Polymorphic Cumulative Inductive T1 := t1 : T1.
Fail Monomorphic Cumulative Inductive T2 := t2 : T2.

Polymorphic Cumulative Record R1 := { r1 : T1 }.
Fail Monomorphic Cumulative Inductive R2 := {r2 : T1}.

Set Universe Polymorphism.
Set Polymorphic Inductive Cumulativity.
Set Printing Universes.

Inductive List (A: Type) := nil | cons : A -> List A -> List A.

Section ListLift.
  Universe i j.

  Constraint i < j.

  Definition LiftL {A} : List@{i} A -> List@{j} A := fun x => x.

End ListLift.

Lemma LiftL_Lem A (l : List A) : l = LiftL l.
Proof. reflexivity. Qed.

Section ListLower.
  Universe i j.

  Constraint i < j.

  Definition LowerL {A : Type@{i}} : List@{j} A -> List@{i} A := fun x => x.

End ListLower.

Lemma LowerL_Lem@{i j} (A : Type@{j}) (l : List@{i} A) : l = LowerL l.
Proof. reflexivity. Qed.

Inductive Tp := tp : Type -> Tp.

Section TpLift.
  Universe i j.

  Constraint i < j.

  Definition LiftTp : Tp@{i} -> Tp@{j} := fun x => x.

End TpLift.

Lemma LiftC_Lem (t : Tp) : LiftTp t = t.
Proof. reflexivity. Qed.

Section TpLower.
  Universe i j.

  Constraint i < j.

  Fail Definition LowerTp : Tp@{j} -> Tp@{i} := fun x => x.

End TpLower.


Section subtyping_test.
  Universe i j.
  Constraint i < j.

  Inductive TP2 := tp2 : Type@{i} -> Type@{j} -> TP2.

End subtyping_test.

Record A : Type := { a :> Type; }.

Record B (X : A) : Type := { b : X; }.

NonCumulative Inductive NCList (A: Type)
  := ncnil | nccons : A -> NCList A -> NCList A.

Section NCListLift.
  Universe i j.

  Constraint i < j.

  Fail Definition LiftNCL {A} : NCList@{i} A -> NCList@{j} A := fun x => x.

End NCListLift.

Inductive eq@{i} {A : Type@{i}} (x : A) : A -> Type@{i} := eq_refl : eq x x.

Definition funext_type@{a b e} (A : Type@{a}) (B : A -> Type@{b})
  := forall f g : (forall a, B a),
    (forall x, eq@{e} (f x) (g x))
    -> eq@{e} f g.

Section down.
  Universes a b e e'.
  Constraint e' < e.
  Lemma funext_down {A B}
    : @funext_type@{a b e} A B -> @funext_type@{a b e'} A B.
  Proof.
    intros H f g Hfg. exact (H f g Hfg).
  Defined.
End down.