| 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 | |