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e63eb7f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 | 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
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