| import Mathlib | |
| import OAI.Probability.GaussianInformation.Kernel.OutgoingLaws | |
| namespace OAI | |
| open MeasureTheory ProbabilityTheory Metric Set | |
| open scoped ENNReal | |
| open MeasureTheory ProbabilityTheory Set | |
| open scoped ENNReal | |
| open MeasureTheory ProbabilityTheory | |
| open MeasureTheory | |
| open scoped ENNReal | |
| namespace KernelPotential.MassPotential | |
| open MeasureTheory ProbabilityTheory Set | |
| open scoped ENNReal | |
| open Coercivity Sphere | |
| noncomputable def normalized {d : ℕ} (ρ : Measure (Ambient d)) : Measure (Ambient d) := (ρ univ)⁻¹ • ρ | |
| noncomputable def energy {d : ℕ} (ρ : Measure (Ambient d)) : ℝ := (ρ univ).toReal*Phi (normalized ρ) | |
| noncomputable def proper {d : ℕ} (ρ : Measure (Ambient d)) : Measure (Ambient d) := | |
| if ρ univ=0 then sphereProbability d else normalized ρ | |
| lemma normalized_probability {d : ℕ} (ρ : Measure (Ambient d)) [IsFiniteMeasure ρ] (hρ : ρ univ≠0) : | |
| IsProbabilityMeasure (normalized ρ) := by | |
| constructor | |
| simp only [normalized,Measure.smul_apply,smul_eq_mul] | |
| exact ENNReal.inv_mul_cancel hρ (measure_ne_top _ _) | |
| lemma proper_probability {d : ℕ} (ρ : Measure (Ambient d)) [IsFiniteMeasure ρ] : IsProbabilityMeasure (proper ρ) := by | |
| unfold proper | |
| split_ifs with h | |
| · infer_instance | |
| · exact normalized_probability ρ h | |
| lemma proper_bounded {d : ℕ} (ρ : Measure (Ambient d)) [IsFiniteMeasure ρ] (hρ : ρ ≤ sphereProbability d) : | |
| BoundedDensity (proper ρ) := by | |
| unfold proper | |
| split_ifs with h | |
| · exact ⟨1,zero_lt_one,by simp⟩ | |
| · have hp : 0 < (ρ univ).toReal := ENNReal.toReal_pos h (measure_ne_top _ _) | |
| refine ⟨1/(ρ univ).toReal,one_div_pos.mpr hp,?_⟩ | |
| unfold normalized | |
| have he : ENNReal.ofReal (1/(ρ univ).toReal)=(ρ univ)⁻¹ := by | |
| rw [one_div,ENNReal.ofReal_inv_of_pos hp,ENNReal.ofReal_toReal (measure_ne_top _ _)] | |
| rw [he] | |
| exact smul_le_smul_left _ hρ | |
| lemma mass_smul_proper {d : ℕ} (ρ : Measure (Ambient d)) [IsFiniteMeasure ρ] : | |
| ENNReal.ofReal (ρ univ).toReal • proper ρ=ρ := by | |
| rw [ENNReal.ofReal_toReal (measure_ne_top _ _)] | |
| by_cases h : ρ univ=0 | |
| · rw [h,zero_smul] | |
| exact (Measure.measure_univ_eq_zero.mp h).symm | |
| · rw [proper,ite_eq_right h,normalized,smul_smul,ENNReal.mul_inv_cancel h (measure_ne_top _ _),one_smul] | |
| lemma energy_proper {d : ℕ} (ρ : Measure (Ambient d)) : energy ρ=(ρ univ).toReal*Phi (proper ρ) := by | |
| unfold energy proper | |
| split_ifs with h | |
| · simp [h] | |
| · rfl | |
| lemma energy_smul_probability {d : ℕ} (ν : Measure (Ambient d)) [IsProbabilityMeasure ν] | |
| (a : ℝ) (ha : 0 ≤ a) : energy (ENNReal.ofReal a • ν)=a*Phi ν := by | |
| by_cases haz : a=0 | |
| · simp [haz,energy,normalized] | |
| · have hp := lt_of_le_of_ne ha (Ne.symm haz) | |
| have hz : ENNReal.ofReal a≠0 := ne_of_gt (ENNReal.ofReal_pos.mpr hp) | |
| have hu : (ENNReal.ofReal a • ν) univ=ENNReal.ofReal a := by simp | |
| unfold energy normalized | |
| rw [hu,ENNReal.toReal_ofReal ha,smul_smul,ENNReal.inv_mul_cancel hz ENNReal.ofReal_ne_top,one_smul] | |
| lemma energy_mixture {d q : ℕ} (hd : 16 ≤ d) (ρ : Fin q → Measure (Ambient d)) | |
| (hρf : ∀ i,IsFiniteMeasure (ρ i)) (hρ : ∀ i,ρ i ≤ sphereProbability d) : | |
| energy (∑ i,ρ i) ≤ ∑ i,energy (ρ i) := by | |
| have (i : Fin q) : IsFiniteMeasure (ρ i) := hρf i | |
| let a := fun i => (ρ i univ).toReal | |
| let A := ∑ i,a i | |
| have ha (i) : 0 ≤ a i := ENNReal.toReal_nonneg | |
| have hA : 0 ≤ A := Finset.sum_nonneg (fun i _ => ha i) | |
| have he : (∑ i,ρ i) univ=ENNReal.ofReal A := by | |
| rw [Measure.finsetSum_apply] | |
| dsimp [A,a] | |
| rw [ENNReal.ofReal_sum_of_nonneg (fun i _ => ENNReal.toReal_nonneg)] | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| exact (ENNReal.ofReal_toReal (measure_ne_top _ _)).symm | |
| have hrepr : (∑ i,ENNReal.ofReal (a i) • proper (ρ i))=∑ i,ρ i := by | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| exact mass_smul_proper (ρ i) | |
| by_cases hAz : A=0 | |
| · have haz (i) : a i=0 := by | |
| have hh : a i ≤ A := Finset.single_le_sum (fun j _ => ha j) (Finset.mem_univ i) | |
| rw [hAz] at hh | |
| exact le_antisymm hh (ha i) | |
| simp only [energy_proper] | |
| rw [he,hAz,ENNReal.ofReal_zero,ENNReal.toReal_zero,zero_mul] | |
| change 0 ≤ ∑ i,a i*Phi (proper (ρ i)) | |
| simp [haz] | |
| · have hAp : 0 < A := lt_of_le_of_ne hA (Ne.symm hAz) | |
| let w := fun i => a i/A | |
| have hw (i) : 0 ≤ w i := div_nonneg (ha i) hA | |
| have hw1 : ∑ i,w i=1 := by simp [w,← Finset.sum_div,A,hAz] | |
| have hnorm : normalized (∑ i,ρ i)=∑ i,ENNReal.ofReal (w i) • proper (ρ i) := by | |
| rw [normalized,he,← hrepr,Finset.smul_sum] | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| rw [smul_smul] | |
| congr 1 | |
| dsimp [w] | |
| rw [ENNReal.ofReal_div_of_pos hAp,div_eq_mul_inv,mul_comm] | |
| have hh := finite_convexity hd q w hw hw1 (fun i => proper (ρ i)) | |
| (fun i => proper_probability _) (fun i => proper_bounded _ (hρ i)) | |
| have := mul_le_mul_of_nonneg_left hh hA | |
| rw [← hnorm] at this | |
| calc energy (∑ i,ρ i) = A*Phi (normalized (∑ i,ρ i)) := by | |
| rw [energy,he,ENNReal.toReal_ofReal hA] | |
| _ ≤ ∑ i,a i*Phi (proper (ρ i)) := by | |
| simpa only [Finset.mul_sum,w,← mul_assoc,mul_div_cancel₀ _ hAz] using this | |
| _ = ∑ i,energy (ρ i) := by simp only [energy_proper,a] | |
| end KernelPotential.MassPotential | |
| end OAI | |