openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Shells /ShellEffectiveScale.lean
| import Mathlib | |
| import OAI.MeasureTheory.FalconerCompletion.Recursion.MaskScale | |
| namespace OAI | |
| /-! Dyadic shells and prefix decompositions. -/ | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport | |
| namespace Attempt4FullFar | |
| open Filter Real | |
| open scoped Topology | |
| noncomputable def shellEffectiveScaleND (l : ℕ) (R σ : ℝ) : ℝ := | |
| R^σ*sourceMaskScaleND l R σ | |
| lemma dyadic_half_sqND (q : ℝ) : ((2:ℝ)^(q/2))^2=(2:ℝ)^q := by | |
| rw [← Real.rpow_natCast ((2:ℝ)^(q/2)) 2,← Real.rpow_mul (by norm_num : (0:ℝ)≤2)] | |
| congr 1 | |
| norm_num | |
| lemma shell_effective_scale_upperND {R σ q : ℝ} (l : ℕ) (hR : 0<R) | |
| (hl : (2:ℝ)^l ≤ (2:ℝ)^(q/2)*R^σ) : | |
| shellEffectiveScaleND l R σ ≤ max (R^σ) ((2:ℝ)^(q/2)*R^(-48*σ)) := by | |
| unfold shellEffectiveScaleND sourceMaskScaleND | |
| rw [mul_max_of_nonneg _ _ (Real.rpow_nonneg hR.le _),mul_one] | |
| apply max_le_max_left | |
| calc | |
| _ ≤ R^σ*(((2:ℝ)^(q/2)*R^σ)*R^(-50*σ)) := by gcongr | |
| _ = (2:ℝ)^(q/2)*(R^σ*R^σ*R^(-50*σ)) := by ring | |
| _ = _ := by rw [← Real.rpow_add hR,← Real.rpow_add hR]; congr 2; ring | |
| lemma shell_effective_scale_sqND {R σ q : ℝ} (l : ℕ) (hR : 0<R) | |
| (hl : (2:ℝ)^l ≤ (2:ℝ)^(q/2)*R^σ) : | |
| (shellEffectiveScaleND l R σ)^2 ≤ max (R^(2*σ)) ((2:ℝ)^q*R^(-96*σ)) := by | |
| have hu := shell_effective_scale_upperND l hR hl | |
| have hn : 0 ≤ shellEffectiveScaleND l R σ := | |
| mul_nonneg (Real.rpow_nonneg hR.le _) (sourceMaskScaleND_pos l R σ).le | |
| have he₁ : (R^σ)^2=R^(2*σ) := by | |
| rw [← Real.rpow_natCast (R^σ) 2,← Real.rpow_mul hR.le] | |
| congr 1 | |
| norm_num | |
| ring | |
| have he₂ : ((2:ℝ)^(q/2)*R^(-48*σ))^2=(2:ℝ)^q*R^(-96*σ) := by | |
| rw [mul_pow,dyadic_half_sqND,← Real.rpow_natCast (R^(-48*σ)) 2,← Real.rpow_mul hR.le] | |
| congr 2 | |
| norm_num | |
| ring | |
| rcases le_total (R^σ) ((2:ℝ)^(q/2)*R^(-48*σ)) with h | h | |
| · rw [max_eq_right h] at hu | |
| exact (pow_le_pow_left₀ hn hu 2).trans (by rw [he₂]; exact le_max_right _ _) | |
| · rw [max_eq_left h] at hu | |
| exact (pow_le_pow_left₀ hn hu 2).trans (by rw [he₁]; exact le_max_left _ _) | |
| lemma shell_scale_regimeND {R σ q c Λ : ℝ} (l : ℕ) | |
| (hR : 1 ≤ R) (hσ : 0 ≤ σ) (hσ₁ : σ ≤ 1) (hc : 0<c) | |
| (hl : (2:ℝ)^l ≤ (2:ℝ)^(q/2)*R^σ) | |
| (hqlo : R^(20*σ) ≤ (2:ℝ)^q) (hqup : (2:ℝ)^q ≤ R^2) | |
| (hΛ : c*(2:ℝ)^q ≤ Λ) : | |
| 1 ≤ shellEffectiveScaleND l R σ ∧ shellEffectiveScaleND l R σ ≤ R^2 ∧ | |
| c*R^(20*σ) ≤ Λ ∧ (shellEffectiveScaleND l R σ)^2/Λ ≤ c⁻¹*R^(-18*σ) ∧ | |
| (2:ℝ)^(-q)*(sourceMaskScaleND l R σ)^2 ≤ 1 := by | |
| have hRp : 0<R := zero_lt_one.trans_le hR | |
| have hQp : 0<(2:ℝ)^q := Real.rpow_pos_of_pos (by norm_num) _ | |
| have hΛp : 0<Λ := (mul_pos hc hQp).trans_le hΛ | |
| have hRs : 1≤R^σ := Real.one_le_rpow hR hσ | |
| have hu := shell_effective_scale_upperND l hRp hl | |
| have hsquare := shell_effective_scale_sqND l hRp hl | |
| have hhalf : (2:ℝ)^(q/2) ≤ R := by | |
| nlinarith [dyadic_half_sqND q, Real.rpow_nonneg (by norm_num : (0:ℝ)≤2) (q/2)] | |
| have hle (u : ℝ) (hu : u ≤ 0) : R^u ≤ 1 := by | |
| simpa using Real.rpow_le_rpow_of_exponent_le hR hu | |
| have hnorm : (shellEffectiveScaleND l R σ)^2 ≤ (2:ℝ)^q*R^(-18*σ) := by | |
| apply hsquare.trans | |
| apply max_le | |
| · calc | |
| R^(2*σ) = R^(20*σ)*R^(-18*σ) := by rw [← Real.rpow_add hRp]; congr 1; ring | |
| _ ≤ _ := mul_le_mul_of_nonneg_right hqlo (Real.rpow_nonneg hRp.le _) | |
| · exact mul_le_mul_of_nonneg_left | |
| (Real.rpow_le_rpow_of_exponent_le hR (by linarith)) hQp.le | |
| refine ⟨?_,?_,?_,?_,?_⟩ | |
| · exact one_le_mul_of_one_le_of_one_le hRs (sourceMaskScaleND_one_le l R σ) | |
| · apply hu.trans | |
| apply max_le | |
| · simpa using Real.rpow_le_rpow_of_exponent_le hR (show σ ≤ (2:ℝ) by linarith) | |
| · calc | |
| _ ≤ (2:ℝ)^(q/2) := mul_le_of_le_one_right (by positivity) (hle (-48*σ) (by linarith)) | |
| _ ≤ R := hhalf | |
| _ ≤ R^2 := by nlinarith | |
| · exact (mul_le_mul_of_nonneg_left hqlo hc.le).trans hΛ | |
| · calc | |
| _ ≤ ((2:ℝ)^q*R^(-18*σ))/(c*(2:ℝ)^q) := | |
| div_le_div₀ (by positivity) hnorm (mul_pos hc hQp) hΛ | |
| _ = _ := by field_simp | |
| · have hQle : sourceMaskScaleND l R σ ≤ shellEffectiveScaleND l R σ := by | |
| exact le_mul_of_one_le_left (sourceMaskScaleND_pos l R σ).le hRs | |
| have hsq : (sourceMaskScaleND l R σ)^2 ≤ (2:ℝ)^q := | |
| (pow_le_pow_left₀ (sourceMaskScaleND_pos l R σ).le hQle 2).trans | |
| (hnorm.trans (mul_le_of_le_one_right hQp.le (hle (-18*σ) (by linarith)))) | |
| rw [Real.rpow_neg (by norm_num : (0:ℝ)≤2)] | |
| exact (inv_mul_le_one₀ hQp).mpr hsq | |
| theorem shell_scale_regime_eventuallyND {σ c : ℝ} (hσ : 0<σ) (hσ₁ : σ≤1) (hc : 0<c) : | |
| ∀ᶠ R : ℝ in atTop, ∀ (l : ℕ) (q Λ : ℝ), | |
| (2:ℝ)^l ≤ (2:ℝ)^(q/2)*R^σ → R^(20*σ) ≤ (2:ℝ)^q → (2:ℝ)^q ≤ R^2 → | |
| c*(2:ℝ)^q ≤ Λ → | |
| 1 ≤ shellEffectiveScaleND l R σ ∧ shellEffectiveScaleND l R σ ≤ R^2 ∧ | |
| R^σ ≤ Λ ∧ (shellEffectiveScaleND l R σ)^2/Λ ≤ R^(-σ) ∧ | |
| (2:ℝ)^(-q)*(sourceMaskScaleND l R σ)^2 ≤ 1 := by | |
| filter_upwards [eventually_ge_atTop (1:ℝ), | |
| (tendsto_rpow_atTop (by linarith : (0:ℝ)<17*σ)).eventually_ge_atTop c⁻¹] with R hR hlarge | |
| intro l q Λ hl hqlo hqup hΛ | |
| have hRp : 0<R := zero_lt_one.trans_le hR | |
| obtain ⟨h₁,h₂,h₃,h₄,h₅⟩ := shell_scale_regimeND l hR hσ.le hσ₁ hc hl hqlo hqup hΛ | |
| refine ⟨h₁,h₂,?_,?_,h₅⟩ | |
| · have hcR : 1 ≤ c*R^(17*σ) := by simpa only [mul_comm] using (inv_le_iff_one_le_mul₀ hc).mp hlarge | |
| have hRs : R^σ ≤ R^(3*σ) := Real.rpow_le_rpow_of_exponent_le hR (by linarith) | |
| apply le_trans _ h₃ | |
| calc | |
| _ ≤ R^(3*σ) := hRs | |
| _ ≤ (c*R^(17*σ))*R^(3*σ) := le_mul_of_one_le_left (Real.rpow_nonneg hRp.le _) hcR | |
| _ = _ := by rw [mul_assoc,← Real.rpow_add hRp]; congr 2; ring | |
| · apply h₄.trans | |
| calc | |
| _ ≤ R^(17*σ)*R^(-18*σ) := mul_le_mul_of_nonneg_right hlarge (Real.rpow_nonneg hRp.le _) | |
| _ = _ := by rw [← Real.rpow_add hRp]; congr 1; ring | |
| lemma shell_source_exponentsND {N l : ℕ} {σ q : ℝ} | |
| (hl : (l:ℝ) ≤ q/2+σ*N) (hqlo : 20*σ*N ≤ q) (hqup : q ≤ 2*N) : | |
| (2:ℝ)^l ≤ (2:ℝ)^(q/2)*((2:ℝ)^N)^σ ∧ | |
| ((2:ℝ)^N)^(20*σ) ≤ (2:ℝ)^q ∧ (2:ℝ)^q ≤ ((2:ℝ)^N)^2 := by | |
| have h₁ : ((2:ℝ)^N)^σ=(2:ℝ)^((N:ℝ)*σ) := by | |
| rw [← Real.rpow_natCast (2:ℝ) N,← Real.rpow_mul (by norm_num : (0:ℝ)≤2)] | |
| have h₂ : ((2:ℝ)^N)^(20*σ)=(2:ℝ)^((N:ℝ)*(20*σ)) := by | |
| rw [← Real.rpow_natCast (2:ℝ) N,← Real.rpow_mul (by norm_num : (0:ℝ)≤2)] | |
| have h₃ : ((2:ℝ)^N)^2=(2:ℝ)^((N:ℝ)*2) := by | |
| rw [← Real.rpow_natCast (2:ℝ) N,← Real.rpow_natCast ((2:ℝ)^(N:ℝ)) 2, | |
| ← Real.rpow_mul (by norm_num : (0:ℝ)≤2)] | |
| norm_num | |
| rw [h₁,h₂,h₃,← Real.rpow_add (by norm_num : (0:ℝ)<2),← Real.rpow_natCast (2:ℝ) l] | |
| exact ⟨Real.rpow_le_rpow_of_exponent_le (by norm_num) (by nlinarith), | |
| Real.rpow_le_rpow_of_exponent_le (by norm_num) (by nlinarith), | |
| Real.rpow_le_rpow_of_exponent_le (by norm_num) (by nlinarith)⟩ | |
| end Attempt4FullFar | |
| end GraphLiteralSupport | |
| end | |
| end | |
| end OAI | |