Theory BVSpec (Isabelle2005: October 2005)

# Theory BVSpec

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theory BVSpec
imports Effect
begin

    (*  Title:      HOL/MicroJava/BV/BVSpec.thy
        ID:         $Id: BVSpec.thy,v 1.28 2005/06/17 14:13:08 haftmann Exp $
        Author:     Cornelia Pusch, Gerwin Klein
        Copyright   1999 Technische Universitaet Muenchen

    *)

    header {* \isaheader{The Bytecode Verifier}\label{sec:BVSpec} *}

    theory BVSpec imports Effect begin

    text {*
      This theory contains a specification of the BV. The specification
      describes correct typings of method bodies; it corresponds
      to type \emph{checking}.
    *}

    constdefs
      -- "The program counter will always be inside the method:"
      check_bounded :: "instr list => exception_table => bool"
      "check_bounded ins et ≡
      (∀pc < length ins. ∀pc' ∈ set (succs (ins!pc) pc). pc' < length ins) ∧
                         (∀e ∈ set et. fst (snd (snd e)) < length ins)"

      -- "The method type only contains declared classes:"
      check_types :: "jvm_prog => nat => nat => state list => bool"
      "check_types G mxs mxr phi ≡ set phi ⊆ states G mxs mxr"

      -- "An instruction is welltyped if it is applicable and its effect"
      -- "is compatible with the type at all successor instructions:"
      wt_instr :: "[instr,jvm_prog,ty,method_type,nat,p_count,
                    exception_table,p_count] => bool"
      "wt_instr i G rT phi mxs max_pc et pc ≡
      app i G mxs rT pc et (phi!pc) ∧
      (∀(pc',s') ∈ set (eff i G pc et (phi!pc)). pc' < max_pc ∧ G \<turnstile> s' <=' phi!pc')"

      -- {* The type at @{text "pc=0"} conforms to the method calling convention: *}
      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"

      -- "A method is welltyped if the body is not empty, if execution does not"
      -- "leave the body, if the method type covers all instructions and mentions"
      -- "declared classes only, if the method calling convention is respected, and"
      -- "if all instructions are welltyped."
      wt_method :: "[jvm_prog,cname,ty list,ty,nat,nat,instr list,
                     exception_table,method_type] => bool"
      "wt_method G C pTs rT mxs mxl ins et phi ≡
      let max_pc = length ins in
      0 < max_pc ∧
      length phi = length ins ∧
      check_bounded ins et ∧
      check_types G mxs (1+length pTs+mxl) (map OK phi) ∧
      wt_start G C pTs mxl phi ∧
      (∀pc. pc<max_pc --> wt_instr (ins!pc) G rT phi mxs max_pc et pc)"

      -- "A program is welltyped if it is wellformed and all methods are welltyped"
      wt_jvm_prog :: "[jvm_prog,prog_type] => bool"
      "wt_jvm_prog G phi ==
      wf_prog (λG C (sig,rT,(maxs,maxl,b,et)).
               wt_method G C (snd sig) rT maxs maxl b et (phi C sig)) G"

    lemma check_boundedD:
      "[| check_bounded ins et; pc < length ins;
        (pc',s') ∈ set (eff (ins!pc) G pc et s)  |] ==>
      pc' < length ins"
      apply (unfold eff_def)
      apply simp
      apply (unfold check_bounded_def)
      apply clarify
      apply (erule disjE)
       apply blast
      apply (erule allE, erule impE, assumption)
      apply (unfold xcpt_eff_def)
      apply clarsimp
      apply (drule xcpt_names_in_et)
      apply clarify
      apply (drule bspec, assumption)
      apply simp
      done

    lemma wt_jvm_progD:
      "wt_jvm_prog G phi ==> (∃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,et); pc < length ins |]
      ==> wt_instr (ins!pc) G rT (phi C sig) maxs (length ins) et pc"
      by (unfold wt_jvm_prog_def, drule method_wf_mdecl,
          simp, simp, simp add: wf_mdecl_def wt_method_def)

    text {*
      We could leave out the check @{term "pc' < max_pc"} in the
      definition of @{term wt_instr} in the context of @{term wt_method}.
    *}
    lemma wt_instr_def2:
      "[| wt_jvm_prog G Phi; is_class G C;
          method (G,C) sig = Some (C,rT,maxs,maxl,ins,et); pc < length ins;
          i = ins!pc; phi = Phi C sig; max_pc = length ins |]
      ==> wt_instr i G rT phi maxs max_pc et pc =
         (app i G maxs rT pc et (phi!pc) ∧
         (∀(pc',s') ∈ set (eff i G pc et (phi!pc)). G \<turnstile> s' <=' phi!pc'))"
    apply (simp add: wt_instr_def)
    apply (unfold wt_jvm_prog_def)
    apply (drule method_wf_mdecl)
    apply (simp, simp, simp add: wf_mdecl_def wt_method_def)
    apply (auto dest: check_boundedD)
    done

    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,et) |] ==>
      0 < (length ins) ∧ 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)

    end

lemma check_boundedD:

      [| check_bounded ins et; pc < length ins;
         (pc', s') ∈ set (eff (ins ! pc) G pc et s) |]
      ==> pc' < length ins

lemma wt_jvm_progD:

      wt_jvm_prog G phi ==> ∃wt. wf_prog wt G

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, et); pc < length ins |]
      ==> wt_instr (ins ! pc) G rT (phi C sig) maxs (length ins) et pc

lemma wt_instr_def2:

      [| wt_jvm_prog G Phi; is_class G C;
         method (G, C) sig = Some (C, rT, maxs, maxl, ins, et); pc < length ins;
         i = ins ! pc; phi = Phi C sig; max_pc = length ins |]
      ==> wt_instr i G rT phi maxs max_pc et pc =
          (app i G maxs rT pc et (phi ! pc) ∧
           (∀(pc', s')∈set (eff i G pc et (phi ! pc)). G |- s' <=' phi ! pc'))

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, et) |]
      ==> 0 < length ins ∧ wt_start G C (snd sig) maxl (phi C sig)