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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
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