Generic bidirectional typing for dependent type theories

Bidirectional typing is a discipline in which the typing judgment is decomposed explicitly into inference and checking modes, allowing to control the flow of type information in typing rules and to specify algorithmically how they should be used. Bidirectional typing has been fruitfully studied and bidirectional systems have been developed for many type theories. However, the formal development of bidirectional typing has until now been kept confined to specific theories, with general guidelines remaining informal. In this work, we give a generic account of bidirectional typing for a general class of dependent type theories. This is done by first giving a general definition of type theories (or equivalently, a logical framework), for which we define declarative and bidirectional type systems. We then show, in a theory-independent fashion, that the two systems are equivalent. This equivalence is then explored to establish the decidability of typing for weak normalizing theories, yielding a generic type-checking algorithm that has been implemented in a prototype and used in practice with many theories

Representing Agda and coinduction in the λΠ-calculus modulo rewriting

International audienceCoinduction is a principle, or a proof technique, dual to induction and which allows to handle possibly infinite objects in a natural way, such as infinite lists, infinite trees, formal languages, non well-founded sets, etc. Because of its usefulness, it is increasingly being added to proof assistants, such as Coq, Isabelle, PVS and Agda. In order to be able to translate proofs by coinduction coming from multiple proof assistants it is thus important to first understand how to encode coinduction in Dedukti, a problem that had never been addressed before. During this internship, we studied the representation of Agda and coinduction in Dedukti. Among the techniques of implementing coinduction in proof assistants, Agda features two presentations: musical coinduction and copattern coinduction. Based on their internal syntax representation in Agda, we proposed an encoding of both presentations in Dedukti. We resumed the development of the Agda2Dedukti translator and extended it with the proposed encoding, allowing it to translate automatically proofs by coinduction into Dedukti

Adequate and Computational Encodings in the Logical Framework Dedukti

Dedukti is a very expressive logical framework which unlike most frameworks, such as the Edinburgh Logical Framework (LF), allows for the representation of computation alongside deduction. However, unlike LF encodings, Dedukti encodings proposed until now do not feature an adequacy theorem - i.e., a bijection between terms in the encoded system and in its encoding. Moreover, many of them also do not have a conservativity result, which compromises the ability of Dedukti to check proofs written in such encodings. We propose a different approach for Dedukti encodings which do not only allow for simpler conservativity proofs, but which also restore the adequacy of encodings. More precisely, we propose in this work adequate (and thus conservative) encodings for Functional Pure Type Systems. However, in contrast with LF encodings, ours is computational - that is, represents computation directly as computation. Therefore, our work is the first to present and prove correct an approach allowing for encodings that are both adequate and computational in Dedukti

Representing Agda and coinduction in the λΠ-calculus modulo rewriting

International audienceCoinduction is a principle, or a proof technique, dual to induction and which allows to handle possibly infinite objects in a natural way, such as infinite lists, infinite trees, formal languages, non well-founded sets, etc. Because of its usefulness, it is increasingly being added to proof assistants, such as Coq, Isabelle, PVS and Agda. In order to be able to translate proofs by coinduction coming from multiple proof assistants it is thus important to first understand how to encode coinduction in Dedukti, a problem that had never been addressed before. During this internship, we studied the representation of Agda and coinduction in Dedukti. Among the techniques of implementing coinduction in proof assistants, Agda features two presentations: musical coinduction and copattern coinduction. Based on their internal syntax representation in Agda, we proposed an encoding of both presentations in Dedukti. We resumed the development of the Agda2Dedukti translator and extended it with the proposed encoding, allowing it to translate automatically proofs by coinduction into Dedukti

Adequate and computational encodings in the logical framework Dedukti

Dedukti is a very expressive logical framework which unlike most frameworks, such as the Edinburgh Logical Framework (LF), allows for the representation of computation alongside deduction. However, unlike LF encodings, Dedukti encodings proposed until now do not feature an adequacy theorem -- i.e., a bijection between terms in the encoded system and in its encoding. Moreover, many of them also do not have a conservativity result, which compromises the ability of Dedukti to check proofs written in such encodings. We propose a different approach for Dedukti encodings which do not only allow for simpler conservativity proofs, but which also restore the adequacy of encodings. More precisely, we propose in this work adequate (and thus conservative) encodings for Functional Pure Type Systems. However, in contrast with LF encodings, ours is computational -- that is, represents computation directly as computation. Therefore, our work is the first to present and prove correct an approach allowing for encodings that are both adequate and computational in Dedukti

Generic bidirectional typing for dependent type theories

Bidirectional typing is a discipline in which the typing judgment is decomposed explicitly into inference and checking modes, allowing to control the flow of type information in typing rules and to specify algorithmically how they should be used. Bidirectional typing has been fruitfully studied and bidirectional systems have been developed for many type theories. However, the formal development of bidirectional typing has until now been kept confined to specific theories, with general guidelines remaining informal. In this work, we give a generic account of bidirectional typing for a general class of dependent type theories. This is done by first giving a general definition of type theories (or equivalently, a logical framework), for which we define declarative and bidirectional type systems. We then show, in a theory-independent fashion, that the two systems are equivalent. This equivalence is then explored to establish the decidability of typing for weak normalizing theories, yielding a generic type-checking algorithm that has been implemented in a prototype and used in practice with many theories

Generic bidirectional typing for dependent type theories

Bidirectional typing is a discipline in which the typing judgment is decomposed explicitly into inference and checking modes, allowing to control the flow of type information in typing rules and to specify algorithmically how they should be used. Bidirectional typing has been fruitfully studied and bidirectional systems have been developed for many type theories. However, the formal development of bidirectional typing has until now been kept confined to specific theories, with general guidelines remaining informal. In this work, we give a generic account of bidirectional typing for a general class of dependent type theories. This is done by first giving a general definition of type theories (or equivalently, a logical framework), for which we define declarative and bidirectional type systems. We then show, in a theory-independent fashion, that the two systems are equivalent. This equivalence is then explored to establish the decidability of typing for weak normalizing theories, yielding a generic type-checking algorithm that has been implemented in a prototype and used in practice with many theories