Proof Mining: A Systematic Way of Analysing Proofs in Mathematics

Abstract

We call proof mining the process of logically analyzing proofs in mathematics with the aim of obtaining new information. In this survey paper we discuss, by means of examples from mathematics, some of the main techniques used in proof mining. We show that those techniques not only apply to proofs based on classical logic, but also to proofs which involve non-effective principles such as the attainment of the infimum of f ∈ C[0, 1] and the convergence for bounded monotone sequences of reals. We also report on recent case studies in approximation theory and fixed point theory where new results were obtained.

Gödel's Functional Interpretation and Its Use in Current Mathematics

“…in E. Bishop's treatment of constructive analysis ( [6,8]). Whereas Bishop himself considered the functional interpretation of implications as 'numerical implication' ( [7]) it is argued in [75] that actually the monotone version is the more natural one. We will come back to this issue at the end of this paper.…”

A Complexity Analysis of Functional Interpretations

“…Using Proposition 4.4, Theorems 3.38 and 3.41, Notation 3.24 and the abbreviations ∂ :≡ ∂(P), S c :≡ S c (P) and S m :≡ S m (P) we can state the following theorem. 37 The axiom versions of IMP and EXP are simply realized with projectors Π. This follows immediately from the fact that (…”

“…This gives functional interpretations the ability of extracting programs and other effective data 4 under certain conditions from ineffective proofs (proof mining). Monotone-functional-interpretationbased proof mining has already produced important results in computational analysis and has helped to obtain new results in mathematical analysis (see, e.g., [26,28,29,31,32,35,36,37,38]). …”

Hybrid Functional Interpretations

“…This generalises Spector's quantifier-free rule of extensionality (see [10]) since it allows us to derive rs τ = rt from s ρ = t in any context of the form ! k Δ, and has the advantage that visibly (2) requires no realizer.…”

“…A real number being positive carries the extra information of a lower bound on how far from zero the limit of the sequence can be (cf. [10]). In order to avoid going into the representation level, when analysing the proof that a certain real function f is positive at x, i.e.…”