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Building decision procedures in the calculus of inductive constructions
 of Lecture Notes in Computer Science
, 2007
"... It is commonly agreed that the success of future proof assistants will rely on their ability to incorporate computations within deduction in order to mimic the mathematician when replacing the proof of a proposition P by the proof of an equivalent proposition P ’ obtained from P thanks to possibly c ..."
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It is commonly agreed that the success of future proof assistants will rely on their ability to incorporate computations within deduction in order to mimic the mathematician when replacing the proof of a proposition P by the proof of an equivalent proposition P ’ obtained from P thanks to possibly complex calculations. In this paper, we investigate a new version of the calculus of inductive constructions which incorporates arbitrary decision procedures into deduction via the conversion rule of the calculus. The novelty of the problem in the context of the calculus of inductive constructions lies in the fact that the computation mechanism varies along proofchecking: goals are sent to the decision procedure together with the set of user hypotheses available from the current context. Our main result shows that this extension of the calculus of constructions does not compromise its main properties: confluence, subject reduction, strong normalization and consistency are all preserved.
Monadic Type Systems: Pure Type Systems for Impure Settings (Preliminary Report)
 In Proceedings of the Second HOOTS Workshop
, 1997
"... Pure type systems and computational monads are two parameterized frameworks that have proved to be quite useful in both theoretical and practical applications. We join the foundational concepts of both of these to obtain monadic type systems. Essentially, monadic type systems inherit the parameteriz ..."
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Cited by 9 (2 self)
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Pure type systems and computational monads are two parameterized frameworks that have proved to be quite useful in both theoretical and practical applications. We join the foundational concepts of both of these to obtain monadic type systems. Essentially, monadic type systems inherit the parameterized higherorder type structure of pure type systems and the monadic term and type structure used to capture computational effects in the theory of computational monads. We demonstrate that monadic type systems nicely characterize previous work and suggest how they can support several new theoretical and practical applications. A technical foundation for monadic type systems is laid by recasting and scaling up the main results from pure type systems (confluence, subject reduction, strong normalisation for particular classes of systems, etc.) and from operational presentations of computational monads (notions of operational equivalence based on applicative similarity, coinduction proof techni...
On the strength of proofirrelevant type theories
 of Lecture Notes in Computer Science
, 2006
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Advisor
, 2002
"... The addition of proof irrelevant types to LF permits more expressive encodings and a richer (but still decidable) denitional equality. These types can also be used to maintain invariants concerning the interchangeability of subproofs of large proofs, which can be useful for proof compression for pro ..."
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The addition of proof irrelevant types to LF permits more expressive encodings and a richer (but still decidable) denitional equality. These types can also be used to maintain invariants concerning the interchangeability of subproofs of large proofs, which can be useful for proof compression for proofcarrying code. We investigate the metatheoretic properties of this language extension and reconcile it with the notion of strictness, a property of notational de nitions which can improve implementation eÆciency of proofchecking.
Type Systems for Dummies
"... We extend Pure Type Systems with a function turning each term M of type A into a dummy ∣M ∣ of the same type ( ∣ ⋅ ∣ is not an identity, in that M ≠ ∣M∣). Intuitively, a dummy represents an unknown, canonical object of the given type: dummies are opaque (cannot be internally inspected), and irrele ..."
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We extend Pure Type Systems with a function turning each term M of type A into a dummy ∣M ∣ of the same type ( ∣ ⋅ ∣ is not an identity, in that M ≠ ∣M∣). Intuitively, a dummy represents an unknown, canonical object of the given type: dummies are opaque (cannot be internally inspected), and irrelevant in the sense that dummies of a same type are convertible to each other. This latter condition makes convertibility in PTS with dummies (DPTS) stronger than usual, hence raising not trivial consistency issues. DPTS offer an alternative approach to (proof) irrelevance, tagging irrelevant information at the level of terms and not of types, and avoiding the annoying syntactical duplication of products, abstractions and applications into an explicit and an implicit version, typical of systems like ICC ∗. Categories and Subject Descriptors F.4.1 [Mathematical Logic
found at the ENTCS Macro Home Page. Inductive and Coinductive Components of Corecursive Functions in Coq
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