| import Mathlib.Analysis.Analytic.Inverse |
| import Mathlib.Analysis.Analytic.IsolatedZeros |
| import Mathlib.Analysis.Calculus.FDeriv.Analytic |
| import Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv |
| import Mathlib.Analysis.Normed.Operator.Bilinear |
| import Mathlib.LinearAlgebra.Determinant |
| import Mathlib.MeasureTheory.Measure.Lebesgue.Basic |
| import Mathlib.MeasureTheory.Topology |
| import Mathlib.Topology.Algebra.Module.FiniteDimension |
|
|
| namespace OAI |
|
|
| section |
|
|
| /-! |
| noncomputable section |
| open MeasureTheory Set Filter |
| namespace BoltzmannNonuniqueness.Published |
|
|
| theorem analytic_line_ae_ne_zero {U : Set ℝ} (hU : MeasurableSet U) |
| (hc : IsPreconnected U) {f : ℝ → ℝ} (hf : AnalyticOnNhd ℝ f U) |
| (hne : ∃ x ∈ U, f x ≠ 0) : ∀ᵐ x ∂volume.restrict U, f x ≠ 0 := by |
| rcases hf.eqOn_zero_or_eventually_ne_zero_of_preconnected hc with hz | hn |
| · obtain ⟨x,hx,hfx⟩ := hne |
| exact False.elim (hfx (hz hx)) |
| · exact ae_restrict_le_codiscreteWithin hU hn |
|
|
| end BoltzmannNonuniqueness.Published |
|
|
| end |
|
|
| end |
|
|
| section |
|
|
| /-! # Analyticity of the existing inverse from the analytic inverse theorem -/ |
|
|
| noncomputable section |
|
|
| open Filter |
| open scoped Topology |
|
|
| namespace BoltzmannNonuniqueness.AnalyticInverse |
|
|
| variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] |
| [NormedAddCommGroup F] [NormedSpace ℝ F] |
|
|
| theorem localInverse_analytic {f : E → F} {x : E} (A : E ≃L[ℝ] F) |
| (ha : AnalyticAt ℝ f x) (hf : HasStrictFDerivAt f A.toContinuousLinearMap x) : |
| AnalyticAt ℝ (hf.localInverse f A x) (f x) := by |
| obtain ⟨p,hp⟩ := ha |
| have hcoeff : p 1 = (continuousMultilinearCurryFin1 ℝ E F).symm A.toContinuousLinearMap := by |
| apply (continuousMultilinearCurryFin1 ℝ E F).injective |
| rw [(continuousMultilinearCurryFin1 ℝ E F).apply_symm_apply] |
| exact hp.hasFDerivAt.unique hf.hasFDerivAt |
| exact ((hf.toOpenPartialHomeomorph f).hasFPowerSeriesAt_symm |
| hf.mem_toOpenPartialHomeomorph_source hp hcoeff).analyticAt |
|
|
| theorem analytic_of_eventually_left_inverse {f : E → F} {g : F → E} {x : E} |
| (A : E ≃L[ℝ] F) (ha : AnalyticAt ℝ f x) |
| (hf : HasStrictFDerivAt f A.toContinuousLinearMap x) |
| (hg : ∀ᶠ y in 𝓝 x, g (f y) = y) : AnalyticAt ℝ g (f x) := by |
| exact (localInverse_analytic A ha hf).congr (Filter.EventuallyEq.symm (hf.localInverse_unique hg)) |
|
|
| end BoltzmannNonuniqueness.AnalyticInverse |
|
|
| end |
|
|
| end |
|
|
| section |
|
|
| /-! # Analyticity of the determinant of a finite-dimensional operator -/ |
|
|
| noncomputable section |
|
|
| namespace BoltzmannNonuniqueness |
|
|
| theorem determinant_analyticAt {X E : Type*} |
| [NormedAddCommGroup X] [NormedSpace ℝ X] |
| [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] |
| {A : X → E →L[ℝ] E} {x : X} (hA : AnalyticAt ℝ A x) : |
| AnalyticAt ℝ (fun y => (A y).toLinearMap.det) x := by |
| classical |
| let b := Module.finBasis ℝ E |
| have hc (i j : Fin (Module.finrank ℝ E)) : |
| AnalyticAt ℝ (fun y => b.coord i (A y (b j))) x := |
| ((b.coord i).toContinuousLinearMap.analyticAt _).comp |
| (((ContinuousLinearMap.apply ℝ E (b j)).analyticAt _).comp hA) |
| have he : (fun y => (A y).toLinearMap.det) = |
| (fun y => (LinearMap.toMatrix b b (A y).toLinearMap).det) := by |
| funext y |
| exact (LinearMap.det_toMatrix b (A y).toLinearMap).symm |
| rw [he] |
| simp only [Matrix.det_apply'] |
| apply Finset.analyticAt_fun_sum |
| intro p _ |
| apply AnalyticAt.mul analyticAt_const |
| apply Finset.analyticAt_fun_prod |
| intro i _ |
| simpa only [LinearMap.toMatrix_apply,Module.Basis.coord_apply] using! hc (p i) i |
|
|
| end BoltzmannNonuniqueness |
|
|
| end |
|
|
| end |
|
|
| end OAI |
|
|
