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Set Universe Polymorphism.
Set 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; }.
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