trailing white spaces

FiniteSets/operations.v

@@ -19,7 +19,6 @@ hrecursion.
 - intros a'. compute. destruct (A_deceq a a'); reflexivity.
 Defined.
 
-Infix "∈" := isIn (at level 9, right associativity).
 
 Definition comprehension : 
   (A -> Bool) -> FSet A -> FSet A.
@@ -54,15 +53,16 @@ Proof.
 intros X Y.
 hrecursion X. 
 - exact true.
-- exact (fun a => (a ∈ Y)).
+- exact (fun a => (isIn a Y)).
 - exact andb.
 - intros. compute. destruct x; reflexivity.
 - intros x y; compute; destruct x, y; reflexivity. 
 - intros x; compute; destruct x; reflexivity.
 - intros x; compute; destruct x; reflexivity.
-- intros x; cbn; destruct (x ∈ Y); reflexivity.
+- intros x; cbn; destruct (isIn x Y); reflexivity.
 Defined.
 
-Notation "⊆" := subset.
-
 End operations.
+
+Infix "∈" := isIn (at level 9, right associativity).
+Infix  "⊆" := subset (at level 10, right associativity).

FiniteSets/properties.v

@@ -9,7 +9,7 @@ Context {A_deceq : DecidablePaths A}.
 (** union properties *)
 Theorem union_idem : forall x: FSet A, U x x = x.
 Proof.
-hinduction; 
+hinduction;
 try (intros ; apply set_path2) ; cbn.
 - apply nl.
 - apply idem.
@@ -24,24 +24,24 @@ try (intros ; apply set_path2) ; cbn.
   reflexivity.
 Defined.
 
-  
+
 (** isIn properties *)
 Lemma isIn_singleton_eq (a b: A) : isIn a  (L b) = true -> a = b.
-Proof. unfold isIn. simpl. 
+Proof. unfold isIn. simpl.
 destruct (dec (a = b)). intro. apply p.
-intro X. 
+intro X.
 contradiction (false_ne_true X).
 Defined.
 
 Lemma isIn_empty_false (a: A) : isIn a E = true -> Empty.
-Proof. 
+Proof.
 cbv. intro X.
-contradiction (false_ne_true X). 
+contradiction (false_ne_true X).
 Defined.
 
-Lemma isIn_union (a: A) (X Y: FSet A) : 
+Lemma isIn_union (a: A) (X Y: FSet A) :
 	    isIn a (U X Y) = (isIn a X || isIn a Y)%Bool.
-Proof. reflexivity. Qed. 
+Proof. reflexivity. Qed.
 
 (** comprehension properties *)
 Lemma comprehension_false Y : comprehension (fun a => isIn a E) Y = E.
@@ -58,20 +58,20 @@ hrecursion Y; try (intros; apply set_path2).
 Defined.
 
 Theorem comprehension_or : forall ϕ ψ (x: FSet A),
-    comprehension (fun a => orb (ϕ a) (ψ a)) x = U (comprehension ϕ x) 
+    comprehension (fun a => orb (ϕ a) (ψ a)) x = U (comprehension ϕ x)
     (comprehension ψ x).
 Proof.
 intros ϕ ψ.
-hinduction; try (intros; apply set_path2). 
+hinduction; try (intros; apply set_path2).
 - cbn. symmetry ; apply nl.
 - cbn. intros.
   destruct (ϕ a) ; destruct (ψ a) ; symmetry.
   * apply idem.
-  * apply nr. 
+  * apply nr.
   * apply nl.
   * apply nl.
 - simpl. intros x y P Q.
-  cbn. 
+  cbn.
   rewrite P.
   rewrite Q.
   rewrite <- assoc.
@@ -105,8 +105,8 @@ Defined.
 
 (** intersection properties *)
 Lemma intersection_0l: forall X: FSet A, intersection E X = E.
-Proof.       
+Proof.
-hinduction; 
+hinduction;
 try (intros ; apply set_path2).
 - reflexivity.
 - intro a.
@@ -144,7 +144,7 @@ hinduction; try (intros ; apply set_path2).
   destruct (isIn a x) ; destruct (isIn a y).
   * apply idem.
   * apply nr.
-  * apply nl. 
+  * apply nl.
   * apply nl.
 Defined.
 
@@ -163,7 +163,7 @@ hrecursion X;  try (intros; apply set_path2).
     * destruct (dec (b = a)) as [pb|]; [|reflexivity].
       by contradiction npa.
   + cbn -[isIn]. intros Y1 Y2 IH1 IH2.
-    rewrite IH1. 
+    rewrite IH1.
     rewrite IH2.
     symmetry.
     apply (comprehension_or (fun a => isIn a Y1) (fun a => isIn a Y2) (L a)).
@@ -171,7 +171,7 @@ hrecursion X;  try (intros; apply set_path2).
   cbn.
   unfold intersection in *.
   rewrite <- IH1.
-  rewrite <- IH2. 
+  rewrite <- IH2.
   apply comprehension_or.
 Defined.
 
@@ -196,7 +196,7 @@ hinduction; try (intros; apply set_path2).
 Defined.
 
 (** assorted lattice laws *)
-Lemma distributive_La (z : FSet A) (a : A) : forall Y : FSet A, 
+Lemma distributive_La (z : FSet A) (a : A) : forall Y : FSet A,
        intersection (U (L a) z) Y = U (intersection (L a) Y) (intersection z Y).
 Proof.
 hinduction; try (intros ; apply set_path2) ; cbn.
@@ -266,7 +266,7 @@ hinduction x; try (intros ; apply set_path2) ; cbn.
   cbn.
   rewrite P.
   rewrite Q.
-  destruct (isIn a X1) ; destruct (isIn a X2) ; destruct (isIn a y) ; 
+  destruct (isIn a X1) ; destruct (isIn a X2) ; destruct (isIn a y) ;
   reflexivity.
 Defined.
 
@@ -292,7 +292,7 @@ hinduction X; try (intros ; apply set_path2).
     + reflexivity.
     + reflexivity.
   * rewrite intersection_0l.
-    reflexivity.      
+    reflexivity.
 - unfold intersection. cbn.
   intros X1 X2 P Q.
   rewrite comprehension_or.
@@ -324,7 +324,7 @@ hinduction; try (intros ; apply set_path2).
   reflexivity.
 Defined.
 
-  
+
 Theorem distributive_U_int (X1 X2 Y : FSet A) :
     U (intersection X1 X2) Y = intersection (U X1 Y) (U X2 Y).
 Proof.
@@ -347,9 +347,9 @@ hinduction X1; try (intros ; apply set_path2) ; cbn.
   cbn.
   intros Z1 Z2 P Q.
   rewrite comprehension_or.
-  assert (U (U (comprehension (fun a : A => isIn a Z1) X2) 
+  assert (U (U (comprehension (fun a : A => isIn a Z1) X2)
   	(comprehension (fun a : A => isIn a Z2) X2))
-    Y = U (U (comprehension (fun a : A => isIn a Z1) X2) 
+    Y = U (U (comprehension (fun a : A => isIn a Z1) X2)
   (comprehension (fun a : A => isIn a Z2) X2))
     (U Y Y)).
     rewrite (union_idem Y).
@@ -358,7 +358,7 @@ hinduction X1; try (intros ; apply set_path2) ; cbn.
   rewrite <- assoc.
   rewrite (assoc (comprehension (fun a : A => isIn a Z2) X2)).
   rewrite Q.
-  rewrite 
+  rewrite
   (comm (U (comprehension (fun a : A => (isIn a Z2 || isIn a Y)%Bool) X2)
            (comprehension (fun a : A => (isIn a Z2 || isIn a Y)%Bool) Y)) Y).
   rewrite assoc.
@@ -369,11 +369,11 @@ hinduction X1; try (intros ; apply set_path2) ; cbn.
   rewrite <- assoc.
   rewrite assoc.
   enough (C : (U (comprehension (fun a : A => (isIn a Z1 || isIn a Y)%Bool) X2)
-             (comprehension (fun a : A => (isIn a Z2 || isIn a Y)%Bool) X2)) 
+             (comprehension (fun a : A => (isIn a Z2 || isIn a Y)%Bool) X2))
  = (comprehension (fun a : A => (isIn a Z1 || isIn a Z2 || isIn a Y)%Bool) X2)).
-  rewrite C. 
+  rewrite C.
   enough (D :  (U (comprehension (fun a : A => (isIn a Z1 || isIn a Y)%Bool) Y)
-                  (comprehension (fun a : A => (isIn a Z2 || isIn a Y)%Bool) Y)) 
+                  (comprehension (fun a : A => (isIn a Z2 || isIn a Y)%Bool) Y))
  = (comprehension (fun a : A => (isIn a Z1 || isIn a Z2 || isIn a Y)%Bool) Y)).
   rewrite D.
   reflexivity.
@@ -438,12 +438,12 @@ Admitted.
 
 
 (* Properties about subset relation. *)
-Lemma subsect_intersection `{Funext} (X Y : FSet A) : 
+Lemma subsect_intersection `{Funext} (X Y : FSet A) :
-			subset X Y = true -> U X Y = Y.
+			X ⊆ Y = true -> U X Y = Y.
 Proof.
 hinduction X; try (intros; apply path_forall; intro; apply set_path2).
 - intros. apply nl.
-- intros a. hinduction Y; 
+- intros a. hinduction Y;
 	try (intros; apply path_forall; intro; apply set_path2).
 	(*intros. apply equiv_hprop_allpath.*)
 	+ intro. cbn.  contradiction (false_ne_true).
@@ -460,10 +460,10 @@ hinduction X; try (intros; apply path_forall; intro; apply set_path2).
 	specialize (IH1 idpath).
 	rewrite assoc. rewrite IH1. reflexivity.
 	specialize (IH2 idpath).
-	rewrite assoc. rewrite (comm (L a)). rewrite <- assoc. rewrite IH2. 
+	rewrite assoc. rewrite (comm (L a)). rewrite <- assoc. rewrite IH2.
 	reflexivity.
-	cbn in Ho. contradiction (false_ne_true). 
+	cbn in Ho. contradiction (false_ne_true).
-- intros X1 X2 IH1 IH2 G. 
+- intros X1 X2 IH1 IH2 G.
 	destruct (subset X1 Y);
 	destruct (subset X2 Y).
 	specialize (IH1 idpath).
@@ -479,4 +479,6 @@ hinduction X; try (intros; apply path_forall; intro; apply set_path2).
 	rewrite <- assoc. rewrite IH2. rewrite IH1. reflexivity.
 Defined.
 
-End properties. 
+Theorem
+
+End properties.