| import OAI.Computability.BinPacking.Packing.PackingScalar |
| import OAI.Computability.BinPacking.Trees.CompletionGeometry |
|
|
| namespace OAI |
|
|
| namespace BinPackingGap.IntegerPackingArithmetic |
|
|
| def widthRadix (D : ℕ) : ℕ := 100 * (D + 1) |
|
|
| def geomDenominator (m R D L : ℕ) : ℕ := |
| 160 * (R + 1) ^ m * widthRadix D ^ L |
|
|
| def sizeDenominator (n geom : ℕ) : ℕ := |
| 10 ^ 9 * primaryBound * 3 ^ n * geom |
|
|
| def signedNumerator (n geom : ℕ) (p q W : ℤ) : ℤ := |
| 200000000 * (primaryBound : ℤ) * 3 ^ n * geom + |
| 1000000 * 3 ^ n * geom * p + 1000 * geom * q + W |
|
|
| def itemNumerator {n : ℕ} (geom : ℕ) (subclass : Subclass) |
| (label : Option (Fin n)) (W : ℤ) : ℤ := |
| signedNumerator n geom (primaryScore subclass) (secondaryScore subclass label) W |
|
|
| theorem widthRadix_pos (D : ℕ) : 0 < widthRadix D := by |
| unfold widthRadix |
| positivity |
|
|
| theorem geomDenominator_pos (m R D L : ℕ) : 0 < geomDenominator m R D L := by |
| unfold geomDenominator |
| exact Nat.mul_pos (by positivity) (pow_pos (widthRadix_pos D) L) |
|
|
| theorem sizeDenominator_pos (n geom : ℕ) (hgeom : 0 < geom) : |
| 0 < sizeDenominator n geom := by |
| unfold sizeDenominator primaryBound |
| positivity |
|
|
| theorem signedNumerator_ratio (n geom : ℕ) (hgeom : 0 < geom) |
| (p q W : ℤ) : |
| (signedNumerator n geom p q W : ℚ) / (sizeDenominator n geom : ℚ) = |
| 1 / 5 + gamma * (p : ℚ) + mu n * (q : ℚ) + |
| nu n * ((W : ℚ) / geom) := by |
| have hg : (geom : ℚ) ≠ 0 := by exact_mod_cast hgeom.ne' |
| have hp : (primaryBound : ℚ) ≠ 0 := by norm_num [primaryBound] |
| have hq : (3 : ℚ) ^ n ≠ 0 := pow_ne_zero n (by norm_num) |
| simp only [signedNumerator, sizeDenominator, nu, mu, gamma] |
| push_cast |
| field_simp [hg, hp, hq] |
| ring |
|
|
| theorem itemNumerator_ratio {n : ℕ} (geom : ℕ) (hgeom : 0 < geom) |
| (subclass : Subclass) (label : Option (Fin n)) (W : ℤ) : |
| (itemNumerator geom subclass label W : ℚ) / (sizeDenominator n geom : ℚ) = |
| encodedSize subclass label ((W : ℚ) / geom) := by |
| exact signedNumerator_ratio n geom hgeom (primaryScore subclass) (secondaryScore subclass label) W |
|
|
| theorem itemNumerator_pos_lt {n : ℕ} (geom : ℕ) (hgeom : 0 < geom) |
| (subclass : Subclass) (label : Option (Fin n)) (W : ℤ) |
| (hW : |(W : ℚ) / geom| ≤ 6) : |
| 0 < itemNumerator geom subclass label W ∧ |
| itemNumerator geom subclass label W < (sizeDenominator n geom : ℤ) := by |
| have hd : (0 : ℚ) < sizeDenominator n geom := by |
| exact_mod_cast sizeDenominator_pos n geom hgeom |
| have hs := encodedSize_mem_interval subclass label ((W : ℚ) / geom) hW |
| rw [← itemNumerator_ratio geom hgeom subclass label W] at hs |
| have hpos : (0 : ℚ) < (itemNumerator geom subclass label W : ℚ) := by |
| have hdiv : (0 : ℚ) < (itemNumerator geom subclass label W : ℚ) / |
| (sizeDenominator n geom : ℚ) := lt_trans (by norm_num) hs.1 |
| rcases div_pos_iff.mp hdiv with h | h |
| · exact h.1 |
| · exact (not_lt_of_ge hd.le h.2).elim |
| have hlt : (itemNumerator geom subclass label W : ℚ) < (sizeDenominator n geom : ℚ) := by |
| have h := (div_lt_iff₀ hd).mp hs.2 |
| simpa only [one_mul] using h |
| constructor |
| · exact_mod_cast hpos |
| · exact_mod_cast hlt |
|
|
| def rawItem {n : ℕ} (geom : ℕ) (subclass : Subclass) |
| (label : Option (Fin n)) (W : ℤ) : ℕ × ℕ := |
| ((itemNumerator geom subclass label W).toNat, sizeDenominator n geom) |
|
|
| theorem rawItem_numerator_cast {n : ℕ} (geom : ℕ) (hgeom : 0 < geom) |
| (subclass : Subclass) (label : Option (Fin n)) (W : ℤ) |
| (hW : |(W : ℚ) / geom| ≤ 6) : |
| ((rawItem geom subclass label W).1 : ℤ) = itemNumerator geom subclass label W := by |
| exact Int.toNat_of_nonneg (itemNumerator_pos_lt geom hgeom subclass label W hW).1.le |
|
|
| theorem rawItem_valid {n : ℕ} (geom : ℕ) (hgeom : 0 < geom) |
| (subclass : Subclass) (label : Option (Fin n)) (W : ℤ) |
| (hW : |(W : ℚ) / geom| ≤ 6) : |
| 0 < (rawItem geom subclass label W).1 ∧ |
| (rawItem geom subclass label W).1 < (rawItem geom subclass label W).2 := by |
| have hs := itemNumerator_pos_lt geom hgeom subclass label W hW |
| rw [← rawItem_numerator_cast geom hgeom subclass label W hW] at hs |
| exact_mod_cast hs |
|
|
| theorem rawItem_ratio {n : ℕ} (geom : ℕ) (hgeom : 0 < geom) |
| (subclass : Subclass) (label : Option (Fin n)) (W : ℤ) |
| (hW : |(W : ℚ) / geom| ≤ 6) : |
| ((rawItem geom subclass label W).1 : ℚ) / (rawItem geom subclass label W).2 = |
| encodedSize subclass label ((W : ℚ) / geom) := by |
| have hc : ((rawItem geom subclass label W).1 : ℚ) = |
| (itemNumerator geom subclass label W : ℚ) := by |
| exact_mod_cast rawItem_numerator_cast geom hgeom subclass label W hW |
| rw [hc] |
| exact itemNumerator_ratio geom hgeom subclass label W |
|
|
| def localNumerator (D L ell : ℕ) : ℕ := 10 * widthRadix D ^ (L - ell) |
|
|
| theorem geomDenominator_mul_gap (m R D L : ℕ) : |
| (geomDenominator m R D L : ℚ) * Geometry.gap m R = |
| 10 * (widthRadix D : ℚ) ^ L := by |
| have hp : ((R : ℚ) + 1) ^ m ≠ 0 := |
| pow_ne_zero m (by positivity) |
| unfold geomDenominator Geometry.gap |
| push_cast |
| field_simp [hp] |
| ring |
|
|
| theorem geomDenominator_mul_lambda (m R D L ell : ℕ) (hell : ell ≤ L) : |
| (geomDenominator m R D L : ℚ) * Geometry.lambda m R D ell = |
| (localNumerator D L ell : ℚ) := by |
| have hc : (widthRadix D : ℚ) ≠ 0 := by |
| exact_mod_cast (widthRadix_pos D).ne' |
| have hpow : (widthRadix D : ℚ) ^ L = |
| (widthRadix D : ℚ) ^ ell * (widthRadix D : ℚ) ^ (L - ell) := by |
| rw [← pow_add, Nat.add_sub_of_le hell] |
| have hrad : (100 * ((D : ℚ) + 1)) = (widthRadix D : ℚ) := by |
| simp [widthRadix] |
| rw [Geometry.lambda, ← mul_div_assoc, geomDenominator_mul_gap, hrad, hpow] |
| simp only [localNumerator, Nat.cast_mul, Nat.cast_ofNat, Nat.cast_pow] |
| field_simp [hc] |
|
|
| theorem geomDenominator_mul_delta (m R D L : ℕ) : |
| (geomDenominator m R D L : ℚ) * Geometry.delta m R D L = 1 := by |
| rw [Geometry.delta, ← mul_div_assoc, geomDenominator_mul_lambda m R D L L le_rfl] |
| norm_num [localNumerator] |
|
|
| def baselineNumerator (m R D L : ℕ) (r : Geometry.Position m R) : ℤ := |
| 80 * (R + 1 : ℤ) ^ m * (widthRadix D : ℤ) ^ L + |
| 10 * (widthRadix D : ℤ) ^ L * |
| (((r.2.val : ℤ) + 1) * (R + 1 : ℤ) ^ r.1.val - 1) |
|
|
| def betaNumerator (m R D L : ℕ) (e : Fin m) : ℤ := |
| 10 * (widthRadix D : ℤ) ^ L * ((R + 1 : ℤ) ^ e.val - 1) |
|
|
| theorem geomDenominator_mul_baseline (m R D L : ℕ) (r : Geometry.Position m R) : |
| (geomDenominator m R D L : ℚ) * Geometry.baseline m R r = |
| (baselineNumerator m R D L r : ℚ) := by |
| rw [Geometry.baseline, mul_add, ← mul_assoc, geomDenominator_mul_gap] |
| simp only [baselineNumerator, Geometry.heightCode, geomDenominator] |
| push_cast |
| ring |
|
|
| theorem geomDenominator_mul_beta (m R D L : ℕ) (e : Fin m) : |
| (geomDenominator m R D L : ℚ) * Geometry.beta m R e = |
| (betaNumerator m R D L e : ℚ) := by |
| rw [Geometry.beta, ← mul_assoc, geomDenominator_mul_gap] |
| simp only [betaNumerator, Geometry.theta] |
| push_cast |
| ring |
|
|
| end BinPackingGap.IntegerPackingArithmetic |
|
|
| end OAI |
|
|
