LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck 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.

Bounded distance is an equivalence relation and is preserved by pre-composition and by post-composition with a coarse Lipschitz map

Statement

Bounded distance is an equivalence relation and is preserved by pre-composition and by post-composition with a coarse Lipschitz map.

Facts & Assumptions

Given: The hypotheses of the Statement.

[F1]

Two maps into a metric space are at bounded distance when the distance between their values is bounded uniformly (Bounded distance between two maps into a metric space).

[L1]

A map is $(L, C)$ -coarse Lipschitz when $d (f (x), f (x^{'})) \leq L d (x, x^{'}) + C$ , and an $(L, C)$ -quasi-isometric embedding when in addition $L^{- 1} d (x, x^{'}) - C \leq d (f (x), f (x^{'}))$ (Coarse Lipschitz maps and quasi-isometric embeddings).

Proof

technique · direct

1.1F1

Reflexivity, symmetry and transitivity follow from the metric axioms with the bounds added.

2.1F1step 1.1

Pre-composition changes no value, so it preserves the bound exactly.

3.1F1L1step 1.1∎

Post-composition with an $(L, C)$ -coarse Lipschitz map multiplies the bound by $L$ and adds $C$ ; without the coarse Lipschitz hypothesis the bound need not survive.

Depends on

Used by

Dependency tree · two levels

4 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

C. Loh, Geometric Group Theory: An Introduction (2015 course version), 264 pp. (standard reference, not scraped)
C. Drutu and M. Kapovich, Geometric Group Theory (with an appendix by B. Nica), 837 pp. (standard reference, not scraped)