ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26

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.

Counting the induced copies of $P_{3}$ in $P_{4}$ by extension sets

Example

The identity that counts induced copies by extension sets can be seen directly in $P_{4}$ : the induced-copy number of $P_{3}$ in $P_{4}$ is $4$ , and the extension-set sum of The induced copies of $H_{1}$ in $G$ are counted by summing, over the induced embeddings of $H_{1} - v$ , the number of vertices that extend them at $v$ gives the same value.

Facts & Assumptions

Given: The path $P_{4}$ with vertices $0, 1, 2, 3$ , and the pattern $P_{3}$ obtained by deleting an endpoint from $P_{4}$ .

[L1]

The induced-copy number counts induced embeddings of the pattern, not only vertex subsets (The induced-embedding count ${ind}_{H} (G)$ , Induced embeddings and induced copies of a graph).

[L2]

The extension lemma expresses the induced-copy number of a graph by summing, over the induced embeddings of the graph with one deleted vertex, the sizes of the corresponding extension sets (The induced copies of $H_{1}$ in $G$ are counted by summing, over the induced embeddings of $H_{1} - v$ , the number of vertices that extend them at $v$ ).

Verification

technique · direct

1.1

Exactly the vertex sets ${0, 1, 2}$ and ${1, 2, 3}$ induce a copy of $P_{3}$ inside $P_{4}$ .

given

2.1

Each of those two vertex sets supports two induced embeddings of $P_{3}$ , one for each automorphism of the path, so [L1] gives ${ind}_{P_{3}} (P_{4}) = 4$ .

step 1.1L1

3.1

Delete the endpoint labelled $0$ from the pattern $0$ - $1$ - $2$ . The six oriented edge embeddings $1 \mapsto a, 2 \mapsto b$ in $P_{4}$ have extension-set sizes $0, 1, 1, 1, 1, 0$ for $(a, b) = (0, 1), (1, 0), (1, 2), (2, 1), (2, 3), (3, 2)$ respectively: the extending image of $0$ must be adjacent to $a$ and nonadjacent to $b$ . Their sum is $4$ , agreeing with step 2.1 and [L2].

step 2.1L2given∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

17 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.

Counting the induced copies of P3P3​ in P4P4​ by extension sets