openAI-math-dataset / lean /OAI /Combinatorics /EditApproximation /Queries /QueryChargedWarmupCost.lean
| import OAI.Combinatorics.EditApproximation.Queries.QueryPhysicalInitialCost | |
| import OAI.Combinatorics.EditApproximation.Queries.QueryChargedWarmup | |
| import OAI.Combinatorics.EditApproximation.Queries.QueryWarmupProgramCost | |
| namespace OAI | |
| namespace EditApproximation | |
| def queryChargedWarmupBitBudget (M P F Amax copies J symbolBits positionBits B C : ℕ) : ℕ := | |
| let I := queryPhysicalInitialBitBudget P F Amax copies J symbolBits positionBits B C | |
| 80 * (B + 1) + I + | |
| BinaryFraction.queryWarmupProgramBitBudget M P F B (I + 7 * J + 33 + C) | |
| section Warmup | |
| variable (source target : List ℕ) {M J passes copies T S : ℕ} | |
| (N P F Amax B C : ℕ) (A a : BinaryFraction) (multiplier : ℕ → ℕ) | |
| (parent : {r : PhysicalTableRequest M J target.length // r.Bounded passes copies T S}) | |
| (occurrence : BinaryMemo PositionCounts) (symbolBits positionBits : ℕ) | |
| (oracle : {child : ChargedPhysicalRequest M J target.length passes copies T S // | |
| child.rank < ChargedPhysicalRequest.rank (.inr parent)} → BinaryFraction) | |
| (readWork : {child : ChargedPhysicalRequest M J target.length passes copies T S // | |
| child.rank < ChargedPhysicalRequest.rank (.inr parent)} → ℕ) | |
| theorem queryChargedWarmup_work_bound | |
| (hB : 10 ≤ B) (hP : 0 < P) (hN : Nat.size N ≤ B) (hPb : Nat.size P ≤ B) | |
| (hF : Nat.size F + 2 ≤ B) (hFpos : 1 ≤ F) | |
| (hs : Nat.size source.length + 2 ≤ B) (ht : Nat.size target.length + 2 ≤ B) | |
| (ht' : Nat.size (target.length + 1) + 2 ≤ B) | |
| (hinit : Nat.size ((32 * F + 1) * N) + 3 ≤ B) | |
| (hM : Nat.size M ≤ B) (hJ : Nat.size J ≤ B) (hi : Nat.size parent.val.index ≤ B) | |
| (hA : A.width ≤ B) (ha : a.width ≤ B) (hAone : 1 ≤ A.value) (hAmax : A.value ≤ Amax) | |
| (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) : | |
| (queryChargedWarmup source target N P F A a multiplier parent occurrence symbolBits positionBits).work | |
| oracle readWork ≤ queryChargedWarmupBitBudget M P F Amax copies J symbolBits positionBits B C := by | |
| let I := queryPhysicalInitialBitBudget P F Amax copies J symbolBits positionBits B C | |
| let reader := queryPhysicalInitial source target N P F A a multiplier parent occurrence symbolBits positionBits | |
| have hiR (q : TargetInterval target.length) : (reader q).work oracle readWork ≤ I := | |
| queryPhysicalInitial_work_bound source target N P F Amax B C A a multiplier parent occurrence | |
| symbolBits positionBits oracle readWork q (by omega) hP hN hPb (by omega) (by omega) (by omega) | |
| hA ha hAone hAmax hp hm hv hc | |
| have hiW (q : TargetInterval target.length) : ((reader q).value oracle).width ≤ B := | |
| queryPhysicalInitial_width source target N P F B A a multiplier parent occurrence symbolBits positionBits | |
| oracle q hP ht' hinit | |
| have hzero := naturalEqualWithWork_cost parent.val.index 0 B hi (by simp) | |
| have hn := (Nat.size_le_size (Nat.le_of_lt_succ parent.val.node.1.isLt)).trans hJ | |
| have hdepth := binaryNaturalCompareWithWork_cost parent.val.node.1.val J | |
| have hdepth' : (binaryNaturalCompareWithWork parent.val.node.1.val J).2 ≤ 8 * B + 10 := by omega | |
| have htest := querySliceTest_cost source (physicalSourceInterval source.length parent.val.node) B (by omega) | |
| unfold queryChargedWarmup | |
| simp only [BitQuery.work] | |
| split | |
| · change (naturalEqualWithWork parent.val.index 0).2 + 1 + (reader parent.val.state).work oracle readWork ≤ _ | |
| have h := hiR parent.val.state | |
| unfold queryChargedWarmupBitBudget | |
| dsimp only | |
| dsimp only [I] at h | |
| omega | |
| · rename_i hnonzero | |
| have hindex : 0 < parent.val.index := | |
| Nat.pos_of_ne_zero (fun h => hnonzero ((naturalEqualWithWork_value _ _).mpr h)) | |
| split | |
| · split | |
| · have h := hiR parent.val.state | |
| change (naturalEqualWithWork parent.val.index 0).2 + 1 + | |
| ((binaryNaturalCompareWithWork parent.val.node.1.val J).2 + 1 + | |
| (_ + (reader parent.val.state).work oracle readWork)) ≤ _ | |
| unfold queryChargedWarmupBitBudget | |
| dsimp only | |
| dsimp only [I] at h | |
| omega | |
| · rename_i hbelow hlong | |
| have hcR (input : Fin M × TargetInterval target.length) : | |
| (queryChargedWarmupChild parent ((binaryNaturalCompareWithWork_lt _ _).mp hbelow) hindex input).work | |
| oracle readWork ≤ I + 7 * J + 33 + C := by | |
| exact (queryChargedRead_work_bound parent oracle readWork C hc _).trans (by omega) | |
| have hw := queryWarmupProgram_work_bound (physicalSourceInterval source.length parent.val.node) | |
| N P F B (I + 7 * J + 33 + C) hP a reader | |
| (queryChargedWarmupChild parent ((binaryNaturalCompareWithWork_lt _ _).mp hbelow) hindex) parent.val.state | |
| oracle readWork hB hN hF hFpos hPb hs ht hM ha hiW (fun input => hv _) | |
| (fun r => (hiR r).trans (by omega)) hcR | |
| change (naturalEqualWithWork parent.val.index 0).2 + 1 + | |
| ((binaryNaturalCompareWithWork parent.val.node.1.val J).2 + 1 + | |
| (_ + (queryWarmupProgram _ N P F a reader _ parent.val.state).work oracle readWork)) ≤ _ | |
| unfold queryChargedWarmupBitBudget | |
| dsimp only | |
| dsimp only [I] at hw | |
| omega | |
| · have h := hiR parent.val.state | |
| change (naturalEqualWithWork parent.val.index 0).2 + 1 + | |
| ((binaryNaturalCompareWithWork parent.val.node.1.val J).2 + 1 + | |
| (reader parent.val.state).work oracle readWork) ≤ _ | |
| unfold queryChargedWarmupBitBudget | |
| dsimp only | |
| dsimp only [I] at h | |
| omega | |
| end Warmup | |
| end EditApproximation | |
| end OAI | |