Proof Mining - Mathematics, Algorithms and Proofs

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Proof Mining where only x =IN y primitive equality predicate but for ρ → τ x X =X y X :≡ dX(x, y) =IR 0IR, s =ρ→τ t :≡ ∀v ρ (s(v) =τ t(v)). The theory A ω [X, d, W ] results by adding constants bX, dX, WX axiom expressing that (X, d, W ) is a nonempty b-bounded hyperbolic space. Definition 11 F ≡ ∀a σ Fqf (a) (resp. F ≡ ∃a σ Fqf (a)) is a ∀-formula (∃-formula) if Fqf is quantifier-free and σ are of the kind IN, IN → IN, X, IN → X, X → X. Ulrich Kohlenbach 16
Proof Mining Definition 12 For x ∈ [0, ∞) ⊂ IR define (x)◦ ∈ IN IN by where (x)◦(n) := j(2k0, 2 n+1 − 1), k k0 := max k ≤ x . 2n+1 Lemma 13 1) If x ∈ [0, ∞), then (x)◦ is a representative of x in the sense of our representation above. 2) If x, x ∗ ∈ [0, ∞) and x ∗ ≥ x (in the sense of IR), then (x ∗ )◦ ≥IR (x)◦ and also (x ∗ )◦ ≥1 (x)◦. 3) x ∈ [0, ∞], then (x)◦ is monotone, i.e. ∀n ∈ IN((x)◦(n) ≤0 (x)◦(n + 1)). 4) If b ∈ IN x, x ∗ ∈ [0, b] with x ∗ ≥ x then (x ∗ )◦ s-maj1 (x)◦. Ulrich Kohlenbach 17
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 and 22: Proof Mining MFI as numerical impli
Page 23 and 24: Proof Mining 2) M.f.i. transforms s
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: A ω,X :=PA ω,X +AC IN , where Pro
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
Page 73 and 74: Proof Mining Proof Mining: Applicat
Page 75 and 76: Proof Mining Proposition 36 (K., NA
Page 77 and 78: Proof Mining The bound in the previ
Page 79 and 80: one has Proof Mining ∀ε > 0∀k
Page 81 and 82: Proof Mining Asymptotically quasi-n
Page 83 and 84: Definition 44 Proof Mining f : C
Page 85 and 86: Then where Proof Mining ∀ε > 0
Page 87 and 88: Comprehensive survey: Proof Mining
Page 89 and 90: Proof Mining 8) Delzell, C., Kreise
Page 91 and 92: Proof Mining Numer. Funct. Anal. an
Page 93 and 94:
Proof Mining 25) Kohlenbach, U., La
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