Alphabeta Math
CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16
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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Binary words avoiding any fixed nonempty factor have a rational length generating function

Statement

Fix a nonempty binary word w. If an is the number of words in {0,1}n that do not contain w as a contiguous factor, then ∑n≥0anxn is a rational formal power series over Z.

Facts & Assumptions

Given: A nonempty word w over the finite alphabet {0,1}.

[L1]

Words over a finite alphabet that avoid a finite set of nonempty factors have a rational length generating function (Words over a finite alphabet avoiding finitely many nonempty factors have a rational length generating function).

Proof

technique · specialization
1.1given

The alphabet {0,1} is finite, and B={w} is a finite set of nonempty words.

2.1step 1.1L1∎

Apply [L1] using step 1.1. Its conclusion is the asserted rationality.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources