ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-02

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.

Three left vertices with only two collective neighbours fail Hall's condition and cannot all be matched

Example

Take left part $X = {x_{1}, x_{2}, x_{3}}$ , right part $Y = {a, b}$ , and all six possible edges between the two parts. This graph has no matching saturating $X$ .

Facts & Assumptions

Given: The displayed finite bipartite graph.

[L1]

A finite bipartite graph has a matching saturating its left part exactly when Hall's condition holds (Hall's marriage theorem for a finite bipartite graph).

Verification

Verification technique: direct.

1.1

For $S = X$ , the neighbourhood is $N (S) = {a, b}$ , so $∣ N (S) ∣ = 2 < 3 = ∣ S ∣$ .

given

1.2

Thus Hall's condition fails, and [L1] proves that no matching can saturate all three left vertices.

2.1

The two available right vertices also show directly that any matching has at most two edges.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 results within two dependency steps 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.