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97c3405 | 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 | import OAI.Combinatorics.ContingencyTables.Dense.DenseBooleanStep
import OAI.Combinatorics.ContingencyTables.Dense.DenseFiniteWalk
import OAI.Combinatorics.ContingencyTables.Machines.FiniteWordWalk
namespace OAI
/-! # The dense bin draw as one bounded unbiased Boolean word -/
namespace ContingencyTables.GridBoundary
open scoped BigOperators Classical
open ResidualMixture FirstSuccess FairBits
noncomputable section
def gridWordEquiv (n k : ℕ) : (Fin (n * k) → Bool) ≃ (Fin n → Fin k → Bool) :=
(Equiv.arrowCongr finProdFinEquiv.symm (Equiv.refl Bool)).trans (Equiv.curry _ _ _)
variable {I J : Type*} [Fintype I] [Fintype J] [DecidableEq I] [DecidableEq J]
variable {q : ℕ}
def denseBooleanWalk (R : Option I → ℕ) (P : Option J → ℕ) (d : ℕ)
(e : I × J ≃ Fin q) (a : Grid (I := Fin q) (4 * d ^ 8)) (n : ℕ)
(bits : Fin (n * denseStepBits d) → Bool) : Grid (I := Fin q) (4 * d ^ 8) :=
wordWalk (denseBitStep R P d e) a n (gridWordEquiv n (denseStepBits d) bits)
def denseBooleanWalkLaw (R : Option I → ℕ) (P : Option J → ℕ) (d : ℕ)
(e : I × J ≃ Fin q) (a : Grid (I := Fin q) (4 * d ^ 8)) (n : ℕ) :
RationalLaw (Grid (I := Fin q) (4 * d ^ 8)) :=
mapLaw (uniformLaw (α := Fin (n * denseStepBits d) → Bool)) (denseBooleanWalk R P d e a n)
/-- Exactly n*(q_bin+8d^5) unbiased bits give the rational n-step law. -/
theorem denseBooleanWalkLaw_eq (R : Option I → ℕ) (P : Option J → ℕ)
(d : ℕ) (hd : 14 ≤ d) (hR : ∀ i, d ^ 12 ≤ R i) (hP : ∀ j, d ^ 12 ≤ P j)
(hRmax : ∀ i, R (some i) ≤ R none) (hPmax : ∀ j, P (some j) ≤ P none)
(hdimR : Fintype.card (Option I) ≤ d) (hdimP : Fintype.card (Option J) ≤ d)
(hfree : Fintype.card (I × J) ≤ d) (htotal : ∑ i, R i = ∑ j, P j)
(e : I × J ≃ Fin q) (a : Grid (I := Fin q) (4 * d ^ 8)) (n : ℕ) :
denseBooleanWalkLaw R P d e a n = rationalWalkLaw R P d hd hfree e n a := by
have hk : (fun s => mapLaw (uniformLaw (α := Fin (denseStepBits d) → Bool)) (denseBitStep R P d e s)) =
finiteDenseStepLaw R P d e := funext (denseBitStepLaw_eq_finite R P d e)
unfold denseBooleanWalkLaw denseBooleanWalk
rw [← map_equiv (gridWordEquiv n (denseStepBits d)), uniform_equiv]
change wordWalkLaw (denseBitStep R P d e) a n = _
rw [wordWalkLaw_eq, hk]
exact finiteDenseWalkLaw_eq R P d hd hR hP hRmax hPmax hdimR hdimP hfree htotal e a n
end
end ContingencyTables.GridBoundary
end OAI
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