File size: 2,886 Bytes
e63eb7f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 | 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
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