| 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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|
