This paper introduces λ, a simply typed lambda calculus supporting inductive types and recursive function definitions with termination ensured by types. The system is shown to enjoy subject reduction, strong normalisation of typable terms and to be stronger than a related system λ G in which termination is ensured by a syntactic guard condition. The system… (More)
We present a non-looping method to construct Kripke trees refuting the non-theorems of intuitionistic propositional logic, using a contraction-free sequent calculus.
(probably none of whom agrees with everything in the chapter).
Bi-intuitionistic logic is a conservative extension of intu-itionistic logic with a connective dual to implication, called exclusion. We present a sound and complete cut-free labelled sequent calculus for bi-intuitionistic propositional logic, BiInt, following S. Negri's general method for devising sequent calculi for normal modal logics. Although it arises… (More)
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-and-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… (More)
We prove a folklore theorem, that two derivations in a cut-free se-quent calculus for intuitionistic propositional logic (based on Kleene's G3) are inter-permutable (using a set of basic \permutation reduction rules" derived from Kleene's work in 1952) ii they determine the same natural deduction. The basic rules form a connuent and weakly normalising… (More)
We describe a sequent calculus, based on work of Herbelin, of which the cut-free derivations are in 1-1 correspondence with the normal natural deduction proofs of intuitionistic logic. We present a simple proof of Herbelin's strong cut-elimination theorem for the calculus, using the recursive path ordering theorem of Dershowitz.
We describe a sequent calculus MJ, based on work of Herbelin, of which the cut-free derivations are in 1-1 correspondence with the normal natural deduction proofs of intuitionistic logic. MJ (without cut) has the sub-formula property and is therefore convenient for automated proof search; it admits no permutations and therefore avoids some of the… (More)
This work presents an extension with cuts of Schwichten-berg's multiary sequent calculus. We identify a set of permutative conversions on it, prove their termination and confluence and establish the permutability theorem. We present our sequent calculus as the typing system of the generalised multiary λ-calculus λJ m , a new calculus introduced in this… (More)
Using some routine properties ([zu], [po]) of the Prawitz translation φ from (sequent calculus) derivations to (natural) deductions, and restricting ourselves to the language of first-order hereditary Harrop formulae, we show (i) that φ maps the simple uniform derivations of Miller et al onto the set of deductions in expanded normal form; and (ii) that φ… (More)