openAI-math-dataset / lean /OAI /Combinatorics /EditApproximation /Queries /QueryChargedSeedCost.lean
| import OAI.Combinatorics.EditApproximation.Queries.QueryChargedSeed | |
| namespace OAI | |
| namespace EditApproximation | |
| section Seed | |
| variable {M J n passes copies T S : ℕ} | |
| (parent : {r : PhysicalTableRequest M J n // r.Bounded passes copies T S}) | |
| (oracle : {child : ChargedPhysicalRequest M J n passes copies T S // | |
| child.rank < ChargedPhysicalRequest.rank (.inr parent)} → BinaryFraction) | |
| (readWork : {child : ChargedPhysicalRequest M J n passes copies T S // | |
| child.rank < ChargedPhysicalRequest.rank (.inr parent)} → ℕ) | |
| theorem queryChargedRead_work_bound (C : ℕ) (hc : ∀ key, readWork key ≤ C) (key) : | |
| (queryChargedRead parent key).work oracle readWork ≤ 7 * J + 33 + C := by | |
| have h := chargedPhysicalCodeWithWork_cost key.val | |
| have hr := hc key | |
| simp only [queryChargedRead, BitQuery.work] | |
| omega | |
| private theorem seed_sum_bound {κ : Type*} (keys : List κ) (cost : κ → ℕ) (C : ℕ) | |
| (hc : ∀ key, cost key ≤ C) : (keys.map cost).sum ≤ keys.length * C := by | |
| induction keys with | |
| | nil => simp | |
| | cons key keys ih => | |
| have h := hc key | |
| simp only [List.map_cons, List.sum_cons, List.length_cons, Nat.add_mul, Nat.one_mul] | |
| omega | |
| def queryChargedSeedBitBudget (copies J B C : ℕ) : ℕ := | |
| 8 * B + 3 + (7 * J + 33 + C) + copies + 1 + | |
| copies * (7 * J + 33 + C + 3) + BinaryFraction.seedMedianBitBudget copies B + B + 1 | |
| theorem queryChargedSeedAt_bounds (N B C : ℕ) (multiplier : ℕ → ℕ) (q : TargetInterval n) | |
| (hB : 1 ≤ B) (hN : Nat.size N ≤ B) (hp : Nat.size parent.val.pass ≤ B) | |
| (hm : Nat.size (multiplier (parent.val.pass - 1)) + 2 ≤ B) | |
| (hv : ∀ key, (oracle key).width ≤ B) (hc : ∀ key, readWork key ≤ C) : | |
| ((queryChargedSeedAt parent N multiplier q).value oracle).width ≤ B ∧ | |
| (queryChargedSeedAt parent N multiplier q).work oracle readWork ≤ | |
| queryChargedSeedBitBudget copies J B C := by | |
| have heq := naturalEqualWithWork_cost parent.val.pass 0 B hp (by simp) | |
| have hr (key) := queryChargedRead_work_bound parent oracle readWork C hc key | |
| unfold queryChargedSeedAt | |
| split | |
| · constructor | |
| · exact hv _ | |
| · simp only [BitQuery.work] | |
| have := hr ⟨.inl (parent.val.node, q), ChargedPhysicalRequest.coarse_rank_lt _ parent rfl⟩ | |
| change (naturalEqualWithWork parent.val.pass 0).2 + 1 + | |
| (queryChargedRead parent _).work oracle readWork ≤ _ | |
| unfold queryChargedSeedBitBudget | |
| omega | |
| · rename_i hpass | |
| let hpos := Nat.pos_of_ne_zero (fun h => hpass ((naturalEqualWithWork_value _ _).mpr h)) | |
| let reader := queryChargedPreviousRead parent hpos q | |
| let words := (List.finRange copies).map fun copy => (reader copy).value oracle | |
| have hlen : words.length = copies := by simp only [words, List.length_map, List.length_finRange] | |
| have hwords : ∀ word ∈ words, word.width ≤ B := by | |
| intro word hw | |
| obtain ⟨copy, _, rfl⟩ := List.mem_map.mp hw | |
| exact hv _ | |
| have hmedian := BinaryFraction.seedMedianWithWork_cost N (multiplier (parent.val.pass - 1)) words B hN hm hwords | |
| have hlength := BinaryFraction.seedMedianWithWork_length N (multiplier (parent.val.pass - 1)) words | |
| have hsum := seed_sum_bound (List.finRange copies) (fun copy => (reader copy).work oracle readWork + 3) | |
| (7 * J + 33 + C + 3) (fun copy => Nat.add_le_add_right (hr _) 3) | |
| simp only [List.length_finRange] at hsum | |
| rw [hlen] at hmedian | |
| constructor | |
| · simp only [BitQuery.value, BitQuery.value_collect, BitQuery.compute, | |
| BinaryFraction.querySeedMedianWithWork, BinaryFraction.ofNaturalWord_width] | |
| exact max_le (hlength.trans hN) hB | |
| · simp only [BitQuery.work, BitQuery.work_collect, BitQuery.compute, Nat.add_zero] | |
| change (naturalEqualWithWork parent.val.pass 0).2 + 1 + | |
| (copies + 1 + (((List.finRange copies).map fun copy => (reader copy).work oracle readWork + 3).sum + | |
| (BinaryFraction.querySeedMedianWithWork N (multiplier (parent.val.pass - 1)) words).2)) ≤ _ | |
| unfold BinaryFraction.querySeedMedianWithWork queryChargedSeedBitBudget | |
| dsimp only | |
| omega | |
| end Seed | |
| end EditApproximation | |
| end OAI | |