Added proof of inclusion/exclusion.

FiniteSets/kuratowski/length.v

@@ -1,5 +1,5 @@
-Require Import HoTT HitTactics prelude.
+Require Import HoTT HitTactics prelude interfaces.lattice_interface interfaces.lattice_examples.
-Require Import kuratowski.operations kuratowski.properties kuratowski.kuratowski_sets.
+Require Import kuratowski.operations kuratowski.properties kuratowski.kuratowski_sets isomorphism.
 
 Section Length.
   Context {A : Type} `{MerelyDecidablePaths A} `{Univalence}.
@@ -17,7 +17,6 @@ Section Length.
       simpl.
       simplify_isIn_d.
       destruct (m_dec_path a b) as [Hab | Hab].
-(*      destruct (dec (Trunc (-1) (a = b))) as [Hab | Hab]. *)
       + strip_truncations.
         rewrite Hab. simplify_isIn_d. reflexivity.
       + rewrite ?singleton_isIn_d_false; auto.
@@ -27,4 +26,133 @@ Section Length.
         ++ intros p.
            apply (Hab (tr p)).          
   Defined.
+
+  Open Scope nat.
+  (** Specification for length. *)
+  Definition length_empty : length ∅ = 0 := idpath.
+  
+  Definition length_singleton a : length {|a|} = 1 := idpath.
+
+  Lemma length_compute (a : A) (X : FSet A) :
+    length ({|a|} ∪ X) = if (a ∈_d X) then length X else S(length X).
+  Proof.
+    unfold length.
+    rewrite FSet_cons_beta_cons.
+    reflexivity.
+  Defined.
+  
+  Definition length_add (a : A) (X : FSet A) (p : a ∈_d X = false)
+             : length ({|a|} ∪ X) = 1 + (length X).
+  Proof.
+    rewrite length_compute.
+    destruct (a ∈_d X).
+    - contradiction (true_ne_false).
+    - reflexivity.
+  Defined.
+
+  Definition disjoint X Y := X ∩ Y = ∅.
+
+  Lemma disjoint_difference X Y : disjoint X (difference Y X).
+  Proof.
+    apply ext.
+    intros a.
+    rewrite intersection_isIn_d, empty_isIn_d, difference_isIn_d.
+    destruct (a ∈_d X), (a ∈_d Y) ; try reflexivity.
+  Defined.
+
+  Lemma disjoint_sub a X Y (H1 : disjoint ({|a|} ∪ X) Y) : disjoint X Y.
+  Proof.
+    unfold disjoint in *.
+    apply ext.
+    intros b.
+    simplify_isIn_d.
+    rewrite empty_isIn_d.
+    pose (ap (fun Z => b ∈_d Z) H1) as p.
+    simpl in p.
+    rewrite intersection_isIn_d, empty_isIn_d, union_isIn_d in p.
+    destruct (b ∈_d X), (b ∈_d Y) ; try reflexivity.
+    - destruct (b ∈_d {|a|}) ; simpl in * ; try (contradiction true_ne_false).
+  Defined.  
+
+  Definition length_disjoint (X Y : FSet A) :
+    forall (HXY : disjoint X Y),
+      length (X ∪ Y) = (length X) + (length Y).
+  Proof.
+    simple refine (FSet_cons_ind (fun Z => _) _ _ _ _ _ X)
+    ; try (intros ; apply path_ishprop) ; simpl.
+    - intros.
+      rewrite nl.
+      reflexivity.
+    - intros a X1 HX1 HX1Y.
+      rewrite <- assoc.
+      rewrite ?length_compute.
+      rewrite ?union_isIn_d in *.
+      unfold disjoint in HX1Y.
+      pose (ap (fun Z => a ∈_d Z) HX1Y) as p.
+      simpl in p.
+      rewrite intersection_isIn_d, union_isIn_d, singleton_isIn_d_aa, empty_isIn_d in p.
+      assert (orb (a ∈_d X1) (a ∈_d Y) = a ∈_d X1) as HaY.
+      { destruct (a ∈_d X1), (a ∈_d Y) ; try reflexivity.
+        contradiction true_ne_false.
+      }
+      rewrite ?HaY, HX1.
+      destruct (a ∈_d X1).
+      * reflexivity.
+      * reflexivity.
+      * apply (disjoint_sub a X1 Y HX1Y).
+  Defined.
+
+  Lemma set_as_difference X Y : X = (difference X Y) ∪ (X ∩ Y).
+  Proof.
+    toBool.
+    generalize (a ∈_d X), (a ∈_d Y).
+    intros b c ; destruct b, c ; reflexivity.
+  Defined.
+  
+  Lemma length_single_disjoint (X Y : FSet A) :
+    length X = length (difference X Y) + length (X ∩ Y).
+  Proof.
+    refine (ap length (set_as_difference X Y) @ _).
+    apply length_disjoint.
+    apply ext.
+    intros a.
+    rewrite ?intersection_isIn_d, empty_isIn_d, difference_isIn_d.
+    destruct (a ∈_d X), (a ∈_d Y) ; try reflexivity.
+  Defined.
+
+  Lemma union_to_disjoint X Y : X ∪ Y = X ∪ (difference Y X).
+  Proof.
+    toBool.
+    generalize (a ∈_d X), (a ∈_d Y).
+    intros b c ; destruct b, c ; reflexivity.
+  Defined.
+  
+  Lemma length_union_1 (X Y : FSet A) :
+    length (X ∪ Y) = length X + length (difference Y X).
+  Proof.
+    refine (ap length (union_to_disjoint X Y) @ _).
+    apply length_disjoint.
+    apply ext.
+    intros a.
+    rewrite ?intersection_isIn_d, empty_isIn_d, difference_isIn_d.
+    destruct (a ∈_d X), (a ∈_d Y) ; try reflexivity.
+  Defined.
+
+  Lemma plus_assoc n m k : n + (m + k) = (n + m) + k.
+  Proof.
+    induction n ; simpl.
+    - reflexivity.
+    - rewrite IHn.
+      reflexivity.
+  Defined.
+  
+  Lemma inclusion_exclusion (X Y : FSet A) :
+    length (X ∪ Y) + length (Y ∩ X) = length X + length Y.
+  Proof.
+    rewrite length_union_1.
+    rewrite (length_single_disjoint Y X).
+    rewrite plus_assoc.
+    reflexivity.
+  Defined.
+  
 End Length.

FiniteSets/kuratowski/operations.v

@@ -203,6 +203,8 @@ Section operations_decidable.
 
   Global Instance fset_intersection : hasIntersection (FSet A)
     := fun X Y => {|X & (fun a => a ∈_d Y)|}.
+
+  Definition difference := fun X Y => {|X & (fun a => negb a ∈_d Y)|}.
 End operations_decidable.
 
 Section FSet_cons_rec.

FiniteSets/kuratowski/properties.v

@@ -315,7 +315,13 @@ Section properties_membership_decidable.
   Defined.
 
   Lemma intersection_isIn_d (X Y: FSet A) (a : A) :
-    a ∈_d (intersection X Y) = andb (a ∈_d X) (a ∈_d Y).
+    a ∈_d (X ∩ Y) = andb (a ∈_d X) (a ∈_d Y).
+  Proof.
+    apply comprehension_isIn_d.
+  Defined.
+
+  Lemma difference_isIn_d (X Y: FSet A) (a : A) :
+    a ∈_d (difference X Y) = andb (a ∈_d X) (negb (a ∈_d Y)).
   Proof.
     apply comprehension_isIn_d.
   Defined.