| import OAI.Analysis.FreeEntropy.PolynomialFunctions |
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| namespace OAI |
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| universe u1 u2 u3 u4 |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory Finset |
| namespace FiniteEntropySeparation |
| variable {ι : Type u1} [Fintype ι] [DecidableEq ι] |
| def IsComplexPolynomial (f : (ι → ℝ) → ℂ) : Prop := |
| IsRealPolynomial (fun x => (f x).re) ∧ IsRealPolynomial (fun x => (f x).im) |
| def complexCoordinateDerivative (f : (ι → ℝ) → ℂ) (i : ι) (x : ι → ℝ) : ℂ := |
| deriv (fun t => f (Function.update x i t)) (x i) |
| theorem complex_integral_re {Ω : Type u2} [MeasurableSpace Ω] {μ : Measure Ω} {f : Ω → ℂ} |
| (hf : Integrable f μ) : (∫ x, f x ∂μ).re = ∫ x, (f x).re ∂μ := (integral_re hf).symm |
| theorem complex_integral_im {Ω : Type u3} [MeasurableSpace Ω] {μ : Measure Ω} {f : Ω → ℂ} |
| (hf : Integrable f μ) : (∫ x, f x ∂μ).im = ∫ x, (f x).im ∂μ := (integral_im hf).symm |
| namespace IsComplexPolynomial |
| theorem const (z : ℂ) : IsComplexPolynomial (fun _ : ι → ℝ => z) := |
| ⟨IsRealPolynomial.const _,IsRealPolynomial.const _⟩ |
| theorem ofReal {f : (ι → ℝ) → ℝ} (hf : IsRealPolynomial f) : |
| IsComplexPolynomial (fun x => (f x : ℂ)) := by |
| exact ⟨hf, IsRealPolynomial.const 0⟩ |
| theorem coord (i : ι) : IsComplexPolynomial (fun x : ι → ℝ => (x i : ℂ)) := ofReal (IsRealPolynomial.coord i) |
| theorem add {f g : (ι → ℝ) → ℂ} (hf : IsComplexPolynomial f) (hg : IsComplexPolynomial g) : |
| IsComplexPolynomial (fun x => f x+g x) := ⟨hf.1.add hg.1,hf.2.add hg.2⟩ |
| theorem sub {f g : (ι → ℝ) → ℂ} (hf : IsComplexPolynomial f) (hg : IsComplexPolynomial g) : |
| IsComplexPolynomial (fun x => f x-g x) := ⟨hf.1.sub hg.1,hf.2.sub hg.2⟩ |
| theorem mul {f g : (ι → ℝ) → ℂ} (hf : IsComplexPolynomial f) (hg : IsComplexPolynomial g) : |
| IsComplexPolynomial (fun x => f x*g x) := |
| ⟨(hf.1.mul hg.1).sub (hf.2.mul hg.2),(hf.1.mul hg.2).add (hf.2.mul hg.1)⟩ |
| theorem const_mul {f : (ι → ℝ) → ℂ} (hf : IsComplexPolynomial f) (z : ℂ) : |
| IsComplexPolynomial (fun x => z*f x) := (const z).mul hf |
| theorem mul_const {f : (ι → ℝ) → ℂ} (hf : IsComplexPolynomial f) (z : ℂ) : |
| IsComplexPolynomial (fun x => f x*z) := hf.mul (const z) |
| theorem finset_sum {κ : Type u4} (s : Finset κ) {f : κ → (ι → ℝ) → ℂ} |
| (hf : ∀ k ∈ s, IsComplexPolynomial (f k)) : IsComplexPolynomial (fun x => ∑ k ∈ s, f k x) := by |
| constructor |
| · simpa using IsRealPolynomial.finset_sum s (fun k hk => (hf k hk).1) |
| · simpa using IsRealPolynomial.finset_sum s (fun k hk => (hf k hk).2) |
| theorem continuous {f : (ι → ℝ) → ℂ} (hf : IsComplexPolynomial f) : Continuous f := by |
| have hc := (Complex.continuous_ofReal.comp hf.1.continuous).add ((Complex.continuous_ofReal.comp hf.2.continuous).mul (continuous_const (y := Complex.I))) |
| convert hc using 1 |
| funext x |
| exact (Complex.re_add_im (f x)).symm |
| theorem integrable {f : (ι → ℝ) → ℂ} (hf : IsComplexPolynomial f) : Integrable f (productGaussian ι) := by |
| have hi := hf.1.integrable.ofReal.add (hf.2.integrable.ofReal.mul_const Complex.I) |
| convert hi using 1 ; try rfl |
| funext x |
| exact (Complex.re_add_im (f x)).symm |
| theorem normSq {f : (ι → ℝ) → ℂ} (hf : IsComplexPolynomial f) : |
| IsRealPolynomial (fun x => Complex.normSq (f x)) := by |
| simpa only [Complex.normSq_apply, pow_two] using (hf.1.pow 2).add (hf.2.pow 2) |
| theorem norm_sq {f : (ι → ℝ) → ℂ} (hf : IsComplexPolynomial f) : |
| IsRealPolynomial (fun x => ‖f x‖^2) := by simpa only [Complex.sq_norm] using hf.normSq |
| theorem hasCoordinateDeriv {f : (ι → ℝ) → ℂ} (hf : IsComplexPolynomial f) (i : ι) (x : ι → ℝ) : |
| HasDerivAt (fun t => f (Function.update x i t)) |
| ((coordinateDerivative (fun y => (f y).re) i x : ℂ) + |
| (coordinateDerivative (fun y => (f y).im) i x : ℂ)*Complex.I) (x i) := by |
| have hd := (hf.1.hasDerivAt i x).ofReal_comp.add ((hf.2.hasDerivAt i x).ofReal_comp.mul_const Complex.I) |
| convert hd using 1 ; try rfl |
| funext t |
| exact (Complex.re_add_im (f (Function.update x i t))).symm |
| theorem coordinateDerivative_eq {f : (ι → ℝ) → ℂ} (hf : IsComplexPolynomial f) (i : ι) (x : ι → ℝ) : |
| complexCoordinateDerivative f i x = |
| (coordinateDerivative (fun y => (f y).re) i x : ℂ) + |
| (coordinateDerivative (fun y => (f y).im) i x : ℂ)*Complex.I := |
| (hf.hasCoordinateDeriv i x).deriv |
| theorem derivative {f : (ι → ℝ) → ℂ} (hf : IsComplexPolynomial f) (i : ι) : |
| IsComplexPolynomial (complexCoordinateDerivative f i) := by |
| have he : complexCoordinateDerivative f i = fun x => |
| (coordinateDerivative (fun y => (f y).re) i x : ℂ) + |
| (coordinateDerivative (fun y => (f y).im) i x : ℂ)*Complex.I := by |
| funext x; exact hf.coordinateDerivative_eq i x |
| rw [he] |
| exact (ofReal (hf.1.derivative i)).add ((ofReal (hf.2.derivative i)).mul_const Complex.I) |
| theorem hasDerivAt {f : (ι → ℝ) → ℂ} (hf : IsComplexPolynomial f) (i : ι) (x : ι → ℝ) : |
| HasDerivAt (fun t => f (Function.update x i t)) (complexCoordinateDerivative f i x) (x i) := by |
| rw [hf.coordinateDerivative_eq] |
| exact hf.hasCoordinateDeriv i x |
| theorem stein {f : (ι → ℝ) → ℂ} (hf : IsComplexPolynomial f) (i : ι) : |
| (∫ x, (x i : ℂ)*f x ∂productGaussian ι) = ∫ x, complexCoordinateDerivative f i x ∂productGaussian ι := by |
| have hi := ((coord i).mul hf).integrable |
| have hd := (hf.derivative i).integrable |
| apply Complex.ext |
| · rw [complex_integral_re hi, complex_integral_re hd] |
| simpa only [Complex.mul_re, Complex.ofReal_re, Complex.ofReal_im, zero_mul, sub_zero, |
| hf.coordinateDerivative_eq, Complex.add_re, Complex.mul_re, Complex.I_re, Complex.I_im, mul_zero, zero_add, add_zero] using hf.1.stein i |
| · rw [complex_integral_im hi, complex_integral_im hd] |
| simpa only [Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im, zero_mul, add_zero, |
| hf.coordinateDerivative_eq, Complex.add_im, Complex.I_re, Complex.I_im, mul_one, mul_zero, zero_add] using hf.2.stein i |
| theorem poincare {f : (ι → ℝ) → ℂ} (hf : IsComplexPolynomial f) : |
| (∫ x, ‖f x-∫ y, f y ∂productGaussian ι‖^2 ∂productGaussian ι) ≤ |
| ∑ i, ∫ x, ‖complexCoordinateDerivative f i x‖^2 ∂productGaussian ι := by |
| have hi := hf.integrable |
| have hnorm (z : ℂ) : ‖z‖^2 = z.re^2+z.im^2 := by rw [Complex.sq_norm]; simp only [Complex.normSq_apply,pow_two] |
| have hr := ((hf.1.sub (IsRealPolynomial.const (∫ y, (f y).re ∂productGaussian ι))).pow 2).integrable |
| have hm := ((hf.2.sub (IsRealPolynomial.const (∫ y, (f y).im ∂productGaussian ι))).pow 2).integrable |
| have he : (∫ x, ‖f x-∫ y, f y ∂productGaussian ι‖^2 ∂productGaussian ι) = |
| (∫ x, ((f x).re-∫ y, (f y).re ∂productGaussian ι)^2 ∂productGaussian ι) + |
| ∫ x, ((f x).im-∫ y, (f y).im ∂productGaussian ι)^2 ∂productGaussian ι := by |
| simp only [hnorm, Complex.sub_re, Complex.sub_im, complex_integral_re hi, complex_integral_im hi] |
| exact integral_add hr hm |
| rw [he] |
| calc |
| _ ≤ (∑ i, ∫ x, (coordinateDerivative (fun y => (f y).re) i x)^2 ∂productGaussian ι) + |
| ∑ i, ∫ x, (coordinateDerivative (fun y => (f y).im) i x)^2 ∂productGaussian ι := |
| add_le_add hf.1.poincare hf.2.poincare |
| _ = _ := by |
| rw [← sum_add_distrib] |
| apply sum_congr rfl |
| intro i hi |
| rw [← integral_add ((hf.1.derivative i).pow 2).integrable ((hf.2.derivative i).pow 2).integrable] |
| apply integral_congr_ae |
| filter_upwards [] with x |
| simp [hnorm, hf.coordinateDerivative_eq] |
| end IsComplexPolynomial |
| end FiniteEntropySeparation |
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| end |
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| end OAI |
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