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+(* Inductive paths {A} (x : A) : A -> Type := idpath : paths x x where "x = y" := (@paths _ x y) : type_scope. *)
+(* Goal forall A B : Set, @paths Type A B -> @paths Set A B. *)
+(*   intros A B H. *)
+(* Fail  exact H. *)
+(* Section . *)
+
+Section lift_strict.
+Polymorphic Definition liftlt := 
+  let t := Type@{i} : Type@{k} in
+  fun A : Type@{i} => A : Type@{k}.
+
+Polymorphic Definition liftle := 
+  fun A : Type@{i} => A : Type@{k}.
+End lift_strict.
+
+
+Set Universe Polymorphism.
+
+(* Inductive option (A : Type) : Type := *)
+(*   | None : option A *)
+(*   | Some : A -> option A. *)
+
+Inductive option (A : Type@{i}) : Type@{i} :=
+  | None : option A
+  | Some : A -> option A.
+
+Definition foo' {A : Type@{i}} (o : option@{i} A) : option@{i} A :=
+  o.
+
+Definition foo'' {A : Type@{i}} (o : option@{j} A) : option@{k} A :=
+  o.
+
+
+Definition testm (A : Type@{i}) : Type@{max(i,j)} := A.
+
+(* Inductive prod (A : Type@{i}) (B : Type@{j}) := *)
+(*   | pair : A -> B -> prod A B. *)
+
+(* Definition snd {A : Type@{i}} (B : Type@{j}) (p : prod A B) : B := *)
+(*   match p with *)
+(*     | pair _ _ a b => b *)
+(*   end. *)
+
+(* Definition snd' {A : Type@{i}} (B : Type@{i}) (p : prod A B) : B := *)
+(*   match p with *)
+(*     | pair _ _ a b => b *)
+(*   end. *)
+
+(* Inductive paths {A : Type} : A -> A -> Type := *)
+(* | idpath (a : A) : paths a a. *)
+
+Inductive paths {A : Type@{i}} : A -> A -> Type@{i} :=
+| idpath (a : A) : paths a a.
+
+Definition Funext := 
+  forall (A : Type) (B : A -> Type),
+  forall f g : (forall a, B a), (forall x : A, paths (f x) (g x)) -> paths f g.
+
+Definition paths_lift_closed (A : Type@{i}) (x y : A) :
+  paths x y -> @paths (liftle@{j Type} A) x y.
+Proof. 
+  intros. destruct X. exact (idpath _).
+Defined.
+
+Definition paths_lift (A : Type@{i}) (x y : A) :
+  paths x y -> paths@{j} x y.
+Proof. 
+  intros. destruct X. exact (idpath _).
+Defined.
+
+Definition paths_lift_closed_strict (A : Type@{i}) (x y : A) :
+  paths x y -> @paths (liftlt@{j Type} A) x y.
+Proof. 
+  intros. destruct X. exact (idpath _).
+Defined.
+
+Definition paths_downward_closed_le (A : Type@{i}) (x y : A) :
+  paths@{j} (A:=liftle@{i j} A) x y -> paths@{i} x y.
+Proof. 
+  intros. destruct X. exact (idpath _).
+Defined.
+
+Definition paths_downward_closed_lt (A : Type@{i}) (x y : A) :
+  @paths (liftlt@{j i} A) x y -> paths x y.
+Proof. 
+  intros. destruct X. exact (idpath _).
+Defined.
+
+Definition paths_downward_closed_lt_nolift (A : Type@{i}) (x y : A) :
+  paths@{j} x y -> paths x y.
+Proof. 
+  intros. destruct X. exact (idpath _).
+Defined.
+
+Definition funext_downward_closed (F : Funext@{i' j' k'}) :
+  Funext@{i j k}. 
+Proof.
+  intros A B f g H. red in F.
+  pose (F A B f g (fun x => paths_lift _ _ _ (H x))).
+  apply paths_downward_closed_lt_nolift. apply p.
+Defined.
+