12854

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(* Title: isabelle/Bali/TypeSafe.thy


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ID: $Id$


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Author: David von Oheimb


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Copyright 1997 Technische Universitaet Muenchen


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*)


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header {* The type soundness proof for Java *}


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theory TypeSafe = Eval + WellForm + Conform:


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section "result conformance"


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constdefs


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assign_conforms :: "st \<Rightarrow> (val \<Rightarrow> state \<Rightarrow> state) \<Rightarrow> ty \<Rightarrow> env_ \<Rightarrow> bool"


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("_\<le>_\<preceq>_\<Colon>\<preceq>_" [71,71,71,71] 70)


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"s\<le>f\<preceq>T\<Colon>\<preceq>E \<equiv>


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\<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"


18 


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rconf :: "prog \<Rightarrow> lenv \<Rightarrow> st \<Rightarrow> term \<Rightarrow> vals \<Rightarrow> tys \<Rightarrow> bool"


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("_,_,_\<turnstile>_\<succ>_\<Colon>\<preceq>_" [71,71,71,71,71,71] 70)


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"G,L,s\<turnstile>t\<succ>v\<Colon>\<preceq>T


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\<equiv> case T of


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Inl T \<Rightarrow> if (\<exists>vf. t=In2 vf)


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then G,s\<turnstile>fst (the_In2 v)\<Colon>\<preceq>T \<and> s\<le>snd (the_In2 v)\<preceq>T\<Colon>\<preceq>(G,L)


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else G,s\<turnstile>the_In1 v\<Colon>\<preceq>T


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 Inr Ts \<Rightarrow> list_all2 (conf G s) (the_In3 v) Ts"


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lemma rconf_In1 [simp]:


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"G,L,s\<turnstile>In1 ec\<succ>In1 v \<Colon>\<preceq>Inl T = G,s\<turnstile>v\<Colon>\<preceq>T"


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apply (unfold rconf_def)


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apply (simp (no_asm))


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done


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lemma rconf_In2 [simp]:


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"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))"


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apply (unfold rconf_def)


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apply (simp (no_asm))


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done


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lemma rconf_In3 [simp]:


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"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"


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apply (unfold rconf_def)


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apply (simp (no_asm))


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done


45 


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section "fits and conf"


47 


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(* unused *)


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lemma conf_fits: "G,s\<turnstile>v\<Colon>\<preceq>T \<Longrightarrow> G,s\<turnstile>v fits T"


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apply (unfold fits_def)


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apply clarify


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apply (erule swap, simp (no_asm_use))


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apply (drule conf_RefTD)


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apply auto


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done


56 


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lemma fits_conf:


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"\<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'"


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apply (auto dest!: fitsD cast_PrimT2 cast_RefT2)


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apply (force dest: conf_RefTD intro: conf_AddrI)


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done


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lemma fits_Array:


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"\<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'"


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apply (auto dest!: fitsD widen_ArrayPrimT widen_ArrayRefT)


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apply (force dest: conf_RefTD intro: conf_AddrI)


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done


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section "gext"


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lemma halloc_gext: "\<And>s1 s2. G\<turnstile>s1 \<midarrow>halloc oi\<succ>a\<rightarrow> s2 \<Longrightarrow> snd s1\<le>snd s2"


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apply (simp (no_asm_simp) only: split_tupled_all)


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apply (erule halloc.induct)


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apply (auto dest!: new_AddrD)


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done


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lemma sxalloc_gext: "\<And>s1 s2. G\<turnstile>s1 \<midarrow>sxalloc\<rightarrow> s2 \<Longrightarrow> snd s1\<le>snd s2"


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apply (simp (no_asm_simp) only: split_tupled_all)


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apply (erule sxalloc.induct)


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apply (auto dest!: halloc_gext)


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done


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lemma eval_gext_lemma [rule_format (no_asm)]:


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"G\<turnstile>s \<midarrow>t\<succ>\<rightarrow> (w,s') \<Longrightarrow> snd s\<le>snd s' \<and> (case w of


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In1 v \<Rightarrow> True


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 In2 vf \<Rightarrow> normal s \<longrightarrow> (\<forall>v x s. s\<le>snd (assign (snd vf) v (x,s)))


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 In3 vs \<Rightarrow> True)"


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apply (erule eval_induct)


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prefer 24


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apply (case_tac "inited C (globs s0)", clarsimp, erule thin_rl) (* Init *)


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apply (auto del: conjI dest!: not_initedD gext_new sxalloc_gext halloc_gext


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simp add: lvar_def fvar_def2 avar_def2 init_lvars_def2


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split del: split_if_asm split add: sum3.split)


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(* 6 subgoals *)


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apply force+


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done


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lemma evar_gext_f:


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"G\<turnstile>Norm s1 \<midarrow>e=\<succ>vf \<rightarrow> s2 \<Longrightarrow> s\<le>snd (assign (snd vf) v (x,s))"


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apply (drule eval_gext_lemma [THEN conjunct2])


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apply auto


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done


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lemmas eval_gext = eval_gext_lemma [THEN conjunct1]


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lemma eval_gext': "G\<turnstile>(x1,s1) \<midarrow>t\<succ>\<rightarrow> (w,x2,s2) \<Longrightarrow> s1\<le>s2"


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apply (drule eval_gext)


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apply auto


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done


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lemma init_yields_initd: "G\<turnstile>Norm s1 \<midarrow>Init C\<rightarrow> s2 \<Longrightarrow> initd C s2"


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apply (erule eval_cases , auto split del: split_if_asm)


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apply (case_tac "inited C (globs s1)")


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apply (clarsimp split del: split_if_asm)+


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apply (drule eval_gext')+


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apply (drule init_class_obj_inited)


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apply (erule inited_gext)


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apply (simp (no_asm_use))


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done


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section "Lemmas"


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lemma obj_ty_obj_class1:


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"\<lbrakk>wf_prog G; is_type G (obj_ty obj)\<rbrakk> \<Longrightarrow> is_class G (obj_class obj)"


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apply (case_tac "tag obj")


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apply (auto simp add: obj_ty_def obj_class_def)


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done


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lemma oconf_init_obj:


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"\<lbrakk>wf_prog G;


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(case r of Heap a \<Rightarrow> is_type G (obj_ty obj)  Stat C \<Rightarrow> is_class G C)


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\<rbrakk> \<Longrightarrow> G,s\<turnstile>obj \<lparr>values:=init_vals (var_tys G (tag obj) r)\<rparr>\<Colon>\<preceq>\<surd>r"


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apply (auto intro!: oconf_init_obj_lemma unique_fields)


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done


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(*


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lemma obj_split: "P obj = (\<forall> oi vs. obj = \<lparr>tag=oi,values=vs\<rparr> \<longrightarrow> ?P \<lparr>tag=oi,values=vs\<rparr>)"


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apply auto


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apply (case_tac "obj")


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apply auto


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*)


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lemma conforms_newG: "\<lbrakk>globs s oref = None; (x, s)\<Colon>\<preceq>(G,L);


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wf_prog G; case oref of Heap a \<Rightarrow> is_type G (obj_ty \<lparr>tag=oi,values=vs\<rparr>)


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 Stat C \<Rightarrow> is_class G C\<rbrakk> \<Longrightarrow>


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(x, init_obj G oi oref s)\<Colon>\<preceq>(G, L)"


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apply (unfold init_obj_def)


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apply (auto elim!: conforms_gupd dest!: oconf_init_obj


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simp add: obj.update_defs)


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done


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lemma conforms_init_class_obj:


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"\<lbrakk>(x,s)\<Colon>\<preceq>(G, L); wf_prog G; class G C=Some y; \<not> inited C (globs s)\<rbrakk> \<Longrightarrow>


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(x,init_class_obj G C s)\<Colon>\<preceq>(G, L)"


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apply (rule not_initedD [THEN conforms_newG])


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apply (auto)


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done


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lemma fst_init_lvars[simp]:


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"fst (init_lvars G C sig (invmode m e) a' pvs (x,s)) =


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(if static m then x else (np a') x)"


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apply (simp (no_asm) add: init_lvars_def2)


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done


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lemma halloc_conforms: "\<And>s1. \<lbrakk>G\<turnstile>s1 \<midarrow>halloc oi\<succ>a\<rightarrow> s2; wf_prog G; s1\<Colon>\<preceq>(G, L);


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is_type G (obj_ty \<lparr>tag=oi,values=fs\<rparr>)\<rbrakk> \<Longrightarrow> s2\<Colon>\<preceq>(G, L)"


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apply (simp (no_asm_simp) only: split_tupled_all)


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apply (case_tac "aa")


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apply (auto elim!: halloc_elim_cases dest!: new_AddrD


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intro!: conforms_newG [THEN conforms_xconf] conf_AddrI)


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done


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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);


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T = obj_ty \<lparr>tag=oi,values=fs\<rparr>; is_type G T\<rbrakk> \<Longrightarrow>


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(x,s)\<Colon>\<preceq>(G, L) \<and> (x = None \<longrightarrow> G,s\<turnstile>Addr a\<Colon>\<preceq>T)"


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apply (auto elim!: halloc_conforms)


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apply (case_tac "aa")


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apply (subst obj_ty_eq)


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apply (auto elim!: halloc_elim_cases dest!: new_AddrD intro!: conf_AddrI)


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done


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lemma sxalloc_type_sound:


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"\<And>s1 s2. \<lbrakk>G\<turnstile>s1 \<midarrow>sxalloc\<rightarrow> s2; wf_prog G\<rbrakk> \<Longrightarrow> case fst s1 of


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None \<Rightarrow> s2 = s1  Some x \<Rightarrow>


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(\<exists>a. fst s2 = Some(Xcpt (Loc a)) \<and> (\<forall>L. s1\<Colon>\<preceq>(G,L) \<longrightarrow> s2\<Colon>\<preceq>(G,L)))"


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apply (simp (no_asm_simp) only: split_tupled_all)


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apply (erule sxalloc.induct)


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apply auto


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apply (rule halloc_conforms [THEN conforms_xconf])


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apply (auto elim!: halloc_elim_cases dest!: new_AddrD intro!: conf_AddrI)


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done


197 


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lemma wt_init_comp_ty:


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"is_acc_type G (pid C) T \<Longrightarrow> \<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>init_comp_ty T\<Colon>\<surd>"


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apply (unfold init_comp_ty_def)


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apply (clarsimp simp add: accessible_in_RefT_simp


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is_acc_type_def is_acc_class_def)


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done


204 


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declare fun_upd_same [simp]


207 


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declare fun_upd_apply [simp del]


209 


210 


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constdefs


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DynT_prop::"[prog,inv_mode,qtname,ref_ty] \<Rightarrow> bool"


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("_\<turnstile>_\<rightarrow>_\<preceq>_"[71,71,71,71]70)


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"G\<turnstile>mode\<rightarrow>D\<preceq>t \<equiv> mode = IntVir \<longrightarrow> is_class G D \<and>


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(if (\<exists>T. t=ArrayT T) then D=Object else G\<turnstile>Class D\<preceq>RefT t)"


216 


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lemma DynT_propI:


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"\<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>


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\<Longrightarrow> G\<turnstile>mode\<rightarrow>invocation_class mode s a' statT\<preceq>statT"


220 
proof (unfold DynT_prop_def)


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assume state_conform: "(x,s)\<Colon>\<preceq>(G, L)"


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and statT_a': "G,s\<turnstile>a'\<Colon>\<preceq>RefT statT"


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and wf: "wf_prog G"


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and mode: "mode = IntVir \<longrightarrow> a' \<noteq> Null"


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let ?invCls = "(invocation_class mode s a' statT)"


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let ?IntVir = "mode = IntVir"


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let ?Concl = "\<lambda>invCls. is_class G invCls \<and>


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(if \<exists>T. statT = ArrayT T


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then invCls = Object


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else G\<turnstile>Class invCls\<preceq>RefT statT)"


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show "?IntVir \<longrightarrow> ?Concl ?invCls"


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proof


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assume modeIntVir: ?IntVir


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with mode have not_Null: "a' \<noteq> Null" ..


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from statT_a' not_Null state_conform


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obtain a obj


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where obj_props: "a' = Addr a" "globs s (Inl a) = Some obj"


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"G\<turnstile>obj_ty obj\<preceq>RefT statT" "is_type G (obj_ty obj)"


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by (blast dest: conforms_RefTD)


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show "?Concl ?invCls"


241 
proof (cases "tag obj")


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case CInst


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with modeIntVir obj_props


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show ?thesis


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by (auto dest!: widen_Array2 split add: split_if)


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next


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case Arr


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from Arr obtain T where "obj_ty obj = T.[]" by (blast dest: obj_ty_Arr1)


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moreover from Arr have "obj_class obj = Object"


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by (blast dest: obj_class_Arr1)


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moreover note modeIntVir obj_props wf


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ultimately show ?thesis by (auto dest!: widen_Array )


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qed


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qed


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qed


256 


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lemma invocation_methd:


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"\<lbrakk>wf_prog G; statT \<noteq> NullT;


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(\<forall> statC. statT = ClassT statC \<longrightarrow> is_class G statC);


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(\<forall> I. statT = IfaceT I \<longrightarrow> is_iface G I \<and> mode \<noteq> SuperM);


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(\<forall> T. statT = ArrayT T \<longrightarrow> mode \<noteq> SuperM);


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G\<turnstile>mode\<rightarrow>invocation_class mode s a' statT\<preceq>statT;


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dynlookup G statT (invocation_class mode s a' statT) sig = Some m \<rbrakk>


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\<Longrightarrow> methd G (invocation_declclass G mode s a' statT sig) sig = Some m"


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proof 


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assume wf: "wf_prog G"


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and not_NullT: "statT \<noteq> NullT"


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and statC_prop: "(\<forall> statC. statT = ClassT statC \<longrightarrow> is_class G statC)"


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and statI_prop: "(\<forall> I. statT = IfaceT I \<longrightarrow> is_iface G I \<and> mode \<noteq> SuperM)"


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and statA_prop: "(\<forall> T. statT = ArrayT T \<longrightarrow> mode \<noteq> SuperM)"


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and invC_prop: "G\<turnstile>mode\<rightarrow>invocation_class mode s a' statT\<preceq>statT"


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and dynlookup: "dynlookup G statT (invocation_class mode s a' statT) sig


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= Some m"


274 
show ?thesis


275 
proof (cases statT)


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case NullT


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with not_NullT show ?thesis by simp


278 
next


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case IfaceT


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with statI_prop obtain I


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where statI: "statT = IfaceT I" and


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is_iface: "is_iface G I" and


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not_SuperM: "mode \<noteq> SuperM" by blast


284 


285 
show ?thesis


286 
proof (cases mode)


287 
case Static


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with wf dynlookup statI is_iface


289 
show ?thesis


290 
by (auto simp add: invocation_declclass_def dynlookup_def


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dynimethd_def dynmethd_C_C


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intro: dynmethd_declclass


293 
dest!: wf_imethdsD


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dest: table_of_map_SomeI


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split: split_if_asm)


296 
next


297 
case SuperM


298 
with not_SuperM show ?thesis ..


299 
next


300 
case IntVir


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with wf dynlookup IfaceT invC_prop show ?thesis


302 
by (auto simp add: invocation_declclass_def dynlookup_def dynimethd_def


303 
DynT_prop_def


304 
intro: methd_declclass dynmethd_declclass


305 
split: split_if_asm)


306 
qed


307 
next


308 
case ClassT


309 
show ?thesis


310 
proof (cases mode)


311 
case Static


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with wf ClassT dynlookup statC_prop


313 
show ?thesis by (auto simp add: invocation_declclass_def dynlookup_def


314 
intro: dynmethd_declclass)


315 
next


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case SuperM


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with wf ClassT dynlookup statC_prop


318 
show ?thesis by (auto simp add: invocation_declclass_def dynlookup_def


319 
intro: dynmethd_declclass)


320 
next


321 
case IntVir


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with wf ClassT dynlookup statC_prop invC_prop


323 
show ?thesis


324 
by (auto simp add: invocation_declclass_def dynlookup_def dynimethd_def


325 
DynT_prop_def


326 
intro: dynmethd_declclass)


327 
qed


328 
next


329 
case ArrayT


330 
show ?thesis


331 
proof (cases mode)


332 
case Static


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with wf ArrayT dynlookup show ?thesis


334 
by (auto simp add: invocation_declclass_def dynlookup_def


335 
dynimethd_def dynmethd_C_C


336 
intro: dynmethd_declclass


337 
dest: table_of_map_SomeI)


338 
next


339 
case SuperM


340 
with ArrayT statA_prop show ?thesis by blast


341 
next


342 
case IntVir


343 
with wf ArrayT dynlookup invC_prop show ?thesis


344 
by (auto simp add: invocation_declclass_def dynlookup_def dynimethd_def


345 
DynT_prop_def dynmethd_C_C


346 
intro: dynmethd_declclass


347 
dest: table_of_map_SomeI)


348 
qed


349 
qed


350 
qed


351 


352 
declare split_paired_All [simp del] split_paired_Ex [simp del]


353 
ML_setup {*


354 
simpset_ref() := simpset() delloop "split_all_tac";


355 
claset_ref () := claset () delSWrapper "split_all_tac"


356 
*}


357 
lemma DynT_mheadsD:


358 
"\<lbrakk>G\<turnstile>invmode (mhd sm) e\<rightarrow>invC\<preceq>statT;


359 
wf_prog G; \<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>e\<Colon>RefT statT;


360 
sm \<in> mheads G C statT sig;


361 
invC = invocation_class (invmode (mhd sm) e) s a' statT;


362 
declC =invocation_declclass G (invmode (mhd sm) e) s a' statT sig


363 
\<rbrakk> \<Longrightarrow>


364 
\<exists> dm.


365 
methd G declC sig = Some dm \<and> G\<turnstile>resTy (mthd dm)\<preceq>resTy (mhd sm) \<and>


366 
wf_mdecl G declC (sig, mthd dm) \<and>


367 
declC = declclass dm \<and>


368 
is_static dm = is_static sm \<and>


369 
is_class G invC \<and> is_class G declC \<and> G\<turnstile>invC\<preceq>\<^sub>C declC \<and>


370 
(if invmode (mhd sm) e = IntVir


371 
then (\<forall> statC. statT=ClassT statC \<longrightarrow> G\<turnstile>invC \<preceq>\<^sub>C statC)


372 
else ( (\<exists> statC. statT=ClassT statC \<and> G\<turnstile>statC\<preceq>\<^sub>C declC)


373 
\<or> (\<forall> statC. statT\<noteq>ClassT statC \<and> declC=Object)) \<and>


374 
(declrefT sm) = ClassT (declclass dm))"


375 
proof 


376 
assume invC_prop: "G\<turnstile>invmode (mhd sm) e\<rightarrow>invC\<preceq>statT"


377 
and wf: "wf_prog G"


378 
and wt_e: "\<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>e\<Colon>RefT statT"


379 
and sm: "sm \<in> mheads G C statT sig"


380 
and invC: "invC = invocation_class (invmode (mhd sm) e) s a' statT"


381 
and declC: "declC =


382 
invocation_declclass G (invmode (mhd sm) e) s a' statT sig"


383 
from wt_e wf have type_statT: "is_type G (RefT statT)"


384 
by (auto dest: ty_expr_is_type)


385 
from sm have not_Null: "statT \<noteq> NullT" by auto


386 
from type_statT


387 
have wf_C: "(\<forall> statC. statT = ClassT statC \<longrightarrow> is_class G statC)"


388 
by (auto)


389 
from type_statT wt_e


390 
have wf_I: "(\<forall>I. statT = IfaceT I \<longrightarrow> is_iface G I \<and>


391 
invmode (mhd sm) e \<noteq> SuperM)"


392 
by (auto dest: invocationTypeExpr_noClassD)


393 
from wt_e


394 
have wf_A: "(\<forall> T. statT = ArrayT T \<longrightarrow> invmode (mhd sm) e \<noteq> SuperM)"


395 
by (auto dest: invocationTypeExpr_noClassD)


396 
show ?thesis


397 
proof (cases "invmode (mhd sm) e = IntVir")


398 
case True


399 
with invC_prop not_Null


400 
have invC_prop': " is_class G invC \<and>


401 
(if (\<exists>T. statT=ArrayT T) then invC=Object


402 
else G\<turnstile>Class invC\<preceq>RefT statT)"


403 
by (auto simp add: DynT_prop_def)


404 
from True


405 
have "\<not> is_static sm"


406 
by (simp add: invmode_IntVir_eq)


407 
with invC_prop' not_Null


408 
have "G,statT \<turnstile> invC valid_lookup_cls_for (is_static sm)"


409 
by (cases statT) auto


410 
with sm wf type_statT obtain dm where


411 
dm: "dynlookup G statT invC sig = Some dm" and


412 
resT_dm: "G\<turnstile>resTy (mthd dm)\<preceq>resTy (mhd sm)" and


413 
static: "is_static dm = is_static sm"


414 
by  (drule dynamic_mheadsD,auto)


415 
with declC invC not_Null


416 
have declC': "declC = (declclass dm)"


417 
by (auto simp add: invocation_declclass_def)


418 
with wf invC declC not_Null wf_C wf_I wf_A invC_prop dm


419 
have dm': "methd G declC sig = Some dm"


420 
by  (drule invocation_methd,auto)


421 
from wf dm invC_prop' declC' type_statT


422 
have declC_prop: "G\<turnstile>invC \<preceq>\<^sub>C declC \<and> is_class G declC"


423 
by (auto dest: dynlookup_declC)


424 
from wf dm' declC_prop declC'


425 
have wf_dm: "wf_mdecl G declC (sig,(mthd dm))"


426 
by (auto dest: methd_wf_mdecl)


427 
from invC_prop'


428 
have statC_prop: "(\<forall> statC. statT=ClassT statC \<longrightarrow> G\<turnstile>invC \<preceq>\<^sub>C statC)"


429 
by auto


430 
from True dm' resT_dm wf_dm invC_prop' declC_prop statC_prop declC' static


431 
show ?thesis by auto


432 
next


433 
case False


434 
with type_statT wf invC not_Null wf_I wf_A


435 
have invC_prop': "is_class G invC \<and>


436 
((\<exists> statC. statT=ClassT statC \<and> invC=statC) \<or>


437 
(\<forall> statC. statT\<noteq>ClassT statC \<and> invC=Object)) "


438 
by (case_tac "statT") (auto simp add: invocation_class_def


439 
split: inv_mode.splits)


440 
with not_Null wf


441 
have dynlookup_static: "dynlookup G statT invC sig = methd G invC sig"


442 
by (case_tac "statT") (auto simp add: dynlookup_def dynmethd_C_C


443 
dynimethd_def)


444 
from sm wf wt_e not_Null False invC_prop' obtain "dm" where


445 
dm: "methd G invC sig = Some dm" and


446 
eq_declC_sm_dm:"declrefT sm = ClassT (declclass dm)" and


447 
eq_mheads:"mhd sm=mhead (mthd dm) "


448 
by  (drule static_mheadsD, auto dest: accmethd_SomeD)


449 
then have static: "is_static dm = is_static sm" by  (case_tac "sm",auto)


450 
with declC invC dynlookup_static dm


451 
have declC': "declC = (declclass dm)"


452 
by (auto simp add: invocation_declclass_def)


453 
from invC_prop' wf declC' dm


454 
have dm': "methd G declC sig = Some dm"


455 
by (auto intro: methd_declclass)


456 
from wf dm invC_prop' declC' type_statT


457 
have declC_prop: "G\<turnstile>invC \<preceq>\<^sub>C declC \<and> is_class G declC"


458 
by (auto dest: methd_declC )


459 
then have declC_prop1: "invC=Object \<longrightarrow> declC=Object" by auto


460 
from wf dm' declC_prop declC'


461 
have wf_dm: "wf_mdecl G declC (sig,(mthd dm))"


462 
by (auto dest: methd_wf_mdecl)


463 
from invC_prop' declC_prop declC_prop1


464 
have statC_prop: "( (\<exists> statC. statT=ClassT statC \<and> G\<turnstile>statC\<preceq>\<^sub>C declC)


465 
\<or> (\<forall> statC. statT\<noteq>ClassT statC \<and> declC=Object))"


466 
by auto


467 
from False dm' wf_dm invC_prop' declC_prop statC_prop declC'


468 
eq_declC_sm_dm eq_mheads static


469 
show ?thesis by auto


470 
qed


471 
qed


472 


473 
(*


474 
lemma DynT_mheadsD:


475 
"\<lbrakk>G\<turnstile>invmode (mhd sm) e\<rightarrow>invC\<preceq>statT;


476 
wf_prog G; \<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>e\<Colon>RefT statT;


477 
sm \<in> mheads G C statT sig;


478 
invC = invocation_class (invmode (mhd sm) e) s a' statT;


479 
declC =invocation_declclass G (invmode (mhd sm) e) s a' statT sig


480 
\<rbrakk> \<Longrightarrow>


481 
\<exists> dm.


482 
methd G declC sig = Some dm \<and> G\<turnstile>resTy (mthd dm)\<preceq>resTy (mhd sm) \<and>


483 
wf_mdecl G declC (sig, mthd dm) \<and>


484 
is_class G invC \<and> is_class G declC \<and> G\<turnstile>invC\<preceq>\<^sub>C declC \<and>


485 
(if invmode (mhd sm) e = IntVir


486 
then (\<forall> statC. statT=ClassT statC \<longrightarrow> G\<turnstile>invC \<preceq>\<^sub>C statC)


487 
else (\<forall> statC. statT=ClassT statC \<longrightarrow> G\<turnstile>statC \<preceq>\<^sub>C declC) \<and>


488 
(declrefT sm) = ClassT (declclass dm))"


489 
proof 


490 
assume invC_prop: "G\<turnstile>invmode (mhd sm) e\<rightarrow>invC\<preceq>statT"


491 
and wf: "wf_prog G"


492 
and wt_e: "\<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>e\<Colon>RefT statT"


493 
and sm: "sm \<in> mheads G C statT sig"


494 
and invC: "invC = invocation_class (invmode (mhd sm) e) s a' statT"


495 
and declC: "declC =


496 
invocation_declclass G (invmode (mhd sm) e) s a' statT sig"


497 
from wt_e wf have type_statT: "is_type G (RefT statT)"


498 
by (auto dest: ty_expr_is_type)


499 
from sm have not_Null: "statT \<noteq> NullT" by auto


500 
from type_statT


501 
have wf_C: "(\<forall> statC. statT = ClassT statC \<longrightarrow> is_class G statC)"


502 
by (auto)


503 
from type_statT wt_e


504 
have wf_I: "(\<forall>I. statT = IfaceT I \<longrightarrow> is_iface G I \<and>


505 
invmode (mhd sm) e \<noteq> SuperM)"


506 
by (auto dest: invocationTypeExpr_noClassD)


507 
from wt_e


508 
have wf_A: "(\<forall> T. statT = ArrayT T \<longrightarrow> invmode (mhd sm) e \<noteq> SuperM)"


509 
by (auto dest: invocationTypeExpr_noClassD)


510 
show ?thesis


511 
proof (cases "invmode (mhd sm) e = IntVir")


512 
case True


513 
with invC_prop not_Null


514 
have invC_prop': "is_class G invC \<and>


515 
(if (\<exists>T. statT=ArrayT T) then invC=Object


516 
else G\<turnstile>Class invC\<preceq>RefT statT)"


517 
by (auto simp add: DynT_prop_def)


518 
with sm wf type_statT not_Null obtain dm where


519 
dm: "dynlookup G statT invC sig = Some dm" and


520 
resT_dm: "G\<turnstile>resTy (mthd dm)\<preceq>resTy (mhd sm)"


521 
by  (clarify,drule dynamic_mheadsD,auto)


522 
with declC invC not_Null


523 
have declC': "declC = (declclass dm)"


524 
by (auto simp add: invocation_declclass_def)


525 
with wf invC declC not_Null wf_C wf_I wf_A invC_prop dm


526 
have dm': "methd G declC sig = Some dm"


527 
by  (drule invocation_methd,auto)


528 
from wf dm invC_prop' declC' type_statT


529 
have declC_prop: "G\<turnstile>invC \<preceq>\<^sub>C declC \<and> is_class G declC"


530 
by (auto dest: dynlookup_declC)


531 
from wf dm' declC_prop declC'


532 
have wf_dm: "wf_mdecl G declC (sig,(mthd dm))"


533 
by (auto dest: methd_wf_mdecl)


534 
from invC_prop'


535 
have statC_prop: "(\<forall> statC. statT=ClassT statC \<longrightarrow> G\<turnstile>invC \<preceq>\<^sub>C statC)"


536 
by auto


537 
from True dm' resT_dm wf_dm invC_prop' declC_prop statC_prop


538 
show ?thesis by auto


539 
next


540 
case False


541 


542 
with type_statT wf invC not_Null wf_I wf_A


543 
have invC_prop': "is_class G invC \<and>


544 
((\<exists> statC. statT=ClassT statC \<and> invC=statC) \<or>


545 
(\<forall> statC. statT\<noteq>ClassT statC \<and> invC=Object)) "


546 


547 
by (case_tac "statT") (auto simp add: invocation_class_def


548 
split: inv_mode.splits)


549 
with not_Null


550 
have dynlookup_static: "dynlookup G statT invC sig = methd G invC sig"


551 
by (case_tac "statT") (auto simp add: dynlookup_def dynmethd_def


552 
dynimethd_def)


553 
from sm wf wt_e not_Null False invC_prop' obtain "dm" where


554 
dm: "methd G invC sig = Some dm" and


555 
eq_declC_sm_dm:"declrefT sm = ClassT (declclass dm)" and


556 
eq_mheads:"mhd sm=mhead (mthd dm) "


557 
by  (drule static_mheadsD, auto dest: accmethd_SomeD)


558 
with declC invC dynlookup_static dm


559 
have declC': "declC = (declclass dm)"


560 
by (auto simp add: invocation_declclass_def)


561 
from invC_prop' wf declC' dm


562 
have dm': "methd G declC sig = Some dm"


563 
by (auto intro: methd_declclass)


564 
from wf dm invC_prop' declC' type_statT


565 
have declC_prop: "G\<turnstile>invC \<preceq>\<^sub>C declC \<and> is_class G declC"


566 
by (auto dest: methd_declC )


567 
from wf dm' declC_prop declC'


568 
have wf_dm: "wf_mdecl G declC (sig,(mthd dm))"


569 
by (auto dest: methd_wf_mdecl)


570 
from invC_prop' declC_prop


571 
have statC_prop: "(\<forall> statC. statT=ClassT statC \<longrightarrow> G\<turnstile>statC \<preceq>\<^sub>C declC)"


572 
by auto


573 
from False dm' wf_dm invC_prop' declC_prop statC_prop


574 
eq_declC_sm_dm eq_mheads


575 
show ?thesis by auto


576 
qed


577 
qed


578 
*)


579 


580 
declare split_paired_All [simp del] split_paired_Ex [simp del]


581 
declare split_if [split del] split_if_asm [split del]


582 
option.split [split del] option.split_asm [split del]


583 
ML_setup {*


584 
simpset_ref() := simpset() delloop "split_all_tac";


585 
claset_ref () := claset () delSWrapper "split_all_tac"


586 
*}


587 


588 
lemma DynT_conf: "\<lbrakk>G\<turnstile>invocation_class mode s a' statT \<preceq>\<^sub>C declC; wf_prog G;


589 
isrtype G (statT);


590 
G,s\<turnstile>a'\<Colon>\<preceq>RefT statT; mode = IntVir \<longrightarrow> a' \<noteq> Null;


591 
mode \<noteq> IntVir \<longrightarrow> (\<exists> statC. statT=ClassT statC \<and> G\<turnstile>statC\<preceq>\<^sub>C declC)


592 
\<or> (\<forall> statC. statT\<noteq>ClassT statC \<and> declC=Object)\<rbrakk>


593 
\<Longrightarrow>G,s\<turnstile>a'\<Colon>\<preceq> Class declC"


594 
apply (case_tac "mode = IntVir")


595 
apply (drule conf_RefTD)


596 
apply (force intro!: conf_AddrI


597 
simp add: obj_class_def split add: obj_tag.split_asm)


598 
apply clarsimp


599 
apply safe


600 
apply (erule (1) widen.subcls [THEN conf_widen])


601 
apply (erule wf_ws_prog)


602 


603 
apply (frule widen_Object) apply (erule wf_ws_prog)


604 
apply (erule (1) conf_widen) apply (erule wf_ws_prog)


605 
done


606 


607 


608 
lemma Ass_lemma:


609 
"\<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';


610 
s1\<le>s2 \<longrightarrow> assign f v (Norm s2)\<Colon>\<preceq>(G, L)


611 
\<rbrakk> \<Longrightarrow> assign f v (Norm s2)\<Colon>\<preceq>(G, L) \<and>


612 
(\<lambda>(x',s'). x' = None \<longrightarrow> G,s'\<turnstile>v\<Colon>\<preceq>T') (assign f v (Norm s2))"


613 
apply (drule_tac x = "None" and s = "s2" and v = "v" in evar_gext_f)


614 
apply (drule eval_gext', clarsimp)


615 
apply (erule conf_gext)


616 
apply simp


617 
done


618 


619 
lemma Throw_lemma: "\<lbrakk>G\<turnstile>tn\<preceq>\<^sub>C SXcpt Throwable; wf_prog G; (x1,s1)\<Colon>\<preceq>(G, L);


620 
x1 = None \<longrightarrow> G,s1\<turnstile>a'\<Colon>\<preceq> Class tn\<rbrakk> \<Longrightarrow> (throw a' x1, s1)\<Colon>\<preceq>(G, L)"


621 
apply (auto split add: split_abrupt_if simp add: throw_def2)


622 
apply (erule conforms_xconf)


623 
apply (frule conf_RefTD)


624 
apply (auto elim: widen.subcls [THEN conf_widen])


625 
done


626 


627 
lemma Try_lemma: "\<lbrakk>G\<turnstile>obj_ty (the (globs s1' (Heap a)))\<preceq> Class tn;


628 
(Some (Xcpt (Loc a)), s1')\<Colon>\<preceq>(G, L); wf_prog G\<rbrakk>


629 
\<Longrightarrow> Norm (lupd(vn\<mapsto>Addr a) s1')\<Colon>\<preceq>(G, L(vn\<mapsto>Class tn))"


630 
apply (rule conforms_allocL)


631 
apply (erule conforms_NormI)


632 
apply (drule conforms_XcptLocD [THEN conf_RefTD],rule HOL.refl)


633 
apply (auto intro!: conf_AddrI)


634 
done


635 


636 
lemma Fin_lemma:


637 
"\<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>


638 
\<Longrightarrow> (abrupt_if True (Some a) x2, s2)\<Colon>\<preceq>(G, L)"


639 
apply (force elim: eval_gext' conforms_xgext split add: split_abrupt_if)


640 
done


641 


642 
lemma FVar_lemma1: "\<lbrakk>table_of (DeclConcepts.fields G Ca) (fn, C) = Some f ;


643 
x2 = None \<longrightarrow> G,s2\<turnstile>a\<Colon>\<preceq> Class Ca; wf_prog G; G\<turnstile>Ca\<preceq>\<^sub>C C; C \<noteq> Object;


644 
class G C = Some y; (x2,s2)\<Colon>\<preceq>(G, L); s1\<le>s2; inited C (globs s1);


645 
(if static f then id else np a) x2 = None\<rbrakk>


646 
\<Longrightarrow>


647 
\<exists>obj. globs s2 (if static f then Inr C else Inl (the_Addr a)) = Some obj \<and>


648 
var_tys G (tag obj) (if static f then Inr C else Inl(the_Addr a))


649 
(Inl(fn,C)) = Some (type f)"


650 
apply (drule initedD)


651 
apply (frule subcls_is_class2, simp (no_asm_simp))


652 
apply (case_tac "static f")


653 
apply clarsimp


654 
apply (drule (1) rev_gext_objD, clarsimp)


655 
apply (frule fields_declC, erule wf_ws_prog, simp (no_asm_simp))


656 
apply (rule var_tys_Some_eq [THEN iffD2])


657 
apply clarsimp


658 
apply (erule fields_table_SomeI, simp (no_asm))


659 
apply clarsimp


660 
apply (drule conf_RefTD, clarsimp, rule var_tys_Some_eq [THEN iffD2])


661 
apply (auto dest!: widen_Array split add: obj_tag.split)


662 
apply (rotate_tac 1) (* to front: tag obja = CInst pid_field_type to enable


663 
conditional rewrite *)


664 
apply (rule fields_table_SomeI)


665 
apply (auto elim!: fields_mono subcls_is_class2)


666 
done


667 


668 
lemma FVar_lemma:


669 
"\<lbrakk>((v, f), Norm s2') = fvar C (static field) fn a (x2, s2); G\<turnstile>Ca\<preceq>\<^sub>C C;


670 
table_of (DeclConcepts.fields G Ca) (fn, C) = Some field; wf_prog G;


671 
x2 = None \<longrightarrow> G,s2\<turnstile>a\<Colon>\<preceq>Class Ca; C \<noteq> Object; class G C = Some y;


672 
(x2, s2)\<Colon>\<preceq>(G, L); s1\<le>s2; inited C (globs s1)\<rbrakk> \<Longrightarrow>


673 
G,s2'\<turnstile>v\<Colon>\<preceq>type field \<and> s2'\<le>f\<preceq>type field\<Colon>\<preceq>(G, L)"


674 
apply (unfold assign_conforms_def)


675 
apply (drule sym)


676 
apply (clarsimp simp add: fvar_def2)


677 
apply (drule (9) FVar_lemma1)


678 
apply (clarsimp)


679 
apply (drule (2) conforms_globsD [THEN oconf_lconf, THEN lconfD])


680 
apply clarsimp


681 
apply (drule (1) rev_gext_objD)


682 
apply (auto elim!: conforms_upd_gobj)


683 
done


684 


685 


686 
lemma AVar_lemma1: "\<lbrakk>globs s (Inl a) = Some obj;tag obj=Arr ty i;


687 
the_Intg i' in_bounds i; wf_prog G; G\<turnstile>ty.[]\<preceq>Tb.[]; Norm s\<Colon>\<preceq>(G, L)


688 
\<rbrakk> \<Longrightarrow> G,s\<turnstile>the ((values obj) (Inr (the_Intg i')))\<Colon>\<preceq>Tb"


689 
apply (erule widen_Array_Array [THEN conf_widen])


690 
apply (erule_tac [2] wf_ws_prog)


691 
apply (drule (1) conforms_globsD [THEN oconf_lconf, THEN lconfD])


692 
defer apply assumption


693 
apply (force intro: var_tys_Some_eq [THEN iffD2])


694 
done


695 


696 
lemma obj_split: "\<And> obj. \<exists> t vs. obj = \<lparr>tag=t,values=vs\<rparr>"


697 
proof record_split


698 
fix tag values more


699 
show "\<exists>t vs. \<lparr>tag = tag, values = values, \<dots> = more\<rparr> = \<lparr>tag = t, values = vs\<rparr>"


700 
by auto


701 
qed


702 


703 
lemma AVar_lemma: "\<lbrakk>wf_prog G; G\<turnstile>(x1, s1) \<midarrow>e2\<succ>i\<rightarrow> (x2, s2);


704 
((v,f), Norm s2') = avar G i a (x2, s2); x1 = None \<longrightarrow> G,s1\<turnstile>a\<Colon>\<preceq>Ta.[];


705 
(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)"


706 
apply (unfold assign_conforms_def)


707 
apply (drule sym)


708 
apply (clarsimp simp add: avar_def2)


709 
apply (drule (1) conf_gext)


710 
apply (drule conf_RefTD, clarsimp)


711 
apply (subgoal_tac "\<exists> t vs. obj = \<lparr>tag=t,values=vs\<rparr>")


712 
defer


713 
apply (rule obj_split)


714 
apply clarify


715 
apply (frule obj_ty_widenD)


716 
apply (auto dest!: widen_Class)


717 
apply (force dest: AVar_lemma1)


718 
apply (auto split add: split_if)


719 
apply (force elim!: fits_Array dest: gext_objD


720 
intro: var_tys_Some_eq [THEN iffD2] conforms_upd_gobj)


721 
done


722 


723 


724 
section "Call"


725 
lemma conforms_init_lvars_lemma: "\<lbrakk>wf_prog G;


726 
wf_mhead G P sig mh;


727 
Ball (set lvars) (split (\<lambda>vn. is_type G));


728 
list_all2 (conf G s) pvs pTsa; G\<turnstile>pTsa[\<preceq>](parTs sig)\<rbrakk> \<Longrightarrow>


729 
G,s\<turnstile>init_vals (table_of lvars)(pars mh[\<mapsto>]pvs)


730 
[\<Colon>\<preceq>]table_of lvars(pars mh[\<mapsto>]parTs sig)"


731 
apply (unfold wf_mhead_def)


732 
apply clarify


733 
apply (erule (2) Ball_set_table [THEN lconf_init_vals, THEN lconf_ext_list])


734 
apply (drule wf_ws_prog)


735 
apply (erule (2) conf_list_widen)


736 
done


737 


738 


739 
lemma lconf_map_lname [simp]:


740 
"G,s\<turnstile>(lname_case l1 l2)[\<Colon>\<preceq>](lname_case L1 L2)


741 
=


742 
(G,s\<turnstile>l1[\<Colon>\<preceq>]L1 \<and> G,s\<turnstile>(\<lambda>x::unit . l2)[\<Colon>\<preceq>](\<lambda>x::unit. L2))"


743 
apply (unfold lconf_def)


744 
apply safe


745 
apply (case_tac "n")


746 
apply (force split add: lname.split)+


747 
done


748 


749 
lemma lconf_map_ename [simp]:


750 
"G,s\<turnstile>(ename_case l1 l2)[\<Colon>\<preceq>](ename_case L1 L2)


751 
=


752 
(G,s\<turnstile>l1[\<Colon>\<preceq>]L1 \<and> G,s\<turnstile>(\<lambda>x::unit. l2)[\<Colon>\<preceq>](\<lambda>x::unit. L2))"


753 
apply (unfold lconf_def)


754 
apply safe


755 
apply (force split add: ename.split)+


756 
done


757 


758 


759 
lemma defval_conf1 [rule_format (no_asm), elim]:


760 
"is_type G T \<longrightarrow> (\<exists>v\<in>Some (default_val T): G,s\<turnstile>v\<Colon>\<preceq>T)"


761 
apply (unfold conf_def)


762 
apply (induct "T")


763 
apply (auto intro: prim_ty.induct)


764 
done


765 


766 


767 
lemma conforms_init_lvars:


768 
"\<lbrakk>wf_mhead G (pid declC) sig (mhead (mthd dm)); wf_prog G;


769 
list_all2 (conf G s) pvs pTsa; G\<turnstile>pTsa[\<preceq>](parTs sig);


770 
(x, s)\<Colon>\<preceq>(G, L);


771 
methd G declC sig = Some dm;


772 
isrtype G statT;


773 
G\<turnstile>invC\<preceq>\<^sub>C declC;


774 
G,s\<turnstile>a'\<Colon>\<preceq>RefT statT;


775 
invmode (mhd sm) e = IntVir \<longrightarrow> a' \<noteq> Null;


776 
invmode (mhd sm) e \<noteq> IntVir \<longrightarrow>


777 
(\<exists> statC. statT=ClassT statC \<and> G\<turnstile>statC\<preceq>\<^sub>C declC)


778 
\<or> (\<forall> statC. statT\<noteq>ClassT statC \<and> declC=Object);


779 
invC = invocation_class (invmode (mhd sm) e) s a' statT;


780 
declC = invocation_declclass G (invmode (mhd sm) e) s a' statT sig;


781 
Ball (set (lcls (mbody (mthd dm))))


782 
(split (\<lambda>vn. is_type G))


783 
\<rbrakk> \<Longrightarrow>


784 
init_lvars G declC sig (invmode (mhd sm) e) a'


785 
pvs (x,s)\<Colon>\<preceq>(G,\<lambda> k.


786 
(case k of


787 
EName e \<Rightarrow> (case e of


788 
VNam v


789 
\<Rightarrow> (table_of (lcls (mbody (mthd dm)))


790 
(pars (mthd dm)[\<mapsto>]parTs sig)) v


791 
 Res \<Rightarrow> Some (resTy (mthd dm)))


792 
 This \<Rightarrow> if (static (mthd sm))


793 
then None else Some (Class declC)))"


794 
apply (simp add: init_lvars_def2)


795 
apply (rule conforms_set_locals)


796 
apply (simp (no_asm_simp) split add: split_if)


797 
apply (drule (4) DynT_conf)


798 
apply clarsimp


799 
(* apply intro *)


800 
apply (drule (4) conforms_init_lvars_lemma)


801 
apply (case_tac "dm",simp)


802 
apply (rule conjI)


803 
apply (unfold lconf_def, clarify)


804 
apply (rule defval_conf1)


805 
apply (clarsimp simp add: wf_mhead_def is_acc_type_def)


806 
apply (case_tac "is_static sm")


807 
apply simp_all


808 
done


809 


810 


811 
lemma Call_type_sound: "\<lbrakk>wf_prog G; G\<turnstile>(x1, s1) \<midarrow>ps\<doteq>\<succ>pvs\<rightarrow> (x2, s2);


812 
declC


813 
= invocation_declclass G (invmode (mhd esm) e) s2 a' statT \<lparr>name=mn,parTs=pTs\<rparr>;


814 
s2'=init_lvars G declC \<lparr>name=mn,parTs=pTs\<rparr> (invmode (mhd esm) e) a' pvs (x2,s2);


815 
G\<turnstile>s2' \<midarrow>Methd declC \<lparr>name=mn,parTs=pTs\<rparr>\<succ>v\<rightarrow> (x3, s3);


816 
\<forall>L. s2'\<Colon>\<preceq>(G, L)


817 
\<longrightarrow> (\<forall>T. \<lparr>prg=G,cls=declC,lcl=L\<rparr>\<turnstile> Methd declC \<lparr>name=mn,parTs=pTs\<rparr>\<Colon>T


818 
\<longrightarrow> (x3, s3)\<Colon>\<preceq>(G, L) \<and> (x3 = None \<longrightarrow> G,s3\<turnstile>v\<Colon>\<preceq>T));


819 
Norm s0\<Colon>\<preceq>(G, L);


820 
\<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;


821 
max_spec G C statT \<lparr>name=mn,parTs=pTsa\<rparr> = {(esm, pTs)};


822 
(x1, s1)\<Colon>\<preceq>(G, L);


823 
x1 = None \<longrightarrow> G,s1\<turnstile>a'\<Colon>\<preceq>RefT statT; (x2, s2)\<Colon>\<preceq>(G, L);


824 
x2 = None \<longrightarrow> list_all2 (conf G s2) pvs pTsa;


825 
sm=(mhd esm)\<rbrakk> \<Longrightarrow>


826 
(x3, set_locals (locals s2) s3)\<Colon>\<preceq>(G, L) \<and>


827 
(x3 = None \<longrightarrow> G,s3\<turnstile>v\<Colon>\<preceq>resTy sm)"


828 
apply clarify


829 
apply (case_tac "x2")


830 
defer


831 
apply (clarsimp split add: split_if_asm simp add: init_lvars_def2)


832 
apply (case_tac "a' = Null \<and> \<not> (static (mhd esm)) \<and> e \<noteq> Super")


833 
apply (clarsimp simp add: init_lvars_def2)


834 
apply clarsimp


835 
apply (drule eval_gext')


836 
apply (frule (1) conf_gext)


837 
apply (drule max_spec2mheads, clarsimp)


838 
apply (subgoal_tac "invmode (mhd esm) e = IntVir \<longrightarrow> a' \<noteq> Null")


839 
defer


840 
apply (clarsimp simp add: invmode_IntVir_eq)


841 
apply (frule (6) DynT_mheadsD [OF DynT_propI,of _ "s2"],(rule HOL.refl)+)


842 
apply clarify


843 
apply (drule wf_mdeclD1, clarsimp)


844 
apply (frule ty_expr_is_type) apply simp


845 
apply (frule (2) conforms_init_lvars)


846 
apply simp


847 
apply assumption+


848 
apply simp


849 
apply assumption+


850 
apply clarsimp


851 
apply (rule HOL.refl)


852 
apply simp


853 
apply (rule Ball_weaken)


854 
apply assumption


855 
apply (force simp add: is_acc_type_def)


856 
apply (tactic "smp_tac 1 1")


857 
apply (frule (2) wt_MethdI, clarsimp)


858 
apply (subgoal_tac "is_static dm = (static (mthd esm))")


859 
apply (simp only:)


860 
apply (tactic "smp_tac 1 1")


861 
apply (rule conjI)


862 
apply (erule conforms_return)


863 
apply blast


864 


865 
apply (force dest!: eval_gext del: impCE simp add: init_lvars_def2)


866 
apply clarsimp


867 
apply (drule (2) widen_trans, erule (1) conf_widen)


868 
apply (erule wf_ws_prog)


869 


870 
apply auto


871 
done


872 


873 


874 
subsection "accessibility"


875 


876 
theorem dynamic_field_access_ok:


877 
(assumes wf: "wf_prog G" and


878 
eval_e: "G\<turnstile>s1 \<midarrow>e\<succ>a\<rightarrow> s2" and


879 
not_Null: "a\<noteq>Null" and


880 
conform_a: "G,(store s2)\<turnstile>a\<Colon>\<preceq> Class statC" and


881 
conform_s2: "s2\<Colon>\<preceq>(G, L)" and


882 
normal_s2: "normal s2" and


883 
wt_e: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>,dt\<Turnstile>e\<Colon>Class statC" and


884 
f: "accfield G accC statC fn = Some f" and


885 
dynC: "if stat then dynC=statC


886 
else dynC=obj_class (lookup_obj (store s2) a)"


887 
) "table_of (DeclConcepts.fields G dynC) (fn,declclass f) = Some (fld f) \<and>


888 
G\<turnstile>Field fn f in dynC dyn_accessible_from accC"


889 
proof (cases "stat")


890 
case True


891 
with dynC


892 
have dynC': "dynC=statC" by simp


893 
with f


894 
have "table_of (DeclConcepts.fields G dynC) (fn,declclass f) = Some (fld f)"


895 
by (auto simp add: accfield_def Let_def intro!: table_of_remap_SomeD)


896 
with dynC' f


897 
show ?thesis


898 
by (auto intro!: static_to_dynamic_accessible_from


899 
dest: accfield_accessibleD accessible_from_commonD)


900 
next


901 
case False


902 
with wf conform_a not_Null conform_s2 dynC


903 
obtain subclseq: "G\<turnstile>dynC \<preceq>\<^sub>C statC" and


904 
"is_class G dynC"


905 
by (auto dest!: conforms_RefTD [of _ _ _ _ "(fst s2)" L]


906 
dest: obj_ty_obj_class1


907 
simp add: obj_ty_obj_class )


908 
with wf f


909 
have "table_of (DeclConcepts.fields G dynC) (fn,declclass f) = Some (fld f)"


910 
by (auto simp add: accfield_def Let_def


911 
dest: fields_mono


912 
dest!: table_of_remap_SomeD)


913 
moreover


914 
from f subclseq


915 
have "G\<turnstile>Field fn f in dynC dyn_accessible_from accC"


916 
by (auto intro!: static_to_dynamic_accessible_from


917 
dest: accfield_accessibleD)


918 
ultimately show ?thesis


919 
by blast


920 
qed


921 


922 
lemma call_access_ok:


923 
(assumes invC_prop: "G\<turnstile>invmode (mhd statM) e\<rightarrow>invC\<preceq>statT"


924 
and wf: "wf_prog G"


925 
and wt_e: "\<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>e\<Colon>RefT statT"


926 
and statM: "statM \<in> mheads G accC statT sig"


927 
and invC: "invC = invocation_class (invmode (mhd statM) e) s a statT"


928 
)"\<exists> dynM. dynlookup G statT invC sig = Some dynM \<and>


929 
G\<turnstile>Methd sig dynM in invC dyn_accessible_from accC"


930 
proof 


931 
from wt_e wf have type_statT: "is_type G (RefT statT)"


932 
by (auto dest: ty_expr_is_type)


933 
from statM have not_Null: "statT \<noteq> NullT" by auto


934 
from type_statT wt_e


935 
have wf_I: "(\<forall>I. statT = IfaceT I \<longrightarrow> is_iface G I \<and>


936 
invmode (mhd statM) e \<noteq> SuperM)"


937 
by (auto dest: invocationTypeExpr_noClassD)


938 
from wt_e


939 
have wf_A: "(\<forall> T. statT = ArrayT T \<longrightarrow> invmode (mhd statM) e \<noteq> SuperM)"


940 
by (auto dest: invocationTypeExpr_noClassD)


941 
show ?thesis


942 
proof (cases "invmode (mhd statM) e = IntVir")


943 
case True


944 
with invC_prop not_Null


945 
have invC_prop': "is_class G invC \<and>


946 
(if (\<exists>T. statT=ArrayT T) then invC=Object


947 
else G\<turnstile>Class invC\<preceq>RefT statT)"


948 
by (auto simp add: DynT_prop_def)


949 
with True not_Null


950 
have "G,statT \<turnstile> invC valid_lookup_cls_for is_static statM"


951 
by (cases statT) (auto simp add: invmode_def


952 
split: split_if split_if_asm) (* was deleted above *)


953 
with statM type_statT wf


954 
show ?thesis


955 
by  (rule dynlookup_access,auto)


956 
next


957 
case False


958 
with type_statT wf invC not_Null wf_I wf_A


959 
have invC_prop': " is_class G invC \<and>


960 
((\<exists> statC. statT=ClassT statC \<and> invC=statC) \<or>


961 
(\<forall> statC. statT\<noteq>ClassT statC \<and> invC=Object)) "


962 
by (case_tac "statT") (auto simp add: invocation_class_def


963 
split: inv_mode.splits)


964 
with not_Null wf


965 
have dynlookup_static: "dynlookup G statT invC sig = methd G invC sig"


966 
by (case_tac "statT") (auto simp add: dynlookup_def dynmethd_C_C


967 
dynimethd_def)


968 
from statM wf wt_e not_Null False invC_prop' obtain dynM where


969 
"accmethd G accC invC sig = Some dynM"


970 
by (auto dest!: static_mheadsD)


971 
from invC_prop' False not_Null wf_I


972 
have "G,statT \<turnstile> invC valid_lookup_cls_for is_static statM"


973 
by (cases statT) (auto simp add: invmode_def


974 
split: split_if split_if_asm) (* was deleted above *)


975 
with statM type_statT wf


976 
show ?thesis


977 
by  (rule dynlookup_access,auto)


978 
qed


979 
qed


980 


981 
section "main proof of type safety"


982 


983 
ML {*


984 
val forward_hyp_tac = EVERY' [smp_tac 1,


985 
FIRST'[mp_tac,etac exI,smp_tac 2,smp_tac 1,EVERY'[etac impE,etac exI]],


986 
REPEAT o (etac conjE)];


987 
val typD_tac = eresolve_tac (thms "wt_elim_cases") THEN_ALL_NEW


988 
EVERY' [full_simp_tac (simpset() setloop (K no_tac)),


989 
clarify_tac(claset() addSEs[])]


990 
*}


991 


992 
lemma conforms_locals [rule_format]:


993 
"(a,b)\<Colon>\<preceq>(G, L) \<longrightarrow> L x = Some T \<longrightarrow> G,b\<turnstile>the (locals b x)\<Colon>\<preceq>T"


994 
apply (force simp: conforms_def Let_def lconf_def)


995 
done


996 


997 
lemma eval_type_sound [rule_format (no_asm)]:


998 
"wf_prog G \<Longrightarrow> G\<turnstile>s0 \<midarrow>t\<succ>\<rightarrow> (v,s1) \<Longrightarrow> (\<forall>L. s0\<Colon>\<preceq>(G,L) \<longrightarrow>


999 
(\<forall>C T. \<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>t\<Colon>T \<longrightarrow> s1\<Colon>\<preceq>(G,L) \<and>


1000 
(let (x,s) = s1 in x = None \<longrightarrow> G,L,s\<turnstile>t\<succ>v\<Colon>\<preceq>T)))"


1001 
apply (erule eval_induct)


1002 


1003 
(* 29 subgoals *)


1004 
(* Xcpt, Inst, Methd, Nil, Skip, Expr, Comp *)


1005 
apply (simp_all (no_asm_use) add: Let_def body_def)


1006 
apply (tactic "ALLGOALS (EVERY'[Clarify_tac, TRY o typD_tac,


1007 
TRY o forward_hyp_tac])")


1008 
apply (tactic"ALLGOALS(EVERY'[asm_simp_tac(simpset()),TRY o Clarify_tac])")


1009 


1010 
(* 20 subgoals *)


1011 


1012 
(* Break *)


1013 
apply (erule conforms_absorb)


1014 


1015 
(* Cons *)


1016 
apply (erule_tac V = "G\<turnstile>Norm s0 \<midarrow>?ea\<succ>\<rightarrow> ?vs1" in thin_rl)


1017 
apply (frule eval_gext')


1018 
apply force


1019 


1020 
(* LVar *)


1021 
apply (force elim: conforms_localD [THEN lconfD] conforms_lupd


1022 
simp add: assign_conforms_def lvar_def)


1023 


1024 
(* Cast *)


1025 
apply (force dest: fits_conf)


1026 


1027 
(* Lit *)


1028 
apply (rule conf_litval)


1029 
apply (simp add: empty_dt_def)


1030 


1031 
(* Super *)


1032 
apply (rule conf_widen)


1033 
apply (erule (1) subcls_direct [THEN widen.subcls])


1034 
apply (erule (1) conforms_localD [THEN lconfD])


1035 
apply (erule wf_ws_prog)


1036 


1037 
(* Acc *)


1038 
apply fast


1039 


1040 
(* Body *)


1041 
apply (rule conjI)


1042 
apply (rule conforms_absorb)


1043 
apply (fast)


1044 
apply (fast intro: conforms_locals)


1045 


1046 
(* Cond *)


1047 
apply (simp split: split_if_asm)


1048 
apply (tactic "forward_hyp_tac 1", force)


1049 
apply (tactic "forward_hyp_tac 1", force)


1050 


1051 
(* If *)


1052 
apply (force split add: split_if_asm)


1053 


1054 
(* Loop *)


1055 
apply (drule (1) wt.Loop)


1056 
apply (clarsimp split: split_if_asm)


1057 
apply (fast intro: conforms_absorb)


1058 


1059 
(* Fin *)


1060 
apply (case_tac "x1", force)


1061 
apply (drule spec, erule impE, erule conforms_NormI)


1062 
apply (erule impE)


1063 
apply blast


1064 
apply (clarsimp)


1065 
apply (erule (3) Fin_lemma)


1066 


1067 
(* Throw *)


1068 
apply (erule (3) Throw_lemma)


1069 


1070 
(* NewC *)


1071 
apply (clarsimp simp add: is_acc_class_def)


1072 
apply (drule (1) halloc_type_sound,blast, rule HOL.refl, simp, simp)


1073 


1074 
(* NewA *)


1075 
apply (tactic "smp_tac 1 1",frule wt_init_comp_ty,erule impE,blast)


1076 
apply (tactic "forward_hyp_tac 1")


1077 
apply (case_tac "check_neg i' ab")


1078 
apply (clarsimp simp add: is_acc_type_def)


1079 
apply (drule (2) halloc_type_sound, rule HOL.refl, simp, simp)


1080 
apply force


1081 


1082 
(* Level 34, 6 subgoals *)


1083 


1084 
(* Init *)


1085 
apply (case_tac "inited C (globs s0)")


1086 
apply (clarsimp)


1087 
apply (clarsimp)


1088 
apply (frule (1) wf_prog_cdecl)


1089 
apply (drule spec, erule impE, erule (3) conforms_init_class_obj)


1090 
apply (drule_tac "psi" = "class G C = ?x" in asm_rl,erule impE,


1091 
force dest!: wf_cdecl_supD split add: split_if simp add: is_acc_class_def)


1092 
apply (drule spec, erule impE, erule conforms_set_locals, rule lconf_empty)


1093 
apply (erule impE) apply (rule exI) apply (erule wf_cdecl_wt_init)


1094 
apply (drule (1) conforms_return, force dest: eval_gext', assumption)


1095 


1096 


1097 
(* Ass *)


1098 
apply (tactic "forward_hyp_tac 1")


1099 
apply (rename_tac x1 s1 x2 s2 v va w L C Ta T', case_tac x1)


1100 
prefer 2 apply force


1101 
apply (case_tac x2)


1102 
prefer 2 apply force


1103 
apply (simp, drule conjunct2)


1104 
apply (drule (1) conf_widen)


1105 
apply (erule wf_ws_prog)


1106 
apply (erule (2) Ass_lemma)


1107 
apply (clarsimp simp add: assign_conforms_def)


1108 


1109 
(* Try *)


1110 
apply (drule (1) sxalloc_type_sound, simp (no_asm_use))


1111 
apply (case_tac a)


1112 
apply clarsimp


1113 
apply clarsimp


1114 
apply (tactic "smp_tac 1 1")


1115 
apply (simp split add: split_if_asm)


1116 
apply (fast dest: conforms_deallocL Try_lemma)


1117 


1118 
(* FVar *)


1119 


1120 
apply (frule accfield_fields)


1121 
apply (frule ty_expr_is_type [THEN type_is_class],simp)


1122 
apply simp


1123 
apply (frule wf_ws_prog)


1124 
apply (frule (1) fields_declC,simp)


1125 
apply clarsimp


1126 
(*b y EVERY'[datac cfield_defpl_is_class 2, Clarsimp_tac] 1; not useful here*)


1127 
apply (tactic "smp_tac 1 1")


1128 
apply (tactic "forward_hyp_tac 1")


1129 
apply (rule conjI, force split add: split_if simp add: fvar_def2)


1130 
apply (drule init_yields_initd, frule eval_gext')


1131 
apply clarsimp


1132 
apply (case_tac "C=Object")


1133 
apply clarsimp


1134 
apply (erule (9) FVar_lemma)


1135 


1136 
(* AVar *)


1137 
apply (tactic "forward_hyp_tac 1")


1138 
apply (erule_tac V = "G\<turnstile>Norm s0 \<midarrow>?e1\<succ>?a'\<rightarrow> (?x1 1, ?s1)" in thin_rl,


1139 
frule eval_gext')


1140 
apply (rule conjI)


1141 
apply (clarsimp simp add: avar_def2)


1142 
apply clarsimp


1143 
apply (erule (5) AVar_lemma)


1144 


1145 
(* Call *)


1146 
apply (tactic "forward_hyp_tac 1")


1147 
apply (rule Call_type_sound)


1148 
apply auto


1149 
done


1150 


1151 


1152 
declare fun_upd_apply [simp]


1153 
declare split_paired_All [simp] split_paired_Ex [simp]


1154 
declare split_if [split] split_if_asm [split]


1155 
option.split [split] option.split_asm [split]


1156 
ML_setup {*


1157 
simpset_ref() := simpset() addloop ("split_all_tac", split_all_tac);


1158 
claset_ref() := claset () addSbefore ("split_all_tac", split_all_tac)


1159 
*}


1160 


1161 
theorem eval_ts:


1162 
"\<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>


1163 
\<Longrightarrow> (x',s')\<Colon>\<preceq>(G,L) \<and> (x'=None \<longrightarrow> G,s'\<turnstile>v\<Colon>\<preceq>T)"


1164 
apply (drule (3) eval_type_sound)


1165 
apply (unfold Let_def)


1166 
apply clarsimp


1167 
done


1168 


1169 
theorem evals_ts:


1170 
"\<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>


1171 
\<Longrightarrow> (x',s')\<Colon>\<preceq>(G,L) \<and> (x'=None \<longrightarrow> list_all2 (conf G s') vs Ts)"


1172 
apply (drule (3) eval_type_sound)


1173 
apply (unfold Let_def)


1174 
apply clarsimp


1175 
done


1176 


1177 
theorem evar_ts:


1178 
"\<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>


1179 
(x',s')\<Colon>\<preceq>(G,L) \<and> (x'=None \<longrightarrow> G,L,s'\<turnstile>In2 v\<succ>In2 vf\<Colon>\<preceq>Inl T)"


1180 
apply (drule (3) eval_type_sound)


1181 
apply (unfold Let_def)


1182 
apply clarsimp


1183 
done


1184 


1185 
theorem exec_ts:


1186 
"\<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)"


1187 
apply (drule (3) eval_type_sound)


1188 
apply (unfold Let_def)


1189 
apply clarsimp


1190 
done


1191 


1192 
(*


1193 
theorem dyn_methods_understood:


1194 
"\<And>s. \<lbrakk>wf_prog G; \<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>{t,md,IntVir}e..mn({pTs'}ps)\<Colon>rT;


1195 
s\<Colon>\<preceq>(G,L); G\<turnstile>s \<midarrow>e\<succ>a'\<rightarrow> Norm s'; a' \<noteq> Null\<rbrakk> \<Longrightarrow>


1196 
\<exists>a obj. a'=Addr a \<and> heap s' a = Some obj \<and>


1197 
cmethd G (obj_class obj) (mn, pTs') \<noteq> None"


1198 
apply (erule wt_elim_cases)


1199 
apply (drule max_spec2mheads)


1200 
apply (drule (3) eval_ts)


1201 
apply (clarsimp split del: split_if split_if_asm)


1202 
apply (drule (2) DynT_propI)


1203 
apply (simp (no_asm_simp))


1204 
apply (tactic *) (* {* exhaust_cmethd_tac "the (cmethd G (target (invmode m e) s' a' md) (mn, pTs'))" 1 *} *)(*)


1205 
apply (drule (4) DynT_mheadsD [THEN conjunct1], rule HOL.refl)


1206 
apply (drule conf_RefTD)


1207 
apply clarsimp


1208 
done


1209 
*)


1210 


1211 
end
