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A mechanically verified, sound and complete theorem prover for first order logic
 In Theorem Proving in Higher Order Logics, 18th International Conference, TPHOLs 2005, volume 3603 of Lecture Notes in Computer Science
, 2005
"... Abstract. We present a system of first order logic, together with soundness and completeness proofs wrt. standard first order semantics. Proofs are mechanised in Isabelle/HOL. Our definitions are computable, allowing us to derive an algorithm to test for first order validity. This algorithm may be e ..."
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Abstract. We present a system of first order logic, together with soundness and completeness proofs wrt. standard first order semantics. Proofs are mechanised in Isabelle/HOL. Our definitions are computable, allowing us to derive an algorithm to test for first order validity. This algorithm may be executed in Isabelle/HOL using the rewrite engine. Alternatively the algorithm has been ported to OCaML. 1
Formal Topologies on the Set of FirstOrder Formulae
 Journal of Symbolic Logic
, 1998
"... this paper that the question has a simple negative answer. This raised further natural questions on what can be said about the points of these two topologies; we give some answers. The observation that topological models for firstorder theories can expressed in the framework of locales appears, for ..."
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Cited by 2 (1 self)
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this paper that the question has a simple negative answer. This raised further natural questions on what can be said about the points of these two topologies; we give some answers. The observation that topological models for firstorder theories can expressed in the framework of locales appears, for instance, in Fourman and Grayson [6], where the analogy between points of a locale and models of a theory is emphasised; the identification of formal points with Henkin sets, gives a precise form to this analogy. We replace the use of locales by formal topology, which can be expressed in a predicative framework such as MartinLof's type theory. Prooftheoretic issues are also considered by Dragalin [4], who presents a topological completeness proof using only finitary inductive definitions. Palmgren and Moerdijk [10] is also concerned with constructions of models: using sheaf semantics, they obtain a stronger conservativity result than the one in [3]. We will first investigate the difference between the DedekindMacNeille cover and the inductive cover. It easy to see that \Delta DM is stronger than \Delta I , that is, OE \Delta I U implies OE \Delta DM U , but the converse does not hold in general. The notion of point is not primitive in formal topology and therefore it is natural to require that a formal topology has some notion of positivity defined on the basic neighbourhoods; that a neighbourhood is positive then corresponds to, in ordinary point based topology, that it is inhabited by some point. We will show several negative results on positivity, both for the inductive topology and the DedekindMacNeille topology. The points of an inductive topology correspond to Henkin sets, but the DedekindMacNeille topology has, in general, no points. Our reasoning is constructi...
A Full Formalization of SLDResolution in the Calculus of Inductive Constructions
"... This paper presents a full formalization of the semantics of definite programs, in the calculus of inductive constructions. First, we describe a formalization of the proof of first order terms unification: this proof is obtained from a similar proof dealing with quasiterms, thus showing how to rela ..."
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This paper presents a full formalization of the semantics of definite programs, in the calculus of inductive constructions. First, we describe a formalization of the proof of first order terms unification: this proof is obtained from a similar proof dealing with quasiterms, thus showing how to relate an inductive set with a subset defined by a predicate. Then, SLDresolution is explicitely defined: the renaming process used in SLDderivations is made explicit, thus introducing complications, usually overlooked, during the proofs of classical results. Last, switching and lifting lemmas and soundness and completeness theorems are formalized. For this, we present two lemmas, usually omitted, which are needed. This development also contains a formalization of basic results on operators and their fixpoints in a general setting. All the proofs of the results, presented here, have been checked with the proof assistant Coq.
Proof Reflection in
 Journal of Automated Reasoning
, 2002
"... We formalise natural deduction for firstorder logic in the proof assistant Coq, using De Bruijn indices for variable binding. The main judgement we model is of the form # # d [:] #, stating that d is a proof term of formula # under hypotheses #; it can be viewed as a typing relation by the Curry ..."
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We formalise natural deduction for firstorder logic in the proof assistant Coq, using De Bruijn indices for variable binding. The main judgement we model is of the form # # d [:] #, stating that d is a proof term of formula # under hypotheses #; it can be viewed as a typing relation by the CurryHoward isomorphism. This relation is proved sound with respect to Coq's native logic and is amenable to the manipulation of formulas and of derivations. As an illustration, we define a reduction relation on proof terms with permutative conversions and prove the property of subject reduction. 1
Formalisation of General Logics in the Calculus of Inductive Constructions: Towards an Abstract . . .
, 1999
"... Formal specifications of logics share many standard concepts and in order to avoid repetitious works, it seems desirable to express these specifications in a uniform framework. General logics à la J. Meseguer provide an uniform and modular way of encoding a logical language, its semantics and its in ..."
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Formal specifications of logics share many standard concepts and in order to avoid repetitious works, it seems desirable to express these specifications in a uniform framework. General logics à la J. Meseguer provide an uniform and modular way of encoding a logical language, its semantics and its inference system. Hence, we describe here a formalisation of general logics in the calculus of inductive constructions thus providing a generic and modular set of speci cations (with the proofs of s...