openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Sampling /ExactGatePolynomial.lean
| import OAI.Combinatorics.ContingencyTables.Sampling.ExactInputBounds | |
| import OAI.Combinatorics.ContingencyTables.Sampling.ExactGatePublic | |
| import OAI.Combinatorics.ContingencyTables.Sampling.PhysicalCostPolynomial | |
| namespace OAI | |
| /-! # A uniform polynomial bound for the cheap public prefix -/ | |
| namespace ContingencyTables.ExactGatePolynomial | |
| open MatchingFPRAS MatchingFPRAS.TreeTyped MatchingFPRAS.PhysicalStream | |
| open ProfilePrograms DensePrograms Polynomial | |
| open Algorithms (samplingSize encodeMargins) | |
| open scoped BigOperators Classical | |
| noncomputable section | |
| attribute [local irreducible] ExactTypedProgram.program exactParsedMargins | |
| ExactGatePrefix.budget ExactGateCost.measure | |
| def measure (as : List Symbol) (B : ℕ) : ℕ := | |
| weight as+ExactGateCost.measure (exactParsedMargins as)+B | |
| theorem budget_polynomial : ∃ p : Polynomial ℕ,∀ as B, | |
| LiteralProgram.packTime as.length+ExactGatePhysical.budget as B+LiteralProgram.postTime B ≤ | |
| p.eval (measure as B) := by | |
| obtain ⟨p,hp⟩ := polynomial_exactParsedMargins | |
| obtain ⟨q,hq⟩ := ExactGateCost.budget_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,?_⟩ | |
| intro as B | |
| let M := measure as B | |
| have hw : weight as ≤ M := by unfold M measure; omega | |
| have hg : ExactGateCost.measure (exactParsedMargins as) ≤ M := by unfold M measure; omega | |
| have hB : B ≤ M := by unfold M measure; omega | |
| have ht : ExactGatePhysical.typedBudget as 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 [ExactGatePhysical.typedBudget,t,eval_add,eval_X] using | |
| Nat.add_le_add (Nat.add_le_add hp' hq') hB | |
| have hb := hv (weight as) (ExactGatePhysical.typedBudget as B) as.length B M hw ht | |
| ((list_length_le_weight as).trans hw) hB | |
| simpa only [ExactGatePhysical.budget,OnlineStack.Executes.factor] using hb | |
| /-- Any fixed polynomial print allowance still gives a uniform public | |
| polynomial bound for the prefix before the rare tabulation starts. -/ | |
| theorem public_bound (print : Polynomial ℕ) : ∃ p : Polynomial ℕ, | |
| ∀ (m n : ℕ) (r : Fin m → ℕ) (c : Fin n → ℕ), | |
| (∑ i,r i)=∑ j,c j → | |
| let B := print.eval (ExactPrintCost.matrixSize (n := n) r) | |
| ExactGatePublic.budget r c B B ≤ p.eval (samplingSize n r 0) := by | |
| obtain ⟨p,hp⟩ := budget_polynomial | |
| obtain ⟨q,hq⟩ := SmallGraphProfiles.exact_gate_measure_polynomial | |
| let u : Polynomial ℕ := 91*X^2+q+print.comp (25*X^3) | |
| refine ⟨p.comp u,?_⟩ | |
| intro m n r c htotal | |
| dsimp only | |
| have hm : measure (encodeMargins r c) (print.eval (ExactPrintCost.matrixSize (n := n) r)) ≤ | |
| u.eval (samplingSize n r 0) := by | |
| simp only [measure,SmallGraphProfiles.exactParsedMargins_spec,u,eval_add,eval_mul, | |
| eval_ofNat,eval_pow,eval_X,eval_comp] | |
| exact Nat.add_le_add (Nat.add_le_add (SmallGraphProfiles.exact_public_weight r c htotal) | |
| (hq m n r c htotal)) | |
| (polynomial_eval_monotone print (SmallGraphProfiles.exact_matrix_size (n := n) r)) | |
| have hb : LiteralProgram.packTime (encodeMargins r c).length+ | |
| ExactGatePhysical.budget (encodeMargins r c) (print.eval (ExactPrintCost.matrixSize (n := n) r))+ | |
| LiteralProgram.postTime (print.eval (ExactPrintCost.matrixSize (n := n) r)) ≤ | |
| (p.comp u).eval (samplingSize n r 0) := | |
| (hp (encodeMargins r c) (print.eval (ExactPrintCost.matrixSize (n := n) r))).trans (by | |
| rw [eval_comp] | |
| exact polynomial_eval_monotone p hm) | |
| simpa only [ExactGatePublic.budget,ExactTablePrefix.input] using hb | |
| end | |
| end ContingencyTables.ExactGatePolynomial | |
| end OAI | |