| import OAI.Combinatorics.ContingencyTables.Sampling.ExactGatePrefix |
|
|
| namespace OAI |
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|
| /-! # Polynomial cost before the rare branch starts -/ |
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|
| namespace ContingencyTables.ExactGateCost |
| open MatchingFPRAS MatchingFPRAS.TreeTyped DensePrograms ProfilePrograms Polynomial |
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|
| def measure (z : CompletionMargins) : ℕ := |
| exactParameterMeasure z+FiniteWordExecution.wordMeasure exactOrdinaryBits z |
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| lemma budget_polynomial : ∃ p : Polynomial ℕ, |
| ∀ z,ExactGatePrefix.budget z ≤ p.eval (measure z) := by |
| obtain ⟨p,hp⟩ := work_rareBranch |
| obtain ⟨q,hq⟩ := work_exactOrdinary |
| refine ⟨p+q+10*X+50,?_⟩ |
| intro z |
| have h1 : rareBranchRealizer.cost z ≤ p.eval (measure z) := |
| (Nat.le_add_right _ _).trans ((hp z).trans (polynomial_eval_monotone p (Nat.le_add_right _ _))) |
| have h2 : exactOrdinaryRealizer.cost z ≤ q.eval (measure z) := |
| (Nat.le_add_right _ _).trans ((hq z).trans (polynomial_eval_monotone q (Nat.le_add_left _ _))) |
| have h3 : weight z ≤ measure z := by unfold measure exactParameterMeasure; omega |
| change rareBranchRealizer.cost z+exactOrdinaryRealizer.cost z+10*weight z+50 ≤ _ |
| simp only [eval_add,eval_mul,eval_ofNat,eval_X] |
| omega |
|
|
| end ContingencyTables.ExactGateCost |
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| end OAI |
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|
