Proof Mining - Mathematics, Algorithms and Proofs

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⎧ ⎪⎨ ⎪⎩ Proof Mining Let y ∈ K be the unique root of f(x, ·), yε an approximate root |f(x, yε)| < ε. Then dK(y, y Φ(x,ε)) < ε). THEOREM 2 (K.,93) For T = BA+HBC(+SEC − ) as before T ⊢ ∀x ∈ X∃!y ∈ K(F (x, y) =IR 0) ∃ BA(+iter.)-definable computable function G : X → K s.t. BA(+IA) ⊢ ∀x ∈ X(F (x, G(x)) =IR 0) (X, K are BA-definable Polish spaces, K compact, F : X × K → IR BA-definable function). Ulrich Kohlenbach 21
Proof Mining 2) M.f.i. transforms statements f : K → K is contractive ∀x, y ∈ K(x = y → d(f(x), f(y)) < d(x, y)) into moduli of contractivity α : IR ∗ + → (0, 1) (Rakotch) ∀x, y ∈ K, ε > 0(d(x, y) > ε → d(f(x), f(y)) < α(ε)d(x, y)). 3) f : K × IN → IR + s.t. (f(x, n))n∈IN is non-increasing for x ∈ K. MFI transforms the statement f(x, n) n→∞ → 0 into a modulus of uniform convergence δ : Q ∗ + → IN ∀x ∈ K∀ε > 0∀n ≥ δ(ε)(f(x, n) < ε). (Numerous papers on such δ e.g. in metric fixed point theory). Ulrich Kohlenbach 22
Page 1 and 2: Proof Mining: Applications of Proof
Page 3 and 4: Proof Mining Logical methods I: Eli
Page 5 and 6: Proof Mining Logical methods II: Pr
Page 7 and 8: Proof Mining Examples 1) P (x, y) d
Page 9 and 10: Proof Mining Again: Modus Ponens
Page 11 and 12: 1) Φ(x, g) := ⎧ ⎨ ⎩ Proof Mi
Page 13 and 14: Proof Mining Proof interpretations
Page 15 and 16: Proof Mining K., 1993-96: P Polish
Page 17 and 18: Proof Mining Limits of Metatheorem
Page 19 and 20: Necessity of A∃ ‘∃-formula’
Page 21: Proof Mining MFI as numerical impli
Page 25 and 26: Proof Mining K., 1998: P Polish spa
Page 27 and 28: Proof Mining Case study: strong uni
Page 29 and 30: Proof Mining Comments on the result
Page 31 and 32: Proof Mining Assume that X is unifo
Page 33 and 34: Proof Mining Observation: If f is g
Page 35 and 36: Proof Mining Our results concern th
Page 37 and 38: Proof Mining Case studies II: gener
Page 39 and 40: Proof Mining • a CAT(0)-spaces (G
Page 41 and 42: A ω,X :=PA ω,X +AC IN , where Pro
Page 43 and 44: Proof Mining Definition 12 For x
Page 45 and 46: Proof Mining Definition 15 A senten
Page 47 and 48: Proof Mining Comments • Also hold
Page 49 and 50: Proof Mining A uniform boundedness
Page 51 and 52: Proof Mining Proof Mining: Applicat
Page 53 and 54: Proof Mining THEOREM 19 (Gerhardy/K
Page 55 and 56: Proof Mining 2) If additional param
Page 57 and 58: Proof Mining Comments • Also hold
Page 59 and 60: Proof Mining Corollary to refined m
Page 61 and 62: Proof Mining Known uniformi.ty resu
Page 63 and 64: or Proof Mining (a) ∀ε > 0∃n
Page 65 and 66: Definition 28 Proof Mining f : X
Page 67 and 68: If P1 : H → u∈M Proof Mining C
Page 69 and 70: Proof Mining COROLLARY 32 (K./Leust
Page 71 and 72: Proof Mining THEOREM 34 (K./Leustea
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Proof Mining Proof Mining: Applicat
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Proof Mining Proposition 36 (K., NA
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Proof Mining The bound in the previ
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one has Proof Mining ∀ε > 0∀k
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Proof Mining Asymptotically quasi-n
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Definition 44 Proof Mining f : C
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Then where Proof Mining ∀ε > 0
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Comprehensive survey: Proof Mining
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Proof Mining 8) Delzell, C., Kreise
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Proof Mining Numer. Funct. Anal. an
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Proof Mining 25) Kohlenbach, U., La
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