Strongly uniform bounds from semi-constructive proofs

Abstract

In [U. Kohlenbach, Some logical metatheorems with applications in functional analysis, Trans. Amer. Math. Soc. 357 (2005) 89–128], the second author obtained metatheorems for the extraction of effective (uniform) bounds from classical, prima facie non-constructive proofs in functional analysis. These metatheorems for the first time cover general classes of structures like arbitrary metric, hyperbolic, CAT(0) and normed linear spaces and guarantee the independence of the bounds from parameters ranging over metrically bounded (not necessarily compact!) spaces. Recently (in [P. Gerhardy, U. Kohlenbach, General logical metatheorems for functional analysis, 2005, p. 42 (submitted for publication)]), the authors obtained generalizations of these metatheorems which allow one to prove similar uniformities even for unbounded spaces as long as certain local boundedness conditions are satisfied. The use of classical logic imposes some severe restrictions on the formulas and proofs for which the extraction can be carried out. In this paper we consider similar metatheorems for semi-intuitionistic proofs, i.e. proofs in an intuitionistic setting enriched with certain non-constructive principles. Contrary to the classical case, there are practically no restrictions on the logical complexity of theorems for which bounds can be extracted. Again, our metatheorems guarantee very general uniformities, even in cases where the existence of uniform bounds is not obtainable by (ineffective) straightforward functional analytic means. Already in the purely intuitionistic case, where the existence of effective bounds is implicit, the metatheorems allow one to derive uniformities that may not be obvious at all from a given constructive proof. Finally, we illustrate our main metatheorem by an example from metric fixed point theory.