openAI-math-dataset / lean /OAI /Combinatorics /EditApproximation /Geometry /AccuracyPrefixInspection.lean
| import OAI.Combinatorics.EditApproximation.Binary.BinaryFractionWork | |
| import OAI.Combinatorics.EditApproximation.Refinement.ComputedNumericalParameters | |
| import OAI.Combinatorics.EditApproximation.Geometry.ParameterAsymptotics | |
| import OAI.Combinatorics.EditApproximation.Definitions | |
| namespace OAI | |
| /-! The uniform program inspects only a logarithmically bounded prefix | |
| of each accuracy word. An encoding exceeding this bound takes the exact | |
| fallback, without first scanning or retaining its unbounded suffix. -/ | |
| namespace EditApproximation | |
| /-- One step reads a constructor of the input word. -/ | |
| theorem inspectBitPrefix_accepts (budget : ℕ) (word : List Bool) : | |
| (inspectBitPrefix budget word).1 = true ↔ word.length ≤ budget := by | |
| induction budget generalizing word with | |
| | zero => cases word <;> simp [inspectBitPrefix] | |
| | succ budget ih => | |
| cases word with | |
| | nil => simp [inspectBitPrefix] | |
| | cons bit rest => simpa only [inspectBitPrefix, List.length_cons, Nat.add_le_add_iff_right] using ih rest | |
| theorem inspectBitPrefix_cost (budget : ℕ) (word : List Bool) : | |
| (inspectBitPrefix budget word).2 ≤ budget + 1 := by | |
| induction budget generalizing word with | |
| | zero => cases word <;> exact le_rfl | |
| | succ budget ih => | |
| cases word with | |
| | nil => exact Nat.succ_le_succ (Nat.zero_le _) | |
| | cons bit rest => exact Nat.add_le_add_right (ih rest) 1 | |
| /-- On acceptance the input words are already bounded; returning the | |
| input value only retains those bounded words. -/ | |
| theorem inspectAccuracyWithWork_spec (budget : ℕ) (epsilon : BinaryFraction) : | |
| (inspectAccuracyWithWork budget epsilon).1 = | |
| if epsilon.width ≤ budget then some epsilon else none := by | |
| simp only [inspectAccuracyWithWork, BinaryFraction.width, max_le_iff] | |
| by_cases hn : epsilon.numerator.bits.length ≤ budget <;> | |
| by_cases hd : epsilon.denominator.length ≤ budget | |
| all_goals | |
| have hnum := inspectBitPrefix_accepts budget epsilon.numerator.bits | |
| have hden := inspectBitPrefix_accepts budget epsilon.denominator | |
| simp_all | |
| theorem inspectAccuracyWithWork_cost (budget : ℕ) (epsilon : BinaryFraction) : | |
| (inspectAccuracyWithWork budget epsilon).2 ≤ 2 * budget + 4 := by | |
| have hn := inspectBitPrefix_cost budget epsilon.numerator.bits | |
| have hd := inspectBitPrefix_cost budget epsilon.denominator | |
| change (inspectBitPrefix budget epsilon.numerator.bits).2 + | |
| (inspectBitPrefix budget epsilon.denominator).2 + 2 ≤ _ | |
| omega | |
| theorem inspectAccuracy_eventually_accepts (epsilon : BinaryFraction) : | |
| ∀ᶠ N in Filter.atTop, (inspectAccuracyWithWork (inputHeight N) epsilon).1 = some epsilon := by | |
| filter_upwards [inputSmallLog_tendsto.eventually (Filter.eventually_ge_atTop epsilon.width)] with N hN | |
| rw [inspectAccuracyWithWork_spec, ite_eq_left (hN.trans (computed_smallLog_le_height N))] | |
| end EditApproximation | |
| end OAI | |