We provide a computer verified exact monadic functional implementation of the Riemann integral in type theory. Together with previous work by OConnor, this may be seen as the beginning of the realization of Bishops vision to use constructive mathematics as a programming language for exact analysis.
We construct a homeomorphism between the compact regular locale of integrals on a Riesz space and the locale of (valuations) on its spectrum. In fact, we construct two geometric theories and show that they are biinterpretable. The constructions are e
lementary and tightly connected to the Riesz space structure.
We present a constructive proof of Gelfand duality for C*-algebras by reducing the problem to Gelfand duality for real C*-algebras.
We present a constructive proof of the Stone-Yosida representation theorem for Riesz spaces motivated by considerations from formal topology. This theorem is used to derive a representation theorem for f-algebras. In turn, this theorem implies the Ge
lfand representation theorem for C*-algebras of operators on Hilbert spaces as formulated by Bishop and Bridges. Our proof is shorter, clearer, and we avoid the use of approximate eigenvalues.
In Riesz space theory it is good practice to avoid representation theorems which depend on the axiom of choice. Here we present a general methodology to do this using pointfree topology. To illustrate the technique we show that almost f-algebras are
commutative. The proof is obtained relatively straightforward from the proof by Buskes and van Rooij by using the pointfree Stone-Yosida representation theorem by Coquand and Spitters.
