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The aim is to provide a machine-checked proof of the Budan-Fourier theorem using Lean 4.

June 19, 2026Open Access

Counting with the Derivative Tower: the Budan-Fourier Theorem, Machine-Checked in Lean 4

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Randomized trial demonstrates real root counting with the Budan-Fourier theorem, indicating a solid approach to polynomial analysis.

The aim is to provide a machine-checked proof of the Budan-Fourier theorem using Lean 4.
Utilized Lean 4 and Mathlib for formalization of the theorem.
Analyzed sign variations in the derivative tower to count real roots.
Developed the proof requiring three standard axioms.
Establishes that the number of real roots is bounded by the difference in sign variations.
Confirms that the surplus in sign variations indicates complex roots not visible in the interval.
Presents the first formal proof of the Budan-Fourier theorem in Lean.

CARLES MARÍN MUÑOZ (2026) studied this question.

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June 19, 2026Open Access

View Full Paper

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Authors

Randomized trial demonstrates real root counting with the Budan-Fourier theorem, indicating a solid approach to polynomial analysis.

The aim is to provide a machine-checked proof of the Budan-Fourier theorem using Lean 4.
Utilized Lean 4 and Mathlib for formalization of the theorem.
Analyzed sign variations in the derivative tower to count real roots.
Developed the proof requiring three standard axioms.
Establishes that the number of real roots is bounded by the difference in sign variations.
Confirms that the surplus in sign variations indicates complex roots not visible in the interval.
Presents the first formal proof of the Budan-Fourier theorem in Lean.

CARLES MARÍN MUÑOZ (2026) studied this question.

View Full Paper

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