Commit 1c48ea12 authored by Robbert Krebbers's avatar Robbert Krebbers

Cofe on the function space A → B where A : Type and B : cofeT.

parent 677aae22
......@@ -207,6 +207,39 @@ Section fixpoint.
Proof. setoid_rewrite equiv_dist; naive_solver eauto using fixpoint_ne. Qed.
End fixpoint.
(** Function space *)
Definition cofe_fun (A : Type) (B : cofeT) := A B.
Section cofe_fun.
Context {A : Type} {B : cofeT}.
Instance cofe_fun_equiv : Equiv (cofe_fun A B) := λ f g, x, f x g x.
Instance cofe_fun_dist : Dist (cofe_fun A B) := λ n f g, x, f x {n} g x.
Program Definition cofe_fun_chain `(c : chain (cofe_fun A B))
(x : A) : chain B := {| chain_car n := c n x |}.
Next Obligation. intros c x n i ?. by apply (chain_cauchy c). Qed.
Program Instance cofe_fun_compl : Compl (cofe_fun A B) := λ c x,
compl (cofe_fun_chain c x).
Definition cofe_fun_cofe_mixin : CofeMixin (cofe_fun A B).
Proof.
split.
- intros f g; split; [intros Hfg n k; apply equiv_dist, Hfg|].
intros Hfg k; apply equiv_dist=> n; apply Hfg.
- intros n; split.
+ by intros f x.
+ by intros f g ? x.
+ by intros f g h ?? x; trans (g x).
- by intros n f g ? x; apply dist_S.
- intros n c x. apply (conv_compl n (cofe_fun_chain c x)).
Qed.
Canonical Structure cofe_funC := CofeT (cofe_fun A B) cofe_fun_cofe_mixin.
End cofe_fun.
Arguments cofe_funC : clear implicits.
Notation "A -c> B" :=
(cofe_funC A B) (at level 99, B at level 200, right associativity).
Instance cofe_fun_inhabited {A} {B : cofeT} `{Inhabited B} :
Inhabited (A -c> B) := populate (λ _, inhabitant).
(** Non-expansive function space *)
Record cofe_mor (A B : cofeT) : Type := CofeMor {
cofe_mor_car :> A B;
......
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