## Proving Equalities in a Commutative Ring Done Right in Coq (2005)

Venue: | Theorem Proving in Higher Order Logics (TPHOLs 2005), LNCS 3603 |

Citations: | 28 - 0 self |

### BibTeX

@INPROCEEDINGS{Gregoire05provingequalities,

author = {Benjamin Gregoire and Assia Mahboubi},

title = {Proving Equalities in a Commutative Ring Done Right in Coq},

booktitle = {Theorem Proving in Higher Order Logics (TPHOLs 2005), LNCS 3603},

year = {2005},

pages = {98--113},

publisher = {Springer}

}

### Years of Citing Articles

### OpenURL

### Abstract

We present a new implementation of a reflexive tactic which solves equalities in a ring structure inside the Coq system. The e#ciency is improved to a point that we can now prove equalities that were previously beyond reach. A special care has been taken to implement e#cient algorithms while keeping the complexity of the correctness proofs low.

### Citations

522 |
Interactive Theorem Proving and Program Development Coq’Art: The Calculus of Inductive Constructions
- BERTOT, CASTÉRAN
- 2004
(Show Context)
Citation Context ...oduce a new axiomatic structure, called almost-ring, which allows to unify the implementations of the procedure for rings and semi-rings. In Section 6 we show how the use of the new metalanguage Ltac =-=[5, 2]-=- allows to completely avoid the use of external Ocaml code. Section 7 is dedicated to examples and benchmarks before we conclude in Section 8. 2 Overall view of the method 2.1 Reflexivity In the Coq s... |

123 |
Quantifier elimination for the elementary theory of real closed fields by cylindrical algebraic decomposition
- Collins
- 1975
(Show Context)
Citation Context ...lgebra. In particular, the second author uses the Horner representation of polynomials to develop a decision procedure for real numbers theory based on G. Collins’ cylindrical algebraic decomposition =-=[4]-=-, which is a quite complex algorithm resting on numerous computations over polynomials (computations of gcd, subresultant coefficients,...). The efficiency of newring overcomes what was before a stron... |

90 | The semantics of reflected proof
- Allen, Constable, et al.
- 1990
(Show Context)
Citation Context ...e having the size of the current goal when the rewriting was performed, hence the size of the proof term heavily depends on the number of these rewriting steps. The reflection technique introduced by =-=[1]-=- takes benefit of the reduction system of the proof assistant to reduce the size of the proof term computed and consequently to speed up its checking. It relies on the following remark:s– Let P : A → ... |

68 | A compiled implementation of strong reduction
- Grégoire, Leroy
- 2002
(Show Context)
Citation Context ...quired. The Coq system has recently been improved by the introduction of a compiler and an abstract machine, which now allows the evaluation of Coq programs with the same efficiency as Ocaml programs =-=[8]-=-. After the experiences of marrying computer algebra systems with theorem provers to get both efficiency and reliability [9], it now seems reasonable to use Coq as a single environment for programming... |

52 | Using reflection to build efficient and certified decision procedures
- Boutin
- 1997
(Show Context)
Citation Context ...ary rewriting steps of the ring axioms. The Coq [12] proof assistant already provides such a tool called ring. It is not based on an automatic rewriting strategy but built using a reflexive technique =-=[3]-=-. The use of reflexivity has already reduced the size of the generatedsproof terms and the time for building and checking them. Nevertheless, the efficiency of ring is not satisfactory. For example, p... |

37 | A Tactic Language for the System Coq
- Delahaye
- 1955
(Show Context)
Citation Context ...oduce a new axiomatic structure, called almost-ring, which allows to unify the implementations of the procedure for rings and semi-rings. In Section 6 we show how the use of the new metalanguage Ltac =-=[5, 2]-=- allows to completely avoid the use of external Ocaml code. Section 7 is dedicated to examples and benchmarks before we conclude in Section 8. 2 Overall view of the method 2.1 Reflexivity In the Coq s... |

32 | Setoids in type theory
- Barthe, Capretta, et al.
(Show Context)
Citation Context ...lity, but an equivalence relation adapted to the ring structure. For example, this is the case for an implementation of Q as Z ×N ∗ . A set equipped with such an equality relation is called a setoid (=-=[7]-=-,[10]). Proving equalities in such a setoid ring requires extra properties stating that all the ring operations are compatibles with the given setoid equality. In the case the equality involved in the... |

28 |
L.: A skeptic’s approach to combining HOL and Maple
- Harrison, Théry
- 1998
(Show Context)
Citation Context ...s the evaluation of Coq programs with the same efficiency as Ocaml programs [8]. After the experiences of marrying computer algebra systems with theorem provers to get both efficiency and reliability =-=[9]-=-, it now seems reasonable to use Coq as a single environment for programming, certifying and evaluating computer algebra algorithms. Our newring decision procedure is one of these efficient tools requ... |

17 | A simple model for quotient types
- Hofmann
- 1995
(Show Context)
Citation Context ..., but an equivalence relation adapted to the ring structure. For example, this is the case for an implementation of Q as Z ×N ∗ . A set equipped with such an equality relation is called a setoid ([7],=-=[10]-=-). Proving equalities in such a setoid ring requires extra properties stating that all the ring operations are compatibles with the given setoid equality. In the case the equality involved in the goal... |

13 |
Field: une procédure de décision pour les nombres réels en Coq
- Delahaye, Mayero
- 2001
(Show Context)
Citation Context ... are the way to separate computations from checking, inside the proof assistant. Furthermore it is the first step for a bunch of other decision procedures, like the simplification of field equalities =-=[6]-=-, or decision methods in geometry [11]. In Section 2, we begin with some general remarks about the reflexive method and its use in our particular context. The Section 3 is dedicated to our choice to g... |

11 | A Decision Procedure for Geometry in Coq
- Narboux
(Show Context)
Citation Context ... from checking, inside the proof assistant. Furthermore it is the first step for a bunch of other decision procedures, like the simplification of field equalities [6], or decision methods in geometry =-=[11]-=-. In Section 2, we begin with some general remarks about the reflexive method and its use in our particular context. The Section 3 is dedicated to our choice to get efficient representation of polynom... |