Datasets:

roseking
/

openAI-math-dataset

import Mathlib
import OAI.Probability.SKGap.Localization.LocalVolumeFull
import OAI.Probability.SKGap.Stability.FieldLogDet
import OAI.Probability.SKGap.Stability.FieldIntegralMeasure
import OAI.Probability.SKGap.Entropy.BadIntegralEndpoints
import OAI.Probability.SKGap.Localization.SmallOverlap

namespace OAI

section
noncomputable section
namespace SKGap
open Matrix Real Set MeasureTheory ProbabilityTheory GaussianDensity
open scoped BigOperators Matrix.Norms.Frobenius ENNReal

theorem small_time_root_stability {j A K q₀ c : ℝ} (hj : 0 < j) (hj1 : j < 1)
    (hA : 1 ≤ A) (hK : 0 ≤ K) (hKj : 0 ≤ K-j) (hq₀ : 0 < q₀)
    (hmargin : c < 1-A*(K-j+6*j*q₀)) :
    ∃ t₀ ρ a : ℝ,0 < t₀ ∧ 0 < ρ ∧ 0 < a ∧ ∃ N : ℕ,0 < N ∧
    ∀ (n : ℕ),N ≤ n → ∀ t ∈ Icc 0 t₀, ∀ ε : ℝ,
    ((gaussianCoordinates (MatrixCoordinates (Fin n))).prod
      (Measure.pi (fun _ : Fin n=>gaussianReal 0 t.toNNReal))).real
      {p | (∀ z : Field n,vectorNorm (plantedInteraction j (goeMatrix (j/(n:ℝ)) p.1)*ᵥz) ≤ K*vectorNorm z) ∧
        ∃ y : Field n, vectorNorm (tapField j (plantedInteraction j (goeMatrix (j/(n:ℝ)) p.1))
          (fun i=>t+p.2 i) y) ≤ ρ*sqrt (n:ℝ) ∧
          (goeMatrix (j/(n:ℝ)) p.1,y)∈badFieldSet n j A ε c} ≤
        4*exp (-min 1 (1/(π^2*j))*(n:ℝ))+exp (-a*(n:ℝ)) := by
  obtain ⟨t₀,σb,a,ht₀,hσb,ha,Nb,hbad⟩ := smalltime_bad_integral hj hj1 hq₀
  let r := q₀/2
  have hr : 0 < r := half_pos hq₀
  have hrq : 2*r ≤ q₀ := by dsimp [r];linarith
  obtain ⟨ε₁,hε₁,hvol⟩ := local_volume_uniform hj.le hK hr (half_pos ha)
  obtain ⟨γ,hγ,Nd,hNd,hdet⟩ := field_logdet_uniform hj hj1 hε₁
  obtain ⟨σ₀,hσ₀,hσvol⟩ := hvol γ hγ
  let σ := min (σ₀/2) σb
  have hσ : 0 < σ := lt_min (by positivity) hσb
  have hσless : σ < σ₀ := (min_le_left _ _).trans_lt (by linarith)
  have hσle : σ ≤ σb := min_le_right _ _
  obtain ⟨ρ,hρ,Nv,hNv,hv⟩ := hσvol σ hσ hσless
  refine ⟨t₀,ρ,a/2,ht₀,hρ,half_pos ha,
    max Nb (max Nd Nv),hNd.trans_le ((le_max_left _ _).trans (le_max_right _ _)),?_⟩
  intro n hn t ht ε
  have hnb : Nb ≤ n := (le_max_left _ _).trans hn
  have hnd : Nd ≤ n := (le_max_left _ _).trans ((le_max_right _ _).trans hn)
  have hnv : Nv ≤ n := (le_max_right _ _).trans ((le_max_right _ _).trans hn)
  have hn0 : 0 < n := hNv.trans_le hnv
  let : NeZero n := ⟨Nat.ne_of_gt hn0⟩
  let μ := gaussianCoordinates (MatrixCoordinates (Fin n))
  let ν := Measure.pi (fun _ : Fin n=>gaussianReal 0 t.toNNReal)
  let D : Set (MatrixCoordinates (Fin n) → ℝ) := {g | ∃ y : Field n,
    (n:ℝ)*(j*varianceAverage y^2+ε₁) <
      log ((fieldJacobian j (plantedInteraction j (goeMatrix (j/(n:ℝ)) g)) y)ᵀ*
        fieldJacobian j (plantedInteraction j (goeMatrix (j/(n:ℝ)) g)) y+γ • 1).det}
  let I : ((MatrixCoordinates (Fin n) → ℝ) × (Fin n → ℝ)) → ℝ≥0∞ := fun p=>
    fieldIntegral j σ (plantedInteraction j (goeMatrix (j/(n:ℝ)) p.1)) (fun i=>t+p.2 i)
      {y | q₀ ≤ overlap y}
  have hE : MeasurableSet {y : Field n | q₀ ≤ overlap y} :=
    (isClosed_le continuous_const (by unfold overlap magnetization;fun_prop)).measurableSet
  have hI : Measurable I := measurable_fieldIntegral_cylinder j t σ _ hE
  have hInt : ∫⁻ p,I p ∂μ.prod ν ≤ ENNReal.ofReal (exp (-a*(n:ℝ))) := by
    rw [lintegral_prod _ hI.aemeasurable]
    exact hbad n hnb t ht σ ⟨hσ,hσle⟩
  have hMarkov := markov_exponential (μ.prod ν) hI (b := a/2) hInt
  have hrate : a-a/2=a/2 := by ring
  rw [hrate] at hMarkov
  have hDP : (μ.prod ν).real (D ×ˢ (Set.univ : Set (Fin n → ℝ))) ≤
      4*exp (-min 1 (1/(π^2*j))*(n:ℝ)) := by
    rw [measureReal_prod_prod]
    simp only [Measure.real,measure_univ,ENNReal.toReal_one,mul_one]
    exact hdet n hnd
  apply measureReal_union_detector (μ.prod ν) _ (D ×ˢ Set.univ) I
    (exp (-(a/2)*(n:ℝ))) _ _ hDP hMarkov
  rintro p ⟨hJ,y,hres,hy⟩ hp
  have hp' : p.1∉D := fun hh=>hp ⟨hh,mem_univ _⟩
  dsimp [I]
  rw [fieldIntegral_eq_localVolume hσ]
  apply hv n hnv _ hJ _ y {z | q₀ ≤ overlap z} hE
  · intro d hd
    exact overlap_neighborhood hn0 hrq y d
      (badField_small_overlap_impossible hn0 hj.le (zero_le_one.trans hA) hKj hmargin _ y hJ hy) hd
  · exact not_lt.mp (fun hh=>hp' ⟨y,hh⟩)
  · exact hres
end SKGap
end
end

end OAI

	import Mathlib
	import OAI.Probability.SKGap.Localization.LocalVolumeFull
	import OAI.Probability.SKGap.Stability.FieldLogDet
	import OAI.Probability.SKGap.Stability.FieldIntegralMeasure
	import OAI.Probability.SKGap.Entropy.BadIntegralEndpoints
	import OAI.Probability.SKGap.Localization.SmallOverlap

	namespace OAI

	section
	noncomputable section
	namespace SKGap
	open Matrix Real Set MeasureTheory ProbabilityTheory GaussianDensity
	open scoped BigOperators Matrix.Norms.Frobenius ENNReal

	theorem small_time_root_stability {j A K q₀ c : ℝ} (hj : 0 < j) (hj1 : j < 1)
	(hA : 1 ≤ A) (hK : 0 ≤ K) (hKj : 0 ≤ K-j) (hq₀ : 0 < q₀)
	(hmargin : c < 1-A(K-j+6j*q₀)) :
	∃ t₀ ρ a : ℝ,0 < t₀ ∧ 0 < ρ ∧ 0 < a ∧ ∃ N : ℕ,0 < N ∧
	∀ (n : ℕ),N ≤ n → ∀ t ∈ Icc 0 t₀, ∀ ε : ℝ,
	((gaussianCoordinates (MatrixCoordinates (Fin n))).prod
	(Measure.pi (fun _ : Fin n=>gaussianReal 0 t.toNNReal))).real
	{p \| (∀ z : Field n,vectorNorm (plantedInteraction j (goeMatrix (j/(n:ℝ)) p.1)ᵥz) ≤ KvectorNorm z) ∧
	∃ y : Field n, vectorNorm (tapField j (plantedInteraction j (goeMatrix (j/(n:ℝ)) p.1))
	(fun i=>t+p.2 i) y) ≤ ρ*sqrt (n:ℝ) ∧
	(goeMatrix (j/(n:ℝ)) p.1,y)∈badFieldSet n j A ε c} ≤
	4exp (-min 1 (1/(π^2j))(n:ℝ))+exp (-a(n:ℝ)) := by
	obtain ⟨t₀,σb,a,ht₀,hσb,ha,Nb,hbad⟩ := smalltime_bad_integral hj hj1 hq₀
	let r := q₀/2
	have hr : 0 < r := half_pos hq₀
	have hrq : 2*r ≤ q₀ := by dsimp [r];linarith
	obtain ⟨ε₁,hε₁,hvol⟩ := local_volume_uniform hj.le hK hr (half_pos ha)
	obtain ⟨γ,hγ,Nd,hNd,hdet⟩ := field_logdet_uniform hj hj1 hε₁
	obtain ⟨σ₀,hσ₀,hσvol⟩ := hvol γ hγ
	let σ := min (σ₀/2) σb
	have hσ : 0 < σ := lt_min (by positivity) hσb
	have hσless : σ < σ₀ := (min_le_left _ _).trans_lt (by linarith)
	have hσle : σ ≤ σb := min_le_right _ _
	obtain ⟨ρ,hρ,Nv,hNv,hv⟩ := hσvol σ hσ hσless
	refine ⟨t₀,ρ,a/2,ht₀,hρ,half_pos ha,
	max Nb (max Nd Nv),hNd.trans_le ((le_max_left _ _).trans (le_max_right _ _)),?_⟩
	intro n hn t ht ε
	have hnb : Nb ≤ n := (le_max_left _ _).trans hn
	have hnd : Nd ≤ n := (le_max_left _ _).trans ((le_max_right _ _).trans hn)
	have hnv : Nv ≤ n := (le_max_right _ _).trans ((le_max_right _ _).trans hn)
	have hn0 : 0 < n := hNv.trans_le hnv
	let : NeZero n := ⟨Nat.ne_of_gt hn0⟩
	let μ := gaussianCoordinates (MatrixCoordinates (Fin n))
	let ν := Measure.pi (fun _ : Fin n=>gaussianReal 0 t.toNNReal)
	let D : Set (MatrixCoordinates (Fin n) → ℝ) := {g \| ∃ y : Field n,
	(n:ℝ)(jvarianceAverage y^2+ε₁) <
	log ((fieldJacobian j (plantedInteraction j (goeMatrix (j/(n:ℝ)) g)) y)ᵀ*
	fieldJacobian j (plantedInteraction j (goeMatrix (j/(n:ℝ)) g)) y+γ • 1).det}
	let I : ((MatrixCoordinates (Fin n) → ℝ) × (Fin n → ℝ)) → ℝ≥0∞ := fun p=>
	fieldIntegral j σ (plantedInteraction j (goeMatrix (j/(n:ℝ)) p.1)) (fun i=>t+p.2 i)
	{y \| q₀ ≤ overlap y}
	have hE : MeasurableSet {y : Field n \| q₀ ≤ overlap y} :=
	(isClosed_le continuous_const (by unfold overlap magnetization;fun_prop)).measurableSet
	have hI : Measurable I := measurable_fieldIntegral_cylinder j t σ _ hE
	have hInt : ∫⁻ p,I p ∂μ.prod ν ≤ ENNReal.ofReal (exp (-a*(n:ℝ))) := by
	rw [lintegral_prod _ hI.aemeasurable]
	exact hbad n hnb t ht σ ⟨hσ,hσle⟩
	have hMarkov := markov_exponential (μ.prod ν) hI (b := a/2) hInt
	have hrate : a-a/2=a/2 := by ring
	rw [hrate] at hMarkov
	have hDP : (μ.prod ν).real (D ×ˢ (Set.univ : Set (Fin n → ℝ))) ≤
	4exp (-min 1 (1/(π^2j))*(n:ℝ)) := by
	rw [measureReal_prod_prod]
	simp only [Measure.real,measure_univ,ENNReal.toReal_one,mul_one]
	exact hdet n hnd
	apply measureReal_union_detector (μ.prod ν) _ (D ×ˢ Set.univ) I
	(exp (-(a/2)*(n:ℝ))) _ _ hDP hMarkov
	rintro p ⟨hJ,y,hres,hy⟩ hp
	have hp' : p.1∉D := fun hh=>hp ⟨hh,mem_univ _⟩
	dsimp [I]
	rw [fieldIntegral_eq_localVolume hσ]
	apply hv n hnv _ hJ _ y {z \| q₀ ≤ overlap z} hE
	· intro d hd
	exact overlap_neighborhood hn0 hrq y d
	(badField_small_overlap_impossible hn0 hj.le (zero_le_one.trans hA) hKj hmargin _ y hJ hy) hd
	· exact not_lt.mp (fun hh=>hp' ⟨y,hh⟩)
	· exact hres
	end SKGap
	end
	end

	end OAI