Abstract
Large scale real number computation is an essential ingredient in several modern mathematical proofs. Because such lengthy computations cannot be verified by hand, some mathematicians want to use software proof assistants to verify the correctness of these proofs. This paper develops a new implementation of the constructive real numbers and elementary functions for such proofs by using the monad properties of the completion operation on metric spaces. Bishop and Bridges's notion (Bishop and Bridges 1985) of regular sequences is generalised to what I call regular functions, which form the completion of any metric space. Using the monad operations, continuous functions on length spaces (which are a common subclass of metric spaces) are created by lifting continuous functions on the original space. A prototype Haskell implementation has been created. I believe that this approach yields a real number library that is reasonably efficient for computation, and still simple enough to verify its correctness easily.
Highlights
Several mathematical theorems rely on numerical computation of real number values in their proofs
The Mertens conjecture claims that the absolute values of the partial sums of the Möbius function are bounded by the square root function
This paper proposes a new constructive implementation of real numbers
Summary
Several mathematical theorems rely on numerical computation of real number values in their proofs. Odlyzko and te Riele’s disproof involves computing the first hundred decimal digits of the first two thousand zeros of the Riemann zeta function Another example is Hales’s famous proof [6] of the Kepler conjecture. I mean a software verified library of continuous functions on R with which computations can be done to arbitrary precision One such library has been created by Cruz-Filipe as a product of his software verified constructive proof of the fundamental theorem of calculus [4]. An implementation that attempts to be simple and elegant, and at the same time practical enough to run the computations needed by the proof of the Kepler conjecture and other mathematical theorems. I will write the reciprocal function as λx.x−1
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