The paper introduces a skolemized variant of focused intuitionistic linear logic (SLJF) that encodes dependencies between quantifier rules and non-invertible propositional rules using explicit substitutions. The main contributions are:
The key idea is to represent dependencies between quantifier rules and non-invertible propositional rules using explicit substitutions. Admissibility of these substitutions is checked during unification at the axioms, ensuring that the proof is valid. This avoids the need for backtracking on quantifier instantiation during proof search.
The paper first introduces focused intuitionistic linear logic (LJF) and its key aspects, such as polarity of connectives and focusing. It then presents the skolemized variant SLJF, defining its syntax, semantics, and the central notion of admissibility.
The skolemization procedure is then defined, showing how LJF formulas are transformed into SLJF formulas and associated substitutions. The soundness and completeness of this skolemization procedure are then proven, establishing the equivalence between LJF and SLJF derivations.
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