| import OAI.Analysis.HenonEmden.LayerSplit |
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| namespace OAI |
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| universe uZ |
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| section LayerAssembly |
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| open MeasureTheory Set |
| open scoped ENNReal |
| namespace HenonLaneEmden.LayerPressure |
| open LineComponents IntervalGeometry LayerTonelli |
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| variable {Z : Type uZ} [NormedAddCommGroup Z] [MeasurableSpace Z] [BorelSpace Z] [SecondCountableTopology Z] |
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| noncomputable def sourceWeight (f h d : Z × ℝ → ℝ) (δ : ℝ) (a : (Z × ℝ) × ℝ) : ℝ := |
| lineWeight {s | ((a.1.1,a.2),s) ∈ parameterLayer f} (fun s => h (a.1.1,s)) |
| (fun s => d (a.1.1,s)) δ a.1.2 |
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| noncomputable def sourceLower (f : Z × ℝ → ℝ) (a : (Z × ℝ) × ℝ) : Z × ℝ := |
| (a.1.1,lowerEndpoint (parameterLayer f) ((a.1.1,a.2),a.1.2)) |
| noncomputable def sourceUpper (f : Z × ℝ → ℝ) (a : (Z × ℝ) × ℝ) : Z × ℝ := |
| (a.1.1,upperEndpoint (parameterLayer f) ((a.1.1,a.2),a.1.2)) |
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| omit [SecondCountableTopology Z] in |
| lemma measurable_sourceWeight {f h d : Z × ℝ → ℝ} (hf : IsOpen (parameterLayer f)) |
| (hh : Continuous h) (hd : Continuous d) (δ : ℝ) : Measurable (sourceWeight f h d δ) := by |
| let h' : (Z × ℝ) × ℝ → ℝ := fun z => h (z.1.1,z.2) |
| let d' : (Z × ℝ) × ℝ → ℝ := fun z => d (z.1.1,z.2) |
| have hh' : Continuous h' := hh.comp (continuous_fst.fst.prodMk continuous_snd) |
| have hd' : Continuous d' := hd.comp (continuous_fst.fst.prodMk continuous_snd) |
| have he : Measurable (fun p : (Z × ℝ) × ℝ => |
| (setWeight (fun s => h' (p.1,s)) (fun s => d' (p.1,s)) δ |
| (connectedComponentIn {s | (p.1,s) ∈ parameterLayer f} p.2)).toReal) := |
| (measurable_parametric_setWeight hf hh' hd' δ).ennreal_toReal |
| have hem := he.comp (tripleSwap : (Z × ℝ) × ℝ ≃ᵐ (Z × ℝ) × ℝ).measurable |
| unfold sourceWeight lineWeight |
| simpa only [Function.comp_def,tripleSwap_apply,h',d'] using hem |
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| omit [SecondCountableTopology Z] in |
| lemma measurable_sourceLower {f : Z × ℝ → ℝ} (hf : IsOpen (parameterLayer f)) : |
| Measurable (sourceLower f) := by |
| exact measurable_fst.fst.prodMk ((measurable_lowerEndpoint hf).comp |
| (measurable_fst.fst.prodMk measurable_snd |>.prodMk measurable_fst.snd)) |
| omit [SecondCountableTopology Z] in |
| lemma measurable_sourceUpper {f : Z × ℝ → ℝ} (hf : IsOpen (parameterLayer f)) : |
| Measurable (sourceUpper f) := by |
| exact measurable_fst.fst.prodMk ((measurable_upperEndpoint hf).comp |
| (measurable_fst.fst.prodMk measurable_snd |>.prodMk measurable_fst.snd)) |
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| omit [NormedAddCommGroup Z] [MeasurableSpace Z] [BorelSpace Z] [SecondCountableTopology Z] in |
| lemma sourceWeight_nonneg (f h d : Z × ℝ → ℝ) (δ : ℝ) (a : (Z × ℝ) × ℝ) : |
| 0 ≤ sourceWeight f h d δ a := ENNReal.toReal_nonneg |
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| omit [NormedAddCommGroup Z] [MeasurableSpace Z] [BorelSpace Z] [SecondCountableTopology Z] in |
| lemma sourceWeight_le {f h d : Z × ℝ → ℝ} {δ : ℝ} {a : (Z × ℝ) × ℝ} |
| (ha : a ∈ sourceSubgraph f) (hh : 0 ≤ h a.1) : sourceWeight f h d δ a ≤ h a.1 := |
| lineWeight_le ha hh |
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| noncomputable def splitKernel (n : ℕ) (C : ℝ) (z : (Z × ℝ) × (Z × ℝ)) : ℝ := |
| kernel n ‖z.1.1-z.2.1‖ C (z.1.2-z.2.2) |
| noncomputable def splitDirection (z : (Z × ℝ) × (Z × ℝ)) : ℝ := |
| (z.1.2-z.2.2)^2/(‖z.1.1-z.2.1‖^2+(z.1.2-z.2.2)^2) |
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| lemma measurable_splitKernel (n : ℕ) (C : ℝ) : Measurable (splitKernel (Z := Z) n C) := by |
| unfold splitKernel kernel |
| fun_prop |
| lemma measurable_splitDirection : Measurable (splitDirection (Z := Z)) := by |
| unfold splitDirection |
| fun_prop |
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| noncomputable def pairLeft (n : ℕ) (C : ℝ) (f g h d : Z × ℝ → ℝ) (δ : ℝ) |
| (z : ((Z × ℝ) × ℝ) × ((Z × ℝ) × ℝ)) : ℝ := |
| (sourceWeight f h d δ z.1+sourceWeight g h d δ z.2)*splitKernel n C (z.1.1,z.2.1) |
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| noncomputable def pairRight (n : ℕ) (C : ℝ) (f g h d : Z × ℝ → ℝ) (δ : ℝ) |
| (z : ((Z × ℝ) × ℝ) × ((Z × ℝ) × ℝ)) : ℝ := |
| sourceWeight f h d δ z.1 * |
| ((splitKernel n C (sourceLower f z.1,z.2.1)+splitKernel n C (sourceUpper f z.1,z.2.1))/2) + |
| sourceWeight g h d δ z.2 * |
| ((splitKernel n C (z.1.1,sourceLower g z.2)+splitKernel n C (z.1.1,sourceUpper g z.2))/2) + |
| (((n:ℝ)-2)*h z.1.1*splitDirection (z.1.1,z.2.1)+ |
| |sourceWeight f h d δ z.1-sourceWeight g h d δ z.2|)*splitKernel n C (z.1.1,z.2.1) |
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| lemma measurable_pairLeft (n : ℕ) (C : ℝ) {f g h d : Z × ℝ → ℝ} |
| (hf : IsOpen (parameterLayer f)) (hg : IsOpen (parameterLayer g)) |
| (hh : Continuous h) (hd : Continuous d) (δ : ℝ) : Measurable (pairLeft n C f g h d δ) := by |
| unfold pairLeft |
| have hχf := measurable_sourceWeight hf hh hd δ |
| have hχg := measurable_sourceWeight hg hh hd δ |
| have hk := measurable_splitKernel (Z := Z) n C |
| fun_prop |
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| lemma measurable_pairRight (n : ℕ) (C : ℝ) {f g h d : Z × ℝ → ℝ} |
| (hf : IsOpen (parameterLayer f)) (hg : IsOpen (parameterLayer g)) |
| (hh : Continuous h) (hd : Continuous d) (δ : ℝ) : Measurable (pairRight n C f g h d δ) := by |
| unfold pairRight |
| have hχf := measurable_sourceWeight hf hh hd δ |
| have hχg := measurable_sourceWeight hg hh hd δ |
| have hlof := measurable_sourceLower hf |
| have hlog := measurable_sourceLower hg |
| have hhif := measurable_sourceUpper hf |
| have hhig := measurable_sourceUpper hg |
| have hk := measurable_splitKernel (Z := Z) n C |
| have hD := measurable_splitDirection (Z := Z) |
| fun_prop |
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| omit [MeasurableSpace Z] [BorelSpace Z] [SecondCountableTopology Z] in |
| lemma weighted_leftAverage_eq {f h d : Z × ℝ → ℝ} (hf : IsOpen (parameterLayer f)) |
| (δ : ℝ) {p : Z × ℝ} {s : ℝ} (hs : (p,s) ∈ parameterLayer f) |
| (k : ℝ → ℝ) (y : ℝ) : |
| lineWeight {s | (p,s) ∈ parameterLayer f} (fun s => h (p.1,s)) (fun s => d (p.1,s)) δ s * |
| leftAverage {s | (p,s) ∈ parameterLayer f} k s y = |
| lineWeight {s | (p,s) ∈ parameterLayer f} (fun s => h (p.1,s)) (fun s => d (p.1,s)) δ s * |
| ((k (lowerEndpoint (parameterLayer f) (p,s)-y)+k (upperEndpoint (parameterLayer f) (p,s)-y))/2) := by |
| by_cases hz : lineWeight {s | (p,s) ∈ parameterLayer f} (fun s => h (p.1,s)) |
| (fun s => d (p.1,s)) δ s = 0 |
| · rw [hz,zero_mul,zero_mul] |
| · have he := relative_endpoints_eq_of_weight_pos hf hs |
| (lt_of_le_of_ne (lineWeight_nonneg _ _ _ _ _) (Ne.symm hz)) |
| simp only [leftAverage,he.1,he.2] |
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| omit [MeasurableSpace Z] [BorelSpace Z] [SecondCountableTopology Z] in |
| lemma weighted_rightAverage_eq {g h d : Z × ℝ → ℝ} (hg : IsOpen (parameterLayer g)) |
| (δ : ℝ) {p : Z × ℝ} {s : ℝ} (hs : (p,s) ∈ parameterLayer g) |
| (k : ℝ → ℝ) (x : ℝ) : |
| lineWeight {s | (p,s) ∈ parameterLayer g} (fun s => h (p.1,s)) (fun s => d (p.1,s)) δ s * |
| rightAverage {s | (p,s) ∈ parameterLayer g} k x s = |
| lineWeight {s | (p,s) ∈ parameterLayer g} (fun s => h (p.1,s)) (fun s => d (p.1,s)) δ s * |
| ((k (x-lowerEndpoint (parameterLayer g) (p,s))+k (x-upperEndpoint (parameterLayer g) (p,s)))/2) := by |
| by_cases hz : lineWeight {s | (p,s) ∈ parameterLayer g} (fun s => h (p.1,s)) |
| (fun s => d (p.1,s)) δ s = 0 |
| · rw [hz,zero_mul,zero_mul] |
| · have he := relative_endpoints_eq_of_weight_pos hg hs |
| (lt_of_le_of_ne (lineWeight_nonneg _ _ _ _ _) (Ne.symm hz)) |
| simp only [rightAverage,he.1,he.2] |
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| omit [MeasurableSpace Z] [BorelSpace Z] [SecondCountableTopology Z] in |
| lemma pair_line_comparison (n : ℕ) (hn : 3 ≤ n) {C : ℝ} (hC : 0 ≤ C) |
| {f g h d : Z × ℝ → ℝ} (hf : IsOpen (parameterLayer f)) (hg : IsOpen (parameterLayer g)) |
| (hhn : ∀ x, 0 ≤ h x) (δ : ℝ) (p : (Z × ℝ) × (Z × ℝ)) (hp : p.1.1 ≠ p.2.1) : |
| (∫⁻ s : ℝ × ℝ in {s | (p.1,s) ∈ parameterLayer f} ×ˢ {s | (p.2,s) ∈ parameterLayer g}, |
| ENNReal.ofReal (pairLeft n C f g h d δ (((p.1.1,s.1),p.1.2),((p.2.1,s.2),p.2.2)))) ≤ |
| ∫⁻ s : ℝ × ℝ in {s | (p.1,s) ∈ parameterLayer f} ×ˢ {s | (p.2,s) ∈ parameterLayer g}, |
| ENNReal.ofReal (pairRight n C f g h d δ (((p.1.1,s.1),p.1.2),((p.2.1,s.2),p.2.2))) := by |
| have hfo := isOpen_section hf p.1 |
| have hgo := isOpen_section hg p.2 |
| have hl : 0 < ‖p.1.1-p.2.1‖ := norm_pos_iff.mpr (sub_ne_zero.mpr hp) |
| have hh := weighted_source_layer_comparison n hn hfo hgo |
| (fun s => h (p.1.1,s)) (fun s => h (p.2.1,s)) |
| (fun s => d (p.1.1,s)) (fun s => d (p.2.1,s)) δ hl hC (fun x _ => hhn _) |
| change (∫⁻ s in _, ENNReal.ofReal _) ≤ _ at hh |
| convert hh using 1 |
| · rfl |
| · apply setLIntegral_congr_fun (hfo.measurableSet.prod hgo.measurableSet) |
| intro s hs |
| unfold pairRight splitKernel splitDirection sourceLower sourceUpper sourceWeight |
| dsimp only |
| rw [weighted_leftAverage_eq hf δ hs.1,weighted_rightAverage_eq hg δ hs.2] |
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| lemma ae_distinct_transverse (μ : Measure Z) [SFinite μ] [NullSingletonClass μ] : |
| ∀ᵐ p : (Z × ℝ) × (Z × ℝ) ∂(μ.prod volume).prod (μ.prod volume), p.1.1 ≠ p.2.1 := by |
| apply (Measure.ae_prod_iff_ae_ae (by measurability)).mpr |
| apply ae_of_all |
| intro x |
| apply (Measure.ae_prod_iff_ae_ae (by measurability)).mpr |
| filter_upwards [μ.ae_ne x.1] with z hz |
| exact ae_of_all _ (fun _ => Ne.symm hz) |
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| /-- The globally integrated nonnegative interval comparison. No source, |
| endpoint, or energy integrability is assumed; all terms are still extended values. -/ |
| lemma global_nonnegative_pressure (μ : Measure Z) [SFinite μ] [NullSingletonClass μ] |
| (n : ℕ) (hn : 3 ≤ n) {C : ℝ} (hC : 0 ≤ C) |
| {f g h d : Z × ℝ → ℝ} (hf : Measurable f) (hg : Measurable g) |
| (hfo : IsOpen (parameterLayer f)) (hgo : IsOpen (parameterLayer g)) |
| (hh : Continuous h) (hd : Continuous d) (hhn : ∀ x, 0 ≤ h x) (δ : ℝ) : |
| (∫⁻ z, ENNReal.ofReal (pairLeft n C f g h d δ z) |
| ∂(layerMeasure (μ.prod volume) f).prod (layerMeasure (μ.prod volume) g)) ≤ |
| ∫⁻ z, ENNReal.ofReal (pairRight n C f g h d δ z) |
| ∂(layerMeasure (μ.prod volume) f).prod (layerMeasure (μ.prod volume) g) := by |
| rw [lintegral_pair_layer_split μ hf hg (measurable_pairLeft n C hfo hgo hh hd δ).ennreal_ofReal, |
| lintegral_pair_layer_split μ hf hg (measurable_pairRight n C hfo hgo hh hd δ).ennreal_ofReal] |
| apply lintegral_mono_ae |
| filter_upwards [ae_distinct_transverse μ] with p hp |
| exact pair_line_comparison n hn hC hfo hgo hhn δ p hp |
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| end HenonLaneEmden.LayerPressure |
| end LayerAssembly |
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| end OAI |
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