| import OAI.Combinatorics.EditApproximation.Coarse.CoarseArraySimulation |
|
|
| namespace OAI |
|
|
| namespace EditApproximation |
| namespace BinaryFraction |
|
|
| / |
| 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 |
|
|
