openAI-math-dataset / lean /OAI /Analysis /BiholderTransport /Regularity /ContDiffUncurryIterated.lean
| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| open Set MeasureTheory Manifold Bundle | |
| open scoped ContDiff Manifold ENNReal NNReal Topology | |
| open Set Filter | |
| open scoped Topology NNReal | |
| open Set Filter | |
| open scoped Topology | |
| open Set Manifold MeasureTheory Bundle | |
| open scoped ENNReal ContDiff Topology | |
| open Set | |
| open scoped Topology | |
| open Set Filter Manifold Bundle ContinuousLinearMap | |
| open scoped Topology ContDiff Manifold Bundle | |
| open Set Filter ContinuousLinearMap InnerProductSpace | |
| open scoped Topology ContDiff | |
| open Set Filter ContinuousLinearMap | |
| open scoped Topology ContDiff | |
| open Set Filter ContinuousLinearMap | |
| open scoped Topology ContDiff | |
| open Set Filter ContinuousLinearMap | |
| open scoped Topology ContDiff | |
| open scoped NNReal | |
| open Set Filter ContinuousLinearMap | |
| open scoped Topology ContDiff | |
| open Set Filter ContinuousLinearMap | |
| open scoped Topology | |
| open MeasureTheory | |
| open scoped ContDiff ENNReal | |
| open Set Filter Manifold Bundle ContinuousLinearMap MeasureTheory | |
| open scoped Topology ContDiff Manifold Bundle ENNReal | |
| open Set Filter Manifold MeasureTheory Bundle | |
| open scoped ENNReal ContDiff Topology Manifold | |
| open Set Filter Manifold Bundle ContinuousLinearMap | |
| open scoped Topology ContDiff Manifold Bundle | |
| open Set Filter Manifold Bundle | |
| open scoped Topology ContDiff Manifold Bundle | |
| open Set Filter Manifold Bundle | |
| open scoped Topology ContDiff Manifold Bundle | |
| open Set Filter Bundle | |
| open scoped Topology Bundle | |
| open scoped Topology | |
| open Function Manifold Set | |
| open Manifold Bundle | |
| open scoped Manifold Bundle | |
| open Set | |
| open Set Filter | |
| open scoped Topology ContDiff | |
| open Set Filter Manifold MeasureTheory Bundle | |
| open scoped ENNReal ContDiff Topology | |
| open Set Filter Manifold MeasureTheory Bundle | |
| open scoped ENNReal ContDiff Topology | |
| open Set Filter Manifold MeasureTheory Bundle | |
| open scoped ENNReal ContDiff Topology | |
| open Set Filter Manifold MeasureTheory Bundle | |
| open scoped ENNReal ContDiff Topology | |
| open Set Filter Manifold MeasureTheory Bundle | |
| open scoped ENNReal ContDiff Topology | |
| open Set Filter Manifold MeasureTheory Bundle | |
| open scoped ENNReal ContDiff Topology | |
| open Set Filter | |
| open scoped ContDiff Topology | |
| namespace WeakMTWTransport | |
| lemma contDiffAt_uncurry_iteratedDeriv {E F : Type*} | |
| [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| [NormedAddCommGroup F] [NormedSpace ℝ F] | |
| {f : E → ℝ → F} {x : E} {t : ℝ} | |
| (hf : ContDiffAt ℝ ∞ (Function.uncurry f) (x,t)) (k : ℕ) : | |
| ContDiffAt ℝ ∞ (fun q : E×ℝ => iteratedDeriv k (f q.1) q.2) (x,t) := by | |
| induction k with | |
| | zero => | |
| apply hf.congr_of_eventuallyEq | |
| exact Filter.Eventually.of_forall (fun q => congrFun (iteratedDeriv_zero (f := f q.1)) q.2) | |
| | succ k ih => | |
| have hcomp : ContDiffAt ℝ ∞ | |
| (Function.uncurry (fun q : E×ℝ => fun s : ℝ => iteratedDeriv k (f q.1) s)) ((x,t),t) := | |
| ih.comp ((x,t),t) ((contDiffAt_fst.fst).prodMk contDiffAt_snd) | |
| have hd : ContDiffAt ℝ ∞ | |
| (fun q : E×ℝ => fderiv ℝ (iteratedDeriv k (f q.1)) q.2) (x,t) := | |
| hcomp.fderiv contDiffAt_snd (by simp) | |
| simpa only [iteratedDeriv_succ,deriv] using hd.clm_apply (contDiffAt_const (c := (1:ℝ))) | |
| lemma contDiffAt_iteratedDeriv_parameter {E F : Type*} | |
| [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| [NormedAddCommGroup F] [NormedSpace ℝ F] | |
| {f : E → ℝ → F} {x : E} {t : ℝ} | |
| (hf : ContDiffAt ℝ ∞ (Function.uncurry f) (x,t)) (k : ℕ) : | |
| ContDiffAt ℝ ∞ (fun q : E => iteratedDeriv k (f q) t) x := | |
| (contDiffAt_uncurry_iteratedDeriv hf k).comp x (contDiffAt_id.prodMk contDiffAt_const) | |
| end WeakMTWTransport | |
| end | |
| end OAI | |