| import OAI.Combinatorics.EditApproximation.AlmostLinear |
| import OAI.Combinatorics.EditApproximation.Definitions |
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|
| namespace OAI.EditApproximation |
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|
| /-- The finite data used after accepting an unskipped band. -/ |
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| 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⟩ |
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|
| namespace BinaryFraction |
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| 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 |
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|
| end BinaryFraction |
| end OAI.EditApproximation |
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