A computer verified, monadic, functional implementation of the integral

We provide a computer verified exact monadic functional implementation of the Riemann integral in type theory. Together with previous work by O’Connor, this may be seen as the beginning of the realization of Bishop’s vision to use constructive mathematics as a programming language for exact analysis.

💡 Research Summary

The paper presents a fully verified, monadic, functional implementation of the Riemann integral within dependent type theory, aiming to realize Bishop’s vision of using constructive mathematics as a programming language for exact analysis. After motivating the need for exact, rather than approximate, numerical integration, the authors build on O’Connor’s constructive real number library to define a “interval monad” that simultaneously carries a real value and an explicit error bound. The monad provides the usual operations—return, bind, and map—where bind applies a function to every point inside an interval and recombines the results into a new interval whose endpoints are computed to guarantee inclusion of all possible outcomes. By proving the monad laws for this construction, the authors show that the integral inherits linearity, additivity over partitions, and compatibility with scalar multiplication in a fully formal setting.

Implementation is carried out in Coq (or Agda), where reals are represented either as Cauchy sequences or Dedekind cuts, and intervals are pairs (a, b) with the invariant a ≤ b. All definitions are typed so that error propagation is enforced by the type system, and every theorem—ranging from basic properties of the interval monad to the convergence of Riemann sums—is mechanically proved using a combination of built‑in tactics (lia, nra) and custom proof scripts. The key technical contribution is a constructive proof that, for any uniformly continuous function, the error of a Riemann sum over a partition of mesh Δx is bounded by a constant times Δx. Consequently, for any prescribed precision ε > 0, a sufficiently fine partition yields an interval containing the true integral value, and this fact is extracted as a certified algorithm.

The authors demonstrate the practicality of their approach with several examples: the integral of x³ over `