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LICS 2012 : Logic In Computer Science, Dubrovnik : Croatie (2012)

Extending Type Theory with Forcing

Guilhem Jaber 1, 2, Nicolas Tabareau 1, 2, Matthieu Sozeau 3

(2012-06-25)

This paper presents an intuitionistic forcing translation for the Calculus of Constructions (CoC), a translation that corresponds to an internalization of the presheaf construction in CoC. Depending on the chosen set of forcing conditions, the resulting type system can be extended with extra logical principles. The translation is proven correct-in the sense that it preserves type checking-and has been implemented in Coq. As a case study, we show how the forcing translation on integers (which corresponds to the internalization of the topos of trees) allows us to define general inductive types in Coq, without the strict positivity condition. Using such general inductive types, we can construct a shallow embedding of the pure \lambda-calculus in Coq, without defining an axiom on the existence of an universal domain. We also build another forcing layer where we prove the negation of the continuum hypothesis.

1:	ASCOLA (INRIA - EMN)
	INRIA – École Nationale Supérieure des Mines - Nantes
2:	Laboratoire d'Informatique de Nantes Atlantique (LINA)
	CNRS : UMR6241 – Université de Nantes – École Nationale Supérieure des Mines - Nantes
3:	PI.R2 (INRIA Paris - Rocquencourt)
	INRIA – Université Paris VII - Paris Diderot – CNRS : UMR7126

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Computer Science/Logic in Computer Science

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From: Guilhem Jaber <>

Submitted on: Wednesday, 4 April 2012 11:49:02

Updated on: Thursday, 12 April 2012 16:17:22