k-finiteness

FiniteSets/variations/k-finite.v

@@ -0,0 +1,46 @@
+Require Import HoTT HitTactics.
+Require Import lattice representations.definition fsets.operations extensionality.
+
+Definition Sub A := A -> hProp.
+
+Section k_finite.
+
+  Context {A : Type}.
+  Context `{Univalence}.
+  
+  Instance subA_set : IsHSet (Sub A).
+  Proof.
+    apply _.
+  Defined.  
+  
+  Definition map : FSet A -> Sub A.
+  Proof.
+    unfold Sub.
+    intros X a.
+    apply (isIn a X).
+  Defined.
+
+  Definition k_finite (B : Sub A) : hProp.
+  Proof.
+    simple refine (@BuildhProp (@sig (FSet A) (fun X => B = map X)) _).
+    apply hprop_allpath.
+    unfold map.
+    intros [X PX] [Y PY].
+    assert (X0 : (fun a : A => a ∈ X) = (fun a : A => a ∈ Y)).
+    {
+      transitivity B.
+      - symmetry ; apply PX.
+      - apply PY.
+    }
+    assert (X = Y).
+    {
+      apply fset_ext.
+      intro a.
+      refine (ap (fun f : A -> hProp => f a) X0).
+    }
+    apply path_sigma_uncurried.
+    exists X1.
+    apply set_path2.
+  Defined.  
+
+End k_finite.