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refactor(PP): reuse the machine strict-majority count lemma #1881

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SamuelSchlesinger merged 1 commit into from
Oct 31, 2026
Merged

refactor(PP): reuse the machine strict-majority count lemma #1881

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12 changes: 2 additions & 10 deletions Complexity/Computability/Complexity/Classes/PP/Time.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -11,6 +11,7 @@ public import Complexity.Computability.Complexity.Classes.PP
public import Complexity.Computability.Complexity.Classes.Time
import Complexity.Computability.Complexity.Classes.SharpP.Probability
import Complexity.Computability.Machines.Turing.MultiTape.Complexity.PolyTime.Plain
import Complexity.Computability.Machines.Turing.MultiTape.Probabilistic.AcceptCount
import Complexity.Computability.Machines.Turing.MultiTape.Probabilistic.CoinTape
import Complexity.Computability.Machines.Turing.MultiTape.Probabilistic.ReplayInput
import Complexity.Computability.Probabilistic.PPT.Capture
Expand Down Expand Up @@ -126,16 +127,7 @@ theorem decidesByMajorityInTime_iff {machine : MultiTapePTM k Bool State}
(fun x : List Bool × List Bool => decide (machine.runClosedCoins x.2 x.1 = [true]))
(enc a) (t a)) := by
refine forall_congr' fun a => and_congr_right' (iff_congr Iff.rfl ?_)
-- the probability of `[true]` is the accepted fraction of the coin strings
have hprob : ProbComp.eval (machine.runClosed (t a) (enc a)) [true] =
((BitString.uniformList (t a)).map fun w =>
decide (machine.runClosedCoins w (enc a) = [true])) true := by
rw [eval_runClosed_eq_map_coins, BitString.uniformList_map_apply,
BitString.uniformList_map_apply]
simp only [decide_eq_true_eq]
rw [hprob]
exact half_lt_uniformList_map_iff
(fun x : List Bool × List Bool => decide (machine.runClosedCoins x.2 x.1 = [true])) _ _
simpa only [one_div] using machine.half_lt_eval_runClosed_apply_iff (enc a) (t a)

end Turing.MultiTapePTM

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