Equivalence induction says that in order to prove
something for all equivalences, it is enough to prove it for all
identity equivalences for all types.
This follows from univalence. But also, conversely,
univalence follows from it:
Is this known? Some years ago it was claimed in this list
that equivalence induction would be strictly weaker than
univalence.
To prove the above, I apply a technique I learned from
Peter Lumsdaine, that given an abstract identity system (Id,
refl , J) with no given "computation rule" for J, produces
another identity system (Id, refl , J' , J'-comp) with
a "propositional computation rule" J'-comp for J'.
Martin
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