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2f887db | 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 | import OAI.Computability.PerfectCompleteness.Foundations.StoppedSharedComparison
import OAI.Computability.PerfectCompleteness.Reduction.FixedComparisonErrorsLemmas
namespace OAI
section
namespace PerfectCompleteness.FixedStoppedError
open FixedParameters FixedRows
noncomputable section
variable {δ : ℚ} {hδ : 0 < δ} (p : Parameters δ hδ)
theorem original_calls_le {root cutoff : Nat}
(hroot : root ≤ p.plan.depth) (hcut : cutoff ≤ root) :
OriginalCutCalls.count (rows p.plan) (repeats p.plan) root cutoff ≤ calls p.plan := by
have h := FixedCallBudget.pairCount_le (rows p.plan) (repeats p.plan)
hroot (hcut.trans hroot)
unfold FixedCallBudget.pairCount at h
unfold calls
omega
theorem original_calls_add_one_le {root cutoff : Nat}
(hroot : root ≤ p.plan.depth) (hcut : cutoff ≤ root) :
OriginalCutCalls.count (rows p.plan) (repeats p.plan) root cutoff + 1 ≤ calls p.plan := by
have h := FixedCallBudget.pairCount_le (rows p.plan) (repeats p.plan)
hroot (hcut.trans hroot)
unfold FixedCallBudget.pairCount at h
unfold calls
omega
theorem hidden_error_lt {retained : Nat} (hcalls : retained ≤ calls p.plan) (height : Nat) :
Real.sqrt ((ChildBlockCardinality.bound (branch p) height (sourceLength p.plan hδ)
retained (rows p.plan) : ℝ) ^ 2 / branch p height) / 2 < p.accuracy := by
have h := (FixedComparisonErrors.comparison_errors p hcalls height).1
simpa only [branch, Nat.cast_pow] using h
theorem stopped_error_lt {root cut : Nat}
(hroot : root ≤ p.plan.depth) (hcut : cut + 1 ≤ root) :
StoppedSharedComparison.error (branch p) (rows p.plan) (repeats p.plan)
root cut (sourceLength p.plan hδ) < p.accuracy :=
hidden_error_lt p (original_calls_le p hroot hcut) cut
theorem collision_error_lt {root cut : Nat}
(hroot : root ≤ p.plan.depth) (hcut : cut + 1 ≤ root) :
Real.sqrt ((ChildBlockCardinality.bound (branch p) cut (sourceLength p.plan hδ)
(OriginalCutCalls.count (rows p.plan) (repeats p.plan) root (cut + 1) + 1)
(rows p.plan) : ℝ) ^ 2 / branch p cut) / 2 < p.accuracy :=
hidden_error_lt p (original_calls_add_one_le p hroot hcut) cut
end
end PerfectCompleteness.FixedStoppedError
end
end OAI
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