src/HOL/MicroJava/BV/BVSpec.thy

merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)

(*  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 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 dest: method_rec [of _ _ C])
qed


end