File size: 3,805 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 68 69 70 | import OAI.Combinatorics.EditApproximation.Binary.BinaryPhysicalContinuation
import OAI.Combinatorics.EditApproximation.Binary.BinaryPowerStorage
import OAI.Combinatorics.EditApproximation.Definitions
namespace OAI
/-! Cache insertion compresses the actual continuation result by executed
Boolean multiplication and division. Its cost is part of the return
counter; no rational normalization operation is assigned unit cost. -/
namespace EditApproximation
namespace BinaryFraction
theorem finishCachedWithWork_value (raw : BinaryFraction × ℕ) (base : List Bool)
(hbase : 0 < bitWordValue base) (remaining : ℕ)
(hden : raw.1.value.den ∣ bitWordValue base ^ remaining) :
(finishCachedWithWork raw base hbase remaining).1.value = raw.1.value :=
powerDenominatorWithWork_value raw.1 base hbase remaining hden
theorem finishCachedWithWork_bounds (raw : BinaryFraction × ℕ) (base : List Bool)
(hbase : 0 < bitWordValue base) (remaining A B C V : ℕ)
(ha : raw.1.width ≤ A) (hb : base.length ≤ B) (hc : raw.2 ≤ C)
(hden : raw.1.value.den ∣ bitWordValue base ^ remaining) (hv : |raw.1.value| ≤ V) :
(finishCachedWithWork raw base hbase remaining).1.width ≤
Nat.size V + 2 * (remaining * B + 1) + 1 ∧
(finishCachedWithWork raw base hbase remaining).2 ≤
C + powerDenominatorBitBudget remaining A B + 1 := by
refine ⟨powerDenominatorWithWork_width raw.1 base hbase remaining V B hb hden hv, ?_⟩
exact Nat.add_le_add_right (Nat.add_le_add hc
(powerDenominatorWithWork_cost raw.1 base hbase remaining A B ha hb)) 1
def cachedPhysicalContinuationWithWork {nx ny M : ℕ} (parent : TargetInterval nx)
(representatives : List (ℕ × TargetInterval ny)) (P F : ℕ)
(delta fallback : BinaryFraction)
(old current : Fin M × TargetInterval ny → BinaryFraction)
(groups online : List (BinaryBellmanAction (TargetInterval ny) (Fin M × TargetInterval ny) M))
(query : TargetInterval ny) (base : List Bool) (hbase : 0 < bitWordValue base) (remaining : ℕ) :
BinaryFraction × ℕ :=
finishCachedWithWork
(physicalContinuationWithWork parent representatives P F delta fallback old current groups online query)
base hbase remaining
theorem cachedPhysicalContinuationWithWork_value {nx ny M : ℕ} (parent : TargetInterval nx)
(representatives : List (ℕ × TargetInterval ny)) (P F : ℕ)
(delta fallback : BinaryFraction)
(old current : Fin M × TargetInterval ny → BinaryFraction)
(groups online : List (BinaryBellmanAction (TargetInterval ny) (Fin M × TargetInterval ny) M))
(query : TargetInterval ny) (base : List Bool) (hbase : 0 < bitWordValue base) (remaining : ℕ)
(hP : 0 < P)
(hden : (rationalWideStep parent representatives P (fun _ => fallback.value) (128 * F) delta.value
(fun s => (old s).value) (fun s => (current s).value)
(groups.map BinaryBellmanAction.decode) (online.map BinaryBellmanAction.decode) query).den ∣
bitWordValue base ^ remaining) :
(cachedPhysicalContinuationWithWork parent representatives P F delta fallback old current groups online
query base hbase remaining).1.value =
rationalWideStep parent representatives P (fun _ => fallback.value) (128 * F) delta.value
(fun s => (old s).value) (fun s => (current s).value)
(groups.map BinaryBellmanAction.decode) (online.map BinaryBellmanAction.decode) query := by
apply (finishCachedWithWork_value _ base hbase remaining ?_).trans
(physicalContinuationWithWork_value parent representatives P F delta fallback old current groups online query hP)
rwa [physicalContinuationWithWork_value parent representatives P F delta fallback old current groups online query hP]
end BinaryFraction
end EditApproximation
end OAI
|