A simple case of universe inconsistency

Question

I can define the following inductive type:

Inductive T : Type -> Type :=
| c1 : forall (A : Type), A -> T A
| c2 : T unit.

But then the command Check (c1 (T nat)) fails with the message: The term T nat has type Type@{max(Set, Top.3+1)} while it is expected to have type Type@{Top.3} (universe inconsistency).

How can I tweak the above inductive definition so that c1 (T nat) does not cause a universe inconsistency, and without setting universe polymorphism on?

The following works, but I would prefer a solution without adding equality:

Inductive T (A : Type) : Type :=
| c1 : A -> T A
| c2' : A = unit -> T A.

Definition c2 : T unit := c2' unit eq_refl.

Check (c1 (T nat)).
(*
c1 (T nat)
     : T nat -> T (T nat)
*)

Answer 1

Let me first answer the question of why we get the universe inconsistency in the first place.

Universe inconsistencies are the errors that Coq reports to avoid proofs of False via Russell's paradox, which results from considering the set of all sets which do not contain themselves.

There's a variant which is more convenient to formalize in type theory called Hurken's Paradox; see Coq.Logic.Hurkens for more details. There is a specialization of Hurken's paradox which proves that no type can retract to a smaller type. That is, given

U := Type@{u}
A : U
down : U -> A
up : A -> U
up_down : forall (X:U), up (down X) = X

we can prove False.

---

This is almost exactly the setup of your Inductive type. Annotating your type with universes, you start with

Inductive T : Type@{i} -> Type@{j} :=
| c1 : forall (A : Type@{i}), A -> T A
| c2 : T unit.

Note that we can invert this inductive; we may write

Definition c1' (A : Type@{i}) (v : T A) : A
  := match v with
     | c1 A x => x
     | c2 => tt
     end.

Lemma c1'_c1 (A : Type@{i}) : forall v, c1' A (c1 A v) = v.
Proof. reflexivity. Qed.

Suppose, for a moment, that c1 (T nat) typechecked. Since T nat : Type@{j}, this would require j <= i. If it gave us that j < i, then we would be set. We could write c1 Type@{j}. And this is exactly the setup for the variant of Hurken's that I mentioned above! We could define

u = j
U := Type@{j}
A := T Type@{j}
down : U -> A := c1 Type@{j}
up : A -> U := c1' Type@{j}
up_down := c1'_c1 Type@{j}

and hence prove False.

Coq needs to implement a rule for avoiding this paradox. As described here, the rule is that for each (non-parameter) argument to a constructor of an inductive, if the type of the argument has a sort in universe u, then the universe of the inductive is constrained to be >= u. In this case, this is stricter than Coq needs to be. As mentioned by SkySkimmer here, Coq could recognize arguments which appear directly in locations which are indices of the inductive, and disregard those in the same way that it disregards parameters.

---

So, to finally answer your question, I believe the following are your only options:

1. You can Set Universe Polymorphism so that in T (T nat), your two Ts take different universe arguments. (Equivalently, you can write Polymorphic Inductive.)
2. You can take advantage of how Coq treats parameters of inductive types specially, which mandates using equality in your case. (The requirement of using equality is a general property of going from indexed inductive types to parameterized inductives types---from moving arguments from after the : to before it.)
3. You can pass Coq the flag -type-in-type to entirely disable universe checking.
4. You can fix bug #7929, which I reported as part of digging into this question, to make Coq handle arguments of constructors which appear in index-position in the inductive in the same way it handles parameters of inductive types.
5. (You can find another edge case of the system, and manage to trick Coq into ignoring the universes you want to slip past it, and probably find a proof of False in the process. (Possibly involving module subtyping, see, e.g., this recent bug in modules with universes.))