openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Angular /CompactContDiffOnDerivativeBound3.lean
| import Mathlib | |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.NearGrid | |
| namespace OAI | |
| /-! Angular cutoffs and derivative estimates. -/ | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport | |
| namespace Attempt4PrefixShell | |
| 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 _ _) | |
| theorem compact_chart_derivative_bound | |
| {f : E → F} {w : E → ℂ} {U K : Set E} | |
| (hU : IsOpen U) (hK : IsCompact K) (hKU : K ⊆ U) | |
| (hf : ContDiffOn ℝ ∞ f U) (hw : ContDiffOn ℝ ∞ w U) | |
| (m : ℕ) (C : ℝ) (hC : 0 ≤ C) : | |
| ∃ D : ℝ, 0 ≤ D ∧ ∀ (b : F → ℂ) (A : ℝ), | |
| ContDiff ℝ ∞ b → 1 ≤ A → | |
| tsupport (fun x => w x * b (f x)) ⊆ K → | |
| (∀ j ≤ m, ∀ y ∈ f '' K, ‖iteratedFDeriv ℝ j b y‖ ≤ C*A^j) → | |
| ∀ j ≤ m, ∀ x, ‖iteratedFDeriv ℝ j (fun x => w x * b (f x)) x‖ ≤ D*A^j := by | |
| obtain ⟨M,hM,hMf⟩ := compact_contDiffOn_derivative_bound hU hK hKU hf m | |
| obtain ⟨W,hW,hWw⟩ := compact_contDiffOn_derivative_bound hU hK hKU hw m | |
| let B : ℝ := (m.factorial:ℝ)*C*M^m | |
| let D : ℝ := 2^m * W * B | |
| have hM0 : 0 ≤ M := le_trans zero_le_one hM | |
| have hW0 : 0 ≤ W := le_trans zero_le_one hW | |
| have hB : 0 ≤ B := by dsimp [B]; positivity | |
| refine ⟨D, by dsimp [D]; positivity, ?_⟩ | |
| intro b A hb hA hsupp hderiv j hj x | |
| have hA0 : 0 ≤ A := le_trans zero_le_one hA | |
| by_cases hx : x ∈ K | |
| · have hxU := hKU hx | |
| have hcomp (k : ℕ) (hkm : k ≤ m) : | |
| ‖iteratedFDerivWithin ℝ k (b ∘ f) U x‖ ≤ B*A^k := by | |
| have hbase := norm_iteratedFDerivWithin_comp_le (N := ∞) (t := univ) | |
| hb.contDiffOn hf (by exact_mod_cast (le_top : (k:ℕ∞) ≤ ⊤)) uniqueDiffOn_univ hU.uniqueDiffOn | |
| (fun _ _ => mem_univ _) hxU (C := C*A^k) (D := M) | |
| (fun i hik => by | |
| rw [iteratedFDerivWithin_univ] | |
| exact (hderiv i (hik.trans hkm) (f x) ⟨x,hx,rfl⟩).trans | |
| (mul_le_mul_of_nonneg_left (pow_le_pow_right₀ hA hik) hC)) | |
| (fun i hi hik => (hMf i (hik.trans hkm) x hx).trans | |
| (by simpa using pow_le_pow_right₀ hM hi)) | |
| apply hbase.trans | |
| calc | |
| (k.factorial:ℝ)*(C*A^k)*M^k ≤ (m.factorial:ℝ)*(C*A^k)*M^m := by | |
| gcongr | |
| _ = B*A^k := by dsimp [B]; ring | |
| have hbg : ContDiffOn ℝ ∞ (b ∘ f) U := hb.comp_contDiffOn hf | |
| have hprod : ContDiffOn ℝ ∞ (fun x => w x * b (f x)) U := hw.mul hbg | |
| rw [← iteratedFDerivWithin_eq_iteratedFDeriv hU.uniqueDiffOn | |
| ((hprod.contDiffAt (hU.mem_nhds hxU)).of_le (by exact_mod_cast (le_top : (j:ℕ∞) ≤ ⊤))) hxU] | |
| have hbase := norm_iteratedFDerivWithin_mul_le hw hbg hU.uniqueDiffOn hxU | |
| (by exact_mod_cast (le_top : (j:ℕ∞) ≤ ⊤)) | |
| apply hbase.trans | |
| have hsum : (∑ i ∈ Finset.range (j+1), (j.choose i:ℝ)) = 2^j := by | |
| exact_mod_cast Nat.sum_range_choose j | |
| calc | |
| (∑ i ∈ Finset.range (j+1), (j.choose i:ℝ) * ‖iteratedFDerivWithin ℝ i w U x‖ * | |
| ‖iteratedFDerivWithin ℝ (j-i) (b ∘ f) U x‖) ≤ | |
| ∑ i ∈ Finset.range (j+1), (j.choose i:ℝ) * W * (B*A^j) := by | |
| apply Finset.sum_le_sum | |
| intro i hi | |
| have hij : i ≤ j := Nat.le_of_lt_succ (Finset.mem_range.mp hi) | |
| have hk : j-i ≤ m := (Nat.sub_le _ _).trans hj | |
| apply mul_le_mul | |
| (mul_le_mul_of_nonneg_left (hWw i (hij.trans hj) x hx) (by positivity)) | |
| ((hcomp (j-i) hk).trans | |
| (mul_le_mul_of_nonneg_left (pow_le_pow_right₀ hA (Nat.sub_le _ _)) hB)) | |
| (norm_nonneg _) (by positivity) | |
| _ = 2^j * W * (B*A^j) := by rw [← Finset.sum_mul, ← Finset.sum_mul, hsum] | |
| _ ≤ 2^m * W * (B*A^j) := | |
| mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right | |
| (pow_le_pow_right₀ (show (1:ℝ) ≤ 2 by norm_num) hj) hW0) | |
| (mul_nonneg hB (pow_nonneg hA0 _)) | |
| _ = D*A^j := by dsimp [D]; ring | |
| · have hx' : x ∉ tsupport (fun x => w x * b (f x)) := fun h => hx (hsupp h) | |
| have hz : iteratedFDeriv ℝ j (fun x => w x * b (f x)) x = 0 := | |
| Function.notMem_support.mp (fun h => hx' (support_iteratedFDeriv_subset j h)) | |
| rw [hz, norm_zero] | |
| dsimp [D] | |
| positivity | |
| end Attempt4PrefixShell | |
| end GraphLiteralSupport | |
| end | |
| end | |
| end OAI | |