cases dest!: new_AddrD intro!: conf_AddrI)
+done
+
+lemma sxalloc_type_sound:
+ "\<And>s1 s2. \<lbrakk>G\<turnstile>s1 \<midarrow>sxalloc\<rightarrow> s2; wf_prog G\<rbrakk> \<Longrightarrow> case fst s1 of
+ None \<Rightarrow> s2 = s1 | Some x \<Rightarrow>
+ (\<exists>a. fst s2 = Some(Xcpt (Loc a)) \<and> (\<forall>L. s1\<Colon>\<preceq>(G,L) \<longrightarrow> s2\<Colon>\<preceq>(G,L)))"
+apply (simp (no_asm_simp) only: split_tupled_all)
+apply (erule sxalloc.induct)
+apply auto
+apply (rule halloc_conforms [THEN conforms_xconf])
+apply (auto elim!: halloc_elim_cases dest!: new_AddrD intro!: conf_AddrI)
+done
+
+lemma wt_init_comp_ty:
+"is_acc_type G (pid C) T \<Longrightarrow> \<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>init_comp_ty T\<Colon>\<surd>"
+apply (unfold init_comp_ty_def)
+apply (clarsimp simp add: accessible_in_RefT_simp
+ is_acc_type_def is_acc_class_def)
+done
+
+
+declare fun_upd_same [simp]
+
+declare fun_upd_apply [simp del]
+
+
+constdefs
+ DynT_prop::"[prog,inv_mode,qtname,ref_ty] \<Rightarrow> bool"
+ ("_\<turnstile>_\<rightarrow>_\<preceq>_"[71,71,71,71]70)
+ "G\<turnstile>mode\<rightarrow>D\<preceq>t \<equiv> mode = IntVir \<longrightarrow> is_class G D \<and>
+ (if (\<exists>T. t=ArrayT T) then D=Object else G\<turnstile>Class D\<preceq>RefT t)"
+
+lemma DynT_propI:
+ "\<lbrakk>(x,s)\<Colon>\<preceq>(G, L); G,s\<turnstile>a'\<Colon>\<preceq>RefT statT; wf_prog G; mode = IntVir \<longrightarrow> a' \<noteq> Null\<rbrakk>
+ \<Longrightarrow> G\<turnstile>mode\<rightarrow>invocation_class mode s a' statT\<preceq>statT"
+proof (unfold DynT_prop_def)
+ assume state_conform: "(x,s)\<Colon>\<preceq>(G, L)"
+ and statT_a': "G,s\<turnstile>a'\<Colon>\<preceq>RefT statT"
+ and wf: "wf_prog G"
+ and mode: "mode = IntVir \<longrightarrow> a' \<noteq> Null"
+ let ?invCls = "(invocation_class mode s a' statT)"
+ let ?IntVir = "mode = IntVir"
+ let ?Concl = "\<lambda>invCls. is_class G invCls \<and>
+ (if \<exists>T. statT = ArrayT T
+ then invCls = Object
+ else G\<turnstile>Class invCls\<preceq>RefT statT)"
+ show "?IntVir \<longrightarrow> ?Concl ?invCls"
+ proof
+ assume modeIntVir: ?IntVir
+ with mode have not_Null: "a' \<noteq> Null" ..
+ from statT_a' not_Null state_conform
+ obtain a obj
+ where obj_props: "a' = Addr a" "globs s (Inl a) = Some obj"
+ "G\<turnstile>obj_ty obj\<preceq>RefT statT" "is_type G (obj_ty obj)"
+ by (blast dest: conforms_RefTD)
+ show "?Concl ?invCls"
+ proof (cases "tag obj")
+ case CInst
+ with modeIntVir obj_props
+ show ?thesis
+ by (auto dest!: widen_Array2 split add: split_if)
+ next
+ case Arr
+ from Arr obtain T where "obj_ty obj = T.[]" by (blast dest: obj_ty_Arr1)
+ moreover from Arr have "obj_class obj = Object"
+ by (blast dest: obj_class_Arr1)
+ moreover note modeIntVir obj_props wf
+ ultimately show ?thesis by (auto dest!: widen_Array )
+ qed
+ qed
+qed
+
+lemma invocation_methd:
+"\<lbrakk>wf_prog G; statT \<noteq> NullT;
+ (\<forall> statC. statT = ClassT statC \<longrightarrow> is_class G statC);
+ (\<forall> I. statT = IfaceT I \<longrightarrow> is_iface G I \<and> mode \<noteq> SuperM);
+ (\<forall> T. statT = ArrayT T \<longrightarrow> mode \<noteq> SuperM);
+ G\<turnstile>mode\<rightarrow>invocation_class mode s a' statT\<preceq>statT;
+ dynlookup G statT (invocation_class mode s a' statT) sig = Some m \<rbrakk>
+\<Longrightarrow> methd G (invocation_declclass G mode s a' statT sig) sig = Some m"
+proof -
+ assume wf: "wf_prog G"
+ and not_NullT: "statT \<noteq> NullT"
+ and statC_prop: "(\<forall> statC. statT = ClassT statC \<longrightarrow> is_class G statC)"
+ and statI_prop: "(\<forall> I. statT = IfaceT I \<longrightarrow> is_iface G I \<and> mode \<noteq> SuperM)"
+ and statA_prop: "(\<forall> T. statT = ArrayT T \<longrightarrow> mode \<noteq> SuperM)"
+ and invC_prop: "G\<turnstile>mode\<rightarrow>invocation_class mode s a' statT\<preceq>statT"
+ and dynlookup: "dynlookup G statT (invocation_class mode s a' statT) sig
+ = Some m"
+ show ?thesis
+ proof (cases statT)
+ case NullT
+ with not_NullT show ?thesis by simp
+ next
+ case IfaceT
+ with statI_prop obtain I
+ where statI: "statT = IfaceT I" and
+ is_iface: "is_iface G I" and
+ not_SuperM: "mode \<noteq> SuperM" by blast
+
+ show ?thesis
+ proof (cases mode)
+ case Static
+ with wf dynlookup statI is_iface
+ show ?thesis
+ by (auto simp add: invocation_declclass_def dynlookup_def
+ dynimethd_def dynmethd_C_C
+ intro: dynmethd_declclass
+ dest!: wf_imethdsD
+ dest: table_of_map_SomeI
+ split: split_if_asm)
+ next
+ case SuperM
+ with not_SuperM show ?thesis ..
+ next
+ case IntVir
+ with wf dynlookup IfaceT invC_prop show ?thesis
+ by (auto simp add: invocation_declclass_def dynlookup_def dynimethd_def
+ DynT_prop_def
+ intro: methd_declclass dynmethd_declclass
+ split: split_if_asm)
+ qed
+ next
+ case ClassT
+ show ?thesis
+ proof (cases mode)
+ case Static
+ with wf ClassT dynlookup statC_prop
+ show ?thesis by (auto simp add: invocation_declclass_def dynlookup_def
+ intro: dynmethd_declclass)
+ next
+ case SuperM
+ with wf ClassT dynlookup statC_prop
+ show ?thesis by (auto simp add: invocation_declclass_def dynlookup_def
+ intro: dynmethd_declclass)
+ next
+ case IntVir
+ with wf ClassT dynlookup statC_prop invC_prop
+ show ?thesis
+ by (auto simp add: invocation_declclass_def dynlookup_def dynimethd_def
+ DynT_prop_def
+ intro: dynmethd_declclass)
+ qed
+ next
+ case ArrayT
+ show ?thesis
+ proof (cases mode)
+ case Static
+ with wf ArrayT dynlookup show ?thesis
+ by (auto simp add: invocation_declclass_def dynlookup_def
+ dynimethd_def dynmethd_C_C
+ intro: dynmethd_declclass
+ dest: table_of_map_SomeI)
+ next
+ case SuperM
+ with ArrayT statA_prop show ?thesis by blast
+ next
+ case IntVir
+ with wf ArrayT dynlookup invC_prop show ?thesis
+ by (auto simp add: invocation_declclass_def dynlookup_def dynimethd_def
+ DynT_prop_def dynmethd_C_C
+ intro: dynmethd_declclass
+ dest: table_of_map_SomeI)
+ qed
+ qed
+qed
+
+declare split_paired_All [simp del] split_paired_Ex [simp del]
+ML_setup {*
+simpset_ref() := simpset() delloop "split_all_tac";
+claset_ref () := claset () delSWrapper "split_all_tac"
+*}
+lemma DynT_mheadsD:
+"\<lbrakk>G\<turnstile>invmode (mhd sm) e\<rightarrow>invC\<preceq>statT;
+ wf_prog G; \<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>e\<Colon>-RefT statT;
+ sm \<in> mheads G C statT sig;
+ invC = invocation_class (invmode (mhd sm) e) s a' statT;
+ declC =invocation_declclass G (invmode (mhd sm) e) s a' statT sig
+ \<rbrakk> \<Longrightarrow>
+ \<exists> dm.
+ methd G declC sig = Some dm \<and> G\<turnstile>resTy (mthd dm)\<preceq>resTy (mhd sm) \<and>
+ wf_mdecl G declC (sig, mthd dm) \<and>
+ declC = declclass dm \<and>
+ is_static dm = is_static sm \<and>
+ is_class G invC \<and> is_class G declC \<and> G\<turnstile>invC\<preceq>\<^sub>C declC \<and>
+ (if invmode (mhd sm) e = IntVir
+ then (\<forall> statC. statT=ClassT statC \<longrightarrow> G\<turnstile>invC \<preceq>\<^sub>C statC)
+ else ( (\<exists> statC. statT=ClassT statC \<and> G\<turnstile>statC\<preceq>\<^sub>C declC)
+ \<or> (\<forall> statC. statT\<noteq>ClassT statC \<and> declC=Object)) \<and>
+ (declrefT sm) = ClassT (declclass dm))"
+proof -
+ assume invC_prop: "G\<turnstile>invmode (mhd sm) e\<rightarrow>invC\<preceq>statT"
+ and wf: "wf_prog G"
+ and wt_e: "\<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>e\<Colon>-RefT statT"
+ and sm: "sm \<in> mheads G C statT sig"
+ and invC: "invC = invocation_class (invmode (mhd sm) e) s a' statT"
+ and declC: "declC =
+ invocation_declclass G (invmode (mhd sm) e) s a' statT sig"
+ from wt_e wf have type_statT: "is_type G (RefT statT)"
+ by (auto dest: ty_expr_is_type)
+ from sm have not_Null: "statT \<noteq> NullT" by auto
+ from type_statT
+ have wf_C: "(\<forall> statC. statT = ClassT statC \<longrightarrow> is_class G statC)"
+ by (auto)
+ from type_statT wt_e
+ have wf_I: "(\<forall>I. statT = IfaceT I \<longrightarrow> is_iface G I \<and>
+ invmode (mhd sm) e \<noteq> SuperM)"
+ by (auto dest: invocationTypeExpr_noClassD)
+ from wt_e
+ have wf_A: "(\<forall> T. statT = ArrayT T \<longrightarrow> invmode (mhd sm) e \<noteq> SuperM)"
+ by (auto dest: invocationTypeExpr_noClassD)
+ show ?thesis
+ proof (cases "invmode (mhd sm) e = IntVir")
+ case True
+ with invC_prop not_Null
+ have invC_prop': " is_class G invC \<and>
+ (if (\<exists>T. statT=ArrayT T) then invC=Object
+ else G\<turnstile>Class invC\<preceq>RefT statT)"
+ by (auto simp add: DynT_prop_def)
+ from True
+ have "\<not> is_static sm"
+ by (simp add: invmode_IntVir_eq)
+ with invC_prop' not_Null
+ have "G,statT \<turnstile> invC valid_lookup_cls_for (is_static sm)"
+ by (cases statT) auto
+ with sm wf type_statT obtain dm where
+ dm: "dynlookup G statT invC sig = Some dm" and
+ resT_dm: "G\<turnstile>resTy (mthd dm)\<preceq>resTy (mhd sm)" and
+ static: "is_static dm = is_static sm"
+ by - (drule dynamic_mheadsD,auto)
+ with declC invC not_Null
+ have declC': "declC = (declclass dm)"
+ by (auto simp add: invocation_declclass_def)
+ with wf invC declC not_Null wf_C wf_I wf_A invC_prop dm
+ have dm': "methd G declC sig = Some dm"
+ by - (drule invocation_methd,auto)
+ from wf dm invC_prop' declC' type_statT
+ have declC_prop: "G\<turnstile>invC \<preceq>\<^sub>C declC \<and> is_class G declC"
+ by (auto dest: dynlookup_declC)
+ from wf dm' declC_prop declC'
+ have wf_dm: "wf_mdecl G declC (sig,(mthd dm))"
+ by (auto dest: methd_wf_mdecl)
+ from invC_prop'
+ have statC_prop: "(\<forall> statC. statT=ClassT statC \<longrightarrow> G\<turnstile>invC \<preceq>\<^sub>C statC)"
+ by auto
+ from True dm' resT_dm wf_dm invC_prop' declC_prop statC_prop declC' static
+ show ?thesis by auto
+ next
+ case False
+ with type_statT wf invC not_Null wf_I wf_A
+ have invC_prop': "is_class G invC \<and>
+ ((\<exists> statC. statT=ClassT statC \<and> invC=statC) \<or>
+ (\<forall> statC. statT\<noteq>ClassT statC \<and> invC=Object)) "
+ by (case_tac "statT") (auto simp add: invocation_class_def
+ split: inv_mode.splits)
+ with not_Null wf
+ have dynlookup_static: "dynlookup G statT invC sig = methd G invC sig"
+ by (case_tac "statT") (auto simp add: dynlookup_def dynmethd_C_C
+ dynimethd_def)
+ from sm wf wt_e not_Null False invC_prop' obtain "dm" where
+ dm: "methd G invC sig = Some dm" and
+ eq_declC_sm_dm:"declrefT sm = ClassT (declclass dm)" and
+ eq_mheads:"mhd sm=mhead (mthd dm) "
+ by - (drule static_mheadsD, auto dest: accmethd_SomeD)
+ then have static: "is_static dm = is_static sm" by - (case_tac "sm",auto)
+ with declC invC dynlookup_static dm
+ have declC': "declC = (declclass dm)"
+ by (auto simp add: invocation_declclass_def)
+ from invC_prop' wf declC' dm
+ have dm': "methd G declC sig = Some dm"
+ by (auto intro: methd_declclass)
+ from wf dm invC_prop' declC' type_statT
+ have declC_prop: "G\<turnstile>invC \<preceq>\<^sub>C declC \<and> is_class G declC"
+ by (auto dest: methd_declC )
+ then have declC_prop1: "invC=Object \<longrightarrow> declC=Object" by auto
+ from wf dm' declC_prop declC'
+ have wf_dm: "wf_mdecl G declC (sig,(mthd dm))"
+ by (auto dest: methd_wf_mdecl)
+ from invC_prop' declC_prop declC_prop1
+ have statC_prop: "( (\<exists> statC. statT=ClassT statC \<and> G\<turnstile>statC\<preceq>\<^sub>C declC)
+ \<or> (\<forall> statC. statT\<noteq>ClassT statC \<and> declC=Object))"
+ by auto
+ from False dm' wf_dm invC_prop' declC_prop statC_prop declC'
+ eq_declC_sm_dm eq_mheads static
+ show ?thesis by auto
+ qed
+qed
+
+(*
+lemma DynT_mheadsD:
+"\<lbrakk>G\<turnstile>invmode (mhd sm) e\<rightarrow>invC\<preceq>statT;
+ wf_prog G; \<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>e\<Colon>-RefT statT;
+ sm \<in> mheads G C statT sig;
+ invC = invocation_class (invmode (mhd sm) e) s a' statT;
+ declC =invocation_declclass G (invmode (mhd sm) e) s a' statT sig
+ \<rbrakk> \<Longrightarrow>
+ \<exists> dm.
+ methd G declC sig = Some dm \<and> G\<turnstile>resTy (mthd dm)\<preceq>resTy (mhd sm) \<and>
+ wf_mdecl G declC (sig, mthd dm) \<and>
+ is_class G invC \<and> is_class G declC \<and> G\<turnstile>invC\<preceq>\<^sub>C declC \<and>
+ (if invmode (mhd sm) e = IntVir
+ then (\<forall> statC. statT=ClassT statC \<longrightarrow> G\<turnstile>invC \<preceq>\<^sub>C statC)
+ else (\<forall> statC. statT=ClassT statC \<longrightarrow> G\<turnstile>statC \<preceq>\<^sub>C declC) \<and>
+ (declrefT sm) = ClassT (declclass dm))"
+proof -
+ assume invC_prop: "G\<turnstile>invmode (mhd sm) e\<rightarrow>invC\<preceq>statT"
+ and wf: "wf_prog G"
+ and wt_e: "\<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>e\<Colon>-RefT statT"
+ and sm: "sm \<in> mheads G C statT sig"
+ and invC: "invC = invocation_class (invmode (mhd sm) e) s a' statT"
+ and declC: "declC =
+ invocation_declclass G (invmode (mhd sm) e) s a' statT sig"
+ from wt_e wf have type_statT: "is_type G (RefT statT)"
+ by (auto dest: ty_expr_is_type)
+ from sm have not_Null: "statT \<noteq> NullT" by auto
+ from type_statT
+ have wf_C: "(\<forall> statC. statT = ClassT statC \<longrightarrow> is_class G statC)"
+ by (auto)
+ from type_statT wt_e
+ have wf_I: "(\<forall>I. statT = IfaceT I \<longrightarrow> is_iface G I \<and>
+ invmode (mhd sm) e \<noteq> SuperM)"
+ by (auto dest: invocationTypeExpr_noClassD)
+ from wt_e
+ have wf_A: "(\<forall> T. statT = ArrayT T \<longrightarrow> invmode (mhd sm) e \<noteq> SuperM)"
+ by (auto dest: invocationTypeExpr_noClassD)
+ show ?thesis
+ proof (cases "invmode (mhd sm) e = IntVir")
+ case True
+ with invC_prop not_Null
+ have invC_prop': "is_class G invC \<and>
+ (if (\<exists>T. statT=ArrayT T) then invC=Object
+ else G\<turnstile>Class invC\<preceq>RefT statT)"
+ by (auto simp add: DynT_prop_def)
+ with sm wf type_statT not_Null obtain dm where
+ dm: "dynlookup G statT invC sig = Some dm" and
+ resT_dm: "G\<turnstile>resTy (mthd dm)\<preceq>resTy (mhd sm)"
+ by - (clarify,drule dynamic_mheadsD,auto)
+ with declC invC not_Null
+ have declC': "declC = (declclass dm)"
+ by (auto simp add: invocation_declclass_def)
+ with wf invC declC not_Null wf_C wf_I wf_A invC_prop dm
+ have dm': "methd G declC sig = Some dm"
+ by - (drule invocation_methd,auto)
+ from wf dm invC_prop' declC' type_statT
+ have declC_prop: "G\<turnstile>invC \<preceq>\<^sub>C declC \<and> is_class G declC"
+ by (auto dest: dynlookup_declC)
+ from wf dm' declC_prop declC'
+ have wf_dm: "wf_mdecl G declC (sig,(mthd dm))"
+ by (auto dest: methd_wf_mdecl)
+ from invC_prop'
+ have statC_prop: "(\<forall> statC. statT=ClassT statC \<longrightarrow> G\<turnstile>invC \<preceq>\<^sub>C statC)"
+ by auto
+ from True dm' resT_dm wf_dm invC_prop' declC_prop statC_prop
+ show ?thesis by auto
+ next
+ case False
+
+ with type_statT wf invC not_Null wf_I wf_A
+ have invC_prop': "is_class G invC \<and>
+ ((\<exists> statC. statT=ClassT statC \<and> invC=statC) \<or>
+ (\<forall> statC. statT\<noteq>ClassT statC \<and> invC=Object)) "
+
+ by (case_tac "statT") (auto simp add: invocation_class_def
+ split: inv_mode.splits)
+ with not_Null
+ have dynlookup_static: "dynlookup G statT invC sig = methd G invC sig"
+ by (case_tac "statT") (auto simp add: dynlookup_def dynmethd_def
+ dynimethd_def)
+ from sm wf wt_e not_Null False invC_prop' obtain "dm" where
+ dm: "methd G invC sig = Some dm" and
+ eq_declC_sm_dm:"declrefT sm = ClassT (declclass dm)" and
+ eq_mheads:"mhd sm=mhead (mthd dm) "
+ by - (drule static_mheadsD, auto dest: accmethd_SomeD)
+ with declC invC dynlookup_static dm
+ have declC': "declC = (declclass dm)"
+ by (auto simp add: invocation_declclass_def)
+ from invC_prop' wf declC' dm
+ have dm': "methd G declC sig = Some dm"
+ by (auto intro: methd_declclass)
+ from wf dm invC_prop' declC' type_statT
+ have declC_prop: "G\<turnstile>invC \<preceq>\<^sub>C declC \<and> is_class G declC"
+ by (auto dest: methd_declC )
+ from wf dm' declC_prop declC'
+ have wf_dm: "wf_mdecl G declC (sig,(mthd dm))"
+ by (auto dest: methd_wf_mdecl)
+ from invC_prop' declC_prop
+ have statC_prop: "(\<forall> statC. statT=ClassT statC \<longrightarrow> G\<turnstile>statC \<preceq>\<^sub>C declC)"
+ by auto
+ from False dm' wf_dm invC_prop' declC_prop statC_prop
+ eq_declC_sm_dm eq_mheads
+ show ?thesis by auto
+ qed
+qed
+*)
+
+declare split_paired_All [simp del] split_paired_Ex [simp del]
+declare split_if [split del] split_if_asm [split del]
+ option.split [split del] option.split_asm [split del]
+ML_setup {*
+simpset_ref() := simpset() delloop "split_all_tac";
+claset_ref () := claset () delSWrapper "split_all_tac"
+*}
+
+lemma DynT_conf: "\<lbrakk>G\<turnstile>invocation_class mode s a' statT \<preceq>\<^sub>C declC; wf_prog G;
+ isrtype G (statT);
+ G,s\<turnstile>a'\<Colon>\<preceq>RefT statT; mode = IntVir \<longrightarrow> a' \<noteq> Null;
+ mode \<noteq> IntVir \<longrightarrow> (\<exists> statC. statT=ClassT statC \<and> G\<turnstile>statC\<preceq>\<^sub>C declC)
+ \<or> (\<forall> statC. statT\<noteq>ClassT statC \<and> declC=Object)\<rbrakk>
+ \<Longrightarrow>G,s\<turnstile>a'\<Colon>\<preceq> Class declC"
+apply (case_tac "mode = IntVir")
+apply (drule conf_RefTD)
+apply (force intro!: conf_AddrI
+ simp add: obj_class_def split add: obj_tag.split_asm)
+apply clarsimp
+apply safe
+apply (erule (1) widen.subcls [THEN conf_widen])
+apply (erule wf_ws_prog)
+
+apply (frule widen_Object) apply (erule wf_ws_prog)
+apply (erule (1) conf_widen) apply (erule wf_ws_prog)
+done
+
+
+lemma Ass_lemma:
+ "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>va=\<succ>(w, f)\<rightarrow> Norm s1; G\<turnstile>Norm s1 \<midarrow>e-\<succ>v\<rightarrow> Norm s2; G,s2\<turnstile>v\<Colon>\<preceq>T';
+ s1\<le>|s2 \<longrightarrow> assign f v (Norm s2)\<Colon>\<preceq>(G, L)
+ \<rbrakk> \<Longrightarrow> assign f v (Norm s2)\<Colon>\<preceq>(G, L) \<and>
+ (\<lambda>(x',s'). x' = None \<longrightarrow> G,s'\<turnstile>v\<Colon>\<preceq>T') (assign f v (Norm s2))"
+apply (drule_tac x = "None" and s = "s2" and v = "v" in evar_gext_f)
+apply (drule eval_gext', clarsimp)
+apply (erule conf_gext)
+apply simp
+done
+
+lemma Throw_lemma: "\<lbrakk>G\<turnstile>tn\<preceq>\<^sub>C SXcpt Throwable; wf_prog G; (x1,s1)\<Colon>\<preceq>(G, L);
+ x1 = None \<longrightarrow> G,s1\<turnstile>a'\<Colon>\<preceq> Class tn\<rbrakk> \<Longrightarrow> (throw a' x1, s1)\<Colon>\<preceq>(G, L)"
+apply (auto split add: split_abrupt_if simp add: throw_def2)
+apply (erule conforms_xconf)
+apply (frule conf_RefTD)
+apply (auto elim: widen.subcls [THEN conf_widen])
+done
+
+lemma Try_lemma: "\<lbrakk>G\<turnstile>obj_ty (the (globs s1' (Heap a)))\<preceq> Class tn;
+ (Some (Xcpt (Loc a)), s1')\<Colon>\<preceq>(G, L); wf_prog G\<rbrakk>
+ \<Longrightarrow> Norm (lupd(vn\<mapsto>Addr a) s1')\<Colon>\<preceq>(G, L(vn\<mapsto>Class tn))"
+apply (rule conforms_allocL)
+apply (erule conforms_NormI)
+apply (drule conforms_XcptLocD [THEN conf_RefTD],rule HOL.refl)
+apply (auto intro!: conf_AddrI)
+done
+
+lemma Fin_lemma:
+"\<lbrakk>G\<turnstile>Norm s1 \<midarrow>c2\<rightarrow> (x2,s2); wf_prog G; (Some a, s1)\<Colon>\<preceq>(G, L); (x2,s2)\<Colon>\<preceq>(G, L)\<rbrakk>
+\<Longrightarrow> (abrupt_if True (Some a) x2, s2)\<Colon>\<preceq>(G, L)"
+apply (force elim: eval_gext' conforms_xgext split add: split_abrupt_if)
+done
+
+lemma FVar_lemma1: "\<lbrakk>table_of (DeclConcepts.fields G Ca) (fn, C) = Some f ;
+ x2 = None \<longrightarrow> G,s2\<turnstile>a\<Colon>\<preceq> Class Ca; wf_prog G; G\<turnstile>Ca\<preceq>\<^sub>C C; C \<noteq> Object;
+ class G C = Some y; (x2,s2)\<Colon>\<preceq>(G, L); s1\<le>|s2; inited C (globs s1);
+ (if static f then id else np a) x2 = None\<rbrakk>
+ \<Longrightarrow>
+ \<exists>obj. globs s2 (if static f then Inr C else Inl (the_Addr a)) = Some obj \<and>
+ var_tys G (tag obj) (if static f then Inr C else Inl(the_Addr a))
+ (Inl(fn,C)) = Some (type f)"
+apply (drule initedD)
+apply (frule subcls_is_class2, simp (no_asm_simp))
+apply (case_tac "static f")
+apply clarsimp
+apply (drule (1) rev_gext_objD, clarsimp)
+apply (frule fields_declC, erule wf_ws_prog, simp (no_asm_simp))
+apply (rule var_tys_Some_eq [THEN iffD2])
+apply clarsimp
+apply (erule fields_table_SomeI, simp (no_asm))
+apply clarsimp
+apply (drule conf_RefTD, clarsimp, rule var_tys_Some_eq [THEN iffD2])
+apply (auto dest!: widen_Array split add: obj_tag.split)
+apply (rotate_tac -1) (* to front: tag obja = CInst pid_field_type to enable
+ conditional rewrite *)
+apply (rule fields_table_SomeI)
+apply (auto elim!: fields_mono subcls_is_class2)
+done
+
+lemma FVar_lemma:
+"\<lbrakk>((v, f), Norm s2') = fvar C (static field) fn a (x2, s2); G\<turnstile>Ca\<preceq>\<^sub>C C;
+ table_of (DeclConcepts.fields G Ca) (fn, C) = Some field; wf_prog G;
+ x2 = None \<longrightarrow> G,s2\<turnstile>a\<Colon>\<preceq>Class Ca; C \<noteq> Object; class G C = Some y;
+ (x2, s2)\<Colon>\<preceq>(G, L); s1\<le>|s2; inited C (globs s1)\<rbrakk> \<Longrightarrow>
+ G,s2'\<turnstile>v\<Colon>\<preceq>type field \<and> s2'\<le>|f\<preceq>type field\<Colon>\<preceq>(G, L)"
+apply (unfold assign_conforms_def)
+apply (drule sym)
+apply (clarsimp simp add: fvar_def2)
+apply (drule (9) FVar_lemma1)
+apply (clarsimp)
+apply (drule (2) conforms_globsD [THEN oconf_lconf, THEN lconfD])
+apply clarsimp
+apply (drule (1) rev_gext_objD)
+apply (auto elim!: conforms_upd_gobj)
+done
+
+
+lemma AVar_lemma1: "\<lbrakk>globs s (Inl a) = Some obj;tag obj=Arr ty i;
+ the_Intg i' in_bounds i; wf_prog G; G\<turnstile>ty.[]\<preceq>Tb.[]; Norm s\<Colon>\<preceq>(G, L)
+\<rbrakk> \<Longrightarrow> G,s\<turnstile>the ((values obj) (Inr (the_Intg i')))\<Colon>\<preceq>Tb"
+apply (erule widen_Array_Array [THEN conf_widen])
+apply (erule_tac [2] wf_ws_prog)
+apply (drule (1) conforms_globsD [THEN oconf_lconf, THEN lconfD])
+defer apply assumption
+apply (force intro: var_tys_Some_eq [THEN iffD2])
+done
+
+lemma obj_split: "\<And> obj. \<exists> t vs. obj = \<lparr>tag=t,values=vs\<rparr>"
+proof record_split
+ fix tag values more
+ show "\<exists>t vs. \<lparr>tag = tag, values = values, \<dots> = more\<rparr> = \<lparr>tag = t, values = vs\<rparr>"
+ by auto
+qed
+
+lemma AVar_lemma: "\<lbrakk>wf_prog G; G\<turnstile>(x1, s1) \<midarrow>e2-\<succ>i\<rightarrow> (x2, s2);
+ ((v,f), Norm s2') = avar G i a (x2, s2); x1 = None \<longrightarrow> G,s1\<turnstile>a\<Colon>\<preceq>Ta.[];
+ (x2, s2)\<Colon>\<preceq>(G, L); s1\<le>|s2\<rbrakk> \<Longrightarrow> G,s2'\<turnstile>v\<Colon>\<preceq>Ta \<and> s2'\<le>|f\<preceq>Ta\<Colon>\<preceq>(G, L)"
+apply (unfold assign_conforms_def)
+apply (drule sym)
+apply (clarsimp simp add: avar_def2)
+apply (drule (1) conf_gext)
+apply (drule conf_RefTD, clarsimp)
+apply (subgoal_tac "\<exists> t vs. obj = \<lparr>tag=t,values=vs\<rparr>")
+defer
+apply (rule obj_split)
+apply clarify
+apply (frule obj_ty_widenD)
+apply (auto dest!: widen_Class)
+apply (force dest: AVar_lemma1)
+apply (auto split add: split_if)
+apply (force elim!: fits_Array dest: gext_objD
+ intro: var_tys_Some_eq [THEN iffD2] conforms_upd_gobj)
+done
+
+
+section "Call"
+lemma conforms_init_lvars_lemma: "\<lbrakk>wf_prog G;
+ wf_mhead G P sig mh;
+ Ball (set lvars) (split (\<lambda>vn. is_type G));
+ list_all2 (conf G s) pvs pTsa; G\<turnstile>pTsa[\<preceq>](parTs sig)\<rbrakk> \<Longrightarrow>
+ G,s\<turnstile>init_vals (table_of lvars)(pars mh[\<mapsto>]pvs)
+ [\<Colon>\<preceq>]table_of lvars(pars mh[\<mapsto>]parTs sig)"
+apply (unfold wf_mhead_def)
+apply clarify
+apply (erule (2) Ball_set_table [THEN lconf_init_vals, THEN lconf_ext_list])
+apply (drule wf_ws_prog)
+apply (erule (2) conf_list_widen)
+done
+
+
+lemma lconf_map_lname [simp]:
+ "G,s\<turnstile>(lname_case l1 l2)[\<Colon>\<preceq>](lname_case L1 L2)
+ =
+ (G,s\<turnstile>l1[\<Colon>\<preceq>]L1 \<and> G,s\<turnstile>(\<lambda>x::unit . l2)[\<Colon>\<preceq>](\<lambda>x::unit. L2))"
+apply (unfold lconf_def)
+apply safe
+apply (case_tac "n")
+apply (force split add: lname.split)+
+done
+
+lemma lconf_map_ename [simp]:
+ "G,s\<turnstile>(ename_case l1 l2)[\<Colon>\<preceq>](ename_case L1 L2)
+ =
+ (G,s\<turnstile>l1[\<Colon>\<preceq>]L1 \<and> G,s\<turnstile>(\<lambda>x::unit. l2)[\<Colon>\<preceq>](\<lambda>x::unit. L2))"
+apply (unfold lconf_def)
+apply safe
+apply (force split add: ename.split)+
+done
+
+
+lemma defval_conf1 [rule_format (no_asm), elim]:
+ "is_type G T \<longrightarrow> (\<exists>v\<in>Some (default_val T): G,s\<turnstile>v\<Colon>\<preceq>T)"
+apply (unfold conf_def)
+apply (induct "T")
+apply (auto intro: prim_ty.induct)
+done
+
+
+lemma conforms_init_lvars:
+"\<lbrakk>wf_mhead G (pid declC) sig (mhead (mthd dm)); wf_prog G;
+ list_all2 (conf G s) pvs pTsa; G\<turnstile>pTsa[\<preceq>](parTs sig);
+ (x, s)\<Colon>\<preceq>(G, L);
+ methd G declC sig = Some dm;
+ isrtype G statT;
+ G\<turnstile>invC\<preceq>\<^sub>C declC;
+ G,s\<turnstile>a'\<Colon>\<preceq>RefT statT;
+ invmode (mhd sm) e = IntVir \<longrightarrow> a' \<noteq> Null;
+ invmode (mhd sm) e \<noteq> IntVir \<longrightarrow>
+ (\<exists> statC. statT=ClassT statC \<and> G\<turnstile>statC\<preceq>\<^sub>C declC)
+ \<or> (\<forall> statC. statT\<noteq>ClassT statC \<and> declC=Object);
+ invC = invocation_class (invmode (mhd sm) e) s a' statT;
+ declC = invocation_declclass G (invmode (mhd sm) e) s a' statT sig;
+ Ball (set (lcls (mbody (mthd dm))))
+ (split (\<lambda>vn. is_type G))
+ \<rbrakk> \<Longrightarrow>
+ init_lvars G declC sig (invmode (mhd sm) e) a'
+ pvs (x,s)\<Colon>\<preceq>(G,\<lambda> k.
+ (case k of
+ EName e \<Rightarrow> (case e of
+ VNam v
+ \<Rightarrow> (table_of (lcls (mbody (mthd dm)))
+ (pars (mthd dm)[\<mapsto>]parTs sig)) v
+ | Res \<Rightarrow> Some (resTy (mthd dm)))
+ | This \<Rightarrow> if (static (mthd sm))
+ then None else Some (Class declC)))"
+apply (simp add: init_lvars_def2)
+apply (rule conforms_set_locals)
+apply (simp (no_asm_simp) split add: split_if)
+apply (drule (4) DynT_conf)
+apply clarsimp
+(* apply intro *)
+apply (drule (4) conforms_init_lvars_lemma)
+apply (case_tac "dm",simp)
+apply (rule conjI)
+apply (unfold lconf_def, clarify)
+apply (rule defval_conf1)
+apply (clarsimp simp add: wf_mhead_def is_acc_type_def)
+apply (case_tac "is_static sm")
+apply simp_all
+done
+
+
+lemma Call_type_sound: "\<lbrakk>wf_prog G; G\<turnstile>(x1, s1) \<midarrow>ps\<doteq>\<succ>pvs\<rightarrow> (x2, s2);
+ declC
+ = invocation_declclass G (invmode (mhd esm) e) s2 a' statT \<lparr>name=mn,parTs=pTs\<rparr>;
+s2'=init_lvars G declC \<lparr>name=mn,parTs=pTs\<rparr> (invmode (mhd esm) e) a' pvs (x2,s2);
+ G\<turnstile>s2' \<midarrow>Methd declC \<lparr>name=mn,parTs=pTs\<rparr>-\<succ>v\<rightarrow> (x3, s3);
+ \<forall>L. s2'\<Colon>\<preceq>(G, L)
+ \<longrightarrow> (\<forall>T. \<lparr>prg=G,cls=declC,lcl=L\<rparr>\<turnstile> Methd declC \<lparr>name=mn,parTs=pTs\<rparr>\<Colon>-T
+ \<longrightarrow> (x3, s3)\<Colon>\<preceq>(G, L) \<and> (x3 = None \<longrightarrow> G,s3\<turnstile>v\<Colon>\<preceq>T));
+ Norm s0\<Colon>\<preceq>(G, L);
+ \<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>e\<Colon>-RefT statT; \<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>ps\<Colon>\<doteq>pTsa;
+ max_spec G C statT \<lparr>name=mn,parTs=pTsa\<rparr> = {(esm, pTs)};
+ (x1, s1)\<Colon>\<preceq>(G, L);
+ x1 = None \<longrightarrow> G,s1\<turnstile>a'\<Colon>\<preceq>RefT statT; (x2, s2)\<Colon>\<preceq>(G, L);
+ x2 = None \<longrightarrow> list_all2 (conf G s2) pvs pTsa;
+ sm=(mhd esm)\<rbrakk> \<Longrightarrow>
+ (x3, set_locals (locals s2) s3)\<Colon>\<preceq>(G, L) \<and>
+ (x3 = None \<longrightarrow> G,s3\<turnstile>v\<Colon>\<preceq>resTy sm)"
+apply clarify
+apply (case_tac "x2")
+defer
+apply (clarsimp split add: split_if_asm simp add: init_lvars_def2)
+apply (case_tac "a' = Null \<and> \<not> (static (mhd esm)) \<and> e \<noteq> Super")
+apply (clarsimp simp add: init_lvars_def2)
+apply clarsimp
+apply (drule eval_gext')
+apply (frule (1) conf_gext)
+apply (drule max_spec2mheads, clarsimp)
+apply (subgoal_tac "invmode (mhd esm) e = IntVir \<longrightarrow> a' \<noteq> Null")
+defer
+apply (clarsimp simp add: invmode_IntVir_eq)
+apply (frule (6) DynT_mheadsD [OF DynT_propI,of _ "s2"],(rule HOL.refl)+)
+apply clarify
+apply (drule wf_mdeclD1, clarsimp)
+apply (frule ty_expr_is_type) apply simp
+apply (frule (2) conforms_init_lvars)
+apply simp
+apply assumption+
+apply simp
+apply assumption+
+apply clarsimp
+apply (rule HOL.refl)
+apply simp
+apply (rule Ball_weaken)
+apply assumption
+apply (force simp add: is_acc_type_def)
+apply (tactic "smp_tac 1 1")
+apply (frule (2) wt_MethdI, clarsimp)
+apply (subgoal_tac "is_static dm = (static (mthd esm))")
+apply (simp only:)
+apply (tactic "smp_tac 1 1")
+apply (rule conjI)
+apply (erule conforms_return)
+apply blast
+
+apply (force dest!: eval_gext del: impCE simp add: init_lvars_def2)
+apply clarsimp
+apply (drule (2) widen_trans, erule (1) conf_widen)
+apply (erule wf_ws_prog)
+
+apply auto
+done
+
+
+subsection "accessibility"
+
+theorem dynamic_field_access_ok:
+ (assumes wf: "wf_prog G" and
+ eval_e: "G\<turnstile>s1 \<midarrow>e-\<succ>a\<rightarrow> s2" and
+ not_Null: "a\<noteq>Null" and
+ conform_a: "G,(store s2)\<turnstile>a\<Colon>\<preceq> Class statC" and
+ conform_s2: "s2\<Colon>\<preceq>(G, L)" and
+ normal_s2: "normal s2" and
+ wt_e: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>,dt\<Turnstile>e\<Colon>-Class statC" and
+ f: "accfield G accC statC fn = Some f" and
+ dynC: "if stat then dynC=statC
+ else dynC=obj_class (lookup_obj (store s2) a)"
+ ) "table_of (DeclConcepts.fields G dynC) (fn,declclass f) = Some (fld f) \<and>
+ G\<turnstile>Field fn f in dynC dyn_accessible_from accC"
+proof (cases "stat")
+ case True
+ with dynC
+ have dynC': "dynC=statC" by simp
+ with f
+ have "table_of (DeclConcepts.fields G dynC) (fn,declclass f) = Some (fld f)"
+ by (auto simp add: accfield_def Let_def intro!: table_of_remap_SomeD)
+ with dynC' f
+ show ?thesis
+ by (auto intro!: static_to_dynamic_accessible_from
+ dest: accfield_accessibleD accessible_from_commonD)
+next
+ case False
+ with wf conform_a not_Null conform_s2 dynC
+ obtain subclseq: "G\<turnstile>dynC \<preceq>\<^sub>C statC" and
+ "is_class G dynC"
+ by (auto dest!: conforms_RefTD [of _ _ _ _ "(fst s2)" L]
+ dest: obj_ty_obj_class1
+ simp add: obj_ty_obj_class )
+ with wf f
+ have "table_of (DeclConcepts.fields G dynC) (fn,declclass f) = Some (fld f)"
+ by (auto simp add: accfield_def Let_def
+ dest: fields_mono
+ dest!: table_of_remap_SomeD)
+ moreover
+ from f subclseq
+ have "G\<turnstile>Field fn f in dynC dyn_accessible_from accC"
+ by (auto intro!: static_to_dynamic_accessible_from
+ dest: accfield_accessibleD)
+ ultimately show ?thesis
+ by blast
+qed
+
+lemma call_access_ok:
+(assumes invC_prop: "G\<turnstile>invmode (mhd statM) e\<rightarrow>invC\<preceq>statT"
+ and wf: "wf_prog G"
+ and wt_e: "\<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>e\<Colon>-RefT statT"
+ and statM: "statM \<in> mheads G accC statT sig"
+ and invC: "invC = invocation_class (invmode (mhd statM) e) s a statT"
+)"\<exists> dynM. dynlookup G statT invC sig = Some dynM \<and>
+ G\<turnstile>Methd sig dynM in invC dyn_accessible_from accC"
+proof -
+ from wt_e wf have type_statT: "is_type G (RefT statT)"
+ by (auto dest: ty_expr_is_type)
+ from statM have not_Null: "statT \<noteq> NullT" by auto
+ from type_statT wt_e
+ have wf_I: "(\<forall>I. statT = IfaceT I \<longrightarrow> is_iface G I \<and>
+ invmode (mhd statM) e \<noteq> SuperM)"
+ by (auto dest: invocationTypeExpr_noClassD)
+ from wt_e
+ have wf_A: "(\<forall> T. statT = ArrayT T \<longrightarrow> invmode (mhd statM) e \<noteq> SuperM)"
+ by (auto dest: invocationTypeExpr_noClassD)
+ show ?thesis
+ proof (cases "invmode (mhd statM) e = IntVir")
+ case True
+ with invC_prop not_Null
+ have invC_prop': "is_class G invC \<and>
+ (if (\<exists>T. statT=ArrayT T) then invC=Object
+ else G\<turnstile>Class invC\<preceq>RefT statT)"
+ by (auto simp add: DynT_prop_def)
+ with True not_Null
+ have "G,statT \<turnstile> invC valid_lookup_cls_for is_static statM"
+ by (cases statT) (auto simp add: invmode_def
+ split: split_if split_if_asm) (* was deleted above *)
+ with statM type_statT wf
+ show ?thesis
+ by - (rule dynlookup_access,auto)
+ next
+ case False
+ with type_statT wf invC not_Null wf_I wf_A
+ have invC_prop': " is_class G invC \<and>
+ ((\<exists> statC. statT=ClassT statC \<and> invC=statC) \<or>
+ (\<forall> statC. statT\<noteq>ClassT statC \<and> invC=Object)) "
+ by (case_tac "statT") (auto simp add: invocation_class_def
+ split: inv_mode.splits)
+ with not_Null wf
+ have dynlookup_static: "dynlookup G statT invC sig = methd G invC sig"
+ by (case_tac "statT") (auto simp add: dynlookup_def dynmethd_C_C
+ dynimethd_def)
+ from statM wf wt_e not_Null False invC_prop' obtain dynM where
+ "accmethd G accC invC sig = Some dynM"
+ by (auto dest!: static_mheadsD)
+ from invC_prop' False not_Null wf_I
+ have "G,statT \<turnstile> invC valid_lookup_cls_for is_static statM"
+ by (cases statT) (auto simp add: invmode_def
+ split: split_if split_if_asm) (* was deleted above *)
+ with statM type_statT wf
+ show ?thesis
+ by - (rule dynlookup_access,auto)
+ qed
+qed
+
+section "main proof of type safety"
+
+ML {*
+val forward_hyp_tac = EVERY' [smp_tac 1,
+ FIRST'[mp_tac,etac exI,smp_tac 2,smp_tac 1,EVERY'[etac impE,etac exI]],
+ REPEAT o (etac conjE)];
+val typD_tac = eresolve_tac (thms "wt_elim_cases") THEN_ALL_NEW
+ EVERY' [full_simp_tac (simpset() setloop (K no_tac)),
+ clarify_tac(claset() addSEs[])]
+*}
+
+lemma conforms_locals [rule_format]:
+ "(a,b)\<Colon>\<preceq>(G, L) \<longrightarrow> L x = Some T \<longrightarrow> G,b\<turnstile>the (locals b x)\<Colon>\<preceq>T"
+apply (force simp: conforms_def Let_def lconf_def)
+done
+
+lemma eval_type_sound [rule_format (no_asm)]:
+ "wf_prog G \<Longrightarrow> G\<turnstile>s0 \<midarrow>t\<succ>\<rightarrow> (v,s1) \<Longrightarrow> (\<forall>L. s0\<Colon>\<preceq>(G,L) \<longrightarrow>
+ (\<forall>C T. \<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>t\<Colon>T \<longrightarrow> s1\<Colon>\<preceq>(G,L) \<and>
+ (let (x,s) = s1 in x = None \<longrightarrow> G,L,s\<turnstile>t\<succ>v\<Colon>\<preceq>T)))"
+apply (erule eval_induct)
+
+(* 29 subgoals *)
+(* Xcpt, Inst, Methd, Nil, Skip, Expr, Comp *)
+apply (simp_all (no_asm_use) add: Let_def body_def)
+apply (tactic "ALLGOALS (EVERY'[Clarify_tac, TRY o typD_tac,
+ TRY o forward_hyp_tac])")
+apply (tactic"ALLGOALS(EVERY'[asm_simp_tac(simpset()),TRY o Clarify_tac])")
+
+(* 20 subgoals *)
+
+(* Break *)
+apply (erule conforms_absorb)
+
+(* Cons *)
+apply (erule_tac V = "G\<turnstile>Norm s0 \<midarrow>?ea\<succ>\<rightarrow> ?vs1" in thin_rl)
+apply (frule eval_gext')
+apply force
+
+(* LVar *)
+apply (force elim: conforms_localD [THEN lconfD] conforms_lupd
+ simp add: assign_conforms_def lvar_def)
+
+(* Cast *)
+apply (force dest: fits_conf)
+
+(* Lit *)
+apply (rule conf_litval)
+apply (simp add: empty_dt_def)
+
+(* Super *)
+apply (rule conf_widen)
+apply (erule (1) subcls_direct [THEN widen.subcls])
+apply (erule (1) conforms_localD [THEN lconfD])
+apply (erule wf_ws_prog)
+
+(* Acc *)
+apply fast
+
+(* Body *)
+apply (rule conjI)
+apply (rule conforms_absorb)
+apply (fast)
+apply (fast intro: conforms_locals)
+
+(* Cond *)
+apply (simp split: split_if_asm)
+apply (tactic "forward_hyp_tac 1", force)
+apply (tactic "forward_hyp_tac 1", force)
+
+(* If *)
+apply (force split add: split_if_asm)
+
+(* Loop *)
+apply (drule (1) wt.Loop)
+apply (clarsimp split: split_if_asm)
+apply (fast intro: conforms_absorb)
+
+(* Fin *)
+apply (case_tac "x1", force)
+apply (drule spec, erule impE, erule conforms_NormI)
+apply (erule impE)
+apply blast
+apply (clarsimp)
+apply (erule (3) Fin_lemma)
+
+(* Throw *)
+apply (erule (3) Throw_lemma)
+
+(* NewC *)
+apply (clarsimp simp add: is_acc_class_def)
+apply (drule (1) halloc_type_sound,blast, rule HOL.refl, simp, simp)
+
+(* NewA *)
+apply (tactic "smp_tac 1 1",frule wt_init_comp_ty,erule impE,blast)
+apply (tactic "forward_hyp_tac 1")
+apply (case_tac "check_neg i' ab")
+apply (clarsimp simp add: is_acc_type_def)
+apply (drule (2) halloc_type_sound, rule HOL.refl, simp, simp)
+apply force
+
+(* Level 34, 6 subgoals *)
+
+(* Init *)
+apply (case_tac "inited C (globs s0)")
+apply (clarsimp)
+apply (clarsimp)
+apply (frule (1) wf_prog_cdecl)
+apply (drule spec, erule impE, erule (3) conforms_init_class_obj)
+apply (drule_tac "psi" = "class G C = ?x" in asm_rl,erule impE,
+ force dest!: wf_cdecl_supD split add: split_if simp add: is_acc_class_def)
+apply (drule spec, erule impE, erule conforms_set_locals, rule lconf_empty)
+apply (erule impE) apply (rule exI) apply (erule wf_cdecl_wt_init)
+apply (drule (1) conforms_return, force dest: eval_gext', assumption)
+
+
+(* Ass *)
+apply (tactic "forward_hyp_tac 1")
+apply (rename_tac x1 s1 x2 s2 v va w L C Ta T', case_tac x1)
+prefer 2 apply force
+apply (case_tac x2)
+prefer 2 apply force
+apply (simp, drule conjunct2)
+apply (drule (1) conf_widen)
+apply (erule wf_ws_prog)
+apply (erule (2) Ass_lemma)
+apply (clarsimp simp add: assign_conforms_def)
+
+(* Try *)
+apply (drule (1) sxalloc_type_sound, simp (no_asm_use))
+apply (case_tac a)
+apply clarsimp
+apply clarsimp
+apply (tactic "smp_tac 1 1")
+apply (simp split add: split_if_asm)
+apply (fast dest: conforms_deallocL Try_lemma)
+
+(* FVar *)
+
+apply (frule accfield_fields)
+apply (frule ty_expr_is_type [THEN type_is_class],simp)
+apply simp
+apply (frule wf_ws_prog)
+apply (frule (1) fields_declC,simp)
+apply clarsimp
+(*b y EVERY'[datac cfield_defpl_is_class 2, Clarsimp_tac] 1; not useful here*)
+apply (tactic "smp_tac 1 1")
+apply (tactic "forward_hyp_tac 1")
+apply (rule conjI, force split add: split_if simp add: fvar_def2)
+apply (drule init_yields_initd, frule eval_gext')
+apply clarsimp
+apply (case_tac "C=Object")
+apply clarsimp
+apply (erule (9) FVar_lemma)
+
+(* AVar *)
+apply (tactic "forward_hyp_tac 1")
+apply (erule_tac V = "G\<turnstile>Norm s0 \<midarrow>?e1-\<succ>?a'\<rightarrow> (?x1 1, ?s1)" in thin_rl,
+ frule eval_gext')
+apply (rule conjI)
+apply (clarsimp simp add: avar_def2)
+apply clarsimp
+apply (erule (5) AVar_lemma)
+
+(* Call *)
+apply (tactic "forward_hyp_tac 1")
+apply (rule Call_type_sound)
+apply auto
+done
+
+
+declare fun_upd_apply [simp]
+declare split_paired_All [simp] split_paired_Ex [simp]
+declare split_if [split] split_if_asm [split]
+ option.split [split] option.split_asm [split]
+ML_setup {*
+simpset_ref() := simpset() addloop ("split_all_tac", split_all_tac);
+claset_ref() := claset () addSbefore ("split_all_tac", split_all_tac)
+*}
+
+theorem eval_ts:
+ "\<lbrakk>G\<turnstile>s \<midarrow>e-\<succ>v \<rightarrow> (x',s'); wf_prog G; s\<Colon>\<preceq>(G,L); \<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>e\<Colon>-T\<rbrakk>
+\<Longrightarrow> (x',s')\<Colon>\<preceq>(G,L) \<and> (x'=None \<longrightarrow> G,s'\<turnstile>v\<Colon>\<preceq>T)"
+apply (drule (3) eval_type_sound)
+apply (unfold Let_def)
+apply clarsimp
+done
+
+theorem evals_ts:
+"\<lbrakk>G\<turnstile>s \<midarrow>es\<doteq>\<succ>vs\<rightarrow> (x',s'); wf_prog G; s\<Colon>\<preceq>(G,L); \<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>es\<Colon>\<doteq>Ts\<rbrakk>
+\<Longrightarrow> (x',s')\<Colon>\<preceq>(G,L) \<and> (x'=None \<longrightarrow> list_all2 (conf G s') vs Ts)"
+apply (drule (3) eval_type_sound)
+apply (unfold Let_def)
+apply clarsimp
+done
+
+theorem evar_ts:
+"\<lbrakk>G\<turnstile>s \<midarrow>v=\<succ>vf\<rightarrow> (x',s'); wf_prog G; s\<Colon>\<preceq>(G,L); \<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>v\<Colon>=T\<rbrakk> \<Longrightarrow>
+ (x',s')\<Colon>\<preceq>(G,L) \<and> (x'=None \<longrightarrow> G,L,s'\<turnstile>In2 v\<succ>In2 vf\<Colon>\<preceq>Inl T)"
+apply (drule (3) eval_type_sound)
+apply (unfold Let_def)
+apply clarsimp
+done
+
+theorem exec_ts:
+"\<lbrakk>G\<turnstile>s \<midarrow>s0\<rightarrow> s'; wf_prog G; s\<Colon>\<preceq>(G,L); \<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>s0\<Colon>\<surd>\<rbrakk> \<Longrightarrow> s'\<Colon>\<preceq>(G,L)"
+apply (drule (3) eval_type_sound)
+apply (unfold Let_def)
+apply clarsimp
+done
+
+(*
+theorem dyn_methods_understood:
+ "\<And>s. \<lbrakk>wf_prog G; \<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>{t,md,IntVir}e..mn({pTs'}ps)\<Colon>-rT;
+ s\<Colon>\<preceq>(G,L); G\<turnstile>s \<midarrow>e-\<succ>a'\<rightarrow> Norm s'; a' \<noteq> Null\<rbrakk> \<Longrightarrow>
+ \<exists>a obj. a'=Addr a \<and> heap s' a = Some obj \<and>
+ cmethd G (obj_class obj) (mn, pTs') \<noteq> None"
+apply (erule wt_elim_cases)
+apply (drule max_spec2mheads)
+apply (drule (3) eval_ts)
+apply (clarsimp split del: split_if split_if_asm)
+apply (drule (2) DynT_propI)
+apply (simp (no_asm_simp))
+apply (tactic *) (* {* exhaust_cmethd_tac "the (cmethd G (target (invmode m e) s' a' md) (mn, pTs'))" 1 *} *)(*)
+apply (drule (4) DynT_mheadsD [THEN conjunct1], rule HOL.refl)
+apply (drule conf_RefTD)
+apply clarsimp
+done
+*)
+
+end