| import OAI.Probability.InvariantIsing.Core.FiniteSpinEntropy |
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| namespace OAI |
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| /-! The finite probability form of Cauchy--Schwarz, including zero atoms. -/ |
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| noncomputable section |
| open MeasureTheory |
| open scoped BigOperators |
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| namespace InvariantIsing |
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| lemma finite_weighted_sum_sq_le {X : Type*} [Fintype X] |
| (p f : X → ℝ) (hp : ∀ x, 0 ≤ p x) (hpSum : ∑ x, p x = 1) : |
| (∑ x, p x * f x) ^ 2 ≤ ∑ x, p x * f x ^ 2 := by |
| have hc := Finset.sum_mul_sq_le_sq_mul_sq Finset.univ |
| (fun x => Real.sqrt (p x)) (fun x => Real.sqrt (p x) * f x) |
| have hm (x : X) : Real.sqrt (p x) * (Real.sqrt (p x) * f x) = p x * f x := by |
| rw [← mul_assoc, ← pow_two, Real.sq_sqrt (hp x)] |
| simp_rw [hm, mul_pow, Real.sq_sqrt (hp _)] at hc |
| simpa only [hpSum, one_mul] using hc |
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| lemma finite_weighted_sum_abs_le_sqrt {X : Type*} [Fintype X] |
| (p f : X → ℝ) (hp : ∀ x, 0 ≤ p x) (hpSum : ∑ x, p x = 1) : |
| |∑ x, p x * f x| ≤ Real.sqrt (∑ x, p x * f x ^ 2) := by |
| have hn : 0 ≤ ∑ x, p x * f x ^ 2 := |
| Finset.sum_nonneg (fun x _ => mul_nonneg (hp x) (sq_nonneg _)) |
| apply (sq_le_sq₀ (abs_nonneg _) (Real.sqrt_nonneg _)).mp |
| rw [sq_abs, Real.sq_sqrt hn] |
| exact finite_weighted_sum_sq_le p f hp hpSum |
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| lemma finite_integral_abs_le_sqrt_sq {X : Type*} [Fintype X] [MeasurableSpace X] |
| [MeasurableSingletonClass X] (μ : Measure X) [IsProbabilityMeasure μ] (F : X → ℝ) : |
| (∫ x, |F x| ∂μ) ≤ Real.sqrt (∫ x, F x ^ 2 ∂μ) := by |
| let p := fun x : X => μ.real {x} |
| have hp : ∀ x, 0 ≤ p x := fun x => measureReal_nonneg |
| have hsum : ∑ x, p x = 1 := by |
| have he := integral_fintype (μ := μ) (f := fun _ : X => (1 : ℝ)) Integrable.of_finite |
| simpa [p, measureReal_def] using he.symm |
| have he := finite_weighted_sum_abs_le_sqrt p (fun x => |F x|) hp hsum |
| have hn : 0 ≤ ∑ x, p x * |F x| := |
| Finset.sum_nonneg (fun x _ => mul_nonneg (hp x) (abs_nonneg _)) |
| simpa only [integral_fintype Integrable.of_finite, smul_eq_mul, p, |
| abs_of_nonneg hn, sq_abs] using he |
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| end InvariantIsing |
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| end |
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| end OAI |
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