| 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 |
|
|
