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e320e82 | 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 | import OAI.Combinatorics.EditApproximation.AlmostLinear
import OAI.Combinatorics.EditApproximation.Definitions
namespace OAI.EditApproximation
/-- The finite data used after accepting an unskipped band. -/
def UnskippedBand.raw {α : Type*} {source target : List α} {M : ℕ}
{parent : TargetInterval source.length} {b P exponent : ℕ}
{tau F : ℚ} {child initial : Fin M → TargetInterval target.length → ℚ}
(member : UnskippedBand source target parent b P exponent tau F child initial) :
RawUnskippedBand M target.length :=
⟨member.center, member.actions, member.nonempty⟩
namespace BinaryFraction
theorem preparedUnskippedReadWithWorkRaw_eq {α : Type*} (source target : List α) {M : ℕ}
(parent : TargetInterval source.length) (b P exponent : ℕ) (hb : 0 < b) (hP : 0 < P)
(tau F : BinaryFraction)
(read initial : Fin M → TargetInterval target.length → BinaryFraction × ℕ)
(R C : ℕ) (hr : ∀ i q, (read i q).2 ≤ R) (hi : ∀ i q, (initial i q).2 ≤ C)
(center : TargetInterval target.length) :
preparedUnskippedReadWithWorkRaw parent b P exponent tau F read initial center =
Prod.map (Option.map UnskippedBand.raw) id
(preparedUnskippedReadWithWork source target parent b P exponent hb hP tau F
read initial R C hr hi center) := by
have hbandValue : (preparedRepresentativeReadWithWork parent center (parent.hi - parent.lo)
b P tau (inversePowerTwo exponent) read).1 =
representativeBandActions source target parent center b P tau.value
((2 : ℚ) ^ (-(exponent : ℤ))) (fun i q => (read i q).1.value) := by
apply (preparedRepresentativeReadWithWork_eq_cost parent center (parent.hi - parent.lo)
b P tau (inversePowerTwo exponent) read R hr).1.trans
simpa only [inversePowerTwo_value] using preparedRepresentativeWithWork_value source target
parent center b P hP tau (inversePowerTwo exponent) (fun i q => (read i q).1)
unfold preparedUnskippedReadWithWorkRaw preparedUnskippedReadWithWork
dsimp only
split_ifs with hempty hguard
· rfl
· simp only [Prod.map, Option.map_some, id_eq, UnskippedBand.raw, UnskippedBand.actions]
congr 3
· rfl
end BinaryFraction
end OAI.EditApproximation
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