2016-05-04 19:29:34 +00:00
|
|
|
(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
|
|
|
|
|
* Chapter 15: Pi-Calculus and Behavioral Refinement
|
|
|
|
|
* Author: Adam Chlipala
|
|
|
|
|
* License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *)
|
|
|
|
|
|
|
|
|
|
Require Import Frap.
|
|
|
|
|
|
|
|
|
|
Set Implicit Arguments.
|
|
|
|
|
Set Asymmetric Patterns.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Notation channel := nat.
|
|
|
|
|
|
|
|
|
|
Inductive proc :=
|
|
|
|
|
| NewChannel (k : channel -> proc)
|
|
|
|
|
| Send (ch : channel) {A : Set} (v : A) (k : proc)
|
|
|
|
|
| Recv (ch : channel) {A : Set} (k : A -> proc)
|
|
|
|
|
| Par (pr1 pr2 : proc)
|
|
|
|
|
| Dup (pr : proc).
|
|
|
|
|
|
|
|
|
|
Notation pid := nat.
|
|
|
|
|
Notation procs := (fmap pid proc).
|
|
|
|
|
Notation channels := (set channel).
|
|
|
|
|
|
|
|
|
|
Inductive step : channels * procs -> channels * procs -> Prop :=
|
|
|
|
|
| StepNew : forall chs ps i k ch,
|
|
|
|
|
ps $? i = Some (NewChannel k)
|
|
|
|
|
-> ~ch \in chs
|
|
|
|
|
-> step (chs, ps) (chs \cup {ch}, ps $+ (i, k ch))
|
|
|
|
|
| StepSendRecv : forall chs ps i ch {A : Set} (v : A) k1 j k2,
|
|
|
|
|
ps $? i = Some (Send ch v k1)
|
|
|
|
|
-> ps $? j = Some (Recv ch k2)
|
|
|
|
|
-> step (chs, ps) (chs, ps $+ (i, k1) $+ (j, k2 v))
|
|
|
|
|
| StepPar : forall chs ps i pr1 pr2 j,
|
|
|
|
|
ps $? i = Some (Par pr1 pr2)
|
|
|
|
|
-> ps $? j = None
|
|
|
|
|
-> step (chs, ps) (chs, ps $+ (i, pr1) $+ (j, pr2))
|
|
|
|
|
| StepDup : forall chs ps i pr j,
|
|
|
|
|
ps $? i = Some (Dup pr)
|
|
|
|
|
-> ps $? j = None
|
|
|
|
|
-> step (chs, ps) (chs, ps $+ (i, Dup pr) $+ (j, pr)).
|
2016-05-04 19:52:42 +00:00
|
|
|
|
|
|
|
|
|
|
|
|
|
(** * Labeled semantics *)
|
|
|
|
|
|
|
|
|
|
Record message := {
|
|
|
|
|
Channel : channel;
|
|
|
|
|
TypeOf : Set;
|
|
|
|
|
Value : TypeOf
|
|
|
|
|
}.
|
|
|
|
|
|
|
|
|
|
Inductive action :=
|
|
|
|
|
| Output (m : message)
|
|
|
|
|
| Input (m : message).
|
|
|
|
|
|
|
|
|
|
Inductive label :=
|
|
|
|
|
| Silent
|
|
|
|
|
| Action (a : action).
|
|
|
|
|
|
|
|
|
|
Coercion Action : action >-> label.
|
|
|
|
|
|
|
|
|
|
Inductive lstep : proc -> label -> proc -> Prop :=
|
|
|
|
|
| LStepSend : forall ch {A : Set} (v : A) k,
|
|
|
|
|
lstep (Send ch v k)
|
|
|
|
|
(Output {| Channel := ch; Value := v |})
|
|
|
|
|
k
|
|
|
|
|
| LStepRecv : forall ch {A : Set} (k : A -> _) v,
|
|
|
|
|
lstep (Recv ch k)
|
|
|
|
|
(Input {| Channel := ch; Value := v |})
|
|
|
|
|
(k v)
|
|
|
|
|
| LStepDup : forall pr,
|
|
|
|
|
lstep (Dup pr) Silent (Par (Dup pr) pr)
|
|
|
|
|
| LStepNew : forall k l k',
|
|
|
|
|
(forall ch, lstep (k ch) l (k' ch))
|
|
|
|
|
-> lstep (NewChannel k) l (NewChannel k')
|
|
|
|
|
| LStepPar1 : forall pr1 pr2 l pr1',
|
|
|
|
|
lstep pr1 l pr1'
|
|
|
|
|
-> lstep (Par pr1 pr2) l (Par pr1' pr2)
|
|
|
|
|
| LStepPar2 : forall pr1 pr2 l pr2',
|
|
|
|
|
lstep pr2 l pr2'
|
|
|
|
|
-> lstep (Par pr1 pr2) l (Par pr1 pr2')
|
|
|
|
|
| LStepRendezvousLeft : forall pr1 ch {A : Set} (v : A) pr1' pr2 pr2',
|
|
|
|
|
lstep pr1 (Input {| Channel := ch; Value := v |}) pr1'
|
|
|
|
|
-> lstep pr2 (Output {| Channel := ch; Value := v |}) pr2'
|
|
|
|
|
-> lstep (Par pr1 pr2) Silent (Par pr1' pr2')
|
|
|
|
|
| LStepRendezvousRight : forall pr1 ch {A : Set} (v : A) pr1' pr2 pr2',
|
|
|
|
|
lstep pr1 (Output {| Channel := ch; Value := v |}) pr1'
|
|
|
|
|
-> lstep pr2 (Input {| Channel := ch; Value := v |}) pr2'
|
|
|
|
|
-> lstep (Par pr1 pr2) Silent (Par pr1' pr2').
|
|
|
|
|
|
|
|
|
|
Inductive couldGenerate : proc -> list action -> Prop :=
|
|
|
|
|
| CgNothing : forall pr,
|
|
|
|
|
couldGenerate pr []
|
|
|
|
|
| CgSilent : forall pr pr' tr,
|
|
|
|
|
lstep pr Silent pr'
|
|
|
|
|
-> couldGenerate pr' tr
|
|
|
|
|
-> couldGenerate pr tr
|
|
|
|
|
| CgAction : forall pr a pr' tr,
|
|
|
|
|
lstep pr (Action a) pr'
|
|
|
|
|
-> couldGenerate pr' tr
|
|
|
|
|
-> couldGenerate pr (a :: tr).
|
|
|
|
|
|
|
|
|
|
Definition refines (pr1 pr2 : proc) :=
|
|
|
|
|
forall tr, couldGenerate pr1 tr -> couldGenerate pr2 tr.
|
