A type that is a member of a universe can be coerced into a higher universe. Is that coercion injective? That is, if two elements of U1 are equal after being coerced to U2, does that imply they are equal before being coerced?
More specifically, is the following provable in Coq?
Universe U1 U2.
Constraint U1 < U2.
Theorem type_coericion_inj:
forall (A B: Type@{U1}),
@eq Type@{U2} A B ->
@eq Type@{U1} A B.
If that isn't provable in Coq, is it at least consistent to add that as an axiom? Is there a better name for this property?