| import OAI.Analysis.HenonEmden.CapLoss |
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| namespace OAI |
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| universe uE |
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| section LayerIntegration |
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| open MeasureTheory Set Metric |
| open scoped ENNReal |
| noncomputable section |
| namespace HenonLaneEmden.SpatialComponents |
| open Cutoff AdaptiveCap LineComponents Angular |
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| lemma low_high_layer_bound {F : ℝ → ℝ} {f w T a : ℝ} |
| (hf : 0 ≤ f) (hw : 0 ≤ w) (hT : 0 ≤ T) (ha : 0 ≤ a) |
| (hglobal : ∀ t ∈ Ioo 0 f, F t ≤ w) |
| (hhigh : ∀ t ∈ Ioo 0 f, T ≤ t → F t ≤ a) : |
| (∫⁻ t in Ioo 0 f, ENNReal.ofReal (F t)) ≤ ENNReal.ofReal (a*f+w*T) := by |
| classical |
| have hpt (t : ℝ) (ht : t ∈ Ioo 0 f) : |
| ENNReal.ofReal (F t) ≤ ENNReal.ofReal a+(Ioo 0 T).indicator (fun _ => ENNReal.ofReal w) t := by |
| by_cases hlt : t < T |
| · rw [indicator_of_mem (show t ∈ Ioo 0 T from ⟨ht.1,hlt⟩)] |
| exact (ENNReal.ofReal_le_ofReal (hglobal t ht)).trans (le_add_left le_rfl) |
| · rw [indicator_of_notMem (fun hh => hlt hh.2),add_zero] |
| exact ENNReal.ofReal_le_ofReal (hhigh t ht (not_lt.mp hlt)) |
| calc |
| _ ≤ ∫⁻ t in Ioo 0 f, ENNReal.ofReal a+(Ioo 0 T).indicator (fun _ => ENNReal.ofReal w) t := |
| lintegral_mono_ae (ae_restrict_mem measurableSet_Ioo |>.mono hpt) |
| _ = ENNReal.ofReal a*ENNReal.ofReal f+ |
| ∫⁻ t in Ioo 0 f, (Ioo 0 T).indicator (fun _ => ENNReal.ofReal w) t := by |
| rw [lintegral_add_left measurable_const,lintegral_const,Measure.restrict_apply_univ,Real.volume_Ioo,sub_zero] |
| _ ≤ ENNReal.ofReal a*ENNReal.ofReal f+ |
| ∫⁻ t, (Ioo 0 T).indicator (fun _ => ENNReal.ofReal w) t := by |
| exact add_le_add le_rfl (lintegral_mono' Measure.restrict_le_self (le_refl _)) |
| _ = _ := by |
| rw [lintegral_indicator measurableSet_Ioo,lintegral_const,Measure.restrict_apply_univ, |
| Real.volume_Ioo,sub_zero,← ENNReal.ofReal_mul ha,← ENNReal.ofReal_mul hw, |
| ← ENNReal.ofReal_add (mul_nonneg ha hf) (mul_nonneg hw hT)] |
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| variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] [Nontrivial E] |
| (μ : Measure E) [μ.IsAddHaarMeasure] |
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| def averagedLoss (f : E → ℝ) (κ δ : ℝ) (z : E × ℝ) : ℝ := |
| ∫ e : sphere (0:E) 1, weight μ κ e z.1-lineWeight μ f κ δ (e,z) ∂μ.toSphere |
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| lemma measurable_averagedLoss {f : E → ℝ} |
| (hf : IsOpen {z : E × ℝ | 0 < z.2 ∧ z.2 < f z.1}) |
| {κ : ℝ} (hκ : 0 < κ) (hκ1 : κ ≤ 1) (δ : ℝ) : |
| Measurable (averagedLoss μ f κ δ) := by |
| have hm : Measurable (fun z : sphere (0:E) 1 × (E × ℝ) => |
| weight μ κ z.1 z.2.1-lineWeight μ f κ δ z) := |
| ((measurable_weight μ hκ hκ1).comp (measurable_fst.prodMk measurable_snd.fst)).sub |
| (measurable_lineWeight μ hf hκ hκ1 δ) |
| exact hm.stronglyMeasurable.integral_prod_left'.measurable |
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| omit [BorelSpace E] [FiniteDimensional ℝ E] [Nontrivial E] [μ.IsAddHaarMeasure] in |
| lemma averagedLoss_nonneg (f : E → ℝ) (κ δ : ℝ) (z : E × ℝ) : |
| 0 ≤ averagedLoss μ f κ δ z := |
| integral_nonneg (fun e => sub_nonneg.mpr (lineWeight_le μ f κ δ (e,z))) |
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| lemma averagedLoss_high {f : E → ℝ} |
| (hf : IsOpen {z : E × ℝ | 0 < z.2 ∧ z.2 < f z.1}) |
| {κ δ : ℝ} (hκ : 0 < κ) (hκ1 : κ ≤ 1) (hδ : 0 < δ) (hδ1 : δ ≤ 1/2) |
| {x : E} (hd : 0 < d x) {t M P : ℝ} (ht : 0 < t) (htf : t < f x) |
| (hM : 0 ≤ M) (hP : 0 < P) (hlevel : P/(d x)^(k (Module.finrank ℝ E)) ≤ t) |
| (hm : (∫⁻ z in ball (0:E) 4, ENNReal.ofReal (f z) ∂μ) ≤ ENNReal.ofReal M) : |
| averagedLoss μ f κ δ (x,t) ≤ |
| (oscillationConstant μ κ*δ*μ.toSphere.real univ+ |
| (boundConstant μ κ*rayConstant (Module.finrank ℝ E)*M)/δ^(Module.finrank ℝ E)/P)* |
| (d x)^(N (Module.finrank ℝ E)) := by |
| let n := Module.finrank ℝ E |
| have hC : 0 ≤ boundConstant μ κ*rayConstant n*M := |
| mul_nonneg (mul_nonneg (boundConstant_nonneg μ hκ.le) (show 0 ≤ rayConstant n by unfold rayConstant; positivity)) hM |
| have hh := high_level_flatness n hd hδ ht hP hC ((div_le_iff₀ (pow_pos hd _)).mp hlevel) |
| have he := averaged_loss_le μ hf hκ hκ1 hδ hδ1 hd ht htf hM hm |
| change averagedLoss μ f κ δ (x,t) ≤ _ at he |
| calc |
| _ ≤ oscillationConstant μ κ*δ*(d x)^(N n)*μ.toSphere.real univ+ |
| (boundConstant μ κ*rayConstant n*M)*(d x)^(N n-n)/(t*(δ*(d x)^(m n))^n) := by |
| convert he using 1; ring |
| _ ≤ oscillationConstant μ κ*δ*(d x)^(N n)*μ.toSphere.real univ+ |
| (boundConstant μ κ*rayConstant n*M)/δ^n/P*(d x)^(N n) := add_le_add le_rfl hh |
| _ = _ := by ring |
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| omit [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] [Nontrivial E] in |
| lemma weight_low_levels_bound (n : ℕ) {x : E} (hd : 0 < d x) {P : ℝ} (hP : 0 ≤ P) : |
| w n x*(P/(d x)^(k n)) ≤ (6*(N n:ℝ)*3^(N n-1))*P := by |
| have hw : w n x ≤ 6*(N n:ℝ)*3^(N n-1)*(d x)^(N n-1) := by |
| rw [w_eq_profile]; exact wProfile_bound n (norm_nonneg x) |
| have hp : (d x)^(N n-1) ≤ (d x)^(k n) := |
| pow_le_pow_of_le_one hd.le (d_le_one x) (by unfold N; omega) |
| calc |
| _ ≤ (6*(N n:ℝ)*3^(N n-1)*(d x)^(N n-1))*(P/(d x)^(k n)) := |
| mul_le_mul_of_nonneg_right hw (div_nonneg hP (pow_nonneg hd.le _)) |
| _ ≤ (6*(N n:ℝ)*3^(N n-1)*(d x)^(k n))*(P/(d x)^(k n)) := by gcongr |
| _ = _ := by field_simp |
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| lemma integrated_averagedLoss {f : E → ℝ} |
| (hf : IsOpen {z : E × ℝ | 0 < z.2 ∧ z.2 < f z.1}) |
| {κ δ : ℝ} (hκ : 0 < κ) (hκ1 : κ ≤ 1) (hδ : 0 < δ) (hδ1 : δ ≤ 1/2) |
| {x : E} (hd : 0 < d x) (hfx : 0 ≤ f x) {M P : ℝ} (hM : 0 ≤ M) (hP : 0 < P) |
| (hm : (∫⁻ z in ball (0:E) 4, ENNReal.ofReal (f z) ∂μ) ≤ ENNReal.ofReal M) : |
| (∫⁻ t in Ioo 0 (f x), ENNReal.ofReal (averagedLoss μ f κ δ (x,t))) ≤ |
| ENNReal.ofReal ((oscillationConstant μ κ*δ*μ.toSphere.real univ+ |
| (boundConstant μ κ*rayConstant (Module.finrank ℝ E)*M)/δ^(Module.finrank ℝ E)/P)* |
| (d x)^(N (Module.finrank ℝ E))*f x + (6*(N (Module.finrank ℝ E):ℝ)*3^(N (Module.finrank ℝ E)-1))*P) := by |
| let n := Module.finrank ℝ E |
| apply (low_high_layer_bound hfx (w_nonneg n x) (div_nonneg hP.le (pow_nonneg hd.le _)) |
| (a := (oscillationConstant μ κ*δ*μ.toSphere.real univ+ |
| (boundConstant μ κ*rayConstant n*M)/δ^n/P)*(d x)^(N n)) |
| (by exact mul_nonneg (add_nonneg |
| (mul_nonneg (mul_nonneg (oscillationConstant_nonneg μ hκ.le) hδ.le) (measureReal_nonneg)) |
| (div_nonneg (div_nonneg (mul_nonneg (mul_nonneg (boundConstant_nonneg μ hκ.le) |
| (show 0 ≤ rayConstant n by unfold rayConstant; positivity)) hM) (pow_nonneg hδ.le _)) hP.le)) (pow_nonneg hd.le _)) |
| (fun t _ => averaged_loss_le_weight μ hf hκ hκ1 δ x t) |
| (fun t ht hlevel => averagedLoss_high μ hf hκ hκ1 hδ hδ1 hd ht.1 ht.2 hM hP hlevel hm)).trans |
| apply ENNReal.ofReal_le_ofReal |
| exact add_le_add le_rfl (weight_low_levels_bound n hd hP.le) |
| |
| end HenonLaneEmden.SpatialComponents |
| end |
| end LayerIntegration |
| |
| end OAI |
| |
