{"paper":{"title":"Computation by infinite descent made explicit","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Every valid proof in this ordinal-annotated non-wellfounded system is computable.","cross_cats":[],"primary_cat":"cs.LO","authors_text":"Sebastian Enqvist","submitted_at":"2025-06-27T13:25:04Z","abstract_excerpt":"We introduce a non-wellfounded proof system for intuitionistic logic extended with inductive and co-inductive definitions, based on a syntax in which fixpoint formulas are annotated with explicit variables for ordinals. We explore the computational content of this system, in particular we introduce a notion of computability and show that every valid proof is computable. As a consequence, we obtain a normalization result for proofs of what we call finitary formulas. A special case of this result is that every proof of a sequent of the appropriate form represents a unique function on natural num"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Every valid proof in the system is computable; as a consequence, every proof of a sequent of the appropriate form represents a unique function on natural numbers for finitary formulas.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The definition of computability for proofs in the non-wellfounded system with ordinal annotations is sufficient to capture the computational content without additional restrictions on the ordinal variables or the interaction between inductive and co-inductive rules.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Presents an ordinal-annotated non-wellfounded proof system for intuitionistic logic with fixpoints, establishes computability and normalization results for finitary formulas, and derives a categorical semantics with algebras and coalgebras.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Every valid proof in this ordinal-annotated non-wellfounded system is computable.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"993859c9b80efd5d360ec902213a12d3defe29a8dd096fcef2b5565fc5bdd098"},"source":{"id":"2506.22206","kind":"arxiv","version":6},"verdict":{"id":"8a7c54d1-7735-42b3-9401-84ab27ac4306","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T08:14:28.907020Z","strongest_claim":"Every valid proof in the system is computable; as a consequence, every proof of a sequent of the appropriate form represents a unique function on natural numbers for finitary formulas.","one_line_summary":"Presents an ordinal-annotated non-wellfounded proof system for intuitionistic logic with fixpoints, establishes computability and normalization results for finitary formulas, and derives a categorical semantics with algebras and coalgebras.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The definition of computability for proofs in the non-wellfounded system with ordinal annotations is sufficient to capture the computational content without additional restrictions on the ordinal variables or the interaction between inductive and co-inductive rules.","pith_extraction_headline":"Every valid proof in this ordinal-annotated non-wellfounded system is computable."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2506.22206/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"18f575a99bf3b97985f03bc77a982822c4a40958fb2b6b0962d925d3e46f3aa2"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}