The Definitional Side of the Forcing

Guilhem Jaber 1 Gabriel Lewertowski 2, 1 Pierre-Marie Pédrot 2 Matthieu Sozeau 3, 1 Nicolas Tabareau 2
2 ASCOLA - Aspect and composition languages
Inria Rennes – Bretagne Atlantique , Département informatique - EMN, LINA - Laboratoire d'Informatique de Nantes Atlantique
3 PI.R2 - Design, study and implementation of languages for proofs and programs
Inria de Paris, CNRS - Centre National de la Recherche Scientifique, UPD7 - Université Paris Diderot - Paris 7, PPS - Preuves, Programmes et Systèmes
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.
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Guilhem Jaber, Gabriel Lewertowski, Pierre-Marie Pédrot, Matthieu Sozeau, Nicolas Tabareau. The Definitional Side of the Forcing. Logics in Computer Science, May 2016, New York, United States. ⟨10.1145/http://dx.doi.org/10.1145/2933575.2935320⟩. ⟨hal-01319066⟩

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