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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 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 | import OAI.Combinatorics.EditApproximation.Binary.BinaryFractionWork
import OAI.Combinatorics.EditApproximation.Binary.BinaryCanonicalWork
import OAI.Combinatorics.EditApproximation.Definitions
namespace OAI
/-! Fractions retain their unreduced numerator and denominator, but discard
leading zero bits after each operation. This exposes actual storage length
without charging or performing rational normalization. -/
namespace EditApproximation
namespace BinaryFraction
theorem canonicalizeWithWork_representation (a : BinaryFraction) :
(canonicalizeWithWork a).1.representation = a.representation := by
apply UnreducedRational.eq_of_num_den
· simp only [canonicalizeWithWork, representation, SignedBinary.value, trimBitWordWithWork_value]
· exact trimBitWordWithWork_value a.denominator
theorem canonicalizeWithWork_width (a : BinaryFraction) :
(canonicalizeWithWork a).1.width =
max (Nat.size a.representation.num.natAbs) (Nat.size a.representation.den) := by
have hnum : a.numerator.value.natAbs = bitWordValue a.numerator.bits := by
cases h : a.numerator.negative <;> simp [SignedBinary.value, signedMagnitude, h]
simp only [canonicalizeWithWork, width, trimBitWordWithWork_length, representation, hnum]
theorem canonicalizeWithWork_cost (a : BinaryFraction) :
(canonicalizeWithWork a).2 ≤ 6 * a.width + 4 := by
have hn : a.numerator.bits.length ≤ a.width := le_max_left _ _
have hd : a.denominator.length ≤ a.width := le_max_right _ _
simp only [canonicalizeWithWork, trimBitWordWithWork_cost]
omega
theorem canonicalAddWithWork_representation (a b : BinaryFraction) :
(canonicalAddWithWork a b).1.representation = a.representation.add b.representation := by
rw [canonicalAddWithWork, canonicalizeWithWork_representation, addWithWork_representation]
theorem canonicalMulWithWork_representation (a b : BinaryFraction) :
(canonicalMulWithWork a b).1.representation = a.representation.mul b.representation := by
rw [canonicalMulWithWork, canonicalizeWithWork_representation, mulWithWork_representation]
theorem rawAdd_width (a b : BinaryFraction) (B : ℕ) (ha : a.width ≤ B) (hb : b.width ≤ B) :
(a.addWithWork b).1.width ≤ 3 * B + 2 := by
have han : a.numerator.bits.length ≤ B := (le_max_left _ _).trans ha
have had : a.denominator.length ≤ B := (le_max_right _ _).trans ha
have hbn : b.numerator.bits.length ≤ B := (le_max_left _ _).trans hb
have hbd : b.denominator.length ≤ B := (le_max_right _ _).trans hb
have hfirst := SignedBinary.mulWithWork_length a.numerator b.denominatorInteger
have hsecond := SignedBinary.mulWithWork_length b.numerator a.denominatorInteger
have hsum := SignedBinary.addWithWork_length (a.numerator.mulWithWork b.denominatorInteger).1
(b.numerator.mulWithWork a.denominatorInteger).1
have hden := bitMulWithWork_length a.denominator b.denominator
simp only [denominatorInteger] at hfirst hsecond
dsimp only [addWithWork, width, denominatorInteger]
simp only [denominatorInteger] at hsum
apply max_le <;> omega
theorem rawMul_width (a b : BinaryFraction) (B : ℕ) (ha : a.width ≤ B) (hb : b.width ≤ B) :
(a.mulWithWork b).1.width ≤ 3 * B + 1 := by
have han : a.numerator.bits.length ≤ B := (le_max_left _ _).trans ha
have had : a.denominator.length ≤ B := (le_max_right _ _).trans ha
have hbn : b.numerator.bits.length ≤ B := (le_max_left _ _).trans hb
have hbd : b.denominator.length ≤ B := (le_max_right _ _).trans hb
have hnum := SignedBinary.mulWithWork_length a.numerator b.numerator
have hden := bitMulWithWork_length a.denominator b.denominator
dsimp only [mulWithWork, width]
apply max_le <;> omega
theorem canonicalAddWithWork_cost (a b : BinaryFraction) (B : ℕ) (ha : a.width ≤ B) (hb : b.width ≤ B) :
(canonicalAddWithWork a b).2 ≤ 6000 * (B + 1) ^ 2 := by
have hr := addWithWork_cost a b B ha hb
have hw := rawAdd_width a b B ha hb
have hc := canonicalizeWithWork_cost (a.addWithWork b).1
change (a.addWithWork b).2 + (canonicalizeWithWork (a.addWithWork b).1).2 ≤ _
nlinarith only [hr, hw, hc]
theorem canonicalMulWithWork_cost (a b : BinaryFraction) (B : ℕ) (ha : a.width ≤ B) (hb : b.width ≤ B) :
(canonicalMulWithWork a b).2 ≤ 2000 * (B + 1) ^ 2 := by
have hr := mulWithWork_cost a b B ha hb
have hw := rawMul_width a b B ha hb
have hc := canonicalizeWithWork_cost (a.mulWithWork b).1
change (a.mulWithWork b).2 + (canonicalizeWithWork (a.mulWithWork b).1).2 ≤ _
nlinarith only [hr, hw, hc]
theorem canonicalizeWithWork_width_le_storageBits (a : BinaryFraction) :
(canonicalizeWithWork a).1.width ≤ a.representation.storageBits := by
rw [canonicalizeWithWork_width]
unfold UnreducedRational.storageBits
exact max_le (by omega) (by omega)
theorem canonicalAddWithWork_width_le_storageBits (a b : BinaryFraction) :
(canonicalAddWithWork a b).1.width ≤
(a.representation.add b.representation).storageBits := by
have h := canonicalizeWithWork_width_le_storageBits (a.addWithWork b).1
rw [addWithWork_representation] at h
exact h
theorem canonicalMulWithWork_width_le_storageBits (a b : BinaryFraction) :
(canonicalMulWithWork a b).1.width ≤
(a.representation.mul b.representation).storageBits := by
have h := canonicalizeWithWork_width_le_storageBits (a.mulWithWork b).1
rw [mulWithWork_representation] at h
exact h
theorem canonicalAddWithWork_self_width (a b : BinaryFraction) :
(canonicalAddWithWork a b).1.width ≤ (canonicalAddWithWork a b).1.representation.storageBits := by
rw [canonicalAddWithWork_representation]
exact canonicalAddWithWork_width_le_storageBits a b
theorem canonicalMulWithWork_self_width (a b : BinaryFraction) :
(canonicalMulWithWork a b).1.width ≤ (canonicalMulWithWork a b).1.representation.storageBits := by
rw [canonicalMulWithWork_representation]
exact canonicalMulWithWork_width_le_storageBits a b
end BinaryFraction
end EditApproximation
end OAI
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