Datasets:

roseking
/

openAI-math-dataset

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


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

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

	namespace OAI

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	section

	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


	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