[HoTT] Re: Why do we need judgmental equality?

[Felix Rech <s9ferech@gmail.com>]

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Thank you all for the discussion!

One of the motivations for my question was that I actually expect usability 
benefits if one worked in a dependent type theory without judgmental 
equality that has good support by a proof assistant. There are primarily 
two reasons for this:

   1. Judgmental equality cannot be taken as assumptions. If one wants to 
   use judgmental equalities then one has to give concrete definitions that 
   satisfy those equalities and cannot hide the definition details. This makes 
   it impossible to change definitions later on without breaking constructions 
   that depend on them.
   2. In nontrivial definitions, judgmental equalities seem more arbitrary 
   than natural. If we define addition of natural numbers for example then we 
   can choose between x + 0 = x and 0 + x = x as judgmental equality but we 
   cannot have both. This makes it hard to find the right definitions and to 
   predict their behavior.

Another motivation was of course that it would simplify the implementation 
of proof checkers and parts of the metatheory.

I would appreciate any comments on this.

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