Commit 86b8e9ed authored by Ralf Jung's avatar Ralf Jung

define the set of low states and prove it closed

parent 1109ca07
...@@ -3,9 +3,11 @@ From heap_lang Require Export derived heap wp_tactics notation. ...@@ -3,9 +3,11 @@ From heap_lang Require Export derived heap wp_tactics notation.
Definition newchan := (λ: "", ref '0)%L. Definition newchan := (λ: "", ref '0)%L.
Definition signal := (λ: "x", "x" <- '1)%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. Module barrier_proto.
Inductive phase := Low | High. Inductive phase := Low | High.
Record stateT := State { state_phase : phase; state_I : gset gname }. Record stateT := State { state_phase : phase; state_I : gset gname }.
...@@ -27,6 +29,7 @@ Module barrier_proto. ...@@ -27,6 +29,7 @@ Module barrier_proto.
Definition sts := sts.STS trans tok. Definition sts := sts.STS trans tok.
(* The set of states containing some particular i *)
Definition i_states (i : gname) : set stateT := Definition i_states (i : gname) : set stateT :=
mkSet (λ s, i state_I s). mkSet (λ s, i state_I s).
...@@ -34,17 +37,21 @@ Module barrier_proto. ...@@ -34,17 +37,21 @@ Module barrier_proto.
sts.closed sts (i_states i) {[ Change i ]}. sts.closed sts (i_states i) {[ Change i ]}.
Proof. Proof.
split. 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. apply lookup_singleton.
- move=>[p I]. rewrite /= /tok !mkSet_elem_of /= =>HI. - move=>[p I]. rewrite /= /tok !mkSet_elem_of /= =>HI.
move=>s' /elem_of_intersection. rewrite !mkSet_elem_of /=. move=>s' /elem_of_intersection. rewrite !mkSet_elem_of /=.
move=>[[Htok|Htok] ? ]; subst s'; first done. move=>[[Htok|Htok] ? ]; subst s'; first done.
destruct p; 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... *) (* We probably want some helper lemmas for this... *)
inversion_clear Hstep as [T1 T2 Hdisj Hstep']. inversion_clear Hstep as [T1 T2 Hdisj Hstep'].
inversion_clear Hstep' as [? ? ? ? Htrans Htok1 Htok2 Htok]. inversion_clear Hstep' as [? ? ? ? Htrans _ _ Htok].
destruct Htrans; last done; move:Hs1 Hdisj Htok1 Htok2 Htok. destruct Htrans; last done; move:Hs1 Hdisj Htok.
rewrite /= /tok /=. rewrite /= /tok /=.
intros. apply dec_stable. intros. apply dec_stable.
assert (Change i change_tokens I1) as HI1 assert (Change i change_tokens I1) as HI1
...@@ -55,6 +62,25 @@ Module barrier_proto. ...@@ -55,6 +62,25 @@ Module barrier_proto.
- solve_elem_of +Htok Hdisj HI1 / discriminate. } - solve_elem_of +Htok Hdisj HI1 / discriminate. }
done. done.
Qed. 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. End barrier_proto.
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