import OAI.Probability.GaussianReplacement.Tilt namespace OAI open MeasureTheory ProbabilityTheory Set Filter open scoped ENNReal NNReal RealInnerProductSpace Topology noncomputable section namespace CurrentProjection open InnerProductSpace open InformationTheory lemma sphere_own_density_moment {d m r : ℕ} (hdim : AdmissibleDimensions d m r 1) (hd : 2 ≤ d) : ∫⁻ z : Rows (m+r) d × Sphere d, exactDensity (uniformSphere d) z.1 (label z.1 (z.2:Vector d)) ∂(gaussianRows (m+r) d).prod (uniformSphere d) ≤ momentConstant 9 d (3*4^d) (3*4^d) := by let : IsProbabilityMeasure (uniformSphere d) := uniformSphere_probability_of_pos (by omega) have hB : (0:ℝ) < 3*4^d := by positivity have hG := uniformSphere_growth (show 1 ≤ d by omega) (show m+r ≤ d-1 by have := hdim.2.2.2.1; omega) have hD := uniformSphere_growth (show 1 ≤ d by omega) (le_refl (d-1)) have ha : ∀ᵐ (s : Sphere d) ∂uniformSphere d, (∫⁻ G, exactDensity (uniformSphere d) G (label G (s:Vector d)) ∂gaussianRows (m+r) d) ≤ momentConstant 9 d (3*4^d) (3*4^d) := by filter_upwards [ae_target_cube_limit (k:=m+r) (uniformSphere d)] with s hs have hfin := extended_ae_finite hdim (uniformSphere d) _ _ hB hB hG hD s have he : ∀ᵐ G ∂gaussianRows (m+r) d, exactDensity (uniformSphere d) G (label G (s:Vector d)) = extendedCubeDensity (uniformSphere d) G (label G (s:Vector d)) := by filter_upwards [hs,hfin] with G hG hf exact exactDensity_eq_of_tendsto _ _ _ hG.1 hf.ne have hm := exact_mixed_moment_of_ae hdim (uniformSphere d) _ _ hB hB hG hD s he simpa only [pow_one, gaussianRows, ←lintegral_pi_append (stdGaussian (Vector d)) m r _ (measurable_fixed_exact (uniformSphere d) s)] using hm rw [lintegral_prod_symm _ (measurable_own_exact (uniformSphere d)).aemeasurable] exact (lintegral_mono_ae ha).trans_eq (by simp) lemma sphere_moment_log_bound {d : ℕ} (hd : 1 ≤ d) : Real.log (1+(momentConstant 9 d (3*4^d) (3*4^d)).toReal) ≤ 20*d := by have hB : (0:ℝ) < 3*4^d := by positivity have he : (momentConstant 9 d (3*4^d) (3*4^d)).toReal = Real.exp (9*d)*(3*4^d) := by simp [momentConstant,ENNReal.toReal_ofReal (by positivity : 0 ≤ Real.exp (9*d)*(3*4^d))] rw [he] have hd1 : (1:ℝ) ≤ d := by exact_mod_cast hd have hpow : (1:ℝ) ≤ 4^d := one_le_pow₀ (by norm_num) have hexp : (1:ℝ) ≤ Real.exp (9*d) := Real.one_le_exp_iff.mpr (by positivity) have ht : 1 ≤ Real.exp (9*d)*(3*4^d) := by nlinarith have hl := Real.log_le_log (by positivity : 0<1+Real.exp (9*d)*(3*4^d)) (show 1+Real.exp (9*d)*(3*4^d) ≤ 2*(Real.exp (9*d)*(3*4^d)) by linarith) rw [Real.log_mul (by norm_num : (2:ℝ) ≠ 0) (by positivity), Real.log_mul (Real.exp_ne_zero _) hB.ne',Real.log_exp, Real.log_mul (by norm_num : (3:ℝ) ≠ 0) (by positivity),Real.log_pow] at hl have h2 := Real.log_le_self (by norm_num : (0:ℝ) ≤ 2) have h3 := Real.log_le_self (by norm_num : (0:ℝ) ≤ 3) have h4 := Real.log_le_self (by norm_num : (0:ℝ) ≤ 4) nlinarith lemma sphere_label_entropy_finite {d m r : ℕ} (hdim : AdmissibleDimensions d m r 1) (hd : 2 ≤ d) : Integrable (fun z : Rows (m+r) d × Sphere d => Real.log (exactDensity (uniformSphere d) z.1 (label z.1 (z.2:Vector d))).toReal) ((gaussianRows (m+r) d).prod (uniformSphere d)) ∧ (∫ z : Rows (m+r) d × Sphere d, Real.log (exactDensity (uniformSphere d) z.1 (label z.1 (z.2:Vector d))).toReal ∂(gaussianRows (m+r) d).prod (uniformSphere d)) ≤ 20*d := by classical let : IsProbabilityMeasure (uniformSphere d) := uniformSphere_probability_of_pos (by omega) let : Nonempty (Sphere d) := nonempty_of_isProbabilityMeasure (uniformSphere d) let μ := (gaussianRows (m+r) d).prod (uniformSphere d) let f : Rows (m+r) d × Sphere d → ℝ≥0∞ := fun z => exactDensity (uniformSphere d) z.1 (label z.1 (z.2:Vector d)) have hf : Measurable f := measurable_own_exact _ have hmoment := sphere_own_density_moment hdim hd have ht : ∫⁻ z, f z ∂μ ≠ ∞ := ne_top_of_le_ne_top ENNReal.ofReal_ne_top hmoment have hJ := log_one_add_jensen μ f hf (fun z => exactDensity_ne_top _ _ _) ht have hM := ENNReal.toReal_mono (show momentConstant 9 d (3*4^d) (3*4^d) ≠ ∞ from ENNReal.ofReal_ne_top) hmoment have hMlog := Real.log_le_log (by positivity : 0<1+(∫⁻ z, f z ∂μ).toReal) (show 1+(∫⁻ z, f z ∂μ).toReal ≤ 1+(momentConstant 9 d (3*4^d) (3*4^d)).toReal from by dsimp [f,μ]; linarith) have hb := sphere_moment_log_bound (show 1 ≤ d by omega) have hJ' : ∫⁻ z, ENNReal.ofReal (Real.log (1+(f z).toReal)) ∂μ ≤ ENNReal.ofReal (20*d) := hJ.trans (ENNReal.ofReal_le_ofReal (hMlog.trans hb)) obtain ⟨⟨hiL,hbL⟩,hiP,_⟩ := log_bound_consequences μ f hf (20*d) (by positivity) hJ' have hB : (0:ℝ) < 3*4^d := by positivity have hAC := ae_labelLaw_ac_of_growth hdim (uniformSphere d) _ _ hB hB (uniformSphere_growth (show 1 ≤ d by omega) (show m+r ≤ d-1 by have := hdim.2.2.2.1; omega)) (uniformSphere_growth (show 1 ≤ d by omega) (le_refl (d-1))) let c : Sphere d := Classical.choice inferInstance obtain ⟨hi,_⟩ := projection_cover_entropy (uniformSphere d) hAC hiP (Finset.singleton_nonempty c) (show (0:ℝ)<1 by norm_num) (ae_of_all _ (fun s => ⟨c,Finset.mem_singleton_self c,by have hdist := (norm_sub_le (s:Vector d) (c:Vector d)) simpa only [Subtype.dist_eq,dist_eq_norm,norm_eq_of_mem_sphere s,norm_eq_of_mem_sphere c,one_add_one_eq_two,mul_one] using hdist⟩)) refine ⟨hi,?_⟩ apply (integral_mono hi hiL (fun z => ?_)).trans hbL exact (le_max_left _ _).trans (logPositive_le_log_one_add (f z)) def conditionalReferenceKL {A B : Type*} [MeasurableSpace A] [MeasurableSpace B] (P : Measure (A×B)) (ν : Measure B) : ℝ≥0∞ := klDiv P (P.fst.prod ν) lemma conditionalReferenceKL_chain {C A B : Type*} [MeasurableSpace C] [MeasurableSpace A] [MeasurableSpace B] [StandardBorelSpace B] [Nonempty B] (P : Measure (C×(A×B))) [IsProbabilityMeasure P] (ν : Measure A) (π : Measure B) [IsProbabilityMeasure ν] [IsProbabilityMeasure π] : conditionalReferenceKL P (ν.prod π) = conditionalReferenceKL (P.map (Prod.map id Prod.fst)) ν + conditionalReferenceKL (P.map MeasurableEquiv.prodAssoc.symm) π := by let R := P.map MeasurableEquiv.prodAssoc.symm let : IsProbabilityMeasure R := inferInstance have hRf : R.fst = P.map (Prod.map id Prod.fst) := by rw [Measure.fst,Measure.map_map measurable_fst MeasurableEquiv.prodAssoc.symm.measurable] rfl have hff : R.fst.fst = P.fst := by rw [hRf,Measure.fst,Measure.map_map measurable_fst (measurable_id.prodMap measurable_fst)] rfl have hr : (P.fst.prod (ν.prod π)).map MeasurableEquiv.prodAssoc.symm = (P.fst.prod ν).prod π := by rw [←Measure.prodAssoc_prod,MeasurableEquiv.map_symm_map] unfold conditionalReferenceKL rw [←klDiv_map_equiv P (P.fst.prod (ν.prod π)) MeasurableEquiv.prodAssoc.symm,hr, klDiv_prod_chain R (P.fst.prod ν) π] rw [←hff,hRf] lemma conditionalReferenceKL_add_condition {C A B : Type*} [MeasurableSpace C] [MeasurableSpace A] [MeasurableSpace B] [StandardBorelSpace A] [StandardBorelSpace B] [Nonempty A] [Nonempty B] (P : Measure (C×(A×B))) [IsProbabilityMeasure P] (π : Measure B) [IsProbabilityMeasure π] : conditionalReferenceKL (P.map MeasurableEquiv.prodAssoc.symm) π = conditionalReferenceKL (P.map (Prod.map id Prod.snd)) π + conditionalMutualInformation P := by let μ := P.fst let κ := P.condKernel have hP : μ ⊗ₘ κ = P := P.disintegrate P.condKernel have hfst : (P.map MeasurableEquiv.prodAssoc.symm).fst = μ ⊗ₘ κ.fst := by rw [←hP] exact fst_map_assoc_compProd μ κ have hsnd : P.map (Prod.map id Prod.snd) = μ ⊗ₘ κ.snd := by rw [Kernel.snd_eq,Measure.compProd_map measurable_snd,hP] have href : (μ ⊗ₘ (κ.fst ×ₖ Kernel.const C π)).map MeasurableEquiv.prodAssoc.symm = (μ ⊗ₘ κ.fst).prod π := by rw [compProd_prod_assoc] have he : (Kernel.const C π).comap Prod.fst measurable_fst = Kernel.const (C×A) π := rfl rw [he,Measure.compProd_const] have h := conditionalKL_chain μ κ π rw [hP] at h have hc := conditionalMutualInformation_compProd μ κ simp only [hP] at hc unfold conditionalReferenceKL rw [hfst,←href,klDiv_map_equiv,h,←hc,hsnd,Measure.fst_compProd] lemma conditionalMutualInformation_symm {C A B : Type*} [MeasurableSpace C] [MeasurableSpace A] [MeasurableSpace B] [StandardBorelSpace A] [StandardBorelSpace B] [Nonempty A] [Nonempty B] (P : Measure (C×(A×B))) [IsProbabilityMeasure P] : conditionalMutualInformation (P.map (Prod.map id Prod.swap)) = conditionalMutualInformation P := by let μ := P.fst let κ := P.condKernel have hP : μ ⊗ₘ κ = P := P.disintegrate P.condKernel have hmap : P.map (Prod.map id Prod.swap) = μ ⊗ₘ κ.map Prod.swap := by rw [Measure.compProd_map measurable_swap,hP] have hfst : (κ.map Prod.swap).fst = κ.snd := by rw [Kernel.fst_eq,Kernel.snd_eq,←Kernel.map_comp_right _ measurable_swap measurable_fst] rfl have hsnd : (κ.map Prod.swap).snd = κ.fst := by rw [Kernel.fst_eq,Kernel.snd_eq,←Kernel.map_comp_right _ measurable_swap measurable_snd] rfl have href : μ ⊗ₘ (κ.snd ×ₖ κ.fst) = (μ ⊗ₘ (κ.fst ×ₖ κ.snd)).map (Prod.map id Prod.swap) := by rw [←Measure.compProd_map measurable_swap,Kernel.map_prod_swap] have : IsMarkovKernel (κ.map Prod.swap) := Kernel.IsMarkovKernel.map κ measurable_swap have hc := conditionalMutualInformation_compProd μ (κ.map Prod.swap) simp only [hfst,hsnd,href,Measure.compProd_map measurable_swap] at hc have he := klDiv_map_equiv (μ ⊗ₘ κ) (μ ⊗ₘ (κ.fst ×ₖ κ.snd)) ((MeasurableEquiv.refl C).prodCongr MeasurableEquiv.prodComm) change klDiv ((μ ⊗ₘ κ).map (Prod.map id Prod.swap)) ((μ ⊗ₘ (κ.fst ×ₖ κ.snd)).map (Prod.map id Prod.swap)) = _ at he rw [he,←conditionalMutualInformation_compProd μ κ] at hc simpa only [hP] using hc lemma conditionalReferenceKL_finite_condition {C A B : Type*} [MeasurableSpace C] [MeasurableSpace A] [MeasurableSpace B] [StandardBorelSpace B] [Nonempty B] [Countable A] [MeasurableSingletonClass A] [Nonempty A] (P : Measure (C×(A×B))) [IsProbabilityMeasure P] (π : Measure B) [IsProbabilityMeasure π] (hbase : conditionalReferenceKL (P.map (Prod.map id Prod.snd)) π ≠ ∞) (hH : shannonEntropy (P.map (fun z => z.2.1)) ≠ ∞) : conditionalReferenceKL (P.map MeasurableEquiv.prodAssoc.symm) π ≠ ∞ ∧ (conditionalReferenceKL (P.map MeasurableEquiv.prodAssoc.symm) π).toReal ≤ (conditionalReferenceKL (P.map (Prod.map id Prod.snd)) π).toReal + (shannonEntropy (P.map (fun z => z.2.1))).toReal := by let Q := P.map (Prod.map id Prod.swap) let : IsProbabilityMeasure Q := inferInstance have hQ : (Q.map MeasurableEquiv.prodAssoc.symm).snd = P.map (fun z => z.2.1) := by simp only [Q,Measure.snd,Measure.map_map measurable_snd MeasurableEquiv.prodAssoc.symm.measurable, Measure.map_map (measurable_snd.comp MeasurableEquiv.prodAssoc.symm.measurable) (measurable_id.prodMap measurable_swap)] rfl obtain ⟨hc,hcb⟩ := conditionalMutualInformation_le_entropy Q (by rwa [hQ]) rw [hQ,conditionalMutualInformation_symm P] at hcb rw [conditionalMutualInformation_symm P] at hc rw [conditionalReferenceKL_add_condition P π] exact ⟨ENNReal.add_ne_top.mpr ⟨hbase,hc⟩,by rw [ENNReal.toReal_add hbase hc]; linarith⟩ def gaussianLabels (k : ℕ) : Measure (Fin k → ℝ) := Measure.pi (fun _ => gaussianReal 0 1) instance gaussianLabels_probability (k : ℕ) : IsProbabilityMeasure (gaussianLabels k) := by unfold gaussianLabels infer_instance lemma gaussianLabels_density (k : ℕ) : (volume : Measure (Fin k → ℝ)).withDensity (fun y => ENNReal.ofReal (gaussianProductPDF 1 0 y)) = gaussianLabels k := by symm apply Measure.pi_eq intro t ht rw [withDensity_apply _ (MeasurableSet.univ_pi ht)] simp only [gaussianProductPDF,radiusVariance,one_pow,Pi.zero_apply] change (∫⁻ y in Set.univ.pi t, ENNReal.ofReal (∏ i, gaussianPDFReal 0 1 (y i))) = _ rw [←ofReal_integral_eq_lintegral_ofReal] · change ENNReal.ofReal (∫ y, ∏ i, gaussianPDFReal 0 1 (y i) ∂(Measure.pi (fun _ : Fin k => (volume : Measure ℝ))).restrict (Set.univ.pi t)) = _ rw [Measure.restrict_pi_pi,integral_fintype_prod_eq_prod, ENNReal.ofReal_prod_of_nonneg (fun i _ => integral_nonneg (fun _ => (gaussianPDFReal_pos 0 1 _ (by norm_num)).le))] congr 1 funext i rw [gaussianReal_apply 0 (by norm_num : (1:ℝ≥0) ≠ 0)] exact ofReal_integral_eq_lintegral_ofReal (integrable_gaussianPDFReal 0 1).integrableOn (ae_of_all _ (fun x => (gaussianPDFReal_pos 0 1 x (by norm_num)).le)) · exact (Integrable.fintype_prod (fun _ : Fin k => integrable_gaussianPDFReal 0 1)).integrableOn · exact ae_of_all _ (fun y => Finset.prod_nonneg (fun i _ => (gaussianPDFReal_pos 0 1 (y i) (by norm_num)).le)) lemma log_gaussianLabelsPDF {k : ℕ} (y : Fin k → ℝ) : Real.log (gaussianProductPDF 1 0 y) = -(k:ℝ)*Real.log (normalizer 1) - (∑ i, (y i)^2)/2 := by simpa using log_gaussianProductPDF (k:=k) (show (0:ℝ)<1 by norm_num) (0:Fin k→ℝ) y lemma projectionSquare_sphere_joint_integrable {d k : ℕ} (ρ : Measure (Sphere d)) [IsProbabilityMeasure ρ] : Integrable (fun z : Rows k d × Sphere d => projectionSquare z.1 (z.2:Vector d)) ((gaussianRows k d).prod ρ) := by have hm : Measurable (fun z : Rows k d × Sphere d => projectionSquare z.1 (z.2:Vector d)) := by unfold projectionSquare label fun_prop apply (integrable_prod_iff' hm.aestronglyMeasurable).mpr refine ⟨ae_of_all _ (fun s => projectionSquare_integrable (s:Vector d)),?_⟩ have he : (fun s : Sphere d => ∫ G, ‖projectionSquare G (s:Vector d)‖ ∂gaussianRows k d) = fun _ => (k:ℝ) := by funext s simp_rw [Real.norm_eq_abs,abs_of_nonneg (projectionSquare_nonneg _ _)] rw [projectionSquare_integral,norm_eq_of_mem_sphere s] simp rw [he] exact integrable_const _ lemma projectionSquare_sphere_joint_integral {d k : ℕ} (ρ : Measure (Sphere d)) [IsProbabilityMeasure ρ] : (∫ z : Rows k d × Sphere d, projectionSquare z.1 (z.2:Vector d) ∂(gaussianRows k d).prod ρ) = k := by rw [integral_prod_symm _ (projectionSquare_sphere_joint_integrable ρ)] simp only [projectionSquare_integral,norm_eq_of_mem_sphere, one_pow,mul_one,integral_const,probReal_univ,one_smul] lemma gaussian_log_reference_projection {d k : ℕ} (ρ : Measure (Sphere d)) [IsProbabilityMeasure ρ] : Integrable (fun z : Rows k d×(Fin k→ℝ) => Real.log (gaussianProductPDF 1 0 z.2)) (projectionJoint ρ (gaussianRows k d)) ∧ (∫ z : Rows k d×(Fin k→ℝ), Real.log (gaussianProductPDF 1 0 z.2) ∂projectionJoint ρ (gaussianRows k d)) = -(k:ℝ)*(Real.log (normalizer 1)+1/2) := by have hf : Measurable (fun z : Rows k d×(Fin k→ℝ) => Real.log (gaussianProductPDF 1 0 z.2)) := by unfold gaussianProductPDF fun_prop have hm : Measurable (fun z : Rows k d×Sphere d => (z.1,label z.1 (z.2:Vector d))) := measurable_fst.prodMk measurable_own_label have hi : Integrable (fun z : Rows k d×Sphere d => Real.log (gaussianProductPDF 1 0 (label z.1 (z.2:Vector d)))) ((gaussianRows k d).prod ρ) := by simp_rw [log_gaussianLabelsPDF] exact (integrable_const _).sub ((projectionSquare_sphere_joint_integrable ρ).div_const 2) refine ⟨(integrable_map_measure hf.aestronglyMeasurable hm.aemeasurable).mpr hi,?_⟩ rw [projectionJoint,integral_map hm.aemeasurable hf.aestronglyMeasurable] simp_rw [log_gaussianLabelsPDF] change (∫ z : Rows k d×Sphere d, -(k:ℝ)*Real.log (normalizer 1) - projectionSquare z.1 (z.2:Vector d)/2 ∂(gaussianRows k d).prod ρ) = _ rw [integral_sub (integrable_const _) ((projectionSquare_sphere_joint_integrable ρ).div_const 2), integral_const,integral_div,projectionSquare_sphere_joint_integral] simp only [probReal_univ,one_smul] ring lemma sphere_projection_reference_finite {d m r : ℕ} (hdim : AdmissibleDimensions d m r 1) (hd : 2 ≤ d) : klDiv (projectionJoint (uniformSphere d) (gaussianRows (m+r) d)) ((gaussianRows (m+r) d).prod (gaussianLabels (m+r))) ≠ ∞ ∧ (klDiv (projectionJoint (uniformSphere d) (gaussianRows (m+r) d)) ((gaussianRows (m+r) d).prod (gaussianLabels (m+r)))).toReal ≤ 20*d + (m+r:ℕ)*(Real.log (normalizer 1)+1/2) := by let : IsProbabilityMeasure (uniformSphere d) := uniformSphere_probability_of_pos (by omega) let P := projectionJoint (uniformSphere d) (gaussianRows (m+r) d) let Q := (gaussianRows (m+r) d).prod (gaussianLabels (m+r)) let Λ := (gaussianRows (m+r) d).prod (volume : Measure (Fin (m+r)→ℝ)) let p : Rows (m+r) d×(Fin (m+r)→ℝ) → ℝ≥0∞ := fun z => exactDensity (uniformSphere d) z.1 z.2 let g : Rows (m+r) d×(Fin (m+r)→ℝ) → ℝ := fun z => gaussianProductPDF 1 0 z.2 have hp : Measurable p := measurable_exactDensity _ have hg : Measurable g := by unfold g gaussianProductPDF; fun_prop have hQ : Λ.withDensity (fun z => ENNReal.ofReal (g z)) = Q := by have hs : Measurable (fun y : Fin (m+r)→ℝ => ENNReal.ofReal (gaussianProductPDF 1 0 y)) := by unfold gaussianProductPDF; fun_prop dsimp [Λ,g,Q] rw [←prod_withDensity_right hs,gaussianLabels_density] have hB : (0:ℝ) < 3*4^d := by positivity have hAC := ae_labelLaw_ac_of_growth hdim (uniformSphere d) _ _ hB hB (uniformSphere_growth (show 1 ≤ d by omega) (show m+r ≤ d-1 by have := hdim.2.2.2.1; omega)) (uniformSphere_growth (show 1 ≤ d by omega) (le_refl (d-1))) obtain ⟨hPQ,he⟩ := llr_of_two_densities P Q Λ p hp (projectionJoint_withDensity _ _ hAC) g hg (fun z => gaussianProductPDF_pos (by norm_num) _ _) hQ obtain ⟨hpi,hpb⟩ := sphere_label_entropy_finite hdim hd have hpm : Measurable (fun z : Rows (m+r) d×Sphere d => (z.1,label z.1 (z.2:Vector d))) := measurable_fst.prodMk measurable_own_label have hpi' : Integrable (fun z => Real.log (p z).toReal) P := (integrable_map_measure hp.ennreal_toReal.log.aestronglyMeasurable hpm.aemeasurable).mpr hpi obtain ⟨hgi,hge⟩ := gaussian_log_reference_projection (k:=m+r) (uniformSphere d) have hll : Integrable (llr P Q) P := (hpi'.sub hgi).congr he.symm refine ⟨klDiv_ne_top hPQ hll,?_⟩ rw [toReal_klDiv_of_measure_eq hPQ (by simp),integral_congr_ae he,integral_sub hpi' hgi,hge] have hpint : (∫ z, Real.log (p z).toReal ∂P) = ∫ z : Rows (m+r) d×Sphere d, Real.log (exactDensity (uniformSphere d) z.1 (label z.1 (z.2:Vector d))).toReal ∂(gaussianRows (m+r) d).prod (uniformSphere d) := integral_map hpm.aemeasurable hp.ennreal_toReal.log.aestronglyMeasurable rw [hpint] linarith lemma weighted_log_integral {T : Type*} [MeasurableSpace T] (μ : Measure T) (w g : T → ℝ) (hw : ∀ t, 0 < w t) (hg : ∀ t, 0 < g t) (hwi : Integrable w μ) (hgi : Integrable g μ) (hli : Integrable (fun t => w t * (Real.log (g t) - Real.log (w t))) μ) (hA : 0 < ∫ t, w t ∂μ) (hM : 0 < ∫ t, g t ∂μ) : (∫ t, w t * (Real.log (g t) - Real.log (w t)) ∂μ) ≤ (∫ t, w t ∂μ) * (Real.log (∫ t, g t ∂μ) - Real.log (∫ t, w t ∂μ)) := by let A := ∫ t, w t ∂μ let M := ∫ t, g t ∂μ change 0 < A at hA change 0 < M at hM have hp (t : T) : w t * (Real.log (g t) - Real.log (w t)) ≤ w t * (Real.log M - Real.log A) + (A/M)*g t - w t := by have hh := Real.log_le_sub_one_of_pos (div_pos (div_pos (hg t) (hw t)) (div_pos hM hA)) rw [Real.log_div (div_ne_zero (hg t).ne' (hw t).ne') (div_ne_zero hM.ne' hA.ne'), Real.log_div (hg t).ne' (hw t).ne',Real.log_div hM.ne' hA.ne'] at hh have hh' := mul_le_mul_of_nonneg_left hh (hw t).le have he : w t * ((g t / w t) / (M / A) - 1) = (A/M)*g t - w t := by field_simp [hA.ne',hM.ne',(hw t).ne'] rw [he] at hh' nlinarith have hi : Integrable (fun t => w t*(Real.log M-Real.log A)+(A/M)*g t-w t) μ := ((hwi.mul_const _).add (hgi.const_mul (A/M))).sub hwi have hb := integral_mono hli hi hp have hv : (∫ t, w t * (Real.log M-Real.log A) + (A/M)*g t - w t ∂μ) = A * (Real.log M-Real.log A) + (A/M)*M - A := by calc _ = (∫ t, w t * (Real.log M-Real.log A) + (A/M)*g t ∂μ) - A := integral_sub ((hwi.mul_const _).add (hgi.const_mul _)) hwi _ = A * (Real.log M-Real.log A) + (A/M)*M - A := by rw [integral_add (hwi.mul_const (Real.log M-Real.log A)) (hgi.const_mul (A/M))] simp only [integral_mul_const, integral_const_mul, A, M] rw [hv,div_mul_cancel₀ _ hM.ne'] at hb linarith def regularizedReciprocal {d : ℕ} (e : ℝ) (s t : Sphere d) : ℝ := (max (dist s t) e)⁻¹ def regularizedScale {d : ℕ} (μ : Measure (Sphere d)) (e : ℝ) (s : Sphere d) : ℝ := ∫ t, regularizedReciprocal e s t ∂μ lemma sphere_subtype_dist_le_two {d : ℕ} (s t : Sphere d) : dist s t ≤ 2 := by simpa only [Subtype.dist_eq,dist_eq_norm,norm_eq_of_mem_sphere s, norm_eq_of_mem_sphere t,one_add_one_eq_two] using norm_sub_le (s:Vector d) (t:Vector d) lemma regularizedReciprocal_pos {d : ℕ} {e : ℝ} (he : 0 < e) (s t : Sphere d) : 0 < regularizedReciprocal e s t := inv_pos.mpr (he.trans_le (le_max_right _ _)) lemma regularizedReciprocal_bounds {d : ℕ} {e : ℝ} (he : 0 < e) (he1 : e ≤ 1) (s t : Sphere d) : 1/2 ≤ regularizedReciprocal e s t ∧ regularizedReciprocal e s t ≤ e⁻¹ := by have hmax : 0 < max (dist s t) e := he.trans_le (le_max_right _ _) constructor · simp only [one_div,regularizedReciprocal] change (2:ℝ)⁻¹ ≤ (max (dist s t) e)⁻¹ exact (inv_le_inv₀ (by norm_num) hmax).mpr (max_le (sphere_subtype_dist_le_two s t) (by linarith)) · exact (inv_le_inv₀ hmax he).mpr (le_max_right _ _) lemma integrable_regularizedReciprocal {d : ℕ} (μ : Measure (Sphere d)) [IsProbabilityMeasure μ] {e : ℝ} (he : 0 < e) (s : Sphere d) : Integrable (regularizedReciprocal e s) μ := by apply (integrable_const (e⁻¹)).mono' (by unfold regularizedReciprocal; fun_prop) apply ae_of_all intro t rw [Real.norm_eq_abs,abs_of_pos (regularizedReciprocal_pos he s t)] exact (inv_le_inv₀ (he.trans_le (le_max_right _ _)) he).mpr (le_max_right _ _) lemma regularizedScale_bounds {d : ℕ} (μ : Measure (Sphere d)) [IsProbabilityMeasure μ] {e : ℝ} (he : 0 < e) (he1 : e ≤ 1) (s : Sphere d) : 1/2 ≤ regularizedScale μ e s ∧ regularizedScale μ e s ≤ e⁻¹ := by have hi := integrable_regularizedReciprocal μ he s constructor · simpa [regularizedScale] using integral_mono (integrable_const (1/2:ℝ)) hi (fun t => (regularizedReciprocal_bounds he he1 s t).1) · simpa [regularizedScale] using integral_mono hi (integrable_const (e⁻¹)) (fun t => (regularizedReciprocal_bounds he he1 s t).2) lemma log_regularizedReciprocal {d : ℕ} (e : ℝ) (s t : Sphere d) : Real.log (regularizedReciprocal e s t) = -Real.log (max (dist s t) e) := Real.log_inv _ end CurrentProjection end end OAI