On Preserving the Computational Content of Mathematical Proofs: Toy Examples for a Formalising Strategy

A computational study of a class of recursive inequalities

“…In practice, the reduction often involves little more than the systematic application of basic algebraic operations along with standard facts about norms or inner products, and it is conceivable that for a given method {x n } converging towards some point x * , a search algorithm could potentially generate recursive inequalities satisfied by ∥x n − x * ∥, thereby providing an automated method for proving the convergence of numerical algorithms, along with quantitative information in the form of a rate of convergence or metastability. The latter would form an instance of the general program of formalised applied proof theory as outlined by Koutsoukou-Argyraki [38].…”

“…While the formal proof interpretations that underlie applied proof theory, such as the Dialectica interpretation, have been implemented in a number of theorem provers (most notably the Minlog system 2 ), the majority of implementations tend to focus on the use of such translations as a verification strategy. In comparison, very little has been done on formalizing concrete applications of proof theory in mathematics, though the potential benefits of creating formal databases of "proof-mined" proofs have been suggested by Koutsoukou-Argyraki over the past few years, and are nicely outlined in her recent article [38].…”