Abstract

. This paper presents two sequent calculi, requiring no clausal form for types, whose typable terms are in 1-1 correspondence with the normal terms of the \Pi- and \Pi\Sigma-calculi. Such sequent calculi allow no permutations in the order in which inference rules occur on derivations of typable terms and are thus appropriate for proof search. In these calculi proof search can be performed in a root-first fashion and type conversions are solely required in axiom formation. 1 The \Pi-calculus The \Pi-calculus [6, 9] is a theory extending the simply typed -calculus [1] with first-order \Pi-types, i.e. with dependent types. It is the type system of the Edinburgh Logical Framework (LF) [6], a logic for presenting logics, and it is the basis of Elf [7, 8], a language giving to types an operational interpretation similar to that given to formulae in logic programming. The \Pi-calculus essentially corresponds to the P system of the -cube [1] and it is also closely related to the \Pi-fr...

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