| import OAI.Combinatorics.EditApproximation.Binary.BinaryScaleGrid |
| import OAI.Combinatorics.EditApproximation.Geometry.LocalCells |
| import OAI.Combinatorics.EditApproximation.Definitions |
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| namespace OAI |
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| /-! The whole-cell grids and nearby cell indices are generated by literal |
| binary quotient, endpoint and bounded-range operations. -/ |
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| namespace EditApproximation |
| namespace BinaryFraction |
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| @[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 |
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| 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]⟩ |
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| @[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] |
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| @[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] |
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| @[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] |
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| @[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] |
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|
| end BinaryFraction |
| end EditApproximation |
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| end OAI |
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|
