Alphabeta Math
ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 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.

Plane trees with at most four vertices from T=Z×SEQ⁡(T)

Example

The rooted plane trees on at most four vertices are counted by the first coefficients of the Catalan series:

1, 1, 2, 5.

Concretely there is one tree on one vertex, one on two vertices, two on three vertices, and five on four vertices.

Facts & Assumptions

Given: The plane-tree generating function satisfies T(x)=x/(1−T(x)) (Rooted plane trees satisfy T(x)=x/(1−T(x))).

Verification

technique · direct
1.1

Solving T=x/(1−T) coefficientwise gives T(x)=x+x2+2x3+5x4+⋯, so the first four counts are 1,1,2,5.

givenalgebra
2.1

These are realized by the evident shapes: a single root; a root with one child; for three vertices, the chain and the root with two children; for four vertices, the chain of length four, the root with three children, the root with one child whose child has two children, and the two left-right orderings of a root with two children one of which has one child.

step 1.1∎

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.