## Ramsey's Theorem in Type Theory (1993)

Citations: | 4 - 1 self |

### BibTeX

@TECHREPORT{Fridlender93ramsey'stheorem,

author = {Daniel Fridlender},

title = {Ramsey's Theorem in Type Theory},

institution = {},

year = {1993}

}

### OpenURL

### Abstract

We present formalizations of constructive proofs of the Intuitionistic Ramsey Theorem and Higman's Lemma in Martin-Lof's Type Theory. We analyze the computational content of these proofs and we compare it with programs extracted out from some classical proofs. Contents 1 Introduction 2 2 The proofs 4 2.1 An inductive formulation of almost-fullness (AF ID ) : : : : : : : : : : 5 2.1.1 Intuitionistic Ramsey Theorem (IRT ID ) : : : : : : : : : : : : 7 2.1.2 Higman's Lemma (HL ID ) : : : : : : : : : : : : : : : : : : : : 12 2.2 A negationless inductive formulation of almost-fullness (AF I ) : : : : : 17 2.2.1 Intuitionistic Ramsey Theorem (IRT I ) : : : : : : : : : : : : : 17 2.3 Equivalence between the various formulations of almost-fullness : : : 20 3 The programs 22 3.1 A higher order program : : : : : : : : : : : : : : : : : : : : : : : : : 24 3.2 A first order program : : : : : : : : : : : : : : : : : : : : : : : : : : : 25 4 Computational content of classical proofs 28 4.1 A cl...

### Citations

261 |
Programming in Martin-Löf’s Type Theory: An Introduction
- Nordstrom, Petersson, et al.
- 1990
(Show Context)
Citation Context ...rovability of IRT in Type Theory. 2 The proofs All the proofs presented in subsections 2.1 and 2.2 have been formalized in ALF (see [1, 6]) which is an implementation of Martin-Lof's Type Theory (see =-=[10]-=-). The ALF system itself is the appropriate tool to read the completely formal proofs. In the presentation of the proofs we write t 0 ; : : : ; t n : T for t 0 ; : : : ; t n are objects of type T . Se... |

134 |
Elements of Intuitionism
- Dummett
- 1977
(Show Context)
Citation Context ...; : : : : A:9n:L s ([a 0 ; : : : ; a n\Gamma1 ]) 8l : A :L s (l) ) P(l) 8l : A :(8a : A:P(l \Delta a)) ) P(l) P([ ]) The principle of bar induction was first introduced by Brouwer and is explained in =-=[4]-=-. It is very useful when proving theorems in which a hypothesis is of the form of its second premiss 8a 0 ; : : : ; a n ; : : : : A:9n:L s ([a 0 ; : : : ; a n\Gamma1 ]) (1) This is in fact the case bo... |

27 |
A short description of another logical framework
- Augustsson, Coquand, et al.
- 1990
(Show Context)
Citation Context ...he Intuitionistic Ramsey Theorem given in [13]. It justifies the unprovability of IRT in Type Theory. 2 The proofs All the proofs presented in subsections 2.1 and 2.2 have been formalized in ALF (see =-=[1, 6]-=-) which is an implementation of Martin-Lof's Type Theory (see [10]). The ALF system itself is the appropriate tool to read the completely formal proofs. In the presentation of the proofs we write t 0 ... |

19 |
A constructive proof of Higman’s lemma
- Murthy, Russell
- 1990
(Show Context)
Citation Context ...this might be given by using the notion of accessibility to prove that in each step the state of the program is reduced with respect to a well-founded relation. Such a way was adopted for instance in =-=[8]-=-. Stefano Berardi has extracted from a classical proof by game interpretation the program presented in subsection 4.2. We conjectured that the classical proof presented in subsection 4.1 has as comput... |

16 |
Pattern matching with dependent types. In Proceeding from the logical framework workshop at B*astad
- Coquand
- 1992
(Show Context)
Citation Context ...serve that no list can be good and bad at the same time, and if A is decidable, any list is either good or bad. In the formalizations we make extensive use of the pattern matching feature of ALF (see =-=[3]-=-) not only to get more readable expressions but also because it avoids introducing universes which would otherwise be necessary. In [3] it is required that for a constant f of arity n defined by patte... |

16 |
The new implementation of ALF
- Magnusson
- 1992
(Show Context)
Citation Context ...he Intuitionistic Ramsey Theorem given in [13]. It justifies the unprovability of IRT in Type Theory. 2 The proofs All the proofs presented in subsections 2.1 and 2.2 have been formalized in ALF (see =-=[1, 6]-=-) which is an implementation of Martin-Lof's Type Theory (see [10]). The ALF system itself is the appropriate tool to read the completely formal proofs. In the presentation of the proofs we write t 0 ... |

16 | Ramsey's theorem and the pigeonhole principle in intuitionistic mathematics
- Veldman, Bezem
- 1992
(Show Context)
Citation Context ...a quasi-ordering. Any almost full relation has to be reflexive (applying the definition to the sequence a; : : : ; a; : : : we get a A a) but may not be transitive. The name almost full is taken from =-=[13]-=- where it stands for a similar notion (see appendix A). 2 The Intuitionistic Ramsey Theorem takes his name from [13], where it stands for an intuitionistically provable reformulation of Ramsey's Theor... |

6 |
Well quasi-ordered sets
- Richman, Stolzenberg
- 1990
(Show Context)
Citation Context ...two) positions 4 . This possibility is mentioned in [3]. 2.1 An inductive formulation of almost-fullness (AF ID ) This subsection describes a formalization of the proof of Higman's Lemma presented in =-=[11]-=-. A difference between that proof and the one presented here is that we use almost full relations instead of well-quasi-orderings, which are just transitive almost 4 The equation: merge (a.as) (b.bs) ... |

2 |
Effectivit'e de calculs polynomiaux
- Lejeune-Jalabert
- 1985
(Show Context)
Citation Context ...= f(m) which is contradictory. This means that (42) does not hold and therefore that (41) does. 2 5 Further work The original aim was to formalize in ALF a program to construct Grobner bases given in =-=[5]-=-, the termination proof of which turned out to be the proof that the component-wise relation on tuples of natural numbers is almost full. We now have formalized proofs that it is almost full but nothi... |

1 |
A semantics of evidence for classical arithmetic. Proceedings of the workshop on types for proofs and programs
- Coquand
- 1991
(Show Context)
Citation Context ...onal content of the proof of IRT I given in subsection 2.2. Its second subsection gives another program for a classical proof which was obtained by Stefano Berardi by game interpretation (we refer to =-=[2]-=- for a description of game interpretation). Finally, appendix A shows the equivalence between IRT and the formulation of the Intuitionistic Ramsey Theorem given in [13]. It justifies the unprovability... |

1 |
Terminating General Recursion. Programming Methodology Group's report 46
- Nordstrom
- 1988
(Show Context)
Citation Context ...l i . This amounts to say that the relation ��Asis well-founded on the subset bad (A s ; A ). To formalize the notion of well-founded relation on a subset we take the notion of accessibility given=-= in [9]. Th-=-e one presented there is useful to express the idea of being a well-founded relation on a set. We reformulate it to express that property on a subset. We define accessibility for any relation ��A ... |

1 |
From ALF to LML
- Sydow
- 1993
(Show Context)
Citation Context ...n LML. 8 In deriving this programs from the proof that the component-wise relation on pairs of natural numbers is almost full two tools were helpful: a translator from ALF proofs to LML programs (see =-=[12]-=-) and the LML tracer. The first one, given in subsection 3.1, is intended to have the same structure as the proof. This is stressed by explaining the correspondence between functions in the program an... |