## Proof Search with Set Variable Instantiation in the Calculus of Constructions (1996)

Venue: | Automated Deduction: CADE-13, volume 1104 of Lecture Notes in Arti Intelligence |

Citations: | 2 - 1 self |

### BibTeX

@INPROCEEDINGS{Felty96proofsearch,

author = {Amy Felty},

title = {Proof Search with Set Variable Instantiation in the Calculus of Constructions},

booktitle = {Automated Deduction: CADE-13, volume 1104 of Lecture Notes in Arti Intelligence},

year = {1996},

pages = {658--672},

publisher = {Springer}

}

### OpenURL

### Abstract

. We show how a procedure developed by Bledsoe for automatically finding substitution instances for set variables in higher-order logic can be adapted to provide increased automation in proof search in the Calculus of Constructions (CC). Bledsoe's procedure operates on an extension of first-order logic that allows existential quantification over set variables. The method finds maximal solutions for this special class of higher-order variables. This class of variables can also be identified in CC. The existence of a correspondence between higher-order logic and higher-order type theories such as CC is well-known. CC can be viewed as an extension of higher-order logic where the basic terms of the language, the simply-typed -terms, are replaced with terms containing dependent types. We adapt Bledsoe's procedure to the corresponding class of variables in CC and extend it to handle terms with dependent types. 1 Introduction Both higher-order logic and higher-order type theories serve as th...

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Citation Context ...at is incomplete but makes several restrictions including the elimination of infinite branching. This procedure is still complete for many interesting sublanguages of CC such as the Logical Framework =-=[11]-=- or higher-order hereditary Harrop (hohh) formulas, which serve as the logical foundation for the Prolog logic programming language [15]. Completeness for full CC can be regained by applying the restr... |

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Citation Context ...variety of interactive tactic-style theorem provers. For example, both HOL [10] and Isabelle [16] implement higher-order logic, while Coq [6] implements the Calculus of Constructions (CC) type theory =-=[5]-=- and Nuprl [4] implements MartinL of type theory [14]. Much work has been carried out in both kinds of systems on building tactics and automating proof search. However, little work has been done on pr... |

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Citation Context ... second-order polymorphic -calculus and secondorder propositional logic, the correspondence between higher-order logic and higherorder type theories is exact and known as the Curry-Howard isomorphism =-=[12]-=-. Although it is less direct for CC, one way to view the correspondence was shown in Felty [9]. Intuitively, a functional type P ! Q corresponds to an implication, while a dependent type 8x : P:Q corr... |

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Citation Context ...ny interesting sublanguages of CC such as the Logical Framework [11] or higher-order hereditary Harrop (hohh) formulas, which serve as the logical foundation for the Prolog logic programming language =-=[15]-=-. Completeness for full CC can be regained by applying the restricted procedure successively, proving a new lemma at each step and adding it to the assumptions before the next pass. Our work can be vi... |

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Citation Context ... For example, both HOL [10] and Isabelle [16] implement higher-order logic, while Coq [6] implements the Calculus of Constructions (CC) type theory [5] and Nuprl [4] implements MartinL of type theory =-=[14]-=-. Much work has been carried out in both kinds of systems on building tactics and automating proof search. However, little work has been done on providing the means for exploiting proof search methods... |

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Citation Context ...1 Introduction Both higher-order logic and higher-order type theories serve as the logical foundation of a variety of interactive tactic-style theorem provers. For example, both HOL [10] and Isabelle =-=[16]-=- implement higher-order logic, while Coq [6] implements the Calculus of Constructions (CC) type theory [5] and Nuprl [4] implements MartinL of type theory [14]. Much work has been carried out in both ... |

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Citation Context ...eractive tactic-style theorem provers. For example, both HOL [10] and Isabelle [16] implement higher-order logic, while Coq [6] implements the Calculus of Constructions (CC) type theory [5] and Nuprl =-=[4]-=- implements MartinL of type theory [14]. Much work has been carried out in both kinds of systems on building tactics and automating proof search. However, little work has been done on providing the me... |

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Citation Context ...s paper we show how the techniques described in Bledsoe [2] for the automatic discovery of substitutions for set variables can be incorporated directly into the search procedure for CC given by Dowek =-=[7, 8]-=-. In doing so, we both adapt these techniques to the type theoretic setting as well as extend them to handle the extra expressivity of dependent types. To incorporate dependent types, we consider not ... |

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Citation Context ...a framework in which to study how theorem proving techniques designed for one kind of system can be applied to proof search in the other. In this paper we show how the techniques described in Bledsoe =-=[2]-=- for the automatic discovery of substitutions for set variables can be incorporated directly into the search procedure for CC given by Dowek [7, 8]. In doing so, we both adapt these techniques to the ... |

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Citation Context ... dependent types. 1 Introduction Both higher-order logic and higher-order type theories serve as the logical foundation of a variety of interactive tactic-style theorem provers. For example, both HOL =-=[10]-=- and Isabelle [16] implement higher-order logic, while Coq [6] implements the Calculus of Constructions (CC) type theory [5] and Nuprl [4] implements MartinL of type theory [14]. Much work has been ca... |

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Citation Context ...ay "if term P has the form Q" to mean that P is fij-convertible to a term of the form Q. This convention is justified by rule (CONV). 3 Set Theory in the Calculus of Constructions It is show=-=n in Huet [13]-=- that higher-order logic is contained within CC (and thus also in Meta). Terms are introduced that encode the connectives and it is shown that the corresponding natural deduction inference rules are p... |

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Citation Context ...entation and installing it as a tactic in a larger system such as Coq. We also plan to adapt other procedures for automating the instantiation of set variables including the Z-match inference rule in =-=[1]-=-. It would also be interesting to extend Bledsoe's procedure directly by transferring our generalization carried out within the type theoretic setting back into the higher-order logic setting. In addi... |

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Citation Context ...gher-order type theories serve as the logical foundation of a variety of interactive tactic-style theorem provers. For example, both HOL [10] and Isabelle [16] implement higher-order logic, while Coq =-=[6]-=- implements the Calculus of Constructions (CC) type theory [5] and Nuprl [4] implements MartinL of type theory [14]. Much work has been carried out in both kinds of systems on building tactics and aut... |

4 | Encoding the calculus of constructions in a higher-order logic
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(Show Context)
Citation Context ...ween higher-order logic and higherorder type theories is exact and known as the Curry-Howard isomorphism [12]. Although it is less direct for CC, one way to view the correspondence was shown in Felty =-=[9]-=-. Intuitively, a functional type P ! Q corresponds to an implication, while a dependent type 8x : P:Q corresponds to universal quantification. An important ? To appear in Proceedings of the 13th Inter... |