Abstract
This paper defines the contextual natural deduction calculus \(\textbf{ND}^\textbf{c}\) for the implicational fragment of intuitionistic logic. \(\textbf{ND}^\textbf{c}\) extends the usual natural deduction calculus (here called \(\textbf{ND}\)) by allowing the implication introduction and elimination rules to operate on formulas that occur inside contexts. In analogy to the Curry-Howard isomorphism between \(\textbf{ND}\) and the simply-typed λ-calculus, an extension of the λ-calculus, here called λ c-calculus, is defined in order to provide compact proof-terms for \(\textbf{ND}^\textbf{c}\) proofs. Soundness and completeness of \(\textbf{ND}^\textbf{c}\) with respect to \(\textbf{ND}\) are proven by defining translations of proofs between these calculi. Furthermore, some \(\textbf{ND}^\textbf{c}\)-proofs are shown to be quadratically smaller than the smallest \(\textbf{ND}\)-proofs of the same theorems.
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Woltzenlogel Paleo, B. (2013). Contextual Natural Deduction. In: Artemov, S., Nerode, A. (eds) Logical Foundations of Computer Science. LFCS 2013. Lecture Notes in Computer Science, vol 7734. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35722-0_27
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DOI: https://doi.org/10.1007/978-3-642-35722-0_27
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