ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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.

One-key tiled attention

Example

Take $n = d = v = 1$ , $Q = [2]$ , $K = [3]$ , $V = [7]$ , scale $c = 1$ , block size $b = 1$ , and allowed-key set $A_{1} = {1}$ . The tiled output is $Y = [7]$ . For upstream scalar $G = [4]$ , the exact reverse output is $d Q = [0]$ , $d K = [0]$ , and $d V = [4]$ .

Example references

Tiled forward and reverse procedures

Exactness of the forward and reverse outputs

Facts & Assumptions

Given: The scalar inputs and procedures in the example statement.

[F1]

Exact tiled softmax attention with block-recomputed vector-Jacobian products: for one allowed key the forward summary is copied as the row state, and the reverse updates use $p_{i j}$ and $γ_{i j}$ .

[F2]

Tiled attention and its recomputed reverse pass are exact: the reverse procedure returns the derivative of the scalarized output for each fixed upstream value.

Verification

1.1F1givenalgebra

The only score is $s_{11} = 2 \cdot 3 = 6$ . Its summary is $(m, ℓ, z) = (6, 1, 7)$ , so the output is $z / ℓ = 7$ .

2.1F1F2step 1.1algebra∎

The probability is $p_{11} = 1$ , the scalar $δ_{1} = G_{1} Y_{1} = 28$ , and $γ_{11} = 1 (4 \cdot 7 - 28) = 0$ . The reverse updates therefore give $d Q = 0$ , $d K = 0$ , and $d V = p_{11} G_{1} = 4$ .

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.

Sources

Tri Dao et al., FlashAttention: Fast and Memory-Efficient Exact Attention with IO-Awareness, §3.1 (standard reference, not scraped)