We formalise natural deduction for first-order logic in the proof assistant
Coq, using De Bruijn indices for variable binding. The main judgement
we model is of the form Γ- d [:] ø, stating that d is a proof term of
formula ø 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
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