Proof Mining

Origin. Kreisel, Kohlenbach (1990s-present). Extract computational content from classical proofs. Functional interpretations. Quantitative results. Applied proof theory.

Models. Classical proofs contain constructive content. Functional interpretation extracts it. Bounds and algorithms. Mathematics applications.

Formalism.

Goal: Classical proof of ∀x∃y.A(x,y). Extract: bound/algorithm for y given x. Quantitative content. Effective existence.

Monotone functional interpretation: Variant of Dialectica. Handles classical logic. Majorizability. Bound extraction.

Majorizability: s* ≥_ρ s ("s* majorizes s of type ρ"). s* bounds functional s. Uniform bounds. Higher types.

Metatheorems: If T proves ∀x∃y.A(x,y) with bounds. Witnessed by functional. Extractable from proof. Kohlenbach's metatheorems.

Applications: Fixed point theory: rates of convergence. Approximation theory: error bounds. Ergodic theory: quantitative results. Metric geometry.

Logical techniques: Proof unwinding. A-translation. Negative translation. Bounded collection.

Results: New quantitative theorems. Sometimes optimal bounds. Algorithmic content. Mathematical applications.

Symbols.

Symbol	Unicode	Meaning
≥_ρ	—	majorization at type ρ
T^ω	—	higher-type extension
Φ	U+03A6	extracted bound
∀∃	—	prenex form

Metatheory. Bound extraction. Functional interpretation. Metatheorems. Applications.

Applies to. Analysis. Fixed point theory. Ergodic theory. Approximation theory.

Limitations. Technical prerequisites. Proof complexity. Case-by-case. Limited automation.