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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 | import OAI.Combinatorics.EditApproximation.Coarse.CoarseIndexedRead
import OAI.Combinatorics.EditApproximation.Binary.BinaryLocalGridWork
import OAI.Combinatorics.EditApproximation.Definitions
namespace OAI
namespace EditApproximation
/-- Read the framed target from its original array and three integer indices. -/
theorem coarseFramedReadWithWork_value {α : Type*} (k n : ℕ) (word : List α) (i : ℕ) :
(coarseFramedReadWithWork k n word i).1 = coarseFramedRead k n word i := by
unfold coarseFramedReadWithWork coarseFramedRead
dsimp only
by_cases hlo : i < k
· rw [ite_eq_left ((binaryNaturalCompareWithWork_lt i k).mpr hlo), ite_eq_right (by omega)]
· rw [ite_eq_right (fun h => hlo ((binaryNaturalCompareWithWork_lt i k).mp h))]
simp only [binaryNaturalAddWithWork_value]
by_cases hhi : i < k + n
· rw [ite_eq_left ((binaryNaturalCompareWithWork_lt i (k + n)).mpr hhi),
ite_eq_left (by omega)]
simp only [saturatingSubtractWithWork_value]
· rw [ite_eq_right (fun h => hhi ((binaryNaturalCompareWithWork_lt i (k + n)).mp h)),
ite_eq_right (by omega)]
theorem coarseFramedReadWithWork_cost {α : Type*} (k n : ℕ) (word : List α) (i A : ℕ)
(hk : Nat.size k ≤ A) (hn : Nat.size n ≤ A) (hi : Nat.size i ≤ A) :
(coarseFramedReadWithWork k n word i).2 ≤ 100 * (A + 1) := by
have hl := binaryNaturalCompareWithWork_cost i k
have ha := binaryNaturalAddWithWork_cost k n
have hs := binaryNaturalAdd_size_le k n
have hmax : max (Nat.size k) (Nat.size n) ≤ A := max_le hk hn
have hu := binaryNaturalCompareWithWork_cost i (bitWordValue (binaryNaturalAddWithWork k n).1)
rw [binaryNaturalAddWithWork_value] at hu
have hd := saturatingSubtractWithWork_cost i k
unfold coarseFramedReadWithWork
dsimp only
simp only [binaryNaturalAddWithWork_value]
split_ifs <;> dsimp only <;> omega
theorem coarseIndexed_leaf_cost {α : Type*} [DecidableEq α] {B : ℕ}
(k n : ℕ) (word : List α) (hn : word.length ≤ n) (letter : Option α)
(start : ℕ) (shift : Fin (2 * k + 1)) :
coarseShiftCost k (coarseTargetFrame k (coarsePadTo n word))
(.leaf (B := B) (some letter)) start shift =
if (coarseFramedReadWithWork k n word (start + shift.val)).1 = some letter then 0 else 1 := by
rw [coarseFramedReadWithWork_value, coarseFramedRead_value k n word hn]
rfl
end EditApproximation
end OAI
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