openAI-math-dataset / lean /OAI /Combinatorics /EditApproximation /Specification /RawBandBridges.lean
| import OAI.Combinatorics.EditApproximation.Specification.RawBandConsumers | |
| import OAI.Combinatorics.EditApproximation.Definitions | |
| namespace OAI.EditApproximation | |
| theorem filterMapWithWork_map_raw {α β γ : Type*} (f : α → Option β × ℕ) | |
| (g : β → γ) (xs : List α) : | |
| filterMapWithWork (fun x => Prod.map (Option.map g) id (f x)) xs = | |
| Prod.map (List.map g) id (filterMapWithWork f xs) := by | |
| induction xs with | |
| | nil => rfl | |
| | cons x xs ih => | |
| simp only [Prod.map, id_eq] at ih | |
| simp only [filterMapWithWork, Prod.map, id_eq, ih] | |
| cases (f x).1 <;> rfl | |
| theorem filterMapAllocation_map_raw {α β γ : Type*} (f : α → Option β × ℕ) | |
| (g : β → γ) (allocation : α → ℕ) (xs : List α) : | |
| filterMapAllocation (fun x => Prod.map (Option.map g) id (f x)) allocation xs = | |
| filterMapAllocation f allocation xs := by | |
| induction xs with | |
| | nil => rfl | |
| | cons x xs ih => | |
| simp only [Prod.map] at ih | |
| simp only [filterMapAllocation, Prod.map, ih, Option.isSome_map] | |
| namespace BinaryFraction | |
| theorem scoredArgmaxWithWork_map_raw {α β : Type*} (mapping : α → β) | |
| (score : β → BinaryFraction × ℕ) (values : List α) : | |
| scoredArgmaxWithWork score (values.map mapping) = | |
| Prod.map (Option.map (fun result => (mapping result.1, result.2))) id | |
| (scoredArgmaxWithWork (fun a => score (mapping a)) values) := by | |
| induction values with | |
| | nil => rfl | |
| | cons head tail ih => | |
| simp only [List.map_cons, scoredArgmaxWithWork, ih, Prod.map, id_eq] | |
| cases (scoredArgmaxWithWork (fun a => score (mapping a)) tail).1 with | |
| | none => rfl | |
| | some best => | |
| simp only [Option.map_some] | |
| split_ifs <;> rfl | |
| theorem selectGroupWithWork_map_raw {α β : Type*} (mapping : α → β) | |
| (score : β → BinaryFraction × ℕ) (values : List α) (hne : values ≠ []) : | |
| selectGroupWithWork (values.map mapping) (by simpa using hne) score = | |
| Prod.map mapping id (selectGroupWithWork values hne (fun a => score (mapping a))) := by | |
| unfold selectGroupWithWork | |
| rw [scoredArgmaxWithWork_map_raw] | |
| simp only [Prod.map, id_eq, Option.map_map, Function.comp_def] | |
| cases values with | |
| | nil => exact False.elim (hne rfl) | |
| | cons head tail => | |
| simp only [List.map_cons, List.head_cons] | |
| cases (scoredArgmaxWithWork (fun a => ((score (mapping a)).1.neg, (score (mapping a)).2 + 1)) | |
| (head :: tail)).1 <;> rfl | |
| theorem scoredArgmaxAllocation_map_raw {α β : Type*} (mapping : α → β) | |
| (score : β → BinaryFraction × ℕ) (allocation : β → ℕ) (values : List α) : | |
| scoredArgmaxAllocation score allocation (values.map mapping) = | |
| scoredArgmaxAllocation (fun a => score (mapping a)) (fun a => allocation (mapping a)) values := by | |
| induction values with | |
| | nil => rfl | |
| | cons head tail ih => | |
| simp only [List.map_cons, scoredArgmaxAllocation, ih, scoredArgmaxWithWork_map_raw, Prod.map] | |
| cases (scoredArgmaxWithWork (fun a => score (mapping a)) tail).1 <;> rfl | |
| theorem selectGroupAllocation_map_raw {α β : Type*} (mapping : α → β) | |
| (score : β → BinaryFraction × ℕ) (allocation : β → ℕ) (values : List α) : | |
| selectGroupAllocation (values.map mapping) score allocation = | |
| selectGroupAllocation values (fun a => score (mapping a)) (fun a => allocation (mapping a)) := by | |
| exact scoredArgmaxAllocation_map_raw mapping _ allocation values | |
| section Group | |
| variable {α : Type*} (source target : List α) {M T : ℕ} | |
| (parent : TargetInterval source.length) (b P exponent Q : ℕ) | |
| (tau F : BinaryFraction) (child : Fin M → TargetInterval target.length → BinaryFraction) | |
| (initial current : Fin M → TargetInterval target.length → BinaryFraction × ℕ) | |
| (initialAllocation currentAllocation : Fin M → TargetInterval target.length → ℕ) | |
| (scales : Fin T → BinaryFraction) | |
| (draw : ∀ t, Fin (groupSampleCount M Q b (scales t).value) → Fin M) | |
| (drawSource : ∀ t, Fin (groupSampleCount M Q b (scales t).value) → Fin M × ℕ) | |
| (drawAllocation : ∀ t, Fin (groupSampleCount M Q b (scales t).value) → ℕ) | |
| local notation "Member" => UnskippedBand source target parent b P exponent (value tau) (value F) | |
| (fun i q => value (child i q)) (fun i q => value (Prod.fst (initial i q))) | |
| theorem unskippedEstimateReadWithWorkRaw_eq (member : Member) : | |
| unskippedEstimateReadWithWorkRaw b exponent Q initial current scales draw member.raw = | |
| unskippedEstimateReadWithWork source target parent b P exponent Q tau F child initial current | |
| scales draw member := rfl | |
| theorem unskippedEstimateSourceWithWorkRaw_eq (member : Member) : | |
| unskippedEstimateSourceWithWorkRaw b exponent Q initial current scales drawSource member.raw = | |
| unskippedEstimateSourceWithWork source target parent b P exponent Q tau F child initial current | |
| scales drawSource member := rfl | |
| theorem chooseUnskippedReadWithWorkRaw_eq (members : List Member) (hne : members ≠ []) : | |
| chooseUnskippedReadWithWorkRaw b exponent Q initial current scales draw | |
| (members.map UnskippedBand.raw) (by simpa using hne) = | |
| Prod.map UnskippedBand.raw id | |
| (chooseUnskippedReadWithWork source target parent b P exponent Q tau F child initial current | |
| scales draw members hne) := by | |
| exact selectGroupWithWork_map_raw UnskippedBand.raw _ members hne | |
| theorem chooseUnskippedSourceWithWorkRaw_eq (members : List Member) (hne : members ≠ []) : | |
| chooseUnskippedSourceWithWorkRaw b exponent Q initial current scales drawSource | |
| (members.map UnskippedBand.raw) (by simpa using hne) = | |
| Prod.map UnskippedBand.raw id | |
| (chooseUnskippedSourceWithWork source target parent b P exponent Q tau F child initial current | |
| scales drawSource members hne) := by | |
| exact selectGroupWithWork_map_raw UnskippedBand.raw _ members hne | |
| theorem sampledGroupActionReadWithWorkRaw_eq (members : List Member) (hne : members ≠ []) : | |
| sampledGroupActionReadWithWorkRaw b exponent Q initial current scales draw | |
| (members.map UnskippedBand.raw) (by simpa using hne) = | |
| sampledGroupActionReadWithWork source target parent b P exponent Q tau F child initial current | |
| scales draw members hne := by | |
| let finish := fun selected : RawUnskippedBand M target.length × ℕ => | |
| let rounded := roundedBandReadWithWork exponent b selected.1.actions selected.1.nonempty initial | |
| let result := endpointActionWithWork selected.1.center rounded.1.get | |
| (result.1, selected.2 + rounded.2 + result.2 + 3) | |
| exact congrArg finish (chooseUnskippedReadWithWorkRaw_eq source target parent b P exponent Q | |
| tau F child initial current scales draw members hne) | |
| theorem sampledGroupActionSourceWithWorkRaw_eq (members : List Member) (hne : members ≠ []) : | |
| sampledGroupActionSourceWithWorkRaw b exponent Q initial current scales drawSource | |
| (members.map UnskippedBand.raw) (by simpa using hne) = | |
| sampledGroupActionSourceWithWork source target parent b P exponent Q tau F child initial current | |
| scales drawSource members hne := by | |
| let finish := fun selected : RawUnskippedBand M target.length × ℕ => | |
| let rounded := roundedBandReadWithWork exponent b selected.1.actions selected.1.nonempty initial | |
| let result := endpointActionWithWork selected.1.center rounded.1.get | |
| (result.1, selected.2 + rounded.2 + result.2 + 3) | |
| exact congrArg finish (chooseUnskippedSourceWithWorkRaw_eq source target parent b P exponent Q | |
| tau F child initial current scales drawSource members hne) | |
| theorem unskippedEstimateSourceAllocationRaw_eq (member : Member) : | |
| unskippedEstimateSourceAllocationRaw b exponent Q initial current initialAllocation currentAllocation | |
| scales drawSource drawAllocation member.raw = | |
| unskippedEstimateSourceAllocation source target parent b P exponent Q tau F child initial current | |
| initialAllocation currentAllocation scales drawSource drawAllocation member := rfl | |
| theorem chooseUnskippedSourceAllocationRaw_eq (members : List Member) : | |
| chooseUnskippedSourceAllocationRaw b exponent Q initial current initialAllocation currentAllocation | |
| scales drawSource drawAllocation (members.map UnskippedBand.raw) = | |
| chooseUnskippedSourceAllocation source target parent b P exponent Q tau F child initial current | |
| initialAllocation currentAllocation scales drawSource drawAllocation members := by | |
| exact selectGroupAllocation_map_raw UnskippedBand.raw _ _ members | |
| theorem sampledGroupActionSourceAllocationRaw_eq (members : List Member) (hne : members ≠ []) : | |
| sampledGroupActionSourceAllocationRaw b exponent Q initial current initialAllocation currentAllocation | |
| scales drawSource drawAllocation (members.map UnskippedBand.raw) (by simpa using hne) = | |
| sampledGroupActionSourceAllocation source target parent b P exponent Q tau F child initial current | |
| initialAllocation currentAllocation scales drawSource drawAllocation members hne := by | |
| unfold sampledGroupActionSourceAllocationRaw sampledGroupActionSourceAllocation | |
| rw [chooseUnskippedSourceWithWorkRaw_eq, chooseUnskippedSourceAllocationRaw_eq] | |
| rfl | |
| end Group | |
| end BinaryFraction | |
| end OAI.EditApproximation | |