The Coq mailing list

[Thorsten Altenkirch <Thorsten.Altenkirch AT nottingham.ac.uk>]

Indeed you need functional extensionality which isn’t provable in coq.

Imho this is a bug, but one which isn’t easy to fix.

We suggested a solution a while ago (called Observational Type Theory) but this was never implemented in any real system. Now there is something better coming up, see Coquand et al’s work on Cubical Type Theory.

Cheers,

Thorsten

From: <coq-club-request AT inria.fr> on behalf of Hugo Carvalho <hugoccomp AT gmail.com>
Reply-To: "coq-club AT inria.fr" <coq-club AT inria.fr>
Date: Tuesday, 26 January 2016 18:56
To: coq-club <coq-club AT inria.fr>
Subject: [Coq-Club] Is it possible to rewrite under binders?

Hi Club!

Consider the following goal:

Example e1 : (fun x => 1 + x) = (fun x => S x).

This is provable, as computation occurs under binders normally, and addition in Coq happens to be defined recursively over the first argument.

This other goal doesn't look as provable:

Example e2: (fun x => x + 1) = (fun x => S x).

Doesn't seem like you can rewrite "x+1" into "S x" (since you can't introduce "x" into the context, you can't refer to it in order to instantiate an equality you'd use to rewrite)... Is this the case? Is this goal unprovable without relying on axioms (functional extensionality does the trick, of course)?

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