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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 | import OAI.Combinatorics.ContingencyTables.Sampling.ExactGatePrefix
namespace OAI
/-! # Polynomial cost before the rare branch starts -/
namespace ContingencyTables.ExactGateCost
open MatchingFPRAS MatchingFPRAS.TreeTyped DensePrograms ProfilePrograms Polynomial
def measure (z : CompletionMargins) : ℕ :=
exactParameterMeasure z+FiniteWordExecution.wordMeasure exactOrdinaryBits z
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
end OAI
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