1.1 Quantification and the subformula property.................. 3 1.2 Ground forward sequent calculus......................... 5 1.3 Lifting to free variables............................... 10

LJQ is a focused sequent calculus for intuitionistic logic, with a simple restriction on the first premisss of the usual left introduction rule for implication. We discuss its history (going back to about 1950, or beyond), present the underlying theory and its applications both to terminating proof-search calculi and to call-by-value reduction in lambda calculus.

We give a short proof-theoretic treatment of a terminating contraction-free calculus G4-LC for the zero-order Gödel-Dummett logic LC. This calculus is a slight variant of a calculus given by Avellone et al, who show its completeness by model-theoretic techniques. In our calculus, all the rules of G4-LC are invertible, thus allowing a deterministic proof-search procedure.

Dragalin in his book on Mathematical Intuitionism has given an outline proof of the admissibility of Contraction and Cut for the multi-succedent intuitionistic sequent calculus. One can give a corresponding (but simpler) proof for the single-succedent calculus. These proofs are given here in detail as a basis for extension and modification elsewhere. 1 Introduction Dragalin's proof [1] of admissibility of Contraction and Cut for a multi-succedent intuitionistic sequent calculus GHPC is a useful basis for similar proofs. One can easily construct a similar proof for the single succedent calculus. The proofs are for calculi in which Contraction is no longer primitive but an admissible rule. There are hidden complexities: our purpose here is just to give the proofs in detail so that claims elsewhere (e.g. [2], [3]) such as "this proof is like Dragalin's: only the following changes and new cases need to be considered" can be easily verified. Similar proofs are outlined in [4], [5]. We main...

Inspired by the Curry-Howard correspondence, we study normalisation procedures in the depth-bounded intuitionistic sequent calculus of Hudelmaier (1988) for the implicational case, thus strengthening existing approaches to Cut-admissibility. We decorate proofs with proofterms and introduce various term-reduction systems representing proof transformations. In contrast to previous papers which gave different arguments for Cut-admissibility suggesting weakly normalising procedures for Cut-elimination, our main reduction system and all its variations are strongly normalising, with the variations corresponding to different optimisations, some of them with good properties such as confluence.

Formalised proofs of Cut admissibility often rely on the invertibility of the rules of a sequent calculus. We will present some sufficient conditions for when a rule is invertible with respect to a calculus, which we illustrate with many examples. Appropriate definitions are given for rarely defined intuitive notions, such as a formula being principal for a rule. It must be noted we give purely syntactic criteria; no guarantees are given as to the suitability of the rules. We also formalise some of the results in the proof assistant Isabelle, as a means to automating Cut admissibility proofs.

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The inversion calculus from the last lecture constitutes a significant step forward, but it still has the problem that in the ⊃L rule, the principal formula has to be copied to the first premise. Therefore, the first premise may not be smaller than the conclusion.

Whilst results from Structural Proof Theory can be couched in many formalisms, it is the sequent calculus which is the most amenable of the formalisms to metamathematical treatment. Constructive syntactic proofs are filled with bureaucratic details; rarely are all cases of a proof completed in the literature. Two intermediate results can be used to drastically reduce the amount of effort needed in proofs of Cut admissibility: Weakening and Invertibility. Indeed, whereas there are proofs of Cut admissibility which do not use Invertibility, Weakening is almost always necessary. Use of these results simply shifts the bureaucracy, however; Weakening and Invertibility, whilst more easy to prove, are still not trivial. We give a framework under which sequent calculi can be codified and analysed, which then allows us to prove various results: for a calculus to admit Weakening and for a rule to be invertible in a calculus. For the latter, even though many calculi are investigated, the general condition is simple and easily verified. The results have been applied to G3ip, G3cp, G3c, G3s, G3-LC and G4ip. Invertibility is important in another respect; that of proof-search. Should all rules in a calculus be invertible, then terminating root-first proof search gives a decision procedure

We will, in this document, give a report of a formalisation, in Isabelle, of Contraction admissibility for the calculus G4ip. Specifically, we formalise sections of [3], which goes on to directly prove Cut admissibility as well. We