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(* Title: HOL/MicroJava/BV/Altern.thy
Author: Martin Strecker
*)
section \<open>Alternative definition of well-typing of bytecode, used in compiler type correctness proof\<close>
theory Altern
imports BVSpec
begin
definition check_type :: "jvm_prog \ nat \ nat \ JVMType.state \ bool" where
"check_type G mxs mxr s \ s \ states G mxs mxr"
definition wt_instr_altern :: "[instr,jvm_prog,ty,method_type,nat,nat,p_count,
exception_table,p_count] \<Rightarrow> bool" where
"wt_instr_altern i G rT phi mxs mxr max_pc et pc \
app i G mxs rT pc et (phi!pc) \<and>
check_type G mxs mxr (OK (phi!pc)) \<and>
(\<forall>(pc',s') \<in> set (eff i G pc et (phi!pc)). pc' < max_pc \<and> G \<turnstile> s' <=' phi!pc')"
definition wt_method_altern :: "[jvm_prog,cname,ty list,ty,nat,nat,instr list,
exception_table,method_type] \<Rightarrow> bool" where
"wt_method_altern G C pTs rT mxs mxl ins et phi \
let max_pc = length ins in
0 < max_pc \<and>
length phi = length ins \<and>
check_bounded ins et \<and>
wt_start G C pTs mxl phi \<and>
(\<forall>pc. pc<max_pc \<longrightarrow> wt_instr_altern (ins!pc) G rT phi mxs (1+length pTs+mxl) max_pc et pc)"
lemma wt_method_wt_method_altern :
"wt_method G C pTs rT mxs mxl ins et phi \ wt_method_altern G C pTs rT mxs mxl ins et phi"
apply (simp add: wt_method_def wt_method_altern_def)
apply (intro strip)
apply clarify
apply (drule spec, drule mp, assumption)
apply (simp add: check_types_def wt_instr_def wt_instr_altern_def check_type_def)
apply (auto intro: imageI)
done
lemma check_type_check_types [rule_format]:
"(\pc. pc < length phi \ check_type G mxs mxr (OK (phi ! pc)))
\<longrightarrow> check_types G mxs mxr (map OK phi)"
apply (induct phi)
apply (simp add: check_types_def)
apply (simp add: check_types_def)
apply clarify
apply (frule_tac x=0 in spec)
apply (simp add: check_type_def)
apply auto
done
lemma wt_method_altern_wt_method [rule_format]:
"wt_method_altern G C pTs rT mxs mxl ins et phi \ wt_method G C pTs rT mxs mxl ins et phi"
apply (simp add: wt_method_def wt_method_altern_def)
apply (intro strip)
apply clarify
apply (rule conjI)
(* show check_types *)
apply (rule check_type_check_types)
apply (simp add: wt_instr_altern_def)
(* show wt_instr *)
apply (intro strip)
apply (drule spec, drule mp, assumption)
apply (simp add: wt_instr_def wt_instr_altern_def)
done
end
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