File size: 1,877 Bytes
ea76aac | 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 46 47 48 49 50 51 52 53 54 55 | import Mathlib
import OAI.Computability.MaxCut.Games.UpperBound
namespace OAI
/-! Exact composition identities for finite distribution maps. -/
namespace MaxCutGames.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 MaxCutGames.Foundations.Games.FiniteDistribution
end OAI
|