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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 | import OAI.Computability.PerfectCompleteness.Reduction.FixedStoppedError
import OAI.Computability.PerfectCompleteness.Reduction.PreliminaryOutputLemmas
namespace OAI
section
namespace PerfectCompleteness.FixedStoppedPair
open FixedParameters FixedRows
noncomputable section
variable {δ : ℚ} {hδ : 0 < δ} (p : Parameters δ hδ)
theorem exists_pair (input : List Bool) (strategy : FixedPreliminaryGame.Strategy p input)
(hsuccess : InitialParameters.epsilon δ ≤
(FixedPreliminaryGame.game p input).success strategy) :
∃ i j : Fin p.plan.depth, i < j ∧
InitialParameters.simultaneous δ + 2 * p.accuracy ≤
(StoppedSharedSampler.stoppedLaw
(PCPSource.clauseFamily (BinaryLanguage.totalRename input))
(rows p.plan) (repeats p.plan) (cut := i.val)
(Nat.succ_le_of_lt i.isLt) (fun k _ => branch_pos p k)
(fun k => FixedPreliminaryGame.rows_pos p (k + 1))).probability
(StoppedSharedEvents.pair
(PCPSource.clauseFamily (BinaryLanguage.totalRename input)) strategy
(Nat.succ_le_of_lt i.isLt) (fun k _ => branch_pos p k) i j) := by
obtain ⟨i, j, hij, hmass⟩ := PreliminaryPairReserve.exists_pair p input strategy hsuccess
have hcut : i.val + 1 ≤ p.plan.depth := Nat.succ_le_of_lt i.isLt
have hi : i.val + 1 ≤ i.val + 1 := le_refl _
have hj : i.val + 1 ≤ j.val + 1 := Nat.add_le_add_right (Nat.le_of_lt hij) 1
have htransfer := StoppedSharedComparison.pair_probability_lower
(PCPSource.clauseFamily (BinaryLanguage.totalRename input)) strategy
hcut (fun k _ => branch_pos p k)
(fun k => FixedPreliminaryGame.rows_pos p (k + 1)) i j hi hj hmass
have herror := FixedStoppedError.stopped_error_lt p (le_refl p.plan.depth) hcut
refine ⟨i, j, hij, ?_⟩
linarith
end
end PerfectCompleteness.FixedStoppedPair
end
end OAI
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