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In particular Agda seems not strong enough to prove that.

Is the predicative Calculus of Inductive Constructions universes (Coq without Prop) sufficient?

How about with the impredicative Prop?

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  • Coq without Prop is not strong enough, because it's basically Martin-Löf type theory with universes.

  • Coq with Prop is strong enough, because you can encode sets of normalizing terms via predicates $S : \mathrm{Term} \to \mathrm{Prop}$, and impredicative universal quantification lets you express arbitrary intersections.

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