ExampleConstruction: AI-generatedVerification: AI-generatedprecheck 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.

Iterating a recursive specification to determine coefficients through degree $6$

Example

For the plane-tree equation

$T = \frac{x}{1 - T},$

start with $T_{0} : = 0$ and define $T_{j + 1} : = x / (1 - T_{j})$ , truncating modulo $x^{7}$ . The iterates are

$T_{1} = x,$

$T_{2} = x + x^{2} + x^{3} + x^{4} + x^{5} + x^{6},$

$T_{3} = x + x^{2} + 2 x^{3} + 4 x^{4} + 8 x^{5} + 16 x^{6},$

$T_{4} = x + x^{2} + 2 x^{3} + 5 x^{4} + 13 x^{5} + 34 x^{6},$

$T_{5} = x + x^{2} + 2 x^{3} + 5 x^{4} + 14 x^{5} + 41 x^{6},$

$T_{6} = x + x^{2} + 2 x^{3} + 5 x^{4} + 14 x^{5} + 42 x^{6} .$

So the coefficients through degree $6$ are $1, 1, 2, 5, 14, 42$ .

Facts & Assumptions

Given: The plane-tree series is the unique fixed point of $Y \mapsto x / (1 - Y)$ (Rooted plane trees satisfy $T (x) = x / (1 - T (x))$ ), and order-raising recursion converges coefficientwise by successive truncation (An order-raising recursive specification has a unique solution).

Verification

technique · direct

1.1givenalgebra

Substituting each displayed iterate into $x / (1 - Y)$ and truncating modulo $x^{7}$ gives the next one in the list.

2.1step 1.1given∎

By the sixth iterate, every coefficient through degree $6$ has stabilized, so the unique fixed point begins $T (x) = x + x^{2} + 2 x^{3} + 5 x^{4} + 14 x^{5} + 42 x^{6} + \dots$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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.

Iterating a recursive specification to determine coefficients through degree 66