feat: add Darboux's theorem eval problem#288
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This PR adds Darboux's theorem (§39 of Knill's "Some Fundamental Theorems
in Mathematics") as a new eval problem: every symplectic form on an open
U ⊆ ℝ^{2n} is locally symplectomorphic to the standard symplectic form
ω₀ = ∑_i dxᵢ ∧ dx_{n+i}. Mathlib has all the supporting differential-form
infrastructure but no symplectic forms, ω₀, or Darboux theorem; no
formalization was found in any other proof assistant.
Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
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This PR adds Darboux's theorem as a new lean-eval challenge problem — §39 of Oliver Knill's Some Fundamental Theorems in Mathematics.
Every symplectic form on an open
U ⊆ ℝ^{2n}is locally symplectomorphic to the standard symplectic formω₀ = ∑_{i=1}^n dxᵢ ∧ dx_{n+i}.The local content lives entirely on open subsets of
ℝ^{2n}, so the theorem is formalized against mathlib's normed-space differential-form machinery — continuous alternating maps, the exterior derivativeextDeriv, alternating-form pullback, andOpenPartialHomeomorph. mathlib has all the supporting infrastructure but no symplectic forms, noω₀, and no Darboux theorem (Analysis/Calculus/Darboux.leanis the unrelated derivative-IVT theorem). The Challenge ships two auxiliary definitions,IsDarbouxNormalandIsSymplecticOn. A search found no formalization of Darboux's theorem in any other proof assistant.🤖 Prepared with Claude Code