Theory ProcedureInterface

(*  Title:      HOL/TLA/Memory/ProcedureInterface.thy
    Author:     Stephan Merz, University of Munich
*)

section ‹Procedure interface for RPC-Memory components›

theory ProcedureInterface
imports "HOL-TLA.TLA" RPCMemoryParams
begin

typedecl ('a,'r) chan
  (* type of channels with argument type 'a and return type 'r.
     we model a channel as an array of variables (of type chan)
     rather than a single array-valued variable because the
     notation gets a little simpler.
  *)
type_synonym ('a,'r) channel =" (PrIds ⇒ ('a,'r) chan) stfun"


(* data-level functions *)

consts
  cbit          :: "('a,'r) chan ⇒ bit"
  rbit          :: "('a,'r) chan ⇒ bit"
  arg           :: "('a,'r) chan ⇒ 'a"
  res           :: "('a,'r) chan ⇒ 'r"


(* state functions *)

definition caller :: "('a,'r) channel ⇒ (PrIds ⇒ (bit * 'a)) stfun"
  where "caller ch == λs p. (cbit (ch s p), arg (ch s p))"

definition rtrner :: "('a,'r) channel ⇒ (PrIds ⇒ (bit * 'r)) stfun"
  where "rtrner ch == λs p. (rbit (ch s p), res (ch s p))"


(* slice through array-valued state function *)

definition slice :: "('a ⇒ 'b) stfun ⇒ 'a ⇒ 'b stfun"
  where "slice x i s ≡ x s i"

syntax
  "_slice" :: "[lift, 'a] ⇒ lift"  ("(_!_)" [70,70] 70)
translations
  "_slice" ⇌ "CONST slice"


(* state predicates *)

definition Calling :: "('a,'r) channel ⇒ PrIds ⇒ stpred"
  where "Calling ch p == PRED cbit< ch!p > ≠ rbit< ch!p >"


(* actions *)

definition ACall :: "('a,'r) channel ⇒ PrIds ⇒ 'a stfun ⇒ action"
  where "ACall ch p v ≡ ACT
      ¬ $Calling ch p
    ∧ (cbit<ch!p>$ ≠ $rbit<ch!p>)
    ∧ (arg<ch!p>$ = $v)"

definition AReturn :: "('a,'r) channel ⇒ PrIds ⇒ 'r stfun ⇒ action"
  where "AReturn ch p v == ACT
      $Calling ch p
    ∧ (rbit<ch!p>$ = $cbit<ch!p>)
    ∧ (res<ch!p>$ = $v)"

syntax
  "_Call" :: "['a, 'b, lift] ⇒ lift"  ("(Call _ _ _)" [90,90,90] 90)
  "_Return" :: "['a, 'b, lift] ⇒ lift"  ("(Return _ _ _)" [90,90,90] 90)
translations
  "_Call" ⇌ "CONST ACall"
  "_Return" ⇌ "CONST AReturn"


(* temporal formulas *)

definition PLegalCaller :: "('a,'r) channel ⇒ PrIds ⇒ temporal"
  where "PLegalCaller ch p == TEMP
     Init(¬ Calling ch p)
     ∧ □[∃a. Call ch p a ]_((caller ch)!p)"

definition LegalCaller :: "('a,'r) channel ⇒ temporal"
  where "LegalCaller ch == TEMP (∀p. PLegalCaller ch p)"

definition PLegalReturner :: "('a,'r) channel ⇒ PrIds ⇒ temporal"
  where "PLegalReturner ch p == TEMP □[∃v. Return ch p v ]_((rtrner ch)!p)"

definition LegalReturner :: "('a,'r) channel ⇒ temporal"
  where "LegalReturner ch == TEMP (∀p. PLegalReturner ch p)"

declare slice_def [simp]

lemmas Procedure_defs = caller_def rtrner_def Calling_def ACall_def AReturn_def
  PLegalCaller_def LegalCaller_def PLegalReturner_def LegalReturner_def

(* Calls and returns change their subchannel *)
lemma Call_changed: "⊢ Call ch p v ⟶ <Call ch p v>_((caller ch)!p)"
  by (auto simp: angle_def ACall_def caller_def Calling_def)

lemma Return_changed: "⊢ Return ch p v ⟶ <Return ch p v>_((rtrner ch)!p)"
  by (auto simp: angle_def AReturn_def rtrner_def Calling_def)

end