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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 | 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
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