chore(library/hott) make funext more general

library/hott/axioms/funext.lean

@@ -7,21 +7,19 @@
 import hott.path hott.equiv
 open path
 
-set_option pp.universes true
-
 -- Funext
 -- ------
 
 -- Define function extensionality as a type class
-inductive funext.{l} [class] : Type.{l+3} :=
+inductive funext [class] : Type  :=
-  mk : (Π (A : Type.{l+1}) (P : A → Type.{l+2}) (f g : Π x, P x), IsEquiv (@apD10 A P f g))
+  mk : (Π (A : Type) (P : A → Type ) (f g : Π x, P x), IsEquiv (@apD10 A P f g))
-         → funext.{l}
+         → funext
 
 namespace funext
 
   context
-    universe l
+    universe variables l k
-    parameters [F : funext.{l}] {A : Type.{l+1}} {P : A → Type.{l+2}} (f g : Π x, P x)
+    parameters [F : funext.{l k}] {A : Type.{l}} {P : A → Type.{k}} (f g : Π x, P x)
 
     protected definition ap [instance] : IsEquiv (@apD10 A P f g) :=
       rec_on F (λ (H : Π A P f g, _), !H)

library/hott/funext_varieties.lean

@@ -16,8 +16,8 @@ open path truncation sigma function
 
 -- Naive funext is the simple assertion that pointwise equal functions are equal.
 -- TODO think about universe levels
-definition naive_funext.{l} :=
+definition naive_funext.{l k} :=
-  Π {A : Type.{l+1}} {P : A → Type.{l+2}} (f g : Πx, P x), (f ∼ g) → f ≈ g
+  Π {A : Type.{l}} {P : A → Type.{k}} (f g : Πx, P x), (f ∼ g) → f ≈ g
 
 -- Weak funext says that a product of contractible types is contractible.
 definition weak_funext.{l} :=
@@ -92,7 +92,7 @@ end
 -- Now the proof is fairly easy; we can just use the same induction principle on both sides.
 universe variable l
 
-theorem weak_funext_implies_funext (wf : weak_funext.{l}) : funext.{l} :=
+theorem weak_funext_implies_funext (wf : weak_funext.{l}) : funext.{l+1 l+2} :=
   funext.mk (λ A B f g,
     let eq_to_f := (λ g' x, f ≈ g') in
     let sim2path := htpy_ind _ f eq_to_f idp in