Commit 1ee007f8 authored by Ralf Jung's avatar Ralf Jung

prelude.collections: add lemma to prove non-emptiness

parent 7827f688
......@@ -7,23 +7,34 @@ Definition wait := (rec: "wait" "x" := if: !"x" = '1 then '() else "wait" "x")%L
(** The STS describing the main barrier protocol. *)
Module barrier_proto.
Inductive state := Low (I : gset gname) | High (I : gset gname).
Inductive phase := Low | High.
Record stateT := State { state_phase : phase; state_I : gset gname }.
Inductive token := Change (i : gname) | Send.
Global Instance stateT_inhabited: Inhabited stateT.
Proof. split. exact (State Low ). Qed.
Definition change_tokens (I : gset gname) : set token :=
mkSet (λ t, match t with Change i => i I | Send => False end).
Inductive trans : relation state :=
| LowChange I1 I2 : trans (Low I1) (Low I2)
| HighChange I2 I1 : trans (High I1) (High I2)
| LowHigh I : trans (Low I) (High I).
Inductive trans : relation stateT :=
| ChangeI p I2 I1 : trans (State p I1) (State p I2)
| ChangePhase I : trans (State Low I) (State High I).
Definition tok (s : state) : set token :=
match s with
| Low I' => change_tokens I'
| High I' => change_tokens I' {[ Send ]}
end.
Definition tok (s : stateT) : set token :=
change_tokens (state_I s)
match state_phase s with Low => | High => {[ Send ]} end.
Definition sts := sts.STS trans tok.
Definition i_states (i : gname) : set stateT :=
mkSet (λ s, i state_I s).
Lemma i_states_closed i :
sts.closed sts (i_states i) {[ Change i ]}.
Proof.
split.
- apply non_empty_inhabited.
End barrier_proto.
......@@ -280,6 +280,8 @@ Section collection.
intros X1 X2 HX Y1 Y2 HY; apply elem_of_equiv; intros x.
by rewrite !elem_of_difference, HX, HY.
Qed.
Lemma non_empty_inhabited x X : x X X .
Proof. solve_elem_of. Qed.
Lemma intersection_singletons x : ({[x]} : C) {[x]} {[x]}.
Proof. solve_elem_of. Qed.
Lemma difference_twice X Y : (X Y) Y X Y.
......
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