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Strong normalization of classical natural deduction with disjunctions. (English) Zbl 1141.03027
In \(\lambda\mu\)-calculus, a vacuous \(\mu\)-term, namely \(\mu aM\) with no occurrence of \(a\) in \(M\), causes trouble, which the authors call erasing-continuation, in connexion with CPS-translation. (‘CPS’ is for ‘continuation passing style’.) This difficulty was overlooked by P. de Groote in his article [“Strong normalization of classical natural deduction with disjunction”, Lect. Notes Comput. Sci. 2044, 182–196 (2001; Zbl 0981.03027)], and so his proof of strong normalization is incomplete. To overcome the difficulty, the authors use the device of augmentations which they introduced in a previous paper [J. Symb. Log. 68, No. 3, 851–859 (2003); Corrigendum ibid. 68, No. 4, 1415–1416 (2003; Zbl 1058.03060)]. To a term \(M\), a set, \(\text{Aug}(M)\), of non-vacuous terms are associated in such a way that any reduction of \(M\) is simulated by terms in \(\text{Aug}(M)\) while avoiding erasure of continuation. Thus, in this paper, the authors complete a proof of strong normalization of the \(\lambda\mu\)-calculus with \(\rightarrow\), \(\wedge\), \(\vee\), and \(\perp\). They extend the strong normalization proof to the calculus that incorporates general elimination rules of J. von Plato [Arch. Math. Logic 40, No. 7, 541–567 (2001; Zbl 1021.03050)]. {The authors’ citation ‘Annals of\hbox…’ is in error.} This calculus allows permutative conversions and thus provides the sub-formula property.

MSC:
03F05 Cut-elimination and normal-form theorems
03B05 Classical propositional logic
03B40 Combinatory logic and lambda calculus
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