CorollaryStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26

How statement and proof provenance work

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 have generating function $1 / (1 - 2 x)$

Statement

Let $B$ be the class of finite binary words, with size equal to word length. Then

$OGF (B) = \frac{1}{1 - 2 x} .$

Facts & Assumptions

Given: Two disjoint copies $Z_{0}$ and $Z_{1}$ of the atomic class $Z$ , and the class $A : = Z_{0} + Z_{1}$ .

[L1]

Disjoint union and Cartesian product translate to addition and multiplication of ordinary generating functions (Disjoint union and Cartesian product translate to addition and multiplication of ordinary generating functions).

[L2]

If $A$ has no size-zero objects then $SEQ (A)$ has generating function $1 / (1 - A (x))$ (If $A$ has no size-zero objects then $SEQ (A)$ has generating function $1 / (1 - A (x))$ ).

Proof

technique · direct

1.1

Each of $Z_{0}$ and $Z_{1}$ has generating function $x$ , so $A$ has generating function $x + x = 2 x$ by [L1]. A binary word is exactly a finite sequence of objects from $A$ .

2.1

The class $A$ has no size-zero objects, so [L2] applies and gives $OGF (B) = OGF (SEQ (A)) = 1 / (1 - 2 x)$ .

step 1.1L2∎

Depends on

Used by

Binary words of length at most three from SEQ(2Z) Example

Dependency tree · two levels

10 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

Robert Sedgewick and Kevin Wayne, Analysis of Algorithms, Section 3.9 (standard reference, not scraped)
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics (standard reference, not scraped)

Binary words have generating function 1/(1−2x)1/(1−2x)