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+Set Universe Polymorphism.
+Definition Lift
+: $(let U1 := constr:(Type) in
+    let U0 := constr:(Type : U1) in
+    exact (U0 -> U1))$
+  := fun T => T.
+
+Fail Check nat:Prop. (* The command has indeed failed with message:
+=> Error:
+The term "nat" has type "Set" while it is expected to have type "Prop". *)
+Set Printing All.
+Set Printing Universes.
+Fail Check Lift nat : Prop. (* Lift (* Top.8 Top.9 Top.10 *) nat:Prop
+     : Prop
+(* Top.10
+   Top.9
+   Top.8 |= Top.10 < Top.9
+             Top.9 < Top.8
+             Top.9 <= Prop
+              *)
+ *)
+Fail Eval compute in Lift nat : Prop.
+(*      = nat
+     : Prop *)
+
+Section Hurkens.
+
+  Monomorphic Definition Type2 := Type.
+  Monomorphic Definition Type1 := Type : Type2.
+
+  (** Assumption of a retract from Type into Prop *)
+
+  Variable down : Type1 -> Prop.
+  Variable up : Prop -> Type1.
+
+  Hypothesis back : forall A, up (down A) -> A.
+
+  Hypothesis forth : forall A, A -> up (down A).
+
+  Hypothesis backforth : forall (A:Type) (P:A->Type) (a:A),
+                           P (back A (forth A a)) -> P a.
+
+  Hypothesis backforth_r : forall (A:Type) (P:A->Type) (a:A),
+                             P a -> P (back A (forth A a)).
+
+  (** Proof *)
+
+  Definition V : Type1 := forall A:Prop, ((up A -> Prop) -> up A -> Prop) -> up A -> Prop.
+  Definition U : Type1 := V -> Prop.
+
+  Definition sb (z:V) : V := fun A r a => r (z A r) a.
+  Definition le (i:U -> Prop) (x:U) : Prop := x (fun A r a => i (fun v => sb v A r a)).
+  Definition le' (i:up (down U) -> Prop) (x:up (down U)) : Prop := le (fun a:U => i (forth _ a)) (back _ x).
+  Definition induct (i:U -> Prop) : Type1 := forall x:U, up (le i x) -> up (i x).
+  Definition WF : U := fun z => down (induct (fun a => z (down U) le' (forth _ a))).
+  Definition I (x:U) : Prop :=
+    (forall i:U -> Prop, up (le i x) -> up (i (fun v => sb v (down U) le' (forth _ x)))) -> False.
+
+  Lemma Omega : forall i:U -> Prop, induct i -> up (i WF).
+  Proof.
+    intros i y.
+    apply y.
+    unfold le, WF, induct.
+    apply forth.
+    intros x H0.
+    apply y.
+    unfold sb, le', le.
+    compute.
+    apply backforth_r.
+    exact H0.
+  Qed.
+
+  Lemma lemma1 : induct (fun u => down (I u)).
+  Proof.
+    unfold induct.
+    intros x p.
+    apply forth.
+    intro q.
+    generalize (q (fun u => down (I u)) p).
+    intro r.
+    apply back in r.
+    apply r.
+    intros i j.
+    unfold le, sb, le', le in j |-.
+                                apply backforth in j.
+    specialize q with (i := fun y => i (fun v:V => sb v (down U) le' (forth _ y))).
+    apply q.
+    exact j.
+  Qed.
+
+  Lemma lemma2 : (forall i:U -> Prop, induct i -> up (i WF)) -> False.
+  Proof.
+    intro x.
+    generalize (x (fun u => down (I u)) lemma1).
+    intro r; apply back in r.
+    apply r.
+    intros i H0.
+    apply (x (fun y => i (fun v => sb v (down U) le' (forth _ y)))).
+    unfold le, WF in H0.
+    apply back in H0.
+    exact H0.
+  Qed.
+
+  Theorem paradox : False.
+  Proof.
+    exact (lemma2 Omega).
+  Qed.
+
+End Hurkens.
+
+Definition informative (x : bool) :=
+  match x with
+    | true => Type
+    | false => Prop
+  end.
+
+Definition depsort (T : Type) (x : bool) : informative x :=
+  match x with
+    | true => T
+    | false => True
+  end.
+
+(** This definition should fail *)
+Definition Box (T : Type1) : Prop := Lift T.
+
+Definition prop {T : Type1} (t : Box T) : T := t.
+Definition wrap {T : Type1} (t : T) : Box T := t.
+
+Definition down (x : Type1) : Prop := Box x.
+Definition up (x : Prop) : Type1 := x.
+
+Fail Definition back A : up (down A) -> A := @prop A.
+
+Fail Definition forth (A : Type1) : A -> up (down A) := @wrap A.
+
+Fail Definition backforth (A:Type1) (P:A->Type) (a:A) :
+      P (back A (forth A a)) -> P a := fun H => H.
+
+Fail Definition backforth_r (A:Type1) (P:A->Type) (a:A) :
+      P a -> P (back A (forth A a)) := fun H => H.
+
+Theorem pandora : False.
+  Fail apply (paradox down up back forth backforth backforth_r).
+  admit.
+Qed.
+
+Print Assumptions pandora.