Results 1  10
of
17
Higher Order Logic
 In Handbook of Logic in Artificial Intelligence and Logic Programming
, 1994
"... Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Definin ..."
Abstract

Cited by 20 (0 self)
 Add to MetaCart
Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Defining data types : : : : : : : : : : : : : : : : : : : : : 6 2.4 Describing processes : : : : : : : : : : : : : : : : : : : : : 8 2.5 Expressing convergence using second order validity : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.6 Truth definitions: the analytical hierarchy : : : : : : : : 10 2.7 Inductive definitions : : : : : : : : : : : : : : : : : : : : : 13 3 Canonical semantics of higher order logic : : : : : : : : : : : : 15 3.1 Tarskian semantics of second order logic : : : : : : : : : 15 3.2 Function and re
Developing certified programs in the system Coq  The Program tactic
, 1993
"... The system Coq is an environment for proof development based on the Calculus of Constructions extended by inductive definitions. Functional programs can be extracted from constructive proofs written in Coq. The extracted program and its corresponding proof are strongly related. The idea in this p ..."
Abstract

Cited by 12 (4 self)
 Add to MetaCart
The system Coq is an environment for proof development based on the Calculus of Constructions extended by inductive definitions. Functional programs can be extracted from constructive proofs written in Coq. The extracted program and its corresponding proof are strongly related. The idea in this paper is to use this link to have another approach: to give a program and to generate automatically the proof from which it could be extracted. Moreover, we introduce a notion of annotated programs.
ProofTerm Synthesis on Dependenttype Systems via Explicit Substitutions
, 1999
"... Typed #terms are used as a compact and linear representation of proofs in intuitionistic logic. This is possible since the CurryHoward isomorphism relates proof trees with typed #terms. The proofsasterms principle can be used to check a proof by type checking the #term extracted from the compl ..."
Abstract

Cited by 8 (1 self)
 Add to MetaCart
Typed #terms are used as a compact and linear representation of proofs in intuitionistic logic. This is possible since the CurryHoward isomorphism relates proof trees with typed #terms. The proofsasterms principle can be used to check a proof by type checking the #term extracted from the complete proof tree. However, proof trees and typed #terms are built differently. Usually, an auxiliary representation of unfinished proofs is needed, where type checking is possible only on complete proofs. In this paper we present a proof synthesis method for dependenttype systems where typed open terms are built incrementally at the same time as proofs are done. This way, every construction step, not just the last one, may be type checked. The method is based on a suitable calculus where substitutions as well as metavariables are firstclass objects.
ThirdOrder Matching in the Presence of Type Constructors
, 1994
"... We show that it is decidable whether a thirdorder matching problem in ! (an extension of the simply typed lambda calculus with type constructors) has a solution or not. We present an algorithm which, given such a problem, returns a solution for this problem if the problem has a solution and returns ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
We show that it is decidable whether a thirdorder matching problem in ! (an extension of the simply typed lambda calculus with type constructors) has a solution or not. We present an algorithm which, given such a problem, returns a solution for this problem if the problem has a solution and returns fail otherwise. We also show that it is undecidable whether a thirdorder matching problem in ! has a closed solution or not. 1 Introduction It is wellknown that type theory is a good basis for the implementation of proof checkers. Although there are various ways to use type theory for proof checking, they all exploit the fact that type theory provides a uniform way to represent and manipulate proofs, formulas and data types. The manmachine interaction of proof checking can be considerably improved if some kind of matching algorithm can be implemented for the terms of the underlying type theory. For if one wants to prove OE(t) for a certain formula OE and term t, and one already has a pr...
Proving Correctness of the Translation from MiniML to the CAM with the Coq Proof Development System
 with the Coq Proof Development System. Research report RR2536, INRIA, Rocquencourt
, 1995
"... In this article we show how we proved correctness of the translation from a small applicative language with recursive definitions (MiniML) to the Categorical abstract machine (CAM) using the Coq system. Our aim was to mechanise the proof of J. Despeyroux [10]. Like her, we use natural semantics to ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
In this article we show how we proved correctness of the translation from a small applicative language with recursive definitions (MiniML) to the Categorical abstract machine (CAM) using the Coq system. Our aim was to mechanise the proof of J. Despeyroux [10]. Like her, we use natural semantics to axiomatise the semantics of our languages. The axiomatisations of inferences systems and of the languages is nicely performed by the mechanism of inductive definitions in the Coq system. Unfortunately both the source and the target semantics involve nested structures that cannot be formalised inductively. We have overcome this problem by making some slight modifications of both the source and target semantics and show how the changes in the source and target semantics are related. For the remaining tranlation we explain how we can use the Coq system to formalize nonterminating programs and incorrect programs, objects that are impossible to explain with only the formalism of natural semantic...
Encoding the Calculus of Constructions in a HigherOrder Logic
 IN EIGHTH ANNUAL SYMPOSIUM ON LOGIC IN COMPUTER SCIENCE
, 1993
"... We present an encoding of the calculus of constructions (CC) in a higherorder intuitionistic logic (I) in a direct way, so that correct typing in CC corresponds to intuitionistic provability in a sequent calculus for I. In addition, we demonstrate a direct correspondence between proofs in these t ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
(Show Context)
We present an encoding of the calculus of constructions (CC) in a higherorder intuitionistic logic (I) in a direct way, so that correct typing in CC corresponds to intuitionistic provability in a sequent calculus for I. In addition, we demonstrate a direct correspondence between proofs in these two systems. The logic I is an extension of hereditary Harrop formulas (hh) which serve as the logical foundation of the logic programming language Prolog. Like hh, I has the uniform proof property, which allows a complete nondeterministic search procedure to be described in a straightforward manner. Via the encoding, this search procedure provides a goal directed description of proof checking and proof search in CC.
The Calculus of Constructions as a Framework for Proof Search with Set Variable Instantiation
, 2000
"... ..."
(Show Context)
The Coq Proof Assistant  Reference Manual Version 6.1
, 1997
"... : Coq is a proof assistant based on a higherorder logic allowing powerful definitions of functions. Coq V6.1 is available by anonymous ftp at ftp.inria.fr:/INRIA/Projects/coq/V6.1 and ftp.enslyon.fr:/pub/LIP/COQ/V6.1 Keywords: Coq, Proof Assistant, Formal Proofs, Calculus of Inductives Constru ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
: Coq is a proof assistant based on a higherorder logic allowing powerful definitions of functions. Coq V6.1 is available by anonymous ftp at ftp.inria.fr:/INRIA/Projects/coq/V6.1 and ftp.enslyon.fr:/pub/LIP/COQ/V6.1 Keywords: Coq, Proof Assistant, Formal Proofs, Calculus of Inductives Constructions (R'esum'e : tsvp) This research was partly supported by ESPRIT Basic Research Action "Types" and by the GDR "Programmation " cofinanced by MREPRC and CNRS. Unit'e de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) T'el'ephone : (33 1) 39 63 55 11  T'el'ecopie : (33 1) 39 63 53 30 Manuel de r'ef'erence du syst`eme Coq version V6.1 R'esum'e : Coq est un syst`eme permettant le d'eveloppement et la v'erification de preuves formelles dans une logique d'ordre sup'erieure incluant un riche langage de d'efinitions de fonctions. Ce document constitue le manuel de r'ef'erence de la version V6.1 qui est distribu 'ee par ftp ...
Towards a formal mathematical vernacular
 Utrecht University
, 1992
"... Contemporary proof veri cators often use a command language to construct proofs. These commands are often called tactics. This new generation of theorem provers is a substantial improvement over earlier ones such asAUTOMATH. Based on experience with these new provers we feel the need to study these ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
Contemporary proof veri cators often use a command language to construct proofs. These commands are often called tactics. This new generation of theorem provers is a substantial improvement over earlier ones such asAUTOMATH. Based on experience with these new provers we feel the need to study these languages further, especially, because we think that these may be improved in their adequateness to express proofs closer to the established mathematical vernacular. We also feel that a systematic treatment of these vernaculars may lead to an improvement towards the automatic inference of trivial proof steps. In any case a systematic treatment will lead to a better understanding of the command languages. This exercise is carried out in the setting of Pure Type Systems (PTSs) in which a whole range of logics can be embedded. We rstidentify a subclass of PTSs, called the PTSs for logic. For this class we de ne a formal mathematical vernacular and we prove elementary sound and completeness. Via an elaborate example we try to assess how easy proofs in mathematics can be written down in our vernacular along the lines of the original proofs. 1
Proof Search with Set Variable Instantiation in the Calculus of Constructions
 Automated Deduction: CADE13, volume 1104 of Lecture Notes in Arti Intelligence
, 1996
"... . We show how a procedure developed by Bledsoe for automatically finding substitution instances for set variables in higherorder 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 firstord ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
(Show Context)
. We show how a procedure developed by Bledsoe for automatically finding substitution instances for set variables in higherorder 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 firstorder logic that allows existential quantification over set variables. The method finds maximal solutions for this special class of higherorder variables. This class of variables can also be identified in CC. The existence of a correspondence between higherorder logic and higherorder type theories such as CC is wellknown. CC can be viewed as an extension of higherorder logic where the basic terms of the language, the simplytyped 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 higherorder logic and higherorder type theories serve as th...