File size: 2,492 Bytes
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 | import OAI.Combinatorics.ContingencyTables.Sampling.ExactSamplingParameters
import OAI.Combinatorics.ContingencyTables.Machines.ListWordProgram
namespace OAI
/-! # The exact dyadic gate that selects the rare correction branch -/
namespace ContingencyTables.ProfilePrograms
open MatchingFPRAS MatchingFPRAS.TreeTyped MatchingFPRAS.PhysicalStream DensePrograms Polynomial
/-- All zero bits select correction; every other word selects the ordinary draw. -/
def rareWord (z : CompletionMargins×List Bool) : Bool := decide (StackAddition.value z.2=0)
def rareWordRealizer : Realizer rareWord :=
composition (f := fun z : CompletionMargins×List Bool => (StackAddition.value z.2,0))
(g := fun p : ℕ×ℕ => decide (p.1=p.2))
(pair (composition second bitValue) (constant 0)) natEqual
lemma polynomial_rareWord : PolynomialTime rareWordRealizer :=
polynomial_composition (polynomial_pair (polynomial_composition polynomial_second polynomial_bitValue)
(polynomial_constant 0)) polynomial_natEqual
def rareBranch (z : CompletionMargins) : Experiment Bool :=
FiniteWordExecution.experiment exactPrecision rareWord z
noncomputable def rareBranchRealizer : OnlineTree.Realizer rareBranch :=
FiniteWordExecution.realizer exactPrecisionRealizer rareWordRealizer
lemma work_rareBranch : OnlineTree.WorkBound rareBranchRealizer exactParameterMeasure := by
apply OnlineTree.work_remeasure
(FiniteWordExecution.work_realizer polynomial_exactPrecision polynomial_rareWord) X
intro z
simp only [FiniteWordExecution.wordMeasure,exactParameterMeasure,eval_X]
omega
lemma rareBranch_probability (z : CompletionMargins) :
(rareBranch z).mean (fun rare => if rare then 1 else 0)=1/(2:ℚ)^(exactPrecision z) := by
rw [rareBranch,FiniteWordExecution.experiment_mean]
simp only [rareWord,decide_eq_true_eq,bitValue_eq_word]
let zeroWord : Fin (exactPrecision z) → Bool :=
(FairBits.binaryWordEquiv (exactPrecision z)).symm ⟨0,by positivity⟩
have he (bits : Fin (exactPrecision z) → Bool) :
(FairBits.binaryWordEquiv (exactPrecision z) bits).val=0 ↔ bits=zeroWord := by
constructor
· intro h
apply (FairBits.binaryWordEquiv (exactPrecision z)).injective
apply Fin.ext
simpa only [zeroWord,Equiv.apply_symm_apply] using h
· intro h
rw [h]
simp only [zeroWord,Equiv.apply_symm_apply]
simp only [he,average,Finset.sum_ite_eq',Finset.mem_univ,ite_true]
end ContingencyTables.ProfilePrograms
end OAI
|