lean/OAI/Combinatorics/IndependentSets/Expansion/PreprocessingLevels.lean

import Mathlib
import OAI.Combinatorics.IndependentSets.Expansion.ExpanderTables

namespace OAI

namespace IndependentSetsGames.Foundations.PCP.PreprocessingLevels

def levelLoop (g k : Nat) : Nat → Nat → Nat → Nat × Nat
  | 0, e, s => (e, s)
  | fuel + 1, e, s =>
      if k ≤ s then (e, s)
      else levelLoop g k fuel (e + 1) (s * g)

theorem levelLoop_level_le (g k fuel e s : Nat) :
    (levelLoop g k fuel e s).1 ≤ e + fuel := by
  induction fuel generalizing e s with
  | zero => simp only [levelLoop, Nat.add_zero, le_refl]
  | succ fuel ih =>
      simp only [levelLoop]
      split
      · omega
      · have h := ih (e + 1) (s * g)
        omega

theorem levelLoop_power (g k fuel e : Nat) :
    (levelLoop g k fuel e (g ^ e)).2 =
      g ^ (levelLoop g k fuel e (g ^ e)).1 := by
  induction fuel generalizing e with
  | zero => rfl
  | succ fuel ih =>
      simp only [levelLoop]
      split
      · rfl
      · simpa only [pow_succ] using ih (e + 1)

theorem levelLoop_covers (g k fuel e s : Nat) (hg : 1 < g)
    (hs : 0 < s) (hbudget : k ≤ s + fuel) :
    k ≤ (levelLoop g k fuel e s).2 := by
  induction fuel generalizing e s with
  | zero => simpa only [levelLoop, Nat.add_zero] using hbudget
  | succ fuel ih =>
      simp only [levelLoop]
      split
      · assumption
      · apply ih
        · exact Nat.mul_pos hs (by omega)
        · have htwo : 2 ≤ g := by omega
          have hmul : s * 2 ≤ s * g := Nat.mul_le_mul_left s htwo
          omega

theorem levelLoop_le_of_covering_power (g k fuel e j : Nat)
    (he : e ≤ j) (hj : k ≤ g ^ j) :
    (levelLoop g k fuel e (g ^ e)).1 ≤ j := by
  induction fuel generalizing e with
  | zero => exact he
  | succ fuel ih =>
      simp only [levelLoop]
      split
      · exact he
      · rename_i hnot
        have hne : e ≠ j := by
          intro h
          subst e
          exact hnot hj
        have hnext : e + 1 ≤ j := by omega
        simpa only [pow_succ] using ih (e + 1) hnext

def searchResult (k : Nat) : Nat × Nat :=
  levelLoop ExpanderFamily.growth k k 0 1

def boundedLevel (k : Nat) : Nat := (searchResult k).1

def paddedSize (k : Nat) : Nat := (searchResult k).2

@[simp] theorem boundedLevel_zero : boundedLevel 0 = 0 := rfl

@[simp] theorem paddedSize_zero : paddedSize 0 = 1 := rfl

theorem boundedLevel_le_input (k : Nat) : boundedLevel k ≤ k := by
  simpa only [boundedLevel, searchResult, Nat.zero_add] using
    levelLoop_level_le ExpanderFamily.growth k k 0 1

theorem paddedSize_power (k : Nat) :
    paddedSize k = ExpanderFamily.growth ^ boundedLevel k := by
  simpa only [paddedSize, boundedLevel, searchResult, pow_zero] using
    levelLoop_power ExpanderFamily.growth k k 0

theorem le_paddedSize (k : Nat) : k ≤ paddedSize k := by
  exact levelLoop_covers ExpanderFamily.growth k k 0 1
    ExpanderFamily.growth_gt_one (by omega) (by omega)

theorem boundedLevel_eq_level (k : Nat) :
    boundedLevel k = ExpanderFamily.level k := by
  apply Nat.le_antisymm
  · simpa only [boundedLevel, searchResult, pow_zero] using
      levelLoop_le_of_covering_power ExpanderFamily.growth k k 0
        (ExpanderFamily.level k) (Nat.zero_le _) (ExpanderFamily.le_size k)
  · apply (Nat.clog_le_iff_le_pow ExpanderFamily.growth_gt_one).2
    rw [← paddedSize_power]
    exact le_paddedSize k

theorem paddedSize_eq_size (k : Nat) : paddedSize k = ExpanderFamily.size k := by
  rw [paddedSize_power, boundedLevel_eq_level]
  rfl

theorem boundedLevel_le_of_le_power {k j : Nat}
    (h : k ≤ ExpanderFamily.growth ^ j) : boundedLevel k ≤ j := by
  rw [boundedLevel_eq_level]
  exact (Nat.clog_le_iff_le_pow ExpanderFamily.growth_gt_one).2 h

theorem paddedSize_positive (k : Nat) : 0 < paddedSize k := by
  rw [paddedSize_power]
  exact Nat.pow_pos (by have h := ExpanderFamily.growth_gt_one; omega)

theorem paddedSize_bounds {k : Nat} (hk : 0 < k) :
    k ≤ paddedSize k ∧ paddedSize k ≤ ExpanderFamily.growth * k := by
  rw [paddedSize_eq_size]
  exact ⟨ExpanderFamily.le_size k, ExpanderFamily.size_le_mul hk⟩

def cloudPaddedSize (k : Nat) : Nat := if k = 0 then 0 else paddedSize k

@[simp] theorem cloudPaddedSize_zero : cloudPaddedSize 0 = 0 := rfl

theorem cloudPaddedSize_of_pos {k : Nat} (hk : 0 < k) :
    cloudPaddedSize k = paddedSize k := by
  simp only [cloudPaddedSize, Nat.ne_of_gt hk, ite_false]

theorem cloudPaddedSize_bounds (k : Nat) :
    k ≤ cloudPaddedSize k ∧ cloudPaddedSize k ≤ ExpanderFamily.growth * k := by
  by_cases hk : k = 0
  · subst k
    simp only [cloudPaddedSize_zero, Nat.mul_zero, le_refl, and_self]
  · rw [cloudPaddedSize_of_pos (Nat.pos_of_ne_zero hk)]
    exact paddedSize_bounds (Nat.pos_of_ne_zero hk)

theorem table_vertexCount_eq_paddedSize (k : Nat) :
    ExpanderTables.vertexCount (Expanders.baseDegree * Expanders.baseDegree)
        (boundedLevel k) = paddedSize k := by
  rw [ExpanderTables.vertexCount_eq]
  have hg : (Expanders.baseDegree * Expanders.baseDegree) *
      (Expanders.baseDegree * Expanders.baseDegree) = ExpanderFamily.growth := by
    unfold ExpanderFamily.growth
    ring
  rw [hg]
  exact (paddedSize_power k).symm

theorem family_row_count_at_size
    (H : ExpanderTables.Table
      ((Expanders.baseDegree * Expanders.baseDegree) *
        (Expanders.baseDegree * Expanders.baseDegree)) Expanders.baseDegree)
    (k : Nat) :
    (ExpanderTables.family H (boundedLevel k)).rows.toList.length =
      paddedSize k * (Expanders.baseDegree * Expanders.baseDegree) := by
  rw [ExpanderTables.row_count, table_vertexCount_eq_paddedSize]

end IndependentSetsGames.Foundations.PCP.PreprocessingLevels

end OAI