jonsterling / agda-bar-induction Goto Github PK

We should be able to prove the fan theorem now

We still can't prove in Agda that every stream processor is continuous (in fact, this principle stated naïvely is false in Type Theory, however a squashed version is consistent).

But from the Fan Theorem, we should be able to prove that continuity implies uniform continuity for functions on the [0,1] continuum.

I think it may be possible to prove the bar theorem directly if we switch from Martin-Löf's interpretation of the quantifiers to Brouwer's—I believe, in the case of quantifying over choice sequences as in forall α. P(α), such a mental construction is no longer merely an arbitrary computational function from streams to the proofs that they satisfy P, but is instead a neighborhood function, which is a function from finite approximations of α to either "I need more info" or a construction that represents the immediate intuition that P holds for every choice sequence in the neighborhood under consideration.

As a result, the demonstration that a node is barred will already be well-ordered, and so the bar theorem itself ought to be an "obvious truth".