feat: add Sturm's theorem eval problem#273
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This PR adds Sturm's theorem (§97 of Knill's "Some Fundamental Theorems in Mathematics") as a new eval problem: the number of distinct real roots of a squarefree real polynomial in an open interval equals the drop in sign variations of its Sturm chain. The Sturm chain, the sign-variation counter, and the variation function are defined in the problem; mathlib has none of them. Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
This was referenced May 19, 2026
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This PR adds Sturm's theorem as a new lean-eval challenge problem — §97 of Oliver Knill's Some Fundamental Theorems in Mathematics.
For a squarefree real polynomial
pand an interval(a, b)whose endpoints are not roots ofp, the number of distinct roots ofpin(a, b)equalsσ(a) − σ(b), the drop in the number of sign variations of the Sturm chain.The problem defines the Sturm chain (
p₀ = p,p₁ = p',pₖ₊₁ = −(pₖ₋₁ mod pₖ)), the sign-change counter, and the variation functionσ— mathlib has none of them. The chain uses the negated-remainder convention, for which the count isσ(a) − σ(b).Cross-checked against the Isabelle/HOL formalization (Manuel Eberl, AFP entry
Sturm_Sequences), which proves the same classical distinct-root form; our statement takesSquarefree pas an explicit hypothesis (the core case).🤖 Prepared with Claude Code