Effective Bounds from ineffective proofs in analysis: An application of functional interpretation and majorization

Journal of Symbolic Logic 57 (4):1239-1273 (1992)
We show how to extract effective bounds Φ for $\bigwedge u^1 \bigwedge v \leq_\gamma tu \bigvee w^\eta G_0$ -sentences which depend on u only (i.e. $\bigwedge u \bigwedge v \leq_\gamma tu \bigvee w \leq_\eta \Phi uG_0$ ) from arithmetical proofs which use analytical assumptions of the form \begin{equation*}\tag{*}\bigwedge x^\delta\bigvee y \leq_\rho sx \bigwedge z^\tau F_0\end{equation*} (γ, δ, ρ, and τ are arbitrary finite types, η ≤ 2, G0 and F0 are quantifier-free, and s and t are closed terms). If τ ≤ 2, (*) can be weakened to $\bigwedge x^\delta, z^\tau\bigvee y \leq_\rho sx \bigwedge \tilde{z} \leq_\tau z F_0$ . This is used to establish new conservation results about weak König's lemma. Applications to proofs in classical analysis, especially uniqueness proofs in approximation theory, will be given in subsequent papers
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    Ulrich Berger (2005). Uniform Heyting Arithmetic. Annals of Pure and Applied Logic 133 (1):125-148.

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