Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11
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.

Polynomial addition and multiplication computed from coefficient convolution

Example

In Z[x]\mathbb Z[x], let f=1+2x+3x2f=1+2x+3x^2 and g=2−xg=2-x. Then

f+g=3+x+3x2,fg=2+3x+4x2−3x3.f+g=3+x+3x^2,\qquad fg=2+3x+4x^2-3x^3.

Facts & Assumptions

Given: The integer polynomials f=1+2x+3x2f=1+2x+3x^2 and g=2−xg=2-x.

[L1]

Polynomial addition is coefficientwise, and the coefficient of xnx^n in a product is the finite convolution sum ∑i+j=naibj\sum_{i+j=n}a_i b_j (The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution).

[L2]

The integers form a commutative ring (The integers form a commutative ring).

Verification

technique · direct
1.1

Coefficientwise addition over the commutative ring [L2] gives the coefficients (1+2,2−1,3+0)=(3,1,3)(1+2,2-1,3+0)=(3,1,3).

givenL1L2algebra
2.1

Convolution gives coefficients 22, −1+4=3-1+4=3, −2+6=4-2+6=4, and −3-3 in degrees 0,1,2,30,1,2,3, respectively, proving the displayed product.

givenL1L2algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 28 results over 8 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources