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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.
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