DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-28

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.

An oriented graph with edge reversal

Definition

An oriented graph consists of a vertex set $V$ , an oriented-edge set $E$ , maps

$o, t : E \to V,$

called the origin and terminus, and a fixed-point-free involution

$E \to E, e \mapsto \overset{ˉ}{e},$

such that

$o (\overset{ˉ}{e}) = t (e), t (\overset{ˉ}{e}) = o (e) .$

Thus each geometric edge is the two-element orbit ${e, \overset{ˉ}{e}}$ . Parallel geometric edges and loops are allowed in a general oriented graph; the tree condition later excludes them.

Used by

Finite simplicial paths, reduced paths, cycles, and connectedness Definition
Graph automorphisms and group actions on a simplicial graph Definition
The quotient graph of an action without inversions Definition

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Jean-Pierre Serre, Trees (standard reference, not scraped)
Yuriy Tumarkin, Groups Acting on Trees (standard reference, not scraped)