File size: 4,040 Bytes
d5b7741 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 | import OAI.Combinatorics.ContingencyTables.Rooted.RootedCatalogStep
namespace OAI
/-! # Mixing of the executed catalog-order oracle walk -/
namespace ContingencyTables.RootedCatalogGrid
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
end OAI
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