feat: add Liouville–Arnold eval problem#304
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§45 of Knill's "Some Fundamental Theorems in Mathematics". On a 2n-
dimensional symplectic manifold with n smooth, pointwise linearly
independent, pairwise Poisson-commuting first integrals F₁, …, F_n,
every compact connected component of a joint level set is diffeomorphic
to the n-torus T^n. Formalized on ℝ^{2n} with the standard symplectic
form ω₀ = ∑ᵢ dpᵢ ∧ dqᵢ. Mathlib has the supporting analysis, the
n-torus via AddCircle, and Homeomorph, but no Poisson brackets, no
symplectic manifolds beyond Matrix.symplecticGroup, no first integrals,
and no Liouville–Arnold theorem in any form.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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This PR adds an eval problem for the Liouville–Arnold theorem on
integrable systems: on a
2n-dimensional symplectic manifold withnsmooth, pointwise linearly independent, pairwise Poisson-commuting first
integrals
F₁, …, F_n, every compact connected component of a jointlevel set is diffeomorphic to the
n-torusT^n. §45 of Knill'sSome Fundamental Theorems in Mathematics.
Formalized on
E n := EuclideanSpace ℝ (Fin (2n))with the standardsymplectic form
ω₀ = ∑ᵢ dpᵢ ∧ dqᵢ. Mathlib hasEuclideanSpace,fderiv,AddCircle,Homeomorph, and the smooth-manifold framework,but no Poisson brackets, no symplectic manifolds beyond
Matrix.symplecticGroup, no first integrals, and no Liouville–Arnoldtheorem in any form (the
Liouvillefiles inMathlib/Analysis/Complex/are Liouville's theorem on bounded entirefunctions, a different result). The Challenge ships ~1 page of helper
definitions (
E,idxP,idxQ,poissonBracket,IsLiouvilleIntegrable,levelSet).No formalization found in any major prover. Not on Freek Wiedijk's
100-theorems list.
🤖 Prepared with Claude Code