Alphabeta Math
ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 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.

A six-element poset of width three and a three-chain cover

Example

Let P={a1,a2,a3,b1,b2,b3}P=\{a_1,a_2,a_3,b_1,b_2,b_3\} with ai<bia_i<b_i for each ii, and no other comparabilities between distinct elements. Then {a1,a2,a3}\{a_1,a_2,a_3\} is an antichain, and

{a1<b1},{a2<b2},{a3<b3}\{a_1<b_1\},\qquad\{a_2<b_2\},\qquad\{a_3<b_3\}

is a chain cover. The width is exactly 33, and this cover is minimum.

a1a2a3b1b2b3C1C2C3maximumantichainfa1;a2;a3g

Facts & Assumptions

Given: The six-element poset PP described in the Example.

[L1]

In a finite poset, the minimum number of chains in a chain cover equals the width (Dilworth's theorem: the minimum number of chains covering a finite poset equals its width).

Verification

technique · direct
1.1

The set {a1,a2,a3}\{a_1,a_2,a_3\} is an antichain, so the width is at least 33.

given
1.2

Every antichain contains at most one element from each comparable pair {ai,bi}\{a_i,b_i\}, so it has at most 33 elements. Hence the width is exactly 33.

given
1.3

The three displayed two-element chains cover all six elements, so they form a chain cover of cardinality 33.

given
2.1

By steps 1.2 and 1.3, and equivalently by [L1], the displayed cover has the minimum possible number of chains.

step 1.2step 1.3L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 14 results over 9 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