| import OAI.Dynamics.ConditionalShuffle.InstrumentTV |
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| namespace OAI |
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| 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] |
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| 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 |
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| end |
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| end OAI |
| |
