author | nipkow |
Mon, 12 Sep 2011 07:55:43 +0200 | |
changeset 44890 | 22f665a2e91c |
parent 42150 | b0c0638c4aad |
child 46226 | e88e980ed735 |
permissions | -rw-r--r-- |
42150 | 1 |
(* Title: HOL/MicroJava/BV/Typing_Framework_JVM.thy |
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Author: Tobias Nipkow, Gerwin Klein |
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Copyright 2000 TUM |
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*) |
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header {* \isaheader{The Typing Framework for the JVM}\label{sec:JVM} *} |
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theory Typing_Framework_JVM |
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imports "../DFA/Abstract_BV" JVMType EffectMono BVSpec |
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begin |
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definition exec :: "jvm_prog \<Rightarrow> nat \<Rightarrow> ty \<Rightarrow> exception_table \<Rightarrow> instr list \<Rightarrow> JVMType.state step_type" where |
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"exec G maxs rT et bs == |
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err_step (size bs) (\<lambda>pc. app (bs!pc) G maxs rT pc et) (\<lambda>pc. eff (bs!pc) G pc et)" |
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definition opt_states :: "'c prog \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> (ty list \<times> ty err list) option set" where |
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"opt_states G maxs maxr \<equiv> opt (\<Union>{list n (types G) |n. n \<le> maxs} \<times> list maxr (err (types G)))" |
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section {* Executability of @{term check_bounded} *} |
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primrec list_all'_rec :: "('a \<Rightarrow> nat \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> 'a list \<Rightarrow> bool" |
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where |
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"list_all'_rec P n [] = True" |
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| "list_all'_rec P n (x#xs) = (P x n \<and> list_all'_rec P (Suc n) xs)" |
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definition list_all' :: "('a \<Rightarrow> nat \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" where |
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"list_all' P xs \<equiv> list_all'_rec P 0 xs" |
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lemma list_all'_rec: |
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"\<And>n. list_all'_rec P n xs = (\<forall>p < size xs. P (xs!p) (p+n))" |
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apply (induct xs) |
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apply auto |
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apply (case_tac p) |
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apply auto |
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done |
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lemma list_all' [iff]: |
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"list_all' P xs = (\<forall>n < size xs. P (xs!n) n)" |
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by (unfold list_all'_def) (simp add: list_all'_rec) |
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lemma [code]: |
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"check_bounded ins et = |
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(list_all' (\<lambda>i pc. list_all (\<lambda>pc'. pc' < length ins) (succs i pc)) ins \<and> |
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list_all (\<lambda>e. fst (snd (snd e)) < length ins) et)" |
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by (simp add: list_all_iff check_bounded_def) |
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section {* Connecting JVM and Framework *} |
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lemma check_bounded_is_bounded: |
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"check_bounded ins et \<Longrightarrow> bounded (\<lambda>pc. eff (ins!pc) G pc et) (length ins)" |
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by (unfold bounded_def) (blast dest: check_boundedD) |
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lemma special_ex_swap_lemma [iff]: |
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"(? X. (? n. X = A n & P n) & Q X) = (? n. Q(A n) & P n)" |
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by blast |
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lemmas [iff del] = not_None_eq |
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theorem exec_pres_type: |
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"wf_prog wf_mb S \<Longrightarrow> |
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pres_type (exec S maxs rT et bs) (size bs) (states S maxs maxr)" |
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apply (unfold exec_def JVM_states_unfold) |
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apply (rule pres_type_lift) |
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apply clarify |
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apply (case_tac s) |
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apply simp |
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apply (drule effNone) |
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apply simp |
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apply (simp add: eff_def xcpt_eff_def norm_eff_def) |
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apply (case_tac "bs!p") |
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apply (clarsimp simp add: not_Err_eq) |
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apply (drule listE_nth_in, assumption) |
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apply fastforce |
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apply (fastforce simp add: not_None_eq) |
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apply (fastforce simp add: not_None_eq typeof_empty_is_type) |
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apply clarsimp |
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apply (erule disjE) |
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apply fastforce |
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apply clarsimp |
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apply (rule_tac x="1" in exI) |
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apply fastforce |
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apply clarsimp |
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apply (erule disjE) |
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apply (fastforce dest: field_fields fields_is_type) |
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apply (simp add: match_some_entry image_iff) |
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apply (rule_tac x=1 in exI) |
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apply fastforce |
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apply clarsimp |
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apply (erule disjE) |
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apply fastforce |
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apply (simp add: match_some_entry image_iff) |
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apply (rule_tac x=1 in exI) |
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apply fastforce |
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apply clarsimp |
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apply (erule disjE) |
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apply fastforce |
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apply clarsimp |
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apply (rule_tac x=1 in exI) |
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apply fastforce |
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defer |
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apply fastforce |
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apply fastforce |
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apply clarsimp |
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apply (rule_tac x="n'+2" in exI) |
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apply simp |
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apply clarsimp |
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apply (rule_tac x="Suc (Suc (Suc (length ST)))" in exI) |
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apply simp |
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apply clarsimp |
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apply (rule_tac x="Suc (Suc (Suc (Suc (length ST))))" in exI) |
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apply simp |
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apply fastforce |
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apply fastforce |
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apply fastforce |
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apply fastforce |
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apply clarsimp |
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apply (erule disjE) |
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apply fastforce |
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apply clarsimp |
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apply (rule_tac x=1 in exI) |
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apply fastforce |
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apply (erule disjE) |
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apply clarsimp |
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apply (drule method_wf_mdecl, assumption+) |
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apply (clarsimp simp add: wf_mdecl_def wf_mhead_def) |
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apply fastforce |
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apply clarsimp |
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apply (rule_tac x=1 in exI) |
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apply fastforce |
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done |
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lemmas [iff] = not_None_eq |
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lemma sup_state_opt_unfold: |
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"sup_state_opt G \<equiv> Opt.le (Product.le (Listn.le (subtype G)) (Listn.le (Err.le (subtype G))))" |
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by (simp add: sup_state_opt_def sup_state_def sup_loc_def sup_ty_opt_def) |
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lemma app_mono: |
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"app_mono (sup_state_opt G) (\<lambda>pc. app (bs!pc) G maxs rT pc et) (length bs) (opt_states G maxs maxr)" |
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by (unfold app_mono_def lesub_def) (blast intro: EffectMono.app_mono) |
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lemma list_appendI: |
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"\<lbrakk>a \<in> list x A; b \<in> list y A\<rbrakk> \<Longrightarrow> a @ b \<in> list (x+y) A" |
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apply (unfold list_def) |
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apply (simp (no_asm)) |
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apply blast |
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done |
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lemma list_map [simp]: |
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"(map f xs \<in> list (length xs) A) = (f ` set xs \<subseteq> A)" |
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apply (unfold list_def) |
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apply simp |
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done |
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lemma [iff]: |
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"(OK ` A \<subseteq> err B) = (A \<subseteq> B)" |
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apply (unfold err_def) |
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apply blast |
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done |
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lemma [intro]: |
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"x \<in> A \<Longrightarrow> replicate n x \<in> list n A" |
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by (induct n, auto) |
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lemma lesubstep_type_simple: |
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"a <=[Product.le (op =) r] b \<Longrightarrow> a <=|r| b" |
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apply (unfold lesubstep_type_def) |
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apply clarify |
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apply (simp add: set_conv_nth) |
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apply clarify |
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apply (drule le_listD, assumption) |
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apply (clarsimp simp add: lesub_def Product.le_def) |
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apply (rule exI) |
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apply (rule conjI) |
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apply (rule exI) |
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apply (rule conjI) |
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apply (rule sym) |
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apply assumption |
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apply assumption |
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apply assumption |
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done |
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lemma eff_mono: |
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"\<lbrakk>p < length bs; s <=_(sup_state_opt G) t; app (bs!p) G maxs rT pc et t\<rbrakk> |
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\<Longrightarrow> eff (bs!p) G p et s <=|sup_state_opt G| eff (bs!p) G p et t" |
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apply (unfold eff_def) |
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apply (rule lesubstep_type_simple) |
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apply (rule le_list_appendI) |
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apply (simp add: norm_eff_def) |
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apply (rule le_listI) |
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apply simp |
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apply simp |
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apply (simp add: lesub_def) |
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apply (case_tac s) |
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apply simp |
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apply (simp del: split_paired_All split_paired_Ex) |
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apply (elim exE conjE) |
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apply simp |
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apply (drule eff'_mono, assumption) |
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apply assumption |
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apply (simp add: xcpt_eff_def) |
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apply (rule le_listI) |
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apply simp |
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apply simp |
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apply (simp add: lesub_def) |
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apply (case_tac s) |
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apply simp |
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apply simp |
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apply (case_tac t) |
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apply simp |
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apply (clarsimp simp add: sup_state_conv) |
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done |
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lemma order_sup_state_opt: |
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"ws_prog G \<Longrightarrow> order (sup_state_opt G)" |
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by (unfold sup_state_opt_unfold) (blast dest: acyclic_subcls1 order_widen) |
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theorem exec_mono: |
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"ws_prog G \<Longrightarrow> bounded (exec G maxs rT et bs) (size bs) \<Longrightarrow> |
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mono (JVMType.le G maxs maxr) (exec G maxs rT et bs) (size bs) (states G maxs maxr)" |
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apply (unfold exec_def JVM_le_unfold JVM_states_unfold) |
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apply (rule mono_lift) |
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apply (fold sup_state_opt_unfold opt_states_def) |
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apply (erule order_sup_state_opt) |
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apply (rule app_mono) |
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apply assumption |
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apply clarify |
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apply (rule eff_mono) |
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apply assumption+ |
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done |
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theorem semilat_JVM_slI: |
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"ws_prog G \<Longrightarrow> semilat (JVMType.sl G maxs maxr)" |
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apply (unfold JVMType.sl_def stk_esl_def reg_sl_def) |
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apply (rule semilat_opt) |
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apply (rule err_semilat_Product_esl) |
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apply (rule err_semilat_upto_esl) |
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apply (rule err_semilat_JType_esl, assumption+) |
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apply (rule err_semilat_eslI) |
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apply (rule Listn_sl) |
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apply (rule err_semilat_JType_esl, assumption+) |
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done |
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lemma sl_triple_conv: |
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"JVMType.sl G maxs maxr == |
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(states G maxs maxr, JVMType.le G maxs maxr, JVMType.sup G maxs maxr)" |
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by (simp (no_asm) add: states_def JVMType.le_def JVMType.sup_def) |
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lemma is_type_pTs: |
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"\<lbrakk> wf_prog wf_mb G; (C,S,fs,mdecls) \<in> set G; ((mn,pTs),rT,code) \<in> set mdecls \<rbrakk> |
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\<Longrightarrow> set pTs \<subseteq> types G" |
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proof |
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assume "wf_prog wf_mb G" |
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"(C,S,fs,mdecls) \<in> set G" |
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"((mn,pTs),rT,code) \<in> set mdecls" |
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hence "wf_mdecl wf_mb G C ((mn,pTs),rT,code)" |
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by (rule wf_prog_wf_mdecl) |
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hence "\<forall>t \<in> set pTs. is_type G t" |
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by (unfold wf_mdecl_def wf_mhead_def) auto |
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moreover |
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fix t assume "t \<in> set pTs" |
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ultimately |
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have "is_type G t" by blast |
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thus "t \<in> types G" .. |
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qed |
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lemma jvm_prog_lift: |
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assumes wf: |
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"wf_prog (\<lambda>G C bd. P G C bd) G" |
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assumes rule: |
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"\<And>wf_mb C mn pTs C rT maxs maxl b et bd. |
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wf_prog wf_mb G \<Longrightarrow> |
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method (G,C) (mn,pTs) = Some (C,rT,maxs,maxl,b,et) \<Longrightarrow> |
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is_class G C \<Longrightarrow> |
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set pTs \<subseteq> types G \<Longrightarrow> |
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bd = ((mn,pTs),rT,maxs,maxl,b,et) \<Longrightarrow> |
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P G C bd \<Longrightarrow> |
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Q G C bd" |
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shows |
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"wf_prog (\<lambda>G C bd. Q G C bd) G" |
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proof - |
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from wf show ?thesis |
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apply (unfold wf_prog_def wf_cdecl_def) |
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apply clarsimp |
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apply (drule bspec, assumption) |
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apply (unfold wf_cdecl_mdecl_def) |
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apply clarsimp |
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apply (drule bspec, assumption) |
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apply (frule methd [OF wf [THEN wf_prog_ws_prog]], assumption+) |
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apply (frule is_type_pTs [OF wf], assumption+) |
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apply clarify |
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apply (drule rule [OF wf], assumption+) |
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apply (rule HOL.refl) |
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apply assumption+ |
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done |
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qed |
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end |