openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Angular /CompactContDiffOnDerivativeBound.lean
| import Mathlib | |
| import OAI.MeasureTheory.FalconerCompletion.Sphere.SphereMeasure | |
| namespace OAI | |
| /-! Angular cutoffs and derivative estimates. -/ | |
| section | |
| section | |
| namespace Attempt4FullFar | |
| open Set Metric | |
| open scoped ContDiff Topology | |
| variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| [NormedAddCommGroup F] [NormedSpace ℝ F] | |
| lemma compact_contDiffOn_derivative_bound {f : E → F} {U K : Set E} | |
| (hU : IsOpen U) (hK : IsCompact K) (hKU : K ⊆ U) (hf : ContDiffOn ℝ ∞ f U) | |
| (m : ℕ) : ∃ C : ℝ, 1 ≤ C ∧ ∀ j ≤ m, ∀ x ∈ K, | |
| ‖iteratedFDerivWithin ℝ j f U x‖ ≤ C := by | |
| have hbound (j : ℕ) : ∃ C : ℝ, ∀ x ∈ K, | |
| ‖iteratedFDerivWithin ℝ j f U x‖ ≤ C := | |
| hK.exists_bound_of_continuousOn | |
| ((hf.continuousOn_iteratedFDerivWithin (by exact_mod_cast (le_top : (j:ℕ∞) ≤ ⊤)) hU.uniqueDiffOn).mono hKU) | |
| induction m with | |
| | zero => | |
| obtain ⟨C,hC⟩ := hbound 0 | |
| refine ⟨max 1 C, le_max_left _ _, ?_⟩ | |
| intro j hj x hx | |
| have : j=0 := Nat.eq_zero_of_le_zero hj | |
| subst j | |
| exact (hC x hx).trans (le_max_right _ _) | |
| | succ m ih => | |
| obtain ⟨D,hD,hboundD⟩ := ih | |
| obtain ⟨C,hC⟩ := hbound (m+1) | |
| refine ⟨max D C, hD.trans (le_max_left _ _), ?_⟩ | |
| intro j hj x hx | |
| by_cases hjm : j ≤ m | |
| · exact (hboundD j hjm x hx).trans (le_max_left _ _) | |
| · have : j=m+1 := by omega | |
| subst j | |
| exact (hC x hx).trans (le_max_right _ _) | |
| end Attempt4FullFar | |
| end | |
| end | |
| end OAI | |