import Mathlib

namespace OAI

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namespace VertexCover.GridCoupling
open MeasureTheory ProbabilityTheory
open scoped ENNReal

noncomputable def midpoint (p : ℕ) (i : Fin p) : ℝ := -1 + (2*(i:ℝ)+1)/p

noncomputable def index (p : ℕ) (hp : 0 < p) (x : ℝ) : Fin p :=
  ⟨min ⌊(p:ℝ)*(x+1)/2⌋₊ (p-1), (min_le_right _ _).trans_lt (by omega)⟩

theorem index_measurable (p : ℕ) (hp : 0 < p) : Measurable (index p hp) := by
  let g : ℕ → Fin p := fun k => ⟨min k (p-1), (min_le_right _ _).trans_lt (by omega)⟩
  exact (measurable_of_countable g).comp
    (((measurable_const.mul (measurable_id.add_const 1)).div_const 2).nat_floor)

theorem floor_lt_on_interval (p : ℕ) (hp : 0 < p) {x : ℝ}
    (hx : x ∈ Set.Ico (-1 : ℝ) 1) : ⌊(p:ℝ)*(x+1)/2⌋₊ < p := by
  apply (Nat.floor_lt' (Nat.ne_of_gt hp)).mpr
  have hpR : (0:ℝ) < p := Nat.cast_pos.mpr hp
  nlinarith [mul_lt_mul_of_pos_left hx.2 hpR]

theorem index_val_on_interval (p : ℕ) (hp : 0 < p) {x : ℝ}
    (hx : x ∈ Set.Ico (-1 : ℝ) 1) : (index p hp x).val = ⌊(p:ℝ)*(x+1)/2⌋₊ := by
  exact min_eq_left (by have h := floor_lt_on_interval p hp hx; omega)

theorem index_eq_iff_on_interval (p : ℕ) (hp : 0 < p) {x : ℝ}
    (hx : x ∈ Set.Ico (-1 : ℝ) 1) (i : Fin p) :
    index p hp x = i ↔ -1 + 2*(i:ℝ)/p ≤ x ∧ x < -1 + 2*((i:ℝ)+1)/p := by
  have hz : 0 ≤ (p:ℝ)*(x+1)/2 := div_nonneg (mul_nonneg (Nat.cast_nonneg _) (by linarith [hx.1])) (by norm_num)
  have hpR : (0:ℝ) < p := Nat.cast_pos.mpr hp
  rw [Fin.ext_iff, index_val_on_interval p hp hx, Nat.floor_eq_iff hz]
  constructor
  · rintro ⟨hl, hu⟩
    constructor
    · have h : 2*(i:ℝ)/p ≤ x+1 := (div_le_iff₀ hpR).mpr (by nlinarith)
      linarith
    · have h : x+1 < 2*((i:ℝ)+1)/p := (lt_div_iff₀ hpR).mpr (by nlinarith)
      linarith
  · rintro ⟨hl, hu⟩
    have hL : 2*(i:ℝ) ≤ (x+1)*p := (div_le_iff₀ hpR).mp (by linarith)
    have hU : (x+1)*p < 2*((i:ℝ)+1) := (lt_div_iff₀ hpR).mp (by linarith)
    constructor <;> nlinarith

end VertexCover.GridCoupling
namespace VertexCover.GridCoupling
open MeasureTheory ProbabilityTheory
open scoped ENNReal

theorem cell_subset_interval (p : ℕ) (hp : 0 < p) (i : Fin p) :
    Set.Ico (-1 + 2*(i:ℝ)/p) (-1 + 2*((i:ℝ)+1)/p) ⊆ Set.Ico (-1:ℝ) 1 := by
  intro x hx
  have hpR : (0:ℝ) < p := Nat.cast_pos.mpr hp
  have hi : (i:ℝ)+1 ≤ p := by exact_mod_cast i.isLt
  have hz : 0 ≤ 2*(i:ℝ)/p := div_nonneg (mul_nonneg (by norm_num) (Nat.cast_nonneg _)) hpR.le
  have hu : 2*((i:ℝ)+1)/p ≤ 2 := (div_le_iff₀ hpR).mpr (by nlinarith)
  exact ⟨by linarith [hx.1], by linarith [hx.2]⟩

theorem preimage_cell (p : ℕ) (hp : 0 < p) (i : Fin p) :
    (index p hp ⁻¹' {i}) ∩ Set.Ico (-1:ℝ) 1 =
      Set.Ico (-1 + 2*(i:ℝ)/p) (-1 + 2*((i:ℝ)+1)/p) := by
  ext x
  constructor
  · rintro ⟨hi, hx⟩
    exact (index_eq_iff_on_interval p hp hx i).mp hi
  · intro hx
    have hx' := cell_subset_interval p hp i hx
    exact ⟨(index_eq_iff_on_interval p hp hx' i).mpr hx, hx'⟩

theorem rounding_error (p : ℕ) (hp : 0 < p) {x : ℝ}
    (hx : x ∈ Set.Ico (-1 : ℝ) 1) : |x - midpoint p (index p hp x)| ≤ 1/p := by
  have h := (index_eq_iff_on_interval p hp hx (index p hp x)).mp rfl
  unfold midpoint
  rw [abs_le]
  simp only [mul_add, mul_one, add_div] at h ⊢
  have he : (2:ℝ)/p = 2*(1/p) := by ring
  rw [he] at h
  constructor <;> linarith

end VertexCover.GridCoupling

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end OAI