[Agda] Agda with excluded middle is inconsistent

[Andreas Abel]

I agree with Thorstens analysis.  There is nothing like a  
"constructor", the machine only nows bits and sequences, so ultimately  
you need to explain everything but just these two principles:  
alternatives (bits) and composition (pairing).  This is realized in so  
called structural type system, where a type is just an expression made  
of constants, +, \times, fixpoints, and function types.  Thorsten  
analyses Agda data declarations into structural expressions, so

   I = \ X. 0

which is not injective.  Named type systems, which use constructors  
(Agda) or class names (Java), restrict the power of structural type  
systems to make subtyping, unification, type checking etc. decidable  
and maybe your code more readable.  But I do not think that named type  
systems are foundational, they need to be explained in terms of  
structure.

So the problem at hand can be solved in conjunction with giving a  
semantics for Agda data types.  Using the semantics one can identify  
sufficient criteria for injectivity.

For a pragmatic solution, I suggest to consider the problem of  
injective type constructors in a simply-typed setting and then  
extrapolate to a dependently typed setting.

Cheers,
Andreas


On Jan 7, 2010, at 4:15 PM, Thorsten Altenkirch wrote:

> 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
>
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Andreas ABEL
INRIA, Projet Pi.R2
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