mirror of https://github.com/nmvdw/HITs-Examples
43 lines
1.2 KiB
Coq
43 lines
1.2 KiB
Coq
Require Import HoTT.
|
|
Require Import FSets.
|
|
|
|
Section structure.
|
|
Variable (T : Type -> Type).
|
|
|
|
Class hasMembership : Type :=
|
|
member : forall A : Type, A -> T A -> hProp.
|
|
|
|
Class hasEmpty : Type :=
|
|
empty : forall A, T A.
|
|
|
|
Class hasSingleton : Type :=
|
|
singleton : forall A, A -> T A.
|
|
|
|
Class hasUnion : Type :=
|
|
union : forall A, T A -> T A -> T A.
|
|
|
|
Class hasComprehension : Type :=
|
|
filter : forall A, (A -> Bool) -> T A -> T A.
|
|
End structure.
|
|
|
|
Arguments member {_} {_} {_} _ _.
|
|
Arguments empty {_} {_} {_}.
|
|
Arguments singleton {_} {_} {_} _.
|
|
Arguments union {_} {_} {_} _ _.
|
|
Arguments filter {_} {_} {_} _ _.
|
|
|
|
Section interface.
|
|
Context `{Univalence}.
|
|
Variable (T : Type -> Type)
|
|
(f : forall A, T A -> FSet A).
|
|
Context `{hasMembership T, hasEmpty T, hasSingleton T, hasUnion T, hasComprehension T}.
|
|
|
|
Class sets :=
|
|
{
|
|
f_empty : forall A, f A empty = E ;
|
|
f_singleton : forall A a, f A (singleton a) = L a;
|
|
f_union : forall A X Y, f A (union X Y) = U (f A X) (f A Y);
|
|
f_filter : forall A ϕ X, f A (filter ϕ X) = comprehension ϕ (f A X);
|
|
f_member : forall A a X, member a X = isIn a (f A X)
|
|
}.
|
|
End interface. |