openAI-math-dataset / lean /OAI /Combinatorics /EditApproximation /Coarse /CoarseChildControlSpace.lean
| import OAI.Combinatorics.EditApproximation.Coarse.CoarseArrayControlSpace | |
| import OAI.Combinatorics.EditApproximation.Coarse.CoarseIndexedChildren | |
| namespace OAI | |
| namespace EditApproximation | |
| namespace BinaryFraction | |
| /-- Invocation count follows the same child sequence, thresholds, recursive calls, | |
| penalty sweeps and accumulated array as the indexed evaluator. -/ | |
| def coarseIndexChildrenInvocations {B : ℕ} (q L : ℕ) (two : BinaryFraction) | |
| (width : Fin B → ℕ) | |
| (evaluate : Fin B → BinaryFraction → ℕ → CoarseBinaryArray × ℕ) | |
| (allocation : Fin B → BinaryFraction → ℕ → ℕ) | |
| (draw : Fin B → Fin B) (allowance : BinaryFraction) : List (Fin B) → ℕ → ℕ | |
| | [], _ => 1 | |
| | i :: rest, start => | |
| let threshold := coarseChildThresholdWithWork B q allowance (draw i) | |
| let childAllowance := coarseChildAllowanceWithWork L threshold.1 | |
| let evaluated := evaluate i childAllowance.1 start | |
| let penalty := coarseBinaryPenaltyWithWork two evaluated.1 | |
| let scale := coarseChildAllowanceWithWork q allowance | |
| let contribution := coarseBinaryMapWithWork (coarseContributionWithWork threshold.1 scale.1) penalty.1 | |
| let remainder := coarseIndexChildren 0 q L two width evaluate draw allowance rest (start + width i) | |
| coarseChildThresholdInvocations B q allowance (draw i) + | |
| coarseChildAllowanceInvocations L threshold.1 + allocation i childAllowance.1 start + | |
| coarseBinaryPenaltyInvocations two evaluated.1 + coarseChildAllowanceInvocations q allowance + | |
| coarseBinaryMapInvocations (coarseContributionInvocations threshold.1 scale.1) penalty.1 + | |
| coarseIndexChildrenInvocations q L two width evaluate allocation draw allowance rest (start + width i) + | |
| coarseBinaryAddInvocations contribution.1 remainder.1 + 1 | |
| /-- The loop's first argument does not contribute to its runtime control space. -/ | |
| private theorem coarseIndexChildren_unused_k {B : ℕ} (k q L : ℕ) (two : BinaryFraction) | |
| (width : Fin B → ℕ) (evaluate : Fin B → BinaryFraction → ℕ → CoarseBinaryArray × ℕ) | |
| (draw : Fin B → Fin B) (allowance : BinaryFraction) (indices : List (Fin B)) (start : ℕ) : | |
| coarseIndexChildren k q L two width evaluate draw allowance indices start = | |
| coarseIndexChildren 0 q L two width evaluate draw allowance indices start := by | |
| induction indices generalizing start with | |
| | nil => rfl | |
| | cons i rest ih => simp only [coarseIndexChildren, ih] | |
| theorem coarseIndexChildrenInvocations_le_work {B : ℕ} (k q L : ℕ) (two : BinaryFraction) | |
| (width : Fin B → ℕ) | |
| (evaluate : Fin B → BinaryFraction → ℕ → CoarseBinaryArray × ℕ) | |
| (allocation : Fin B → BinaryFraction → ℕ → ℕ) | |
| (hallocation : ∀ i a start, allocation i a start ≤ 2 * (evaluate i a start).2 + 1) | |
| (draw : Fin B → Fin B) (allowance : BinaryFraction) (indices : List (Fin B)) (start : ℕ) : | |
| coarseIndexChildrenInvocations q L two width evaluate allocation draw allowance indices start ≤ | |
| 2 * (coarseIndexChildren k q L two width evaluate draw allowance indices start).2 + 1 := by | |
| rw [coarseIndexChildren_unused_k] | |
| induction indices generalizing start with | |
| | nil => exact Nat.le_refl 1 | |
| | cons i rest ih => | |
| let threshold := coarseChildThresholdWithWork B q allowance (draw i) | |
| let childAllowance := coarseChildAllowanceWithWork L threshold.1 | |
| let evaluated := evaluate i childAllowance.1 start | |
| let penalty := coarseBinaryPenaltyWithWork two evaluated.1 | |
| let scale := coarseChildAllowanceWithWork q allowance | |
| let contribution := coarseBinaryMapWithWork (coarseContributionWithWork threshold.1 scale.1) penalty.1 | |
| let remainder := coarseIndexChildren 0 q L two width evaluate draw allowance rest (start + width i) | |
| have ht := coarseChildThresholdInvocations_le_work B q allowance (draw i) | |
| have ha := coarseChildAllowanceInvocations_le_work L threshold.1 | |
| have he := hallocation i childAllowance.1 start | |
| have hp := coarseBinaryPenaltyInvocations_le_work two evaluated.1 | |
| have hs := coarseChildAllowanceInvocations_le_work q allowance | |
| have hm := coarseBinaryMapInvocations_le_work | |
| (coarseContributionWithWork threshold.1 scale.1) | |
| (coarseContributionInvocations threshold.1 scale.1) | |
| (coarseContributionInvocations_le_work threshold.1 scale.1) penalty.1 | |
| have hr := ih (start + width i) | |
| have had := coarseBinaryAddInvocations_le_work contribution.1 remainder.1 | |
| change coarseChildThresholdInvocations B q allowance (draw i) + | |
| coarseChildAllowanceInvocations L threshold.1 + allocation i childAllowance.1 start + | |
| coarseBinaryPenaltyInvocations two evaluated.1 + coarseChildAllowanceInvocations q allowance + | |
| coarseBinaryMapInvocations (coarseContributionInvocations threshold.1 scale.1) penalty.1 + | |
| coarseIndexChildrenInvocations q L two width evaluate allocation draw allowance rest (start + width i) + | |
| coarseBinaryAddInvocations contribution.1 remainder.1 + 1 ≤ | |
| 2 * (threshold.2 + childAllowance.2 + evaluated.2 + penalty.2 + scale.2 + contribution.2 + | |
| remainder.2 + (coarseBinaryAddWithWork contribution.1 remainder.1).2 + | |
| Nat.size start + Nat.size (width i) + 9) + 1 | |
| change coarseChildThresholdInvocations B q allowance (draw i) ≤ threshold.2 at ht | |
| change coarseChildAllowanceInvocations L threshold.1 ≤ childAllowance.2 at ha | |
| change allocation i childAllowance.1 start ≤ 2 * evaluated.2 + 1 at he | |
| change coarseBinaryPenaltyInvocations two evaluated.1 ≤ 2 * penalty.2 + 9 at hp | |
| change coarseChildAllowanceInvocations q allowance ≤ scale.2 at hs | |
| change coarseBinaryMapInvocations (coarseContributionInvocations threshold.1 scale.1) penalty.1 ≤ | |
| contribution.2 + 2 at hm | |
| change coarseIndexChildrenInvocations q L two width evaluate allocation draw allowance rest (start + width i) ≤ | |
| 2 * remainder.2 + 1 at hr | |
| omega | |
| end BinaryFraction | |
| end EditApproximation | |
| end OAI | |