Program Extraction from Proofs: The Fan Theorem for Uniformly Coconvex Bars

2019 ◽  
pp. 333-341
Author(s):  
Helmut Schwichtenberg
Keyword(s):  
Program Extraction ◽  
Fan Theorem

A TYPE-FREE THEORY OF HALF-MONOTONE INDUCTIVE DEFINITIONS

1995 ◽  
Vol 06 (03) ◽  
pp. 203-234 ◽  
Author(s):  
YUKIYOSHI KAMEYAMA
Keyword(s):  
Type Theory ◽  
Characteristic Point ◽  
Logical Theory ◽  
Inductive Definition ◽  
Program Extraction ◽  
First Order ◽  
Inductive Definitions ◽  
Free Theory ◽  
Program Proof ◽  
Definition Of

This paper studies an extension of inductive definitions in the context of a type-free theory. It is a kind of simultaneous inductive definition of two predicates where the defining formulas are monotone with respect to the first predicate, but not monotone with respect to the second predicate. We call this inductive definition half-monotone in analogy of Allen’s term half-positive. We can regard this definition as a variant of monotone inductive definitions by introducing a refined order between tuples of predicates. We give a general theory for half-monotone inductive definitions in a type-free first-order logic. We then give a realizability interpretation to our theory, and prove its soundness by extending Tatsuta’s technique. The mechanism of half-monotone inductive definitions is shown to be useful in interpreting many theories, including the Logical Theory of Constructions, and Martin-Löf’s Type Theory. We can also formalize the provability relation “a term p is a proof of a proposition P” naturally. As an application of this formalization, several techniques of program/proof-improvement can be formalized in our theory, and we can make use of this fact to develop programs in the paradigm of Constructive Programming. A characteristic point of our approach is that we can extract an optimization program since our theory enjoys the program extraction theorem.

scholarly journals An improvement of Ky Fan theorem for matrix eigenvalues

2005 ◽  
Vol 49 (5-6) ◽  
pp. 789-803 ◽  
Author(s):  
Hou-Biao Li ◽  
Ting-Zhu Huang
Keyword(s):  
Fan Theorem ◽  
Matrix Eigenvalues ◽  
Ky Fan ◽  
Ky Fan Theorem

scholarly journals Program Extraction from Normalization Proofs

Studia Logica ◽  
2006 ◽  
Vol 82 (1) ◽  
pp. 25-49 ◽  
Author(s):  
Ulrich Berger ◽  
Stefan Berghofer ◽  
Pierre Letouzey ◽  
Helmut Schwichtenberg
Keyword(s):  
Program Extraction

scholarly journals SEPARATING THE FAN THEOREM AND ITS WEAKENINGS

2014 ◽  
Vol 79 (3) ◽  
pp. 792-813 ◽  
Author(s):  
ROBERT S. LUBARSKY ◽  
HANNES DIENER
Keyword(s):  
Constructive Mathematics ◽  
Kripke Models ◽  
Fan Theorem

AbstractVarieties of the Fan Theorem have recently been developed in reverse constructive mathematics, corresponding to different continuity principles. They form a natural implicational hierarchy. Some of the implications have been shown to be strict, others strict in a weak context, and yet others not at all, using disparate techniques. Here we present a family of related Kripke models which separates all of the as yet identified fan theorems.

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