File size: 4,213 Bytes
59d1c80 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 | import OAI.Geometry.Kahler.BaseCorrectionDifferential
namespace OAI
universe uKahler14283_1
open Complex
open scoped ContDiff Matrix Matrix.Norms.Elementwise
open scoped ContDiff Matrix Matrix.Norms.Elementwise ComplexOrder
open scoped ContDiff ComplexOrder
open scoped ContDiff ENNReal
open Set Filter Topology MeasureTheory
open scoped ContDiff ENNReal Pointwise
open Set Filter Topology
open scoped ContDiff
noncomputable section
open Set Filter Topology
open scoped ContDiff
namespace PinchedHartogs.BaseConstruction
lemma disjoint_sum_abs_bound {α : Type uKahler14283_1} (s : Finset α) (active : α → Prop)
(hu : ∀ p ∈ s, ∀ q ∈ s, active p → active q → p=q)
(A : α → ℝ) {B : ℝ} (hB : 0 ≤ B)
(hzero : ∀ p ∈ s, ¬ active p → A p=0)
(hbound : ∀ p ∈ s, active p → |A p| ≤ B) :
|∑ p ∈ s, A p| ≤ B := by
classical
by_cases ha : ∃ p ∈ s, active p
· obtain ⟨p,hp,hap⟩ := ha
rw [Finset.sum_eq_single p]
· exact hbound p hp hap
· intro q hq hqp
exact hzero q hq (fun haq => hqp (hu q hq p hp haq hap))
· exact fun h => (h hp).elim
· have hz : ∑ p ∈ s, A p=0 := Finset.sum_eq_zero (fun p hp => hzero p hp (fun h => ha ⟨p,hp,h⟩))
rw [hz,abs_zero]
exact hB
lemma densityCorrectionRaw_value_bound {k : ℕ} (hk : 0 < k)
(f b : ℝ → ℝ) (W : Base → ℝ) (p : Sphere) (ξ : Base) (hξ : bracket ξ (p:Base) ≠ 0)
{F B r : ℝ} (hF : 0 ≤ F) (hB : 0 ≤ B) (hr : 0 ≤ r)
(hS : 0 ≤ W (centralPoint p ξ))
(hT : |phaseDerivative W (centralPoint p ξ)| ≤ (k:ℝ)*r*W (centralPoint p ξ))
(hf : |f (densityHeight k p ξ)| ≤ F) (hb : |b (densityHeight k p ξ)| ≤ B) :
|densityCorrectionRaw k f b W p ξ| ≤
Real.exp (2*densityHeight k p ξ/k)*W (centralPoint p ξ)*(F+B*r) := by
have hk0 : (0:ℝ) < k := by exact_mod_cast hk
let z := (bracket ξ (p:Base)/(‖bracket ξ (p:Base)‖:ℂ))^k
have hn : ‖z‖=1 := by
dsimp [z]
rw [norm_pow,norm_div,Complex.norm_real,Real.norm_eq_abs,abs_norm,div_self (norm_ne_zero_iff.mpr hξ),one_pow]
have hzre : |z.re| ≤ 1 := (Complex.abs_re_le_norm z).trans_eq hn
have hzim : |z.im| ≤ 1 := (Complex.abs_im_le_norm z).trans_eq hn
rw [densityCorrectionRaw_model hk f b W p hξ]
unfold correctionModel
rw [abs_mul,abs_of_pos (Real.exp_pos _)]
calc
_ ≤ Real.exp (2*densityHeight k p ξ/k)*
(|f (densityHeight k p ξ)| * |W (centralPoint p ξ)| * |z.re| +
|b (densityHeight k p ξ)/k| * |phaseDerivative W (centralPoint p ξ)| * |z.im|) := by
apply mul_le_mul_of_nonneg_left _ (Real.exp_pos _).le
simpa only [abs_mul] using abs_sub_triangle
(f (densityHeight k p ξ)*W (centralPoint p ξ)*z.re)
(b (densityHeight k p ξ)/k*phaseDerivative W (centralPoint p ξ)*z.im)
_ ≤ Real.exp (2*densityHeight k p ξ/k)*(F*W (centralPoint p ξ)*1+
(B/k)*((k:ℝ)*r*W (centralPoint p ξ))*1) := by
rw [abs_of_nonneg hS,abs_div,abs_of_pos hk0]
gcongr
_ = _ := by field_simp
lemma horizontal_direction_ratio {k : ℕ} {R D : ℝ} (hk : 0 < k) (hR : 0 < R)
(hkR : 2*R ≤ k) (hD : 2*Real.sqrt (2*R) ≤ D)
(p ξ : Sphere) (hξ : ξ ∈ peakPatch k R p) (v : Base) (hv : Horizontal ξ v) :
‖bracket v (p:Base)/bracket (ξ:Base) (p:Base)‖ ≤ D*Real.sqrt (k:ℝ)*‖v‖/k := by
have hk0 : (0:ℝ) < k := by exact_mod_cast hk
have hb := horizontal_patch_log_derivative_bound hk hR hkR p ξ hξ v hv
rw [Real.sqrt_div (by positivity)] at hb
apply hb.trans
have hroot := Real.sq_sqrt hk0.le
have hroot0 : 0 < Real.sqrt (k:ℝ) := Real.sqrt_pos.mpr hk0
apply (le_div_iff₀ hk0).mpr
have he : 2*(Real.sqrt (2*R)/Real.sqrt (k:ℝ))*‖v‖*(k:ℝ)=
(2*Real.sqrt (2*R))*Real.sqrt (k:ℝ)*‖v‖ := by
rw [← hroot]
rw [Real.sqrt_sq (Real.sqrt_nonneg _)]
field_simp
rw [he]
gcongr
lemma phase_direction_ratio (p ξ : Sphere) (hξ : bracket (ξ:Base) (p:Base) ≠ 0) :
‖bracket (Complex.I • (ξ:Base)) (p:Base)/bracket (ξ:Base) (p:Base)‖=1 := by
rw [bracket_smul,mul_div_cancel_right₀ Complex.I hξ,Complex.norm_I]
end PinchedHartogs.BaseConstruction
end
end OAI
|