## A computational approach to pocklington certificates in type theory (2006)

Venue: | In Proc. of the 8th Int. Symp. on Functional and Logic Programming, volume 3945 of LNCS |

Citations: | 15 - 5 self |

### BibTeX

@INPROCEEDINGS{Grégoire06acomputational,

author = {Benjamin Grégoire and Laurent Théry and Benjamin Werner},

title = {A computational approach to pocklington certificates in type theory},

booktitle = {In Proc. of the 8th Int. Symp. on Functional and Logic Programming, volume 3945 of LNCS},

year = {2006},

pages = {97--113},

publisher = {Springer}

}

### OpenURL

### Abstract

Abstract. Pocklington certificates are known to provide short proofs of primality. We show how to perform this in the framework of formal, mechanically checked, proofs. We present an encoding of certificates for the proof system Coq which yields radically improved performances by relying heavily on computations inside and outside of the system (twolevel approach). 1 Formal Computational Proofs 1.1 Machines and the Quest for Correctness It is generally considered that modern mathematical logic was born towards the end of 19 th century, with the work of logicians like Frege, Peano, Russell or Zermelo, which lead to the precise definition of the notion of logical deduction and to formalisms like arithmetic, set theory or early type theory. From then on, a mathematical proof could be understood as a mathematical object itself, whose correction obeys some well-defined syntactical rules. In most formalisms, a formal proof is viewed as some tree-structure; in natural deduction for instance, given to formal proofs σA and σB respectively of propositions A and B, these can be combined in order to build a proof of A ∧ B: σA σB ⊢ A ⊢ B ⊢ A ∧ B To sum things up, the logical point of view is that a mathematical statement holds in a given formalism if there exists a formal proof of this statement which follows the syntactical rules of the formalism. A traditional mathematical text can then be understood as an informal description of the formal proof. Things changed in the 1960-ties, when N.G. de Bruijn’s team started to use computers to actually build formal proofs and verify their correctness. Using the fact that data-structures like formal proofs are very naturally represented in a computer’s memory, they delegated the proof-verification work to the machine; their software Automath is considered as the first proof-system and is the common

### Citations

68 | A compiled implementation of strong reduction
- Grégoire, Leroy
- 2002
(Show Context)
Citation Context ...es (here 3 and 2). So, the verification routine will verify many times these certificates. It order to share certificates, we drop trees by flattening them to lists. In this case this yields: [{127,3,=-=[7;3;2]-=-};{7,2,[3;2]};{3,2,[2]};(2,prime2)]. We have replaced recursive certificates by their corresponding prime number in decompositions. These certificates appear in the tail of the list. Note that doing s... |

61 |
Algebra, volume 211 of Graduate Texts in Mathematics
- Lang
- 2002
(Show Context)
Citation Context ... property of cyclotomic polynomial 5 : x n +1=(x 2n − 1)/(x n − 1) = � Φd(x)/ � Φd(x) d|2n where Φd(x) is the cyclotomic polynomial, given by: Φd(x)= � (x δ − 1) µ(d/δ) where µ is the Möbius function =-=[9]-=-. Setting 2n =2tm with m odd, leads to: x n +1= � Φ2td(x) δ|d d|m where all the Φ2 t d(x) are divisors of x n + 1 that we can factorize. Using this trick we have been able to compute enough prime divi... |

23 | Autarkic computations in formal proofs
- Barendregt, Barendsen
- 2002
(Show Context)
Citation Context ...ker and slower computations which are part of the proof-checking process, one has to choose between two possibilities. A first one is to perform all computations inside the system, in an autarkic way =-=[3]-=-; this is what happens with the trivial algorithm mentioned above. Another one, is to remark it is easier to find a way out of a labyrinth, when some outside source provides one with an Ariadne’s thre... |

13 |
The Determination of the Prime or Composite Nature of Large Numbers by Fermat's Theorem
- Pocklington
- 1914
(Show Context)
Citation Context ...Pocklingtoncertificatecomputedbyoutsidemeanswasfirstused by Caprotti and Oostdijk [6] whose work was the starting base for our effort. 2 Pocklington Certificates 2.1 The Theorem Pocklington’s theorem =-=[10]-=- dates back to 1914 and provides a sufficient condition for primality of natural numbers: Theorem 1. Given a natural numbers n>1 andawitnessaand some pairs (p1,α1),...,(pk,αk), it is sufficient for n ... |

10 |
A skeptic’s approach to combining HOL
- Harrison, Théry
- 1998
(Show Context)
Citation Context ...ormation on how computations should be performed and is thus a part of the resulting proof-object. This approach is called skeptical computations by Barendregt and was pioneered by Harrison and Théry =-=[8]-=-, although in a very different context. This work is typical for the skeptical computational approach. The idea of buildingaproofusingaPocklingtoncertificatecomputedbyoutsidemeanswasfirstused by Capro... |

9 |
New primality criteria and factorizations of 2 m 1
- Brillhart, Lehmer, et al.
- 1975
(Show Context)
Citation Context ...erate a certificate one needs to be able to partially factorize n − 1 at least till its square root. For our experiment we are using an improved version proposed by Brillhart, Lehmer and Selfridge in =-=[4]-=-. With this new version, we only need to partially factorize till the cube root. As factorizing n−1 is the time-consuming part of finding a certificate, this is a considerable improvement. The theorem... |

7 |
Formal and efficient primality proofs by use of computer algebra oracles
- Caprotti, Oostdijk
- 2001
(Show Context)
Citation Context ...ery different context. This work is typical for the skeptical computational approach. The idea of buildingaproofusingaPocklingtoncertificatecomputedbyoutsidemeanswasfirstused by Caprotti and Oostdijk =-=[6]-=- whose work was the starting base for our effort. 2 Pocklington Certificates 2.1 The Theorem Pocklington’s theorem [10] dates back to 1914 and provides a sufficient condition for primality of natural ... |

1 |
Is There Endogenous Long-Run Growth? Evidence from the U.S
- unknown authors
- 1997
(Show Context)
Citation Context ...es as argument a file containing a prime number and a partial decomposition of its predecessor. The syntax is the following: pocklington -dec file. We can use the trick described below and the tables =-=[5]-=- to build the file. These tables contain most of the factorizations of b n ±1forb =2,3,5,6,7,10,11,12 and n a high power (for b =2,n should be less than 1200). Those tables are available directly on t... |