Alphabeta Math
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 exact CSR product and its stored-entry count

Example

Use the CSR record with r=(0,2,5,8,10), c=(0,1,0,1,2,1,2,3,2,3), and v=(2,−1,−1,2,−1,−1,2,−1,−1,2) for the four-row tridiagonal matrix. For x=(1,2,3,4), the algorithm returns y=(0,0,0,5). It uses 10 field multiplications, 10 field additions, and 4 zero initializations; the input CSR arrays occupy 25 data slots.

Facts & Assumptions

Given: The displayed CSR data, the vector x=(1,2,3,4) over Q, and exact rational arithmetic.

[L1]

Compressed sparse row matrices with stored-entry semantics: the row pointers delimit the stored entries in each row and each entry contributes its value at its stored column.

[L2]

Validated compressed sparse row matrix-vector product: on valid input, the algorithm initializes each output coordinate to zero and adds vtxct for each stored entry t in its row.

[L3]

CSR matrix-vector multiplication returns the represented product: for valid data, the returned vector is the represented matrix times x.

[L4]

CSR storage and exact matrix-vector operation counts: a valid record with s stored entries uses s multiplications, s additions, m initializations, and 2s+m+1 CSR input slots.

[L5]

The rationals form a field: the rational numbers with their standard operations form a field.

Verification

1.1givenL1L2L5algebra

The row formulas give y0=2(1)−1(2)=0, y1=−1(1)+2(2)−1(3)=0, y2=−1(2)+2(3)−1(4)=0, and y3=−1(3)+2(4)=5 in Q.

2.1L3step 1.1

By [L3], these row values are the exact matrix-vector output, so the algorithm returns (0,0,0,5).

3.1givenL4algebra∎

The CSR has s=10 entries and m=4 rows. [L4] gives 10 multiplications, 10 additions, 4 initializations, and 2(10)+4+1=25 input slots.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 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