From 81ed73437f6d8b002ceeffb806d7c141ea3a5bc9 Mon Sep 17 00:00:00 2001 From: Ralf Jung Date: Thu, 5 Jan 2017 13:18:13 +0100 Subject: [PATCH] COFE for sigma types --- theories/algebra/ofe.v | 53 ++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 53 insertions(+) diff --git a/theories/algebra/ofe.v b/theories/algebra/ofe.v index fbec685e..6b38ce23 100644 --- a/theories/algebra/ofe.v +++ b/theories/algebra/ofe.v @@ -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. -- 2.26.2