| import OAI.Combinatorics.ContingencyTables.Rooted.RootedCatalogStep |
|
|
| namespace OAI |
|
|
| /-! # Mixing of the executed catalog-order oracle walk -/ |
|
|
| namespace ContingencyTables.RootedCatalogGrid |
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|
| open CompanionFactorialApplication ForestAssignmentSplit GridBoundary |
| open RootedCompletion ResidualMixture FirstSuccess |
| open CompanionOracleDensity (rationalHeight oracleTime) |
| open scoped BigOperators Classical |
|
|
| noncomputable section |
| variable {m n : ℕ} |
|
|
| theorem walkLaw_eq (A : Instance (Fin m) (Fin n)) (C a : ℕ) (hC : 0 < C) |
| (haH : a≤rationalHeight A C) (z : View A → ℕ) (hz : BalancedBox.InBox (bounds A) z) |
| {K : ℕ} (hK : 1≤K) (hKB : (K:ℝ)≤CompanionOracleDensity.binWidth A C) |
| (v : Grid (I := Fin (freeCount A)) K) (N : ℕ) : |
| FirstSuccess.walkLaw (bitStepLaw A C a z) v N= |
| equivLaw (gridEquiv A K).symm |
| (RootedOracleDensity.oracleWalkLaw A (rowColumn A.T) C a z hK (gridEquiv A K v) N) := by |
| induction N with |
| | zero => |
| apply RationalLaw.ext |
| funext w |
| simp only [RootedOracleDensity.oracleWalkLaw,FirstSuccess.walkLaw,equivLaw,pointLaw, |
| Equiv.symm_symm,Equiv.apply_eq_iff_eq] |
| | succ N ih => |
| have hm (b : Grid (I := Fin (freeCount A)) K) : |
| (FirstSuccess.walkLaw (bitStepLaw A C a z) v N).mass b= |
| (RootedOracleDensity.oracleWalkLaw A (rowColumn A.T) C a z hK |
| (gridEquiv A K v) N).mass (gridEquiv A K b) := |
| congrArg (fun p => p.mass b) ih |
| have hk (b w : Grid (I := Fin (freeCount A)) K) : |
| (bitStepLaw A C a z b).mass w= |
| (RootedOracleDensity.oracleTransitionLaw A (rowColumn A.T) C a z hK |
| (gridEquiv A K b)).mass (gridEquiv A K w) := |
| congrArg (fun p => p.mass w) (bitStepLaw_eq A C a hC haH z hz hK hKB b) |
| apply RationalLaw.ext |
| funext w |
| change (∑ b,(FirstSuccess.walkLaw (bitStepLaw A C a z) v N).mass b* |
| (bitStepLaw A C a z b).mass w)= |
| ∑ b,(RootedOracleDensity.oracleWalkLaw A (rowColumn A.T) C a z hK |
| (gridEquiv A K v) N).mass b* |
| (RootedOracleDensity.oracleTransitionLaw A (rowColumn A.T) C a z hK b).mass |
| (gridEquiv A K w) |
| simp_rw [hm,hk] |
| exact Fintype.sum_equiv (gridEquiv A K) _ _ (fun _ => rfl) |
|
|
| theorem wordWalkLaw_eq (A : Instance (Fin m) (Fin n)) (C a : ℕ) (hC : 0 < C) |
| (haH : a≤rationalHeight A C) (z : View A → ℕ) (hz : BalancedBox.InBox (bounds A) z) |
| {K : ℕ} (hK : 1≤K) (hKB : (K:ℝ)≤CompanionOracleDensity.binWidth A C) |
| (v : Grid (I := Fin (freeCount A)) K) (N : ℕ) : |
| FirstSuccess.wordWalkLaw (bitStep A C a z) v N= |
| equivLaw (gridEquiv A K).symm |
| (RootedOracleDensity.oracleWalkLaw A (rowColumn A.T) C a z hK (gridEquiv A K v) N) := by |
| rw [FirstSuccess.wordWalkLaw_eq] |
| exact walkLaw_eq A C a hC haH z hz hK hKB v N |
|
|
| /-- The original quantitative mixing estimate holds for the actual |
| catalog-order word walk, with the identical number of steps. -/ |
| theorem wordWalkLaw_variation (A : Instance (Fin m) (Fin n)) (C a : ℕ) (theta : ℝ) |
| (hC : 0 < C) (haH : a≤rationalHeight A C) (htheta : 0 < theta) |
| (z : View A → ℕ) (hz : BalancedBox.InBox (bounds A) z) |
| (K Q : ℕ) (hK : 1≤K) (hQ : 0 < Q) (hKB : (K:ℝ)≤CompanionOracleDensity.binWidth A C) |
| (hPL : PublishedInputs.FiniteBoxPrekopaLeindler (CompanionOracleDensity.dimension A)) |
| (hCheeger : PublishedInputs.FiniteCheeger |
| (Grid (I := Fin (CompanionOracleDensity.dimension A)) K)) |
| (v : Grid (I := Fin (freeCount A)) K) : |
| (variation (FirstSuccess.wordWalkLaw (bitStep A C a z) v (oracleTime A C K Q theta)) |
| (equivLaw (gridEquiv A K).symm |
| (RootedOracleDensity.oracleStationaryLaw A (rowColumn A.T) C a z hK)):ℝ)≤theta/(8*Q) := by |
| rw [wordWalkLaw_eq A C a hC haH z hz hK hKB,variation_equiv] |
| apply RootedOracleDensity.oracleWalkLaw_variation A (rowColumn A.T) C a theta |
| hC _ htheta z hz K Q hK hQ hKB hPL hCheeger (gridEquiv A K v) |
| rw [←CompanionOracleDensity.rationalHeight_cast] |
| exact_mod_cast haH |
|
|
| end |
| end ContingencyTables.RootedCatalogGrid |
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| end OAI |
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|
