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.

A mixed chain of blocks collapses to one quotient block

Example

Let $L = (A, B, C, D)$ be a blockade in which $(A, B)$ and $(B, C)$ are mixed, while every pair involving $D$ is anticomplete and $(A, C)$ is also anticomplete. Then the quotient blockade $L / M$ has exactly two blocks, namely $A \cup B \cup C$ and $D$ .

Facts & Assumptions

Given: The four-block configuration in the Statement.

[L1]

The mixed-block reachability relation is an equivalence relation, so its equivalence classes are the blocks of the quotient blockade (Mixed-block reachability is an equivalence relation, The quotient blockade obtained from mixed-block reachability).

Verification

technique · direct

1.1

Because $(A, B)$ and $(B, C)$ are mixed, there is a mixed chain from $A$ to $C$ through $B$ . Thus $A$ , $B$ , and $C$ lie in the same $M$ -class.

givenL1

1.2

No pair involving $D$ is mixed, so there is no mixed chain from $D$ to any of $A$ , $B$ , or $C$ . Therefore $D$ lies in a different $M$ -class.

givenL1

2.1

By [L1], the quotient blockade has one block equal to the union $A \cup B \cup C$ of the first class and one block equal to $D$ from the second class.

step 1.1step 1.2L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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.