File size: 4,077 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 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 | import OAI.Combinatorics.EditApproximation.Numerics.ScalarAllocation
import OAI.Combinatorics.EditApproximation.Coarse.CoarseReadAccess
import OAI.Combinatorics.EditApproximation.Coarse.CoarseReadLeafWork
namespace OAI
namespace EditApproximation
/-- Comparisons and minimum reuse their input bit arrays. -/
def coarseSaturatingSubtractAllocation (a b : ℕ) : ℕ :=
if (binaryNaturalCompareWithWork a b).1 = .lt then 0
else bitSubtractAllocation a.bits b.bits false
theorem coarseSaturatingSubtractAllocation_le_work (a b : ℕ) :
coarseSaturatingSubtractAllocation a b ≤ (saturatingSubtractWithWork a b).2 := by
have hs := bitSubtractAllocation_le_work a.bits b.bits false
unfold coarseSaturatingSubtractAllocation saturatingSubtractWithWork
dsimp only
split_ifs <;> simp only [binaryNaturalSubtractWithWork] <;> omega
def coarseDescriptorAllocation (B d n start : ℕ) : ℕ :=
bitPowerAllocation B.bits d + coarseSaturatingSubtractAllocation n start +
Nat.size B + Nat.size d + Nat.size n + Nat.size start
theorem coarseDescriptorAllocation_le_work (B d n start : ℕ) :
coarseDescriptorAllocation B d n start ≤ (coarseDescriptorWithWork B d n start).2 := by
have hp := bitPowerAllocation_le_work B.bits d
have hs := coarseSaturatingSubtractAllocation_le_work n start
unfold coarseDescriptorAllocation coarseDescriptorWithWork
dsimp only
omega
def coarseChildOffsetAllocation (offset capacity index : ℕ) : ℕ :=
bitMulAllocation index.bits capacity.bits +
bitAddAllocation offset.bits (bitWordValue (binaryNaturalMulWithWork index capacity).1).bits false +
Nat.size index + Nat.size capacity + Nat.size offset
theorem coarseChildOffsetAllocation_le_work (offset capacity index : ℕ) :
coarseChildOffsetAllocation offset capacity index ≤ (coarseChildOffsetWithWork offset capacity index).2 := by
have hm := bitMulAllocation_le_work index.bits capacity.bits
have ha := bitAddAllocation_le_work offset.bits
(bitWordValue (binaryNaturalMulWithWork index capacity).1).bits false
unfold coarseChildOffsetAllocation coarseChildOffsetWithWork
simp only [binaryNaturalMulWithWork, binaryNaturalAddWithWork] at ha ⊢
omega
def coarseFramedReadAllocation (k n i : ℕ) : ℕ :=
Nat.size k + Nat.size n + Nat.size i +
if (binaryNaturalCompareWithWork i k).1 = .lt then 0 else
bitAddAllocation k.bits n.bits false +
if (binaryNaturalCompareWithWork i (bitWordValue (binaryNaturalAddWithWork k n).1)).1 = .lt
then coarseSaturatingSubtractAllocation i k else 0
theorem coarseFramedReadAllocation_le_work {α : Type*} (k n : ℕ) (word : List α) (i : ℕ) :
coarseFramedReadAllocation k n i ≤ (coarseFramedReadWithWork k n word i).2 := by
have ha := bitAddAllocation_le_work k.bits n.bits false
have hs := coarseSaturatingSubtractAllocation_le_work i k
unfold coarseFramedReadAllocation coarseFramedReadWithWork
dsimp only
split_ifs <;> simp only [binaryNaturalAddWithWork] at * <;> omega
namespace BinaryFraction
def coarseReadShiftAllocation (k n : ℕ) (letter : Option (Option ℕ))
(start : ℕ) (shift : Fin (2 * k + 1)) : ℕ :=
match letter with
| none => 0
| some _ =>
bitAddAllocation start.bits shift.val.bits false +
coarseFramedReadAllocation k n (bitWordValue (binaryNaturalAddWithWork start shift.val).1)
theorem coarseReadShiftAllocation_le_work (k n : ℕ) (target : List ℕ)
(letter : Option (Option ℕ)) (start : ℕ) (shift : Fin (2 * k + 1)) :
coarseReadShiftAllocation k n letter start shift ≤ (coarseReadShiftWithWork k n target letter start shift).2 := by
cases letter with
| none => exact Nat.zero_le _
| some letter =>
have ha := bitAddAllocation_le_work start.bits shift.val.bits false
have hf := coarseFramedReadAllocation_le_work k n target
(bitWordValue (binaryNaturalAddWithWork start shift.val).1)
unfold coarseReadShiftAllocation coarseReadShiftWithWork
simp only [binaryNaturalAddWithWork] at hf ⊢
omega
end BinaryFraction
end EditApproximation
end OAI
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