(* Title: HOL/MicroJava/BV/BVSpec.thy
ID: $Id$
Author: Cornelia Pusch
Copyright 1999 Technische Universitaet Muenchen
*)
header "The Bytecode Verifier"
theory BVSpec = Step:
constdefs
wt_instr :: "[instr,jvm_prog,ty,method_type,nat,p_count,p_count] => bool"
"wt_instr i G rT phi mxs max_pc pc ==
app i G mxs rT (phi!pc) \\<and>
(\\<forall> pc' \\<in> set (succs i pc). pc' < max_pc \\<and> (G \\<turnstile> step i G (phi!pc) <=' phi!pc'))"
wt_start :: "[jvm_prog,cname,ty list,nat,method_type] => bool"
"wt_start G C pTs mxl phi ==
G \\<turnstile> Some ([],(OK (Class C))#((map OK pTs))@(replicate mxl Err)) <=' phi!0"
wt_method :: "[jvm_prog,cname,ty list,ty,nat,nat,instr list,method_type] => bool"
"wt_method G C pTs rT mxs mxl ins phi ==
let max_pc = length ins
in
0 < max_pc \\<and> wt_start G C pTs mxl phi \\<and>
(\\<forall>pc. pc<max_pc --> wt_instr (ins ! pc) G rT phi mxs max_pc pc)"
wt_jvm_prog :: "[jvm_prog,prog_type] => bool"
"wt_jvm_prog G phi ==
wf_prog (\\<lambda>G C (sig,rT,(maxs,maxl,b)).
wt_method G C (snd sig) rT maxs maxl b (phi C sig)) G"
lemma wt_jvm_progD:
"wt_jvm_prog G phi ==> (\\<exists>wt. wf_prog wt G)"
by (unfold wt_jvm_prog_def, blast)
lemma wt_jvm_prog_impl_wt_instr:
"[| wt_jvm_prog G phi; is_class G C;
method (G,C) sig = Some (C,rT,maxs,maxl,ins); pc < length ins |]
==> wt_instr (ins!pc) G rT (phi C sig) maxs (length ins) pc";
by (unfold wt_jvm_prog_def, drule method_wf_mdecl,
simp, simp, simp add: wf_mdecl_def wt_method_def)
lemma wt_jvm_prog_impl_wt_start:
"[| wt_jvm_prog G phi; is_class G C;
method (G,C) sig = Some (C,rT,maxs,maxl,ins) |] ==>
0 < (length ins) \\<and> wt_start G C (snd sig) maxl (phi C sig)"
by (unfold wt_jvm_prog_def, drule method_wf_mdecl,
simp, simp, simp add: wf_mdecl_def wt_method_def)
text {* for most instructions wt\_instr collapses: *}
lemma
"succs i pc = [pc+1] ==> wt_instr i G rT phi mxs max_pc pc =
(app i G mxs rT (phi!pc) \\<and> pc+1 < max_pc \\<and> (G \\<turnstile> step i G (phi!pc) <=' phi!(pc+1)))"
by (simp add: wt_instr_def)
(* ### move to WellForm *)
lemma methd:
"[| wf_prog wf_mb G; (C,S,fs,mdecls) \\<in> set G; (sig,rT,code) \\<in> set mdecls |]
==> method (G,C) sig = Some(C,rT,code) \\<and> is_class G C"
proof -
assume wf: "wf_prog wf_mb G"
assume C: "(C,S,fs,mdecls) \\<in> set G"
assume m: "(sig,rT,code) \\<in> set mdecls"
moreover
from wf
have "class G Object = Some (arbitrary, [], [])"
by simp
moreover
from wf C
have c: "class G C = Some (S,fs,mdecls)"
by (auto simp add: wf_prog_def class_def is_class_def intro: map_of_SomeI)
ultimately
have O: "C \\<noteq> Object"
by auto
from wf C
have "unique mdecls"
by (unfold wf_prog_def wf_cdecl_def) auto
hence "unique (map (\\<lambda>(s,m). (s,C,m)) mdecls)"
by - (induct mdecls, auto)
with m
have "map_of (map (\\<lambda>(s,m). (s,C,m)) mdecls) sig = Some (C,rT,code)"
by (force intro: map_of_SomeI)
with wf C m c O
show ?thesis
by (auto simp add: is_class_def dest: method_rec [of _ _ C])
qed
end