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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 | import OAI.Combinatorics.EditApproximation.Coarse.CoarseReadTree
import OAI.Combinatorics.EditApproximation.Coarse.CoarseIndexedFrameWork
import OAI.Combinatorics.EditApproximation.Binary.BinaryPowerStorage
import OAI.Combinatorics.EditApproximation.Definitions
namespace OAI
namespace EditApproximation
theorem saturatingSubtractWithWork_length (a b : ℕ) :
(saturatingSubtractWithWork a b).1.length ≤ max (Nat.size a) (Nat.size b) := by
unfold saturatingSubtractWithWork
dsimp only
split_ifs
· exact Nat.zero_le _
· simpa only [binaryNaturalSubtractWithWork, bitSubtractWithWork_length,
Nat.size_eq_bits_len] using (le_refl (max (Nat.size a) (Nat.size b)))
/-- Construct a descriptor's clipped length by Boolean exponentiation,
subtraction and minimum, retaining the resulting bits. -/
theorem coarseDescriptorWithWork_value (B d n start : ℕ) :
bitWordValue (coarseDescriptorWithWork B d n start).1 = coarseDescriptorLength B d n start := by
simp only [coarseDescriptorWithWork, wordMinWithWork_value,
bitPowerWithWork_value, bitWordValue_bits, saturatingSubtractWithWork_value,
coarseDescriptorLength]
theorem coarseDescriptorWithWork_cost (B d n start A : ℕ)
(hB : Nat.size B ≤ A) (hn : Nat.size n ≤ A) (hs : Nat.size start ≤ A) :
(coarseDescriptorWithWork B d n start).2 ≤
bitPowerBitBudget d A + 100 * (d + 2) * (A + 1) := by
have hp := bitPowerWithWork_cost B.bits d A (by simpa only [Nat.size_eq_bits_len] using hB)
have hpl := bitPowerWithWork_length_le B.bits d A (by simpa only [Nat.size_eq_bits_len] using hB)
have hr := saturatingSubtractWithWork_cost n start
have hrl := saturatingSubtractWithWork_length n start
have hm := wordLEWithWork_cost (bitPowerWithWork B.bits d).1
(saturatingSubtractWithWork n start).1
have hd : Nat.size d ≤ d := Nat.size_le.mpr (Nat.lt_pow_self (by decide : 1 < (2 : ℕ)))
simp only [coarseDescriptorWithWork, wordMinWithWork]
nlinarith only [hp, hpl, hr, hrl, hm, hd, hB, hn, hs,
max_le hn hs, Nat.zero_le d, Nat.zero_le A]
end EditApproximation
end OAI
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