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.

Composites of coarse Lipschitz maps and of quasi-isometric embeddings are again such, with explicit constants

Statement

Composites of coarse Lipschitz maps and of quasi-isometric embeddings are again such, with explicit constants.

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).

Proof

technique · direct

1.1

Substituting the inequality for the inner map into the one for the outer map bounds the composite above with constants $L_{1} L_{2}$ and $L_{2} C_{1} + C_{2}$ .

F1algebra

2.1

The same substitution on the lower bounds gives the other inequality, with the constants recorded rather than absorbed.

F1step 1.1∎

Depends on

Coarse Lipschitz maps and quasi-isometric embeddings

Used by

Dependency tree · two levels

3 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)