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import OAI.Computability.VertexCover.Games.Stochastic
import OAI.Computability.VertexCover.Games.Mixture
namespace OAI
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namespace UniqueGames.Foundations.Games.FiniteDistribution
open scoped BigOperators
noncomputable section
variable {A B C : Type*} [Fintype A] [Fintype B] [Fintype C]
theorem weight_eq_probability_singleton [DecidableEq A] (μ : FiniteDistribution A) (a : A) :
μ.weight a = μ.probability (fun x => decide (x = a)) := by
classical
simp [probability]
theorem pushforward_comp (μ : FiniteDistribution A) (f : A → B) (g : B → C) :
(μ.pushforward f).pushforward g = μ.pushforward (fun a => g (f a)) := by
classical
apply eq_of_weight_eq
intro «c»
rw [weight_eq_probability_singleton, weight_eq_probability_singleton]
simp only [probability_pushforward]
theorem transport_eq_pushforward (μ : FiniteDistribution A) (e : A ≃ B) :
μ.transport e = μ.pushforward e := by
classical
apply eq_of_weight_eq
intro b
rw [weight_eq_probability_singleton, weight_eq_probability_singleton,
probability_transport, probability_pushforward]
theorem expectation_pushforward (μ : FiniteDistribution A) (f : A → B) (h : B → ℝ) :
(μ.pushforward f).expectation h = μ.expectation (fun a => h (f a)) := by
classical
simp only [expectation, pushforward, Finset.sum_mul]
rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro a _
simp [ite_mul]
theorem pushforward_mixture (μ : FiniteDistribution A) (ν : A → FiniteDistribution B)
(f : B → C) :
(μ.mixture ν).pushforward f = μ.mixture (fun a => (ν a).pushforward f) := by
classical
apply eq_of_weight_eq
intro «c»
rw [weight_eq_probability_singleton, weight_eq_probability_singleton]
simp only [probability_pushforward, probability_mixture]
end
end UniqueGames.Foundations.Games.FiniteDistribution
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end OAI
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