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Showing posts from July, 2016

Some thoughts on the meaning of equality and cyclic proof - can CTT help were ITT hinders?

The CTT (Computational Type Theory) approach to type theory, as represented in NuPRL and described in [1][2][3], provides a contrasting methodology to what appears to be in vogue at the moment - an approach known as ITT (Intensional Type Theory) as represented in modern type-theory based proof assistants such as Coq and Agda.

There is a very good exposition by Jon Sterling which describes the more fundamental philosophical differences here [4]. It's somewhat difficult to convey the subtleties without such a complete exposition, but since its precisely this that I want to talk about, I'll try to recapitulate the essential differences.

When we talk about the elements of a type in ITT, we give the process of their construction in such a way that the elements which inhabit the type are made entirely explicit in the definition of the type.  So, for instance, in defining the data type for the natural numbers we give zero and succ explicitly as the elements of this type.  The type re…