File size: 3,624 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 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 | import OAI.Combinatorics.ContingencyTables.Sampling.ExactGatePolynomial
namespace OAI
/-! # Polynomial dependence of the correction cost on its finite tabulation measure -/
namespace ContingencyTables.ExactRareCost
open MatchingFPRAS MatchingFPRAS.TreeTyped MatchingFPRAS.PhysicalStream
open ProfilePrograms DensePrograms Polynomial
noncomputable section
attribute [local irreducible] ExactTypedProgram.program exactParsedMargins
ExactGateCost.measure exactRareMeasure exactMatrixInput
def measure (z : CompletionMargins) : ℕ := ExactGateCost.measure z+exactRareMeasure z
theorem core_polynomial : ∃ p : Polynomial ℕ,∀ z,
ExactCoreCost.budget z 1 ≤ p.eval (measure z) := by
obtain ⟨g,hg⟩ := ExactGateCost.budget_polynomial
obtain ⟨p,hp⟩ := work_exactMatrixInput
obtain ⟨q,hq⟩ := ExactMatrixTrial.polynomial_trial
refine ⟨g+p+q.comp p+12*p+C (weight (default : MatrixCode)+40),?_⟩
intro z
have hgate : ExactGatePrefix.budget z ≤ g.eval (measure z) :=
(hg z).trans (polynomial_eval_monotone g (Nat.le_add_right _ _))
have hmatrix : exactMatrixInputRealizer.cost z+weight (exactMatrixInput z) ≤ p.eval (measure z) :=
(hp z).trans (polynomial_eval_monotone p (Nat.le_add_left _ _))
have hw : weight (exactMatrixInput z) ≤ p.eval (measure z) :=
(Nat.le_add_left _ _).trans hmatrix
have htrial := (Nat.le_add_right _ _).trans
((hq (exactMatrixInput z)).trans (polynomial_eval_monotone q hw))
change ExactMatrixTrial.trialRealizer.cost (exactMatrixInput z) ≤ q.eval (p.eval (measure z)) at htrial
change rareBranchRealizer.cost z+exactOrdinaryRealizer.cost z+exactMatrixInputRealizer.cost z+
1*(ExactMatrixTrial.trialRealizer.cost (exactMatrixInput z)+12*weight (exactMatrixInput z)+
weight (default : MatrixCode)+40)+10*weight z+50 ≤ _
change rareBranchRealizer.cost z+exactOrdinaryRealizer.cost z+10*weight z+50 ≤ _ at hgate
simp only [eval_add,eval_mul,eval_ofNat,eval_C,eval_comp]
omega
def publicMeasure (as : List Symbol) (B : ℕ) : ℕ :=
weight as+measure (exactParsedMargins as)+B
/-- All copying and physical simulation costs are polynomial in the
already explicit finite-tabulation measure. -/
theorem public_polynomial : ∃ p : Polynomial ℕ,∀ as B,
LiteralProgram.packTime as.length+ExactPhysicalCost.budget as 1 B+
LiteralProgram.postTime B+1 ≤ p.eval (publicMeasure as B) := by
obtain ⟨p,hp⟩ := polynomial_exactParsedMargins
obtain ⟨q,hq⟩ := core_polynomial
let t : Polynomial ℕ := p+q+X
obtain ⟨v,hv⟩ := PhysicalCostPolynomial.bound
(OnlineStack.accesses ExactTypedProgram.program.code)
(OnlineStack.overhead ExactTypedProgram.program.code) t
refine ⟨v+1,?_⟩
intro as B
let M := publicMeasure as B
have hw : weight as ≤ M := by unfold M publicMeasure; omega
have hg : measure (exactParsedMargins as) ≤ M := by unfold M publicMeasure; omega
have hB : B ≤ M := by unfold M publicMeasure; omega
have ht : ExactPhysicalCost.typedBudget as 1 B ≤ t.eval M := by
have hp' := (Nat.le_add_right _ _).trans ((hp as).trans (polynomial_eval_monotone p hw))
have hq' := (hq _).trans (polynomial_eval_monotone q hg)
simpa only [ExactPhysicalCost.typedBudget,t,eval_add,eval_X] using
Nat.add_le_add (Nat.add_le_add hp' hq') hB
have hb := hv (weight as) (ExactPhysicalCost.typedBudget as 1 B) as.length B M hw ht
((list_length_le_weight as).trans hw) hB
simpa only [ExactPhysicalCost.budget,OnlineStack.Executes.factor,eval_add,eval_one]
using Nat.add_le_add_right hb 1
end
end ContingencyTables.ExactRareCost
end OAI
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