DefinitionDefinition: AI-adaptedProof: Not applicableaudited 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 $ρ : P \to N$ such that every minimal element has rank $0$ and

$ρ (y) = ρ (x) + 1$

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

$P_{k} : = {x \in P : ρ (x) = k} .$

Depends on

Used by

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

Dependency tree · one level

2 results within one dependency step 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

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