Simplified independence proof

FiniteSets/Sub.v

@@ -43,18 +43,11 @@ Section isIn.
   Context `{Univalence}.
   Context {HS : closedSingleton C} {HIn : forall X, C X -> forall a, Decidable (X a)}.
 
-  Theorem decidable_A_isIn : forall a b : A, Decidable (Trunc (-1) (b = a)).
+  Theorem decidable_A_isIn (a b : A) : Decidable (Trunc (-1) (b = a)).
   Proof.
-    intros.
+    destruct (HIn {|a|} (HS a) b).
-    unfold Decidable, closedSingleton in *.
+    - apply (inl t).
-    pose (HIn {|a|} (HS a) b).
+    - refine (inr(fun p => _)).
-    destruct s.
-    - unfold singleton in t.
-      left.
-      apply t.
-    - right.
-      intro p.
-      unfold singleton in n.
       strip_truncations.
       contradiction (n (tr p)).
   Defined.
@@ -78,24 +71,17 @@ Section intersect.
     {HI : closedIntersection C} {HE : closedEmpty C}
     {HS : closedSingleton C} {HDE : hasDecidableEmpty C}.
 
-  Theorem decidable_A_intersect : forall a b : A, Decidable (Trunc (-1) (b = a)).
+  Theorem decidable_A_intersect (a b : A) : Decidable (Trunc (-1) (b = a)).
   Proof.
-    intros.
+    unfold Decidable.
-    unfold Decidable, closedEmpty, closedIntersection, closedSingleton, hasDecidableEmpty in *.
     pose (HI {|a|} {|b|} (HS a) (HS b)) as IntAB.
     pose (HDE ({|a|} ∪ {|b|}) IntAB) as IntE.
-    refine (@Trunc_rec _ _ _  _ _ IntE) ; intros [p | p] ; unfold min_fun, singleton in p.
+    refine (Trunc_rec _ IntE) ; intros [p | p].
-    - right.
+    - refine (inr(fun q => _)).
-      intro q.
       strip_truncations.
-      rewrite q in p.
+      refine (transport (fun Z => a ∈ Z) p (tr idpath, tr q^)).
-      enough (a ∈ ∅) as X.
-      { apply X. }
-      rewrite <- p.
-      cbn.
-      split ; apply (tr idpath).
     - strip_truncations.
-      destruct p as [a0 [t1 t2]].
+      destruct p as [? [t1 t2]].
       strip_truncations.
       apply (inl (tr (t2^ @ t1))).
   Defined.