openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Sampling /ExactSamplingMachine.lean
| import OAI.Combinatorics.ContingencyTables.Sampling.ExactExpectedTime | |
| import OAI.Combinatorics.ContingencyTables.Sampling.ExactRarePublicBound | |
| namespace OAI | |
| /-! # The exact public sampler in uniform expected polynomial bit time -/ | |
| namespace ContingencyTables.SmallGraphProfiles | |
| open MatchingFPRAS MatchingFPRAS.TreeTyped MatchingFPRAS.PhysicalStream Turing | |
| open ProfilePrograms DensePrograms Polynomial Algorithms | |
| open scoped BigOperators Classical | |
| noncomputable section | |
| attribute [local irreducible] ExactPublicPrefix.program | |
| theorem exact_expected_polynomial : ∃ p : Polynomial ℕ, | |
| ∀ (_hPL : ∀ q,PublishedInputs.FiniteBoxPrekopaLeindler q) | |
| (_hCheeger : ∀ q k,PublishedInputs.FiniteCheeger (GridBoundary.Grid (I := Fin q) (4*k^8))), | |
| ∀ (m n : ℕ) (r : Fin m → ℕ) (c : Fin n → ℕ), (∑ i,r i)=∑ j,c j → | |
| (∑' (t : ℕ), (1 - haltMass ExactPublicPrefix.machine (encodeMargins r c) t)) ≤ | |
| ((p.eval (samplingSize n r 0) : ℕ) : ℝ) := by | |
| obtain ⟨print,hprint⟩ := ExactPrintCost.polynomial_print_bound | |
| obtain ⟨p,hp⟩ := ExactGatePolynomial.public_bound print | |
| obtain ⟨q,hq⟩ := exact_public_correction_bound print | |
| refine ⟨p+11*q,?_⟩ | |
| intro hPL hCheeger m n r c htotal | |
| let B := print.eval (ExactPrintCost.matrixSize (n := n) r) | |
| let S := ExactGatePublic.budget r c B B | |
| let T := ExactPublicCost.budget r c 1 B B+1 | |
| let den : ℝ := (2:ℝ)^correctionPrecision r c | |
| have hB (X : Table r c) : GreedyFeasibleTable.printMatrixRealizer.cost (matrixCode X.val) ≤ B := | |
| (Nat.le_add_right _ _).trans (hprint m n r c X) | |
| have hL (X : Table r c) : (ExactPublicPrefix.word r c X).length ≤ B := | |
| (list_length_le_weight _).trans ((Nat.le_add_left _ _).trans (hprint m n r c X)) | |
| have htail := (ExactExpectedTime.bound r c hPL hCheeger htotal B B hB hL).2 | |
| have hgate : (S:ℝ) ≤ ((p.eval (samplingSize n r 0):ℕ):ℝ) := by exact_mod_cast hp m n r c htotal | |
| have hT : (T:ℝ) ≤ den*((q.eval (samplingSize n r 0):ℕ):ℝ) := by | |
| have hn : T ≤ 2^(correctionPrecision r c)*q.eval (samplingSize n r 0) := hq m n r c htotal | |
| have ht : (T:ℝ) ≤ ((2^(correctionPrecision r c)*q.eval (samplingSize n r 0):ℕ):ℝ) := by | |
| exact_mod_cast hn | |
| simpa only [Nat.cast_mul,Nat.cast_pow,Nat.cast_ofNat,den] using ht | |
| have hrare : (1/den)*T ≤ ((q.eval (samplingSize n r 0):ℕ):ℝ) := by | |
| have hden : 0 < den := by unfold den; positivity | |
| have ht : (T:ℝ)/den ≤ ((q.eval (samplingSize n r 0):ℕ):ℝ) := | |
| (div_le_iff₀ hden).mpr (by simpa only [mul_comm] using hT) | |
| simpa only [one_div,div_eq_mul_inv,mul_comm,one_mul] using ht | |
| have he : (fun t => 1-haltMass ExactPublicPrefix.machine (encodeMargins r c) t)= | |
| PhysicalRetryTail.tail ExactPublicPrefix.program.table | |
| (ExactPublicPrefix.program.initial (Tape.mk₁ (encodeMargins r c))) := by | |
| funext t | |
| exact PhysicalPublicLaw.tail_eq _ _ _ | |
| rw [←he] at htail | |
| change _ ≤ (S:ℝ)+11*(1/den)*T at htail | |
| simp only [eval_add,eval_mul,eval_ofNat,Nat.cast_add,Nat.cast_mul,Nat.cast_ofNat] | |
| nlinarith | |
| /-- Paper 200-01, main theorem (ii), conditional on the cited finite-box | |
| Prékopa--Leindler and finite-grid Cheeger inputs. The machine is fixed, | |
| including parsing, printing, and the unbiased-bit implementation. -/ | |
| theorem exactSampling_conditional | |
| (hPL : ∀ q,PublishedInputs.FiniteBoxPrekopaLeindler q) | |
| (hCheeger : ∀ q k,PublishedInputs.FiniteCheeger (GridBoundary.Grid (I := Fin q) (4*k^8))) : | |
| ExactSamplingStatement := by | |
| obtain ⟨p,hp⟩ := exact_expected_polynomial | |
| refine ⟨ExactPublicPrefix.machine,p.eval 1+1,p.natDegree,by omega,?_⟩ | |
| intro m n r c htotal | |
| dsimp only | |
| refine ⟨ExactPublicLaw.halted_feasible r c hPL hCheeger htotal, | |
| ExactPublicLaw.tableMass_limit r c hPL hCheeger htotal, | |
| ExactPublicLaw.haltMass_limit r c hPL hCheeger htotal, | |
| ExactExpectedTime.summable r c hPL hCheeger htotal,?_⟩ | |
| have hS : 1 ≤ samplingSize n r 0 := by unfold samplingSize; omega | |
| have hpoly : p.eval (samplingSize n r 0) ≤ p.eval 1*(samplingSize n r 0)^p.natDegree := by | |
| simpa only [one_mul,pow_one] using polynomial_scaled_power p 1 (samplingSize n r 0) 1 hS | |
| have hnat := hpoly.trans (Nat.mul_le_mul_right _ (Nat.le_succ _)) | |
| have hreal : ((p.eval (samplingSize n r 0):ℕ):ℝ) ≤ | |
| (p.eval 1+1:ℕ)*(samplingSize n r 0:ℝ)^p.natDegree := by exact_mod_cast hnat | |
| exact (hp hPL hCheeger m n r c htotal).trans hreal | |
| end | |
| end ContingencyTables.SmallGraphProfiles | |
| end OAI | |