We prove a folklore theorem, that two derivations in a cut-free sequent 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) iff they determine the same natural deduction. The basic rules form a confluent and weakly normalising rewriting system. We refer to Schwichtenberg's proof elsewhere that a modification of this system is strongly normalising. Key words: intuitionistic logic, proof theory, natural deduction, sequent calculus. 1 Introduction There is a folklore theorem that two intuitionistic sequent calculus derivations are "really the same" iff they are inter-permutable, using permutations as described by Kleene in [13]. Our purpose here is to make precise and prove such a "permutability theorem". Prawitz [18] showed how intuitionistic sequent calculus derivations determine natural deductions, via a mapping ' from LJ to NJ (here we consider only ...

rule). This is also the case for NJ and LJ as defined in this formalisation. This is due to the particular nature of the logics in question, and does not necessarily generalise to other logics. In particular, a formalisation of linear logic would not work in this fashion, and a more complex variable-referencing mechanism would be required. See Section 6 for a further discussion of this problem. Other operations, such as substitutions (sub in Table 2) and weakening, require lift and drop operations as defined in [27] to ensure the correctness of the de Bruijn indexing.

Introduction In this note we wish to draw attention to certain points of detail, concerning the subtle differences between several possible versions of Gentzen systems and systems of natural deduction. In notation and terminology we conform to [TS96]. Gentzen([Gen35]) introduced the calculi LJ, LK with left- and right introduction rules, operating on sequents; systems of this kind are in the literature often called "sequent calculi". Contrary to what many people think, natural deduction did not originate with Gentzen (there is, for example, the earlier work by J'askowski, [J'as34]) although Gentzen's work made it wellknown. Versions of natural deduction are sometimes also presented as calculi operating on sequents (as in this note). For these reasons I have rejected the widely used designation "sequent calculi" for systems of the LJ,LK-type and call them "Gentzen systems" instead. (One might object that "Gentzen systems" might be interpreted as referring to all the formalisms

We describe a formalisation of proof theory about sequent-style calculi, based on informal work in [DP96]. The formalisation uses de Bruijn nameless dummy variables (also called de Bruijn indices) [dB72], and is performed within the proof assistant Coq [BB + 96]. We also present a description of some of the other possible approaches to formal meta-theory, particularly an abstract named syntax and higher order abstract syntax. 1 Introduction Formal proof has developed into a significant area of mathematics and logic. Until recently, however, such proofs have concentrated on proofs within logical systems, and meta-theoretic work has continued to be done informally. Recent developments in proof assistants and automated theorem provers have opened up the possibilities for machine-supported meta-theory. This paper presents a formalisation of a large theory comprising of over 200 definitions and more than 500 individual theorems about three different deductive system. 1 The central dif...