## Complete integer decision procedures as derived rules in HOL (2003)

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Venue: | Theorem Proving in Higher Order Logics, TPHOLs 2003, volume 2758 of Lect. Notes in Comp. Sci |

Citations: | 10 - 0 self |

### BibTeX

@INPROCEEDINGS{Norrish03completeinteger,

author = {Michael Norrish},

title = {Complete integer decision procedures as derived rules in HOL},

booktitle = {Theorem Proving in Higher Order Logics, TPHOLs 2003, volume 2758 of Lect. Notes in Comp. Sci},

year = {2003},

pages = {71--86},

publisher = {Springer-Verlag}

}

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### Abstract

Abstract. I describe the implementation of two complete decision procedures for integer Presburger arithmetic in the HOL theorem-proving system. The first procedure is Cooper’s algorithm, the second, the Omega Test. Between them, the algorithms illustrate three different implementation techniques in a fully expansive system. 1

### Citations

575 | PVS: a prototype verification system
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(Show Context)
Citation Context ...plicitly invoked by the user, they remove a great deal of tedium from the task of proving goals. Modern interactive systems, including ACL2 [10], Coq [1], HOL [4,11], Isabelle [13], Nuprl [8] and PVS =-=[12]-=-, all implement such decision procedures. There are essentially three procedures implemented in the systems mentioned above: Fourier-Motzkin variable elimination (in HOL, Isabelle and Coq 1 ), SUP-INF... |

203 |
Melham, editors. Introduction to HOL: A Theorem Proving Environment for Higher Order Logic
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Citation Context ...routines and running automatically, or explicitly invoked by the user, they remove a great deal of tedium from the task of proving goals. Modern interactive systems, including ACL2 [10], Coq [1], HOL =-=[4,11]-=-, Isabelle [13], Nuprl [8] and PVS [12], all implement such decision procedures. There are essentially three procedures implemented in the systems mentioned above: Fourier-Motzkin variable elimination... |

136 |
Edinburgh LCF – A Mechanised Logic of Computation. LNCS 78
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(Show Context)
Citation Context ...tisfiable. 2. Check the dark shadow. If it is satisfiable, so is the original. 3. Check the splinters. �s78 M. Norrish 4 Implementations in HOL HOL is a theorem-proving system in the tradition of LCF =-=[5]-=-. Theorems are implemented as an abstract data type in the language SML, exploiting that language’s strong typing system, which guarantees that a type’s abstraction barrier can not be subverted. The i... |

110 | An industrial strength theorem prover for a logic based on common lisp
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(Show Context)
Citation Context ... in simplification routines and running automatically, or explicitly invoked by the user, they remove a great deal of tedium from the task of proving goals. Modern interactive systems, including ACL2 =-=[10]-=-, Coq [1], HOL [4,11], Isabelle [13], Nuprl [8] and PVS [12], all implement such decision procedures. There are essentially three procedures implemented in the systems mentioned above: Fourier-Motzkin... |

97 |
Theorem proving in arithmetic without multiplication
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- 1972
(Show Context)
Citation Context ...In this paper, I describe two other algorithms. The Omega Test [14] is an extension of Fourier-Motzkin variable elimination, which makes it complete over Z. The second procedure is Cooper’s algorithm =-=[3]-=-, which is unlike the other algorithms in not requiring formulas to be in DNF before it eliminates a quantifier. Both of the algorithms I describe differ in scope from the others mentioned: they are b... |

46 | Integrating Gandalf and HOL
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(Show Context)
Citation Context ...very short in comparison to the work done in exploring all of the possible paths to a negated goal’s refutation. This approach is exemplified by Hurd’s linking of HOL to the Gandalf resolution prover =-=[7]-=-. There the external tool is external not just to the logical kernel, but to HOL itself. On proving a goal, Gandalf provides a log of the successful resolution and modulation steps required, and Hurd’... |

24 | Efficiency in a fully-expansive theorem prover
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- 1993
(Show Context)
Citation Context ...d Hurd’s HOL code then interprets this proof-log “back into” the logical kernel. Another example of the technique is Boulton’s implementation of his procedure for universal Presburger arithmetic on N =-=[2]-=-. On this domain, Fourier-Motzkin variable elimination can be seen as a refutation procedure. Boulton translates the negated goal into a special data structure representing the set of known constraint... |

24 |
The Nuprl proof development system, version 4.2 reference manual and user’s guide
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(Show Context)
Citation Context ...cally, or explicitly invoked by the user, they remove a great deal of tedium from the task of proving goals. Modern interactive systems, including ACL2 [10], Coq [1], HOL [4,11], Isabelle [13], Nuprl =-=[8]-=- and PVS [12], all implement such decision procedures. There are essentially three procedures implemented in the systems mentioned above: Fourier-Motzkin variable elimination (in HOL, Isabelle and Coq... |

13 | A thread of HOL development
- Norrish, Slind
(Show Context)
Citation Context ...routines and running automatically, or explicitly invoked by the user, they remove a great deal of tedium from the task of proving goals. Modern interactive systems, including ACL2 [10], Coq [1], HOL =-=[4,11]-=-, Isabelle [13], Nuprl [8] and PVS [12], all implement such decision procedures. There are essentially three procedures implemented in the systems mentioned above: Fourier-Motzkin variable elimination... |

6 | A comparison of decision procedures in Presburger arithmetic. Research paper no. 872, Division of Informatics
- Janicic, Green, et al.
- 1997
(Show Context)
Citation Context ...rously would need careful construction. In recent work, Janičić, Green and Bundy compared implementations of Cooper’s algorithm and Hodes’s method applied to universal natural number Presburger goals =-=[9]-=-. The tests used for this work were generated randomly, and suggested that Cooper’s algorithm could perform as well as Hodes’s method. 5 Extensions After implementing the basic algorithms, it is possi... |

4 |
Theorem Proving with the Real Numbers. CPHC/BCS Distinguished Dissertations
- Harrison
- 1998
(Show Context)
Citation Context ... Q and R. If, for example, P is a predicate on Z, with HOL type int -> bool, it is difficult to directly define a function to calculate P−∞. 5 Instead, Harrison’s “shadow syntax” approach can be used =-=[6]-=-: a concrete type is used to implement a syntax for the formula, and an interpretation function relates this new syntax back to the original domain. Thus, in his implementation of Kreisel and Krivine’... |

1 |
See the Coq home-page at http://coq.inria.fr
- Barras, Boutin, et al.
- 1997
(Show Context)
Citation Context ...fication routines and running automatically, or explicitly invoked by the user, they remove a great deal of tedium from the task of proving goals. Modern interactive systems, including ACL2 [10], Coq =-=[1]-=-, HOL [4,11], Isabelle [13], Nuprl [8] and PVS [12], all implement such decision procedures. There are essentially three procedures implemented in the systems mentioned above: Fourier-Motzkin variable... |