| 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) |
|
|
| / |
| 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 |
|
|
