define the set of low states and prove it closed

barrier/barrier.v

@@ -3,9 +3,11 @@ From heap_lang Require Export derived heap wp_tactics notation.
 
 Definition newchan := (λ: "", ref '0)%L.
 Definition signal := (λ: "x", "x" <- '1)%L.
-Definition wait := (rec: "wait" "x" := if: !"x" = '1 then '() else "wait" "x")%L.
+Definition wait := (rec: "wait" "x" :=if: !"x" = '1 then '() else "wait" "x")%L.
 
-(** The STS describing the main barrier protocol. *)
+(** The STS describing the main barrier protocol. Every state has an index-set
+    associated with it. These indices are actually [gname], because we use them
+    with saved propositions. *)
 Module barrier_proto.
   Inductive phase := Low | High.
   Record stateT := State { state_phase : phase; state_I : gset gname }.
@@ -27,6 +29,7 @@ Module barrier_proto.
 
   Definition sts := sts.STS trans tok.
 
+  (* The set of states containing some particular i *)
   Definition i_states (i : gname) : set stateT :=
     mkSet (λ s, i ∈ state_I s).
 
@@ -34,17 +37,21 @@ Module barrier_proto.
     sts.closed sts (i_states i) {[ Change i ]}.
   Proof.
     split.
-    - apply (non_empty_inhabited (State Low {[ i ]})). rewrite !mkSet_elem_of /=.
+    - apply (non_empty_inhabited(State Low {[ i ]})). rewrite !mkSet_elem_of /=.
       apply lookup_singleton.
     - move=>[p I]. rewrite /= /tok !mkSet_elem_of /= =>HI.
       move=>s' /elem_of_intersection. rewrite !mkSet_elem_of /=.
       move=>[[Htok|Htok] ? ]; subst s'; first done.
       destruct p; done.
-    - move=>s1 s2. rewrite !mkSet_elem_of /==> Hs1 Hstep.
+    - (* If we do the destruct of the states early, and then inversion
+         on the proof of a transition, it doesn't work - we do not obtain
+         the equalities we need. So we destruct the states late, because this
+         means we can use "destruct" instead of "inversion". *)
+      move=>s1 s2. rewrite !mkSet_elem_of /==> Hs1 Hstep.
       (* We probably want some helper lemmas for this... *)
       inversion_clear Hstep as [T1 T2 Hdisj Hstep'].
-      inversion_clear Hstep' as [? ? ? ? Htrans Htok1 Htok2 Htok].
-      destruct Htrans; last done; move:Hs1 Hdisj Htok1 Htok2 Htok.
+      inversion_clear Hstep' as [? ? ? ? Htrans _ _ Htok].
+      destruct Htrans; last done; move:Hs1 Hdisj Htok.
       rewrite /= /tok /=.
       intros. apply dec_stable. 
       assert (Change i ∉ change_tokens I1) as HI1
@@ -55,6 +62,25 @@ Module barrier_proto.
         - solve_elem_of +Htok Hdisj HI1 / discriminate. }
       done.
   Qed.
-    
+
+  (* The set of low states *)
+  Definition low_states : set stateT :=
+    mkSet (λ s, if state_phase s is Low then True else False).
+  
+  Lemma low_states_closed :
+    sts.closed sts low_states {[ Send ]}.
+  Proof.
+    split.
+    - apply (non_empty_inhabited(State Low ∅)). by rewrite !mkSet_elem_of /=.
+    - move=>[p I]. rewrite /= /tok !mkSet_elem_of /= =>HI.
+      destruct p; last done. solve_elem_of+ /discriminate.
+    - move=>s1 s2. rewrite !mkSet_elem_of /==> Hs1 Hstep.
+      inversion_clear Hstep as [T1 T2 Hdisj Hstep'].
+      inversion_clear Hstep' as [? ? ? ? Htrans _ _ Htok].
+      destruct Htrans; move:Hs1 Hdisj Htok =>/=;
+                                first by destruct p.
+      rewrite /= /tok /=. intros. solve_elem_of +Hdisj Htok.
+  Qed.
+
 End barrier_proto.