| import Mathlib |
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| namespace OAI |
|
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| section |
| noncomputable section |
| open scoped BigOperators Topology |
| open MeasureTheory ProbabilityTheory Real |
| namespace SKRatio.Scalar |
|
|
| def m (h : ℝ) : ℝ := tanh h |
| def v (h : ℝ) : ℝ := 1 - m h ^ 2 |
| def w (h : ℝ) : ℝ := 1 - m h |
| def kernel (h t : ℝ) : ℝ := v h * v t / (w h + w t) |
| def g (h : ℝ) : ℝ := v h / (w h + 3/4) |
| def q (h : ℝ) : ℝ := (w h - 3/4) / (w h + 3/4) |
| def rho (h : ℝ) : ℝ := (1+m h)*v h/2 - (3/2)*g h^2 |
|
|
| lemma m_bounds (h : ℝ) : -1 < m h ∧ m h < 1 := |
| ⟨neg_one_lt_tanh h, tanh_lt_one h⟩ |
| lemma v_pos (h : ℝ) : 0 < v h := sub_pos.mpr (tanh_sq_lt_one h) |
| lemma v_le_one (h : ℝ) : v h ≤ 1 := by dsimp [v]; nlinarith [sq_nonneg (m h)] |
| lemma w_pos (h : ℝ) : 0 < w h := sub_pos.mpr (m_bounds h).2 |
| lemma v_factor (h : ℝ) : v h = w h * (1+m h) := by unfold v w; ring |
| lemma kernel_nonneg (h t : ℝ) : 0 ≤ kernel h t := |
| div_nonneg (mul_nonneg (v_pos h).le (v_pos t).le) (add_pos (w_pos h) (w_pos t)).le |
| lemma kernel_diag (h : ℝ) : kernel h h = (1+m h)*v h/2 := by |
| unfold kernel |
| nth_rw 1 [v_factor h] |
| have hw := (w_pos h).ne' |
| field_simp |
| ring |
|
|
| lemma kernel_diag_le_one (h : ℝ) : kernel h h ≤ 1 := by |
| rw [kernel_diag] |
| have hm := (m_bounds h).2.le |
| have hv := v_le_one h |
| have hv0 := (v_pos h).le |
| nlinarith [(mul_le_mul_of_nonneg_right (show 1+m h ≤ 2 by linarith) hv0)] |
|
|
| lemma kernel_square_le (h t : ℝ) : kernel h t^2 ≤ kernel h h * kernel t t := by |
| have ha := w_pos h |
| have hb := w_pos t |
| have hab : 0 < w h+w t := add_pos ha hb |
| unfold kernel |
| rw [div_pow,div_mul_div_comm] |
| apply (div_le_div_iff₀ (sq_pos_of_pos hab) (mul_pos (add_pos ha ha) (add_pos hb hb))).mpr |
| have hh := mul_nonneg (sq_nonneg (v h*v t)) (sq_nonneg (w h-w t)) |
| nlinarith only [hh] |
|
|
| lemma abs_kernel_le_one (h t : ℝ) : |kernel h t| ≤ 1 := by |
| rw [abs_of_nonneg (kernel_nonneg h t)] |
| have hs := kernel_square_le h t |
| have hd := mul_le_mul (kernel_diag_le_one h) (kernel_diag_le_one t) |
| (kernel_nonneg t t) (by norm_num : (0:ℝ) ≤ 1) |
| nlinarith [sq_nonneg (kernel h t-1)] |
|
|
| lemma residual_identity (h t : ℝ) : |
| kernel h t - (3/2)*g h*g t = kernel h t*q h*q t := by |
| have hht := (add_pos (w_pos h) (w_pos t)).ne' |
| have hh : w h+3/4 ≠ 0 := by linarith [w_pos h] |
| have ht : w t+3/4 ≠ 0 := by linarith [w_pos t] |
| unfold kernel g q |
| have hh4 : w h*4+3 ≠ 0 := by linarith [w_pos h] |
| have ht4 : w t*4+3 ≠ 0 := by linarith [w_pos t] |
| field_simp [hht,hh4,ht4] |
| ring |
|
|
| lemma rho_eq (h : ℝ) : rho h = kernel h h * q h^2 := by |
| have hh := residual_identity h h |
| rw [kernel_diag] at hh |
| unfold rho |
| rw [kernel_diag] |
| nlinarith only [hh] |
|
|
| lemma rho_nonneg (h : ℝ) : 0 ≤ rho h := by |
| rw [rho_eq] |
| exact mul_nonneg (kernel_nonneg h h) (sq_nonneg (q h)) |
|
|
| lemma rho_le_one (h : ℝ) : rho h ≤ 1 := by |
| have hi := kernel_diag_le_one h |
| rw [kernel_diag] at hi |
| unfold rho |
| nlinarith [sq_nonneg (g h)] |
|
|
| lemma abs_g_le_one (h : ℝ) : |g h| ≤ 1 := by |
| have hi := kernel_diag_le_one h |
| rw [kernel_diag] at hi |
| have hr := rho_nonneg h |
| unfold rho at hr |
| rw [abs_le] |
| constructor <;> nlinarith [sq_nonneg (g h+1),sq_nonneg (g h-1)] |
|
|
| lemma abs_residual_le (h t : ℝ) : |
| |kernel h t - (3/2)*g h*g t| ≤ sqrt (rho h)*sqrt (rho t) := by |
| have hs := mul_le_mul_of_nonneg_right (kernel_square_le h t) |
| (mul_nonneg (sq_nonneg (q h)) (sq_nonneg (q t))) |
| have hsq : (kernel h t - (3/2)*g h*g t)^2 ≤ rho h*rho t := by |
| rw [residual_identity,rho_eq h,rho_eq t] |
| nlinarith only [hs] |
| have hprod : 0 ≤ sqrt (rho h)*sqrt (rho t) := mul_nonneg (sqrt_nonneg _) (sqrt_nonneg _) |
| apply (sq_le_sq₀ (abs_nonneg _) hprod).mp |
| rw [sq_abs,mul_pow,sq_sqrt (rho_nonneg h),sq_sqrt (rho_nonneg t)] |
| exact hsq |
|
|
| lemma continuous_m : Continuous m := by |
| unfold m |
| simp_rw [tanh_eq_sinh_div_cosh] |
| exact Continuous.div Real.continuous_sinh Real.continuous_cosh (fun h => ne_of_gt (Real.cosh_pos h)) |
| lemma continuous_v : Continuous v := continuous_const.sub (continuous_m.pow 2) |
| lemma continuous_w : Continuous w := continuous_const.sub continuous_m |
| lemma continuous_g : Continuous g := by |
| apply continuous_v.div (continuous_w.add continuous_const) |
| intro h |
| change w h+3/4 ≠ 0 |
| linarith [w_pos h] |
| lemma continuous_kernel : Continuous (Function.uncurry kernel) := by |
| apply ((continuous_v.comp continuous_fst).mul (continuous_v.comp continuous_snd)).div |
| ((continuous_w.comp continuous_fst).add (continuous_w.comp continuous_snd)) |
| intro x |
| exact (add_pos (w_pos x.1) (w_pos x.2)).ne' |
| lemma continuous_rho : Continuous rho := |
| (((continuous_const.add continuous_m).mul continuous_v).div_const 2).sub |
| ((continuous_g.pow 2).const_mul (3/2)) |
|
|
| section Integration |
| variable {μ : Measure ℝ} [IsProbabilityMeasure μ] |
|
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| lemma memLp_g : MemLp g 2 μ := by |
| apply (memLp_const (1:ℝ)).of_le_mul continuous_g.aestronglyMeasurable |
| (c := (1:ℝ)) |
| exact Filter.Eventually.of_forall (fun h => by simpa using abs_g_le_one h) |
|
|
| lemma memLp_sqrt_rho : MemLp (fun h => sqrt (rho h)) 2 μ := by |
| apply (memLp_const (1:ℝ)).of_le_mul |
| (continuous_sqrt.comp continuous_rho).aestronglyMeasurable (c := (1:ℝ)) |
| apply Filter.Eventually.of_forall |
| intro h |
| simp only [Function.comp_apply,Real.norm_eq_abs,abs_one,mul_one,abs_of_nonneg (sqrt_nonneg _)] |
| exact (sqrt_le_one).mpr (rho_le_one h) |
|
|
| lemma integrable_rho : Integrable rho μ := by |
| exact ((integrable_const (1:ℝ)).mono' continuous_rho.aestronglyMeasurable |
| (Filter.Eventually.of_forall (fun h => by |
| simpa only [Real.norm_eq_abs,abs_of_nonneg (rho_nonneg h)] using rho_le_one h))) |
|
|
| lemma integral_abs_mul_sq_le {α : Type*} [MeasurableSpace α] {ν : Measure α} |
| {f z : α → ℝ} (hf : MemLp f 2 ν) (hz : MemLp z 2 ν) : |
| (∫ x, |f x| *|z x| ∂ν)^2 ≤ (∫ x, f x^2 ∂ν)*(∫ x, z x^2 ∂ν) := by |
| have hb := integral_mul_norm_le_Lp_mul_Lq Real.HolderConjugate.two_two |
| (by simpa using hf) (by simpa using hz) |
| simp only [Real.rpow_two,Real.norm_eq_abs,sq_abs,←Real.sqrt_eq_rpow] at hb |
| have ha : 0 ≤ ∫ x, |f x| *|z x| ∂ν := integral_nonneg (fun x => mul_nonneg (abs_nonneg _) (abs_nonneg _)) |
| have hs := sq_le_sq₀ ha (mul_nonneg (sqrt_nonneg _) (sqrt_nonneg _)) |>.mpr hb |
| simpa only [mul_pow,sq_sqrt (integral_nonneg (fun x => sq_nonneg (f x))), |
| sq_sqrt (integral_nonneg (fun x => sq_nonneg (z x)))] using hs |
|
|
| lemma kernel_pair_integrable {b : ℝ → ℝ} (hb : MemLp b 2 μ) : |
| Integrable (fun z : ℝ×ℝ => b z.1 * kernel z.1 z.2 * b z.2) (μ.prod μ) := by |
| have hI := hb.integrable (by norm_num : (1:ENNReal) ≤ 2) |
| apply (hI.norm.mul_prod hI.norm).mono' |
| ((hb.aestronglyMeasurable.comp_fst.mul continuous_kernel.aestronglyMeasurable).mul |
| hb.aestronglyMeasurable.comp_snd) |
| apply Filter.Eventually.of_forall |
| intro z |
| change |b z.1*kernel z.1 z.2*b z.2| ≤ |b z.1| * |b z.2| |
| simp only [abs_mul] |
| calc |
| |b z.1| *|kernel z.1 z.2| *|b z.2| ≤ |b z.1| *1*|b z.2| := by |
| gcongr |
| exact abs_kernel_le_one _ _ |
| _ = |b z.1| *|b z.2| := by ring |
|
|
| theorem kernel_bound {b : ℝ → ℝ} (hb : MemLp b 2 μ) : |
| ∫ z : ℝ×ℝ, b z.1*kernel z.1 z.2*b z.2 ∂μ.prod μ ≤ |
| (3/2)*(∫ h, b h*g h ∂μ)^2 + |
| (∫ h, rho h ∂μ)*(∫ h, b h^2 ∂μ) := by |
| have hbg : Integrable (fun h => b h*g h) μ := hb.integrable_mul memLp_g |
| have hbr : Integrable (fun h => |b h| *sqrt (rho h)) μ := |
| hb.norm.integrable_mul memLp_sqrt_rho |
| have hmain := (hbg.mul_prod hbg).const_mul (3/2:ℝ) |
| have hrem := hbr.mul_prod hbr |
| have him := integral_mono (kernel_pair_integrable hb) (hmain.add hrem) |
| (fun z => show b z.1*kernel z.1 z.2*b z.2 ≤ |
| (3/2)*((b z.1*g z.1)*(b z.2*g z.2)) + |
| (|b z.1| *sqrt (rho z.1))*(|b z.2| *sqrt (rho z.2)) from by |
| have ha := mul_le_mul_of_nonneg_left (abs_residual_le z.1 z.2) |
| (mul_nonneg (abs_nonneg (b z.1)) (abs_nonneg (b z.2))) |
| have hh := le_abs_self (b z.1*(kernel z.1 z.2-(3/2)*g z.1*g z.2)*b z.2) |
| simp only [abs_mul] at hh |
| nlinarith only [ha,hh]) |
| simp only [Pi.add_apply] at him |
| rw [integral_add hmain hrem,integral_const_mul, |
| integral_prod_mul (fun h => b h*g h) (fun h => b h*g h), |
| integral_prod_mul (fun h => |b h| * sqrt (rho h)) (fun h => |b h| * sqrt (rho h))] at him |
| have hcs := integral_abs_mul_sq_le hb (memLp_sqrt_rho (μ := μ)) |
| simp only [abs_of_nonneg (sqrt_nonneg _),sq_sqrt (rho_nonneg _)] at hcs |
| nlinarith only [him,hcs] |
|
|
| omit [IsProbabilityMeasure μ] in |
| lemma integral_rho_nonneg : 0 ≤ ∫ h, rho h ∂μ := integral_nonneg rho_nonneg |
|
|
| end Integration |
| end SKRatio.Scalar |
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| end |
| end |
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| end OAI |
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