DefinitionDefinition: AI-adaptedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-07-31

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.

Graded poset, rank function, and rank levels

Definition

Let $P$ be a finite poset. An element is minimal if no element is strictly below it, the dual of the maximal-element notion (Maximal element and greatest element). An element $y$ covers $x$ if $x<y$ and there is no $z\in P$ with $x<z<y$ . A rank function is a map $\rho:P\to\mathbb N$ such that every minimal element has rank $0$ and

$\rho(y)=\rho(x)+1$

whenever $y$ covers $x$ . A poset admitting a rank function is graded. Its rank- $k$ level is

$P_k:=\{x\in P:\rho(x)=k\}.$

Depends on

Used by

The Boolean lattice of subsets of a finite set and its rank levels Definition

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 2 results over 2 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

M. Keller and W. T. Trotter, Applied Combinatorics, §6.2 (standard reference, not scraped)