Semantify the proofs of type inversion #48
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…sInj. We do not use it before and both from the point of view of practicality and theoretical considerations, it is easier to postpone it after the availability of the fundamental lemma.
We leverage the knowledge that convertibility implies well-typedness instead of performing syntactic inversions.
MevenBertrand
approved these changes
Sep 14, 2023
| apply TypeRefl, id_ty_inj in Hty as [HA HB]. | ||
| prod_splitter; boundary. | ||
| Qed. | ||
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| Lemma termGen' Γ t A : |
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I guess you did not want to prove termGen' directly, rename it termGen and grep the whole dev? That would probably be nicer in the long run…
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I'm not sure how to easily prove termGen directly, actually. I'd like to avoid doing an induction on the typing derivation and reason exclusively from the logical relation but I don't see how to do that. But in any case at some point we should indeed perform a renaming (maybe not now).
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What I meant is removing the annoying equality case, which is only needed before you have boundaries (which let you show that if Γ |- t : A then Γ |- A ≅A and avoid that case).
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We move the generation lemma after the fundamental lemma and prove the various type inversions through the latter rather than the former. This makes the development more robust to tweaks to the declarative typing rules.