## Analyzing Proofs in Analysis (1993)

Venue: | LOGIC: FROM FOUNDATIONS TO APPLICATIONS. EUROPEAN LOGIC COLLOQUIUM (KEELE |

Citations: | 50 - 31 self |

### BibTeX

@INPROCEEDINGS{Kohlenbach93analyzingproofs,

author = {Ulrich Kohlenbach},

title = {Analyzing Proofs in Analysis},

booktitle = {LOGIC: FROM FOUNDATIONS TO APPLICATIONS. EUROPEAN LOGIC COLLOQUIUM (KEELE},

year = {1993},

pages = {225--260},

publisher = {University Press}

}

### Years of Citing Articles

### OpenURL

### Abstract

### Citations

418 | Constructive analysis - BISHOP, BRIDGES - 1985 |

302 |
Introduction to approximation theory
- Cheney
- 1966
(Show Context)
Citation Context ... ∗ + such that ∧ (∗) p ∈ H(‖f − p‖∞ ≥ ‖f − pb‖∞ + γ · ‖p − pb‖∞) holds. (Here H ⊂ C[0, 1] denotes a Haar space.) The existence of a γ satisfying (∗) was proved (ineffectively) first in [28] (see also =-=[10]-=- ). A proof of this fact is already implicit in [12] (see [5] ). For more information on strong unicity see [24] . 5.3.2 shows that the concept ‘modulus of uniqueness’ generalizes the concept of stron... |

146 |
Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes
- Gödel
- 1958
(Show Context)
Citation Context ...ood behaviour. Proof of theorem 2.7 : Description of the algorithm for extracting uniform bounds by monotone functional interpretation We use (as in [26] and [31] ) the formalization of WE–HAω logic (=-=[14]-=- ). in Gödel’s calculus of intuitionistic 1) The most complicated axioms for the usual functional interpretation are A ∨ A → A and A → A ∧ A. The later one is even more complicated in requiring the ex... |

122 |
Metamathematical investigations of intuitionistic arithmetic and analysis
- Troelstra
(Show Context)
Citation Context ...e (2) ∧1∨0 is essentially a sentence having the form A0. ∧ If a sentence A ≡ f 1 , x0∨y0A0(f, x, y) is proved e.g. in a subsystem A of classical extensional arithmetic in all finite types E–PAω (from =-=[31]-=-), then one can use (after elimination of extensionality and negative translation) Gödel’s method of functional interpretation to extract from the proof a ∨ computable functional Ψ, which realizes x0 ... |

44 |
Theories of finite type related to mathematical practice
- Feferman
- 1977
(Show Context)
Citation Context ...13] ) this interpretation also applies to classical systems e.g. E–PA ω +AC 0,1 –qf. We stress that our interpretation also works for various subsystems of WE–HA ω , e.g. the system WE− HA ω |\ from =-=[11]-=- with quantifier–free induction and elementary recursor constants only and 3also to much weaker systems (w.r.t. the growth of provably functionals but not necessarily w.r.t. to proof–theoretic streng... |

43 |
Zur intuitionistischen arithmetik und zahlentheorie. Ergebnisse eines mathematischen Kolloquiums, 4:34–38
- Gödel
- 1933
(Show Context)
Citation Context ... Y τ(ρ)∧ x ρ A0(x, Y x) (A0 quantifier–free). We now carry out our monotone functional interpretation for WE-HA ω . By doing first elimination of extensionality ([26] ) and then negative translation (=-=[13]-=- ) this interpretation also applies to classical systems e.g. E–PA ω +AC 0,1 –qf. We stress that our interpretation also works for various subsystems of WE–HA ω , e.g. the system WE− HA ω |\ from [11... |

41 |
Hereditarily majorizable functionals of finite type
- Howard
- 1973
(Show Context)
Citation Context ...x ∗ s–maj0 x :≡ x ∗ ≥0 x, x ∗ s–maj τ(ρ) x :≡ ∧ y ∗ , y ( y ∗ s–maj ρ y → x ∗ y ∗ s–maj τ x ∗ y, xy ) . Remark 2.2 The addition of the clause ‘x ∗ y’ in definition 2.1.2 is a modification of Howard’s =-=[16]-=- original relation majρ which is due to Bezem [2]. Although we could use also Howard’s notion we prefer Bezem’s variant since it has the natural property that x ∗ s–maj x → x ∗ s–maj x ∗ , which e.g. ... |

36 | Strongly majorizable functionals of finite type: a model of bar recursion containing discontinuous functionals - Bezem - 1985 |

36 |
Effective bounds from ineffective proofs in analysis: an application of functional interpretation and majorization
- Kohlenbach
- 1992
(Show Context)
Citation Context ...s we obtain as a special case a new and very perspicuous proof for the conservativity of WKL over A with respect ∧ to sentences u1∧ ∨ v ≤γ tu wτ B0(u, v, w) (γ, τ arbitrary). This was first proved in =-=[21]-=- in a more complicated way. Interesting mathematical examples of sentences (1) with compact X := K are uniqueness theorems ∧ (5) x1, x2 ∈ K ( ) F (x1) = 0 = F (x2) → x1 = x2 . Here the uniform bound Φ... |

27 |
Complexity theory of real functions. Birkhäuser
- Ko
- 1991
(Show Context)
Citation Context ...The moduli of uniqueness and pointwise continuity from Bridges [8] and [9] allow a similar improvement. Furthermore our modulus Φ3 improves a modulus of uniqueness for Pn which is implicit in Ko [18],=-=[19]-=- (see [24] for details). 22Notes ∧ ∧ 1) Here and in the following we may also have tuples x1 ∈ X1, . . . , xn ∈ Xn(. . .). 2) By A0, B0, C0, . . . we denote always quantifier–free formulas. 3) Let us... |

27 |
Extensional Gödel Functional Interpretation
- Luckhardt
- 1973
(Show Context)
Citation Context ...ogical form in which certain key–lemmas (e.g. the alternation theorem) are used in these proofs. 2 A monotone functional interpretation The usual Gödel functional interpretation (as developed in e.g. =-=[26]-=- or [31]) can be simplified both with respect to the extraction algorithm and with respect to the functionals needed if only the ∧ ∨ extraction of (good) bounds for x yA0–sentences is wanted. Such bou... |

19 |
Foundations of Constructive Mathematics. Ergebnisse der Mathematik und ihrer Grenzgebeite
- Beeson
- 1985
(Show Context)
Citation Context ...short) and F, G : X → IR are constructively definable (and therefore continuous) functions. As an example of such a theorem we mention the uniqueness theorem for best Chebycheff approximation of f ∈ C=-=[0, 1]-=- by (algebraic) polynomials p ∈ Pn (over IR) of degree ≤ n. (This example will be studied in detail in section 5 below): ∧ ( f ∈ C[0, 1], p1, p2 ∈ Pn 2 ∧ (‖pi − f‖∞ = dist(f, Pn)) → ‖p1 − p2‖∞ = 0 ) ;... |

19 |
Foundations of constructive analysis (McGraw-Hill
- BISHOP
- 1967
(Show Context)
Citation Context ...(see [24] ). In particular we can improve significantly estimates for general Haar spaces obtained by D. Bridges in [7], [8] (who works entirely within the framework of Bishop’s constructive analysis =-=[4]-=- ): Definition 5.6 (D. Bridges) Let φ := {φ1, . . . , φn} be a Chebycheff system over [0, 1], φ(x) := ( φ1(x), . . . , φn(x) ) ∈ IR n , ‖φ‖ := sup ‖φ(x)‖2, where ‖ · ‖2 denotes the Euclidean norm on I... |

15 | New effective moduli of uniqueness and uniform a–priori estimates for constants of strong unicity by logical analysis of known proofs in best approximation theory
- Kohlenbach
- 1993
(Show Context)
Citation Context ... great importance as the next section shows. 5 Applications to uniqueness proofs in approximation theory In this section we give a survey of our proof–theoretic applications to analysis from [23] and =-=[24]-=- and analyze them from the perspective of the present paper. In [23] and [24] we applied a combination of functional interpretation and majorization which was developed in [21] to concrete proofs from... |

14 | Pointwise hereditary majorization and some applications - Kohlenbach - 1992 |

13 |
Theorie der majorisierbaren und stetigen Funktionale und ihre Anwendung bei der Extraktion von Schranken aus inkonstruktiven Beweisen: Effektive Eindeutigkeitsmodule bei besten Approximationen aus ineffektiven Eindeutigkeitsbeweisen. Dissertation, Frankfu
- Kohlenbach
- 1990
(Show Context)
Citation Context ...ard [16] is proved e.g. in Bezem [2]. 4Remark 2.4 In [21] (and also in [23] ) we used a pointwise variant majρ of the relation s–majρ with ∧ y 0 (x ∗ y majρ xy). This variant which was introduced in =-=[20]-=-, [22] (and the clause x∗ majρ0 x :≡ which is particular useful in the context of bar recursive functionals of finite and infinite types, see [20] ) has the advantage of being more closely related to ... |

12 | E#ective bounds from ine#ective proofs in analysis: an application of functional interpretation and majorization - Kohlenbach - 1992 |

10 |
Lecons sur les Fonctions de Variables Réelles. Gauthier–Villars
- Borel
- 1905
(Show Context)
Citation Context ...(the third one also for general Haar spaces): 1) the most common proof from de La Vallée Poussin [29] ( 56) (as presented with all details e.g. in [27] ), 2) a proof due to Kirchberger [17] and Borel =-=[6]-=- and 3) a simplification of a proof sketched by Young [34] (and worked out in Rice [30] ). From all three proofs i = 1, 2, 3 we obtained moduli of uniqueness Φi which are linear in q if the data f ∈ C... |

10 | E#ective moduli from ine#ective uniqueness proofs. An unwinding of de La Vallee Poussin's proof for Chebyche# approximation - Kohlenbach - 1993 |

10 |
Note on the fan theorem
- Troelstra
- 1974
(Show Context)
Citation Context ... ⊢ u 1∧ ∨ v ≤1 tu Since E–HAω ∧ ⊢ u1∧ ∨ v ≤1 tu w ≤0 ΦuA0(u, v, w) ↔ ∨ χuv =0 0 ↔ where w ≤0 Φu A0(u, v, w). ∧ u 1 , v 1 (χuv =0 0) where χ ∈ T is such that w ≤0 ΦuA0(u, min(v, tu), w), it follows by =-=[32]-=- (Thm.4(a)) that ∧ u 1 , v 1 [χuv =0 0] ECF , ∧ u1 , v1 [χuv =0 0] ECF ∈ L(EL) is prenex. EL+FAN+AC 0,1 ⊢ Thus by [32] (Thm.2) EL+AC 0,1 ⊢ ∧ u 1 , v 1 [χuv = 0] ECF . Provably (in WE–HA ω +AC 0,0 ) th... |

8 | New e#ective moduli of uniqueness and uniform a--priori estimates for constants of strong unicity by logical analysis of known proofs in best approximation theory - Kohlenbach - 1993 |

6 |
A constructive developement of Chebychev approximation theory
- Bridges
- 1980
(Show Context)
Citation Context ...explicit moduli of uniqueness also for other (constructively definable) Haar spaces (see [24] ). In particular we can improve significantly estimates for general Haar spaces obtained by D. Bridges in =-=[7]-=-, [8] (who works entirely within the framework of Bishop’s constructive analysis [4] ): Definition 5.6 (D. Bridges) Let φ := {φ1, . . . , φn} be a Chebycheff system over [0, 1], φ(x) := ( φ1(x), . . .... |

6 |
Lecons sur l’Approximation des Fonctions d’une Variable Réelle. Gauthier–Villars
- Poussin, de
- 1919
(Show Context)
Citation Context ...erent proofs of the uniqueness of the best Chebycheff approximation of f ∈ C[0, 1] by polynomials ∈ Pn (the third one also for general Haar spaces): 1) the most common proof from de La Vallée Poussin =-=[29]-=- ( 56) (as presented with all details e.g. in [27] ), 2) a proof due to Kirchberger [17] and Borel [6] and 3) a simplification of a proof sketched by Young [34] (and worked out in Rice [30] ). From al... |

5 |
Lipschitz constants and moduli of continuity for the Chebyshev projection
- Bridges
- 1982
(Show Context)
Citation Context ...cit moduli of uniqueness also for other (constructively definable) Haar spaces (see [24] ). In particular we can improve significantly estimates for general Haar spaces obtained by D. Bridges in [7], =-=[8]-=- (who works entirely within the framework of Bishop’s constructive analysis [4] ): Definition 5.6 (D. Bridges) Let φ := {φ1, . . . , φn} be a Chebycheff system over [0, 1], φ(x) := ( φ1(x), . . . , φn... |

5 |
Recent progress in constructive approximation theory
- Bridges
- 1982
(Show Context)
Citation Context ...ωA,H(lH,A) ) and 0 < lH,A ≤ EH,A. These estimates are much weaker than ours since γ(α) ‖φ‖ (≤ 1) is very close to 0 in practice. The moduli of uniqueness and pointwise continuity from Bridges [8] and =-=[9]-=- allow a similar improvement. Furthermore our modulus Φ3 improves a modulus of uniqueness for Pn which is implicit in Ko [18],[19] (see [24] for details). 22Notes ∧ ∧ 1) Here and in the following we ... |

5 |
On the computational complexity of best Chebyshev approximation
- Ko
- 1986
(Show Context)
Citation Context ...ice. The moduli of uniqueness and pointwise continuity from Bridges [8] and [9] allow a similar improvement. Furthermore our modulus Φ3 improves a modulus of uniqueness for Pn which is implicit in Ko =-=[18]-=-,[19] (see [24] for details). 22Notes ∧ ∧ 1) Here and in the following we may also have tuples x1 ∈ X1, . . . , xn ∈ Xn(. . .). 2) By A0, B0, C0, . . . we denote always quantifier–free formulas. 3) L... |

5 |
Some theorems on Cebysev approximation
- Newman, S
- 1963
(Show Context)
Citation Context ...greatest γ ∈ IR ∗ + such that ∧ (∗) p ∈ H(‖f − p‖∞ ≥ ‖f − pb‖∞ + γ · ‖p − pb‖∞) holds. (Here H ⊂ C[0, 1] denotes a Haar space.) The existence of a γ satisfying (∗) was proved (ineffectively) first in =-=[28]-=- (see also [10] ). A proof of this fact is already implicit in [12] (see [5] ). For more information on strong unicity see [24] . 5.3.2 shows that the concept ‘modulus of uniqueness’ generalizes the c... |

4 |
Equivalence of bar recursors in the theory of functionals of finite type
- Bezem
- 1988
(Show Context)
Citation Context ...dden in the implicative premise ‘f represents a Cauchy sequence of rationals with modulus 2 −k ’ are eliminated by the use of f ↦→ f. See Kohlenbach [23] 3 (and also [1] ) for details on this. 4) In =-=[3]-=- it is shown that this form of ER–qf is in fact derivable from the simpler one without A0. However for the formalization of given proofs our version is more convenient. 5) Instead of Y, Y ′ , X ′′ , x... |

4 |
General theory of approximation by functions involving a given number of arbitrary parameters
- Young
- 1907
(Show Context)
Citation Context ... common proof from de La Vallée Poussin [29] ( 56) (as presented with all details e.g. in [27] ), 2) a proof due to Kirchberger [17] and Borel [6] and 3) a simplification of a proof sketched by Young =-=[34]-=- (and worked out in Rice [30] ). From all three proofs i = 1, 2, 3 we obtained moduli of uniqueness Φi which are linear in q if the data f ∈ C[0, 1], n ∈ IN are enriched by a lower estimate 0 < lf,n ≤... |

4 |
Eine Ungleichung für Tschebyscheffsche Approximationspolynome
- Freud
- 1958
(Show Context)
Citation Context ...· ‖p − pb‖∞) holds. (Here H ⊂ C[0, 1] denotes a Haar space.) The existence of a γ satisfying (∗) was proved (ineffectively) first in [28] (see also [10] ). A proof of this fact is already implicit in =-=[12]-=- (see [5] ). For more information on strong unicity see [24] . 5.3.2 shows that the concept ‘modulus of uniqueness’ generalizes the concept of strong unicity. In [23],[24] we analyze three different p... |

3 |
Konstruktive Funktionentheorie. Akademie–Verlag, Berlin (German translations of the original russian edition
- Natanson
- 1949
(Show Context)
Citation Context ...eff approximation of f ∈ C[0, 1] by polynomials ∈ Pn (the third one also for general Haar spaces): 1) the most common proof from de La Vallée Poussin [29] ( 56) (as presented with all details e.g. in =-=[27]-=- ), 2) a proof due to Kirchberger [17] and Borel [6] and 3) a simplification of a proof sketched by Young [34] (and worked out in Rice [30] ). From all three proofs i = 1, 2, 3 we obtained moduli of u... |

2 | Lipschitz continuity and strong unicity in G. Freud’s work - Blatt - 1986 |

2 | Eine Ungleichung fur Tschebysche#sche Approximationspolynome - Freud - 1958 |

2 |
Über Tschebychefsche Annäherungsmethoden. Dissertation
- Kirchberger
- 1902
(Show Context)
Citation Context ...lynomials ∈ Pn (the third one also for general Haar spaces): 1) the most common proof from de La Vallée Poussin [29] ( 56) (as presented with all details e.g. in [27] ), 2) a proof due to Kirchberger =-=[17]-=- and Borel [6] and 3) a simplification of a proof sketched by Young [34] (and worked out in Rice [30] ). From all three proofs i = 1, 2, 3 we obtained moduli of uniqueness Φi which are linear in q if ... |

2 |
The approximation of functions, vol.1
- Rice
- 1964
(Show Context)
Citation Context ...ée Poussin [29] ( 56) (as presented with all details e.g. in [27] ), 2) a proof due to Kirchberger [17] and Borel [6] and 3) a simplification of a proof sketched by Young [34] (and worked out in Rice =-=[30]-=- ). From all three proofs i = 1, 2, 3 we obtained moduli of uniqueness Φi which are linear in q if the data f ∈ C[0, 1], n ∈ IN are enriched by a lower estimate 0 < lf,n ≤ dist(f, Pn) (lf,n ∈ Q ∗ +), ... |

1 |
Collected Works, Vol.II
- Godel
- 1937
(Show Context)
Citation Context ...h does not depend on x ∈ K, i.e. ∧ (3) x ∈ K, k ∈ IN ( |F (x)| ≤ 2 −Φk → |G(x)| < 2 −k) . (For a very nice introduction to functional interpretation we refer to Troelstra’s introductory notes [33] in =-=[15]-=-. Most parts of the present paper presuppose only information on functional interpretation which can be found in these notes). In 2 we present a new monotone version of Gödel’s functional interpretati... |

1 | growth in standard parts of analysis. Preprint xv+166 pp - Kohlenbach, Real - 1995 |

1 |
Introductory note to 1958
- Troelstra
- 1990
(Show Context)
Citation Context ...∨ n which does not depend on x ∈ K, i.e. ∧ (3) x ∈ K, k ∈ IN ( |F (x)| ≤ 2 −Φk → |G(x)| < 2 −k) . (For a very nice introduction to functional interpretation we refer to Troelstra’s introductory notes =-=[33]-=- in [15]. Most parts of the present paper presuppose only information on functional interpretation which can be found in these notes). In 2 we present a new monotone version of Gödel’s functional inte... |

1 | Pointwise hereditary majorization and some - Kohlenbach - 1992 |