openAI-math-dataset / lean /OAI /Combinatorics /EditApproximation /Numerics /RationalIntermediateBits.lean
| import OAI.Combinatorics.EditApproximation.Numerics.NumericalWork | |
| namespace OAI | |
| /-! Bounds on the reduced integer representations produced by the rational | |
| operations in the materialized numerical routines. -/ | |
| namespace EditApproximation | |
| theorem rational_add_numerator_le (a b : ℚ) : | |
| (a + b).num.natAbs ≤ a.num.natAbs * b.den + b.num.natAbs * a.den := by | |
| have heq := congrArg Int.natAbs (Rat.add_num_den' a b) | |
| simp only [Int.natAbs_mul, Int.natAbs_natCast] at heq | |
| have hadd := Int.natAbs_add_le (a.num * (b.den : ℤ)) (b.num * (a.den : ℤ)) | |
| simp only [Int.natAbs_mul, Int.natAbs_natCast] at hadd | |
| have hden := Nat.le_of_dvd (Nat.mul_pos a.den_pos b.den_pos) (Rat.add_den_dvd a b) | |
| have hmul := Nat.mul_le_mul hadd hden | |
| have hpos := Nat.mul_pos a.den_pos b.den_pos | |
| nlinarith only [heq, hmul, hpos] | |
| theorem rational_mul_numerator_le (a b : ℚ) : | |
| (a * b).num.natAbs ≤ a.num.natAbs * b.num.natAbs := by | |
| have heq := congrArg Int.natAbs (Rat.mul_num_den' a b) | |
| simp only [Int.natAbs_mul, Int.natAbs_natCast] at heq | |
| have hden := Nat.le_of_dvd (Nat.mul_pos a.den_pos b.den_pos) (Rat.mul_den_dvd a b) | |
| have hmul := Nat.mul_le_mul_left (a.num.natAbs * b.num.natAbs) hden | |
| have hpos := Nat.mul_pos a.den_pos b.den_pos | |
| nlinarith only [heq, hmul, hpos] | |
| theorem rationalEncodingBits_add (a b : ℚ) : | |
| rationalEncodingBits (a + b) ≤ 2 * rationalEncodingBits a + 2 * rationalEncodingBits b + 2 := by | |
| have hnum := (Nat.size_le_size (rational_add_numerator_le a b)).trans | |
| (natural_sum_bit_length (a.num.natAbs * b.den) (b.num.natAbs * a.den)) | |
| have hnum1 := natural_product_bit_length a.num.natAbs b.den | |
| have hnum2 := natural_product_bit_length b.num.natAbs a.den | |
| have hden := (Nat.size_le_size (Nat.le_of_dvd (Nat.mul_pos a.den_pos b.den_pos) | |
| (Rat.add_den_dvd a b))).trans (natural_product_bit_length a.den b.den) | |
| unfold rationalEncodingBits | |
| omega | |
| theorem rationalEncodingBits_mul (a b : ℚ) : | |
| rationalEncodingBits (a * b) ≤ rationalEncodingBits a + rationalEncodingBits b + 1 := by | |
| have hnum := (Nat.size_le_size (rational_mul_numerator_le a b)).trans | |
| (natural_product_bit_length a.num.natAbs b.num.natAbs) | |
| have hden := (Nat.size_le_size (Nat.le_of_dvd (Nat.mul_pos a.den_pos b.den_pos) | |
| (Rat.mul_den_dvd a b))).trans (natural_product_bit_length a.den b.den) | |
| unfold rationalEncodingBits | |
| omega | |
| theorem rationalEncodingBits_neg (a : ℚ) : rationalEncodingBits (-a) = rationalEncodingBits a := by | |
| simp [rationalEncodingBits] | |
| theorem rationalEncodingBits_sub (a b : ℚ) : | |
| rationalEncodingBits (a - b) ≤ 2 * rationalEncodingBits a + 2 * rationalEncodingBits b + 2 := by | |
| simpa only [sub_eq_add_neg, rationalEncodingBits_neg] using rationalEncodingBits_add a (-b) | |
| theorem rationalEncodingBits_inv (a : ℚ) : | |
| rationalEncodingBits a⁻¹ ≤ rationalEncodingBits a + 2 := by | |
| by_cases ha : a.num = 0 | |
| · have hzero : a = 0 := Rat.zero_of_num_zero ha | |
| subst a | |
| norm_num [rationalEncodingBits] | |
| · simp only [rationalEncodingBits, Rat.num_inv, Rat.den_inv, ha, ↓reduceIte, | |
| Int.natAbs_mul, Int.natAbs_natCast, Int.natAbs_sign_of_ne_zero ha, Nat.one_mul] | |
| omega | |
| theorem rationalEncodingBits_div (a b : ℚ) : | |
| rationalEncodingBits (a / b) ≤ rationalEncodingBits a + rationalEncodingBits b + 3 := by | |
| have hmul := rationalEncodingBits_mul a b⁻¹ | |
| have hinv := rationalEncodingBits_inv b | |
| rw [div_eq_mul_inv] | |
| omega | |
| theorem rationalEncodingBits_natCast (n : ℕ) : | |
| rationalEncodingBits (n : ℚ) = Nat.size n + 2 := by | |
| simp [rationalEncodingBits] | |
| theorem rationalEncodingBits_pow (a : ℚ) (n : ℕ) : | |
| rationalEncodingBits (a ^ n) ≤ n * (rationalEncodingBits a + 1) + 3 := by | |
| induction n with | |
| | zero => norm_num [rationalEncodingBits] | |
| | succ n ih => | |
| rw [pow_succ] | |
| have hmul := rationalEncodingBits_mul (a ^ n) a | |
| nlinarith only [ih, hmul] | |
| end EditApproximation | |
| end OAI | |