Quelle  Objects.thy

(*  Title:      HOL/MicroJava/J/State.thy

    Author:     David von Oheimb
    Copyright   1999 Technische Universitaet Muenchen
*)

section ‹Objects and the Heap›

theory Objects imports TypeRel Value begin

subsection‹Objects›

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_fields :: "((vname × cname) × ty) list → fields"
where
  "init_fields ≡ map_of ∘ map (λ(F,T). (F,default_val T))"
  
  ― ‹a new, blank object with default values in all fields:›
definition blank :: "'m prog → cname → obj"
where
  "blank P C ≡ (C,init_fields (fields P C))" 

lemma [simp]: "obj_ty (C,fs) = Class C"
(*<*)by (simp add: obj_ty_def)(*>*)

subsection‹Heap›

type_synonym heap  = "addr ⇀ obj"

abbreviation
  cname_of :: "heap → addr → cname" where
  "cname_of hp a == fst (the (hp a))"

definition new_Addr  :: "heap → addr option"
where
  "new_Addr h ≡ if ∃a. h a = None then Some(LEAST a. h a = None) else None"

definition cast_ok :: "'m prog → cname → heap → val → bool"
where
  "cast_ok P C h v ≡ v = Null ∨ P ⊨ cname_of h (the_Addr v) ⪯^* C"

definition hext :: "heap → heap → bool" (‹_ ⊴ _› [51,51] 50)
where
  "h ⊴ h' ≡ ∀a C fs. h a = Some(C,fs) ⟶ (∃fs'. h' a = Some(C,fs'))"

primrec typeof_h :: "heap → val → ty option"  (‹typeof›)
where
  "typeof Unit = Some Void"
| "typeof Null = Some NT"
| "typeof (Bool b) = Some Boolean"
| "typeof (Intg i) = Some Integer"
| "typeof (Addr a) = (case h a of None → None | Some(C,fs) → Some(Class C))"

lemma new_Addr_SomeD:
  "new_Addr h = Some a ==> h a = None"
 (*<*)by(fastforce simp add:new_Addr_def split:if_splits intro:LeastI)(*>*)

lemma [simp]: "(typeof v = Some Boolean) = (∃b. v = Bool b)"
 (*<*)by(induct v) auto(*>*)

lemma [simp]: "(typeof v = Some Integer) = (∃i. v = Intg i)"
(*<*)by(cases v) auto(*>*)

lemma [simp]: "(typeof v = Some NT) = (v = Null)"
 (*<*)by(cases v) auto(*>*)

lemma [simp]: "(typeof v = Some(Class C)) = (∃a fs. v = Addr a ∧ h a = Some(C,fs))"
 (*<*)by(cases v) auto(*>*)

lemma [simp]: "h a = Some(C,fs) ==> typeof_{h(a↦(C,fs')))} v = typeof v"
 (*<*)by(induct v) (auto simp:fun_upd_apply)(*>*)

text‹For literal values the first parameter of @{term typeof} can be
  to @{term Map.empty} because they do not contain addresses:›

abbreviation
  typeof :: "val → ty option" where
  "typeof v == typeof_h Map.empty v"

lemma typeof_lit_typeof:
  "typeof v = Some T ==> typeof v = Some T"
 (*<*)by(cases v) auto(*>*)

lemma typeof_lit_is_type: 
  "typeof v = Some T ==> is_type P T"
 (*<*)by (induct v) (auto simp:is_type_def)(*>*)


subsection ‹Heap extension ‹⊴››

lemma hextI: "∀a C fs. h a = Some(C,fs) ⟶ (∃fs'. h' a = Some(C,fs')) ==> h ⊴ h'"
(*<*)by(auto simp: hext_def)(*>*)

lemma hext_objD: "[ h ⊴ h'; h a = Some(C,fs) ] ==> ∃fs'. h' a = Some(C,fs')"
(*<*)by(auto simp: hext_def)(*>*)

lemma hext_refl [iff]: "h ⊴ h"
(*<*)by (rule hextI) fast(*>*)

lemma hext_new [simp]: "h a = None ==> h ⊴ h(a↦x)"
(*<*)by (rule hextI) (auto simp:fun_upd_apply)(*>*)

lemma hext_trans: "[ h ⊴ h'; h' ⊴ h'' ] ==> h ⊴ h''"
(*<*)by (rule hextI) (fast dest: hext_objD)(*>*)

lemma hext_upd_obj: "h a = Some (C,fs) ==> h ⊴ h(a↦(C,fs'))"
(*<*)by (rule hextI) (auto simp:fun_upd_apply)(*>*)

lemma hext_typeof_mono: "[ h ⊴ h'; typeof v = Some T ] ==> typeof_' v = Some T"
(*<*)
proof(cases v)
  case Addr assume "h ⊴ h'" and "typeof v = ⌊T⌋"
  then show ?thesis using Addr by(fastforce simp:hext_def)
qed simp_all
(*>*)

text ‹Code generator setup for @{term "new_Addr"}›

definition gen_new_Addr :: "heap → addr → addr option"
where "gen_new_Addr h n ≡ if ∃a. a ≥ n ∧ h a = None then Some(LEAST a. a ≥ n ∧ h a = None) else None"

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

lemma gen_new_Addr_code [code]:
  "gen_new_Addr h n = (if h n = None then Some n else gen_new_Addr h (Suc n))"
apply(simp add: gen_new_Addr_def)
apply(rule impI)
apply(rule conjI)
 apply safe[1]
  apply(fastforce intro: Least_equality)
 apply(rule arg_cong[where f=Least])
 apply(rule ext)
 apply(case_tac "n = ac")
  apply simp
 apply(auto)[1]
apply clarify
apply(subgoal_tac "a = n")
 apply simp
 apply(rule Least_equality)
 apply auto[2]
apply(rule ccontr)
apply(erule_tac x="a" in allE)
apply simp
done

end