Abstract. Deciding the typing judgement of type theories with dependent types such as the Logical Framework relies on deciding the equality judgement for the same theory. Implementing the conversion algorithm for βη-equality and justifying this algorithm is therefore an important problem for applications such as proof assistants and modules systems. This article gives a proof of decidability, correctness and completeness of the conversion algorithms for βη-equality defined by Coquand [3] and Harper and Pfenning [8] for the Logical Framework, relying on established metatheoretic results for the type theory. Proofs are also given of the same properties for a typed algorithm for conversion for System F, a new result.
Abstract. We present four constructions for standard equipment which can be generated for every inductive datatype: case analysis, structural recursion, no confusion, acyclicity. Our constructions follow a two-level approach-they require less work than the standard techniques which inspired them [11,8]. Moreover, given a suitably heterogeneous notion of equality, they extend without difficulty to inductive families of datatypes. These constructions are vital components of the translation from dependently typed programs in pattern matching style [7] to the equivalent programs expressed in terms of induction principles [21] and as such play a crucial behind-the-scenes rôle in Epigram [25].
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