| import Mathlib | |
| import OAI.Probability.SKRatio.Variational.ScalarCoherent | |
| import OAI.Probability.SKRatio.Variational.ScalarTemperature | |
| namespace OAI | |
| noncomputable section | |
| open scoped Topology ENNReal NNReal | |
| open MeasureTheory ProbabilityTheory Real | |
| namespace SKRatio.Scalar | |
| section Norm | |
| variable {μ : Measure ℝ} | |
| lemma l2Norm_add {f g : ℝ → ℝ} (hf : MemLp f 2 μ) (hg : MemLp g 2 μ) : | |
| l2Norm μ (fun x => f x+g x) ≤ l2Norm μ f+l2Norm μ g := by | |
| have he := integral_square_add hf hg 1 1 | |
| simp only [one_mul,one_pow,mul_one] at he | |
| have hcs := (le_abs_self _).trans (abs_integral_mul_le hf hg) | |
| apply (sq_le_sq₀ (l2Norm_nonneg _) (add_nonneg (l2Norm_nonneg _) (l2Norm_nonneg _))).mp | |
| rw [l2Norm_sq,he] | |
| nlinarith only [hcs,l2Norm_sq (μ := μ) f,l2Norm_sq (μ := μ) g] | |
| lemma l2Norm_const_mul (f : ℝ → ℝ) (c : ℝ) : | |
| l2Norm μ (fun x => c*f x) = |c| *l2Norm μ f := by | |
| apply (sq_eq_sq₀ (l2Norm_nonneg _) (mul_nonneg (abs_nonneg _) (l2Norm_nonneg _))).mp | |
| simp only [l2Norm_sq,mul_pow,integral_const_mul,sq_abs] | |
| end Norm | |
| section Probability | |
| variable {μ ν : Measure ℝ} [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] | |
| omit [IsProbabilityMeasure μ] in | |
| lemma sd_sq (f : ℝ → ℝ) : sd μ f^2 = ∫ x, centered μ f x^2 ∂μ := l2Norm_sq _ | |
| lemma sd_sq_eq {f : ℝ → ℝ} (hf : MemLp f 2 μ) : | |
| sd μ f^2 = (∫ x, f x^2 ∂μ)-(∫ x, f x ∂μ)^2 := by | |
| rw [sd_sq] | |
| exact integral_centered_sq hf | |
| lemma square_center_expansion {f : ℝ → ℝ} (hf : MemLp f 2 μ) (c : ℝ) : | |
| ∫ x, (f x-c)^2 ∂μ = sd μ f^2+((∫ x, f x ∂μ)-c)^2 := by | |
| have he := integral_square_add hf (memLp_const (1:ℝ)) 1 (-c) | |
| simp only [one_mul,one_pow,mul_one,one_mul,←sub_eq_add_neg,neg_sq, | |
| integral_const,probReal_univ,smul_eq_mul,one_mul] at he | |
| rw [he,sd_sq_eq hf] | |
| ring | |
| lemma sd_min_center {f : ℝ → ℝ} (hf : MemLp f 2 μ) (c : ℝ) : | |
| sd μ f^2 ≤ ∫ x, (f x-c)^2 ∂μ := by | |
| rw [square_center_expansion hf c] | |
| exact le_add_of_nonneg_right (sq_nonneg _) | |
| omit [IsProbabilityMeasure μ] in | |
| lemma centered_add {f g : ℝ → ℝ} (hf : Integrable f μ) (hg : Integrable g μ) : | |
| centered μ (fun x => f x+g x) = fun x => centered μ f x+centered μ g x := by | |
| ext x | |
| simp only [centered,integral_add hf hg] | |
| ring | |
| omit [IsProbabilityMeasure μ] in | |
| lemma centered_const_mul (f : ℝ → ℝ) (c : ℝ) : | |
| centered μ (fun x => c*f x) = fun x => c*centered μ f x := by | |
| ext x | |
| simp only [centered,integral_const_mul] | |
| ring | |
| lemma centered_sub_const {f : ℝ → ℝ} (hf : Integrable f μ) (c : ℝ) : | |
| centered μ (fun x => f x-c) = centered μ f := by | |
| ext x | |
| simp only [centered,integral_sub hf (integrable_const _),integral_const, | |
| probReal_univ,smul_eq_mul,one_mul] | |
| ring | |
| lemma sd_add {f g : ℝ → ℝ} (hf : MemLp f 2 μ) (hg : MemLp g 2 μ) : | |
| sd μ (fun x => f x+g x) ≤ sd μ f+sd μ g := by | |
| unfold sd | |
| rw [centered_add (hf.integrable (by norm_num)) (hg.integrable (by norm_num))] | |
| exact l2Norm_add (memLp_centered hf) (memLp_centered hg) | |
| omit [IsProbabilityMeasure μ] in | |
| lemma sd_const_mul (f : ℝ → ℝ) (c : ℝ) : | |
| sd μ (fun x => c*f x) = |c| *sd μ f := by | |
| unfold sd | |
| rw [centered_const_mul,l2Norm_const_mul] | |
| lemma sd_sub_const {f : ℝ → ℝ} (hf : MemLp f 2 μ) (c : ℝ) : | |
| sd μ (fun x => f x-c) = sd μ f := by | |
| unfold sd | |
| rw [centered_sub_const (hf.integrable (by norm_num))] | |
| lemma sd_sub {f g : ℝ → ℝ} (hf : MemLp f 2 μ) (hg : MemLp g 2 μ) : | |
| sd μ (fun x => f x-g x) ≤ sd μ f+sd μ g := by | |
| have he (x : ℝ) : f x-g x = f x+(-1)*g x := by ring | |
| simp_rw [he] | |
| have hh := sd_add hf (hg.const_mul (-1)) | |
| simpa only [sd_const_mul,abs_neg,abs_one,one_mul] using hh | |
| lemma abs_sd_sub_le {f g : ℝ → ℝ} (hf : MemLp f 2 μ) (hg : MemLp g 2 μ) : | |
| |sd μ f-sd μ g| ≤ sd μ (fun x => f x-g x) := by | |
| have h1 := sd_add (hf.sub hg) hg | |
| have h2 := sd_sub hf (hf.sub hg) | |
| simp only [Pi.sub_apply,sub_add_cancel,sub_sub_cancel] at h1 h2 | |
| change sd μ f ≤ sd μ (fun x => f x-g x)+sd μ g at h1 | |
| change sd μ g ≤ sd μ f+sd μ (fun x => f x-g x) at h2 | |
| exact abs_le.mpr ⟨by linarith,by linarith⟩ | |
| lemma sd_mono_measure {C : ℝ} (hC : 0 ≤ C) | |
| (hμν : μ ≤ ENNReal.ofReal C • ν) | |
| {f : ℝ → ℝ} (hf : MemLp f 2 μ) (hf' : MemLp f 2 ν) : | |
| sd μ f^2 ≤ C*sd ν f^2 := by | |
| calc | |
| sd μ f^2 ≤ ∫ x, (f x-(∫ y, f y ∂ν))^2 ∂μ := sd_min_center hf _ | |
| _ ≤ ∫ x, (f x-(∫ y, f y ∂ν))^2 ∂(ENNReal.ofReal C • ν) := | |
| integral_mono_measure hμν (ae_of_all _ (fun _ => sq_nonneg _)) | |
| ((memLp_centered hf').integrable_sq.smul_measure ENNReal.ofReal_ne_top) | |
| _ = C*sd ν f^2 := by | |
| rw [integral_smul_measure,ENNReal.toReal_ofReal hC,smul_eq_mul,sd_sq] | |
| rfl | |
| lemma sd_mono_measure_sqrt {C : ℝ} (hC : 0 ≤ C) | |
| (hμν : μ ≤ ENNReal.ofReal C • ν) | |
| {f : ℝ → ℝ} (hf : MemLp f 2 μ) (hf' : MemLp f 2 ν) : | |
| sd μ f ≤ sqrt C*sd ν f := by | |
| apply (sq_le_sq₀ (sd_nonneg _) (mul_nonneg (sqrt_nonneg _) (sd_nonneg _))).mp | |
| rw [mul_pow,sq_sqrt hC] | |
| exact sd_mono_measure hC hμν hf hf' | |
| lemma weighted_sd_mono_measure {B Z : ℝ} (hB : 0 ≤ B) (hZ : 0 ≤ Z) | |
| (hμν : ENNReal.ofReal B • μ ≤ ENNReal.ofReal Z • ν) | |
| {f : ℝ → ℝ} (hf : MemLp f 2 μ) (hf' : MemLp f 2 ν) : | |
| B*sd μ f^2 ≤ Z*sd ν f^2 := by | |
| calc | |
| B*sd μ f^2 ≤ B*(∫ x, (f x-(∫ y, f y ∂ν))^2 ∂μ) := | |
| mul_le_mul_of_nonneg_left (sd_min_center hf _) hB | |
| _ = ∫ x, (f x-(∫ y, f y ∂ν))^2 ∂(ENNReal.ofReal B • μ) := by | |
| rw [integral_smul_measure,ENNReal.toReal_ofReal hB,smul_eq_mul] | |
| _ ≤ ∫ x, (f x-(∫ y, f y ∂ν))^2 ∂(ENNReal.ofReal Z • ν) := | |
| integral_mono_measure hμν (ae_of_all _ (fun _ => sq_nonneg _)) | |
| ((memLp_centered hf').integrable_sq.smul_measure ENNReal.ofReal_ne_top) | |
| _ = Z*sd ν f^2 := by | |
| rw [integral_smul_measure,ENNReal.toReal_ofReal hZ,smul_eq_mul,sd_sq] | |
| rfl | |
| end Probability | |
| lemma weighted_sd_temperature_mono {β z : ℝ} (hβ : 0 < β) (hβz : β ≤ z) (hz : z ≤ 1) | |
| {f : ℝ → ℝ} (hf : MemLp f 2 (fieldLaw β)) (hf' : MemLp f 2 (fieldLaw z)) : | |
| β^2*sd (fieldLaw β) f^2 ≤ z^2*sd (fieldLaw z) f^2 := | |
| weighted_sd_mono_measure (sq_nonneg _) (sq_nonneg _) | |
| (scaled_fieldLaw_mono hβ hβz hz) hf hf' | |
| lemma weighted_sd_temperature_mono' {β z : ℝ} (hβ : 0 < β) (hβz : β ≤ z) (hz : z ≤ 1) | |
| {f : ℝ → ℝ} (hf : MemLp f 2 (fieldLaw β)) (hf' : MemLp f 2 (fieldLaw z)) : | |
| β*sd (fieldLaw β) f ≤ z*sd (fieldLaw z) f := by | |
| apply (sq_le_sq₀ (mul_nonneg hβ.le (sd_nonneg _)) | |
| (mul_nonneg (hβ.le.trans hβz) (sd_nonneg _))).mp | |
| simpa only [mul_pow] using weighted_sd_temperature_mono hβ hβz hz hf hf' | |
| lemma weighted_mv_temperature_mono {β z : ℝ} (hβ : 0 < β) (hβz : β ≤ z) (hz : z ≤ 1) : | |
| β^2*(∫ h, m h*v h ∂fieldLaw β) ≤ z^2*(∫ h, m h*v h ∂fieldLaw z) := by | |
| simp_rw [mean_mv_eq_mean_m_sq_v] | |
| apply scaled_integral_mono hβ hβz hz (fun h => mul_nonneg (sq_nonneg _) (v_pos h).le) | |
| apply (integrable_const (1:ℝ)).mono' | |
| ((continuous_m.pow 2).mul continuous_v).aestronglyMeasurable | |
| apply ae_of_all | |
| intro h | |
| change |m h^2*v h| ≤ 1 | |
| rw [abs_of_nonneg (mul_nonneg (sq_nonneg _) (v_pos h).le)] | |
| exact (mul_le_mul_of_nonneg_right (tanh_sq_lt_one h).le (v_pos h).le).trans (by simpa using v_le_one h) | |
| lemma weighted_R_temperature_mono {β z : ℝ} (hβ : 0 < β) (hβz : β ≤ z) (hz : z ≤ 1) : | |
| β^2*R β ≤ z^2*R z := by | |
| apply scaled_integral_mono hβ hβz hz rho_nonneg | |
| exact integrable_rho | |
| end SKRatio.Scalar | |
| end | |
| end OAI | |