fix(hott): minor changes to merge ulrik and my commits

hott/types/equiv.hlean

@@ -110,17 +110,13 @@ namespace is_equiv
   variables {A : Type} {P Q : A → Type}
   variable (f : Πa, P a → Q a)
 
-  definition is_fiberwise_equiv [class] := Πa, is_equiv (f a)
+  definition is_fiberwise_equiv [reducible] := Πa, is_equiv (f a)
 
-  definition is_equiv_total_of_is_fiberwise_equiv [H : is_fiberwise_equiv f] : is_equiv (sigma_functor id f) :=
-  begin
-    apply is_equiv_of_is_contr_fun, apply sigma.rec, intros a q,
-    apply @is_contr_equiv_closed _ _ (fiber_total_equiv f q)⁻¹ᵉ,
-    apply @is_contr_fun_of_is_equiv _ _ (f a),
-    exact H a
-  end
+  definition is_equiv_total_of_is_fiberwise_equiv [H : is_fiberwise_equiv f] : is_equiv (total f) :=
+  is_equiv_sigma_functor id f
 
-  definition is_fiberwise_equiv_of_is_equiv_total [H : is_equiv (sigma_functor id f)] : is_fiberwise_equiv f :=
+  definition is_fiberwise_equiv_of_is_equiv_total [H : is_equiv (sigma_functor id f)]
+    : is_fiberwise_equiv f :=
   begin
     intro a,
     apply is_equiv_of_is_contr_fun, intro q,

hott/types/fiber.hlean

@@ -97,7 +97,7 @@ namespace fiber
   calc
     fiber (sigma_functor id f) ⟨a , q⟩
           ≃ Σ(w : Σx, P x), ⟨w.1 , f w.1 w.2 ⟩ = ⟨a , q⟩
-            : sigma_char
+            : fiber.sigma_char
       ... ≃ Σ(x : A), Σ(p : P x), ⟨x , f x p⟩ = ⟨a , q⟩
             : sigma_assoc_equiv
       ... ≃ Σ(x : A), Σ(p : P x), Σ(H : x = a), f x p =[H] q
@@ -126,6 +126,6 @@ namespace fiber
               apply pathover_idp
             end
       ... ≃ fiber (f a) q
-            : sigma_char
+            : fiber.sigma_char
 
 end fiber

hott/types/sigma.hlean

@@ -199,6 +199,10 @@ namespace sigma
   definition sigma_functor [unfold 7] (u : Σa, B a) : Σa', B' a' :=
   ⟨f u.1, g u.1 u.2⟩
 
+  definition total [reducible] [unfold 5] {B' : A → Type} (g : Πa, B a → B' a) (u : Σa, B a)
+    : Σa', B' a' :=
+  sigma_functor id g u
+
   /- Equivalences -/
   definition is_equiv_sigma_functor [H1 : is_equiv f] [H2 : Π a, is_equiv (g a)]
       : is_equiv (sigma_functor f g) :=