The first-order hypothetical logic of proofs

The Propositional Logic of Proofs (LP) is a modal logic in which the modality □A is revisited as [[t]]A, t being an expression that bears witness to the validity of A. It enjoys arithmetical soundness and completeness, can realize all S4 theorems and is capable of reflecting its own proofs (⊢A implies ⊢[[t]]A, for some t). A presentation of first-order LP has recently been proposed, FOLP, which enjoys arithmetical soundness and has an exact provability semantics. A key notion in this presentation is how free variables are dealt with in a formula of the form [[t]]A(i). We revisit this notion in the setting of a Natural Deduction presentation and propose a Curry-Howard correspondence for FOLP. A term assignment is provided and a proof of strong normalization is given.