A short proof of the strong normalization of classical natural deduction with disjunction. (English) Zbl 1066.03056

The authors give “a direct proof of the strong normalization of the cut-elimination procedure for full propositional classical logic”. By ‘full classical’ they mean to take all connectives as primitives; and by ‘direct’: no reduction to known cases, or a frontal attack. The work is carried out in the format of the \(\lambda\mu\)-calculus, which is capable of referring to subterms directly. The main result is a proof of the strong normalization property: every reduction of a typed term terminates in a finite number of steps. The chief task is, of course, to show that this property is preserved under substituition. This is, indeed, a travail, as always. In carrying out the inductive proof, the authors enlarge the format, with an explanation of its necessity, This paper is written very clearly. The authors explain the context, aims, difficulties, strategies, etc. very well.


03F05 Cut-elimination and normal-form theorems
03B05 Classical propositional logic
03B40 Combinatory logic and lambda calculus
68N18 Functional programming and lambda calculus
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