The Definitional Side of the ForcingReport as inadecuate




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1 IRIF - Institut de Recherche en Informatique Fondamentale 2 ASCOLA - Aspect and composition languages LINA - Laboratoire d-Informatique de Nantes Atlantique, Département informatique - EMN, Inria Rennes – Bretagne Atlantique 3 PI.R2 - Design, study and implementation of languages for proofs and programs PPS - Preuves, Programmes et Systèmes, UPD7 - Université Paris Diderot - Paris 7, CNRS - Centre National de la Recherche Scientifique, Inria de Paris

Abstract : This paper studies forcing translations of proofs in dependent type theory, through the Curry-Howard correspondence. Based on a call-by-push-value decomposition, we synthesize two simply-typed translations: i one call-by-value, corresponding to the translation derived from the presheaf construction as studied in a previous paper ; ii one call-by-name, whose intuitions already appear in Kriv-ine and Miquel-s work. Focusing on the call-by-name translation, we adapt it to the dependent case and prove that it is compatible with the definitional equality of our system, thus avoiding coherence problems. This allows us to use any category as forcing conditions , which is out of reach with the call-by-value translation. Our construction also exploits the notion of storage operators in order to interpret dependent elimination for inductive types. This is a novel example of a dependent theory with side-effects, clarifying how dependent elimination for inductive types must be restricted in a non-pure setting. Being implemented as a Coq plugin, this work gives the possibility to formalize easily consistency results, for instance the consistency of the negation of Voevodsky-s univalence axiom.

Keywords : Effects Coq Forcing Dependent type theory Inductive types





Author: Guilhem Jaber - Gabriel Lewertowski - Pierre-Marie Pédrot - Matthieu Sozeau - Nicolas Tabareau -

Source: https://hal.archives-ouvertes.fr/



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