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.

A divided difference with negative Laurent exponents

Example

In rank two, with $A = Z [Q, Q^{- 1}] [X_{1}^{\pm 1}, X_{2}^{\pm 1}]$ , the divided difference of $f = X_{1}^{- 2} X_{2}$ is

D_{1} (f) = \frac{X_{2}}{X_{1}^{2}} + \frac{1}{X_{1}} + \frac{1}{X_{2}} .

Facts & Assumptions

Given: The Laurent algebra and divided difference from The type-A weight lattice, Laurent ring, and divided differences with $n = 2$ .

[F1]

The type-A weight lattice, Laurent ring, and divided differences: $D_{1} (f) = (f - s_{1} (f)) / (1 - X_{1} / X_{2})$ , and the quotient is the unique Laurent polynomial with this property.

Verification

Given: $f = X_{1}^{- 2} X_{2}$ and $s_{1}$ interchanges $X_{1}$ and $X_{2}$ . Verify the displayed value of $D_{1} (f)$ .

1.1givenF1algebra

Applying the transposition gives $s_{1} (f) = X_{2}^{- 2} X_{1}$ , so $f - s_{1} (f) = X_{2} / X_{1}^{2} - X_{1} / X_{2}^{2}$ .

2.1givenF1step 1.1algebra

Multiplication shows $(1 - X_{1} / X_{2}) (X_{2} / X_{1}^{2} + 1 / X_{1} + 1 / X_{2}) = X_{2} / X_{1}^{2} - X_{1} / X_{2}^{2} = f - s_{1} (f)$ . Each cancellation is in the Laurent ring, where $X_{1}$ and $X_{2}$ are units.

3.1F1step 2.1∎

The quotient in $A$ is unique by [F1], so $D_{1} (f) = X_{2} / X_{1}^{2} + 1 / X_{1} + 1 / X_{2}$ , as asserted.

Depends on

The type-A weight lattice, Laurent ring, and divided differences

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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