Commit 81ed7343 authored by Ralf Jung's avatar Ralf Jung

COFE for sigma types

parent 6bbc6b49
......@@ -955,6 +955,59 @@ Proof.
destruct n as [|n]; simpl in *; first done. apply cFunctor_ne, Hfg.
Qed.
(** Sigma *)
Section sigma.
Context {A : ofeT} {f : A Prop}.
(* TODO: Find a better place for this Equiv instance. It also
should not depend on A being an OFE. *)
Instance sig_equiv : Equiv (sig f) :=
λ x1 x2, (proj1_sig x1) (proj1_sig x2).
Instance sig_dist : Dist (sig f) :=
λ n x1 x2, (proj1_sig x1) {n} (proj1_sig x2).
Global Lemma exist_ne :
n x1 x2, x1 {n} x2
(H1 : f x1) (H2 : f x2), (exist f x1 H1) {n} (exist f x2 H2).
Proof. intros n ?? Hx ??. exact Hx. Qed.
Global Instance proj1_sig_ne : Proper (dist n ==> dist n) (@proj1_sig _ f).
Proof. intros n [] [] ?. done. Qed.
Definition sig_ofe_mixin : OfeMixin (sig f).
Proof.
split.
- intros x y. unfold dist, sig_dist, equiv, sig_equiv.
destruct x, y. apply equiv_dist.
- unfold dist, sig_dist. intros n.
split; [intros [] | intros [] [] | intros [] [] []]; simpl; try done.
intros. by etrans.
- intros n [] []. unfold dist, sig_dist. apply dist_S.
Qed.
Canonical Structure sigC : ofeT := OfeT (sig f) sig_ofe_mixin.
Global Class LimitPreserving `{Cofe A} : Prop :=
limit_preserving : c : chain A, ( n, f (c n)) f (compl c).
Program Definition sig_compl `{LimitPreserving} : Compl sigC :=
λ c, exist f (compl (chain_map proj1_sig c)) _.
Next Obligation.
intros ? Hlim c. apply Hlim. move=>n /=. destruct (c n). done.
Qed.
Program Definition sig_cofe `{LimitPreserving} : Cofe sigC :=
{| compl := sig_compl |}.
Next Obligation.
intros ? Hlim n c. apply (conv_compl n (chain_map proj1_sig c)).
Qed.
Global Instance sig_timeless (x : sig f) :
Timeless (proj1_sig x) Timeless x.
Proof. intros ? y. destruct x, y. unfold dist, sig_dist, equiv, sig_equiv. apply (timeless _). Qed.
Global Instance sig_discrete_cofe : Discrete A Discrete sigC.
Proof.
intros ? [??] [??]. rewrite /dist /equiv /ofe_dist /ofe_equiv /=.
rewrite /sig_dist /sig_equiv /=. apply discrete_timeless.
Qed.
End sigma.
Arguments sigC {A} f.
(** Notation for writing functors *)
Notation "∙" := idCF : cFunctor_scope.
Notation "T -c> F" := (ofe_funCF T%type F%CF) : cFunctor_scope.
......
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