| import OAI.Analysis.HenonEmden.FarInteraction |
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| namespace OAI |
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| section FarLocalization |
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| open MeasureTheory Set Metric |
| open scoped InnerProductSpace |
| noncomputable section |
| namespace HenonLaneEmden |
| open Cutoff AdaptiveCap |
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| def farLocalConstant (n : ℕ) : ℝ := 2*3^(N n)+2*derivativeConstant n |
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| 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] |
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| 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))] |
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| 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 |
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| 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] |
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| 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⟩))) |
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| 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)) |
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| end HenonLaneEmden |
| end |
| end FarLocalization |
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| end OAI |
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