Fixes

FiniteSets/subobjects/sub.v

@@ -0,0 +1,78 @@
+Require Import HoTT.
+Require Import set_names lattice_interface lattice_examples prelude.
+
+Section subobjects.
+  Variable A : Type.
+
+  Definition Sub := A -> hProp.
+
+  Global Instance sub_empty : hasEmpty Sub := fun _ => False_hp.
+  Global Instance sub_union : hasUnion Sub := max_fun.
+  Global Instance sub_intersection : hasIntersection Sub := min_fun.
+  Global Instance sub_singleton : hasSingleton Sub A
+    := fun a b => BuildhProp (Trunc (-1) (b = a)).
+  Global Instance sub_membership : hasMembership Sub A := fun a X => X a.
+  Global Instance sub_comprehension : hasComprehension Sub A
+    := fun ϕ X a => BuildhProp (X a * (ϕ a = true)).
+  Global Instance sub_subset `{Univalence} : hasSubset Sub
+    := fun X Y => BuildhProp (forall a, X a -> Y a).
+
+End subobjects.
+
+Section sub_classes.
+  Context {A : Type}.
+  Variable C : (A -> hProp) -> hProp.
+  Context `{Univalence}.
+
+  Instance subobject_lattice : Lattice (Sub A).
+  Proof.
+    apply _.
+  Defined.  
+
+  Definition closedUnion := forall X Y, C X -> C Y -> C (X ∪ Y).
+  Definition closedIntersection := forall X Y, C X -> C Y -> C (X ∩ Y).
+  Definition closedEmpty := C ∅.
+  Definition closedSingleton := forall a, C {|a|}.
+  Definition hasDecidableEmpty := forall X, C X -> hor (X = ∅) (hexists (fun a => a ∈ X)).
+End sub_classes.
+
+Section isIn.
+  Variable A : Type.
+  Variable C : (A -> hProp) -> hProp.
+
+  Context `{Univalence}.
+  Context {HS : closedSingleton C} {HIn : forall X, C X -> forall a, Decidable (X a)}.
+
+  Theorem decidable_A_isIn (a b : A) : Decidable (Trunc (-1) (b = a)).
+  Proof.
+    destruct (HIn {|a|} (HS a) b).
+    - apply (inl t).
+    - refine (inr(fun p => _)).
+      strip_truncations.
+      contradiction (n (tr p)).
+  Defined.
+
+End isIn.
+
+Section intersect.
+  Variable A : Type.
+  Variable C : (Sub A) -> hProp.
+  Context `{Univalence}
+    {HI : closedIntersection C} {HE : closedEmpty C}
+    {HS : closedSingleton C} {HDE : hasDecidableEmpty C}.
+
+  Theorem decidable_A_intersect (a b : A) : Decidable (Trunc (-1) (b = a)).
+  Proof.
+    unfold Decidable.
+    pose (HI {|a|} {|b|} (HS a) (HS b)) as IntAB.
+    pose (HDE ({|a|} ∪ {|b|}) IntAB) as IntE.
+    refine (Trunc_rec _ IntE) ; intros [p | p].
+    - refine (inr(fun q => _)).
+      strip_truncations.
+      refine (transport (fun Z => a ∈ Z) p (tr idpath, tr q^)).
+    - strip_truncations.
+      destruct p as [? [t1 t2]].
+      strip_truncations.
+      apply (inl (tr (t2^ @ t1))).
+  Defined.
+End intersect.