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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.
+
+Definition LiftL@{k i j|k <= i, k <= j} {A:Type@{k}} : List@{i} A -> List@{j} A := fun x => x.
+
+Lemma LiftL_Lem A (l : List A) : l = LiftL l.
+Proof. reflexivity. Qed.
+
+Inductive Tp := tp : Type -> Tp.
+
+Definition LiftTp@{i j|i <= j} : Tp@{i} -> Tp@{j} := fun x => x.
+
+Fail Definition LowerTp@{i j|j < i} : Tp@{i} -> Tp@{j} := fun x => x.
+
+Record Tp' := { tp' : Tp }.
+
+Definition CTp := Tp.
+(* here we have to reduce a constant to infer the correct subtyping. *)
+Record Tp'' := { tp'' : CTp }.
+
+Definition LiftTp'@{i j|i <= j} : Tp'@{i} -> Tp'@{j} := fun x => x.
+Definition LiftTp''@{i j|i <= j} : Tp''@{i} -> Tp''@{j} := fun x => x.
+
+Lemma LiftC_Lem (t : Tp) : LiftTp t = t.
+Proof. reflexivity. Qed.
+
+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.
+
+Fail Definition LiftNCL@{k i j|k <= i, k <= j} {A:Type@{k}}
+  : NCList@{i} A -> NCList@{j} A := fun x => x.
+
+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.
+
+Record Arrow@{i j} := { arrow : Type@{i} -> Type@{j} }.
+
+Fail Definition arrow_lift@{i i' j j' | i' < i, j < j'}
+  : Arrow@{i j} -> Arrow@{i' j'}
+  := fun x => x.
+
+Definition arrow_lift@{i i' j j' | i' = i, j <= j'}
+  : Arrow@{i j} -> Arrow@{i' j'}
+  := fun x => x.
+
+Inductive Mut1 A :=
+| Base1 : Type -> Mut1 A
+| Node1 : (A -> Mut2 A) -> Mut1 A
+with Mut2 A :=
+     | Base2 : Type -> Mut2 A
+     | Node2 : Mut1 A -> Mut2 A.
+
+(* If we don't reduce T while inferring cumulativity for the
+   constructor we will see a Rel and believe i is irrelevant. *)
+Inductive withparams@{i j} (T:=Type@{i}:Type@{j}) := mkwithparams : T -> withparams.
+
+Definition withparams_co@{i i' j|i < i', i' < j} : withparams@{i j} -> withparams@{i' j}
+  := fun x => x.
+
+Fail Definition withparams_not_irr@{i i' j|i' < i, i' < j} : withparams@{i j} -> withparams@{i' j}
+  := fun x => x.
+
+(** Cumulative constructors *)
+
+
+Record twotys@{u v w} : Type@{w} :=
+  twoconstr { fstty : Type@{u}; sndty : Type@{v} }.
+
+Monomorphic Universes i j k l.
+
+Monomorphic Constraint i < j.
+Monomorphic Constraint j < k.
+Monomorphic Constraint k < l.
+
+Parameter Tyi : Type@{i}.
+
+Definition checkcumul :=
+  eq_refl _ : @eq twotys@{k k l} (twoconstr@{i j k} Tyi Tyi) (twoconstr@{j i k} Tyi Tyi).
+
+(* They can only be compared at the highest type *)
+Fail Definition checkcumul' :=
+  eq_refl _ : @eq twotys@{i k l} (twoconstr@{i j k} Tyi Tyi) (twoconstr@{j i k} Tyi Tyi).
+
+(* An inductive type with an irrelevant universe *)
+Inductive foo@{i} : Type@{i} := mkfoo { }.
+
+Definition bar := foo.
+
+(* The universe on mkfoo is flexible and should be unified with i. *)
+Definition foo1@{i} : foo@{i} := let x := mkfoo in x. (* fast path for conversion *)
+Definition foo2@{i} : bar@{i} := let x := mkfoo in x. (* must reduce *)
+
+(* Rigid universes however should not be unified unnecessarily. *)
+Definition foo3@{i j|} : foo@{i} := let x := mkfoo@{j} in x.
+Definition foo4@{i j|} : bar@{i} := let x := mkfoo@{j} in x.
+
+(* Constructors for an inductive with indices *)
+Module WithIndex.
+  Inductive foo@{i} : (Prop -> Prop) -> Prop := mkfoo: foo (fun x => x).
+
+  Monomorphic Universes i j.
+  Monomorphic Constraint i < j.
+  Definition bar : eq mkfoo@{i} mkfoo@{j} := eq_refl _.
+End WithIndex.