import Mathlib
import OAI.Computability.VertexCover.Games.Stochastic

namespace OAI

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namespace UniqueGames.Foundations.Games.FiniteDistribution
open scoped BigOperators
noncomputable section

theorem iid_pushforward {Ω Γ : Type*} [Fintype Ω] [Fintype Γ]
    (μ : FiniteDistribution Ω) (f : Ω → Γ) (n : Nat) :
    (μ.pushforward f).iid n =
      (μ.iid n).pushforward (fun answers i => f (answers i)) := by
  classical
  apply eq_of_weight_eq
  intro y
  simp only [iid, pushforward]
  rw [Fintype.prod_sum]
  apply Finset.sum_congr rfl
  intro x _
  by_cases h : (fun i => f (x i)) = y
  · have hp : ∀ i, f (x i) = y i := congrFun h
    simp [hp]
  · rw [ite_eq_right h]
    have hn : ¬ ∀ i, f (x i) = y i := fun hp => h (funext hp)
    obtain ⟨i, hi⟩ := not_forall.mp hn
    apply Finset.prod_eq_zero (Finset.mem_univ i)
    simp [hi]

def uniform (Ω : Type*) [Fintype Ω] [Nonempty Ω] : FiniteDistribution Ω where
  weight _ := 1 / (Fintype.card Ω : ℝ)
  nonnegative _ := div_nonneg zero_le_one (Nat.cast_nonneg _)
  normalized := by
    have hn : (Fintype.card Ω : ℝ) ≠ 0 :=
      Nat.cast_ne_zero.mpr Fintype.card_ne_zero
    simp only [Finset.sum_const, Finset.card_univ, nsmul_eq_mul, one_div]
    exact mul_inv_cancel₀ hn

theorem iid_uniform {Ω : Type*} [Fintype Ω] [Nonempty Ω] (n : Nat) :
    (uniform Ω).iid n = uniform (Fin n → Ω) := by
  classical
  apply eq_of_weight_eq
  intro x
  simp [iid, uniform, Nat.cast_pow, one_div]

theorem expectation_uniform {Ω : Type*} [Fintype Ω] [Nonempty Ω] (f : Ω → ℝ) :
    (uniform Ω).expectation f = (∑ x, f x) / (Fintype.card Ω : ℝ) := by
  unfold expectation uniform
  rw [← Finset.mul_sum]
  simp [div_eq_mul_inv, mul_comm]

end
end UniqueGames.Foundations.Games.FiniteDistribution

end


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end OAI