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Datei: State.thy Sprache: Isabelle

Original von: Isabelle^©

(*  Title:      HOL/MicroJava/J/State.thy
    Author:     David von Oheimb
    Copyright   1999 Technische Universitaet Muenchen
*)

section \<open>Program State\<close>

theory State
imports TypeRel Value
begin

type_synonym
  fields' = "(vname \ cname \ val)" \ \field name, defining class, value\

type_synonym
  obj = "cname \ fields'" \ \class instance with class name and fields\

definition obj_ty :: "obj => ty" where
"obj_ty obj == Class (fst obj)"

definition init_vars :: "('a \ ty) list => ('a \ val)" where
"init_vars == map_of o map (\(n,T). (n,default_val T))"

type_synonym aheap = "loc \ obj" \ \"\heap\" used in a translation below\
type_synonym locals = "vname \ val" \ \simple state, i.e. variable contents\

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

abbreviation (input)
  heap :: "state => aheap"
  where "heap == fst"

abbreviation (input)
  locals :: "state => locals"
  where "locals == snd"

abbreviation "Norm s == (None, s)"

abbreviation (input)
  abrupt :: "xstate \ val option"
  where "abrupt == fst"

abbreviation (input)
  store :: "xstate \ state"
  where "store == snd"

abbreviation
  lookup_obj :: "state \ val \ obj"
  where "lookup_obj s a' == the (heap s (the_Addr a'))"

definition raise_if :: "bool \ xcpt \ val option \ val option" where
  "raise_if b x xo \ if b \ (xo = None) then Some (Addr (XcptRef x)) else xo"

text \<open>Make \<open>new_Addr\<close> completely specified (at least for the code generator)\<close>
(*
definition new_Addr  :: "aheap => loc \<times> val option" where
  "new_Addr h \<equiv> SOME (a,x). (h a = None \<and>  x = None) |  x = Some (Addr (XcptRef OutOfMemory))"
*)
consts nat_to_loc' :: "nat => loc'"
code_datatype nat_to_loc'
definition new_Addr  :: "aheap => loc \ val option" where
  "new_Addr h \
   if \<exists>n. h (Loc (nat_to_loc' n)) = None
   then (Loc (nat_to_loc' (LEAST n. h (Loc (nat_to_loc' n)) = None)), None)
   else (Loc (nat_to_loc' 0), Some (Addr (XcptRef OutOfMemory)))"

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

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

definition cast_ok :: "'c prog => cname => aheap => val => bool" where
"cast_ok G C h v == v = Null \ G\obj_ty (the (h (the_Addr v)))\ 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 \<and> xcp = None \<or> xcp = Some (Addr (XcptRef OutOfMemory))"
apply (drule sym)
apply (unfold new_Addr_def)
apply (simp split: if_split_asm)
apply (erule LeastI)
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"
unfolding raise_if_def by (induct x) auto

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)

text \<open>Naive implementation for \<^term>\<open>new_Addr\<close> by exhaustive search\<close>

definition gen_new_Addr :: "aheap => nat \ loc \ val option" where
  "gen_new_Addr h n \
   if \<exists>a. a \<ge> n \<and> h (Loc (nat_to_loc' a)) = None
   then (Loc (nat_to_loc' (LEAST a. a \ n \ h (Loc (nat_to_loc' a)) = None)), None)
   else (Loc (nat_to_loc' 0), Some (Addr (XcptRef OutOfMemory)))"

lemma new_Addr_code_code [code]:
  "new_Addr h = gen_new_Addr h 0"
by(simp only: new_Addr_def gen_new_Addr_def split: if_split) simp

lemma gen_new_Addr_code [code]:
  "gen_new_Addr h n = (if h (Loc (nat_to_loc' n)) = None then (Loc (nat_to_loc' n), None) else gen_new_Addr h (Suc n))"
apply(simp add: gen_new_Addr_def)
apply(rule impI)
apply(rule conjI)
apply safe[1]
  apply(auto intro: arg_cong[where f=nat_to_loc'] Least_equality)[1]
apply(rule arg_cong[where f=nat_to_loc'])
apply(rule arg_cong[where f=Least])
apply(rule ext)
apply(safe, simp_all)[1]
apply(rename_tac "n'")
apply(case_tac "n = n'", simp_all)[1]
apply clarify
apply(subgoal_tac "a = n")
apply(auto intro: Least_equality arg_cong[where f=nat_to_loc'])[1]
apply(rule ccontr)
apply(erule_tac x="a" in allE)
apply simp
done

instantiation loc' :: equal
begin

definition "HOL.equal (l :: loc') l' \ l = l'"
instance by standard (simp add: equal_loc'_def)

end

end

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