openAI-math-dataset / lean /OAI /Computability /MinUncut /Analysis /CenteredPointCorrelationPower.lean
| import Mathlib | |
| import OAI.Computability.MinUncut.Estimates.PointField | |
| namespace OAI | |
| noncomputable section | |
| open scoped BigOperators | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped Topology NNReal | |
| open scoped BigOperators | |
| open MeasureTheory ProbabilityTheory Polynomial Filter | |
| open scoped BigOperators Topology | |
| open MeasureTheory ProbabilityTheory WithLp | |
| open scoped BigOperators RealInnerProductSpace | |
| open scoped BigOperators | |
| namespace MinUncut.Inner | |
| open MeasureTheory ProbabilityTheory | |
| open scoped BigOperators | |
| variable {m n : ℕ} | |
| theorem centered_point_correlation_power (μ : Measure ℝ) [IsProbabilityMeasure μ] | |
| (V : Point m n → ℝ → ℝ) (hcont : ∀ x, Continuous (V x)) {C : ℝ} (hC : 0≤C) | |
| (hV : ∀ x t, |V x t|≤C) (hmean : ∀ x, (∫ t, V x t ∂μ)=0) | |
| (hm : 0 < m) (hn : 0 < n) : | |
| (∫ c, faceCorrelation (pointField V c) ∂Measure.pi (fun _ : Point m n => μ))^(2^m) ≤ | |
| C^(2^m)*((m:ℝ)/n) := by | |
| have : Nonempty (Fin n) := Fin.pos_iff_nonempty.mp hn | |
| have hb (c : Point m n → ℝ) : |faceCorrelation (pointField V c)|≤C := by | |
| rw [abs_of_nonneg (faceCorrelation_nonneg _)] | |
| exact (faceCorrelation_le_l1 _).trans ((Finset.expect_le_expect (fun x _ => hV x (c x))).trans_eq (Fintype.expect_const C)) | |
| have hi : Integrable (fun c => faceCorrelation (pointField V c)^(2^m)) (Measure.pi (fun _ : Point m n => μ)) := by | |
| apply Box.bounded_integrable (C := C^(2^m)) ((point_correlation_continuous V hcont).pow _).aestronglyMeasurable | |
| intro c | |
| simp only [Pi.pow_apply, abs_pow] | |
| exact pow_le_pow_left₀ (abs_nonneg _) (hb c) _ | |
| have hmom : Integrable (fun c => Box.moment (pointField V c)) (Measure.pi (fun _ : Point m n => μ)) := by | |
| unfold Box.moment | |
| simp only [Finset.expect_eq_sum_div_card] | |
| apply Integrable.div_const | |
| apply integrable_finsetSum | |
| intro x _ | |
| apply Integrable.div_const | |
| exact integrable_finsetSum _ (fun y _ => point_cube_integrable μ V hcont hC hV x y) | |
| calc | |
| _ ≤ ∫ c, faceCorrelation (pointField V c)^(2^m) ∂Measure.pi (fun _ : Point m n => μ) := | |
| Box.integral_pow_bound (point_correlation_continuous V hcont).aestronglyMeasurable hb | |
| (fun c => faceCorrelation_nonneg _) _ | |
| _ ≤ ∫ c, Box.moment (pointField V c) ∂Measure.pi (fun _ : Point m n => μ) := | |
| integral_mono hi hmom (fun c => faceCorrelation_box_cs hm hn _) | |
| _ ≤ _ := centered_point_moment μ V hcont hC hV hmean hn | |
| end MinUncut.Inner | |
| namespace MinUncut.Inner | |
| open BinaryFourier RowNoise GaussianHermite | |
| open scoped BigOperators | |
| attribute [local instance] Classical.propDecidable | |
| variable {m n : ℕ} | |
| variable {V A : Type*} [AddCommGroup V] [Module F₂ V] [AddTorsor V A] [Fintype A] | |
| lemma pairSlice_odd (f : FoldedProof A) {σ : ℝ} (hσ : σ≠0) (η : ℝ) | |
| (x : Point m n) {k : ℕ} (Y : Subbox.PointedBox x k) (p : PairClass m) | |
| (J : Finset (Finset (Row m n) × (Point m n → ℕ))) | |
| (B : FaceArray A m n) (c : Point m n → ℝ) (b : Forms A) : | |
| pairSlice f σ η x Y p J B c (b+AffineMap.const F₂ A 1) = | |
| -pairSlice f σ η x Y p J B c b := by | |
| apply localSlice_odd | |
| intro j hj α hm hc | |
| exact (joint_pair_matching f hσ η x j.2 j.1 p | |
| (retained_pair_admissible x Y p J j hj) α hm hc).2 | |
| lemma pairSlice_coefficient_zero (f : FoldedProof A) {σ : ℝ} (hσ : σ≠0) (η : ℝ) | |
| (x : Point m n) {k : ℕ} (Y : Subbox.PointedBox x k) (p : PairClass m) | |
| (J : Finset (Finset (Row m n) × (Point m n → ℕ))) | |
| (B : FaceArray A m n) (c : Point m n → ℝ) (α : Module.Dual F₂ (Forms A)) | |
| (ha : α (AffineMap.const F₂ A 1)≠1) : coefficient (pairSlice f σ η x Y p J B c) α=0 := | |
| odd_coefficient_zero _ _ (pairSlice_odd f hσ η x Y p J B c) α ha | |
| theorem processed_pair_correlation_power (f : FoldedProof A) {σ : ℝ} (hσ : σ≠0) (η : ℝ) | |
| (hm : 0 < m) (hn : 0 < n) {k : ℕ} (Y : ∀ x : Point m n, Subbox.PointedBox x k) | |
| (p : PairClass m) (J : Finset (Finset (Row m n) × (Point m n → ℕ))) | |
| (B : FaceArray A m n) (c : Point m n → ℝ) {H γ A₀ : ℝ} (hA : 0 < A₀) (hγ : 0≤γ) : | |
| (𝔼 Bj, 𝔼 Bi, faceCorrelation (pairField p.val.1 p.val.2 | |
| (fun x => Slice.processed H γ A₀ (AffineMap.const F₂ A 1) (pairSlice f σ η x (Y x) p J B c)) Bj Bi))^(2^m) ≤ | |
| A₀^(2^m)*((m:ℝ)/n) + ((γ+H^2/A₀)^2*H^2)^(2^Fintype.card (CubeRows.Other p.val.1 p.val.2)) := by | |
| apply pair_correlation_power hm hn _ _ (ne_of_lt p.property) _ (γ+H^2/A₀) H A₀ | |
| (by positivity) hA.le | |
| · intro x α | |
| exact Slice.processed_coefficient_le hA hγ _ _ (pairSlice_coefficient_zero f hσ η x (Y x) p J B c) α | |
| · intro x | |
| exact Slice.processed_energy_le hA.le _ _ | |
| · intro x b | |
| exact Slice.processed_abs_le hA.le _ _ b | |
| end MinUncut.Inner | |
| end | |
| end OAI | |