Further simplifications in interface

FiniteSets/implementations/interface.v

@@ -33,14 +33,10 @@ Section interface.
       * refine (tr({|a|};_)).
         apply f_singleton.
       * intros ; apply path_ishprop.
-    - intros Y1 Y2 HY1 HY2.
-      destruct HY1 as [X1' HX1].
-      destruct HY2 as [X2' HX2].
+    - intros Y1 Y2 [X1' ?] [X2' ?].
       simple refine (BuildContr _ _ _).
-      * simple refine (Trunc_rec _ X1') ; intro X1.
-        simple refine (Trunc_rec _ X2') ; intro X2.
-        destruct X1 as [X1 fX1].
-        destruct X2 as [X2 fX2].
+      * simple refine (Trunc_rec _ X1') ; intros [X1 fX1].
+        simple refine (Trunc_rec _ X2') ; intros [X2 fX2].
         refine (tr(X1 ∪ X2;_)).
         rewrite f_union, fX1, fX2.
         reflexivity.
@@ -97,7 +93,6 @@ Section quotient_surjection.
   Proof.
     apply isembedding_isinj_hset.
     unfold isinj.
-    simpl.
     hrecursion ; [ | intros; apply path_ishprop ].
     intro.
     hrecursion ; [ | intros; apply path_ishprop ].
@@ -110,16 +105,14 @@ Section quotient_surjection.
     apply BuildIsSurjection.
     intros Y.
     destruct (H Y).
-    simple refine (Trunc_rec _ center) ; intro X.
-    apply tr.
-    destruct X as [a fa].
-    apply (class_of _ a;fa).
+    simple refine (Trunc_rec _ center) ; intros [a fa].
+    apply (tr(class_of _ a;fa)).
   Defined.
 
   Definition quotient_iso : quotientB <~> B.
   Proof.
     refine (BuildEquiv _ _ quotientB_to_B _).
-    apply isequiv_surj_emb; apply _.
+    apply isequiv_surj_emb ; apply _.
   Defined.
 
   Definition reflect_eq : forall (X Y : A),
@@ -210,8 +203,7 @@ Section quotient_properties.
     - apply (member a).
     - intros X Y HXY.
       reduce T.
-      rewrite HXY.
-      reflexivity.
+      apply (ap _ HXY).
   Defined.
 
   Global Instance View_empty A: hasEmpty (View A).
@@ -232,7 +224,7 @@ Section quotient_properties.
       apply (class_of _ (union a b)).
     - intros x x' Hxx' y y' Hyy' ; simpl.
       apply related_classes_eq.
-      eapply well_defined_union; eauto.
+      eapply well_defined_union ; eauto.
   Defined.
 
   Global Instance View_union A: hasUnion (View A).
@@ -248,7 +240,7 @@ Section quotient_properties.
       apply (class_of _ (filter ϕ X)).
     - intros X X' HXX' ; simpl.
       apply related_classes_eq.
-      eapply well_defined_filter; eauto.
+      eapply well_defined_filter ; eauto.
   Defined.
 
   Hint Unfold Commutative Associative Idempotent NeutralL NeutralR.