| import Mathlib |
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| namespace OAI |
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| noncomputable section |
| open Set Filter MeasureTheory ProbabilityTheory |
| open scoped Topology ENNReal |
| open MeasureTheory ProbabilityTheory |
| open scoped RealInnerProductSpace |
| namespace YauCounterexamples |
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| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
|
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| theorem stdGaussian_map_inner_unit (a : E) (ha : ‖a‖ = 1) : |
| (stdGaussian E).map (fun x => ⟪a, x⟫) = gaussianReal 0 1 := by |
| change (stdGaussian E).map (innerSL ℝ a) = _ |
| rw [IsGaussian.map_eq_gaussianReal, integral_strongDual_stdGaussian, |
| variance_dual_stdGaussian] |
| simp [ha] |
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| theorem stdGaussian_map_orthonormal_pair (a c : E) (ha : ‖a‖ = 1) (hc : ‖c‖ = 1) |
| (hac : ⟪a, c⟫ = 0) : |
| (stdGaussian E).map (fun x => (⟪a, x⟫, ⟪c, x⟫)) = |
| (gaussianReal 0 1).prod (gaussianReal 0 1) := by |
| have hlaw : HasGaussianLaw (fun x : E => (⟪a, x⟫, ⟪c, x⟫)) (stdGaussian E) := by |
| exact (IsGaussian.hasGaussianLaw_id (μ := stdGaussian E)).map_fun |
| ((innerSL ℝ a).prod (innerSL ℝ c)) |
| have hcov : cov[fun x : E => ⟪a, x⟫, fun x : E => ⟪c, x⟫; stdGaussian E] = 0 := by |
| rw [← covarianceBilin_apply_eq_cov IsGaussian.memLp_two_id, |
| covarianceBilin_stdGaussian] |
| exact hac |
| rw [(hlaw.indepFun_of_covariance_eq_zero hcov).map_prod_eq_prod_map_map |
| (by fun_prop) (by fun_prop), stdGaussian_map_inner_unit a ha, |
| stdGaussian_map_inner_unit c hc] |
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| omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in |
|
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| theorem correlated_pair_orthogonal (a b : E) (ha : ‖a‖ = 1) (hb : ‖b‖ = 1) |
| (hρ : |⟪a, b⟫| < 1) : |
| let ρ := ⟪a, b⟫ |
| let s := Real.sqrt (1 - ρ ^ 2) |
| let c := s⁻¹ • (b - ρ • a) |
| ‖c‖ = 1 ∧ ⟪a, c⟫ = 0 ∧ b = ρ • a + s • c := by |
| dsimp only |
| let ρ := ⟪a, b⟫ |
| let s := Real.sqrt (1 - ρ ^ 2) |
| have hr : 0 < 1 - ρ ^ 2 := by |
| dsimp only [ρ] |
| nlinarith [abs_lt.mp hρ] |
| have hs : 0 < s := Real.sqrt_pos.mpr hr |
| have hs2 : s ^ 2 = 1 - ρ ^ 2 := Real.sq_sqrt hr.le |
| have haa : ⟪a, a⟫ = 1 := by simp [ha] |
| have hbb : ⟪b, b⟫ = 1 := by simp [hb] |
| have hnorm : ‖b - ρ • a‖ ^ 2 = s ^ 2 := by |
| rw [← real_inner_self_eq_norm_sq] |
| simp only [inner_sub_left, inner_sub_right, real_inner_smul_left, |
| real_inner_smul_right, haa, hbb] |
| rw [real_inner_comm a b] |
| change 1 - ρ * ρ - (ρ * ρ - ρ * (ρ * 1)) = s ^ 2 |
| nlinarith |
| have hnorm' : ‖b - ρ • a‖ = s := by |
| nlinarith [norm_nonneg (b - ρ • a)] |
| refine ⟨?_, ?_, ?_⟩ |
| · change ‖s⁻¹ • (b - ρ • a)‖ = 1 |
| rw [norm_smul, Real.norm_eq_abs, abs_inv, abs_of_pos hs, hnorm', inv_mul_cancel₀ hs.ne'] |
| · change ⟪a, s⁻¹ • (b - ρ • a)⟫ = 0 |
| simp only [real_inner_smul_right, inner_sub_right, haa] |
| change s⁻¹ * (ρ - ρ * 1) = 0 |
| ring |
| · change b = ρ • a + s • (s⁻¹ • (b - ρ • a)) |
| rw [smul_smul, mul_inv_cancel₀ hs.ne', one_smul] |
| abel |
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|
| end YauCounterexamples |
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| end |
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| end OAI |
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