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