LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck 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.

A map at bounded distance from a quasi-isometric embedding is one, with the additive constant enlarged

Statement

A map at bounded distance from a quasi-isometric embedding is one, with the additive constant enlarged.

Facts & Assumptions

Given: The hypotheses of the Statement.

[F1]

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).

[L1]

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).

[L2]

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

Proof

technique · direct

1.1

Two applications of the triangle inequality, one at each argument, relate the two maps’ distances up to twice the bound.

F1L1

2.1

Enlarging the additive constant by twice the bound gives both inequalities, with the multiplicative constant unchanged.

F1L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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)