Proof Reflection in Coq

Abstract

We formalize natural deduction for first-order logic in the proof assistant Coq, using de Bruijn indices for variable binding. The main judgment we model is of the form Γvd [:] φ, stating that d is a proof term of formula φ under hypotheses Γ; it can be viewed as a typing relation by the Curry–Howard isomorphism. This relation is proved sound with respect to Coq's native logic and is amenable to the manipulation of formulas and of derivations. As an illustration, we define a reduction relation on proof terms with permutative conversions and prove the property of subject reduction.