The foundation of a generic theorem prover
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Handbook of logic in computer science (vol. 2)
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LFCS '94 Proceedings of the Third International Symposium on Logical Foundations of Computer Science
Handbook of automated reasoning
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Handbook of automated reasoning
PAL+: a lambda-free logical framework
Journal of Functional Programming
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Electronic Notes in Theoretical Computer Science (ENTCS)
Electronic Notes in Theoretical Computer Science (ENTCS)
ACM Transactions on Computational Logic (TOCL)
Spurious Disambiguation Error Detection
Calculemus '07 / MKM '07 Proceedings of the 14th symposium on Towards Mechanized Mathematical Assistants: 6th International Conference
Electronic Notes in Theoretical Computer Science (ENTCS)
The lambda-context calculus (extended version)
Information and Computation
Curry-Howard for incomplete first-order logic derivations using one-and-a-half level terms
Information and Computation
TYPES'02 Proceedings of the 2002 international conference on Types for proofs and programs
TYPES'06 Proceedings of the 2006 international conference on Types for proofs and programs
Mikibeta: a general GUI library for visualizing proof trees system description and demonstration
LOPSTR'10 Proceedings of the 20th international conference on Logic-based program synthesis and transformation
Translating a fragment of weak type theory into type theory with open terms
MKM'05 Proceedings of the 4th international conference on Mathematical Knowledge Management
CPP'12 Proceedings of the Second international conference on Certified Programs and Proofs
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When proving a theorem, one makes intermediate claims, leaving parts temporarily unspecified. These 'open' parts may be proofs but also terms. In interactive theorem proving systems, one prominently deals with these 'unfinished proofs' and 'open terms'. We study these 'open phenomena' from the point of view of logic. This amounts to finding a correctness criterion for 'unfinished proofs' (where some parts may be left open, but the logical steps that have been made are still correct). Furthermore we want to capture the notion of 'proof state'. Proof states are the objects that interactive theorem provers operate on and we want to understand them in terms of logic.In this paper we define 'open higher order predicate logic', an extension of higher order logic with unfinished (open) proofs and open terms. Then we define a type theoretic variant of this open higher order logic together with a formulas-as-types embedding from open higher order logic to this type theory. We show how this type theory nicely captures the notion of 'proof state', which is now a type-theoretic context.