Alphabeta Math
ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-29
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.

The Bass-Serre tree of an amalgamated free product

Example

For an amalgamated free product A∗CB, the Bass-Serre tree has vertices the left cosets of A and B and edges the left cosets of C.

Facts & Assumptions

Given: The one-segment graph-of-groups description of an amalgamated free product.

[L1]

A one-segment graph of groups yields an amalgamated free product. (A one-segment graph of groups gives an amalgamated free product)

[L2]

In the Bass-Serre tree, vertices are cosets of vertex groups and edges are cosets of edge groups. (The Bass-Serre tree of a graph of groups)

Verification

technique · direct
1.1

Realize A∗CB by the one-segment graph of groups from [L1]. Then [L2] says the two vertex orbits are (A∗CB)/A and (A∗CB)/B, while the edge orbit is (A∗CB)/C.

L1L2given
2.1

The edge coset γC joins the vertices γA and γB, so the tree records exactly how the amalgamating subgroup sits in the two factors.

step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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