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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 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 90 91 92 93 94 95 96 97 98 99 100 | import OAI.Combinatorics.EditApproximation.Binary.BinaryOptimizerSpace
import OAI.Combinatorics.EditApproximation.Binary.BinaryReductionSpace
import OAI.Combinatorics.EditApproximation.Binary.BinaryBandInputs
namespace OAI
namespace EditApproximation.BinaryFraction
def bandCoordinateAllocation {M : ℕ} {σ : Type*} [DecidableEq σ]
(mass : Vector BinaryFraction M) (total : BinaryFraction)
(action : Fin M → σ) (coordinate : Fin M × σ) : ℕ :=
if coordinate.2 = action coordinate.1 then divAllocation (mass.get coordinate.1) total else 0
def bandCoordinateInvocations {M : ℕ} {σ : Type*} [DecidableEq σ]
(mass : Vector BinaryFraction M) (total : BinaryFraction)
(action : Fin M → σ) (coordinate : Fin M × σ) : ℕ :=
if coordinate.2 = action coordinate.1 then divInvocations (mass.get coordinate.1) total else 1
theorem bandCoordinateAllocation_le_work {M : ℕ} {σ : Type*} [DecidableEq σ]
(mass : Vector BinaryFraction M) (total : BinaryFraction)
(action : Fin M → σ) (coordinate : Fin M × σ) :
bandCoordinateAllocation mass total action coordinate ≤ (bandCoordinateWithWork mass total action coordinate).2 := by
unfold bandCoordinateAllocation bandCoordinateWithWork
split_ifs
· exact divAllocation_le_work _ _
· exact Nat.zero_le _
theorem bandCoordinateInvocations_le_work {M : ℕ} {σ : Type*} [DecidableEq σ]
(mass : Vector BinaryFraction M) (total : BinaryFraction)
(action : Fin M → σ) (coordinate : Fin M × σ) :
bandCoordinateInvocations mass total action coordinate ≤ (bandCoordinateWithWork mass total action coordinate).2 := by
unfold bandCoordinateInvocations bandCoordinateWithWork
split_ifs
· exact divInvocations_le_work _ _
· exact le_refl _
def bandVerticesAllocation {M d : ℕ} {σ : Type*} [DecidableEq σ]
(coordinates : Vector (Fin M × σ) d) (mass : Vector BinaryFraction M)
(band : List (Fin M → σ)) : ℕ :=
bandMassAllocation mass +
(band.map fun action => vectorMapAllocation d fun i =>
bandCoordinateAllocation mass (bandMassTotalWithWork mass).1 action (coordinates.get i)).sum +
3 * band.length
def bandVerticesInvocations {M d : ℕ} {σ : Type*} [DecidableEq σ]
(coordinates : Vector (Fin M × σ) d) (mass : Vector BinaryFraction M)
(band : List (Fin M → σ)) : ℕ :=
bandMassInvocations mass +
(band.map fun action => vectorMapInvocations d fun i =>
bandCoordinateInvocations mass (bandMassTotalWithWork mass).1 action (coordinates.get i)).sum +
3 * band.length + 1
theorem bandVerticesAllocation_le_work {M d : ℕ} {σ : Type*} [DecidableEq σ]
(coordinates : Vector (Fin M × σ) d) (mass : Vector BinaryFraction M)
(band : List (Fin M → σ)) :
bandVerticesAllocation coordinates mass band ≤ (bandVerticesWithWork coordinates mass band).2 := by
have hm := bandMassAllocation_le_work mass
have hs : (band.map fun action => vectorMapAllocation d fun i =>
bandCoordinateAllocation mass (bandMassTotalWithWork mass).1 action (coordinates.get i)).sum ≤
(band.map fun action => (vectorMapWithWork d fun i =>
bandCoordinateWithWork mass (bandMassTotalWithWork mass).1 action (coordinates.get i)).2).sum := by
induction band with
| nil => simp
| cons a rest ih =>
have h := vectorMapAllocation_le_work d
(fun i => bandCoordinateWithWork mass (bandMassTotalWithWork mass).1 a (coordinates.get i))
(fun i => bandCoordinateAllocation mass (bandMassTotalWithWork mass).1 a (coordinates.get i))
(fun i => bandCoordinateAllocation_le_work _ _ _ _)
simp only [List.map_cons, List.sum_cons]
omega
unfold bandVerticesAllocation bandVerticesWithWork
simp only [List.map_map, Function.comp_def]
omega
theorem bandVerticesInvocations_le_work_add_one {M d : ℕ} {σ : Type*} [DecidableEq σ]
(coordinates : Vector (Fin M × σ) d) (mass : Vector BinaryFraction M)
(band : List (Fin M → σ)) :
bandVerticesInvocations coordinates mass band ≤ (bandVerticesWithWork coordinates mass band).2 + 1 := by
have hm := bandMassInvocations_le_work mass
have hs : (band.map fun action => vectorMapInvocations d fun i =>
bandCoordinateInvocations mass (bandMassTotalWithWork mass).1 action (coordinates.get i)).sum ≤
(band.map fun action => (vectorMapWithWork d fun i =>
bandCoordinateWithWork mass (bandMassTotalWithWork mass).1 action (coordinates.get i)).2).sum + band.length := by
induction band with
| nil => simp
| cons a rest ih =>
have h := vectorMapInvocations_le_work_add_one d
(fun i => bandCoordinateWithWork mass (bandMassTotalWithWork mass).1 a (coordinates.get i))
(fun i => bandCoordinateInvocations mass (bandMassTotalWithWork mass).1 a (coordinates.get i))
(fun i => bandCoordinateInvocations_le_work _ _ _ _)
simp only [List.map_cons, List.sum_cons, List.length_cons]
omega
unfold bandVerticesInvocations bandVerticesWithWork
simp only [List.map_map, Function.comp_def]
omega
end EditApproximation.BinaryFraction
end OAI
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