openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Sampling /ExactIntegerPhysical.lean
| 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 | |