The Coq mailing list

[Thorsten Altenkirch <txa AT Cs.Nott.AC.UK>]

Hi Noam,

>> Surely, type constructors should not be injective in general. A definition
>> of the form
>> data I : (Set -> Set) -> Set where
>> should be expandable by an annonymous declaration
>> I : (Set -> Set) -> Set
>> I F = data {}
>> in an analogous way a named function definition can be expanded by
>> definition and a lambda abstraction
> 
> ...I'm not sure about this expansion, and the analogy.  I think the
> question is whether we view the declaration of type constructors in
> the same way that we view the declaration of term constructors (i.e.,
> as stipulating a new way of constructing things, in this case
> inhabitants of Set).  If we do, then type constructors are by
> definition injective.

I don't see the analogy. A term constructor is injective because an inductive 
type is freely generated from constructors. Indeed, we can show that the 
structure map of an initial algebra is an isomorphism and hence injective. 
However the universe of sets is not freely constructed from type constructors 
but they are really definitions.

We should justify principles like this (i.e. all type constructors are 
injective) by a semantic explanation or by a translation into a well 
understood core language e.g. (extensional) Marttin-Loef Type Theory with 
W-types and universes. I don't know any such explanation which would justify 
this principle. Moreover, Chung's example shows that this is not possible 
(without giving up something else) since pure type theory is consistent with 
EM.

Cheers,
Thorsten