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In this paper we introduce a term calculus ${\cal B}$ which adds to theaffine $\lambda$-calculus with pairing a new construct allowing for arestricted form of contraction. We obtain a Curry-Howard correspondence between${\cal B}$ and the sub-structural logical system which we call "minimal{\L}ukasiewicz logic", also known in the literature as the logic of hoops (ageneralisation of MV-algebras). This logic lies strictly in between affineminimal logic and standard minimal logic. We prove that ${\cal B}$ is stronglynormalising and has the Church-Rosser property. We also give examples of termsin ${\cal B}$ corresponding to some important derivations from our work and theliterature. Finally, we discuss the relation between normalisation in ${\calB}$ and cut-elimination for a Gentzen-style formulation of minimal{\L}ukasiewicz logic.

A Curry-Howard Correspondence for the Minimal Fragment of {\L}ukasiewicz Logic