File size: 2,336 Bytes
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 | import OAI.Combinatorics.EditApproximation.Coarse.CoarseArraySimulation
namespace OAI
namespace EditApproximation
namespace BinaryFraction
/-- Leading-zero removal is enough: this property does not assert that a
stored rational is reduced by a gcd. -/
def CoarseCompact (a : BinaryFraction) : Prop :=
a.width ≤ a.representation.storageBits
theorem coarseCompact_canonical (a : BinaryFraction) :
CoarseCompact (canonicalizeWithWork a).1 := by
change (canonicalizeWithWork a).1.width ≤ _
rw [canonicalizeWithWork_representation]
exact canonicalizeWithWork_width_le_storageBits a
theorem coarseCompact_gridNat (den : List Bool) (hden : 0 < bitWordValue den) (n : ℕ) :
CoarseCompact (coarseGridNatWithWork den hden n).1 := by
unfold coarseGridNatWithWork
exact coarseCompact_canonical _
theorem coarseCompact_gridAdd (a b : BinaryFraction) :
CoarseCompact (coarseGridAddWithWork a b).1 := by
unfold coarseGridAddWithWork
exact coarseCompact_canonical _
theorem coarseCompact_min (a b : BinaryFraction) (ha : CoarseCompact a) (hb : CoarseCompact b) :
CoarseCompact (minWithWork a b).1 := by
unfold minWithWork
dsimp only
split_ifs <;> assumption
theorem coarseCompact_max (a b : BinaryFraction) (ha : CoarseCompact a) (hb : CoarseCompact b) :
CoarseCompact (maxWithWork a b).1 := by
unfold maxWithWork
dsimp only
split_ifs <;> assumption
theorem coarseCompact_divide (a b : BinaryFraction) :
CoarseCompact (coarseDivideWithWork a b).1 := by
unfold coarseDivideWithWork fixedDenominatorWithWork
exact coarseCompact_canonical _
theorem coarseCompact_contribution (threshold scale a : BinaryFraction)
(hs : CoarseCompact scale) (ha : CoarseCompact a) :
CoarseCompact (coarseContributionWithWork threshold scale a).1 := by
unfold coarseContributionWithWork
dsimp only
split_ifs
· exact coarseCompact_max a scale ha hs
· exact coarseCompact_gridNat _ _ 0
theorem coarseCompact_width (a : BinaryFraction) (V D : ℕ)
(ha : CoarseCompact a) (hv : |a.value| ≤ V)
(hd : Nat.size a.representation.den ≤ D) :
a.width ≤ Nat.size V + 2 * D + 1 := by
apply ha.trans
exact UnreducedRational.storageBits_le a.representation V D
(by simpa only [representation_value] using hv) hd
end BinaryFraction
end EditApproximation
end OAI
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