This paper presents a measure of inference in classical and intuitionistic logics in the Gentzenstyle sequent calculi. The measure for a proof of a sequent is the width of the proof tree, that is, the number of leaves of the proof tree. Then the measure for a sequent is the minimum value of the widths of possible proofs of the sequent; if it is unprovable, the assigned value is +∞・It counts the...

The focusing theorem identifies a complete class of sequent proofs that have no inessential nondeterministic choices and restrict the essential choices to a particular normal form. Focused proofs are therefore well suited both for the search and for the representation of sequent proofs. The calculus of structures is a proof formalism that allows rules to be applied deep inside a formula. Throug...

The identity rule of the sequent calculus exhibits one connection between the judgments A left and A right : If we assume A left we can prove A right . In other words, the left rules of the sequent calculus are strong enough so that we can reconstitute a proof of A from the assumption A. So the identity theorem (see Section 5) is a global version of the local completeness property for the elimi...

In 1994 Herbelin started and partially achieved the programme of showing that, for intuitionistic implicational logic, there is a Curry-Howard interpretation of sequent calculus into a variant of the λ-calculus, specifically a variant which manipulates formally “applicative contexts” and inverts the associativity of “applicative terms”. Herbelin worked with a fragment of sequent calculus with c...

This paper presents a measure of inference in classical and intuitionistic logics in the Gentzen-style sequent calculi. The measure for a proof of a sequent is the width of the proof tree, that is, the number of leaves of the proof tree. Then the measure for a sequent is the minimum value of the widths of possible proofs of the sequent; if it is unprovable, the assigned value is +∞ It counts t...

The identity theorem of the sequent calculus exhibits one connection between the judgments A left and A right : If we assume A left we can prove A right . In other words, the left rules of the sequent calculus are strong enough so that we can reconstitute a proof of A from the assumption A. So the identity theorem is a global version of the local completeness property for the elimination rules....

This paper proposes a definition of categorical model of the deep inference system BV, defined by Guglielmi. Deep inference introduces the idea of performing a deduction in the interior of a formula, at any depth. Traditional sequent calculus rules only see the roots of formulae. However in these new systems, one can rewrite at any position in the formula tree. Deep inference in particular allo...