| 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 | |