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[Matthieu Sozeau <mattam AT mattam.org>]

I agree with you Andrej, and the (well founded) transitive closure of the subterm relation can easily be defined for computational inductive families (all inductive types if you remove prop), avoiding the computation of ordinals. That's precisely the "semantic" (maybe "evidence-based"?) explanation that C. Paulin used in her habilitation thesis to justify recursive definitions and the most general one for users (it does not even need to be attached to an inductive type). Equations can derive this subterm relation automatically for (non-mutual, non-nested) inductive families, and prove its wellfoundedness. Extending this to the other cases is a matter of thinking and engineering. The Below predicate of Epigram gives you similar access to every subterm you can recurse on "logically". The only culprit is reduction behavior/efficiency using this machinery, but that should be optimizable.

Best,

-- Matthieu

Le mercredi 8 janvier 2014, Andrej Bauer a écrit :

I would just like to point out that ordinals are an inherently
classical notion. The correct constructive and computationally
meaningful replacement is that of a well-founded relation, i.e., a
relation < on a set X satisfying, for all properties P,

 (forall y, ((forall x < y, P x) -> P y)) -> forall z, P z.

This is all well known, and of course you can recognize the
recursor/eliminator in the above formula. So if we are to take
computation seriously, we ought to think about inductive definitions
which are justified by a more general notion of well foundedness, not
just ordinals. The ordinals are bound to go wrong when we push them a
little bit.

Also, the HoTT experience has thought us (at least me) the value of
semantic notions over syntactic ones. I am referring to HoTT hProp vs.
CiC Prop. The former delineates the concept of "proposition" with a
semantic condition, while the latter does it formalistically. It would
seem sensible to me to go the same route with inductive definitions,
namely, rely on semantic justifications rather than syntactic ones. [I
may be misusing the words "semantic" and "syntactic" here, but I
cannot think of better ones.]

With kind regards,

Andrej

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-- 
-- Matthieu