Constructive decidability of classical continuity

Constructive decidability of classical continuity

We show that the following instance of the principle of excluded middle holds: any function on the one-point compactification of the natural numbers with values on the natural numbers is either classically continuous or classically discontinuous. The proof doesn’t require choice and can be understood in any of the usual varieties of constructive mathematics. Classical (dis)continuity is a weake...

Continuity Properties in Constructive Analysis

Constructive Equivalents of the Uniform Continuity Theorem

For the purpose of constructive reverse mathematics, we show the equivalence of the uniform continuity theorem to a series of propositions; this illuminates the relationship between Brouwer’s fan theorem and the uniform continuity theorem.

A Constructive Model of Uniform Continuity

We construct a continuous model of Gödel’s system T and its logic HA in which all functions from the Cantor space 2N to the natural numbers are uniformly continuous. Our development is constructive, and has been carried out in intensional type theory in Agda notation, so that, in particular, we can compute moduli of uniform continuity of T-definable functions 2N → N. Moreover, the model has a c...

Constructive homomorphisms for classical groups

Let Ω ≤ GLd(q) be a quasisimple classical group in its natural representation and let ∆ = NGLd(q)(Ω). We construct the projection from ∆ to ∆/Ω and provide fast, polynomialtime algorithms for computing the image of an element. Given a discrete logarithm oracle, we also represent ∆/Ω as a group with at most 3 generators and 6 relations. We then compute canonical representatives for the cosets of...