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.

An empty first masked key block

Example

Take $n = 2$ , $d = v = 1$ , $Q = K = [\begin{array}{c} 0 \\ 0 \end{array}]$ , $V = [\begin{array}{c} 3 \\ 5 \end{array}]$ , scale $c = 1$ , block size $b = 1$ , and allowed-key sets $A_{1} = {2}$ and $A_{2} = {1, 2}$ . The first key block is empty for query row $1$ but nonempty for query row $2$ . The tiled output is $Y = [\begin{array}{c} 5 \\ 4 \end{array}]$ .

Example references

Empty-block and nonempty-block handling

Exactness for every nonempty allowed row

Facts & Assumptions

Given: The two-row inputs, fixed allowed-key sets, and block size in the example statement.

[F1]

Exact tiled softmax attention with block-recomputed vector-Jacobian products: each row begins in state $⊥$ ; an empty allowed block is skipped, while the first nonempty block summary is copied.

[F2]

Tiled attention and its recomputed reverse pass are exact: a nonempty row's final weighted summary divided by its positive normalizer is its exact attention output.

Verification

1.1F1givenalgebra

For query row $1$ , the first block has $J_{1} \cap A_{1} = \emptyset$ , so the state remains $⊥$ . The second block contains key $2$ , whose score is $0$ and value is $5$ , giving summary $(0, 1, 5)$ and output $5$ .

2.1F1F2step 1.1algebra∎

For query row $2$ , both blocks are allowed and have score $0$ . The first summary is $(0, 1, 3)$ and merging the second gives $(0, 2, 8)$ , so the output is $8 / 2 = 4$ . Both allowed-key sets are nonempty, so these are defined rows of the claimed attention map.

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 and Appendix C (standard reference, not scraped)