lean/OAI/Geometry/PinnedDistances/ArithmeticVariance.lean

import OAI.Geometry.PinnedDistances.Variance

namespace OAI

universe u1 u2 u3 u4 u5 u6 u7

section
noncomputable section
open NumberField
open scoped BigOperators
noncomputable section
open MeasureTheory
open scoped Topology
noncomputable section
open MeasureTheory
open scoped Topology ENNReal
noncomputable section
open MeasureTheory NumberField
open scoped BigOperators
noncomputable section
open scoped BigOperators
open MeasureTheory

namespace WeakPinned.Global
open Hierarchy
variable {P : Type u1} [Fintype P] [DecidableEq P]
variable {K : Type u2} [Field K] [NumberField K]
variable (Z : P → K) (hZ : Function.Injective Z)

def cellMass (C : Finset P) : ℝ :=
  (∑ v : InfinitePlace K, Arithmetic.archWeight v * ∫ ω, (arch Z v ω).mass C ∂Grid.fullLaw) +
    (Module.finrank ℚ K : ℝ)⁻¹ * ∑ v ∈ badPlaces Z hZ, (finite Z hZ v).mass C

lemma cellMass_nonneg (C : Finset P) : 0 ≤ cellMass Z hZ C := by
  exact add_nonneg
    (Finset.sum_nonneg (fun v _ => mul_nonneg (Arithmetic.archWeight_nonneg v)
      (integral_nonneg (fun ω => (arch Z v ω).mass_nonneg C))))
    (mul_nonneg (by positivity) (Finset.sum_nonneg (fun v _ => (finite Z hZ v).mass_nonneg C)))

lemma cellMass_singleton (C : Finset P) (hC : C.card ≤ 1) : cellMass Z hZ C = 0 := by
  simp only [cellMass, Depth.mass_singleton _ C hC, integral_zero, mul_zero, Finset.sum_const_zero, add_zero]

lemma cellMass_overlap (hn : 2 ≤ Fintype.card P) (x y : P) (hxy : x ≠ y) :
    (∑ C, cellMass Z hZ C * (if x ∈ C ∧ y ∈ C then (1 : ℝ) else 0)) = L Z hZ x y hxy := by
  simp only [cellMass, add_mul, Finset.sum_add_distrib, Finset.sum_mul,
    mul_assoc, L]
  rw [← Finset.mul_sum]
  congr 1
  · rw [Finset.sum_comm]
    apply Finset.sum_congr rfl
    intro v _
    rw [← Finset.mul_sum, Depth.averaged_mass_overlap (arch Z v) (arch_measurable Z hZ v)
      (arch_integrable Z hZ v) hn x y hxy]
  · congr 1
    rw [Finset.sum_comm]
    apply Finset.sum_congr rfl
    intro v _
    exact (finite Z hZ v).mass_overlap hn x y hxy

end WeakPinned.Global

namespace WeakPinned.Variance
variable {P : Type u3} [Fintype P] [DecidableEq P]

/-- Finite linear model of the actual occurrence integral on expressions
    vanishing on singleton and empty cells. Multiplicities are in m. -/
def weightedIntegral (m : Finset P → ℝ) (f : Finset P → ℝ) : ℝ := ∑ C, m C * f C

def weightedOverlap (m : Finset P → ℝ) (x y : P) : ℝ :=
  weightedIntegral m (fun C => if x ∈ C ∧ y ∈ C then 1 else 0)

lemma weighted_cancel (m : Finset P → ℝ) (b : ℝ) (S : P → ℝ)
    (hL : ∀ x y, x ≠ y → weightedOverlap m x y = b+S x+S y)
    (B : Finset P) (hB : 2 ≤ B.card) : weightedIntegral m (signedCell B) = 0 := by
  have he : weightedIntegral m (signedCell B) =
      ∑ x, ∑ y, signedPair B x y * weightedOverlap m x y := by
    simp only [weightedIntegral, signedCell_eq]
    have hc (C : Finset P) : (∑ x ∈ C, ∑ y ∈ C, signedPair B x y) =
        ∑ x, ∑ y, signedPair B x y * (if x ∈ C ∧ y ∈ C then (1:ℝ) else 0) := by
      simp only [mul_ite, mul_one, mul_zero, ite_and]
      simp only [Finset.sum_ite_irrel, Finset.sum_const_zero]
      simp
    simp_rw [hc, Finset.mul_sum]
    rw [Finset.sum_comm]
    apply Finset.sum_congr rfl
    intro x _
    rw [Finset.sum_comm]
    apply Finset.sum_congr rfl
    intro y _
    simp only [weightedOverlap, weightedIntegral, Finset.mul_sum]
    apply Finset.sum_congr rfl
    intro C _
    ring
  rw [he]
  exact variance_cancel B hB _ b S hL

lemma weighted_variance {I : Type u4} [Fintype I]
    (m : Finset P → ℝ) (hm : ∀ C, 0 ≤ m C) (b : ℝ) (S : P → ℝ)
    (hL : ∀ x y, x ≠ y → weightedOverlap m x y = b+S x+S y)
    (w : I → ℝ) (B : I → Finset P) (T M ε τ : ℝ)
    (hw : ∀ i, 0 ≤ w i) (hW : (∑ i, w i) ≤ 1)
    (hT : 1 ≤ T) (hM : 0 ≤ M) (hε : 0 ≤ ε) (hτ : 0 < τ)
    (hscale : τ * Fintype.card P ≤ T)
    (hB : ∀ i, 2 ≤ (B i).card)
    (hk : ∀ i, 1 / (((B i).card : ℝ)-1) ≤ ε)
    (hn : 1 / ((Fintype.card P : ℝ)-1) ≤ ε)
    (hdom : ∀ x, (∑ i, w i * prob (B i) x) ≤ M * prob Finset.univ x) :
    weightedIntegral m (varianceCell w B T) ≤
      (2*ε*(M+1)/τ)*weightedIntegral m (pairCell Finset.univ) +
      4*M*weightedIntegral m (fun C => if (C.card : ℝ) ≤ T then pairCell Finset.univ C else 0) := by
  have hh := Finset.sum_le_sum (s := Finset.univ) (fun C _ =>
    mul_le_mul_of_nonneg_left
      (varianceCell_bound w B T M ε τ hw hW hT hM hε hτ hscale hB hk hn hdom C) (hm C))
  simp only [mul_add, Finset.sum_add_distrib] at hh
  have hz : (∑ C, m C * ∑ i, w i * signedCell (B i) C) = 0 := by
    simp_rw [Finset.mul_sum]
    rw [Finset.sum_comm]
    have hi (i : I) : (∑ C, m C * (w i * signedCell (B i) C)) = 0 := by
      calc
        _ = w i * weightedIntegral m (signedCell (B i)) := by
          simp only [weightedIntegral, Finset.mul_sum]
          apply Finset.sum_congr rfl
          intro C _
          ring
        _ = 0 := by rw [weighted_cancel m b S hL _ (hB i),mul_zero]
    simp_rw [hi]
    exact Finset.sum_const_zero
  rw [hz,zero_add] at hh
  simp only [← mul_add] at hh
  change weightedIntegral m (varianceCell w B T) ≤ _ at hh
  have he₁ : (∑ C, m C * (2*ε*(M+1)/τ * pairCell Finset.univ C)) =
      (2*ε*(M+1)/τ)*weightedIntegral m (pairCell Finset.univ) := by
    simp only [weightedIntegral, Finset.mul_sum]
    apply Finset.sum_congr rfl
    intro C _
    ring
  have he₂ : (∑ C, m C * (4*M*(if (C.card : ℝ) ≤ T then pairCell Finset.univ C else 0))) =
      4*M*weightedIntegral m (fun C => if (C.card : ℝ) ≤ T then pairCell Finset.univ C else 0) := by
    simp only [weightedIntegral, Finset.mul_sum]
    apply Finset.sum_congr rfl
    intro C _
    ring
  rwa [he₁,he₂] at hh
end WeakPinned.Variance

namespace WeakPinned.Global
open Hierarchy Variance
variable {P : Type u5} [Fintype P] [DecidableEq P]
variable {K : Type u6} [Field K] [NumberField K]
variable (Z₁ Z₂ : P → K) (h₁ : Function.Injective Z₁) (h₂ : Function.Injective Z₂)

def totalMass (C : Finset P) : ℝ := cellMass Z₁ h₁ C + cellMass Z₂ h₂ C

def totalPotential (x : P) : ℝ := S Z₁ h₁ x + S Z₂ h₂ x

def baseConstant : ℝ := 2*Grid.kappa - M Z₁ h₁ - M Z₂ h₂

lemma totalMass_nonneg (C : Finset P) : 0 ≤ totalMass Z₁ Z₂ h₁ h₂ C :=
  add_nonneg (cellMass_nonneg Z₁ h₁ C) (cellMass_nonneg Z₂ h₂ C)

lemma totalPotential_nonneg (x : P) : 0 ≤ totalPotential Z₁ Z₂ h₁ h₂ x :=
  add_nonneg (S_nonneg Z₁ h₁ x) (S_nonneg Z₂ h₂ x)

lemma totalMass_overlap (hn : 2 ≤ Fintype.card P) (x y : P) (hxy : x ≠ y) :
    weightedOverlap (totalMass Z₁ Z₂ h₁ h₂) x y =
      baseConstant Z₁ Z₂ h₁ h₂ + totalPotential Z₁ Z₂ h₁ h₂ x + totalPotential Z₁ Z₂ h₁ h₂ y := by
  simp only [weightedOverlap,weightedIntegral,totalMass,add_mul,Finset.sum_add_distrib]
  rw [cellMass_overlap Z₁ h₁ hn x y hxy,cellMass_overlap Z₂ h₂ hn x y hxy,
    coordinate_overlap_identity Z₁ h₁ hn x y hxy,coordinate_overlap_identity Z₂ h₂ hn x y hxy]
  simp only [baseConstant,totalPotential]
  ring

/-- Unconditional arithmetic-hierarchy instance of the quantitative variance
    bound. The overlap identity follows from the product formula, and the
    family hypotheses are exactly finite sampling data. -/
theorem arithmetic_variance {I : Type u7} [Fintype I]
    (hnP : 2 ≤ Fintype.card P)
    (w : I → ℝ) (B : I → Finset P) (T M₀ ε τ : ℝ)
    (hw : ∀ i, 0 ≤ w i) (hW : (∑ i, w i) ≤ 1)
    (hT : 1 ≤ T) (hM : 0 ≤ M₀) (hε : 0 ≤ ε) (hτ : 0 < τ)
    (hscale : τ * Fintype.card P ≤ T)
    (hB : ∀ i, 2 ≤ (B i).card)
    (hk : ∀ i, 1 / (((B i).card : ℝ)-1) ≤ ε)
    (hn : 1 / ((Fintype.card P : ℝ)-1) ≤ ε)
    (hdom : ∀ x, (∑ i, w i * prob (B i) x) ≤ M₀ * prob Finset.univ x) :
    weightedIntegral (totalMass Z₁ Z₂ h₁ h₂) (varianceCell w B T) ≤
      (2*ε*(M₀+1)/τ)*weightedIntegral (totalMass Z₁ Z₂ h₁ h₂) (pairCell Finset.univ) +
      4*M₀*weightedIntegral (totalMass Z₁ Z₂ h₁ h₂)
        (fun C => if (C.card : ℝ) ≤ T then pairCell Finset.univ C else 0) :=
  weighted_variance _ (totalMass_nonneg Z₁ Z₂ h₁ h₂) _ _
    (totalMass_overlap Z₁ Z₂ h₁ h₂ hnP) w B T M₀ ε τ hw hW hT hM hε hτ hscale hB hk hn hdom
end WeakPinned.Global

end
end
end
end
end
end

end OAI