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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 | 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
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