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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 | import OAI.Combinatorics.EditApproximation.Binary.BinaryFixedDenominator
import OAI.Combinatorics.EditApproximation.Binary.BinaryFiniteLoops
import OAI.Combinatorics.EditApproximation.Coarse.CoarsePrecision
import OAI.Combinatorics.EditApproximation.Definitions
namespace OAI
/-! Exact arithmetic on a shared coarse-array denominator. Addition touches
only the signed numerators and keeps the denominator, preventing denominator
growth across the child-array sums. All counters run Boolean primitives. -/
namespace EditApproximation
namespace BinaryFraction
/-- Store an integer on the common denominator by a literal bit product. -/
theorem coarseGridNatWithWork_value (denominator : List Bool)
(hden : 0 < bitWordValue denominator) (n : ℕ) :
(coarseGridNatWithWork denominator hden n).1.value = n := by
unfold coarseGridNatWithWork
dsimp only
rw [← representation_value, canonicalizeWithWork_representation, representation_value]
simp only [value, SignedBinary.value, signedMagnitude, Bool.false_eq_true, ↓reduceIte,
bitMulWithWork_value, bitWordValue_bits]
push_cast
exact mul_div_cancel_right₀ _ (by exact_mod_cast hden.ne')
theorem coarseGridNatWithWork_den (denominator : List Bool)
(hden : 0 < bitWordValue denominator) (n : ℕ) :
(coarseGridNatWithWork denominator hden n).1.representation.den =
bitWordValue denominator := by
unfold coarseGridNatWithWork
dsimp only
rw [canonicalizeWithWork_representation]
rfl
theorem coarseGridNatWithWork_cost (denominator : List Bool)
(hden : 0 < bitWordValue denominator) (n B : ℕ)
(hn : Nat.size n ≤ B) (hd : denominator.length ≤ B) :
(coarseGridNatWithWork denominator hden n).2 ≤ 100 * (B + 1) ^ 2 := by
have hc := bitMulWithWork_cost n.bits denominator
have hl := bitMulWithWork_length n.bits denominator
have hnb : n.bits.length ≤ B := by simpa only [← Nat.size_eq_bits_len] using hn
have hw : (⟨⟨false, (bitMulWithWork n.bits denominator).1⟩, denominator,
hden⟩ : BinaryFraction).width ≤ 3 * B + 1 := by
change max (bitMulWithWork n.bits denominator).1.length denominator.length ≤ _
apply max_le <;> omega
have ht := canonicalizeWithWork_cost
(⟨⟨false, (bitMulWithWork n.bits denominator).1⟩, denominator, hden⟩ : BinaryFraction)
have hp : (bitMulWithWork n.bits denominator).2 ≤ 40 * (B + 1) ^ 2 := by
apply hc.trans
have hm := Nat.mul_le_mul (show n.bits.length + 1 ≤ B + 1 by omega)
(show n.bits.length + denominator.length + 1 ≤ 2 * (B + 1) by omega)
nlinarith only [hm]
change (bitMulWithWork n.bits denominator).2 + _ + 3 ≤ _
nlinarith only [hp, ht, hw, Nat.zero_le B]
/-- Add two stored cells after their denominators have been made identical. -/
theorem coarseGridAddWithWork_value (a b : BinaryFraction)
(hden : bitWordValue a.denominator = bitWordValue b.denominator) :
(coarseGridAddWithWork a b).1.value = a.value + b.value := by
unfold coarseGridAddWithWork
dsimp only
rw [← representation_value, canonicalizeWithWork_representation, representation_value]
simp only [value, SignedBinary.addWithWork_value, Int.cast_add, ← hden]
ring
theorem coarseGridAddWithWork_den (a b : BinaryFraction) :
(coarseGridAddWithWork a b).1.representation.den = a.representation.den := by
unfold coarseGridAddWithWork
dsimp only
rw [canonicalizeWithWork_representation]
rfl
theorem coarseGridAddWithWork_cost (a b : BinaryFraction) (B : ℕ)
(ha : a.width ≤ B) (hb : b.width ≤ B) :
(coarseGridAddWithWork a b).2 ≤ 100 * (B + 1) := by
have han : a.numerator.bits.length ≤ B := (le_max_left _ _).trans ha
have hbn : b.numerator.bits.length ≤ B := (le_max_left _ _).trans hb
have had : a.denominator.length ≤ B := (le_max_right _ _).trans ha
have hc := SignedBinary.addWithWork_cost a.numerator b.numerator
have hl := SignedBinary.addWithWork_length a.numerator b.numerator
have hw : (⟨(a.numerator.addWithWork b.numerator).1, a.denominator,
a.denominator_pos⟩ : BinaryFraction).width ≤ B + 1 := by
change max (a.numerator.addWithWork b.numerator).1.bits.length a.denominator.length ≤ _
apply max_le <;> omega
have ht := canonicalizeWithWork_cost
(⟨(a.numerator.addWithWork b.numerator).1, a.denominator, a.denominator_pos⟩ : BinaryFraction)
change (a.numerator.addWithWork b.numerator).2 + _ + 3 ≤ _
omega
theorem coarseGridAddWithWork_width (a b : BinaryFraction) (V B : ℕ)
(hden : bitWordValue a.denominator = bitWordValue b.denominator)
(hvalue : |a.value + b.value| ≤ V) (hbits : a.denominator.length ≤ B) :
(coarseGridAddWithWork a b).1.width ≤ Nat.size V + 2 * B + 1 := by
let answer := (coarseGridAddWithWork a b).1
have hv : |answer.representation.value| ≤ V := by
rw [representation_value]
exact (coarseGridAddWithWork_value a b hden) ▸ hvalue
have hd : Nat.size answer.representation.den ≤ B := by
rw [coarseGridAddWithWork_den]
exact (Nat.size_le.mpr (bitWordValue_lt_pow_length a.denominator)).trans hbits
have hs := UnreducedRational.storageBits_le answer.representation V B hv hd
have hw : answer.width ≤ answer.representation.storageBits := by
unfold answer coarseGridAddWithWork
dsimp only
rw [canonicalizeWithWork_representation]
exact canonicalizeWithWork_width_le_storageBits _
exact hw.trans hs
end BinaryFraction
end EditApproximation
end OAI
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