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d5b7741 | 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 | import OAI.Combinatorics.ContingencyTables.Machines.OnlineRetryRuntime
import OAI.Combinatorics.ContingencyTables.Sampling.ExactIntegerOnline
namespace OAI
/-! # Finite expected machine time for exact binary weighted choice -/
namespace ContingencyTables.ExactIntegerPhysical
open MatchingFPRAS MatchingFPRAS.PhysicalStream ExactIntegerChoice
theorem prefix_failure_bound {r : ℕ} (w : Fin r → ℕ) (h : 0<totalWeight w) (N : ℕ) :
(OnlineIntegerRetry.prefixExperiment (f := ExactIntegerOnline.trial) (List.ofFn w) N).mean
(fun b => if b=none then 1 else 0) ≤ (1/2:ℚ)^N := by
rw [ExactIntegerOnline.integer_prefix_failure w h]
exact pow_le_pow_left₀
(failure_nonnegative w (Nat.le_pow_clog (by decide : 1<2) (totalWeight w)))
(failure_lt_half w (clog_power_lt_twice (totalWeight w) h)).le N
/-- The repeating finite-state bit program has summable physical halting
tail for every positive list of integer weights. -/
theorem expected_time {r : ℕ} (w : Fin r → ℕ) (h : 0<totalWeight w)
(s : TreeData.Data) (H : ℕ)
(hH : StackCost.StackBound
(OnlineTree.initial (OnlineIntegerRetry.program ExactIntegerOnline.trialRealizer)
(TreeTyped.encode (List.ofFn w)) s).stk H) :
Summable (PhysicalRetryTail.tail
(OnlineRetryRuntime.machineTable ExactIntegerOnline.trialRealizer)
(OnlineRetryRuntime.initialCfg ExactIntegerOnline.trialRealizer (List.ofFn w) s)) ∧
(∑' t,PhysicalRetryTail.tail
(OnlineRetryRuntime.machineTable ExactIntegerOnline.trialRealizer)
(OnlineRetryRuntime.initialCfg ExactIntegerOnline.trialRealizer (List.ofFn w) s) t) ≤
2*(OnlineRetryRuntime.attemptBound ExactIntegerOnline.trialRealizer (List.ofFn w) H:ℝ) :=
OnlineRetryRuntime.expected_time ExactIntegerOnline.trialRealizer (List.ofFn w) s H hH
(prefix_failure_bound w h)
end ContingencyTables.ExactIntegerPhysical
end OAI
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