Theory State (Isabelle2005: October 2005)

# Theory State

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theory State
imports TypeRel Value
begin

    (*  Title:      HOL/MicroJava/J/State.thy
        ID:         $Id: State.thy,v 1.18 2005/06/17 14:13:09 haftmann Exp $
        Author:     David von Oheimb
        Copyright   1999 Technische Universitaet Muenchen
    *)

    header {* \isaheader{Program State} *}

    theory State imports TypeRel Value begin

    types
      fields_ = "(vname × cname \<rightharpoonup> val)"  -- "field name, defining class, value"

      obj = "cname × fields_"    -- "class instance with class name and fields"

    constdefs
      obj_ty  :: "obj => ty"
     "obj_ty obj  == Class (fst obj)"

      init_vars :: "('a × ty) list => ('a \<rightharpoonup> val)"
     "init_vars == map_of o map (λ(n,T). (n,default_val T))"

    types aheap  = "loc \<rightharpoonup> obj"    -- {* "@{text heap}" used in a translation below *}
          locals = "vname \<rightharpoonup> val"  -- "simple state, i.e. variable contents"

          state  = "aheap × locals"      -- "heap, local parameter including This"
          xstate = "val option × state" -- "state including exception information"

    syntax
      heap    :: "state => aheap"
      locals  :: "state => locals"
      Norm    :: "state => xstate"
      abrupt  :: "xstate => val option"
      store   :: "xstate => state"
      lookup_obj   :: "state => val => obj"

    translations
      "heap"   => "fst"
      "locals" => "snd"
      "Norm s" == "(None,s)"
      "abrupt"     => "fst"
      "store"      => "snd"
     "lookup_obj s a'"  == "the (heap s (the_Addr a'))"

    constdefs
      raise_if :: "bool => xcpt => val option => val option"
      "raise_if b x xo ≡ if b ∧  (xo = None) then Some (Addr (XcptRef x)) else xo"

      new_Addr  :: "aheap => loc × val option"
      "new_Addr h ≡ SOME (a,x). (h a = None ∧  x = None) |  x = Some (Addr (XcptRef OutOfMemory))"

      np    :: "val => val option => val option"
     "np v == raise_if (v = Null) NullPointer"

      c_hupd  :: "aheap => xstate => xstate"
     "c_hupd h'== λ(xo,(h,l)). if xo = None then (None,(h',l)) else (xo,(h,l))"

      cast_ok :: "'c prog => cname => aheap => val => bool"
     "cast_ok G C h v == v = Null ∨ G\<turnstile>obj_ty (the (h (the_Addr v)))\<preceq> Class C"

    lemma obj_ty_def2 [simp]: "obj_ty (C,fs) = Class C"
    apply (unfold obj_ty_def)
    apply (simp (no_asm))
    done

    lemma new_AddrD: "new_Addr hp = (ref, xcp) ==>
      hp ref = None ∧ xcp = None ∨ xcp = Some (Addr (XcptRef OutOfMemory))"
    apply (drule sym)
    apply (unfold new_Addr_def)
    apply(simp add: Pair_fst_snd_eq Eps_split)
    apply(rule someI)
    apply(rule disjI2)
    apply(rule_tac r = "snd (?a,Some (Addr (XcptRef OutOfMemory)))" in trans)
    apply auto
    done

    lemma raise_if_True [simp]: "raise_if True x y ≠ None"
    apply (unfold raise_if_def)
    apply auto
    done

    lemma raise_if_False [simp]: "raise_if False x y = y"
    apply (unfold raise_if_def)
    apply auto
    done

    lemma raise_if_Some [simp]: "raise_if c x (Some y) ≠ None"
    apply (unfold raise_if_def)
    apply auto
    done

    lemma raise_if_Some2 [simp]:
      "raise_if c z (if x = None then Some y else x) ≠ None"
    apply (unfold raise_if_def)
    apply(induct_tac "x")
    apply auto
    done

    lemma raise_if_SomeD [rule_format (no_asm)]:
      "raise_if c x y = Some z --> c ∧  Some z = Some (Addr (XcptRef x)) |  y = Some z"
    apply (unfold raise_if_def)
    apply auto
    done

    lemma raise_if_NoneD [rule_format (no_asm)]:
      "raise_if c x y = None --> ¬ c ∧  y = None"
    apply (unfold raise_if_def)
    apply auto
    done

    lemma np_NoneD [rule_format (no_asm)]:
      "np a' x' = None --> x' = None ∧  a' ≠ Null"
    apply (unfold np_def raise_if_def)
    apply auto
    done

    lemma np_None [rule_format (no_asm), simp]: "a' ≠ Null --> np a' x' = x'"
    apply (unfold np_def raise_if_def)
    apply auto
    done

    lemma np_Some [simp]: "np a' (Some xc) = Some xc"
    apply (unfold np_def raise_if_def)
    apply auto
    done

    lemma np_Null [simp]: "np Null None = Some (Addr (XcptRef NullPointer))"
    apply (unfold np_def raise_if_def)
    apply auto
    done

    lemma np_Addr [simp]: "np (Addr a) None = None"
    apply (unfold np_def raise_if_def)
    apply auto
    done

    lemma np_raise_if [simp]: "(np Null (raise_if c xc None)) =
      Some (Addr (XcptRef (if c then  xc else NullPointer)))"
    apply (unfold raise_if_def)
    apply (simp (no_asm))
    done

    lemma c_hupd_fst [simp]: "fst (c_hupd h (x, s)) = x"
    by (simp add: c_hupd_def split_beta)

    end

lemma obj_ty_def2:

      obj_ty (C, fs) = Class C

lemma new_AddrD:

      new_Addr hp = (ref, xcp)
      ==> hp ref = None ∧ xcp = None ∨ xcp = Some (Addr (XcptRef OutOfMemory))

lemma raise_if_True:

      raise_if True x y ≠ None

lemma raise_if_False:

      raise_if False x y = y

lemma raise_if_Some:

      raise_if c x (Some y) ≠ None

lemma raise_if_Some2:

      raise_if c z (if x = None then Some y else x) ≠ None

lemma raise_if_SomeD:

      raise_if c x y = Some z ==> c ∧ Some z = Some (Addr (XcptRef x)) ∨ y = Some z

lemma raise_if_NoneD:

      raise_if c x y = None ==> ¬ c ∧ y = None

lemma np_NoneD:

      np a' x' = None ==> x' = None ∧ a' ≠ Null

lemma np_None:

      a' ≠ Null ==> np a' x' = x'

lemma np_Some:

      np a' (Some xc) = Some xc

lemma np_Null:

      np Null None = Some (Addr (XcptRef NullPointer))

lemma np_Addr:

      np (Addr a) None = None

lemma np_raise_if:

      np Null (raise_if c xc None) =
      Some (Addr (XcptRef (if c then xc else NullPointer)))

lemma c_hupd_fst:

      fst (c_hupd h (x, s)) = x