ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-28

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 quotient-witness reduction in a three-block configuration

Example

Take original blocks $A_{1}, A_{2}, B$ and outside vertices $x, y, u$ with the following adjacency pattern:

each of $A_{1}$ , $A_{2}$ , and $B$ induces a connected subgraph;
$A_{1}$ and $A_{2}$ are mixed;
$B$ is complete to $A_{1}$ and anticomplete to $A_{2}$ ;
$x$ and $y$ are nonadjacent and both are complete to $A_{1} \cup A_{2} \cup B$ ;
$u$ is adjacent to $x$ , nonadjacent to $y$ , complete to $A_{1} \cup A_{2}$ , and anticomplete to $B$ .

Then the quotient blocks are $D_{1} : = A_{1} \cup A_{2}$ and $D_{2} : = B$ , and the quotient-level witness descends to the original mixed pair $A_{1}, A_{2}$ .

Facts & Assumptions

Given: The three-block configuration described in the Statement.

[L1]

Mixed original blocks lie in one quotient block, so $A_{1}$ and $A_{2}$ merge into $D_{1}$ , while $B$ stays separate because it is pure to each of $A_{1}$ and $A_{2}$ (The quotient blockade obtained from mixed-block reachability).

[L2]

The descent lemma turns a quotient-level witness on $D_{1}, D_{2}$ into a witness on mixed original blocks inside $D_{1}$ (A quotient-level mixed-block witness descends to two mixed member blocks).

Verification

technique · direct

1.1

By item 2 of the configuration and [L1], the two original blocks $A_{1}, A_{2}$ form one quotient block $D_{1} = A_{1} \cup A_{2}$ , while $D_{2} = B$ is the other quotient block. Because $B$ is complete to $A_{1}$ and anticomplete to $A_{2}$ , the quotient blocks $D_{1}$ and $D_{2}$ are mixed.

givenL1

2.1

Item 1 of the configuration says that every original block of the blockade is connected, so the connectivity hypothesis of [L2] holds. No vertex of $D_{1}$ is mixed on $D_{2}$ : every vertex of $A_{1}$ is complete to $B$ , and every vertex of $A_{2}$ is anticomplete to $B$ . The outside vertices satisfy the remaining hypotheses of [L2]: $x$ and $y$ are nonadjacent and complete to $D_{1} \cup D_{2}$ , while $u \in N (x) ∖ N (y)$ is complete to $D_{1}$ and anticomplete to $D_{2}$ .

step 1.1given

3.1

Applying [L2] therefore yields mixed original blocks $A_{1}, A_{2}$ inside $D_{1}$ and vertices $x^{'}, y^{'}, u^{'}$ outside $A_{1} \cup A_{2}$ with the required adjacency pattern. Thus the quotient-level witness has been pushed down to a witness on the original mixed pair inside $D_{1}$ .

step 2.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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.