author  schirmer 
Fri, 22 Feb 2002 11:26:44 +0100  
changeset 12925  99131847fb93 
parent 12859  f63315dfffd4 
child 13337  f75dfc606ac7 
permissions  rwrr 
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(* Title: HOL/Bali/AxCompl.thy 
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ID: $Id$ 
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Author: David von Oheimb 

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License: GPL (GNU GENERAL PUBLIC LICENSE) 
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*) 
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header {* 

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Completeness proof for Axiomatic semantics of Java expressions and statements 

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*} 

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theory AxCompl = AxSem: 

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text {* 

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design issues: 

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\begin{itemize} 

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\item proof structured by Most General Formulas (> Thomas Kleymann) 

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\end{itemize} 

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*} 

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section "set of not yet initialzed classes" 
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constdefs 

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nyinitcls :: "prog \<Rightarrow> state \<Rightarrow> qtname set" 

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"nyinitcls G s \<equiv> {C. is_class G C \<and> \<not> initd C s}" 

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lemma nyinitcls_subset_class: "nyinitcls G s \<subseteq> {C. is_class G C}" 

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apply (unfold nyinitcls_def) 

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apply fast 

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done 

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lemmas finite_nyinitcls [simp] = 

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finite_is_class [THEN nyinitcls_subset_class [THEN finite_subset], standard] 

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lemma card_nyinitcls_bound: "card (nyinitcls G s) \<le> card {C. is_class G C}" 

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apply (rule nyinitcls_subset_class [THEN finite_is_class [THEN card_mono]]) 

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done 

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lemma nyinitcls_set_locals_cong [simp]: 

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"nyinitcls G (x,set_locals l s) = nyinitcls G (x,s)" 

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apply (unfold nyinitcls_def) 

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apply (simp (no_asm)) 

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done 

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lemma nyinitcls_abrupt_cong [simp]: "nyinitcls G (f x, y) = nyinitcls G (x, y)" 

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apply (unfold nyinitcls_def) 

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apply (simp (no_asm)) 

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done 

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lemma nyinitcls_abupd_cong [simp]:"!!s. nyinitcls G (abupd f s) = nyinitcls G s" 

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apply (unfold nyinitcls_def) 

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apply (simp (no_asm_simp) only: split_tupled_all) 

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apply (simp (no_asm)) 

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done 

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lemma card_nyinitcls_abrupt_congE [elim!]: 

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"card (nyinitcls G (x, s)) \<le> n \<Longrightarrow> card (nyinitcls G (y, s)) \<le> n" 

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apply (unfold nyinitcls_def) 

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apply auto 

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done 

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lemma nyinitcls_new_xcpt_var [simp]: 

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"nyinitcls G (new_xcpt_var vn s) = nyinitcls G s" 

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apply (unfold nyinitcls_def) 

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apply (induct_tac "s") 

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apply (simp (no_asm)) 

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done 

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lemma nyinitcls_init_lvars [simp]: 

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"nyinitcls G ((init_lvars G C sig mode a' pvs) s) = nyinitcls G s" 

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apply (induct_tac "s") 

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apply (simp (no_asm) add: init_lvars_def2 split add: split_if) 

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done 

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lemma nyinitcls_emptyD: "\<lbrakk>nyinitcls G s = {}; is_class G C\<rbrakk> \<Longrightarrow> initd C s" 

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apply (unfold nyinitcls_def) 

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apply fast 

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done 

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lemma card_Suc_lemma: "\<lbrakk>card (insert a A) \<le> Suc n; a\<notin>A; finite A\<rbrakk> \<Longrightarrow> card A \<le> n" 

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apply (rotate_tac 1) 

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apply clarsimp 

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done 

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lemma nyinitcls_le_SucD: 

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"\<lbrakk>card (nyinitcls G (x,s)) \<le> Suc n; \<not>inited C (globs s); class G C=Some y\<rbrakk> \<Longrightarrow> 

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card (nyinitcls G (x,init_class_obj G C s)) \<le> n" 

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apply (subgoal_tac 

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"nyinitcls G (x,s) = insert C (nyinitcls G (x,init_class_obj G C s))") 

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apply clarsimp 

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apply (erule thin_rl) 

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apply (rule card_Suc_lemma [OF _ _ finite_nyinitcls]) 

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apply (auto dest!: not_initedD elim!: 

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simp add: nyinitcls_def inited_def split add: split_if_asm) 

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done 

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ML {* bind_thm("inited_gext'",permute_prems 0 1 (thm "inited_gext")) *} 

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lemma nyinitcls_gext: "snd s\<le>snd s' \<Longrightarrow> nyinitcls G s' \<subseteq> nyinitcls G s" 

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apply (unfold nyinitcls_def) 

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apply (force dest!: inited_gext') 

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done 

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lemma card_nyinitcls_gext: 

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"\<lbrakk>snd s\<le>snd s'; card (nyinitcls G s) \<le> n\<rbrakk>\<Longrightarrow> card (nyinitcls G s') \<le> n" 

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apply (rule le_trans) 

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apply (rule card_mono) 

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apply (rule finite_nyinitcls) 

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apply (erule nyinitcls_gext) 

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apply assumption 

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done 

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section "initle" 

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constdefs 

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init_le :: "prog \<Rightarrow> nat \<Rightarrow> state \<Rightarrow> bool" ("_\<turnstile>init\<le>_" [51,51] 50) 

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"G\<turnstile>init\<le>n \<equiv> \<lambda>s. card (nyinitcls G s) \<le> n" 

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lemma init_le_def2 [simp]: "(G\<turnstile>init\<le>n) s = (card (nyinitcls G s)\<le>n)" 

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apply (unfold init_le_def) 

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apply auto 

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done 

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lemma All_init_leD: "\<forall>n::nat. G,A\<turnstile>{P \<and>. G\<turnstile>init\<le>n} t\<succ> {Q} \<Longrightarrow> G,A\<turnstile>{P} t\<succ> {Q}" 

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apply (drule spec) 

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apply (erule conseq1) 

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apply clarsimp 

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apply (rule card_nyinitcls_bound) 

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done 

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section "Most General Triples and Formulas" 

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constdefs 

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remember_init_state :: "state assn" ("\<doteq>") 

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"\<doteq> \<equiv> \<lambda>Y s Z. s = Z" 

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lemma remember_init_state_def2 [simp]: "\<doteq> Y = op =" 

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apply (unfold remember_init_state_def) 

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apply (simp (no_asm)) 

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done 

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consts 

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MGF ::"[state assn, term, prog] \<Rightarrow> state triple" ("{_} _\<succ> {_\<rightarrow>}"[3,65,3]62) 

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MGFn::"[nat , term, prog] \<Rightarrow> state triple" ("{=:_} _\<succ> {_\<rightarrow>}"[3,65,3]62) 

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defs 

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MGF_def: 

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"{P} t\<succ> {G\<rightarrow>} \<equiv> {P} t\<succ> {\<lambda>Y s' s. G\<turnstile>s \<midarrow>t\<succ>\<rightarrow> (Y,s')}" 

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MGFn_def: 

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"{=:n} t\<succ> {G\<rightarrow>} \<equiv> {\<doteq> \<and>. G\<turnstile>init\<le>n} t\<succ> {G\<rightarrow>}" 

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(* unused *) 

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lemma MGF_valid: "wf_prog G \<Longrightarrow> G,{}\<Turnstile>{\<doteq>} t\<succ> {G\<rightarrow>}" 
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apply (unfold MGF_def) 
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apply (simp add: ax_valids_def triple_valid_def2) 
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apply (auto elim: evaln_eval) 
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done 
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lemma MGF_res_eq_lemma [simp]: 
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"(\<forall>Y' Y s. Y = Y' \<and> P s \<longrightarrow> Q s) = (\<forall>s. P s \<longrightarrow> Q s)" 

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apply auto 

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done 

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lemma MGFn_def2: 

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"G,A\<turnstile>{=:n} t\<succ> {G\<rightarrow>} = G,A\<turnstile>{\<doteq> \<and>. G\<turnstile>init\<le>n} 

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t\<succ> {\<lambda>Y s' s. G\<turnstile>s \<midarrow>t\<succ>\<rightarrow> (Y,s')}" 

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apply (unfold MGFn_def MGF_def) 

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apply fast 

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done 

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lemma MGF_MGFn_iff: "G,A\<turnstile>{\<doteq>} t\<succ> {G\<rightarrow>} = (\<forall>n. G,A\<turnstile>{=:n} t\<succ> {G\<rightarrow>})" 

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apply (simp (no_asm_use) add: MGFn_def2 MGF_def) 

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apply safe 

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apply (erule_tac [2] All_init_leD) 

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apply (erule conseq1) 

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apply clarsimp 

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done 

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lemma MGFnD: 

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"G,A\<turnstile>{=:n} t\<succ> {G\<rightarrow>} \<Longrightarrow> 

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G,A\<turnstile>{(\<lambda>Y' s' s. s' = s \<and> P s) \<and>. G\<turnstile>init\<le>n} 

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t\<succ> {(\<lambda>Y' s' s. G\<turnstile>s\<midarrow>t\<succ>\<rightarrow>(Y',s') \<and> P s) \<and>. G\<turnstile>init\<le>n}" 

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apply (unfold init_le_def) 

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apply (simp (no_asm_use) add: MGFn_def2) 

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apply (erule conseq12) 

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apply clarsimp 

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apply (erule (1) eval_gext [THEN card_nyinitcls_gext]) 

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done 

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lemmas MGFnD' = MGFnD [of _ _ _ _ "\<lambda>x. True"] 

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lemma MGFNormalI: "G,A\<turnstile>{Normal \<doteq>} t\<succ> {G\<rightarrow>} \<Longrightarrow> 

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G,(A::state triple set)\<turnstile>{\<doteq>::state assn} t\<succ> {G\<rightarrow>}" 

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apply (unfold MGF_def) 

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apply (rule ax_Normal_cases) 

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apply (erule conseq1) 

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apply clarsimp 

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apply (rule ax_derivs.Abrupt [THEN conseq1]) 

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apply (clarsimp simp add: Let_def) 

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done 

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lemma MGFNormalD: "G,A\<turnstile>{\<doteq>} t\<succ> {G\<rightarrow>} \<Longrightarrow> G,A\<turnstile>{Normal \<doteq>} t\<succ> {G\<rightarrow>}" 

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apply (unfold MGF_def) 

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apply (erule conseq1) 

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apply clarsimp 

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done 

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lemma MGFn_NormalI: 

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"G,(A::state triple set)\<turnstile>{Normal((\<lambda>Y' s' s. s'=s \<and> normal s) \<and>. G\<turnstile>init\<le>n)}t\<succ> 

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{\<lambda>Y s' s. G\<turnstile>s \<midarrow>t\<succ>\<rightarrow> (Y,s')} \<Longrightarrow> G,A\<turnstile>{=:n}t\<succ>{G\<rightarrow>}" 

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apply (simp (no_asm_use) add: MGFn_def2) 

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apply (rule ax_Normal_cases) 

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apply (erule conseq1) 

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apply clarsimp 

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apply (rule ax_derivs.Abrupt [THEN conseq1]) 

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apply (clarsimp simp add: Let_def) 

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done 

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lemma MGFn_free_wt: 

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"(\<exists>T L C. \<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>t\<Colon>T) 

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\<longrightarrow> G,(A::state triple set)\<turnstile>{=:n} t\<succ> {G\<rightarrow>} 

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\<Longrightarrow> G,A\<turnstile>{=:n} t\<succ> {G\<rightarrow>}" 

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apply (rule MGFn_NormalI) 

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apply (rule ax_free_wt) 

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apply (auto elim: conseq12 simp add: MGFn_def MGF_def) 

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done 

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lemma MGFn_free_wt_NormalConformI: 
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"(\<forall> T L C. \<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>t\<Colon>T 
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\<longrightarrow> G,(A::state triple set) 
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\<turnstile>{Normal((\<lambda>Y' s' s. s'=s \<and> normal s) \<and>. G\<turnstile>init\<le>n) \<and>. (\<lambda> s. s\<Colon>\<preceq>(G, L))} 
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t\<succ> 
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{\<lambda>Y s' s. G\<turnstile>s \<midarrow>t\<succ>\<rightarrow> (Y,s')}) 
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\<Longrightarrow> G,A\<turnstile>{=:n}t\<succ>{G\<rightarrow>}" 
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apply (rule MGFn_NormalI) 
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apply (rule ax_no_hazard) 
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apply (rule ax_escape) 
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apply (intro strip) 
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apply (simp only: type_ok_def peek_and_def) 
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apply (erule conjE)+ 
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apply (erule exE,erule exE, erule exE,erule conjE,drule (1) mp) 
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apply (drule spec,drule spec, drule spec, drule (1) mp) 
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apply (erule conseq12) 
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apply blast 
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done 
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section "main lemmas" 

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declare fun_upd_apply [simp del] 

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declare splitI2 [rule del] (*prevents ugly renaming of state variables*) 

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ML_setup {* 

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Delsimprocs [eval_expr_proc, eval_var_proc, eval_exprs_proc, eval_stmt_proc] 

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*} (*prevents modifying rhs of MGF*) 

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ML {* 

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val eval_css = (claset() delrules [thm "eval.Abrupt"] addSIs (thms "eval.intros") 

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delrules[thm "eval.Expr", thm "eval.Init", thm "eval.Try"] 

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addIs [thm "eval.Expr", thm "eval.Init"] 

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addSEs[thm "eval.Try"] delrules[equalityCE], 

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simpset() addsimps [split_paired_all,Let_def] 

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addsimprocs [eval_expr_proc,eval_var_proc,eval_exprs_proc,eval_stmt_proc]); 

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val eval_Force_tac = force_tac eval_css; 

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val wt_prepare_tac = EVERY'[ 

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rtac (thm "MGFn_free_wt"), 

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clarsimp_tac (claset() addSEs (thms "wt_elim_cases"), simpset())] 

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val compl_prepare_tac = EVERY'[rtac (thm "MGFn_NormalI"), Simp_tac] 

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val wt_conf_prepare_tac = EVERY'[ 
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rtac (thm "MGFn_free_wt_NormalConformI"), 
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clarsimp_tac (claset() addSEs (thms "wt_elim_cases"), simpset())] 
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val forw_hyp_tac = EVERY'[etac (thm "MGFnD'" RS thm "conseq12"), Clarsimp_tac] 
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val forw_hyp_eval_Force_tac = 

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EVERY'[TRY o rtac allI, forw_hyp_tac, eval_Force_tac] 

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*} 

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lemma MGFn_Init: "\<forall>m. Suc m\<le>n \<longrightarrow> (\<forall>t. G,A\<turnstile>{=:m} t\<succ> {G\<rightarrow>}) \<Longrightarrow> 

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G,(A::state triple set)\<turnstile>{=:n} In1r (Init C)\<succ> {G\<rightarrow>}" 

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apply (tactic "wt_prepare_tac 1") 

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(* requires is_class G C two times for nyinitcls *) 

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apply (tactic "compl_prepare_tac 1") 

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apply (rule_tac C = "initd C" in ax_cases) 

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apply (rule ax_derivs.Done [THEN conseq1]) 

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apply (clarsimp intro!: init_done) 

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apply (rule_tac y = n in nat.exhaust, clarsimp) 

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apply (rule ax_impossible [THEN conseq1]) 

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apply (force dest!: nyinitcls_emptyD) 

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apply clarsimp 

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apply (drule_tac x = "nat" in spec) 

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apply clarsimp 

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apply (rule_tac Q = " (\<lambda>Y s' (x,s) . G\<turnstile> (x,init_class_obj G C s) \<midarrow> (if C=Object then Skip else Init (super (the (class G C))))\<rightarrow> s' \<and> x=None \<and> \<not>inited C (globs s)) \<and>. G\<turnstile>init\<le>nat" in ax_derivs.Init) 

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apply simp 

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apply (rule_tac P' = "Normal ((\<lambda>Y s' s. s' = supd (init_class_obj G C) s \<and> normal s \<and> \<not> initd C s) \<and>. G\<turnstile>init\<le>nat) " in conseq1) 

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prefer 2 

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apply (force elim!: nyinitcls_le_SucD) 

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apply (simp split add: split_if, rule conjI, clarify) 

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apply (rule ax_derivs.Skip [THEN conseq1], tactic "eval_Force_tac 1") 

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apply clarify 

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apply (drule spec) 

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apply (erule MGFnD' [THEN conseq12]) 

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apply (tactic "force_tac (claset(), simpset() addsimprocs[eval_stmt_proc]) 1") 

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apply (rule allI) 

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apply (drule spec) 

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apply (erule MGFnD' [THEN conseq12]) 

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apply clarsimp 

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apply (tactic {* pair_tac "sa" 1 *}) 

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apply (tactic"clarsimp_tac (claset(), simpset() addsimprocs[eval_stmt_proc]) 1") 

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apply (rule eval_Init, force+) 

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done 

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lemmas MGFn_InitD = MGFn_Init [THEN MGFnD, THEN ax_NormalD] 

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text {* For @{text MGFn_Call} we need the wellformedness of the program to 
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switch from the evalnsemantics to the evalsemantics *} 
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lemma MGFn_Call: 
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"\<lbrakk>\<forall>C sig. G,(A::state triple set)\<turnstile>{=:n} In1l (Methd C sig)\<succ> {G\<rightarrow>}; 

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G,A\<turnstile>{=:n} In1l e\<succ> {G\<rightarrow>}; G,A\<turnstile>{=:n} In3 ps\<succ> {G\<rightarrow>};wf_prog G\<rbrakk> \<Longrightarrow> 
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G,A\<turnstile>{=:n} In1l ({accC,statT,mode}e\<cdot>mn({pTs'}ps))\<succ> {G\<rightarrow>}" 
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apply (tactic "wt_conf_prepare_tac 1") 
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apply (rule_tac 
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Q="(\<lambda>Y s1 (x,s) . x = None \<and> 
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(\<exists>a. G\<turnstile>Norm s \<midarrow>e\<succ>a\<rightarrow> s1 \<and> (normal s1 \<longrightarrow> G, store s1\<turnstile>a\<Colon>\<preceq>RefT statT) 
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\<and> Y = In1 a)) 
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332 
\<and>. G\<turnstile>init\<le>n \<and>. (\<lambda> s. s\<Colon>\<preceq>(G, L))" and 
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333 
R = "\<lambda>a'. (\<lambda>Y (x2,s2) (x,s) . x = None \<and> 
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334 
(\<exists>s1 pvs. G\<turnstile>Norm s \<midarrow>e\<succ>a'\<rightarrow> s1 \<and> 
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335 
(normal s1 \<longrightarrow> G, store s1\<turnstile>a'\<Colon>\<preceq>RefT statT)\<and> 
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336 
Y = In3 pvs \<and> G\<turnstile>s1 \<midarrow>ps\<doteq>\<succ>pvs\<rightarrow> (x2,s2))) 
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337 
\<and>. G\<turnstile>init\<le>n \<and>. (\<lambda> s. s\<Colon>\<preceq>(G, L))" in ax_derivs.Call) 
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338 
apply (tactic "forw_hyp_tac 1") 
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339 
apply (tactic "clarsimp_tac eval_css 1") 
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340 
apply (frule (3) eval_type_sound) 
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341 
apply force 
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342 

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343 
apply safe 
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344 
apply (tactic "forw_hyp_tac 1") 
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345 
apply (tactic "clarsimp_tac eval_css 1") 
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346 
apply (frule (3) eval_type_sound) 
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347 
apply (rule conjI) 
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348 
apply (rule exI,rule conjI) 
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349 
apply (assumption) 
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350 

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351 
apply (rule conjI) 
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352 
apply simp 
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353 
apply assumption 
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354 
apply blast 
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355 

12854  356 
apply (drule spec, drule spec) 
357 
apply (erule MGFnD' [THEN conseq12]) 

358 
apply (tactic "clarsimp_tac eval_css 1") 

359 
apply (erule (1) eval_Call) 

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360 
apply (rule HOL.refl)+ 
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361 
apply (subgoal_tac "check_method_access G C statT (invmode m e) 
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362 
\<lparr>name = mn, parTs = pTs'\<rparr> a 
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363 
(init_lvars G 
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364 
(invocation_declclass G (invmode m e) (snd (ab, ba)) a statT 
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365 
\<lparr>name = mn, parTs = pTs'\<rparr>) 
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366 
\<lparr>name = mn, parTs = pTs'\<rparr> (invmode m e) a vs 
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367 
(ab, 
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368 
ba)) = (init_lvars G 
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369 
(invocation_declclass G (invmode m e) (snd (ab, ba)) a statT 
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370 
\<lparr>name = mn, parTs = pTs'\<rparr>) 
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371 
\<lparr>name = mn, parTs = pTs'\<rparr> (invmode m e) a vs 
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372 
(ab, 
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373 
ba))") 
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374 
apply simp 
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375 
defer 
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376 
apply simp 
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377 
apply (erule (3) error_free_call_access) (* now showing the subgoal *) 
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378 
apply auto 
12854  379 
done 
380 

381 
lemma MGF_altern: "G,A\<turnstile>{Normal (\<doteq> \<and>. p)} t\<succ> {G\<rightarrow>} = 

382 
G,A\<turnstile>{Normal ((\<lambda>Y s Z. \<forall>w s'. G\<turnstile>s \<midarrow>t\<succ>\<rightarrow> (w,s') \<longrightarrow> (w,s') = Z) \<and>. p)} 

383 
t\<succ> {\<lambda>Y s Z. (Y,s) = Z}" 

384 
apply (unfold MGF_def) 

385 
apply (auto del: conjI elim!: conseq12) 

386 
apply (case_tac "\<exists>w s. G\<turnstile>Norm sa \<midarrow>t\<succ>\<rightarrow> (w,s) ") 

387 
apply (fast dest: unique_eval) 

388 
apply clarsimp 

389 
apply (erule thin_rl) 

390 
apply (erule thin_rl) 

391 
apply (drule split_paired_All [THEN subst]) 

392 
apply (clarsimp elim!: state_not_single) 

393 
done 

394 

395 

396 
lemma MGFn_Loop: 

397 
"\<lbrakk>G,(A::state triple set)\<turnstile>{=:n} In1l expr\<succ> {G\<rightarrow>};G,A\<turnstile>{=:n} In1r stmnt\<succ> {G\<rightarrow>} \<rbrakk> 

398 
\<Longrightarrow> 

399 
G,A\<turnstile>{=:n} In1r (l\<bullet> While(expr) stmnt)\<succ> {G\<rightarrow>}" 

400 
apply (rule MGFn_NormalI, simp) 

401 
apply (rule_tac p2 = "\<lambda>s. card (nyinitcls G s) \<le> n" in 

402 
MGF_altern [unfolded MGF_def, THEN iffD2, THEN conseq1]) 

403 
prefer 2 

404 
apply clarsimp 

405 
apply (rule_tac P' = 

406 
"((\<lambda>Y s Z. \<forall>w s'. G\<turnstile>s \<midarrow>In1r (l\<bullet> While(expr) stmnt) \<succ>\<rightarrow> (w,s') \<longrightarrow> (w,s') = Z) 

407 
\<and>. (\<lambda>s. card (nyinitcls G s) \<le> n))" in conseq12) 

408 
prefer 2 

409 
apply clarsimp 

410 
apply (tactic "smp_tac 1 1", erule_tac V = "All ?P" in thin_rl) 

411 
apply (rule_tac [2] P' = " (\<lambda>b s (Y',s') . (\<exists>s0. G\<turnstile>s0 \<midarrow>In1l expr\<succ>\<rightarrow> (b,s)) \<and> (if normal s \<and> the_Bool (the_In1 b) then (\<forall>s'' w s0. G\<turnstile>s \<midarrow>stmnt\<rightarrow> s'' \<and> G\<turnstile>(abupd (absorb (Cont l)) s'') \<midarrow>In1r (l\<bullet> While(expr) stmnt) \<succ>\<rightarrow> (w,s0) \<longrightarrow> (w,s0) = (Y',s')) else (\<diamondsuit>,s) = (Y',s'))) \<and>. G\<turnstile>init\<le>n" in polymorphic_Loop) 

412 
apply (force dest!: eval.Loop split add: split_if_asm) 

413 
prefer 2 

414 
apply (erule MGFnD' [THEN conseq12]) 

415 
apply clarsimp 

416 
apply (erule_tac V = "card (nyinitcls G s') \<le> n" in thin_rl) 

417 
apply (tactic "eval_Force_tac 1") 

418 
apply (erule MGFnD' [THEN conseq12] , clarsimp) 

419 
apply (rule conjI, erule exI) 

420 
apply (tactic "clarsimp_tac eval_css 1") 

421 
apply (case_tac "a") 

422 
prefer 2 

423 
apply (clarsimp) 

424 
apply (clarsimp split add: split_if) 

425 
apply (rule conjI, (tactic {* force_tac (claset() addSDs [thm "eval.Loop"], 

426 
simpset() addsimps [split_paired_all] addsimprocs [eval_stmt_proc]) 1*})+) 

427 
done 

428 

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429 
text {* For @{text MGFn_FVar} we need the wellformedness of the program to 
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430 
switch from the evalnsemantics to the evalsemantics *} 
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431 
lemma MGFn_FVar: 
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432 
"\<lbrakk>G,A\<turnstile>{=:n} In1r (Init statDeclC)\<succ> {G\<rightarrow>}; G,A\<turnstile>{=:n} In1l e\<succ> {G\<rightarrow>}; wf_prog G\<rbrakk> 
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433 
\<Longrightarrow> G,(A\<Colon>state triple set)\<turnstile>{=:n} In2 ({accC,statDeclC,stat}e..fn)\<succ> {G\<rightarrow>}" 
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434 
apply (tactic "wt_conf_prepare_tac 1") 
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435 
apply (rule_tac 
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436 
Q="(\<lambda>Y s1 (x,s) . x = None \<and> 
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437 
(G\<turnstile>Norm s \<midarrow>Init statDeclC\<rightarrow> s1 
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438 
)) \<and>. G\<turnstile>init\<le>n \<and>. (\<lambda> s. s\<Colon>\<preceq>(G, L))" 
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439 
in ax_derivs.FVar) 
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440 
apply (tactic "forw_hyp_tac 1") 
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441 
apply (tactic "clarsimp_tac eval_css 1") 
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442 
apply (subgoal_tac "is_class G statDeclC") 
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443 
apply (force dest: eval_type_sound) 
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444 
apply (force dest: ty_expr_is_type [THEN type_is_class] 
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445 
accfield_fields [THEN fields_declC]) 
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446 
apply (tactic "forw_hyp_tac 1") 
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447 
apply (tactic "clarsimp_tac eval_css 1") 
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448 
apply (subgoal_tac "(\<exists> v' s2' s3. 
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449 
( fvar statDeclC (is_static f) fn v (aa, ba) = (v',s2') ) \<and> 
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450 
(s3 = check_field_access G C statDeclC fn (is_static f) v s2') \<and> 
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451 
(s3 = s2'))") 
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452 
apply (erule exE)+ 
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453 
apply (erule conjE)+ 
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454 
apply (erule (1) eval.FVar) 
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455 
apply simp 
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456 
apply simp 
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457 

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458 
apply (case_tac "fvar statDeclC (is_static f) fn v (aa, ba)") 
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459 
apply (rule exI)+ 
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460 
apply (rule context_conjI) 
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461 
apply force 
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462 

99131847fb93
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463 
apply (rule context_conjI) 
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464 
apply simp 
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465 

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466 
apply (erule (3) error_free_field_access) 
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467 
apply (auto dest: eval_type_sound) 
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468 
done 
99131847fb93
Added check for field/method access to operational semantics and proved the acesses valid.
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changeset

469 

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470 
text {* For @{text MGFn_lemma} we need the wellformedness of the program to 
99131847fb93
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471 
switch from the evalnsemantics to the evalsemantics cf. @{text MGFn_call}, 
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472 
@{text MGFn_FVar}*} 
12854  473 
lemma MGFn_lemma [rule_format (no_asm)]: 
12925
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474 
"\<lbrakk>\<forall>n C sig. G,(A::state triple set)\<turnstile>{=:n} In1l (Methd C sig)\<succ> {G\<rightarrow>}; 
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475 
wf_prog G\<rbrakk> 
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476 
\<Longrightarrow> \<forall>t. G,A\<turnstile>{=:n} t\<succ> {G\<rightarrow>}" 
12854  477 
apply (rule full_nat_induct) 
478 
apply (rule allI) 

479 
apply (drule_tac x = n in spec) 

480 
apply (drule_tac psi = "All ?P" in asm_rl) 

481 
apply (subgoal_tac "\<forall>v e c es. G,A\<turnstile>{=:n} In2 v\<succ> {G\<rightarrow>} \<and> G,A\<turnstile>{=:n} In1l e\<succ> {G\<rightarrow>} \<and> G,A\<turnstile>{=:n} In1r c\<succ> {G\<rightarrow>} \<and> G,A\<turnstile>{=:n} In3 es\<succ> {G\<rightarrow>}") 

482 
apply (tactic "Clarify_tac 2") 

483 
apply (induct_tac "t") 

484 
apply (induct_tac "a") 

485 
apply fast+ 

486 
apply (rule var_expr_stmt.induct) 

487 
(* 28 subgoals *) 

488 
prefer 14 apply fast (* Methd *) 

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489 
prefer 13 apply (erule (3) MGFn_Call) 
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490 
prefer 2 apply (drule MGFn_Init,erule (2) MGFn_FVar) 
12854  491 
apply (erule_tac [!] V = "All ?P" in thin_rl) (* assumptions on Methd *) 
12925
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492 
apply (erule_tac [23] MGFn_Init) 
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493 
prefer 18 apply (erule (1) MGFn_Loop) 
12854  494 
apply (tactic "ALLGOALS compl_prepare_tac") 
495 

496 
apply (rule ax_derivs.LVar [THEN conseq1], tactic "eval_Force_tac 1") 

497 

498 
apply (rule ax_derivs.AVar) 

499 
apply (erule MGFnD [THEN ax_NormalD]) 

500 
apply (tactic "forw_hyp_eval_Force_tac 1") 

501 

502 
apply (rule ax_derivs.NewC) 

503 
apply (erule MGFn_InitD [THEN conseq2]) 

504 
apply (tactic "eval_Force_tac 1") 

505 

506 
apply (rule_tac Q = "(\<lambda>Y' s' s. normal s \<and> G\<turnstile>s \<midarrow>In1r (init_comp_ty ty) \<succ>\<rightarrow> (Y',s')) \<and>. G\<turnstile>init\<le>n" in ax_derivs.NewA) 

507 
apply (simp add: init_comp_ty_def split add: split_if) 

508 
apply (rule conjI, clarsimp) 

509 
apply (erule MGFn_InitD [THEN conseq2]) 

510 
apply (tactic "clarsimp_tac eval_css 1") 

511 
apply clarsimp 

512 
apply (rule ax_derivs.Skip [THEN conseq1], tactic "eval_Force_tac 1") 

513 
apply (tactic "forw_hyp_eval_Force_tac 1") 

514 

515 
apply (erule MGFnD'[THEN conseq12,THEN ax_derivs.Cast],tactic"eval_Force_tac 1") 

516 

517 
apply (erule MGFnD'[THEN conseq12,THEN ax_derivs.Inst],tactic"eval_Force_tac 1") 

518 
apply (rule ax_derivs.Lit [THEN conseq1], tactic "eval_Force_tac 1") 

519 
apply (rule ax_derivs.Super [THEN conseq1], tactic "eval_Force_tac 1") 

520 
apply (erule MGFnD'[THEN conseq12,THEN ax_derivs.Acc],tactic"eval_Force_tac 1") 

521 

522 
apply (rule ax_derivs.Ass) 

523 
apply (erule MGFnD [THEN ax_NormalD]) 

524 
apply (tactic "forw_hyp_eval_Force_tac 1") 

525 

526 
apply (rule ax_derivs.Cond) 

527 
apply (erule MGFnD [THEN ax_NormalD]) 

528 
apply (rule allI) 

529 
apply (rule ax_Normal_cases) 

530 
prefer 2 

531 
apply (rule ax_derivs.Abrupt [THEN conseq1], clarsimp simp add: Let_def) 

532 
apply (tactic "eval_Force_tac 1") 

533 
apply (case_tac "b") 

534 
apply (simp, tactic "forw_hyp_eval_Force_tac 1") 

535 
apply (simp, tactic "forw_hyp_eval_Force_tac 1") 

536 

537 
apply (rule_tac Q = " (\<lambda>Y' s' s. normal s \<and> G\<turnstile>s \<midarrow>Init pid_field_type\<rightarrow> s') \<and>. G\<turnstile>init\<le>n" in ax_derivs.Body) 

538 
apply (erule MGFn_InitD [THEN conseq2]) 

539 
apply (tactic "eval_Force_tac 1") 

540 
apply (tactic "forw_hyp_tac 1") 

541 
apply (tactic {* clarsimp_tac (eval_css delsimps2 [split_paired_all]) 1 *}) 

542 
apply (erule (1) eval.Body) 

543 

544 
apply (rule ax_derivs.Skip [THEN conseq1], tactic "eval_Force_tac 1") 

545 

546 
apply (erule MGFnD'[THEN conseq12,THEN ax_derivs.Expr],tactic"eval_Force_tac 1") 

547 

548 
apply (erule MGFnD' [THEN conseq12, THEN ax_derivs.Lab]) 

549 
apply (tactic "clarsimp_tac eval_css 1") 

550 

551 
apply (rule ax_derivs.Comp) 

552 
apply (erule MGFnD [THEN ax_NormalD]) 

553 
apply (tactic "forw_hyp_eval_Force_tac 1") 

554 

555 
apply (rule ax_derivs.If) 

556 
apply (erule MGFnD [THEN ax_NormalD]) 

557 
apply (rule allI) 

558 
apply (rule ax_Normal_cases) 

559 
prefer 2 

560 
apply (rule ax_derivs.Abrupt [THEN conseq1], clarsimp simp add: Let_def) 

561 
apply (tactic "eval_Force_tac 1") 

562 
apply (case_tac "b") 

563 
apply (simp, tactic "forw_hyp_eval_Force_tac 1") 

564 
apply (simp, tactic "forw_hyp_eval_Force_tac 1") 

565 

566 
apply (rule ax_derivs.Do [THEN conseq1]) 

567 
apply (tactic {* force_tac (eval_css addsimps2 [thm "abupd_def2"]) 1 *}) 

568 

569 
apply (erule MGFnD' [THEN conseq12, THEN ax_derivs.Throw]) 

570 
apply (tactic "clarsimp_tac eval_css 1") 

571 

572 
apply (rule_tac Q = " (\<lambda>Y' s' s. normal s \<and> (\<exists>s''. G\<turnstile>s \<midarrow>In1r stmt1\<succ>\<rightarrow> (Y',s'') \<and> G\<turnstile>s'' \<midarrow>sxalloc\<rightarrow> s')) \<and>. G\<turnstile>init\<le>n" in ax_derivs.Try) 

573 
apply (tactic "eval_Force_tac 3") 

574 
apply (tactic "forw_hyp_eval_Force_tac 2") 

575 
apply (erule MGFnD [THEN ax_NormalD, THEN conseq2]) 

576 
apply (tactic "clarsimp_tac eval_css 1") 

577 
apply (force elim: sxalloc_gext [THEN card_nyinitcls_gext]) 

578 

579 
apply (rule_tac Q = " (\<lambda>Y' s' s. normal s \<and> G\<turnstile>s \<midarrow>stmt1\<rightarrow> s') \<and>. G\<turnstile>init\<le>n" in ax_derivs.Fin) 

580 
apply (tactic "forw_hyp_eval_Force_tac 1") 

581 
apply (rule allI) 

582 
apply (tactic "forw_hyp_tac 1") 

583 
apply (tactic {* pair_tac "sb" 1 *}) 

584 
apply (tactic"clarsimp_tac (claset(),simpset() addsimprocs [eval_stmt_proc]) 1") 

585 
apply (drule (1) eval.Fin) 

586 
apply clarsimp 

587 

588 
apply (rule ax_derivs.Nil [THEN conseq1], tactic "eval_Force_tac 1") 

589 

590 
apply (rule ax_derivs.Cons) 

591 
apply (erule MGFnD [THEN ax_NormalD]) 

592 
apply (tactic "forw_hyp_eval_Force_tac 1") 

593 
done 

594 

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Added check for field/method access to operational semantics and proved the acesses valid.
schirmer
parents:
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diff
changeset

595 
lemma MGF_asm: 
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parents:
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diff
changeset

596 
"\<lbrakk>\<forall>C sig. is_methd G C sig \<longrightarrow> G,A\<turnstile>{\<doteq>} In1l (Methd C sig)\<succ> {G\<rightarrow>}; wf_prog G\<rbrakk> 
99131847fb93
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parents:
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diff
changeset

597 
\<Longrightarrow> G,(A::state triple set)\<turnstile>{\<doteq>} t\<succ> {G\<rightarrow>}" 
12854  598 
apply (simp (no_asm_use) add: MGF_MGFn_iff) 
599 
apply (rule allI) 

600 
apply (rule MGFn_lemma) 

601 
apply (intro strip) 

602 
apply (rule MGFn_free_wt) 

603 
apply (force dest: wt_Methd_is_methd) 

12925
99131847fb93
Added check for field/method access to operational semantics and proved the acesses valid.
schirmer
parents:
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diff
changeset

604 
apply assumption (* wf_prog G *) 
12854  605 
done 
606 

607 
declare splitI2 [intro!] 

608 
ML_setup {* 

609 
Addsimprocs [ eval_expr_proc, eval_var_proc, eval_exprs_proc, eval_stmt_proc] 

610 
*} 

611 

612 

613 
section "nested version" 

614 

615 
lemma nesting_lemma' [rule_format (no_asm)]: "[ !!A ts. ts <= A ==> P A ts; 

616 
!!A pn. !b:bdy pn. P (insert (mgf_call pn) A) {mgf b} ==> P A {mgf_call pn}; 

617 
!!A t. !pn:U. P A {mgf_call pn} ==> P A {mgf t}; 

618 
finite U; uA = mgf_call`U ] ==> 

619 
!A. A <= uA > n <= card uA > card A = card uA  n > (!t. P A {mgf t})" 

620 
proof  

621 
assume ax_derivs_asm: "!!A ts. ts <= A ==> P A ts" 

622 
assume MGF_nested_Methd: "!!A pn. !b:bdy pn. P (insert (mgf_call pn) A) 

623 
{mgf b} ==> P A {mgf_call pn}" 

624 
assume MGF_asm: "!!A t. !pn:U. P A {mgf_call pn} ==> P A {mgf t}" 

625 
assume "finite U" "uA = mgf_call`U" 

626 
then show ?thesis 

627 
apply  

628 
apply (induct_tac "n") 

629 
apply (tactic "ALLGOALS Clarsimp_tac") 

630 
apply (tactic "dtac (permute_prems 0 1 card_seteq) 1") 

631 
apply simp 

632 
apply (erule finite_imageI) 

633 
apply (simp add: MGF_asm ax_derivs_asm) 

634 
apply (rule MGF_asm) 

635 
apply (rule ballI) 

636 
apply (case_tac "mgf_call pn : A") 

637 
apply (fast intro: ax_derivs_asm) 

638 
apply (rule MGF_nested_Methd) 

639 
apply (rule ballI) 

640 
apply (drule spec, erule impE, erule_tac [2] impE, erule_tac [3] impE, 

641 
erule_tac [4] spec) 

642 
apply fast 

643 
apply (erule Suc_leD) 

644 
apply (drule finite_subset) 

645 
apply (erule finite_imageI) 

646 
apply auto 

647 
apply arith 

648 
done 

649 
qed 

650 

651 
lemma nesting_lemma [rule_format (no_asm)]: "[ !!A ts. ts <= A ==> P A ts; 

652 
!!A pn. !b:bdy pn. P (insert (mgf (f pn)) A) {mgf b} ==> P A {mgf (f pn)}; 

653 
!!A t. !pn:U. P A {mgf (f pn)} ==> P A {mgf t}; 

654 
finite U ] ==> P {} {mgf t}" 

655 
proof  

656 
assume 2: "!!A pn. !b:bdy pn. P (insert (mgf (f pn)) A) {mgf b} ==> P A {mgf (f pn)}" 

657 
assume 3: "!!A t. !pn:U. P A {mgf (f pn)} ==> P A {mgf t}" 

658 
assume "!!A ts. ts <= A ==> P A ts" "finite U" 

659 
then show ?thesis 

660 
apply  

661 
apply (rule_tac mgf = "mgf" in nesting_lemma') 

662 
apply (erule_tac [2] 2) 

663 
apply (rule_tac [2] 3) 

664 
apply (rule_tac [6] le_refl) 

665 
apply auto 

666 
done 

667 
qed 

668 

669 
lemma MGF_nested_Methd: "\<lbrakk> 

670 
G,insert ({Normal \<doteq>} In1l (Methd C sig) \<succ>{G\<rightarrow>}) A\<turnstile> 

671 
{Normal \<doteq>} In1l (body G C sig) \<succ>{G\<rightarrow>} 

672 
\<rbrakk> \<Longrightarrow> G,A\<turnstile>{Normal \<doteq>} In1l (Methd C sig) \<succ>{G\<rightarrow>}" 

673 
apply (unfold MGF_def) 

674 
apply (rule ax_MethdN) 

675 
apply (erule conseq2) 

676 
apply clarsimp 

677 
apply (erule MethdI) 

678 
done 

679 

12925
99131847fb93
Added check for field/method access to operational semantics and proved the acesses valid.
schirmer
parents:
12859
diff
changeset

680 
lemma MGF_deriv: "wf_prog G \<Longrightarrow> G,({}::state triple set)\<turnstile>{\<doteq>} t\<succ> {G\<rightarrow>}" 
12854  681 
apply (rule MGFNormalI) 
682 
apply (rule_tac mgf = "\<lambda>t. {Normal \<doteq>} t\<succ> {G\<rightarrow>}" and 

683 
bdy = "\<lambda> (C,sig) .{In1l (body G C sig) }" and 

684 
f = "\<lambda> (C,sig) . In1l (Methd C sig) " in nesting_lemma) 

685 
apply (erule ax_derivs.asm) 

686 
apply (clarsimp simp add: split_tupled_all) 

687 
apply (erule MGF_nested_Methd) 

12925
99131847fb93
Added check for field/method access to operational semantics and proved the acesses valid.
schirmer
parents:
12859
diff
changeset

688 
apply (erule_tac [2] finite_is_methd [OF wf_ws_prog]) 
12854  689 
apply (rule MGF_asm [THEN MGFNormalD]) 
12925
99131847fb93
Added check for field/method access to operational semantics and proved the acesses valid.
schirmer
parents:
12859
diff
changeset

690 
apply (auto intro: MGFNormalI) 
12854  691 
done 
692 

693 

694 
section "simultaneous version" 

695 

696 
lemma MGF_simult_Methd_lemma: "finite ms \<Longrightarrow> 

697 
G,A\<union> (\<lambda>(C,sig). {Normal \<doteq>} In1l (Methd C sig)\<succ> {G\<rightarrow>}) ` ms 

698 
\<turnstile>(\<lambda>(C,sig). {Normal \<doteq>} In1l (body G C sig)\<succ> {G\<rightarrow>}) ` ms \<Longrightarrow> 

699 
G,A\<turnstile>(\<lambda>(C,sig). {Normal \<doteq>} In1l (Methd C sig)\<succ> {G\<rightarrow>}) ` ms" 

700 
apply (unfold MGF_def) 

701 
apply (rule ax_derivs.Methd [unfolded mtriples_def]) 

702 
apply (erule ax_finite_pointwise) 

703 
prefer 2 

704 
apply (rule ax_derivs.asm) 

705 
apply fast 

706 
apply clarsimp 

707 
apply (rule conseq2) 

708 
apply (erule (1) ax_methods_spec) 

709 
apply clarsimp 

710 
apply (erule eval_Methd) 

711 
done 

712 

12925
99131847fb93
Added check for field/method access to operational semantics and proved the acesses valid.
schirmer
parents:
12859
diff
changeset

713 
lemma MGF_simult_Methd: "wf_prog G \<Longrightarrow> 
12854  714 
G,({}::state triple set)\<turnstile>(\<lambda>(C,sig). {Normal \<doteq>} In1l (Methd C sig)\<succ> {G\<rightarrow>}) 
715 
` Collect (split (is_methd G)) " 

12925
99131847fb93
Added check for field/method access to operational semantics and proved the acesses valid.
schirmer
parents:
12859
diff
changeset

716 
apply (frule finite_is_methd [OF wf_ws_prog]) 
12854  717 
apply (rule MGF_simult_Methd_lemma) 
718 
apply assumption 

719 
apply (erule ax_finite_pointwise) 

720 
prefer 2 

721 
apply (rule ax_derivs.asm) 

722 
apply blast 

723 
apply clarsimp 

724 
apply (rule MGF_asm [THEN MGFNormalD]) 

12925
99131847fb93
Added check for field/method access to operational semantics and proved the acesses valid.
schirmer
parents:
12859
diff
changeset

725 
apply (auto intro: MGFNormalI) 
12854  726 
done 
727 

12925
99131847fb93
Added check for field/method access to operational semantics and proved the acesses valid.
schirmer
parents:
12859
diff
changeset

728 
lemma MGF_deriv: "wf_prog G \<Longrightarrow> G,({}::state triple set)\<turnstile>{\<doteq>} t\<succ> {G\<rightarrow>}" 
12854  729 
apply (rule MGF_asm) 
730 
apply (intro strip) 

731 
apply (rule MGFNormalI) 

732 
apply (rule ax_derivs.weaken) 

733 
apply (erule MGF_simult_Methd) 

12925
99131847fb93
Added check for field/method access to operational semantics and proved the acesses valid.
schirmer
parents:
12859
diff
changeset

734 
apply auto 
12854  735 
done 
736 

737 

738 
section "corollaries" 

739 

12925
99131847fb93
Added check for field/method access to operational semantics and proved the acesses valid.
schirmer
parents:
12859
diff
changeset

740 
lemma eval_to_evaln: "\<lbrakk>G\<turnstile>s \<midarrow>t\<succ>\<rightarrow> (Y', s');type_ok G t s; wf_prog G\<rbrakk> 
99131847fb93
Added check for field/method access to operational semantics and proved the acesses valid.
schirmer
parents:
12859
diff
changeset

741 
\<Longrightarrow> \<exists>n. G\<turnstile>s \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (Y', s')" 
99131847fb93
Added check for field/method access to operational semantics and proved the acesses valid.
schirmer
parents:
12859
diff
changeset

742 
apply (cases "normal s") 
99131847fb93
Added check for field/method access to operational semantics and proved the acesses valid.
schirmer
parents:
12859
diff
changeset

743 
apply (force simp add: type_ok_def intro: eval_evaln) 
99131847fb93
Added check for field/method access to operational semantics and proved the acesses valid.
schirmer
parents:
12859
diff
changeset

744 
apply (force intro: evaln.Abrupt) 
99131847fb93
Added check for field/method access to operational semantics and proved the acesses valid.
schirmer
parents:
12859
diff
changeset

745 
done 
99131847fb93
Added check for field/method access to operational semantics and proved the acesses valid.
schirmer
parents:
12859
diff
changeset

746 

99131847fb93
Added check for field/method access to operational semantics and proved the acesses valid.
schirmer
parents:
12859
diff
changeset

747 
lemma MGF_complete: 
99131847fb93
Added check for field/method access to operational semantics and proved the acesses valid.
schirmer
parents:
12859
diff
changeset

748 
"\<lbrakk>G,{}\<Turnstile>{P} t\<succ> {Q}; G,({}::state triple set)\<turnstile>{\<doteq>} t\<succ> {G\<rightarrow>}; wf_prog G\<rbrakk> 
99131847fb93
Added check for field/method access to operational semantics and proved the acesses valid.
schirmer
parents:
12859
diff
changeset

749 
\<Longrightarrow> G,({}::state triple set)\<turnstile>{P::state assn} t\<succ> {Q}" 
12854  750 
apply (rule ax_no_hazard) 
751 
apply (unfold MGF_def) 

752 
apply (erule conseq12) 

753 
apply (simp (no_asm_use) add: ax_valids_def triple_valid_def) 

12925
99131847fb93
Added check for field/method access to operational semantics and proved the acesses valid.
schirmer
parents:
12859
diff
changeset

754 
apply (blast dest: eval_to_evaln) 
12854  755 
done 
756 

12925
99131847fb93
Added check for field/method access to operational semantics and proved the acesses valid.
schirmer
parents:
12859
diff
changeset

757 
theorem ax_complete: "wf_prog G \<Longrightarrow> 
12854  758 
G,{}\<Turnstile>{P::state assn} t\<succ> {Q} \<Longrightarrow> G,({}::state triple set)\<turnstile>{P} t\<succ> {Q}" 
759 
apply (erule MGF_complete) 

12925
99131847fb93
Added check for field/method access to operational semantics and proved the acesses valid.
schirmer
parents:
12859
diff
changeset

760 
apply (erule (1) MGF_deriv) 
12854  761 
done 
762 

763 
end 