| import OAI.Combinatorics.ContingencyTables.Dense.DenseBooleanWalk |
| import OAI.Combinatorics.ContingencyTables.Dense.DenseBooleanOffsets |
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|
| namespace OAI |
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|
| /-! # The full dense trial as one bounded unbiased Boolean word -/ |
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|
| namespace ContingencyTables.GridBoundary |
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|
| open scoped BigOperators Classical |
| open ResidualMixture FirstSuccess FairBits |
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|
| noncomputable section |
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| def dependentWordEquiv {q : ℕ} (k : Fin q → ℕ) : |
| (Fin (∑ j, k j) → Bool) ≃ (∀ j, Fin (k j) → Bool) := |
| (Equiv.arrowCongr finSigmaFinEquiv.symm (Equiv.refl Bool)).trans (Equiv.piCurry (fun _ _ => Bool)) |
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| variable {I J : Type*} [Fintype I] [Fintype J] [DecidableEq I] [DecidableEq J] |
| variable {q : ℕ} |
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| def denseTrialBits (R : Option I → ℕ) (P : Option J → ℕ) (d : ℕ) |
| (e : I × J ≃ Fin q) (h : ℕ) : ℕ := |
| (d ^ 50 * (h + 1) ^ 2) * denseStepBits d + |
| ∑ j : Fin q, offsetBitLength (integerWidth R P d (e.symm j)) h d |
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| def trialWordEquiv (R : Option I → ℕ) (P : Option J → ℕ) (d : ℕ) |
| (e : I × J ≃ Fin q) (h : ℕ) : (Fin (denseTrialBits R P d e h) → Bool) ≃ |
| ((Fin ((d ^ 50 * (h + 1) ^ 2) * denseStepBits d) → Bool) × |
| OffsetWords e (integerWidth R P d) h d) := |
| (splitWordEquiv _ _).trans (Equiv.prodCongr (Equiv.refl _) |
| (dependentWordEquiv (fun j => offsetBitLength (integerWidth R P d (e.symm j)) h d))) |
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| def denseBooleanTrial (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) |
| (e : I × J ≃ Fin q) (h : ℕ) (a : Grid (I := Fin q) (4 * d ^ 8)) |
| (bits : Fin (denseTrialBits R P d e h) → Bool) : DenseTrial R P d q := |
| let z := trialWordEquiv R P d e h bits |
| (denseBooleanWalk R P d e a (d ^ 50 * (h + 1) ^ 2) z.1, |
| drawOffsetWords e (integerWidth R P d) (integerWidth_pos R P d hd hR hP) h d z.2) |
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|
| / |
| all independent quotient offsets, before its integer acceptance test. -/ |
| theorem denseBooleanTrialLaw_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) (h : ℕ) (a : Grid (I := Fin q) (4 * d ^ 8)) : |
| mapLaw (uniformLaw (α := Fin (denseTrialBits R P d e h) → Bool)) |
| (denseBooleanTrial R P d hd hR hP e h a) = boundedTrialLaw R P d hd hR hP hfree e h a := by |
| unfold denseBooleanTrial |
| rw [← map_equiv (trialWordEquiv R P d e h) |
| (uniformLaw (α := Fin (denseTrialBits R P d e h) → Bool)) |
| (fun z => (denseBooleanWalk R P d e a (d ^ 50 * (h+1)^2) z.1, |
| drawOffsetWords e (integerWidth R P d) (integerWidth_pos R P d hd hR hP) h d z.2)), |
| uniform_equiv, ← uniform_product, mapLaw_product] |
| rw [show mapLaw (uniformLaw (α := Fin ((d ^ 50 * (h + 1) ^ 2) * denseStepBits d) → Bool)) |
| (denseBooleanWalk R P d e a (d ^ 50 * (h + 1) ^ 2)) = |
| rationalWalkLaw R P d hd hfree e (d ^ 50 * (h + 1) ^ 2) a from |
| denseBooleanWalkLaw_eq R P d hd hR hP hRmax hPmax hdimR hdimP hfree htotal e a _] |
| unfold boundedTrialLaw |
| congr 1 |
| exact drawOffsetWordsLaw_eq e (integerWidth R P d) (integerWidth_pos R P d hd hR hP) h d |
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|
| end |
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| end ContingencyTables.GridBoundary |
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| end OAI |
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