MessagesAndRefinement: trace refinement

MessagesAndRefinement.v

@@ -39,3 +39,66 @@ Inductive step : channels * procs -> channels * procs -> Prop :=
     ps $? i = Some (Dup pr)
     -> ps $? j = None
     -> step (chs, ps) (chs, ps $+ (i, Dup pr) $+ (j, pr)).
+
+
+(** * 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.