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