| import OAI.Combinatorics.ContingencyTables.Transport.FiniteObservableTransport |
| import OAI.Combinatorics.ContingencyTables.Machines.FiniteWordDependent |
|
|
| namespace OAI |
|
|
| /-! # Composing exact laws through a noninjective state encoding -/ |
| namespace ContingencyTables.FiniteObservableTransport |
| open CommonBasesFPRAS FairBits |
| open scoped Classical |
| noncomputable section |
|
|
| theorem encoded_bind {A B U V C Z W : Type*} |
| [Fintype A] [Fintype B] [Fintype U] [Fintype V] |
| (f : A → C) (g : B → Z) (e : Z → C) (F : C → U → W) (G : Z → V → W) |
| (hinit : SameLaw f (fun b => e (g b))) (hstep : ∀ z,SameLaw (F (e z)) (G z)) : |
| SameLaw (fun p : A×U => F (f p.1) p.2) (fun p : B×V => G (g p.1) p.2) := by |
| intro phi |
| rw [uniformAverage_prod,uniformAverage_prod] |
| have hi := hinit (fun c => uniformAverage (fun u : U => phi (F c u))) |
| refine hi.trans ?_ |
| apply congrArg uniformAverage |
| funext b |
| exact hstep (g b) phi |
|
|
| theorem flat_encoded_bind {C Z W : Type*} (a b c d : ℕ) |
| (f : (Fin a → Bool) → C) (g : (Fin c → Bool) → Z) (e : Z → C) |
| (F : C → (Fin b → Bool) → W) (G : Z → (Fin d → Bool) → W) |
| (hinit : SameLaw f (fun bits => e (g bits))) (hstep : ∀ z,SameLaw (F (e z)) (G z)) : |
| SameLaw (fun bits : Fin (a+b) → Bool => |
| F (f (splitWordEquiv a b bits).1) (splitWordEquiv a b bits).2) |
| (fun bits : Fin (c+d) → Bool => |
| G (g (splitWordEquiv c d bits).1) (splitWordEquiv c d bits).2) := by |
| have h := encoded_bind f g e F G hinit hstep |
| intro phi |
| have hl := uniformAverage_equiv (splitWordEquiv a b) |
| (fun p : (Fin a → Bool)×(Fin b → Bool) => phi (F (f p.1) p.2)) |
| have hr := uniformAverage_equiv (splitWordEquiv c d) |
| (fun p : (Fin c → Bool)×(Fin d → Bool) => phi (G (g p.1) p.2)) |
| exact hl.trans ((h phi).trans hr.symm) |
|
|
| end |
| end ContingencyTables.FiniteObservableTransport |
|
|
| end OAI |
|
|
