Theory MemClerk

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

section ‹RPC-Memory example: specification of the memory clerk›

theory MemClerk
imports Memory RPC MemClerkParameters
begin

(* The clerk takes the same arguments as the memory and sends requests to the RPC *)
type_synonym mClkSndChType = "memChType"
type_synonym mClkRcvChType = "rpcSndChType"
type_synonym mClkStType = "(PrIds ⇒ mClkState) stfun"

definition
  (* state predicates *)
  MClkInit      :: "mClkRcvChType ⇒ mClkStType ⇒ PrIds ⇒ stpred"
  where "MClkInit rcv cst p = PRED (cst!p = #clkA  ∧  ¬Calling rcv p)"

definition
  (* actions *)
  MClkFwd       :: "mClkSndChType ⇒ mClkRcvChType ⇒ mClkStType ⇒ PrIds ⇒ action"
  where "MClkFwd send rcv cst p = ACT
                           $Calling send p
                         ∧ $(cst!p) = #clkA
                         ∧ Call rcv p MClkRelayArg<arg<send!p>>
                         ∧ (cst!p)$ = #clkB
                         ∧ unchanged (rtrner send!p)"

definition
  MClkRetry     :: "mClkSndChType ⇒ mClkRcvChType ⇒ mClkStType ⇒ PrIds ⇒ action"
  where "MClkRetry send rcv cst p = ACT
                           $(cst!p) = #clkB
                         ∧ res<$(rcv!p)> = #RPCFailure
                         ∧ Call rcv p MClkRelayArg<arg<send!p>>
                         ∧ unchanged (cst!p, rtrner send!p)"

definition
  MClkReply     :: "mClkSndChType ⇒ mClkRcvChType ⇒ mClkStType ⇒ PrIds ⇒ action"
  where "MClkReply send rcv cst p = ACT
                           ¬$Calling rcv p
                         ∧ $(cst!p) = #clkB
                         ∧ Return send p MClkReplyVal<res<rcv!p>>
                         ∧ (cst!p)$ = #clkA
                         ∧ unchanged (caller rcv!p)"

definition
  MClkNext      :: "mClkSndChType ⇒ mClkRcvChType ⇒ mClkStType ⇒ PrIds ⇒ action"
  where "MClkNext send rcv cst p = ACT
                        (  MClkFwd send rcv cst p
                         ∨ MClkRetry send rcv cst p
                         ∨ MClkReply send rcv cst p)"

definition
  (* temporal *)
  MClkIPSpec    :: "mClkSndChType ⇒ mClkRcvChType ⇒ mClkStType ⇒ PrIds ⇒ temporal"
  where "MClkIPSpec send rcv cst p = TEMP
                           Init MClkInit rcv cst p
                         ∧ □[ MClkNext send rcv cst p ]_(cst!p, rtrner send!p, caller rcv!p)
                         ∧ WF(MClkFwd send rcv cst p)_(cst!p, rtrner send!p, caller rcv!p)
                         ∧ SF(MClkReply send rcv cst p)_(cst!p, rtrner send!p, caller rcv!p)"

definition
  MClkISpec     :: "mClkSndChType ⇒ mClkRcvChType ⇒ mClkStType ⇒ temporal"
  where "MClkISpec send rcv cst = TEMP (∀p. MClkIPSpec send rcv cst p)"

lemmas MC_action_defs =
  MClkInit_def MClkFwd_def MClkRetry_def MClkReply_def MClkNext_def

lemmas MC_temp_defs = MClkIPSpec_def MClkISpec_def

(* The Clerk engages in an action for process p only if there is an outstanding,
   unanswered call for that process.
*)
lemma MClkidle: "⊢ ¬$Calling send p ∧ $(cst!p) = #clkA ⟶ ¬MClkNext send rcv cst p"
  by (auto simp: AReturn_def MC_action_defs)

lemma MClkbusy: "⊢ $Calling rcv p ⟶ ¬MClkNext send rcv cst p"
  by (auto simp: ACall_def MC_action_defs)


(* Enabledness of actions *)

lemma MClkFwd_enabled: "⋀p. basevars (rtrner send!p, caller rcv!p, cst!p) ⟹  
      ⊢ Calling send p ∧ ¬Calling rcv p ∧ cst!p = #clkA   
         ⟶ Enabled (MClkFwd send rcv cst p)"
  by (tactic ‹action_simp_tac (\<^context> addsimps [@{thm MClkFwd_def},
    @{thm ACall_def}, @{thm caller_def}, @{thm rtrner_def}]) [exI]
    [@{thm base_enabled}, @{thm Pair_inject}] 1›)

lemma MClkFwd_ch_enabled: "⊢ Enabled (MClkFwd send rcv cst p)  ⟶   
         Enabled (<MClkFwd send rcv cst p>_(cst!p, rtrner send!p, caller rcv!p))"
  by (auto elim!: enabled_mono simp: angle_def MClkFwd_def)

lemma MClkReply_change: "⊢ MClkReply send rcv cst p ⟶  
         <MClkReply send rcv cst p>_(cst!p, rtrner send!p, caller rcv!p)"
  by (auto simp: angle_def MClkReply_def elim: Return_changed [temp_use])

lemma MClkReply_enabled: "⋀p. basevars (rtrner send!p, caller rcv!p, cst!p) ⟹  
      ⊢ Calling send p ∧ ¬Calling rcv p ∧ cst!p = #clkB   
         ⟶ Enabled (<MClkReply send rcv cst p>_(cst!p, rtrner send!p, caller rcv!p))"
  apply (tactic ‹action_simp_tac \<^context>
    [@{thm MClkReply_change} RSN (2, @{thm enabled_mono})] [] 1›)
  apply (tactic ‹action_simp_tac (\<^context> addsimps
    [@{thm MClkReply_def}, @{thm AReturn_def}, @{thm caller_def}, @{thm rtrner_def}])
    [exI] [@{thm base_enabled}, @{thm Pair_inject}] 1›)
  done

lemma MClkReplyNotRetry: "⊢ MClkReply send rcv cst p ⟶ ¬MClkRetry send rcv cst p"
  by (auto simp: MClkReply_def MClkRetry_def)

end