
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.
focusing, FOS: Computer and information sciences, Computer Science - Logic in Computer Science, intuitionstic logic, Structure of proofs, polarized logic, proof search, cut admissibility, Logic in Computer Science (cs.LO), normalization, proof terms, identity expansion, F.4.1, 89999 Information and Computing Sciences not elsewhere classified, Cut-elimination and normal-form theorems, 03F05, 03F07, 03F65, Subsystems of classical logic (including intuitionistic logic)
focusing, FOS: Computer and information sciences, Computer Science - Logic in Computer Science, intuitionstic logic, Structure of proofs, polarized logic, proof search, cut admissibility, Logic in Computer Science (cs.LO), normalization, proof terms, identity expansion, F.4.1, 89999 Information and Computing Sciences not elsewhere classified, Cut-elimination and normal-form theorems, 03F05, 03F07, 03F65, Subsystems of classical logic (including intuitionistic logic)
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