Theory Objects

(*  Title:      JinjaDCI/Common/Objects.thy

    Author:     David von Oheimb, Susannah Mansky
    Copyright   1999 Technische Universitaet Muenchen, 2019-20 UIUC

    Based on the Jinja theory Common/Objects.thy by David von Oheimb

section ‹ Objects and the Heap ›

theory Objects imports TypeRel Value begin

subsection‹ Objects ›

  fields = "vname × cname  val"  ― ‹field name, defining class, value›
  obj = "cname × fields"    ― ‹class instance with class name and fields›
  sfields = "vname  val"  ― ‹field name to value›

definition obj_ty  :: "obj  ty"
  "obj_ty obj    Class (fst obj)"

 ― ‹ initializes a given list of fields ›
definition init_fields :: "((vname × cname) × staticb × ty) list  fields"
  "init_fields FDTs    (map_of  map (λ((F,D),b,T). ((F,D),default_val T))) FDTs"

definition init_sfields :: "((vname × cname) × staticb × ty) list  sfields"
  "init_sfields FDTs    (map_of  map (λ((F,D),b,T). (F,default_val T))) FDTs"
  ― ‹a new, blank object with default values for instance fields:›
definition blank :: "'m prog  cname  obj"
  "blank P C   (C,init_fields (ifields P C))"

  ― ‹a new, blank object with default values for static fields:›
definition sblank :: "'m prog  cname  sfields"
  "sblank P C  init_sfields (isfields P C)"

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

(* replaced all vname, cname in below with `char list' and ⇀ with returned option
  so that pretty printing works  -SM *)
  (type) "fields" <= (type) "char list × char list  val option"
  (type) "obj" <= (type) "char list × fields"
  (type) "sfields" <= (type) "char list  val option"

subsection‹ Heap ›

type_synonym heap  = "addr  obj"

(* replaced addr with nat and ⇀ with returned option so that pretty printing works  -SM *)
 (type) "heap" <= (type) "nat  obj option"

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

definition new_Addr  :: "heap  addr option"
  "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"
  "cast_ok P C h v    v = Null  P  cname_of h (the_Addr v) * C"

definition hext :: "heap  heap  bool" ("_  _" [51,51] 50)
  "h  h'    a C fs. h a = Some(C,fs)  (fs'. h' a = Some(C,fs'))"

primrec typeof_h :: "heap  val  ty option"  ("typeof⇘_")
  "typeof⇘hUnit    = Some Void"
| "typeof⇘hNull    = Some NT"
| "typeof⇘h(Bool b) = Some Boolean"
| "typeof⇘h(Intg i) = Some Integer"
| "typeof⇘h(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: new_Addr_def split:if_splits intro:LeastI)(*>*)

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

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

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

lemma [simp]: "(typeof⇘hv = 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⇘hv"
 (*<*)by(induct v) (auto simp:fun_upd_apply)(*>*)

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

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

lemma typeof_lit_typeof:
  "typeof v = Some T  typeof⇘hv = 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 @{text"⊴"}

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(ax)"
(*<*)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⇘hv = Some T   typeof⇘h'v = Some T"
proof(cases v)
  case Addr assume "h  h'" and "typeof⇘hv = T"
  then show ?thesis using Addr by(fastforce simp:hext_def)
qed simp_all

subsection‹ Static field information function ›

datatype init_state = Done | Processing | Prepared | Error
	― ‹@{term Done} = initialized›
	― ‹@{term Processing} = currently being initialized›
	― ‹@{term Prepared} = uninitialized and not currently being initialized›
	― ‹@{term Error} = previous initialization attempt resulted in erroneous state›

inductive iprog :: "init_state  init_state  bool" ("_ i _" [51,51] 50)
  [simp]: "Prepared i i"
| [simp]: "Processing i Done"
| [simp]: "Processing i Error"
| [simp]: "i i i"

lemma iprog_Done[simp]: "(Done i i) = (i = Done)"
 by(simp only: iprog.simps, simp)

lemma iprog_Error[simp]: "(Error i i) = (i = Error)"
 by(simp only: iprog.simps, simp)

lemma iprog_Processing[simp]: "(Processing i i) = (i = Done  i = Error  i = Processing)"
 by(simp only: iprog.simps, simp)

lemma iprog_trans: " i i i'; i' i i''   i i i''"
(*<*)by(case_tac i; case_tac i') simp_all(*>*)

subsection‹ Static Heap ›

text ‹The static heap (sheap) is used for storing information about static
 field values and initialization status for classes.›

  sheap = "cname  sfields × init_state"

 (type) "sheap" <= (type) "char list  (sfields × init_state) option"

definition shext :: "sheap  sheap  bool" ("_ s _" [51,51] 50)
  "sh s sh'    C sfs i. sh C = Some(sfs,i)  (sfs' i'. sh' C = Some(sfs',i')  i i i')"

lemma shextI: "C sfs i. sh C = Some(sfs,i)  (sfs' i'. sh' C = Some(sfs',i')  i i i')  sh s sh'"
(*<*)by(auto simp: shext_def)(*>*)

lemma shext_objD: " sh s sh'; sh C = Some(sfs,i)   sfs' i'. sh' C = Some(sfs', i')  i i i'"
(*<*)by(auto simp: shext_def)(*>*)

lemma shext_refl [iff]: "sh s sh"
(*<*)by (rule shextI) auto(*>*)

lemma shext_new [simp]: "sh C = None  sh s sh(Cx)"
(*<*)by (rule shextI) (auto simp:fun_upd_apply)(*>*)

lemma shext_trans: " sh s sh'; sh' s sh''   sh s sh''"
(*<*)by (rule shextI) (fast dest: iprog_trans shext_objD)(*>*)

lemma shext_upd_obj: " sh C = Some (sfs,i); i i i'   sh s sh(C(sfs',i'))"
(*<*)by (rule shextI) (auto simp:fun_upd_apply)(*>*)