Admissibility of Substitution Rule in Cyclic-Proof Systems
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| Format: | Preprint |
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2025
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| _version_ | 1866915558585270272 |
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| author | Saotome, Kenji Nakazawa, Koji |
| author_facet | Saotome, Kenji Nakazawa, Koji |
| contents | This paper investigates the admissibility of the substitution rule in cyclic-proof systems. The substitution rule complicates theoretical case analysis and increases computational cost in proof search since every sequent can be a conclusion of an instance of the substitution rule; hence, admissibility is desirable on both fronts. While admissibility is often shown by local proof transformations in non-cyclic systems, such transformations may disrupt cyclic structure and do not readily apply. Prior remarks suggested that the substitution rule is likely nonadmissible in the cyclic-proof system CLKID^omega for first-order logic with inductive predicates. In this paper, we prove admissibility in CLKID^omega, assuming the presence of the cut rule. Our approach unfolds a cyclic proof into an infinitary form, lifts the substitution rules, and places back edges to construct a cyclic proof without the substitution rule. If we restrict substitutions to exclude function symbols, the result extends to a broader class of systems, including cut-free CLKID^omega and cyclic-proof systems for the separation logic. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_14749 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Admissibility of Substitution Rule in Cyclic-Proof Systems Saotome, Kenji Nakazawa, Koji Logic in Computer Science This paper investigates the admissibility of the substitution rule in cyclic-proof systems. The substitution rule complicates theoretical case analysis and increases computational cost in proof search since every sequent can be a conclusion of an instance of the substitution rule; hence, admissibility is desirable on both fronts. While admissibility is often shown by local proof transformations in non-cyclic systems, such transformations may disrupt cyclic structure and do not readily apply. Prior remarks suggested that the substitution rule is likely nonadmissible in the cyclic-proof system CLKID^omega for first-order logic with inductive predicates. In this paper, we prove admissibility in CLKID^omega, assuming the presence of the cut rule. Our approach unfolds a cyclic proof into an infinitary form, lifts the substitution rules, and places back edges to construct a cyclic proof without the substitution rule. If we restrict substitutions to exclude function symbols, the result extends to a broader class of systems, including cut-free CLKID^omega and cyclic-proof systems for the separation logic. |
| title | Admissibility of Substitution Rule in Cyclic-Proof Systems |
| topic | Logic in Computer Science |
| url | https://arxiv.org/abs/2510.14749 |