Authors: Barendregt, Henk
Ghilezan, Silvia 
Title: Lambda terms for natural deduction, sequent calculus and cut elimination
Journal: Journal of Functional Programming
Volume: 10
Issue: 1
First page: 121
Last page: 134
Issue Date: 1-Jan-2000
ISSN: 0956-7968
DOI: 10.1017/S0956796899003524
Abstract: 
It is well known that there is an isomorphism between natural deduction derivations and typed lambda terms. Moreover, normalising these terms corresponds to eliminating cuts in the equivalent sequent calculus derivations. Several papers have been written on this topic. The correspondence between sequent calculus derivations and natural deduction derivations is, however, not a one-one map, which causes some syntactic technicalities. The correspondence is best explained by two extensionally equivalent type assignment systems for untyped lambda terms, one corresponding to natural deduction (λN) and the other to sequent calculus (λL). These two systems constitute different grammars for generating the same (type assignment relation for untyped) lambda terms. The second grammar is ambiguous, but the first one is not. This fact explains the many-one correspondence mentioned above. Moreover, the second type assignment system has a 'cut-free' fragment (λLcf). This fragment generates exactly the typeable lambda terms in normal form. The cut elimination theorem becomes a simple consequence of the fact that typed lambda terms posses a normal form.
Publisher: Cambridge University Press

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