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16
Holes with Binding Power
 In Types for Proofs and Programs, Second International Workshop, TYPES 2002, Berg en Dal, The Netherlands, April 2428, 2002, Selected Papers, H. Geuvers and F. Wiedijk, Eds. Lecture Notes in Computer Science (LNCS 2646
, 2002
"... Incomplete logical proofs are the logical counterpart of the incomplete #terms that one usually works with in an interactive theorem prover based on type theory. In this paper we extend the formalization of such incomplete proofs given in [5] by introducing unknowns that are allowed to provide ..."
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Incomplete logical proofs are the logical counterpart of the incomplete #terms that one usually works with in an interactive theorem prover based on type theory. In this paper we extend the formalization of such incomplete proofs given in [5] by introducing unknowns that are allowed to provide temporary bindings for variables that are supposed to be bound, but whose binders are not constructed yet  a situation that typically occurs when one does forward reasoning.
Spurious Disambiguation Error Detection
"... Abstract. The disambiguation approach to the input of formulae enables the user to type correct formulae in a terse syntax close to the usual ambiguous mathematical notation. When it comes to incorrect formulae we want to present only errors related to the interpretation meant by the user, hiding er ..."
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Abstract. The disambiguation approach to the input of formulae enables the user to type correct formulae in a terse syntax close to the usual ambiguous mathematical notation. When it comes to incorrect formulae we want to present only errors related to the interpretation meant by the user, hiding errors related to other interpretations (spurious errors). We propose a heuristic to recognize spurious errors, which has been integrated with the disambiguation algorithm of [6]. 1
The lambdacontext calculus
 In: LFMTP’07: International Workshop on Logical Frameworks and MetaLanguages
"... We present a simple lambdacalculus whose syntax is populated by variables which behave like metavariables. It can express both captureavoiding and capturing substitution (instantiation). To do this requires several innovations, including a key insight in the confluence proof and a set of reductio ..."
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We present a simple lambdacalculus whose syntax is populated by variables which behave like metavariables. It can express both captureavoiding and capturing substitution (instantiation). To do this requires several innovations, including a key insight in the confluence proof and a set of reduction rules which manages the complexity of a calculus of contexts over the ‘vanilla ’ lambdacalculus in a very simple and modular way. This calculus remains extremely close in look and feel to a standard lambdacalculus with explicit substitutions, and good properties of the lambdacalculus are preserved. These include a HindleyMilner type system with principal typings and subject reduction, and an applicative characterisation of contextual equivalence. Key words: Lambdacalculus, calculi of contexts, functional programming,
The lambdacontext calculus (extended version
 Information and computation
, 2009
"... We present the Lambda Context Calculus. This simple lambdacalculus features variables arranged in a hierarchy of strengths such that substitution of a strong variable does not avoid capture with respect to abstraction by a weaker variable. This allows the calculus to express both captureavoiding ..."
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We present the Lambda Context Calculus. This simple lambdacalculus features variables arranged in a hierarchy of strengths such that substitution of a strong variable does not avoid capture with respect to abstraction by a weaker variable. This allows the calculus to express both captureavoiding and capturing substitution (instantiation). The reduction rules extend the ‘vanilla ’ lambdacalculus in a simple and modular way and preserve the look and feel of a standard lambdacalculus with explicit substitutions. Good properties of the lambdacalculus are preserved. The LamCC is confluent, and a natural injection into the LamCC of the untyped lambdacalculus exists and preserves strong normalisation. We discuss the calculus and its design with full proofs. In the presence of the hierarchy of variables, functional binding splits into a functional abstraction λ (lambda) and a namebinder N(new). We investigate how the components of this calculus interact with each other and with the reduction rules, with examples. In two more extended case studies we demonstrate how global state can be expressed, and how contexts and contextual equivalence can be naturally internalised using function application.
found at the
"... this file with prentcsmacro.sty for your meeting, or with entcsmacro.sty for your meeting. Both can be ..."
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this file with prentcsmacro.sty for your meeting, or with entcsmacro.sty for your meeting. Both can be
CurryHoward
"... for incomplete firstorder logic derivations using oneandahalf level terms ..."
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for incomplete firstorder logic derivations using oneandahalf level terms
Rating Disambiguation Errors ⋆
"... Abstract. Ambiguous notation is a powerful tool developed to deal with the complexity of mathematics without sacrificing clarity or conciseness. In the mechanized parsing of ambiguous terms, a disambiguation algorithm can be used to provide the system with the intelligence necessary to select valid ..."
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Abstract. Ambiguous notation is a powerful tool developed to deal with the complexity of mathematics without sacrificing clarity or conciseness. In the mechanized parsing of ambiguous terms, a disambiguation algorithm can be used to provide the system with the intelligence necessary to select valid interpretations for the overloaded symbols received in input. Disambiguation works by means of an incremental analysis of the input term, progressively discarding all invalid interpretations. As a result, if the input term cannot be disambiguated, many errors will be produced, only a handful of which are truly meaningful to the user. In this paper, we improve the existing technique to classify disambiguation errors by introducing a new heuristic to sort errors from the most meaningful to the least, showing that it can be implemented in a natural way in the existing disambiguation algorithm. We also describe a neat interface to present disambiguation errors to the user, suitable for the use in interactive theorem proving applications. 1
The λcontext Calculus
"... We present a simple but expressive lambdacalculus whose syntax is populated by variables which behave like metavariables. It can express both captureavoiding and capturing substitution (instantiation). To do this requires several innovations, including a key insight in the confluence proof and a ..."
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We present a simple but expressive lambdacalculus whose syntax is populated by variables which behave like metavariables. It can express both captureavoiding and capturing substitution (instantiation). To do this requires several innovations, including a key insight in the confluence proof and a set of reduction rules which manages the complexity of a calculus of contexts over the ‘vanilla ’ lambdacalculus in a very simple and modular way. This calculus remains extremely close in look and feel to a standard lambdacalculus with explicit substitutions, and good properties of the lambdacalculus are preserved. Keywords: Lambdacalculus, contexts, metavariables, captureavoiding substitution, capturing substitution, instantiation, confluence, nominal techniques, calculus of explicit substitutions.
VeriML: A dependentlytyped, userextensible and languagecentric approach to proof assistants
, 2013
"... Software certification is a promising approach to producing programs which are virtually free of bugs. It requires the construction of a formal proof which establishes that the code in question will behave according to its specification – a higherlevel description of its functionality. The construc ..."
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Software certification is a promising approach to producing programs which are virtually free of bugs. It requires the construction of a formal proof which establishes that the code in question will behave according to its specification – a higherlevel description of its functionality. The construction of such formal proofs is carried out in tools called proof assistants. Advances in the current stateoftheart proof assistants have enabled the certification of a number of complex and realistic systems software. Despite such success stories, largescale proof development is an arcane art that requires significant manual effort and is extremely timeconsuming. The widely accepted best practice for limiting this effort is to develop domainspecific automation procedures to handle all but the most essential steps of proofs. Yet this practice is rarely followed or needs comparable development effort as well. This is due to a profound architectural shortcoming of existing proof assistants: developing automation procedures is currently overly complicated and errorprone. It involves the use of an amalgam of extension languages, each with a different programming model and a set of limitations, and with significant interfacing problems between them. This thesis posits that this situation can be significantly improved by designing a proof assistant with extensibility as the central focus. Towards that effect, I have designed a novel programming language called
Placeholder Calculus for FirstOrder Logic
"... Abstract. In this paper we present an extension of firstorder predicate logic with placeholders. These placeholders allow the construction of proofs for incomplete theorems. These theorems can be completed during the proof construction process. By using special definitions of substitutions and rep ..."
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Abstract. In this paper we present an extension of firstorder predicate logic with placeholders. These placeholders allow the construction of proofs for incomplete theorems. These theorems can be completed during the proof construction process. By using special definitions of substitutions and replacements, we obtain an unexpectedly simple calculus. Furthermore, we avoid the need of additional rules for explicit substitutions to deal with postponed substitutions in placeholders, since the definitions of substitution and replacement deal with them directly. 1