Alphabeta Math
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.

The lower and upper shadows of a uniform set family

Definition

Let AA be a finite set and let F⊆[A]k\mathcal F\subseteq[A]^k be a kk-uniform family (The set [A]k[A]^{k} of kk-element subsets and the binomial coefficient (nk):=∣[n]k∣\binom{n}{k} := \lvert [n]^{k}\rvert). Its lower shadow is

∂F:={ R∈[A]k−1:R⊂S for some S∈F }\partial\mathcal F:=\{\,R\in[A]^{k-1}:R\subset S\text{ for some }S\in\mathcal F\,\}

when k≥1k\ge1, and ∂F:=∅\partial\mathcal F:=\varnothing when k=0k=0. Its upper shadow is

∇F:={ T∈[A]k+1:S⊂T for some S∈F }.\nabla\mathcal F:=\{\,T\in[A]^{k+1}:S\subset T\text{ for some }S\in\mathcal F\,\}.

If k=∣A∣k=|A|, the upper shadow is empty. Both shadows consist of the immediate neighbours of F\mathcal F one rank below or above it in the Boolean lattice.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 37 results over 17 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