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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 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 | import OAI.Combinatorics.ContingencyTables.Sampling.ExactCoreCost
import OAI.Combinatorics.ContingencyTables.Sampling.PhysicalPartialSequence
namespace OAI
/-! # A cheap prefix ending before rare-branch tabulation -/
namespace ContingencyTables.ExactGatePrefix
open MatchingFPRAS MatchingFPRAS.TreeData MatchingFPRAS.TreeFunction
open MatchingFPRAS.TreeMachine (Env K encodeEnv)
open MatchingFPRAS.PhysicalStream MatchingFPRAS.OnlineStack
open ProfilePrograms DensePrograms ExactCoreProgram
noncomputable def output (z : CompletionMargins) (s : Data) :
Option MatrixCode → OnlinePartialCall.Config instructions
| none => OnlineControl.cfg instructions (some .prepareRare) (env (TreeTyped.encode z) .nil s)
| some X => OnlineControl.cfg instructions none (env (TreeTyped.encode X) .nil s)
def experiment (z : CompletionMargins) : Experiment (Option MatrixCode) :=
(rareBranch z).bind (fun b => if b then .done none else (exactOrdinary z).map some)
noncomputable def budget (z : CompletionMargins) : ℕ :=
rareBranchRealizer.cost z+exactOrdinaryRealizer.cost z+10*TreeFunction.size (TreeTyped.encode z)+50
/-- The rare branch is selected but its potentially large tabulation has
not started at the failure endpoint of this bounded prefix. -/
theorem prefix_correct_cost (z : CompletionMargins) (s : Data) :
∃ p,Executes program.code program.coin (output z s)
(OnlineTree.initial program (TreeTyped.encode z) s) p ∧
(∀ g : Option MatrixCode → ℚ,p.mean g=(experiment z).mean g) ∧ p.cost≤budget z := by
let out := output z s
have halted (X : MatrixCode) :
OnlinePartialCall.At instructions out (some .halt) (env (TreeTyped.encode X) .nil s)
(.delay 1 (.done (some X))) :=
OnlinePartialCall.halt instructions out .halt rfl (some X) _ rfl
have ordinary : ∃ p,OnlinePartialCall.At instructions out (some .ordinary)
(env (TreeTyped.encode z) .nil s) p ∧
(∀ g : Option MatrixCode → ℚ,p.mean g=((exactOrdinary z).map some).mean g) ∧
p.cost≤exactOrdinaryRealizer.cost z+2 := by
obtain ⟨u,hu,hum,huc⟩ := exactOrdinaryRealizer.correct z s
refine ⟨.delay 1 (u.bind (fun X => .delay 1 (.done (some X)))),?_,?_,?_⟩
· exact OnlinePartialCall.call instructions out .ordinary .halt exactOrdinaryRealizer.program
rfl _ (fun X => env (TreeTyped.encode X) .nil s) u hu _ halted
· intro g
simp only [Experiment.mean,Experiment.mean_bind,Experiment.mean_map]
exact hum _
· have hc := Experiment.cost_bind_le u (fun X => .delay 1 (.done (some X))) 1
(by intro X; rfl)
simp only [Experiment.cost] at hc ⊢
omega
have branch (b : Bool) : ∃ p,OnlinePartialCall.At instructions out (some .choose)
(env (TreeTyped.encode b) .nil (.cons (TreeTyped.encode z) s)) p ∧
(∀ g : Option MatrixCode → ℚ,p.mean g=
(if b then (.done none : Experiment (Option MatrixCode)) else (exactOrdinary z).map some).mean g) ∧
p.cost≤exactOrdinaryRealizer.cost z+2*TreeFunction.size (TreeTyped.encode z)+20 := by
cases b with
| false =>
obtain ⟨p,hp,hpm,hpc⟩ := ordinary
obtain ⟨v,hv,hvm,hvc⟩ := OnlinePartialCall.deterministicCall instructions out
.ordinaryInput .ordinary (split 0 2 (by decide)) rfl _ _ _
(split02 (TreeTyped.encode false) .nil (TreeTyped.encode z) s) p hp
refine ⟨.delay 1 v,?_,(fun g => (hvm g).trans (hpm g)),?_⟩
· exact OnlinePartialCall.nilBranch instructions out .choose .ordinaryInput .rareInput 0 rfl _ rfl v hv
· have hb : TreeFunction.size (TreeTyped.encode false)≤3 := by decide
simp only [Experiment.cost]
omega
| true =>
have hp : OnlinePartialCall.At instructions out (some .prepareRare)
(env (TreeTyped.encode z) .nil s) (.done none) := .done none
obtain ⟨v,hv,hvm,hvc⟩ := OnlinePartialCall.deterministicCall instructions out
.rareInput .prepareRare (split 0 2 (by decide)) rfl _ _ _
(split02 (TreeTyped.encode true) .nil (TreeTyped.encode z) s) (.done none) hp
refine ⟨.delay 1 v,?_,hvm,?_⟩
· exact OnlinePartialCall.consBranch instructions out .choose .ordinaryInput .rareInput 0 rfl _
.nil .nil rfl v hv
· have hb : TreeFunction.size (TreeTyped.encode true)≤3 := by decide
simp only [Experiment.cost] at hvc ⊢
omega
choose v hv hvm hvc using branch
obtain ⟨u,hu,hum,huc⟩ := rareBranchRealizer.correct z (.cons (TreeTyped.encode z) s)
have hgate := OnlinePartialCall.call instructions out .gate .choose rareBranchRealizer.program
rfl _ (fun b => env (TreeTyped.encode b) .nil (.cons (TreeTyped.encode z) s)) u hu v hv
obtain ⟨p,hp,hpm,hpc⟩ := OnlinePartialCall.deterministicCall instructions out .prepare .gate
OnlineIntegerRetry.prepareProc rfl _ _ _ (OnlineIntegerRetry.prepare_exec (TreeTyped.encode z) s)
(.delay 1 (u.bind v)) hgate
refine ⟨p,hp,?_,?_⟩
· intro g
rw [hpm]
simp only [Experiment.mean,Experiment.mean_bind]
simp_rw [hvm]
rw [hum,experiment,Experiment.mean_bind]
· have hc := Experiment.cost_bind_le u v
(exactOrdinaryRealizer.cost z+2*TreeFunction.size (TreeTyped.encode z)+20) hvc
simp only [Experiment.cost] at hpc
unfold budget
omega
lemma failure (z : CompletionMargins) :
(experiment z).mean (fun o => if o=none then 1 else 0)=1/(2:ℚ)^(exactPrecision z) := by
rw [experiment,Experiment.mean_bind]
have he : (fun b : Bool => (if b then (.done none : Experiment (Option MatrixCode))
else (exactOrdinary z).map some).mean (fun o => if o=none then 1 else 0))=
(fun b => if b then 1 else 0) := by
funext b
cases b with
| true => rfl
| false =>
simp only [Bool.false_eq_true,ite_false,Experiment.mean_map,Option.some_ne_none]
exact PhysicalPartialSequence.mean_const _ _
rw [he,rareBranch_probability]
end ContingencyTables.ExactGatePrefix
end OAI
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