ExampleConstruction: Literature-sourcedVerification: AI-adaptedverified 2026-09-26 (gpt-6-sol)

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 hyperbolic plane is hyperbolic

Example

The hyperbolic plane $H^{2}$ is a hyperbolic geodesic metric space.

Facts & Assumptions

Given: The standard geodesic metric on $H^{2}$ .

[F1]

Löh's cited lecture notes state in Example 4.3.2 that $H^{2}$ is geodesic and state on p. 153 that all geodesic triangles in $H^{2}$ are uniformly slim, citing Theorem A.3.27 of Löh's Geometric Group Theory: An Introduction. Thus some single $δ \geq 0$ works for every geodesic triangle in $H^{2}$ .

[L1]

A geodesic metric space is hyperbolic exactly when all geodesic triangles are $δ$ -slim for some $δ \geq 0$ (Delta-slim triangles and hyperbolic spaces).

Verification

technique · direct

1.1F1

By the external geometric result [F1], $H^{2}$ is geodesic and all its geodesic triangles are $δ$ -slim for a single $δ \geq 0$ .

2.1L1step 1.1∎

Therefore [L1] shows that $H^{2}$ is hyperbolic.

Depends on

Delta-slim triangles and hyperbolic spaces

Used by

Closed surface groups are hyperbolic Example

Dependency tree · two levels

2 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

Clara Löh, Geometric Group Theory lecture notes (2022), Example 4.3.2, Section 6.1 p. 153, and Example 6.2.3 (citing Löh, Geometric Group Theory: An Introduction, Theorem A.3.27) (standard reference, not scraped)