import OAI.Combinatorics.EditApproximation.Binary.BinaryScaleGrid
import OAI.Combinatorics.EditApproximation.Geometry.LocalCells
import OAI.Combinatorics.EditApproximation.Definitions


namespace OAI

/-! The whole-cell grids and nearby cell indices are generated by literal
binary quotient, endpoint and bounded-range operations. -/

namespace EditApproximation
namespace BinaryFraction



@[simp] theorem naturalFloorWithWork_value (a : BinaryFraction) :
    bitWordValue (naturalFloorWithWork a).1 = ⌊a.value⌋₊ := by
  cases hs : a.numerator.negative
  · have hdiv := bitDivModWithWork_value a.denominator a.numerator.bits a.denominator_pos
    have hq : bitWordValue (bitDivModWithWork a.denominator a.numerator.bits).1 =
        bitWordValue a.numerator.bits / bitWordValue a.denominator := by
      conv_rhs => rw [hdiv.1]
      rw [Nat.mul_add_div a.denominator_pos, Nat.div_eq_of_lt hdiv.2, Nat.add_zero]
    simpa only [naturalFloorWithWork, hs, Bool.false_eq_true, ↓reduceIte, value,
      SignedBinary.value, signedMagnitude, Int.cast_natCast, Nat.floor_div_eq_div] using hq
  · have hn : a.value ≤ 0 := by
      simp only [value, SignedBinary.value, signedMagnitude, hs, ↓reduceIte,
        Int.cast_neg, Int.cast_natCast]
      exact div_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr (Nat.cast_nonneg _)) (Nat.cast_nonneg _)
    simp only [naturalFloorWithWork, hs, ↓reduceIte, bitWordValue]
    exact (Nat.floor_eq_zero.mpr (lt_of_le_of_lt hn (by norm_num))).symm

theorem naturalFloorWithWork_bounds (a : BinaryFraction) :
    (naturalFloorWithWork a).1.length ≤ a.width ∧
      (naturalFloorWithWork a).2 ≤ 200 * (a.width + 1) ^ 2 := by
  have hn : a.numerator.bits.length ≤ a.width := le_max_left _ _
  have hd : a.denominator.length ≤ a.width := le_max_right _ _
  have hlen := (bitDivModWithWork_lengths a.denominator a.numerator.bits).1
  have hc := bitDivModWithWork_cost a.denominator a.numerator.bits
  unfold naturalFloorWithWork
  split_ifs <;> dsimp only
  · exact ⟨Nat.zero_le _, by nlinarith only [Nat.zero_le a.width]⟩
  · exact ⟨hlen.le.trans hn, by nlinarith only [hc, hn, hd, Nat.zero_le a.width]⟩



@[simp] theorem cellCoordinatesWithWork_value (side : BinaryFraction) (p radius : ℕ) :
    (cellCoordinatesWithWork side p radius).1 = localCellCoordinates side.value p radius := by
  simp only [cellCoordinatesWithWork, naturalFloorWithWork_value, divWithWork_value,
    nat_value, saturatingSubtractWithWork_value, binaryNaturalAddWithWork_value,
    localCellCoordinates]



@[simp] theorem targetCellsWithWork_value {n : ℕ} (side : BinaryFraction) (radius : ℕ)
    (q : TargetInterval n) :
    (targetCellsWithWork side radius q).1 = localTargetCells side.value radius q := by
  simp only [targetCellsWithWork, arithmeticFlatMapWithWork_value, arithmeticMapWithWork_value,
    cellCoordinatesWithWork_value, localTargetCells]



@[simp] theorem cellGridPointsWithWork_value (side : BinaryFraction) (g cell : ℕ) :
    (cellGridPointsWithWork side g cell).1 = cellGridPoints side.value g cell := by
  simp only [cellGridPointsWithWork, arithmeticMapWithWork_value, binaryNaturalMulWithWork_value,
    binaryNaturalAddWithWork_value, canonicalMulWithWork_value, divWithWork_value,
    naturalCeilingWithWork_value, nat_value, saturatingSubtractWithWork_value, cellGridPoints,
    Nat.cast_add, Nat.cast_one]



@[simp] theorem cellStatesWithWork_value (n g : ℕ) (side : BinaryFraction) (cell : ℕ × ℕ) :
    (cellStatesWithWork n g side cell).1 = cellTargetStates n g side.value cell := by
  simp only [cellStatesWithWork, arithmeticFlatMapWithWork_value, filterMapWithWork_value,
    cellGridPointsWithWork_value, makeStateWithWork_value, cellTargetStates]

end BinaryFraction
end EditApproximation

end OAI