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695bd70 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 | import Mathlib
namespace OAI
noncomputable section
open Set Filter MeasureTheory ProbabilityTheory
open scoped Topology ENNReal
open MeasureTheory ProbabilityTheory
open scoped RealInnerProductSpace
namespace YauCounterexamples
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
[FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
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]
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]
omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in
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
end YauCounterexamples
end
end OAI
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