openAI-math-dataset / lean /OAI /Combinatorics /EditApproximation /Coarse /CoarseIndexedControlSpace.lean
| import OAI.Combinatorics.EditApproximation.Coarse.CoarseAccessControlSpace | |
| import OAI.Combinatorics.EditApproximation.Coarse.CoarseChildControlSpace | |
| import OAI.Combinatorics.EditApproximation.Coarse.CoarseIndexedExecution | |
| namespace OAI | |
| namespace EditApproximation | |
| namespace BinaryFraction | |
| /-- Conservative recursive invocation count for the indexed evaluator, | |
| including scalar callees, index traversals and repeated descriptor reads. -/ | |
| def coarseIndexedInvocations {B : ℕ} (k q L n : ℕ) (source target : List ℕ) | |
| (two : BinaryFraction) : (d offset : ℕ) → CoarseThresholdDraws B d → | |
| BinaryFraction → ℕ → ℕ | |
| | 0, offset, _, allowance, start => | |
| let descriptor := coarseDescriptorWithWork B 0 n offset | |
| let size := bitWordValue descriptor.1 | |
| let prune := leWithWork (nat size) allowance | |
| let setup := coarseDescriptorInvocations B 0 n offset + compareInvocations allowance (nat size) + 2 | |
| if prune.1 then setup else | |
| let letter := coarsePaddedReadWithWork n source offset | |
| setup + coarsePaddedReadInvocations n offset + | |
| arithmeticMapInvocations (fun shift : Fin (2 * k + 1) => | |
| let mismatch := coarseReadShiftWithWork k n target letter.1 start shift | |
| coarseReadShiftInvocations k n target letter.1 start shift + | |
| coarseGridNatInvocations allowance.denominator allowance.denominator_pos mismatch.1 + 1) | |
| (List.finRange (2 * k + 1)) + (2 * k + 1) + 1 | |
| | d + 1, offset, (draws, below), allowance, start => | |
| let descriptor := coarseDescriptorWithWork B (d + 1) n offset | |
| let size := bitWordValue descriptor.1 | |
| let prune := leWithWork (nat size) allowance | |
| let setup := coarseDescriptorInvocations B (d + 1) n offset + compareInvocations allowance (nat size) + 2 | |
| if prune.1 then setup else | |
| let capacity := bitPowerWithWork B.bits d | |
| let childOffset := fun i : Fin B => | |
| coarseChildOffsetWithWork offset (bitWordValue capacity.1) i.val | |
| let childSize := fun i : Fin B => | |
| coarseDescriptorWithWork B d n (bitWordValue (childOffset i).1) | |
| let evaluate := fun (i : Fin B) a s => | |
| let evaluated := coarseIndexedWithWork k q L n source target two d | |
| (bitWordValue (childOffset i).1) (below i) a s | |
| let next := binaryNaturalAddWithWork s (bitWordValue (childSize i).1) | |
| (evaluated.1, evaluated.2 + 3 * ((childOffset i).2 + (childSize i).2 + next.2 + 3)) | |
| let allocation := fun (i : Fin B) a s => | |
| coarseIndexedInvocations k q L n source target two d | |
| (bitWordValue (childOffset i).1) (below i) a s + | |
| 3 * (coarseChildOffsetInvocations offset (bitWordValue capacity.1) i.val + | |
| coarseDescriptorInvocations B d n (bitWordValue (childOffset i).1) + | |
| bitAddInvocations s.bits (bitWordValue (childSize i).1).bits false) + 1 | |
| setup + bitPowerInvocations B.bits d + | |
| coarseIndexChildrenInvocations q L two (fun i => bitWordValue (childSize i).1) | |
| evaluate allocation draws allowance (List.finRange B) start + B + 1 | |
| theorem arithmeticMapInvocations_with_indices_le_work {α β : Type*} (f : α → β × ℕ) | |
| (invocations : α → ℕ) (values : List α) | |
| (h : ∀ a ∈ values, invocations a ≤ 2 * (f a).2 + 1) : | |
| arithmeticMapInvocations invocations values + values.length + 1 ≤ | |
| 2 * (arithmeticMapWithWork f values).2 + 2 := by | |
| induction values with | |
| | nil => exact Nat.le_refl 2 | |
| | cons a rest ih => | |
| have hh := h a List.mem_cons_self | |
| have ht := ih (fun b hb => h b (List.mem_cons_of_mem _ hb)) | |
| simp only [arithmeticMapInvocations, arithmeticMapWithWork, List.length_cons] | |
| omega | |
| theorem coarseIndexedInvocations_le_work {B : ℕ} (k q L n : ℕ) (source target : List ℕ) | |
| (two : BinaryFraction) (d offset : ℕ) (draws : CoarseThresholdDraws B d) | |
| (allowance : BinaryFraction) (start : ℕ) : | |
| coarseIndexedInvocations k q L n source target two d offset draws allowance start ≤ | |
| 2 * (coarseIndexedWithWork k q L n source target two d offset draws allowance start).2 + 1 := by | |
| induction d generalizing offset allowance start with | |
| | zero => | |
| have hd := coarseDescriptorInvocations_le_work B 0 n offset | |
| have hp := compareInvocations_le_work allowance (nat (bitWordValue (coarseDescriptorWithWork B 0 n offset).1)) | |
| simp only [coarseIndexedInvocations, coarseIndexedWithWork] | |
| split_ifs | |
| · simp only [leWithWork] | |
| omega | |
| · have hmap := arithmeticMapInvocations_with_indices_le_work | |
| (fun shift : Fin (2 * k + 1) => | |
| let mismatch := coarseReadShiftWithWork k n target (coarsePaddedReadWithWork n source offset).1 start shift | |
| let value := coarseGridNatWithWork allowance.denominator allowance.denominator_pos mismatch.1 | |
| (value.1, value.2 + mismatch.2)) | |
| (fun shift : Fin (2 * k + 1) => | |
| let mismatch := coarseReadShiftWithWork k n target (coarsePaddedReadWithWork n source offset).1 start shift | |
| coarseReadShiftInvocations k n target (coarsePaddedReadWithWork n source offset).1 start shift + | |
| coarseGridNatInvocations allowance.denominator allowance.denominator_pos mismatch.1 + 1) | |
| (List.finRange (2 * k + 1)) (by | |
| intro shift _ | |
| have hs := coarseReadShiftInvocations_le_work k n target | |
| (coarsePaddedReadWithWork n source offset).1 start shift | |
| have hg := coarseGridNatInvocations_le_work allowance.denominator allowance.denominator_pos | |
| (coarseReadShiftWithWork k n target (coarsePaddedReadWithWork n source offset).1 start shift).1 | |
| dsimp only | |
| omega) | |
| have hpadded := coarsePaddedReadInvocations_le_work n source offset | |
| simp only [List.length_finRange] at hmap | |
| simp only [leWithWork, coarsePaddedReadWithWork] at hpadded ⊢ | |
| simp only [coarsePaddedReadWithWork] at hmap | |
| omega | |
| | succ d ih => | |
| obtain ⟨draw, below⟩ := draws | |
| let capacity := bitPowerWithWork B.bits d | |
| let childOffset := fun i : Fin B => coarseChildOffsetWithWork offset (bitWordValue capacity.1) i.val | |
| let childSize := fun i : Fin B => coarseDescriptorWithWork B d n (bitWordValue (childOffset i).1) | |
| let evaluate := fun (i : Fin B) a s => | |
| let evaluated := coarseIndexedWithWork k q L n source target two d | |
| (bitWordValue (childOffset i).1) (below i) a s | |
| let next := binaryNaturalAddWithWork s (bitWordValue (childSize i).1) | |
| (evaluated.1, evaluated.2 + 3 * ((childOffset i).2 + (childSize i).2 + next.2 + 3)) | |
| let allocation := fun (i : Fin B) a s => | |
| coarseIndexedInvocations k q L n source target two d | |
| (bitWordValue (childOffset i).1) (below i) a s + | |
| 3 * (coarseChildOffsetInvocations offset (bitWordValue capacity.1) i.val + | |
| coarseDescriptorInvocations B d n (bitWordValue (childOffset i).1) + | |
| bitAddInvocations s.bits (bitWordValue (childSize i).1).bits false) + 1 | |
| have he : ∀ i a s, allocation i a s ≤ 2 * (evaluate i a s).2 + 1 := by | |
| intro i a s | |
| have hr := ih (bitWordValue (childOffset i).1) (below i) a s | |
| have ho := coarseChildOffsetInvocations_le_work offset (bitWordValue capacity.1) i.val | |
| have hs := coarseDescriptorInvocations_le_work B d n (bitWordValue (childOffset i).1) | |
| have ha := bitAddInvocations_le_work s.bits (bitWordValue (childSize i).1).bits false | |
| change bitAddInvocations s.bits (bitWordValue (childSize i).1).bits false ≤ | |
| (binaryNaturalAddWithWork s (bitWordValue (childSize i).1)).2 at ha | |
| change coarseChildOffsetInvocations offset (bitWordValue capacity.1) i.val ≤ (childOffset i).2 at ho | |
| change coarseDescriptorInvocations B d n (bitWordValue (childOffset i).1) ≤ (childSize i).2 at hs | |
| dsimp only [allocation, evaluate] | |
| omega | |
| have hc := coarseIndexChildrenInvocations_le_work k q L two (fun i => bitWordValue (childSize i).1) | |
| evaluate allocation he draw allowance (List.finRange B) start | |
| have hd := coarseDescriptorInvocations_le_work B (d + 1) n offset | |
| have hp := compareInvocations_le_work allowance | |
| (nat (bitWordValue (coarseDescriptorWithWork B (d + 1) n offset).1)) | |
| have hb := bitPowerInvocations_le_work B.bits d | |
| simp only [coarseIndexedInvocations, coarseIndexedWithWork] | |
| split_ifs | |
| · simp only [leWithWork] | |
| omega | |
| · change coarseDescriptorInvocations B (d + 1) n offset + | |
| compareInvocations allowance (nat (bitWordValue (coarseDescriptorWithWork B (d + 1) n offset).1)) + 2 + | |
| bitPowerInvocations B.bits d + | |
| coarseIndexChildrenInvocations q L two (fun i => bitWordValue (childSize i).1) | |
| evaluate allocation draw allowance (List.finRange B) start + B + 1 ≤ | |
| 2 * ((coarseDescriptorWithWork B (d + 1) n offset).2 + capacity.2 + | |
| (coarseIndexChildren k q L two (fun i => bitWordValue (childSize i).1) | |
| evaluate draw allowance (List.finRange B) start).2 + | |
| (leWithWork (nat (bitWordValue (coarseDescriptorWithWork B (d + 1) n offset).1)) allowance).2 + | |
| Nat.size (bitWordValue (coarseDescriptorWithWork B (d + 1) n offset).1) + B + 1) + 1 | |
| change bitPowerInvocations B.bits d ≤ capacity.2 at hb | |
| simp only [leWithWork] | |
| omega | |
| end BinaryFraction | |
| end EditApproximation | |
| end OAI | |