| import Mathlib |
| import OAI.Computability.MinUncut.Estimates.FaceUpdateSelf |
| import OAI.Computability.MinUncut.Estimates.Away |
|
|
| 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.CubeRows |
| open BinaryFourier IndependentSquares |
| open scoped BigOperators |
| attribute [local instance] Classical.propDecidable |
| variable {ι A W : Type*} [Fintype ι] [DecidableEq ι] [Fintype A] [DecidableEq A] [Nonempty A] |
| [Fintype W] [AddCommGroup W] [Module F₂ W] |
|
|
| omit [DecidableEq ι] [Fintype A] [Nonempty A] in |
| lemma collision_indicator (x y : ι → A) : |
| (if ∃ i, x i=y i then (1:ℝ) else 0) ≤ ∑ i, if x i=y i then (1:ℝ) else 0 := by |
| by_cases h : ∃ i, x i=y i |
| · obtain ⟨i,hi⟩ := h |
| simpa only [ite_eq_left (show ∃ j, x j=y j from ⟨i,hi⟩),ite_eq_left hi] using |
| (Finset.single_le_sum (s := Finset.univ) (f := fun i => if x i=y i then (1:ℝ) else 0) |
| (fun _ _ => by positivity) (Finset.mem_univ i)) |
| · simp only [ite_eq_right h]; exact Finset.sum_nonneg (fun _ _ => by positivity) |
|
|
| lemma collision_probability : |
| (𝔼 x : ι → A, 𝔼 y : ι → A, if ∃ i, x i=y i then (1:ℝ) else 0) ≤ |
| (Fintype.card ι:ℝ)/Fintype.card A := by |
| have hf (x : ι → A) : |
| (𝔼 y : ι → A, if ∃ i, x i=y i then (1:ℝ) else 0) ≤ |
| (Fintype.card ι:ℝ)/Fintype.card A := by |
| calc |
| _ ≤ 𝔼 y : ι → A, ∑ i, if x i=y i then (1:ℝ) else 0 := |
| Finset.expect_le_expect (fun y _ => collision_indicator x y) |
| _ = ∑ i : ι, 𝔼 a : A, if x i=a then (1:ℝ) else 0 := by |
| rw [Finset.expect_sum_comm] |
| apply Finset.sum_congr rfl |
| intro i _ |
| exact Subbox.expect_coordinate (B := fun _ : ι => A) i (fun a => if x i=a then (1:ℝ) else 0) |
| _ = _ := by simp [div_eq_mul_inv] |
| exact (Finset.expect_le_expect (fun x _ => hf x)).trans_eq (Fintype.expect_const _) |
|
|
| omit [Nonempty A] [Module F₂ W] in |
| lemma cubeExpectation_abs_bound (i j : ι) (x y : ι → A) |
| (V : (ι → Bool) → W → ℝ) {C : ℝ} (hC : 0≤C) (hV : ∀ ν w, |V ν w|≤C) : |
| |cubeExpectation i j x y V| ≤ C^(2^Fintype.card ι) := by |
| unfold cubeExpectation |
| refine (Finset.abs_expect_le _ _).trans ?_ |
| apply (Finset.expect_le_expect (fun Bj _ => ?_)).trans_eq (Fintype.expect_const _) |
| refine (Finset.abs_expect_le _ _).trans ?_ |
| apply (Finset.expect_le_expect (fun Bi _ => ?_)).trans_eq (Fintype.expect_const _) |
| rw [Finset.abs_prod] |
| calc |
| _ ≤ ∏ ν : ι → Bool, |C| := Finset.prod_le_prod₀ (fun _ _ => abs_nonneg _) |
| (fun ν _ => (hV ν _).trans (le_abs_self C)) |
| _ = _ := by simp [abs_of_nonneg hC] |
|
|
| lemma full_cube_bound (i j : ι) (hij : i≠j) (V : (ι → A) → W → ℝ) |
| (ε H C : ℝ) (hε : 0≤ε) (hC : 0≤C) |
| (hcoef : ∀ x α, |coefficient (V x) α|≤ε) |
| (henergy : ∀ x, (𝔼 w, (V x w)^2)≤H^2) (hV : ∀ x w, |V x w|≤C) : |
| (𝔼 x : ι → A, 𝔼 y : ι → A, |
| cubeExpectation i j x y (fun ν => V (vertex ν x y))) ≤ |
| C^(2^Fintype.card ι)*((Fintype.card ι:ℝ)/Fintype.card A) + |
| (ε^2*H^2)^(2^Fintype.card (Other i j)) := by |
| let D := (ε^2*H^2)^(2^Fintype.card (Other i j)) |
| have hD : 0≤D := by dsimp [D]; positivity |
| have hpoint (x y : ι → A) : |
| cubeExpectation i j x y (fun ν => V (vertex ν x y)) ≤ |
| C^(2^Fintype.card ι)*(if ∃ k, x k=y k then (1:ℝ) else 0)+D := by |
| by_cases hxy : ∃ k, x k=y k |
| · simp only [ite_eq_left hxy,mul_one] |
| exact (le_abs_self _).trans ((cubeExpectation_abs_bound i j x y _ hC (fun ν => hV _)).trans (le_add_of_nonneg_right hD)) |
| · simp only [ite_eq_right hxy,mul_zero,zero_add] |
| exact (le_abs_self _).trans (cubeExpectation_bound i j hij x y (by simpa only [not_exists] using hxy) |
| _ ε H hε (fun ν => hcoef _) (fun ν => henergy _)) |
| calc |
| _ ≤ 𝔼 x : ι → A, 𝔼 y : ι → A, |
| (C^(2^Fintype.card ι)*(if ∃ k, x k=y k then (1:ℝ) else 0)+D) := |
| Finset.expect_le_expect (fun x _ => Finset.expect_le_expect (fun y _ => hpoint x y)) |
| _ = C^(2^Fintype.card ι)*(𝔼 x : ι → A, 𝔼 y : ι → A, if ∃ k, x k=y k then (1:ℝ) else 0)+D := by |
| simp only [Finset.expect_add_distrib,← Finset.mul_expect,Fintype.expect_const] |
| _ ≤ _ := by |
| simpa only [D,add_comm] using add_le_add_right (mul_le_mul_of_nonneg_left (collision_probability (ι := ι) (A := A)) (pow_nonneg hC (2^Fintype.card ι))) D |
| end MinUncut.CubeRows |
|
|
| namespace MinUncut.Inner |
| open BinaryFourier |
| open scoped BigOperators |
| attribute [local instance] Classical.propDecidable |
| variable {m n : ℕ} |
| variable {W : Type*} [Fintype W] [AddCommGroup W] [Module F₂ W] |
|
|
| def pairField (i j : Fin m) (V : Point m n → W → ℝ) |
| (Bj : ({k : Fin m |
| (Bi : ({k : Fin m |
| V x (Bj (face x j)+Bi (face x i)) |
|
|
| omit [Module F₂ W] in |
| lemma pair_moment (hn : 0 < n) (i j : Fin m) (V : Point m n → W → ℝ) : |
| (𝔼 Bj, 𝔼 Bi, Box.moment (pairField i j V Bj Bi)) = |
| 𝔼 x : Point m n, 𝔼 y : Point m n, CubeRows.cubeExpectation i j x y (fun ν => V (Box.vertex ν x y)) := by |
| have : Nonempty (Fin n) := Fin.pos_iff_nonempty.mp hn |
| unfold Box.moment CubeRows.cubeExpectation pairField |
| conv_lhs => arg 2; ext Bj; rw [Finset.expect_comm] |
| rw [Finset.expect_comm] |
| apply Finset.expect_congr rfl |
| intro x _ |
| conv_lhs => arg 2; ext Bj; rw [Finset.expect_comm] |
| rw [Finset.expect_comm] |
| rfl |
|
|
| theorem pair_correlation_power (hm : 0 < m) (hn : 0 < n) (i j : Fin m) (hij : i≠j) |
| (V : Point m n → W → ℝ) (ε H C : ℝ) (hε : 0≤ε) (hC : 0≤C) |
| (hcoef : ∀ x α, |coefficient (V x) α|≤ε) |
| (henergy : ∀ x, (𝔼 w, (V x w)^2)≤H^2) (hV : ∀ x w, |V x w|≤C) : |
| (𝔼 Bj, 𝔼 Bi, faceCorrelation (pairField i j V Bj Bi))^(2^m) ≤ |
| C^(2^m)*((m:ℝ)/n)+(ε^2*H^2)^(2^Fintype.card (CubeRows.Other i j)) := by |
| have : Nonempty (Fin n) := Fin.pos_iff_nonempty.mp hn |
| calc |
| _ ≤ 𝔼 Bj, (𝔼 Bi, faceCorrelation (pairField i j V Bj Bi))^(2^m) := Box.expect_pow_two_pow _ m |
| _ ≤ 𝔼 Bj, 𝔼 Bi, faceCorrelation (pairField i j V Bj Bi)^(2^m) := |
| Finset.expect_le_expect (fun Bj _ => Box.expect_pow_two_pow _ m) |
| _ ≤ 𝔼 Bj, 𝔼 Bi, Box.moment (pairField i j V Bj Bi) := |
| Finset.expect_le_expect (fun Bj _ => Finset.expect_le_expect (fun Bi _ => faceCorrelation_box_cs hm hn _)) |
| _ = _ := pair_moment hn i j V |
| _ ≤ _ := by |
| have hc := CubeRows.full_cube_bound i j hij V ε H C hε hC hcoef henergy hV |
| simp only [Fintype.card_fin] at hc |
| exact hc |
| end MinUncut.Inner |
| namespace MinUncut.Box |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped BigOperators |
| variable {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ] |
|
|
| lemma bounded_integrable {f : Ω → ℝ} {C : ℝ} (hf : AEStronglyMeasurable f μ) |
| (hC : ∀ x, |f x|≤C) : Integrable f μ := |
| (MemLp.of_bound hf C (Eventually.of_forall (fun x => by simpa only [Real.norm_eq_abs] using hC x)) : MemLp f 2 μ).integrable (by norm_num) |
|
|
| lemma integral_pow_bound {f : Ω → ℝ} {C : ℝ} (hf : AEStronglyMeasurable f μ) |
| (hC : ∀ x, |f x|≤C) (hpos : ∀ x, 0≤f x) (k : ℕ) : |
| (∫ x, f x ∂μ)^k ≤ ∫ x, (f x)^k ∂μ := by |
| apply (convexOn_pow k).map_integral_le (by fun_prop) isClosed_Ici |
| (Eventually.of_forall hpos) (bounded_integrable hf hC) |
| apply bounded_integrable (C := C^k) (hf.pow k) |
| intro x |
| simpa only [Function.comp_apply,Pi.pow_apply,abs_pow] using pow_le_pow_left₀ (abs_nonneg (f x)) (hC x) k |
|
|
| omit [IsProbabilityMeasure μ] in |
| lemma integral_expect {T : Type*} [Fintype T] (f : T → Ω → ℝ) (hf : ∀ t, Integrable (f t) μ) : |
| (∫ x, (𝔼 t, f t x) ∂μ) = 𝔼 t, ∫ x, f t x ∂μ := by |
| simp only [Finset.expect_eq_sum_div_card] |
| rw [integral_div, integral_finsetSum _ (fun t _ => hf t)] |
| end MinUncut.Box |
|
|
| end |
|
|
| end OAI |
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|
