openAI-math-dataset / lean /OAI /Combinatorics /EditApproximation /Queries /QueryChargedSeedSpace.lean
| import OAI.Combinatorics.EditApproximation.Queries.QueryPhysicalReaderSpace | |
| import OAI.Combinatorics.EditApproximation.Queries.QueryChargedSeed | |
| import OAI.Combinatorics.EditApproximation.Definitions | |
| namespace OAI | |
| namespace EditApproximation | |
| /-- Scalar cells of the actual coarse/table address construction. -/ | |
| theorem chargedPhysicalCodeAllocation_le_work {M J n passes copies T S : ℕ} | |
| (request : ChargedPhysicalRequest M J n passes copies T S) : | |
| chargedPhysicalCodeAllocation request ≤ (chargedPhysicalCodeWithWork request).2 := by | |
| have entryBound (d J : ℕ) : | |
| 2 * d + J + 12 ≤ 3 * d + 1 + (2 * J + 1) + 3 + 8 + 2 := by | |
| omega | |
| cases request with | |
| | inl entry => | |
| simp only [chargedPhysicalCodeAllocation, chargedPhysicalCodeWithWork, physicalEntryCodeWithWork, | |
| physicalNodeCodeWithWork, treeLayerCodeWithWork_cost, parameterPowerWithWork_spec] | |
| exact entryBound _ _ | |
| | inr request => | |
| have h := physicalTableCodeAllocation_le_work copies T S request.val | |
| change physicalTableCodeAllocation request.val + 2 ≤ (physicalTableCodeWithWork copies T S request.val).2 + 3 | |
| exact Nat.add_le_add h (by decide : 2 ≤ 3) | |
| open BitQuery | |
| section Seed | |
| variable {M J n passes copies T S : ℕ} | |
| (parent : {r : PhysicalTableRequest M J n // r.Bounded passes copies T S}) | |
| def queryChargedReadHeap | |
| (key : {child : ChargedPhysicalRequest M J n passes copies T S // | |
| child.rank < ChargedPhysicalRequest.rank (.inr parent)}) : Heap (queryChargedRead parent key) := | |
| Heap.chargeOf (chargedPhysicalCodeAllocation key.val + 1) | |
| (Nat.add_le_add_right (chargedPhysicalCodeAllocation_le_work key.val) 1) (.read key _ fun word => .done word) | |
| def queryChargedSeedAtHeap (N : ℕ) (multiplier : ℕ → ℕ) (q : TargetInterval n) : | |
| Heap (queryChargedSeedAt parent N multiplier q) := by | |
| refine Heap.chargeOf (naturalEqualAllocation parent.val.pass 0) | |
| ((naturalEqualAllocation_le_work _ _).trans (Nat.le_succ _)) ?_ | |
| split | |
| · exact queryChargedReadHeap parent _ | |
| · refine Heap.chargeOf copies (Nat.le_succ _) ?_ | |
| refine Heap.collect _ (fun _ => queryChargedReadHeap parent _) _ _ fun words => ?_ | |
| let factor := multiplier (parent.val.pass - 1) | |
| refine Heap.computeOf (BinaryFraction.querySeedMedianWithWork N factor words) | |
| (BinaryFraction.seedMedianAllocation N factor words + (BinaryFraction.seedMedianWithWork N factor words).1.length + 1) ?_ | |
| have hm := BinaryFraction.seedMedianAllocation_le_work N factor words | |
| unfold BinaryFraction.querySeedMedianWithWork | |
| dsimp only | |
| omega | |
| end Seed | |
| end EditApproximation | |
| end OAI | |