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

A three-set family with an explicit system of distinct representatives

Example

For $A_1=\{a,b\}$ , $A_2=\{b,c\}$ , and $A_3=\{a,c\}$ , the assignment $r(1)=a$ , $r(2)=b$ , $r(3)=c$ is an SDR.

Facts & Assumptions

Given: The three finite sets $A_1,A_2,A_3$ .

[L1]

A family with finite index set and finite union has an SDR exactly when every indexed subfamily has union at least as large as its index set (A finite family has an SDR if and only if every subfamily has a union at least as large as its index set).

Verification

Verification technique: constructive.

1.1

Each singleton union has size two, each two-set union is $\{a,b,c\}$ , and the three-set union is $\{a,b,c\}$ ; the Hall union inequalities all hold.

givenconstruct

1.2

By [L1], the family has an SDR.

2.1

The displayed assignment chooses an element of each $A_i$ and has three distinct values, so it is the promised SDR.

step 1.1step 1.2discharge-construct∎

Depends on

A finite family has an SDR if and only if every subfamily has a union at least as large as its index set

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: 10 results over 7 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.