Proof Mining - Mathematics, Algorithms and Proofs

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Proof Mining Special case of the Modus Ponens-Problem A:≡∀x ∃y ∀z Aqf (x,y,z) ∀x ∃y ∀z Aqf (x,y,z)→∀u ∃v Bqf (u,v) ∀u ∃v Bqf (u,v) 1. Attempt: Explicit realization of existential quantifiers: ∀x,z Aqf (x,ϕ(x),z) ∀f(∀x,z Aqf (x,f(x),z)→∀u Bqf(u,Φ(u,f))) ∀u Bqf(u,Φ(u,ϕ)) Discussion • works for intuitionistic proofs (‘m-realizability’). • for classical proofs of A: i.g. no computable ϕ! Ulrich Kohlenbach 5 . .
Proof Mining Examples 1) P (x, y) decidable, but Q(x) := ∃y P (x, y) undecidable. ∀x∃y∀z(P (x, y) ∨ ¬P (x, z)) logically true, but no computable ϕ (x, y, z ∈ IN). 2) (an)n∈IN nonincreasing sequence in [0, 1] ∩ Q. Then PCM(an) :≡ ∀x∃y∀z ≥ y(|ay − az| ≤ 2 −x ). Even for prim.rec. (an) i.g. no computable bound for y (Specker 1947). Ulrich Kohlenbach 6
Page 1 and 2: Proof Mining: Applications of Proof
Page 3 and 4: Proof Mining Logical methods I: Eli
Page 5: Proof Mining Logical methods II: Pr
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 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
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Proof Mining Comments • Also hold
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Proof Mining Corollary to refined m
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Proof Mining Known uniformi.ty resu
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or Proof Mining (a) ∀ε > 0∃n
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Definition 28 Proof Mining f : X
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If P1 : H → u∈M Proof Mining C
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Proof Mining COROLLARY 32 (K./Leust
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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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Proof Mining - Mathematics, Algorithms and Proofs

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