Improved notatio

FiniteSets/fsets/properties_cons_repr.v

@@ -6,38 +6,30 @@ From fsets Require Import operations_cons_repr.
 Section properties.
   Context {A : Type}.
 
-  Lemma append_nl: 
-    forall (x: FSetC A), append ∅ x = x. 
-  Proof.
-    intro. reflexivity.
-  Defined.
+  Definition append_nl : forall (x: FSetC A), ∅ ∪ x = x
+    := fun _ => idpath. 
 
-  Lemma append_nr: 
-    forall (x: FSetC A), append x ∅ = x.
+  Lemma append_nr : forall (x: FSetC A), x ∪ ∅ = x.
   Proof.
     hinduction; try (intros; apply set_path2).
     -  reflexivity.
-    -  intros. apply (ap (fun y => a ;; y) X).
+    -  intros. apply (ap (fun y => a;;y) X).
   Defined.
   
   Lemma append_assoc {H: Funext}:  
-    forall (x y z: FSetC A), append x (append y z) = append (append x y) z.
+    forall (x y z: FSetC A), x ∪ (y ∪ z) = (x ∪ y) ∪ z.
   Proof.
-    intro x; hinduction x; try (intros; apply set_path2).
+    hinduction
+    ; try (intros ; apply path_forall ; intro
+           ; apply path_forall ; intro ;  apply set_path2).
     - reflexivity.
     - intros a x HR y z. 
       specialize (HR y z).
-      apply (ap (fun y => a ;; y) HR). 
-    - intros. apply path_forall. 
-      intro. apply path_forall. 
-      intro. apply set_path2.
-    - intros. apply path_forall.
-      intro. apply path_forall. 
-      intro. apply set_path2.
+      apply (ap (fun y => a;;y) HR). 
   Defined.
   
   Lemma append_singleton: forall (a: A) (x: FSetC A), 
-      a ;; x = append x (a ;; ∅).
+      a ;; x = x ∪ (a ;; ∅).
   Proof.
     intro a. hinduction; try (intros; apply set_path2).
     - reflexivity.
@@ -47,29 +39,26 @@ Section properties.
   Defined.
 
   Lemma append_comm {H: Funext}: 
-    forall (x1 x2: FSetC A), append x1 x2 = append x2 x1.
+    forall (x1 x2: FSetC A), x1 ∪ x2 = x2 ∪ x1.
   Proof.
-    intro x1. 
-    hinduction x1;  try (intros; apply set_path2).
+    hinduction ;  try (intros ; apply path_forall ; intro ; apply set_path2).
     - intros. symmetry. apply append_nr. 
     - intros a x1 HR x2.
       etransitivity.
       apply (ap (fun y => a;;y) (HR x2)).  
-      transitivity  (append (append x2 x1) (a;;∅)).
+      transitivity  ((x2 ∪ x1) ∪ (a;;∅)).
       + apply append_singleton. 
       + etransitivity.
     	* symmetry. apply append_assoc.
-    	* simple refine (ap (fun y => append x2 y) (append_singleton _ _)^).
-    - intros. apply path_forall.
-      intro. apply set_path2.
-    - intros. apply path_forall.
-      intro. apply set_path2.  
+    	* simple refine (ap (x2 ∪) (append_singleton _ _)^).
   Defined.
 
   Lemma singleton_idem: forall (a: A), 
-      append (singleton a) (singleton a) = singleton a.
+      {|a|} ∪ {|a|} = {|a|}.
   Proof.
-    unfold singleton. intro. cbn. apply dupl.
+    unfold singleton.
+    intro.
+    apply dupl.
   Defined.
 
 End properties.