File size: 5,799 Bytes
9eeb9c7 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 | import OAI.Analysis.HenonEmden.FarInteraction
namespace OAI
section FarLocalization
open MeasureTheory Set Metric
open scoped InnerProductSpace
noncomputable section
namespace HenonLaneEmden
open Cutoff AdaptiveCap
def farLocalConstant (n : ℕ) : ℝ := 2*3^(N n)+2*derivativeConstant n
lemma norm_X_bound {n : ℕ} (x : Space n) : ‖X n x‖ ≤ 2*eta n x := by
by_cases hx : ‖x‖ ≤ 2
· rw [X,norm_smul,Real.norm_eq_abs,abs_of_nonneg (eta_nonneg n x)]
nlinarith [eta_nonneg n x]
· simp only [X,eta_eq_zero (le_of_not_ge hx),zero_smul,norm_zero,mul_zero,le_refl]
lemma fluxFactor_bound {n : ℕ} (x y : Space n) :
|fluxFactor n (x,y)| ≤ 2*(eta n x+eta n y)/‖x-y‖ := by
by_cases hxy : x=y
· subst y
simp only [fluxFactor,sub_self,inner_zero_left,zero_div,abs_zero,norm_zero,div_zero,le_refl]
have hL : 0 < ‖x-y‖ := norm_pos_iff.mpr (sub_ne_zero.mpr hxy)
rw [fluxFactor,abs_div,abs_of_nonneg (sq_nonneg ‖x-y‖),div_le_div_iff₀ (sq_pos_of_pos hL) hL]
have hh := (abs_real_inner_le_norm (x-y) (X n x-X n y)).trans
(mul_le_mul_of_nonneg_left (norm_sub_le _ _) (norm_nonneg _))
have hm := add_le_add (norm_X_bound x) (norm_X_bound y)
have hh' := mul_le_mul_of_nonneg_left hm (norm_nonneg (x-y))
nlinarith [mul_le_mul_of_nonneg_right (hh.trans hh') (norm_nonneg (x-y))]
lemma localError_rough {n : ℕ} (x y : Space n) :
localError (x,y) ≤ (2*eta n x/‖x-y‖+2*eta n x+w n x)+
(2*eta n y/‖x-y‖+2*eta n y+w n y) := by
have hηx := eta_nonneg n x
have hηy := eta_nonneg n y
have hwx := w_nonneg n x
have hwy := w_nonneg n y
have hD := radialFactor_bounds (x,y)
have hh : |fluxFactor n (x,y)-eta n x+w n x*radialFactor (x,y)| ≤ |fluxFactor n (x,y)|+eta n x+w n x := by
have hs : |fluxFactor n (x,y)-eta n x| ≤ |fluxFactor n (x,y)|+eta n x := by
simpa only [sub_zero,zero_sub,abs_neg,abs_of_nonneg hηx] using abs_sub_le (fluxFactor n (x,y)) 0 (eta n x)
have hw : |w n x*radialFactor (x,y)| ≤ w n x := by
rw [abs_of_nonneg (mul_nonneg hwx hD.1)]
exact mul_le_of_le_one_right hwx hD.2
exact (abs_add_le _ _).trans (by linarith)
have he : |eta n x-eta n y| ≤ eta n x+eta n y := by
simpa only [sub_zero,zero_sub,abs_neg,abs_of_nonneg hηx,abs_of_nonneg hηy] using abs_sub_le (eta n x) 0 (eta n y)
have hf := fluxFactor_bound x y
unfold localError
rw [mul_add,add_div] at hf
linarith
lemma endpoint_far_bound {n : ℕ} {δ : ℝ} (hδ : 0 < δ) {x y : Space n} (hne : x ≠ y)
(hfar : 0 < d x → δ*d x^(m n) ≤ ‖x-y‖) :
2*eta n x/‖x-y‖+2*eta n x+w n x ≤ farLocalConstant n*
(farWeight δ (N n-1:ℕ) (x,y)+δ⁻¹*farWeight δ ((N n:ℝ)-(m n:ℝ)) (x,y)) := by
have hd := d_nonneg x
have hN := N_ge_two n
have hC : 0 ≤ derivativeConstant n := (constants_nonneg n).1
by_cases hx : 0 < d x
· have hf := hfar hx
simp only [farWeight,ite_eq_left (show 0 < d x ∧ δ*d x^(m n) ≤ ‖x-y‖ from ⟨hx,hf⟩),Real.rpow_natCast]
have hη := (eta_gap_bounds n x).2
have hw : w n x ≤ 2*derivativeConstant n*d x^(N n-1) := by
convert wProfile_bound n (norm_nonneg x) using 1
· exact w_eq_profile n x
· dsimp [derivativeConstant,d]; ring
have hpow : d x^(N n) ≤ d x^(N n-1) := pow_le_pow_of_le_one hd (d_le_one x) (by omega)
have hη' : eta n x ≤ 3^(N n)*d x^(N n-1) := hη.trans (mul_le_mul_of_nonneg_left hpow (by positivity))
have hL : 0 < ‖x-y‖ := norm_pos_iff.mpr (sub_ne_zero.mpr hne)
have hηL : 2*eta n x/‖x-y‖ ≤ 2*3^(N n)*δ⁻¹*d x^((N n:ℝ)-(m n:ℝ)) := by
have hh := div_le_div₀ (by positivity : 0 ≤ 2*(3^(N n)*d x^(N n))) (show 2*eta n x ≤ 2*(3^(N n)*d x^(N n)) by linarith)
(mul_pos hδ (pow_pos hx _)) hf
apply hh.trans_eq
rw [Real.rpow_sub hx,Real.rpow_natCast,Real.rpow_natCast]
field_simp
have hp0 : 0 ≤ δ⁻¹*d x^((N n:ℝ)-(m n:ℝ)) := mul_nonneg (inv_nonneg.mpr hδ.le) (Real.rpow_nonneg hd _)
unfold farLocalConstant
have hm := mul_nonneg hC hp0
nlinarith [hηL,hw,hη']
· have hx2 : 2 ≤ ‖x‖ := not_lt.mp (fun hh => hx ((d_pos_iff x).mpr hh))
simp only [eta_eq_zero hx2,w_outer n hx2,mul_zero,zero_div,farWeight,hx,false_and,
ite_false,add_zero,le_refl]
lemma far_pair_endpoint {n : ℕ} {δ : ℝ} (hδ : 0 ≤ δ) {x y : Space n}
(hf : ¬(0 < pairGap (x,y) ∧ ‖x-y‖ ≤ δ*pairGap (x,y)^(m n))) :
(0 < d x → δ*d x^(m n) ≤ ‖x-y‖) ∧ (0 < d y → δ*d y^(m n) ≤ ‖x-y‖) := by
have hsx : d x ≤ pairGap (x,y) := le_max_left _ _
have hsy : d y ≤ pairGap (x,y) := le_max_right _ _
constructor
· intro hx
have hs := hx.trans_le hsx
exact (mul_le_mul_of_nonneg_left (pow_le_pow_left₀ (d_nonneg x) hsx _) hδ).trans
(le_of_lt (not_le.mp (fun hL => hf ⟨hs,hL⟩)))
· intro hy
have hs := hy.trans_le hsy
exact (mul_le_mul_of_nonneg_left (pow_le_pow_left₀ (d_nonneg y) hsy _) hδ).trans
(le_of_lt (not_le.mp (fun hL => hf ⟨hs,hL⟩)))
lemma localError_far {n : ℕ} {δ : ℝ} (hδ : 0 < δ) {x y : Space n} (hne : x ≠ y)
(hf : ¬(0 < pairGap (x,y) ∧ ‖x-y‖ ≤ δ*pairGap (x,y)^(m n))) :
localError (x,y) ≤ farLocalConstant n*
(farWeight δ (N n-1:ℕ) (x,y)+δ⁻¹*farWeight δ ((N n:ℝ)-(m n:ℝ)) (x,y)+
farWeight δ (N n-1:ℕ) (y,x)+δ⁻¹*farWeight δ ((N n:ℝ)-(m n:ℝ)) (y,x)) := by
obtain ⟨hx,hy⟩ := far_pair_endpoint hδ.le hf
have hh := endpoint_far_bound hδ hne hx
have hh' := endpoint_far_bound hδ hne.symm (fun hy' => by simpa only [norm_sub_rev y x] using hy hy')
rw [norm_sub_rev y x] at hh'
exact (localError_rough x y).trans ((add_le_add hh hh').trans_eq (by ring))
end HenonLaneEmden
end
end FarLocalization
end OAI
|