Commit 1e519f29 authored by Ralf Jung's avatar Ralf Jung

prove dist_later_dist; define constant chain

parent 711bead3
......@@ -130,6 +130,14 @@ Lemma compl_chain_map `{Cofe A, Cofe B} (f : A → B) c `(NonExpansive f) :
compl (chain_map f c) f (compl c).
Proof. apply equiv_dist=>n. by rewrite !conv_compl. Qed.
Program Definition chain_const {A : ofeT} (a : A) : chain A :=
{| chain_car n := a |}.
Next Obligation. by intros A a n i _. Qed.
Lemma compl_chain_const {A : ofeT} `{!Cofe A} (a : A) :
compl (chain_const a) a.
Proof. apply equiv_dist=>n. by rewrite conv_compl. Qed.
(** General properties *)
Section ofe.
Context {A : ofeT}.
......@@ -192,6 +200,17 @@ Proof. destruct n as [|n]. by split. apply dist_equivalence. Qed.
Lemma dist_dist_later {A : ofeT} n (x y : A) : dist n x y dist_later n x y.
Proof. intros Heq. destruct n; first done. exact: dist_S. Qed.
Lemma dist_later_dist {A : ofeT} n (x y : A) : dist_later (S n) x y dist n x y.
Proof. done. Qed.
(* We don't actually need this lemma (as our tactics deal with this through
other means), but technically speaking, this is the reason why
pre-composing a non-expansive function to a contractive function
preserves contractivity. *)
Lemma ne_dist_later {A B : ofeT} (f : A B) :
NonExpansive f n, Proper (dist_later n ==> dist_later n) f.
Proof. intros Hf [|n]; last exact: Hf. hnf. by intros. Qed.
Notation Contractive f := ( n, Proper (dist_later n ==> dist n) f).
Instance const_contractive {A B : ofeT} (x : A) : Contractive (@const A B x).
......@@ -221,7 +240,7 @@ Ltac f_contractive :=
try match goal with
| |- @dist_later ?A ?n ?x ?y =>
destruct n as [|n]; [done|change (@dist A _ n x y)]
destruct n as [|n]; [exact I|change (@dist A _ n x y)]
try reflexivity.
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