Adapted to new simplifier.
(* Title: HOL/MicroJava/BV/JVM.thy
ID: $Id$
Author: Tobias Nipkow, Gerwin Klein
Copyright 2000 TUM
*)
header {* \isaheader{The Typing Framework for the JVM}\label{sec:JVM} *}
theory Typing_Framework_JVM = Typing_Framework_err + JVMType + EffectMono + BVSpec:
constdefs
exec :: "jvm_prog \<Rightarrow> nat \<Rightarrow> ty \<Rightarrow> exception_table \<Rightarrow> instr list \<Rightarrow> state step_type"
"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)"
constdefs
opt_states :: "'c prog \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> (ty list \<times> ty err list) option set"
"opt_states G maxs maxr \<equiv> opt (\<Union>{list n (types G) |n. n \<le> maxs} \<times> list maxr (err (types G)))"
section {* Executability of @{term check_bounded} *}
consts
list_all'_rec :: "('a \<Rightarrow> nat \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> 'a list \<Rightarrow> bool"
primrec
"list_all'_rec P n [] = True"
"list_all'_rec P n (x#xs) = (P x n \<and> list_all'_rec P (Suc n) xs)"
constdefs
list_all' :: "('a \<Rightarrow> nat \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool"
"list_all' P xs \<equiv> list_all'_rec P 0 xs"
lemma list_all'_rec:
"\<And>n. list_all'_rec P n xs = (\<forall>p < size xs. P (xs!p) (p+n))"
apply (induct xs)
apply auto
apply (case_tac p)
apply auto
done
lemma list_all' [iff]:
"list_all' P xs = (\<forall>n < size xs. P (xs!n) n)"
by (unfold list_all'_def) (simp add: list_all'_rec)
lemma list_all_ball:
"list_all P xs = (\<forall>x \<in> set xs. P x)"
by (induct xs) auto
lemma [code]:
"check_bounded ins et =
(list_all' (\<lambda>i pc. list_all (\<lambda>pc'. pc' < length ins) (succs i pc)) ins \<and>
list_all (\<lambda>e. fst (snd (snd e)) < length ins) et)"
by (simp add: list_all_ball check_bounded_def)
section {* Connecting JVM and Framework *}
lemma check_bounded_is_bounded:
"check_bounded ins et \<Longrightarrow> bounded (\<lambda>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 \<Longrightarrow>
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 simp add: not_Err_eq)
apply (drule listE_nth_in, assumption)
apply fastsimp
apply (fastsimp simp add: not_None_eq)
apply (fastsimp simp add: not_None_eq typeof_empty_is_type)
apply clarsimp
apply (erule disjE)
apply fastsimp
apply clarsimp
apply (rule_tac x="1" in exI)
apply fastsimp
apply clarsimp
apply (erule disjE)
apply (fastsimp dest: field_fields fields_is_type)
apply (simp add: match_some_entry split: split_if_asm)
apply (rule_tac x=1 in exI)
apply fastsimp
apply clarsimp
apply (erule disjE)
apply fastsimp
apply (simp add: match_some_entry split: split_if_asm)
apply (rule_tac x=1 in exI)
apply fastsimp
apply clarsimp
apply (erule disjE)
apply fastsimp
apply clarsimp
apply (rule_tac x=1 in exI)
apply fastsimp
defer
apply fastsimp
apply fastsimp
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 fastsimp
apply fastsimp
apply fastsimp
apply fastsimp
apply clarsimp
apply (erule disjE)
apply fastsimp
apply clarsimp
apply (rule_tac x=1 in exI)
apply fastsimp
apply (erule disjE)
apply (clarsimp simp add: Un_subset_iff)
apply (drule method_wf_mdecl, assumption+)
apply (clarsimp simp add: wf_mdecl_def wf_mhead_def)
apply fastsimp
apply clarsimp
apply (rule_tac x=1 in exI)
apply fastsimp
done
lemmas [iff] = not_None_eq
lemma sup_state_opt_unfold:
"sup_state_opt G \<equiv> 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) (\<lambda>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:
"\<lbrakk>a \<in> list x A; b \<in> list y A\<rbrakk> \<Longrightarrow> a @ b \<in> list (x+y) A"
apply (unfold list_def)
apply (simp (no_asm))
apply blast
done
lemma list_map [simp]:
"(map f xs \<in> list (length xs) A) = (f ` set xs \<subseteq> A)"
apply (unfold list_def)
apply simp
done
lemma [iff]:
"(OK ` A \<subseteq> err B) = (A \<subseteq> B)"
apply (unfold err_def)
apply blast
done
lemma [intro]:
"x \<in> A \<Longrightarrow> replicate n x \<in> list n A"
by (induct n, auto)
lemma lesubstep_type_simple:
"a <=[Product.le (op =) r] b \<Longrightarrow> 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:
"\<lbrakk>p < length bs; s <=_(sup_state_opt G) t; app (bs!p) G maxs rT pc et t\<rbrakk>
\<Longrightarrow> eff (bs!p) G p et s <=|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:
"wf_prog wf_mb G \<Longrightarrow> order (sup_state_opt G)"
by (unfold sup_state_opt_unfold) (blast dest: acyclic_subcls1 order_widen)
theorem exec_mono:
"wf_prog wf_mb G \<Longrightarrow> bounded (exec G maxs rT et bs) (size bs) \<Longrightarrow>
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:
"wf_prog wf_mb G \<Longrightarrow> 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 map_id [rule_format]:
"(\<forall>n < length xs. f (g (xs!n)) = xs!n) \<longrightarrow> map f (map g xs) = xs"
by (induct xs, auto)
lemma is_type_pTs:
"\<lbrakk> wf_prog wf_mb G; (C,S,fs,mdecls) \<in> set G; ((mn,pTs),rT,code) \<in> set mdecls \<rbrakk>
\<Longrightarrow> set pTs \<subseteq> types G"
proof
assume "wf_prog wf_mb G"
"(C,S,fs,mdecls) \<in> set G"
"((mn,pTs),rT,code) \<in> set mdecls"
hence "wf_mdecl wf_mb G C ((mn,pTs),rT,code)"
by (unfold wf_prog_def wf_cdecl_def) auto
hence "\<forall>t \<in> set pTs. is_type G t"
by (unfold wf_mdecl_def wf_mhead_def) auto
moreover
fix t assume "t \<in> set pTs"
ultimately
have "is_type G t" by blast
thus "t \<in> types G" ..
qed
lemma jvm_prog_lift:
assumes wf:
"wf_prog (\<lambda>G C bd. P G C bd) G"
assumes rule:
"\<And>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 (\<lambda>G C bd. Q G C bd) G"
proof -
from wf show ?thesis
apply (unfold wf_prog_def wf_cdecl_def)
apply clarsimp
apply (drule bspec, assumption)
apply (unfold wf_mdecl_def)
apply clarsimp
apply (drule bspec, assumption)
apply clarsimp
apply (frule methd [OF wf], assumption+)
apply (frule is_type_pTs [OF wf], assumption+)
apply clarify
apply (drule rule [OF wf], assumption+)
apply (rule refl)
apply assumption+
done
qed
end