But why would anybody say that there is a *need* for that? Is it impossible to define a sensible dependent type theory (say for the purpose of serving as a foundation for univalent mathematics) that doesn't mention anything like definitional equality? If not, why not? And notice that I am not talking about *usability* of a proof assistant such as the *existing* ones (say Coq, Agda, Lean) were definitional equalities to be removed. I don't care if such hypothetical proof assistants would be impossibly difficult to use for a dependent type theory lacking definitional equalities (if such a thing exists).

The question asked by Felix is a very sensible one: why is it claimed that definitional equalities are essential to dependent type theories?

(I do understand that they are used to compute, and so if you are interested in constructive mathematics (like I am) then they are useful. But, again, in principle we can think of a dependent type theory with no definitional equalities and instead an existence property like e.g. in Lambek and Scott's "introduction to higher-order categorical logic". And like was discussed in a relatively recent message by Thierry Coquand in this list,)

Martin 

On Wednesday, 30 January 2019 11:54:07 UTC, Felix Rech wrote:

In section 1.1 of the HoTT book it says "In type theory there is also a need for an equality judgment." Currently it seems to me like one could, in principle, replace substitution along judgmental equality with explicit transports if one added a few sensible rules to the type theory. Is there a fundamental reason why the equality judgment is still necessary?

Thanks,

Felix Rech