openAI-math-dataset / lean /OAI /Combinatorics /EditApproximation /Binary /BinaryBandActionReads.lean
| import OAI.Combinatorics.EditApproximation.Binary.BinaryOnlineReads | |
| import OAI.Combinatorics.EditApproximation.Binary.BinaryBandAction | |
| import OAI.Combinatorics.EditApproximation.Binary.BinaryCoordinateAction | |
| import OAI.Combinatorics.EditApproximation.Definitions | |
| namespace OAI | |
| /-! The actual labelled online action, with charged initial and history reads. -/ | |
| namespace EditApproximation | |
| namespace BinaryFraction | |
| def preparedBandActionReadWithWork {M : ℕ} {σ : Type*} [DecidableEq σ] | |
| (mass : Vector BinaryFraction M) (read : ℕ → Fin M × σ → BinaryFraction × ℕ) | |
| (eta kappa : BinaryFraction) (band : List (Fin M → σ)) (G j t : ℕ) : | |
| Option (Fin M → σ) × ℕ := | |
| let coordinates := listedBandCoordinateVector band | |
| let vertices := labeledBandVerticesWithWork coordinates mass band | |
| let linear := vectorMapWithWork _ fun i => | |
| onlineLinearReadWithWork (fun j => mass.get j) read eta kappa t (coordinates.get i) | |
| let selected := selectionWithWork vertices.1 linear.1 G j | |
| (selected.1, vertices.2 + linear.2 + selected.2 + 2) | |
| theorem preparedBandActionReadWithWork_eq_cost {M : ℕ} {σ : Type*} [DecidableEq σ] | |
| (mass : Vector BinaryFraction M) (read : ℕ → Fin M × σ → BinaryFraction × ℕ) | |
| (eta kappa : BinaryFraction) (band : List (Fin M → σ)) (G j t R : ℕ) | |
| (hr : ∀ s ≤ t + 1, ∀ i : Fin (listedBandCoordinates band).length, | |
| (read s ((listedBandCoordinateVector band).get i)).2 ≤ R) : | |
| (preparedBandActionReadWithWork mass read eta kappa band G j t).1 = | |
| (preparedBandActionWithWork mass (fun s i => (read s i).1) eta kappa band G j t).1 ∧ | |
| (preparedBandActionReadWithWork mass read eta kappa band G j t).2 ≤ | |
| (preparedBandActionWithWork mass (fun s i => (read s i).1) eta kappa band G j t).2 + | |
| (listedBandCoordinates band).length * ((3 * t + 1) * R) := by | |
| have hl := vectorMapWithWork_extra (listedBandCoordinates band).length | |
| (fun i => onlineLinearReadWithWork (fun j => mass.get j) read eta kappa t | |
| ((listedBandCoordinateVector band).get i)) | |
| (fun i => onlineLinearWithWork (fun j => mass.get j) (fun s k => (read s k).1) eta kappa t | |
| ((listedBandCoordinateVector band).get i)) | |
| ((3 * t + 1) * R) | |
| (fun i => (onlineLinearReadWithWork_eq_cost _ read eta kappa t _ R | |
| (fun s hs => hr s hs i)).1) | |
| (fun i => (onlineLinearReadWithWork_eq_cost _ read eta kappa t _ R | |
| (fun s hs => hr s hs i)).2) | |
| unfold preparedBandActionReadWithWork preparedBandActionWithWork | |
| dsimp only | |
| rw [hl.1] | |
| exact ⟨rfl, by omega⟩ | |
| def bandActionReadWithWork {M n : ℕ} (b : ℕ) | |
| (initial : Fin M → TargetInterval n → BinaryFraction × ℕ) | |
| (read : ℕ → Fin M × TargetInterval n → BinaryFraction × ℕ) | |
| (eta kappa : BinaryFraction) (band : List (Fin M → TargetInterval n)) (G j t : ℕ) : | |
| Option (Fin M → TargetInterval n) × ℕ := | |
| let mass := envelopeVectorReadWithWork b band initial | |
| let selected := preparedBandActionReadWithWork mass.1 read eta kappa band G j t | |
| (selected.1, mass.2 + selected.2 + 1) | |
| theorem bandActionReadWithWork_eq_cost {M n : ℕ} (b : ℕ) | |
| (initial : Fin M → TargetInterval n → BinaryFraction × ℕ) | |
| (read : ℕ → Fin M × TargetInterval n → BinaryFraction × ℕ) | |
| (eta kappa : BinaryFraction) (band : List (Fin M → TargetInterval n)) (G j t R C : ℕ) | |
| (hi : ∀ i q, (initial i q).2 ≤ C) | |
| (hr : ∀ s ≤ t + 1, ∀ i : Fin (listedBandCoordinates band).length, | |
| (read s ((listedBandCoordinateVector band).get i)).2 ≤ R) : | |
| (bandActionReadWithWork b initial read eta kappa band G j t).1 = | |
| (bandActionWithWork b (fun i q => (initial i q).1) | |
| (fun s i => (read s i).1) eta kappa band G j t).1 ∧ | |
| (bandActionReadWithWork b initial read eta kappa band G j t).2 ≤ | |
| (bandActionWithWork b (fun i q => (initial i q).1) | |
| (fun s i => (read s i).1) eta kappa band G j t).2 + | |
| M * (band.length * (C + 3)) + | |
| (listedBandCoordinates band).length * ((3 * t + 1) * R) := by | |
| have hm := envelopeVectorReadWithWork_eq_cost b band initial C hi | |
| have hs := preparedBandActionReadWithWork_eq_cost | |
| (envelopeVectorWithWork b band (fun i q => (initial i q).1)).1 read | |
| eta kappa band G j t R hr | |
| unfold bandActionReadWithWork bandActionWithWork | |
| dsimp only | |
| rw [hm.1] | |
| exact ⟨hs.1, by omega⟩ | |
| def sampledOnlineActionReadWithWork {n M : ℕ} (center : TargetInterval n) (b : ℕ) | |
| (initial : Fin M → TargetInterval n → BinaryFraction × ℕ) | |
| (read : ℕ → Fin M × TargetInterval n → BinaryFraction × ℕ) | |
| (eta kappa : BinaryFraction) (band : List (Fin M → TargetInterval n)) (hne : band ≠ []) | |
| (G K t : ℕ) (integer : Fin (optimizerTriangularMass K)) : | |
| BinaryBellmanAction (TargetInterval n) (Fin M × TargetInterval n) M × ℕ := | |
| let chosen := triangularChoiceWithWork K integer | |
| let selected := bandActionReadWithWork b initial read eta kappa band G chosen.1.val t | |
| let states := selected.1.getD (band.head hne) | |
| (⟨center, zero, fun i => (i, states i)⟩, chosen.2 + selected.2 + M + 2) | |
| theorem sampledOnlineActionReadWithWork_eq_cost {n M : ℕ} (center : TargetInterval n) (b : ℕ) | |
| (initial : Fin M → TargetInterval n → BinaryFraction × ℕ) | |
| (read : ℕ → Fin M × TargetInterval n → BinaryFraction × ℕ) | |
| (eta kappa : BinaryFraction) (band : List (Fin M → TargetInterval n)) (hne : band ≠ []) | |
| (G K t R C : ℕ) (integer : Fin (optimizerTriangularMass K)) | |
| (hi : ∀ i q, (initial i q).2 ≤ C) | |
| (hr : ∀ s ≤ t + 1, ∀ i : Fin (listedBandCoordinates band).length, | |
| (read s ((listedBandCoordinateVector band).get i)).2 ≤ R) : | |
| (sampledOnlineActionReadWithWork center b initial read eta kappa band hne G K t integer).1 = | |
| (sampledOnlineActionWithWork center b (fun i q => (initial i q).1) | |
| (fun s i => (read s i).1) eta kappa band hne G K t integer).1 ∧ | |
| (sampledOnlineActionReadWithWork center b initial read eta kappa band hne G K t integer).2 ≤ | |
| (sampledOnlineActionWithWork center b (fun i q => (initial i q).1) | |
| (fun s i => (read s i).1) eta kappa band hne G K t integer).2 + | |
| M * (band.length * (C + 3)) + | |
| (listedBandCoordinates band).length * ((3 * t + 1) * R) := by | |
| have hs := bandActionReadWithWork_eq_cost b initial read eta kappa band G | |
| (triangularChoiceWithWork K integer).1.val t R C hi hr | |
| unfold sampledOnlineActionReadWithWork sampledOnlineActionWithWork | |
| dsimp only | |
| rw [hs.1] | |
| exact ⟨rfl, by omega⟩ | |
| theorem countedPreparedActionReadWithWork_eq_cost {M n : ℕ} (mass : Vector BinaryFraction M) | |
| (read : ℕ → Fin M × TargetInterval n → BinaryFraction × ℕ) | |
| (eta kappa : BinaryFraction) (band : List (Fin M → TargetInterval n)) (G j t R : ℕ) | |
| (hr : ∀ s ≤ t + 1, ∀ i : Fin (listedBandCoordinates band).length, | |
| (read s ((listedBandCoordinateVector band).get i)).2 ≤ R) : | |
| (countedPreparedActionReadWithWork mass read eta kappa band G j t).1 = | |
| (countedPreparedActionWithWork mass (fun s i => (read s i).1) eta kappa band G j t).1 ∧ | |
| (countedPreparedActionReadWithWork mass read eta kappa band G j t).2 ≤ | |
| (countedPreparedActionWithWork mass (fun s i => (read s i).1) eta kappa band G j t).2 + | |
| (listedBandCoordinates band).length * ((3 * t + 1) * R) := by | |
| have hl := vectorMapWithWork_extra (listedBandCoordinates band).length | |
| (fun i => onlineLinearReadWithWork (fun j => mass.get j) read eta kappa t | |
| ((listedBandCoordinateVector band).get i)) | |
| (fun i => onlineLinearWithWork (fun j => mass.get j) (fun s k => (read s k).1) eta kappa t | |
| ((listedBandCoordinateVector band).get i)) | |
| ((3 * t + 1) * R) | |
| (fun i => (onlineLinearReadWithWork_eq_cost _ read eta kappa t _ R | |
| (fun s hs => hr s hs i)).1) | |
| (fun i => (onlineLinearReadWithWork_eq_cost _ read eta kappa t _ R | |
| (fun s hs => hr s hs i)).2) | |
| unfold countedPreparedActionReadWithWork countedPreparedActionWithWork | |
| simp only [coordinateVectorWithWork_value] | |
| rw [hl.1] | |
| exact ⟨rfl, by omega⟩ | |
| theorem countedBandActionReadWithWork_eq_cost {M n : ℕ} (b : ℕ) | |
| (initial : Fin M → TargetInterval n → BinaryFraction × ℕ) | |
| (read : ℕ → Fin M × TargetInterval n → BinaryFraction × ℕ) | |
| (eta kappa : BinaryFraction) (band : List (Fin M → TargetInterval n)) (G j t R C : ℕ) | |
| (hi : ∀ i q, (initial i q).2 ≤ C) | |
| (hr : ∀ s ≤ t + 1, ∀ i : Fin (listedBandCoordinates band).length, | |
| (read s ((listedBandCoordinateVector band).get i)).2 ≤ R) : | |
| (countedBandActionReadWithWork b initial read eta kappa band G j t).1 = | |
| (countedBandActionWithWork b (fun i q => (initial i q).1) | |
| (fun s i => (read s i).1) eta kappa band G j t).1 ∧ | |
| (countedBandActionReadWithWork b initial read eta kappa band G j t).2 ≤ | |
| (countedBandActionWithWork b (fun i q => (initial i q).1) | |
| (fun s i => (read s i).1) eta kappa band G j t).2 + | |
| M * (band.length * (C + 3)) + | |
| (listedBandCoordinates band).length * ((3 * t + 1) * R) := by | |
| have hm := envelopeVectorReadWithWork_eq_cost b band initial C hi | |
| have hs := countedPreparedActionReadWithWork_eq_cost | |
| (envelopeVectorWithWork b band (fun i q => (initial i q).1)).1 read | |
| eta kappa band G j t R hr | |
| unfold countedBandActionReadWithWork countedBandActionWithWork | |
| dsimp only | |
| rw [hm.1] | |
| exact ⟨hs.1, by omega⟩ | |
| theorem countedSampledOnlineReadWithWork_eq_cost {n M : ℕ} (center : TargetInterval n) (b : ℕ) | |
| (initial : Fin M → TargetInterval n → BinaryFraction × ℕ) | |
| (read : ℕ → Fin M × TargetInterval n → BinaryFraction × ℕ) | |
| (eta kappa : BinaryFraction) (band : List (Fin M → TargetInterval n)) (hne : band ≠ []) | |
| (G K t R C : ℕ) (integer : Fin (optimizerTriangularMass K)) | |
| (hi : ∀ i q, (initial i q).2 ≤ C) | |
| (hr : ∀ s ≤ t + 1, ∀ i : Fin (listedBandCoordinates band).length, | |
| (read s ((listedBandCoordinateVector band).get i)).2 ≤ R) : | |
| (countedSampledOnlineReadWithWork center b initial read eta kappa band hne G K t integer).1 = | |
| (countedSampledOnlineWithWork center b (fun i q => (initial i q).1) | |
| (fun s i => (read s i).1) eta kappa band hne G K t integer).1 ∧ | |
| (countedSampledOnlineReadWithWork center b initial read eta kappa band hne G K t integer).2 ≤ | |
| (countedSampledOnlineWithWork center b (fun i q => (initial i q).1) | |
| (fun s i => (read s i).1) eta kappa band hne G K t integer).2 + | |
| M * (band.length * (C + 3)) + | |
| (listedBandCoordinates band).length * ((3 * t + 1) * R) := by | |
| have hs := countedBandActionReadWithWork_eq_cost b initial read eta kappa band G | |
| (triangularChoiceWithWork K integer).1.val t R C hi hr | |
| unfold countedSampledOnlineReadWithWork countedSampledOnlineWithWork | |
| dsimp only | |
| rw [hs.1] | |
| exact ⟨rfl, by omega⟩ | |
| end BinaryFraction | |
| end EditApproximation | |
| end OAI | |