Theory Typing_Framework_JVM

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


header {* \isaheader{The Typing Framework for the JVM}\label{sec:JVM} *}

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) (λpc. app (bs!pc) G maxs rT pc et) (λpc. 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 (\<Union>{list n (types G) |n. n ≤ maxs} × list maxr (err (types G)))"


section {* Executability of @{term check_bounded} *}

primrec list_all'_rec :: "('a => nat => bool) => nat => 'a list => 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 => nat => bool) => 'a list => 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' (λi pc. list_all (λpc'. pc' < length ins) (succs i pc)) ins ∧
list_all (λe. fst (snd (snd e)) < length ins) et)"

by (simp add: list_all_iff check_bounded_def)


section {* Connecting JVM and Framework *}

lemma check_bounded_is_bounded:
"check_bounded ins et ==> bounded (λpc. 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) (λpc. 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 (op =) 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|]
==> 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:
"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 |]
==> set pTs ⊆ 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 (λG 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 ==>
method (G,C) (mn,pTs) = Some (C,rT,maxs,maxl,b,et) ==>
is_class G C ==>
set pTs ⊆ types G ==>
bd = ((mn,pTs),rT,maxs,maxl,b,et) ==>
P G C bd ==>
Q G C bd"


shows
"wf_prog (λG 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