ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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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.

The lexicographic reduction algorithm in three variables

Example

Let

$P = \sum_{s y m} x^{3} + 2 \sum_{s y m} x^{2} y + 3 x y z,$

where the first sum is $x^{3} + y^{3} + z^{3}$ and the second is the sum of the six distinct monomials with exponent pattern $(2, 1, 0)$ . Then

$P = e_{1}^{3} - e_{1} e_{2} .$

Facts & Assumptions

Given: The polynomial $P$ above, with lexicographic order $x > y > z$ .

[L1]

Lexicographic reduction cancels the leading partition by the corresponding monomial in $e_{1}, e_{2}, e_{3}$ and strictly decreases the leading multidegree (Every symmetric polynomial over a commutative ring is a polynomial in the elementary symmetric polynomials).

[L2]

Lexicographic order compares exponent tuples at their first differing coordinate (Lexicographic order on exponent tuples and the multidegree of a nonzero polynomial).

Verification

technique · direct

1.1

The leading multidegree of $P$ is $(3, 0, 0)$ with coefficient $1$ , so [L1] first subtracts $e_{1}^{3}$ . Its expansion is $\sum_{s y m} x^{3} + 3 \sum_{s y m} x^{2} y + 6 x y z$ .

givenL1L2algebra

2.1

The remainder is $- \sum_{s y m} x^{2} y - 3 x y z$ , whose leading multidegree is $(2, 1, 0)$ . Since $e_{1} e_{2} = \sum_{s y m} x^{2} y + 3 x y z$ , the next prescribed subtraction is $- e_{1} e_{2}$ and the remainder becomes zero.

step 1.1L1L2algebra

3.1

Therefore $P = e_{1}^{3} - e_{1} e_{2}$ . Direct expansion verifies the equality and exhibits both lexicographic decreases.

step 2.1algebra∎

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: 11 results over 7 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

K. Conrad, Symmetric Polynomials, Section 2 (standard reference, not scraped)