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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 | import OAI.Combinatorics.ContingencyTables.Dense.DenseBoundedTrial
import OAI.Combinatorics.ContingencyTables.Transport.FiniteScaledDraw
import OAI.Combinatorics.ContingencyTables.Transport.RationalMapProducts
namespace OAI
/-! # Independent actual Boolean offset draws realize the bounded offset law -/
namespace ContingencyTables.GridBoundary
open scoped BigOperators Classical
open ResidualMixture FirstSuccess FairBits
noncomputable section
variable {I J : Type*} [Fintype I] [Fintype J] [DecidableEq I] [DecidableEq J] {q : ℕ}
abbrev OffsetWords (e : I × J ≃ Fin q) (w : I × J → ℕ) (h d : ℕ) :=
∀ j : Fin q, Fin (offsetBitLength (w (e.symm j)) h d) → Bool
def drawOffsetWords (e : I × J ≃ Fin q) (w : I × J → ℕ)
(hw : ∀ p, 0 < w p) (h d : ℕ) (bits : OffsetWords e w h d) : Offset w :=
offsetCoordinateEquiv e w (fun j =>
scaledBitInteger (w (e.symm j)) (offsetBitLength (w (e.symm j)) h d) (hw _) (bits j))
/-- Every coordinate uses its declared independent bounded Boolean word;
the resulting offset vector has the bounded quotient-offset law. -/
theorem drawOffsetWordsLaw_eq (e : I × J ≃ Fin q) (w : I × J → ℕ)
[DecidableEq (Offset w)]
(hw : ∀ p, 0 < w p) (h d : ℕ) :
mapLaw (uniformLaw (α := OffsetWords e w h d)) (drawOffsetWords e w hw h d) =
boundedOffsetLaw e w hw h d := by
unfold drawOffsetWords boundedOffsetLaw
refine (mapLaw_equiv_output (offsetCoordinateEquiv e w)
(uniformLaw (α := OffsetWords e w h d)) (fun bits j =>
scaledBitInteger (w (e.symm j)) (offsetBitLength (w (e.symm j)) h d) (hw _)
(bits j))).trans ?_
congr 1
rw [← pi_uniformLaw, pi_mapLaw]
congr 1
funext j
exact scaledBitIntegerLaw_eq (w (e.symm j)) (offsetBitLength (w (e.symm j)) h d) (hw _)
end
end ContingencyTables.GridBoundary
end OAI
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