| import OAI.Combinatorics.EditApproximation.Binary.BinaryArithmeticWork |
| import OAI.Combinatorics.EditApproximation.Definitions |
|
|
|
|
| namespace OAI |
|
|
| /-! Removing only leading zero bits of an integer word. This is unrelated |
| to rational reduction: no common divisor is computed. -/ |
|
|
| namespace EditApproximation |
|
|
|
|
|
|
| theorem trimBitWordWithWork_value (bits : List Bool) : |
| bitWordValue (trimBitWordWithWork bits).1 = bitWordValue bits := by |
| induction bits with |
| | nil => rfl |
| | cons bit bits ih => |
| simp only [trimBitWordWithWork] |
| split_ifs with h |
| · have ht : bitWordValue bits = 0 := by rw [← ih, h.1]; rfl |
| simp [h.2, bitWordValue, ht] |
| · simp only [bitWordValue, ih] |
|
|
| theorem trimBitWordWithWork_canonical (bits : List Bool) : |
| (trimBitWordWithWork bits).1 = (bitWordValue bits).bits := by |
| induction bits with |
| | nil => rfl |
| | cons bit bits ih => |
| by_cases hzero : bitWordValue bits = 0 |
| · rw [trimBitWordWithWork, ih, hzero] |
| cases bit <;> simp [bitWordValue, hzero] |
| · have hnonempty : (bitWordValue bits).bits ≠ [] := by |
| intro h |
| have hv := bitWordValue_bits (bitWordValue bits) |
| rw [h] at hv |
| exact hzero hv.symm |
| have hvalue : bitWordValue (bit :: bits) = Nat.bit bit (bitWordValue bits) := by |
| cases bit <;> simp [bitWordValue, Nat.bit, Nat.add_comm] |
| rw [trimBitWordWithWork, ih, ite_eq_right (by simp [hnonempty]), hvalue, |
| Nat.bits_append_bit _ _ (fun h => False.elim (hzero h))] |
|
|
| theorem trimBitWordWithWork_length (bits : List Bool) : |
| (trimBitWordWithWork bits).1.length = Nat.size (bitWordValue bits) := by |
| rw [trimBitWordWithWork_canonical, Nat.size_eq_bits_len] |
|
|
| theorem trimBitWordWithWork_cost (bits : List Bool) : |
| (trimBitWordWithWork bits).2 = 3 * bits.length + 1 := by |
| induction bits with |
| | nil => rfl |
| | cons bit bits ih => |
| simp only [trimBitWordWithWork, List.length_cons, ih] |
| omega |
|
|
| end EditApproximation |
|
|
| end OAI |
|
|
