Abstract
Focusing, introduced by Jean-Marc Andreoli in the context of classical linear logic [Andreoli 1992], defines a normal form for sequent calculus derivations that cuts down on the number of possible derivations by eagerly applying invertible rules and grouping sequences of non-invertible rules. A focused sequent calculus is defined relative to some nonfocused sequent calculus; focalization is the property that every nonfocused derivation can be transformed into a focused derivation. In this article, we present a focused sequent calculus for propositional intuitionistic logic and prove the focalization property relative to a standard presentation of propositional intuitionistic logic. Compared to existing approaches, the proof is quite concise, depending only on the internal soundness and completeness of the focused logic. In turn, both of these properties can be established (and mechanically verified) by structural induction in the style of Pfenning's structural cut elimination without the need for any tedious and repetitious invertibility lemmas. The proof of cut admissibility for the focused system, which establishes internal soundness, is not particularly novel. The proof of identity expansion, which establishes internal completeness, is a major contribution of this work.
Highlights
The propositions of intuitionistic propositional logic are recognizable and standard: we will consider a logic with atomic propositions, falsehood, disjunction, truth, conjunction, and implication.P, Q ::= p | ⊥ | P1 ∨ P2 | | P1 ∧ P2 | P1 ⊃ P2The sequent calculus presentation for intuitionistic logic is standard; the system in Figure 1 is precisely the propositional fragment of Kleene’s sequent system G3 as presented in [Pfenning 2000]
Sequent calculi are a nice way of presenting logics, and a logic’s sequent calculus presentation is a convenient setting in which to establish the logic’s metatheory in a way that is straightforwardly mechanizable in proof assistants that are organized around the idea of structural induction
The first, cut admissibility, justifies the use of lemmas: if we know P and we know that Q follows from assuming P, we can come to know Q without the additional assumption of P. 1 A proof of the cut admissibility property establishes the internal soundness of a logic – it
Summary
The propositions of intuitionistic propositional logic are recognizable and standard: we will consider a logic with atomic propositions, falsehood, disjunction, truth, conjunction, and implication. The sequent calculus presentation for intuitionistic logic is standard; the system in Figure 1 is precisely the propositional fragment of Kleene’s sequent system G3 as presented in [Pfenning 2000]. A proof of the identity property establishes the internal completeness of a logic. We call these properties internal, following Pfenning [2010], to emphasize that these are properties of the deductive system itself and not a comment on the system’s relationship to any external semantics. There is a tradition in logic, dating back to Gentzen [1935], that views the sequent calculus as a convenient formalism for proving a logic’s metatheoretic properties while viewing natural deduction proofs as the “true proof objects.” One reason for this bias towards natural deduction is that natural deduction proofs have nice normalization properties. A natural deduction proof is normal if if there are no instances of an introduction rule immediately followed by an elimination rule of the same connective; such detours give rise to local reductions which eliminate the detour, such as this one: D1
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