How does infinite descent work using a proof assistant? It's been quite a while, so I may be remembering incorrectly, but this is my understanding:

Coq uses an inductive type like "N = 0 : N | S : N → N", and a first-order theory with integers axiomatized this way admits nonstandard models (whose prefixes are isomorphic to the naturals but have elements not reachable by repeated application of S, defeating infinite descent). But universal quantification in system F (inherited by the calculus of inductive constructions) corresponds to a fragment of second-order logic, where there is only one model (– the theory is categorical).

Is that right?