## Generic proof synthesis for Presburger arithmetic (2003)

Citations: | 4 - 3 self |

### BibTeX

@TECHREPORT{Chaieb03genericproof,

author = {Amine Chaieb and Tobias Nipkow},

title = {Generic proof synthesis for Presburger arithmetic},

institution = {},

year = {2003}

}

### OpenURL

### Abstract

We develop in complete detail an extension of Cooper’s decision procedure for Presburger arithmetic that returns a proof of the equivalence of the input formula to a quantifier-free formula. For closed input formulae this is a proof of their validity or unsatisfiability. The algorithm is formulated as a functional program that makes only very minimal assumptions w.r.t. the underlying logical system and is therefore easily adaptable to specific theorem provers.

### Citations

802 |
Isabelle/HOL – A Proof Assistant for Higher-Order Logic. LNCS 2283
- Nipkow, Paulson, et al.
- 2002
(Show Context)
Citation Context ...e we very specific about the underlying logic, except that we assume it is classical. As evidence of the genericity of our code we can offer that it is derived from an implementation for Isabelle/HOL =-=[15]-=- by the first author [3] yet is quite close to an implementation in the HOL system [6]. Although we are fully aware of the computational complexity of Presburger arithmetic, we have not tried to optim... |

207 |
A proof generating system for higher-order logic
- Gordon
- 1988
(Show Context)
Citation Context ...s evidence of the genericity of our code we can offer that it is derived from an implementation for Isabelle/HOL [15] by the first author [3] yet is quite close to an implementation in the HOL system =-=[6]-=-. Although we are fully aware of the computational complexity of Presburger arithmetic, we have not tried to optimize our implementation. It is meant as an easy to understand starting point for other ... |

96 |
Theorem proving in arithmetic without multiplication
- Cooper
- 1972
(Show Context)
Citation Context ...n for PA Presburger arithmetic Presburger arithmetic (PA) is the first-order logic over the integers with + and <. Presburger [19] first showed its decidability. We extend Cooper’s decision procedure =-=[4]-=- such that a successful run returns a proof of equivalence of the input formula to a quantifier-free formula. The atomic PA-formulae are 0 = t, 0 < t and d dvd t, where d ∈ Z and t is buit up from var... |

94 | Theorem Proving with the real Numbers
- Harrison
- 1996
(Show Context)
Citation Context ...tinct formulae may be logically equivalent. Hence the relevant fragment of formulae must be represented (reflected) inside the logic as a datatype. Sometimes this datatype is called the shadow syntax =-=[9]-=-. For concreteness we call it rep. The two levels of formulae must be connected by two functions: interp, a function in the logic, maps an element of rep to the formula it represents. convert, a funct... |

88 |
C.P.: Edinburgh LCF: A Mechanised Logic
- Gordon, Milner, et al.
- 1979
(Show Context)
Citation Context ...e to be proved, solutions are theorems, and termination will be guaranteed because all decompositions yield syntactically smaller terms. This style of theorem proving was invented with the LCF system =-=[7, 18]-=-, where decomp is called a tactic. Hence we refer to it as tactic-style theorem proving. Note that the interface of our theorem data type is quite abstract. Theorems may be implemented as full-blown p... |

63 |
Functional unification of higher-order patterns
- Nipkow
- 1993
(Show Context)
Citation Context ... always be one occurrence of P (x) among the premises of th such that x is a bound variable. In this case there is at most one matching substitution because it is a special case of patternunification =-=[14]-=-. Further occurrences of P among the premises cannot lead to further matching substitutions but can only rule some out. Hence it is justified to speak of the matcher. Furthermore this kind of higher-o... |

60 | Metatheory and reflection in theorem proving: A survey and critique
- Harrison
- 1995
(Show Context)
Citation Context ...on follows a second implementation by Harrison in HOL Light [10]. Thus our work can partly be seen as one of lifting Harrison’s implementations from HOL to a more abstract and generic level. Harrison =-=[8]-=- has also studied the general issue of reflection in LCF-like 2stheorem provers and comes to the conclusion that a truly convincing example of the superiority (in terms of efficiency) of reflection is... |

14 |
Programming and computing in HOL
- Barras
(Show Context)
Citation Context ...e theorem θ(A). The free variables in a theorem th can be instantiated from left to right with terms t1, . . . , tn by writing th[t1, . . . , tn]. For example, if th is the theorem m ≤ m + n·n then th=-=[1, 2]-=- is the theorem 1 ≤ 1 + 2·2. Function gen performs ∀-introduction: it takes a variable x and a theorem P (x) and returns the theorem ∀x.P (x). We assume that the underlying theorem prover provides a f... |

11 | Verifying and reflecting quantifier elimination for Presburger arithmetic
- Chaieb, Nipkow
- 2005
(Show Context)
Citation Context ...e theorem θ(A). The free variables in a theorem th can be instantiated from left to right with terms t1, . . . , tn by writing th[t1, . . . , tn]. For example, if th is the theorem m ≤ m + n·n then th=-=[1, 2]-=- is the theorem 1 ≤ 1 + 2·2. Function gen performs ∀-introduction: it takes a variable x and a theorem P (x) and returns the theorem ∀x.P (x). We assume that the underlying theorem prover provides a f... |

10 | Validated Proof-Producing Decision Procedures
- Klapper, Stump
- 2004
(Show Context)
Citation Context ...urk in the code for a long time because they are not caught by a standard static type system which cannot express the precise form the theorem produced or expected by some function must have (but see =-=[12]-=-). This problem is exacerbated by the fact that decision procedures are re-implemented time and again for different systems, and that in the literature these implementations 1sare only sketched if dis... |

10 | Complete integer decision procedures as derived rules in HOL - Norrish - 2003 |

5 |
Une procédure de décision réflexive pour un fragment de l’arithmétique de Presburger
- Crégut
- 2004
(Show Context)
Citation Context ... principles on an abstract level but omits the details of proof synthesis. For that he refers to the actual code, which we would argue is much too system specific to be easily portable. Pierre Crégut =-=[5]-=- presents a reflective version of the Omega test written for Coq, where an optimized proof trace is interpreted to solve the goal. Unlike the other references his implementation only deals with quanti... |

2 |
Isabelle trifft Presburger Arithmetik
- Chaieb
(Show Context)
Citation Context ...the underlying logic, except that we assume it is classical. As evidence of the genericity of our code we can offer that it is derived from an implementation for Isabelle/HOL [15] by the first author =-=[3]-=- yet is quite close to an implementation in the HOL system [6]. Although we are fully aware of the computational complexity of Presburger arithmetic, we have not tried to optimize our implementation. ... |

2 |
An Interpretation of Isabelle/HOL in HOL Light
- McLauglin
- 2006
(Show Context)
Citation Context ... surprises at runtime. The explicitness of reflection pays off even more during maintenance, where tactics can be awkward to modify. Due to the progress in sharing theorems with other theorem provers =-=[13, 17]-=-, reflection provide these decision procedures for free. A final advantage of reflection is that it allows to formalize notions like duality, cf. § 4.6, which reduces the size of the background theory... |