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Conference Papers Year : 2024

Observational Equality Meets CIC

Abstract

Equality is at the heart of dependent type theory, as it plays a fundamental role in specifications and mathematical reasoning. The standard way to handle it in mainstream proof assistants such as Agda , Lean or Coq is based on Martin-Löf’s identity type, which comes straight out of the ’70s—its elegance and simplicity have earned it a long-standing use, despite a major discrepancy with traditional mathematical formulations: it does not satisfy any extensionality principles. Recently, the work on observational equality has regained interest as a new way to encode equality in proof assistants that support a universe of definitionally proof-irrelevant propositions; however it has yet to be integrated in any major proof assistant, because it is not fully compatible with another important feature of type theory: indexed inductive types. In this paper, we propose a systematic integration of indexed inductive types with an observational equality, and show that this integration can only be completely satisfactory if the observational equality satisfies the computational rule of Martin-Löf’s identity type. The second contribution of this paper is a formal proof that this additional computation rule, although not present in previous works on observational equality, can be integrated to the system without compromising the decidability of conversion.
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hal-04535982 , version 1 (07-04-2024)

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Loïc Pujet, Nicolas Tabareau. Observational Equality Meets CIC. ESOP 2024 - 33rd European Symposium on Programming, Apr 2024, Luxembourg, Luxembourg. pp.275-301, ⟨10.1007/978-3-031-57262-3_12⟩. ⟨hal-04535982⟩
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