mirror of https://github.com/nmvdw/HITs-Examples
47 lines
959 B
Coq
47 lines
959 B
Coq
Require Import HoTT HitTactics.
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Require Import lattice representations.definition fsets.operations extensionality.
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Definition Sub A := A -> hProp.
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Section k_finite.
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Context {A : Type}.
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Context `{Univalence}.
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Instance subA_set : IsHSet (Sub A).
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Proof.
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apply _.
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Defined.
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Definition map : FSet A -> Sub A.
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Proof.
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unfold Sub.
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intros X a.
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apply (isIn a X).
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Defined.
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Definition k_finite (B : Sub A) : hProp.
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Proof.
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simple refine (@BuildhProp (@sig (FSet A) (fun X => B = map X)) _).
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apply hprop_allpath.
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unfold map.
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intros [X PX] [Y PY].
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assert (X0 : (fun a : A => a ∈ X) = (fun a : A => a ∈ Y)).
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{
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transitivity B.
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- symmetry ; apply PX.
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- apply PY.
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}
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assert (X = Y).
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{
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apply fset_ext.
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intro a.
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refine (ap (fun f : A -> hProp => f a) X0).
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}
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apply path_sigma_uncurried.
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exists X1.
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apply set_path2.
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Defined.
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End k_finite.
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