| import Mathlib | |
| import OAI.Geometry.SmoothYau.Estimates.RealPhaseLinear | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter | |
| open scoped Topology ContDiff | |
| open Set Filter | |
| open scoped Topology ContDiff | |
| open MvPolynomial | |
| open Set Filter | |
| open scoped ContDiff | |
| open Set Filter | |
| open scoped Topology ContDiff | |
| open Set Filter MvPolynomial | |
| open scoped Topology ContDiff | |
| open Set Filter Function MvPolynomial | |
| open scoped Topology ContDiff | |
| open Set Filter Function MvPolynomial | |
| open scoped Topology ContDiff | |
| open Set Filter | |
| open scoped Topology ContDiff | |
| open Set Filter | |
| open scoped Topology ContDiff | |
| open Set Filter Function | |
| open scoped Topology ContDiff | |
| open Set Filter Function | |
| open scoped Topology ContDiff | |
| open scoped Topology | |
| open Set Filter Manifold Bundle MeasureTheory | |
| open scoped Topology ContDiff ENNReal | |
| open Matrix | |
| open scoped Topology Matrix.Norms.Elementwise | |
| open Set Filter Manifold Bundle | |
| open scoped Topology ContDiff | |
| open Set Filter | |
| open scoped ContDiff Topology | |
| namespace YauCounterexamples | |
| lemma continuous_gradient_of_iterated {X : Type*} [TopologicalSpace X] | |
| (φ : X → PhaseSpace → ℝ) | |
| (hφ1 : Continuous (fun q : X × PhaseSpace => iteratedFDeriv ℝ 1 (φ q.1) q.2)) : | |
| Continuous (fun p => gradient (φ p) 0) := by | |
| have he (p : X) : (continuousMultilinearCurryFin1 ℝ PhaseSpace ℝ) | |
| (iteratedFDeriv ℝ 1 (φ p) 0) = fderiv ℝ (φ p) 0 := by | |
| ext v | |
| exact iteratedFDeriv_one_apply (fun _ => v) | |
| unfold gradient | |
| exact (InnerProductSpace.toDual ℝ PhaseSpace).symm.continuous.comp (by | |
| simpa only [Function.comp_def,id_eq,he] using | |
| (continuousMultilinearCurryFin1 ℝ PhaseSpace ℝ).continuous.comp | |
| (hφ1.comp (continuous_id.prodMk (continuous_const : Continuous (fun _ : X => (0 : PhaseSpace)))))) | |
| end YauCounterexamples | |
| end | |
| end OAI | |