A note on the proof theory the λII-calculus

Studia Logica 54 (2):199 - 230 (1995)
Abstract
The II-calculus, a theory of first-order dependent function types in Curry-Howard-de Bruijn correspondence with a fragment of minimal first-order logic, is defined as a system of (linearized) natural deduction. In this paper, we present a Gentzen-style sequent calculus for the II-calculus and prove the cut-elimination theorem.The cut-elimination result builds upon the existence of normal forms for the natural deduction system and can be considered to be analogous to a proof provided by Prawitz for first-order logic. The type-theoretic setting considered here elegantly illustrates the distinction between the processes of normalization in a natural deduction system and cut-elimination in a Gentzen-style sequent calculus.
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DOI 10.1007/BF01063152
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First-Order Logic.Raymond M. Smullyan - 1968 - New York [Etc.]Springer-Verlag.
A Natural Extension of Natural Deduction.Peter Schroeder-Heister - 1984 - Journal of Symbolic Logic 49 (4):1284-1300.

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