File size: 3,478 Bytes
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 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 | import OAI.Combinatorics.ContingencyTables.Dense.DenseBooleanWalk
import OAI.Combinatorics.ContingencyTables.Dense.DenseBooleanOffsets
namespace OAI
/-! # The full dense trial as one bounded unbiased Boolean word -/
namespace ContingencyTables.GridBoundary
open scoped BigOperators Classical
open ResidualMixture FirstSuccess FairBits
noncomputable section
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))
variable {I J : Type*} [Fintype I] [Fintype J] [DecidableEq I] [DecidableEq J]
variable {q : ℕ}
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
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)))
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)
/-- This is the exact law of the full declared Boolean trial, including
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
end
end ContingencyTables.GridBoundary
end OAI
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