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92416a2 | 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 | import OAI.Dynamics.ConditionalShuffle.InstrumentTV
namespace OAI
noncomputable section
namespace Revealed.Disintegration
open scoped Classical
open Thorp Thorp.Conditional
variable {E E' G Ω : Type} [Fintype E] [Fintype E'] [Fintype G] [Nonempty G] [Fintype Ω]
lemma conditional_map_injective (a : Ω → E) (b : Ω → G) (f : E → E')
(hf : Function.Injective f) (e : E) :
conditional (fairMass (fun ω => (f (a ω), b ω))) (f e) =
conditional (fairMass (fun ω => (a ω, b ω))) e := by
funext g
have hm : marginal (fairMass (fun ω => (f (a ω), b ω))) (f e) =
marginal (fairMass (fun ω => (a ω, b ω))) e := by
simp only [marginal_fairMass, fairMass, hf.eq_iff]
have hj : fairMass (fun ω => (f (a ω), b ω)) (f e,g) =
fairMass (fun ω => (a ω, b ω)) (e,g) := by
simp only [fairMass, Prod.mk.injEq, hf.eq_iff]
simp only [conditional, hm, hj]
lemma weighted_conditional_map (a : Ω → E) (b : Ω → G) (f : E → E')
(hf : Function.Injective f) (c : (G → ℝ) → ℝ) :
(∑ e', marginal (fairMass (fun ω => (f (a ω), b ω))) e' *
c (conditional (fairMass (fun ω => (f (a ω), b ω))) e')) =
∑ e, marginal (fairMass (fun ω => (a ω, b ω))) e *
c (conditional (fairMass (fun ω => (a ω, b ω))) e) := by
simp only [marginal_fairMass, fairMass_test]
simp_rw [conditional_map_injective a b f hf]
end Revealed.Disintegration
end
end OAI
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