Clean up trailing whitespaces and an unused definition.

FiniteSets/fsets/properties_cons_repr.v

@@ -7,7 +7,7 @@ Section properties.
   Context {A : Type}.
 
   Definition append_nl : forall (x: FSetC A), ∅ ∪ x = x
-    := fun _ => idpath. 
+    := fun _ => idpath.
 
   Lemma append_nr : forall (x: FSetC A), x ∪ ∅ = x.
   Proof.
@@ -15,20 +15,20 @@ Section properties.
     -  reflexivity.
     -  intros. apply (ap (fun y => a;;y) X).
   Defined.
-  
-  Lemma append_assoc {H: Funext}:  
+
+  Lemma append_assoc {H: Funext}:
     forall (x y z: FSetC A), x ∪ (y ∪ z) = (x ∪ y) ∪ z.
   Proof.
     hinduction
     ; try (intros ; apply path_forall ; intro
            ; apply path_forall ; intro ;  apply set_path2).
     - reflexivity.
-    - intros a x HR y z. 
+    - intros a x HR y z.
       specialize (HR y z).
-      apply (ap (fun y => a;;y) HR). 
+      apply (ap (fun y => a;;y) HR).
   Defined.
-  
-  Lemma append_singleton: forall (a: A) (x: FSetC A), 
+
+  Lemma append_singleton: forall (a: A) (x: FSetC A),
       a ;; x = x ∪ (a ;; ∅).
   Proof.
     intro a. hinduction; try (intros; apply set_path2).
@@ -38,22 +38,22 @@ Section properties.
       + apply (ap (fun y => b ;; y) HR).
   Defined.
 
-  Lemma append_comm {H: Funext}: 
+  Lemma append_comm {H: Funext}:
     forall (x1 x2: FSetC A), x1 ∪ x2 = x2 ∪ x1.
   Proof.
     hinduction ;  try (intros ; apply path_forall ; intro ; apply set_path2).
-    - intros. symmetry. apply append_nr. 
+    - intros. symmetry. apply append_nr.
     - intros a x1 HR x2.
       etransitivity.
-      apply (ap (fun y => a;;y) (HR x2)).  
+      apply (ap (fun y => a;;y) (HR x2)).
       transitivity  ((x2 ∪ x1) ∪ (a;;∅)).
-      + apply append_singleton. 
+      + apply append_singleton.
       + etransitivity.
     	* symmetry. apply append_assoc.
     	* simple refine (ap (x2 ∪) (append_singleton _ _)^).
   Defined.
 
-  Lemma singleton_idem: forall (a: A), 
+  Lemma singleton_idem: forall (a: A),
       {|a|} ∪ {|a|} = {|a|}.
   Proof.
     unfold singleton.