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  • $\begingroup$ In the recursive case, can't you get essentially the same induction principle you would get from the inductive definition by just doing induction on the indices? In particular, in Agda you could just split directly on the indices rather than going to the pain of explicitly calling an induction function. $\endgroup$
    – Mike Shulman
    Commented Feb 5 at 19:27
  • $\begingroup$ I think this is true only if your inductive type has the same structure as your indexing type (thinking about ornaments here). But if you have a more complicated relation, this might not be as simple. For instance, if you have a "catch-all" constructor that can be used to construct a value for any index (think conversion rule for inductively defined typing), then you'll have a corresponding case for each index, and you'll end up repeating that case. $\endgroup$
    – Meven Lennon-Bertrand ♦
    Commented Feb 6 at 9:41
  • $\begingroup$ I feel like this might be similar to the use of functional induction, ie an induction principle that follows the structure of a recursive functions, which might be more complicated than the one corresponding to the induction principle for the inductive type. Here too you want an induction principle "of the right shape", which is not the same as that of the underlying datastructure. But maybe all this is easier to do with Agda's pattern-matching. $\endgroup$
    – Meven Lennon-Bertrand ♦
    Commented Feb 6 at 9:44
  • 1
    $\begingroup$ I didn't mean to include an inductive family with a "catch-all" constructor among the "situations where both definitions are possible" that I said in the question I wanted to restrict to. I suppose in that case you could get something like a recursive version by duplicating the catch-all constructor at each index, as you suggest, but it wouldn't be as strongly equivalent since, for instance, you wouldn't be able to apply the catch-all constructor without first destructing the index. $\endgroup$
    – Mike Shulman
    Commented Feb 6 at 21:11