lean/OAI/Computability/VertexCover/Sampling/ThresholdPartition.lean

import Mathlib

namespace OAI

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namespace UniqueGames.Foundations.CorrelatedSampling

noncomputable section

structure ThresholdInterval where
  lo : ℝ
  hi : ℝ
  valid : lo ≤ hi

def splitPoint (I : ThresholdInterval) (t : ℝ) : ℝ := max I.lo (min t I.hi)

def splitLeft (I : ThresholdInterval) (t : ℝ) : ThresholdInterval :=
  ⟨I.lo, splitPoint I t, le_max_left _ _⟩

def splitRight (I : ThresholdInterval) (t : ℝ) : ThresholdInterval :=
  ⟨splitPoint I t, I.hi, max_le I.valid (min_le_right _ _)⟩

def cutAt (r : ℝ) (I : ThresholdInterval) : Prop := I.hi ≤ r ∨ r ≤ I.lo

def intervalCDF (r : ℝ) (I : ThresholdInterval) : ℝ := min r I.hi - min r I.lo

theorem split_cdf (I : ThresholdInterval) (t r : ℝ) :
    intervalCDF r (splitLeft I t) + intervalCDF r (splitRight I t) = intervalCDF r I := by
  unfold intervalCDF splitLeft splitRight
  ring

theorem accepted_interval_mass (I : ThresholdInterval) (r : ℝ) (hc : cutAt r I) :
    (if I.lo < r then I.hi - I.lo else 0) = intervalCDF r I := by
  by_cases hl : I.lo < r
  · have hu : I.hi ≤ r := hc.resolve_right (not_le_of_gt hl)
    simp [hl, intervalCDF, min_eq_right hu, min_eq_right (le_of_lt hl)]
  · have hr : r ≤ I.lo := le_of_not_gt hl
    simp [hl, intervalCDF, min_eq_left hr, min_eq_left (hr.trans I.valid)]

theorem splitLeft_cut (I : ThresholdInterval) (t : ℝ) : cutAt t (splitLeft I t) := by
  by_cases h : I.lo ≤ t
  · exact Or.inl (max_le h (min_le_left _ _))
  · exact Or.inr (le_of_lt (lt_of_not_ge h))

theorem splitRight_cut (I : ThresholdInterval) (t : ℝ) : cutAt t (splitRight I t) := by
  by_cases h : t ≤ I.hi
  · right
    change t ≤ max I.lo (min t I.hi)
    rw [min_eq_left h]
    exact le_max_right _ _
  · exact Or.inl (le_of_lt (lt_of_not_ge h))

theorem splitLeft_preserves_cut (I : ThresholdInterval) (t r : ℝ) (h : cutAt r I) :
    cutAt r (splitLeft I t) := by
  rcases h with h | h
  · exact Or.inl ((max_le I.valid (min_le_right _ _)).trans h)
  · exact Or.inr h

theorem splitRight_preserves_cut (I : ThresholdInterval) (t r : ℝ) (h : cutAt r I) :
    cutAt r (splitRight I t) := by
  rcases h with h | h
  · exact Or.inl h
  · exact Or.inr (h.trans (le_max_left _ _))

def splitAll (t : ℝ) : List ThresholdInterval → List ThresholdInterval
  | [] => []
  | I :: rest => splitLeft I t :: splitRight I t :: splitAll t rest

def unitThresholdInterval : ThresholdInterval := ⟨0, 1, by norm_num⟩

def thresholdPartition : List ℝ → List ThresholdInterval
  | [] => [unitThresholdInterval]
  | t :: rest => splitAll t (thresholdPartition rest)

def sumOnIntervals (f : ThresholdInterval → ℝ) : List ThresholdInterval → ℝ
  | [] => 0
  | I :: rest => f I + sumOnIntervals f rest

theorem sumOn_splitAll (f : ThresholdInterval → ℝ) (t : ℝ)
    (hf : ∀ I, f (splitLeft I t) + f (splitRight I t) = f I)
    (intervals : List ThresholdInterval) :
    sumOnIntervals f (splitAll t intervals) = sumOnIntervals f intervals := by
  induction intervals with
  | nil => rfl
  | cons I rest ih =>
      simp only [splitAll, sumOnIntervals, ih]
      rw [← add_assoc, hf]

theorem partition_sum (f : ThresholdInterval → ℝ)
    (hf : ∀ t I, f (splitLeft I t) + f (splitRight I t) = f I)
    (thresholds : List ℝ) :
    sumOnIntervals f (thresholdPartition thresholds) = f unitThresholdInterval := by
  induction thresholds with
  | nil => simp [thresholdPartition, sumOnIntervals]
  | cons t rest ih =>
      rw [thresholdPartition, sumOn_splitAll f t (hf t), ih]

theorem partition_total_mass (thresholds : List ℝ) :
    sumOnIntervals (fun I => I.hi - I.lo) (thresholdPartition thresholds) = 1 := by
  have h := partition_sum (fun I => I.hi - I.lo)
    (fun t I => by simp only [splitLeft, splitRight]; ring) thresholds
  simpa [unitThresholdInterval] using h

theorem partition_cdf (thresholds : List ℝ) (r : ℝ) (h0 : 0 ≤ r) (h1 : r ≤ 1) :
    sumOnIntervals (intervalCDF r) (thresholdPartition thresholds) = r := by
  have h := partition_sum (intervalCDF r) (fun t I => split_cdf I t r) thresholds
  simpa [intervalCDF, unitThresholdInterval, min_eq_left h1, min_eq_right h0] using h

theorem splitAll_new_cuts (intervals : List ThresholdInterval) (t : ℝ) :
    ∀ I ∈ splitAll t intervals, cutAt t I := by
  induction intervals with
  | nil => intro I h; simp [splitAll] at h
  | cons J rest ih =>
      intro I h
      rcases List.mem_cons.mp h with h | h
      · subst I; exact splitLeft_cut J t
      · rcases List.mem_cons.mp h with h | h
        · subst I; exact splitRight_cut J t
        · exact ih I h

theorem splitAll_preserves_cuts (intervals : List ThresholdInterval) (t r : ℝ) :
    (∀ I ∈ intervals, cutAt r I) → ∀ I ∈ splitAll t intervals, cutAt r I := by
  induction intervals with
  | nil => intro _ I h; simp [splitAll] at h
  | cons J rest ih =>
      intro hc I h
      have hJ : cutAt r J := hc J (by simp)
      have hrest : ∀ K ∈ rest, cutAt r K := fun K hK => hc K (by simp [hK])
      rcases List.mem_cons.mp h with h | h
      · subst I; exact splitLeft_preserves_cut J t r hJ
      · rcases List.mem_cons.mp h with h | h
        · subst I; exact splitRight_preserves_cut J t r hJ
        · exact ih hrest I h

theorem partition_cuts (thresholds : List ℝ) (r : ℝ) :
    r ∈ thresholds → ∀ I ∈ thresholdPartition thresholds, cutAt r I := by
  induction thresholds with
  | nil => intro hr; simp at hr
  | cons t rest ih =>
      intro hr
      rcases List.mem_cons.mp hr with h | h
      · subst r; exact splitAll_new_cuts (thresholdPartition rest) t
      · exact splitAll_preserves_cuts (thresholdPartition rest) t r (ih h)

theorem sumOnIntervals_congr (f g : ThresholdInterval → ℝ) (intervals : List ThresholdInterval) :
    (∀ I ∈ intervals, f I = g I) → sumOnIntervals f intervals = sumOnIntervals g intervals := by
  induction intervals with
  | nil => intro _; rfl
  | cons J rest ih =>
      intro h
      have hJ := h J (by simp)
      have hr : ∀ I ∈ rest, f I = g I := fun I hI => h I (by simp [hI])
      simp only [sumOnIntervals, hJ, ih hr]

theorem partition_acceptance_mass (thresholds : List ℝ) (r : ℝ)
    (hr : r ∈ thresholds) (h0 : 0 ≤ r) (h1 : r ≤ 1) :
    sumOnIntervals (fun I => if I.lo < r then I.hi - I.lo else 0)
      (thresholdPartition thresholds) = r := by
  calc
    _ = sumOnIntervals (intervalCDF r) (thresholdPartition thresholds) :=
      sumOnIntervals_congr _ _ _ (fun I hI =>
        accepted_interval_mass I r (partition_cuts thresholds r hr I hI))
    _ = r := partition_cdf thresholds r h0 h1

theorem sum_interval_get (f : ThresholdInterval → ℝ) (intervals : List ThresholdInterval) :
    (∑ i : Fin intervals.length, f intervals[i]) = sumOnIntervals f intervals := by
  induction intervals with
  | nil => simp [sumOnIntervals]
  | cons I rest ih =>
      simpa [Fin.sum_univ_succ, sumOnIntervals] using congrArg (fun x : ℝ => f I + x) ih

end

end UniqueGames.Foundations.CorrelatedSampling

end


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end OAI