| import OAI.Combinatorics.EditApproximation.Queries.QuerySourceWidths | |
| import OAI.Combinatorics.EditApproximation.Queries.QueryRestrictionCost | |
| namespace OAI | |
| namespace EditApproximation | |
| section Body | |
| variable {σ : Type*} {M J n : ℕ} (passes copies T S : ℕ) | |
| (parent : {r : PhysicalTableRequest M J n // r.Bounded passes copies T S}) | |
| abbrev QueryBoundedChild := Sum σ {child : {r : PhysicalTableRequest M J n // | |
| r.Bounded passes copies T S} // child.val.combinedRank T S < parent.val.combinedRank T S} | |
| def queryRestrictedPhysicalOracle (oracle : QueryBoundedChild (σ := σ) passes copies T S parent → BinaryFraction) : | |
| QueryPhysicalChild (σ := σ) passes copies T S parent → BinaryFraction := | |
| BitQuery.restrictedOracle _ (querySupportedChildWithWork passes copies T S parent) | |
| (querySupportedChildWithWork_value passes copies T S parent) | |
| (fun key => oracle (SourceRanked.supportedChildMap (PhysicalTableRequest.combinedRank T S) | |
| (PhysicalTableRequest.Bounded passes copies T S) parent key)) | |
| def queryRestrictedPhysicalReadWork (readWork : QueryBoundedChild (σ := σ) passes copies T S parent → ℕ) : | |
| QueryPhysicalChild (σ := σ) passes copies T S parent → ℕ := | |
| BitQuery.restrictedReadWork _ (querySupportedChildWithWork passes copies T S parent) | |
| (querySupportedChildWithWork_value passes copies T S parent) | |
| (fun key => 1 + readWork (SourceRanked.supportedChildMap (PhysicalTableRequest.combinedRank T S) | |
| (PhysicalTableRequest.Bounded passes copies T S) parent key)) | |
| theorem queryRestrictPhysicalBody_work | |
| (program : BitQuery (QueryPhysicalChild (σ := σ) passes copies T S parent) BinaryFraction) | |
| (oracle : QueryBoundedChild (σ := σ) passes copies T S parent → BinaryFraction) | |
| (readWork : QueryBoundedChild (σ := σ) passes copies T S parent → ℕ) : | |
| (queryRestrictPhysicalBody passes copies T S parent program).work oracle readWork = | |
| program.work (queryRestrictedPhysicalOracle passes copies T S parent oracle) | |
| (queryRestrictedPhysicalReadWork passes copies T S parent readWork) := by | |
| rw [queryRestrictPhysicalBody, BitQuery.mapKeysWithWork_work, BitQuery.restrictWithWork_work] | |
| rfl | |
| theorem queryRestrictPhysicalBody_value | |
| (program : BitQuery (QueryPhysicalChild (σ := σ) passes copies T S parent) BinaryFraction) | |
| (oracle : QueryBoundedChild (σ := σ) passes copies T S parent → BinaryFraction) : | |
| (queryRestrictPhysicalBody passes copies T S parent program).value oracle = | |
| program.value (queryRestrictedPhysicalOracle passes copies T S parent oracle) := by | |
| rw [queryRestrictPhysicalBody, BitQuery.mapKeysWithWork_value, BitQuery.restrictWithWork_value] | |
| rfl | |
| theorem queryRestrictedPhysicalOracle_table_width | |
| (oracle : QueryBoundedChild (σ := σ) passes copies T S parent → BinaryFraction) | |
| (B : ℕ) (hB : 2 ≤ B) (hw : ∀ key, (oracle (Sum.inr key)).width ≤ B) (child) : | |
| (queryRestrictedPhysicalOracle passes copies T S parent oracle (Sum.inr child)).width ≤ B := by | |
| unfold queryRestrictedPhysicalOracle BitQuery.restrictedOracle | |
| split | |
| · exact hw _ | |
| · exact (BinaryFraction.nat_width_le 0).trans (by simpa only [Nat.size_zero, Nat.zero_add] using hB) | |
| theorem queryRestrictedPhysicalOracle_table_nonneg | |
| (oracle : QueryBoundedChild (σ := σ) passes copies T S parent → BinaryFraction) | |
| (hn : ∀ key, 0 ≤ (oracle (Sum.inr key)).value) (child) : | |
| 0 ≤ (queryRestrictedPhysicalOracle passes copies T S parent oracle (Sum.inr child)).value := by | |
| unfold queryRestrictedPhysicalOracle BitQuery.restrictedOracle | |
| split | |
| · exact hn _ | |
| · simp only [BinaryFraction.nat_value, Nat.cast_zero, le_refl] | |
| theorem queryRestrictedPhysicalOracle_width | |
| (oracle : QueryBoundedChild (σ := σ) passes copies T S parent → BinaryFraction) | |
| (B : ℕ) (hB : 2 ≤ B) (hw : ∀ key, (oracle key).width ≤ B) (key) : | |
| (queryRestrictedPhysicalOracle passes copies T S parent oracle key).width ≤ B := | |
| BitQuery.restrictedOracle_width _ _ _ _ B hB (fun _ => hw _) key | |
| @[simp] theorem queryRestrictedPhysicalOracle_source | |
| (oracle : QueryBoundedChild (σ := σ) passes copies T S parent → BinaryFraction) (source : σ) : | |
| queryRestrictedPhysicalOracle passes copies T S parent oracle (Sum.inl source) = oracle (Sum.inl source) := rfl | |
| theorem queryRestrictedPhysicalOracle_table | |
| (oracle : QueryBoundedChild (σ := σ) passes copies T S parent → BinaryFraction) | |
| (child : {r : PhysicalTableRequest M J n // r.combinedRank T S < parent.val.combinedRank T S}) | |
| (hb : child.val.Bounded passes copies T S) : | |
| queryRestrictedPhysicalOracle passes copies T S parent oracle (Sum.inr child) = | |
| oracle (Sum.inr ⟨⟨child.val, hb⟩, child.property⟩) := by | |
| unfold queryRestrictedPhysicalOracle BitQuery.restrictedOracle | |
| rw [dite_eq_left ((querySupportedChildWithWork_value passes copies T S parent (Sum.inr child)).mpr hb)] | |
| rfl | |
| private theorem index_le_combinedRank (request : PhysicalTableRequest M J n) : | |
| request.index ≤ request.combinedRank T S := by | |
| unfold PhysicalTableRequest.combinedRank PhysicalTableRequest.rank | |
| omega | |
| private theorem pass_le_combinedRank (request : PhysicalTableRequest M J n) : | |
| request.pass ≤ request.combinedRank T S := by | |
| have hmul : request.pass ≤ request.pass * (T + S + 1) := Nat.le_mul_of_pos_right _ (by omega) | |
| unfold PhysicalTableRequest.combinedRank PhysicalTableRequest.rank | |
| omega | |
| theorem queryRestrictedPhysicalReadWork_bound | |
| (readWork : QueryBoundedChild (σ := σ) passes copies T S parent → ℕ) (B W : ℕ) | |
| (hp : Nat.size passes ≤ B) (hc : Nat.size copies ≤ B) (hT : Nat.size T ≤ B) (hS : Nat.size S ≤ B) | |
| (hr : Nat.size (parent.val.combinedRank T S) ≤ B) (hcopy : Nat.size parent.val.copy ≤ B) | |
| (hW : ∀ key, readWork key ≤ W) | |
| (key : QueryPhysicalChild (σ := σ) passes copies T S parent) | |
| (hk : ∀ child, key = Sum.inr child → child.val.copy = parent.val.copy) : | |
| queryRestrictedPhysicalReadWork passes copies T S parent readWork key ≤ 50 * B + 21 + W := by | |
| have hread (k) := hW (SourceRanked.supportedChildMap (PhysicalTableRequest.combinedRank T S) | |
| (PhysicalTableRequest.Bounded passes copies T S) parent k) | |
| cases key with | |
| | inl source => | |
| change 1 + (1 + readWork _) ≤ _ | |
| have := hread ⟨Sum.inl source, trivial⟩ | |
| omega | |
| | inr child => | |
| have hpass : Nat.size child.val.pass ≤ B := | |
| (Nat.size_le_size ((pass_le_combinedRank T S child.val).trans child.property.le)).trans hr | |
| have hindex : Nat.size child.val.index ≤ B := | |
| (Nat.size_le_size ((index_le_combinedRank T S child.val).trans child.property.le)).trans hr | |
| have hcopy' : Nat.size child.val.copy ≤ B := by rw [hk child rfl]; exact hcopy | |
| have hguard := queryPhysicalBoundedWithWork_cost passes copies T S B child.val hp hc hT hS hpass hcopy' hindex | |
| unfold queryRestrictedPhysicalReadWork BitQuery.restrictedReadWork | |
| change (querySupportedChildWithWork (σ := σ) passes copies T S parent (Sum.inr child)).2 ≤ 50 * B + 20 at hguard | |
| by_cases hgate : (querySupportedChildWithWork (σ := σ) passes copies T S parent (Sum.inr child)).1 = true | |
| · rw [dite_eq_left hgate] | |
| dsimp only | |
| have := hread ⟨Sum.inr child, (querySupportedChildWithWork_value passes copies T S parent _).mp hgate⟩ | |
| omega | |
| · rw [dite_eq_right hgate] | |
| omega | |
| end Body | |
| end EditApproximation | |
| end OAI | |