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Datei: Typing_Framework_JVM.thy Sprache: Isabelle

Original von: Isabelle^©

(*  Title:      HOL/MicroJava/BV/Typing_Framework_JVM.thy
    Author:     Tobias Nipkow, Gerwin Klein
    Copyright   2000 TUM
*)

section \<open>The Typing Framework for the JVM \label{sec:JVM}\<close>

theory Typing_Framework_JVM
imports "../DFA/Abstract_BV" JVMType EffectMono BVSpec
begin

definition exec :: "jvm_prog \ nat \ ty \ exception_table \ instr list \ JVMType.state step_type" where
  "exec G maxs rT et bs ==
  err_step (size bs) (\<lambda>pc. app (bs!pc) G maxs rT pc et) (\<lambda>pc. eff (bs!pc) G pc et)"

definition opt_states :: "'c prog \ nat \ nat \ (ty list \ ty err list) option set" where
  "opt_states G maxs maxr \ opt (\{list n (types G) |n. n \ maxs} \ list maxr (err (types G)))"

subsection \<open>Executability of \<^term>\<open>check_bounded\<close>\<close>

primrec list_all'_rec :: "('a \<Rightarrow> nat \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> 'a list \<Rightarrow> bool"
where
  "list_all'_rec P n [] = True"
| "list_all'_rec P n (x#xs) = (P x n \ list_all'_rec P (Suc n) xs)"

definition list_all' :: "('a \<Rightarrow> nat \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" where
  "list_all' P xs \ list_all'_rec P 0 xs"

lemma list_all'_rec:
  "list_all'_rec P n xs = (\p < size xs. P (xs!p) (p+n))"
  apply (induct xs arbitrary: n)
  apply auto
  apply (case_tac p)
  apply auto
  done

lemma list_all' [iff]:
  "list_all' P xs = (\n < size xs. P (xs!n) n)"
  by (unfold list_all'_def) (simp add: list_all'_rec)

lemma [code]:
  "check_bounded ins et =
  (list_all' (\i pc. list_all (\pc'. pc' < length ins) (succs i pc)) ins \
   list_all (\<lambda>e. fst (snd (snd e)) < length ins) et)"
  by (simp add: list_all_iff check_bounded_def)


subsection \<open>Connecting JVM and Framework\<close>

lemma check_bounded_is_bounded:
  "check_bounded ins et \ bounded (\pc. eff (ins!pc) G pc et) (length ins)"
  by (unfold bounded_def) (blast dest: check_boundedD)

lemma special_ex_swap_lemma [iff]:
  "(\X. (\n. X = A n \ P n) & Q X) = (\n. Q(A n) \ P n)"
  by blast

lemmas [iff del] = not_None_eq

theorem exec_pres_type:
  "wf_prog wf_mb S \
  pres_type (exec S maxs rT et bs) (size bs) (states S maxs maxr)"
  apply (unfold exec_def JVM_states_unfold)
  apply (rule pres_type_lift)
  apply clarify
  apply (case_tac s)
   apply simp
   apply (drule effNone)
   apply simp
  apply (simp add: eff_def xcpt_eff_def norm_eff_def)
  apply (case_tac "bs!p")

  apply clarsimp
  apply (drule listE_nth_in, assumption)
  apply fastforce

  apply (fastforce simp add: not_None_eq)

  apply (fastforce simp add: not_None_eq typeof_empty_is_type)

  apply clarsimp
  apply (erule disjE)
   apply fastforce
  apply clarsimp
  apply (rule_tac x="1" in exI)
  apply fastforce

  apply clarsimp
  apply (erule disjE)
   apply (fastforce dest: field_fields fields_is_type)
  apply (simp add: match_some_entry image_iff)
  apply (rule_tac x=1 in exI)
  apply fastforce

  apply clarsimp
  apply (erule disjE)
   apply fastforce
  apply (simp add: match_some_entry image_iff)
  apply (rule_tac x=1 in exI)
  apply fastforce

  apply clarsimp
  apply (erule disjE)
   apply fastforce
  apply clarsimp
  apply (rule_tac x=1 in exI)
  apply fastforce

  defer

  apply fastforce
  apply fastforce

  apply clarsimp
  apply (rule_tac x="n'+2" in exI)
  apply simp

  apply clarsimp
  apply (rule_tac x="Suc (Suc (Suc (length ST)))" in exI)
  apply simp

  apply clarsimp
  apply (rule_tac x="Suc (Suc (Suc (Suc (length ST))))" in exI)
  apply simp

  apply fastforce
  apply fastforce
  apply fastforce
  apply fastforce

  apply clarsimp
  apply (erule disjE)
   apply fastforce
  apply clarsimp
  apply (rule_tac x=1 in exI)
  apply fastforce

  apply (erule disjE)
   apply clarsimp
   apply (drule method_wf_mdecl, assumption+)
   apply (clarsimp simp add: wf_mdecl_def wf_mhead_def)
   apply fastforce
  apply clarsimp
  apply (rule_tac x=1 in exI)
  apply fastforce
  done

lemmas [iff] = not_None_eq

lemma sup_state_opt_unfold:
  "sup_state_opt G \ Opt.le (Product.le (Listn.le (subtype G)) (Listn.le (Err.le (subtype G))))"
  by (simp add: sup_state_opt_def sup_state_def sup_loc_def sup_ty_opt_def)

lemma app_mono:
  "app_mono (sup_state_opt G) (\pc. app (bs!pc) G maxs rT pc et) (length bs) (opt_states G maxs maxr)"
  by (unfold app_mono_def lesub_def) (blast intro: EffectMono.app_mono)


lemma list_appendI:
  "\a \ list x A; b \ list y A\ \ a @ b \ list (x+y) A"
  apply (unfold list_def)
  apply (simp (no_asm))
  apply blast
  done

lemma list_map [simp]:
  "(map f xs \ list (length xs) A) = (f ` set xs \ A)"
  apply (unfold list_def)
  apply simp
  done

lemma [iff]:
  "(OK ` A \ err B) = (A \ B)"
  apply (unfold err_def)
  apply blast
  done

lemma [intro]:
  "x \ A \ replicate n x \ list n A"
  by (induct n, auto)

lemma lesubstep_type_simple:
  "a <=[Product.le (=) r] b \ a \|r| b"
  apply (unfold lesubstep_type_def)
  apply clarify
  apply (simp add: set_conv_nth)
  apply clarify
  apply (drule le_listD, assumption)
  apply (clarsimp simp add: lesub_def Product.le_def)
  apply (rule exI)
  apply (rule conjI)
   apply (rule exI)
   apply (rule conjI)
    apply (rule sym)
    apply assumption
   apply assumption
  apply assumption
  done


lemma eff_mono:
  "\p < length bs; s <=_(sup_state_opt G) t; app (bs!p) G maxs rT pc et t\
  \<Longrightarrow> eff (bs!p) G p et s \<le>|sup_state_opt G| eff (bs!p) G p et t"
  apply (unfold eff_def)
  apply (rule lesubstep_type_simple)
  apply (rule le_list_appendI)
   apply (simp add: norm_eff_def)
   apply (rule le_listI)
    apply simp
   apply simp
   apply (simp add: lesub_def)
   apply (case_tac s)
    apply simp
   apply (simp del: split_paired_All split_paired_Ex)
   apply (elim exE conjE)
   apply simp
   apply (drule eff'_mono, assumption)
   apply assumption
  apply (simp add: xcpt_eff_def)
  apply (rule le_listI)
    apply simp
  apply simp
  apply (simp add: lesub_def)
  apply (case_tac s)
   apply simp
  apply simp
  apply (case_tac t)
   apply simp
  apply (clarsimp simp add: sup_state_conv)
  done

lemma order_sup_state_opt:
  "ws_prog G \ order (sup_state_opt G)"
  by (unfold sup_state_opt_unfold) (blast dest: acyclic_subcls1 order_widen)

theorem exec_mono:
  "ws_prog G \ bounded (exec G maxs rT et bs) (size bs) \
  mono (JVMType.le G maxs maxr) (exec G maxs rT et bs) (size bs) (states G maxs maxr)"
  apply (unfold exec_def JVM_le_unfold JVM_states_unfold)
  apply (rule mono_lift)
     apply (fold sup_state_opt_unfold opt_states_def)
     apply (erule order_sup_state_opt)
    apply (rule app_mono)
   apply assumption
  apply clarify
  apply (rule eff_mono)
  apply assumption+
  done

theorem semilat_JVM_slI:
  "ws_prog G \ semilat (JVMType.sl G maxs maxr)"
  apply (unfold JVMType.sl_def stk_esl_def reg_sl_def)
  apply (rule semilat_opt)
  apply (rule err_semilat_Product_esl)
  apply (rule err_semilat_upto_esl)
  apply (rule err_semilat_JType_esl, assumption+)
  apply (rule err_semilat_eslI)
  apply (rule Listn_sl)
  apply (rule err_semilat_JType_esl, assumption+)
  done

lemma sl_triple_conv:
  "JVMType.sl G maxs maxr ==
  (states G maxs maxr, JVMType.le G maxs maxr, JVMType.sup G maxs maxr)"
  by (simp (no_asm) add: states_def JVMType.le_def JVMType.sup_def)

lemma is_type_pTs:
  "\ wf_prog wf_mb G; (C,S,fs,mdecls) \ set G; ((mn,pTs),rT,code) \ set mdecls \
  \<Longrightarrow> set pTs \<subseteq> types G"
proof
  assume "wf_prog wf_mb G"
         "(C,S,fs,mdecls) \ set G"
         "((mn,pTs),rT,code) \ set mdecls"
  hence "wf_mdecl wf_mb G C ((mn,pTs),rT,code)"
    by (rule wf_prog_wf_mdecl)
  hence "\t \ set pTs. is_type G t"
    by (unfold wf_mdecl_def wf_mhead_def) auto
  moreover
  fix t assume "t \ set pTs"
  ultimately
  have "is_type G t" by blast
  thus "t \ types G" ..
qed

lemma jvm_prog_lift:
  assumes wf:
  "wf_prog (\G C bd. P G C bd) G"

  assumes rule:
  "\wf_mb C mn pTs C rT maxs maxl b et bd.
   wf_prog wf_mb G \<Longrightarrow>
   method (G,C) (mn,pTs) = Some (C,rT,maxs,maxl,b,et) \<Longrightarrow>
   is_class G C \<Longrightarrow>
   set pTs \<subseteq> types G \<Longrightarrow>
   bd = ((mn,pTs),rT,maxs,maxl,b,et) \<Longrightarrow>
   P G C bd \<Longrightarrow>
   Q G C bd"

  shows
  "wf_prog (\G C bd. Q G C bd) G"
  using wf
  apply (unfold wf_prog_def wf_cdecl_def)
  apply clarsimp
  apply (drule bspec, assumption)
  apply (unfold wf_cdecl_mdecl_def)
  apply clarsimp
  apply (drule bspec, assumption)
  apply (frule methd [OF wf [THEN wf_prog_ws_prog]], assumption+)
  apply (frule is_type_pTs [OF wf], assumption+)
  apply clarify
  apply (drule rule [OF wf], assumption+)
  apply (rule HOL.refl)
  apply assumption+
  done

end

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