openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Sampling /AllSamplingMachine.lean
| import OAI.Combinatorics.ContingencyTables.Sampling.AllSamplingAccuracy | |
| import OAI.Combinatorics.ContingencyTables.Sampling.ProfileRuntimeBound | |
| import OAI.Combinatorics.ContingencyTables.Machines.FiniteWordTableLaw | |
| namespace OAI | |
| /-! # The bounded public sampler on every margin pair -/ | |
| namespace ContingencyTables.ProfilePrograms | |
| open MatchingFPRAS MatchingFPRAS.TreeTyped | |
| noncomputable def allSamplingMachine : RandomMachine := | |
| FiniteWordExecution.machine publicWordCountRealizer publicSamplingAllRealizer | |
| noncomputable def allSamplingMachineTime (as : List MatchingFPRAS.Symbol) : ℕ := | |
| FiniteWordExecution.machineTime publicWordCountRealizer publicSamplingAllRealizer as | |
| lemma publicSamplingAll_alphabet (as : List MatchingFPRAS.Symbol) (bits : List Bool) : | |
| ∀ a∈publicSamplingAll (as,bits),a≠6 ∧ a≠7 := printMatrix_alphabet _ | |
| lemma allSamplingMachineTime_polynomial : | |
| ∃ p : Polynomial ℕ,∀ as,allSamplingMachineTime as≤p.eval | |
| (FiniteWordExecution.wordMeasure publicWordCount as) := | |
| FiniteWordExecution.machineTime_polynomial polynomial_publicWordCount polynomial_publicSamplingAll | |
| end ContingencyTables.ProfilePrograms | |
| namespace ContingencyTables.SmallGraphProfiles | |
| open ProfilePrograms DensePrograms CompletionCounts FirstPaperPhysicalMarginal | |
| open MatchingFPRAS MatchingFPRAS.PhysicalStream MatchingFPRAS.TreeTyped Turing Algorithms Polynomial | |
| open ResidualMixture FirstSuccess FairBits | |
| open scoped BigOperators Classical | |
| noncomputable section | |
| lemma allSamplingMachineTime_public_bound : | |
| ∃ C d : ℕ,0<C ∧ ∀ (m n : ℕ) (r : Fin m → ℕ) (c : Fin n → ℕ), | |
| (∑ i,r i)=∑ j,c j → ∀ k, | |
| allSamplingMachineTime (encodeSamplingInput r c k)≤C*(samplingSize n r k)^d := by | |
| obtain ⟨p,hp⟩ := allSamplingMachineTime_polynomial | |
| refine ⟨p.eval (344^644+91)+1,1288*p.natDegree,by omega,?_⟩ | |
| intro m n r c htotal k | |
| have hS : 1 ≤ samplingSize n r k := by unfold samplingSize; omega | |
| calc | |
| _ ≤ p.eval (FiniteWordExecution.wordMeasure publicWordCount (encodeSamplingInput r c k)) := hp _ | |
| _ ≤ p.eval ((344^644+91)*(samplingSize n r k)^1288) := | |
| polynomial_eval_monotone p (public_wordMeasure_bound r c htotal k) | |
| _ ≤ p.eval (344^644+91)*(samplingSize n r k)^(1288*p.natDegree) := | |
| polynomial_scaled_power p (344^644+91) (samplingSize n r k) 1288 hS | |
| _ ≤ _ := Nat.mul_le_mul_right _ (Nat.le_succ _) | |
| variable {m n : ℕ} (r : Fin m → ℕ) (c : Fin n → ℕ) | |
| lemma allSamplingMachine_outputs (htotal : (∑ i,r i)=∑ j,c j) (k t : ℕ) | |
| (ht : allSamplingMachineTime (encodeSamplingInput r c k)≤t) : | |
| ∀ bits : Bits t,∃ X : Table r c, | |
| OutputsTable allSamplingMachine (encodeSamplingInput r c k) bits X.val := by | |
| let as := encodeSamplingInput r c k | |
| let P : Tape MatchingFPRAS.Symbol → Prop := fun tape => ∃ X : Table r c, | |
| tape.right₀=ListBlank.mk (encodeMatrix X.val) | |
| have hf : ∀ bits : Bits (publicWordCount as),P (Tape.mk₁ (publicSamplingAll (as,List.ofFn bits))) := by | |
| dsimp only [as] | |
| rw [publicWordCount_spec] | |
| intro bits | |
| refine ⟨scheduledTableDraw r c htotal k bits,?_⟩ | |
| rw [publicSamplingAll_spec r c htotal] | |
| simp only [Tape.mk₁,Tape.mk₂,Tape.mk'_right₀] | |
| have hP := FiniteWordExecution.machine_all publicWordCountRealizer publicSamplingAllRealizer as | |
| (samplingInput_nonzero r c k) (publicSamplingAll_alphabet as) P hf t ht | |
| have hhalt := (FiniteWordExecution.machine_law publicWordCountRealizer publicSamplingAllRealizer as | |
| (samplingInput_nonzero r c k) (publicSamplingAll_alphabet as) t ht).1 | |
| intro bits | |
| obtain ⟨X,hX⟩ := hP bits | |
| exact ⟨X,hhalt bits,hX⟩ | |
| lemma allSamplingMachine_mass (htotal : (∑ i,r i)=∑ j,c j) (k t : ℕ) | |
| (ht : allSamplingMachineTime (encodeSamplingInput r c k)≤t) (X : Table r c) : | |
| tableMass allSamplingMachine (encodeSamplingInput r c k) t X.val= | |
| ((mapLaw (uniformLaw (α := Bits (finiteSamplerBits r c k))) | |
| (scheduledTableDraw r c htotal k)).mass X:ℝ) := | |
| FiniteWordExecution.machine_tableMass publicWordCountRealizer publicSamplingAllRealizer | |
| (encodeSamplingInput r c k) (samplingInput_nonzero r c k) | |
| (publicSamplingAll_alphabet _) (publicWordCount_spec r c k) (scheduledTableDraw r c htotal k) | |
| (publicSamplingAll_spec r c htotal k) t ht X | |
| /-- Paper 200-01, main theorem (i), conditional only on Prékopa (1973) | |
| Theorem 3/(2.3) on finite boxes and Levin--Peres (2017) Theorem 13.10, | |
| (13.6), Lemma 13.7 on the completion grids. -/ | |
| theorem boundedSampling_conditional | |
| (hPL : ∀ q,PublishedInputs.FiniteBoxPrekopaLeindler q) | |
| (hCheeger : ∀ q d,PublishedInputs.FiniteCheeger | |
| (GridBoundary.Grid (I := Fin q) (4*d^8))) : BoundedSamplingStatement := by | |
| obtain ⟨C,D,hC,hbound⟩ := allSamplingMachineTime_public_bound | |
| refine ⟨allSamplingMachine,C,D,hC,?_⟩ | |
| intro m n r c htotal k hk | |
| dsimp only | |
| have ht := hbound m n r c htotal k | |
| refine ⟨allSamplingMachine_outputs r c htotal k _ ht,?_⟩ | |
| let : Nonempty (Table r c) := ⟨GreedyFeasibleTable.table r c htotal⟩ | |
| have hv := scheduledTableDraw_accuracy r c (hPL _) | |
| (hCheeger _ (dimensionAllowance (I := Fin m) (J := Fin n))) htotal k hk | |
| dsimp only at hv | |
| simp_rw [allSamplingMachine_mass r c htotal k _ ht] | |
| simpa only [variation,Rat.cast_div,Rat.cast_sum,Rat.cast_abs,Rat.cast_sub, | |
| Rat.cast_ofNat,Rat.cast_inv,uniformLaw,uniformMass,Rat.cast_one,Rat.cast_natCast,one_div] using hv | |
| end | |
| end ContingencyTables.SmallGraphProfiles | |
| end OAI | |