openAI-math-dataset / lean /OAI /Combinatorics /EditApproximation /Coarse /CoarseCompleteArrayWork.lean
| import OAI.Combinatorics.EditApproximation.Coarse.CoarseCompleteWork | |
| namespace OAI | |
| namespace EditApproximation | |
| namespace BinaryFraction | |
| theorem coarseActiveChildrenWork_const_bound {B d : ℕ} {α : Type*} (Q lam allowance : ℚ) (C : ℕ) | |
| (work : CoarseSourceTree B α d → CoarseThresholdDraws B d → ℚ → ℕ) | |
| (children : List (CoarseSourceTree B α d × Fin B × CoarseThresholdDraws B d)) | |
| (hc : ∀ child draw below, (child, draw, below) ∈ children → | |
| work child below (lam * coarseChildThreshold B Q allowance draw) ≤ C) : | |
| coarseActiveChildrenWork Q lam allowance work children ≤ children.length * C := by | |
| induction children with | |
| | nil => simp only [coarseActiveChildrenWork, List.length_nil, Nat.zero_mul, le_refl] | |
| | cons data rest ih => | |
| obtain ⟨child, draw, below⟩ := data | |
| have hi := hc child draw below List.mem_cons_self | |
| have hr := ih (fun c r b h => hc c r b (List.mem_cons_of_mem _ h)) | |
| rw [coarseActiveChildrenWork, List.length_cons, Nat.add_mul, Nat.one_mul] | |
| omega | |
| attribute [local irreducible] coarseActiveArrayWork | |
| theorem coarseActiveArrayWork_le_complete {B d : ℕ} {α : Type*} (k : ℕ) (Q lam : ℚ) | |
| (tree : CoarseSourceTree B α d) (draws : CoarseThresholdDraws B d) (allowance : ℚ) : | |
| coarseActiveArrayWork k Q lam tree draws allowance ≤ | |
| (B * (2 * k + 1)) * coarseCompleteNodes B d := by | |
| induction tree generalizing allowance with | |
| | leaf letter => | |
| rw [coarseActiveArrayWork] | |
| split_ifs <;> simp only [coarseCompleteNodes, Nat.mul_one] <;> omega | |
| | @branch d children ih => | |
| obtain ⟨draws, below⟩ := draws | |
| have hr := coarseActiveChildrenWork_const_bound Q lam allowance | |
| ((B * (2 * k + 1)) * coarseCompleteNodes B d) | |
| (fun child => coarseActiveArrayWork k Q lam child) | |
| (List.ofFn fun i => (children i, draws i, below i)) (by | |
| intro child draw b hb | |
| obtain ⟨i, hi⟩ := List.mem_ofFn.mp hb | |
| cases hi | |
| exact ih i (below i) _) | |
| simp only [List.length_ofFn] at hr | |
| rw [coarseActiveArrayWork] | |
| split_ifs | |
| · exact Nat.zero_le _ | |
| · rw [coarseCompleteNodes] | |
| nlinarith only [hr] | |
| end BinaryFraction | |
| end EditApproximation | |
| end OAI | |