| import OAI.Analysis.HenonEmden.RadialTaylor |
|
|
| namespace OAI |
|
|
| universe uE uE2 |
|
|
| section NearLocalization |
|
|
| open Set Metric |
| open scoped InnerProductSpace |
| noncomputable section |
| namespace HenonLaneEmden |
| open Cutoff AdaptiveCap |
|
|
| @[fun_prop] lemma measurable_radialFactor (n : ℕ) : Measurable (radialFactor : Space n × Space n → ℝ) := by |
| unfold radialFactor; fun_prop |
| @[fun_prop] lemma measurable_fluxFactor (n : ℕ) : Measurable (fluxFactor n : Space n × Space n → ℝ) := by |
| unfold fluxFactor X |
| have hη := (continuous_eta n (E := Space n)).measurable |
| fun_prop |
|
|
| lemma eta_nonneg {E : Type uE} [NormedAddCommGroup E] (n : ℕ) (x : E) : 0 ≤ eta n x := |
| pow_nonneg (baseProfile_nonneg ‖x‖) _ |
|
|
| lemma eta_gap_bounds {E : Type uE2} [NormedAddCommGroup E] (n : ℕ) (x : E) : |
| d x^(N n) ≤ eta n x ∧ eta n x ≤ 3^(N n)*d x^(N n) := etaProfile_bounds n ‖x‖ |
|
|
| def pairGap {n : ℕ} (z : Space n × Space n) : ℝ := max (d z.1) (d z.2) |
| def localError {n : ℕ} (z : Space n × Space n) : ℝ := |
| |fluxFactor n z-eta n z.1+w n z.1*radialFactor z|+|eta n z.1-eta n z.2| |
| def closeConstant (n : ℕ) : ℝ := (2*secondConstant n+4*derivativeConstant n)*2^(N n) |
|
|
| lemma pairGap_bounds {n : ℕ} (z : Space n × Space n) : 0 ≤ pairGap z ∧ pairGap z ≤ 1 := |
| ⟨(d_nonneg z.1).trans (le_max_left _ _),max_le (d_le_one z.1) (d_le_one z.2)⟩ |
| lemma pairGap_power_le {n : ℕ} (z : Space n × Space n) : |
| pairGap z^(N n) ≤ eta n z.1+eta n z.2 := by |
| rcases le_total (d z.1) (d z.2) with h | h |
| · rw [pairGap,max_eq_right h] |
| exact (eta_gap_bounds n z.2).1.trans (le_add_of_nonneg_left (eta_nonneg n z.1)) |
| · rw [pairGap,max_eq_left h] |
| exact (eta_gap_bounds n z.1).1.trans (le_add_of_nonneg_right (eta_nonneg n z.2)) |
|
|
| lemma close_pair_gaps {n : ℕ} {δ : ℝ} (hδ : 0 ≤ δ) (hδ1 : δ ≤ 1/4) |
| {x y : Space n} (hs : 0 < pairGap (x,y)) (hxy : ‖x-y‖ ≤ δ*pairGap (x,y)^(m n)) : |
| 0 < d x ∧ 0 < d y ∧ d x+dist x y ≤ 2*pairGap (x,y) ∧ ‖x-y‖ ≤ δ*pairGap (x,y) := by |
| have hb := pairGap_bounds (x,y) |
| have hp : pairGap (x,y)^(m n) ≤ pairGap (x,y) := by |
| simpa only [pow_one] using pow_le_pow_of_le_one hb.1 hb.2 (by simp [m] : 1 ≤ m n) |
| have hL := hxy.trans (mul_le_mul_of_nonneg_left hp hδ) |
| have hd := abs_le.mp (d_oscillation x y) |
| rw [dist_eq_norm] at hd |
| have hmax : pairGap (x,y) = d x ∨ pairGap (x,y) = d y := max_choice _ _ |
| have hx := le_max_left (d x) (d y) |
| have hy := le_max_right (d x) (d y) |
| change d x ≤ pairGap (x,y) at hx |
| change d y ≤ pairGap (x,y) at hy |
| have hm : δ*pairGap (x,y) ≤ pairGap (x,y)/4 := by nlinarith |
| refine ⟨?_,?_,?_,hL⟩ |
| · rcases hmax with h | h <;> nlinarith |
| · rcases hmax with h | h <;> nlinarith |
| · rw [dist_eq_norm]; linarith |
|
|
| lemma close_power_bound {n j : ℕ} {s δ L : ℝ} (hs : 0 ≤ s) (hs1 : s ≤ 1) |
| (hδ : 0 ≤ δ) (hL : L ≤ δ*s^(m n)) (hj : j ≤ N n) (hjm : N n ≤ j+m n) : |
| (2*s)^j*L ≤ 2^(N n)*δ*s^(N n) := by |
| rw [mul_pow] |
| calc |
| _ ≤ (2^j*s^j)*(δ*s^(m n)) := mul_le_mul_of_nonneg_left hL (by positivity) |
| _ = 2^j*δ*s^(j+m n) := by rw [pow_add]; ring |
| _ ≤ 2^(N n)*δ*s^(N n) := by |
| apply mul_le_mul |
| · exact mul_le_mul_of_nonneg_right (pow_le_pow_right₀ (by norm_num : (1:ℝ) ≤ 2) hj) hδ |
| · exact pow_le_pow_of_le_one hs hs1 hjm |
| · positivity |
| · positivity |
|
|
| lemma flux_close_estimate {n : ℕ} (hn : 1 ≤ n) {x y : Space n} (hx : (1:ℝ)/2 ≤ ‖x‖) |
| (hy : ‖y‖ ≤ 2) (hxy : x ≠ y) {D : ℝ} (hD : 0 ≤ D) |
| (hseg : ∀ t ∈ uIcc ‖x‖ ‖y‖, gapProfile t ≤ D) : |
| |fluxFactor n (x,y)-eta n x+w n x*radialFactor (x,y)| ≤ |
| (2*secondConstant n*D^(N n-2)+3*derivativeConstant n*D^(N n-1))*‖x-y‖ := by |
| obtain ⟨hDc,hSc,hWc⟩ := constants_nonneg n |
| have hr : 0 < ‖x‖ := by linarith |
| have hL : 0 < ‖x-y‖ := norm_pos_iff.mpr (sub_ne_zero.mpr hxy) |
| have hD1 : |derivativeProfile n ‖x‖| ≤ derivativeConstant n*D^(N n-1) := |
| (derivativeProfile_bound n ‖x‖).trans (mul_le_mul_of_nonneg_left |
| (pow_le_pow_left₀ (gapProfile_nonneg _) (hseg _ left_mem_uIcc) _) (constants_nonneg n).1) |
| have hJ : |inner ℝ (x-y) y| ≤ 2*‖x-y‖ := |
| (abs_real_inner_le_norm _ _).trans (by nlinarith [norm_nonneg (x-y)]) |
| have hI : |inner ℝ x (x-y)| ≤ ‖x‖*‖x-y‖ := abs_real_inner_le_norm _ _ |
| have he := eta_radial_taylor hn hx y hD hseg |
| rw [flux_linearization_identity (norm_pos_iff.mp hr) hxy] |
| apply (abs_sub_le _ 0 _).trans |
| simp only [sub_zero,zero_sub,abs_neg,abs_div,abs_mul,abs_of_nonneg (norm_nonneg x),abs_of_nonneg (sq_nonneg ‖x-y‖)] |
| have hfirst : |eta n x-eta n y-derivativeProfile n ‖x‖*inner ℝ x (x-y)/‖x‖| * |
| |inner ℝ (x-y) y|/‖x-y‖^2 ≤ |
| 2*(secondConstant n*D^(N n-2)+derivativeConstant n*D^(N n-1))*‖x-y‖ := by |
| apply (div_le_iff₀ (sq_pos_of_pos hL)).mpr |
| have hh := mul_le_mul he hJ (abs_nonneg _) (by positivity) |
| nlinarith [hh] |
| have hsecond : |derivativeProfile n ‖x‖| *|inner ℝ x (x-y)|/‖x‖ ≤ derivativeConstant n*D^(N n-1)*‖x-y‖ := by |
| apply (div_le_iff₀ hr).mpr |
| have hh := mul_le_mul hD1 hI (abs_nonneg _) (by positivity) |
| nlinarith [hh] |
| linarith |
|
|
| lemma localError_close {n : ℕ} (hn : 1 ≤ n) {δ : ℝ} (hδ : 0 ≤ δ) (hδ1 : δ ≤ 1/4) |
| {x y : Space n} (hs : 0 < pairGap (x,y)) (hxy : ‖x-y‖ ≤ δ*pairGap (x,y)^(m n)) (hne : x ≠ y) : |
| localError (x,y) ≤ closeConstant n*δ*(eta n x+eta n y) := by |
| obtain ⟨hxpos,hypos,hseg,hL⟩ := close_pair_gaps hδ hδ1 hs hxy |
| have hs0 := pairGap_bounds (x,y) |
| obtain ⟨hDc,hSc,hWc⟩ := constants_nonneg n |
| have hC : 0 ≤ closeConstant n := by unfold closeConstant; positivity |
| by_cases hx : ‖x‖ ≤ 1/2 |
| · have hy : ‖y‖ ≤ 1 := by |
| have hh := norm_le_norm_add_norm_sub x y |
| have hm : δ*pairGap (x,y) ≤ 1/4 := mul_le_of_le_one_right hδ (hs0.2) |>.trans hδ1 |
| linarith |
| have hx1 : ‖x‖ ≤ 1 := by linarith |
| have hflux : fluxFactor n (x,y)=1 := by |
| simp only [fluxFactor,X,eta_eq_one hx1,eta_eq_one hy,one_smul,real_inner_self_eq_norm_sq] |
| exact div_self (ne_of_gt (sq_pos_of_pos (norm_pos_iff.mpr (sub_ne_zero.mpr hne)))) |
| simp only [localError,hflux,eta_eq_one hx1,eta_eq_one hy,w_inner n hx1,zero_mul, |
| sub_self,add_zero,abs_zero] |
| positivity |
| · have hD : 0 ≤ 2*pairGap (x,y) := by positivity |
| have hf := flux_close_estimate hn (le_of_not_ge hx) ((d_pos_iff y).mp hypos).le hne hD |
| (fun _ ht => (norm_segment_gap hseg ht).2) |
| have he := eta_oscillation n hD hseg |
| rw [abs_sub_comm,dist_eq_norm] at he |
| have hN := N_ge_two n |
| have hP := close_power_bound hs0.1 hs0.2 hδ hxy (by omega : N n-2 ≤ N n) (by simp [m]; omega : N n ≤ N n-2+m n) |
| have hP' := close_power_bound hs0.1 hs0.2 hδ hxy (by omega : N n-1 ≤ N n) (by simp [m]; omega : N n ≤ N n-1+m n) |
| unfold localError |
| calc |
| _ ≤ (2*secondConstant n)*((2*pairGap (x,y))^(N n-2)*‖x-y‖)+ |
| (4*derivativeConstant n)*((2*pairGap (x,y))^(N n-1)*‖x-y‖) := by linarith |
| _ ≤ (2*secondConstant n+4*derivativeConstant n)*(2^(N n)*δ*pairGap (x,y)^(N n)) := by |
| have hh := mul_le_mul_of_nonneg_left hP (by positivity : 0 ≤ 2*secondConstant n) |
| have hh' := mul_le_mul_of_nonneg_left hP' (by positivity : 0 ≤ 4*derivativeConstant n) |
| nlinarith |
| _ = closeConstant n*δ*pairGap (x,y)^(N n) := by unfold closeConstant; ring |
| _ ≤ _ := mul_le_mul_of_nonneg_left (pairGap_power_le (x,y)) (mul_nonneg hC hδ) |
|
|
| end HenonLaneEmden |
| end |
| end NearLocalization |
|
|
| end OAI |
|
|
