openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Foundations /FiniteProduct.lean
| import Mathlib.Data.Fintype.BigOperators | |
| import OAI.Computability.PerfectCompleteness.Foundations.RecursiveSpaceEquivLemmas | |
| import OAI.Computability.UniqueGames.Foundations.ValueLemmas | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.FiniteProduct | |
| open scoped BigOperators | |
| open UniqueGamesTheorem.Foundations.Games | |
| noncomputable section | |
| variable {I : Type*} [Fintype I] [DecidableEq I] | |
| {Ω : I → Type*} [∀ i, Fintype (Ω i)] | |
| def law (P : (i : I) → FiniteDistribution (Ω i)) : | |
| FiniteDistribution ((i : I) → Ω i) where | |
| weight x := ∏ i, (P i).weight (x i) | |
| nonnegative x := Finset.prod_nonneg fun i _ => (P i).nonnegative (x i) | |
| normalized := by | |
| rw [← Fintype.prod_sum] | |
| simp only [FiniteDistribution.normalized, Finset.prod_const_one] | |
| theorem expectation_product (P : (i : I) → FiniteDistribution (Ω i)) | |
| (f : (i : I) → Ω i → ℝ) : | |
| (law P).expectation (fun x => ∏ i, f i (x i)) = | |
| ∏ i, (P i).expectation (f i) := by | |
| change (∑ x : (i : I) → Ω i, (∏ i, (P i).weight (x i)) * | |
| (∏ i, f i (x i))) = ∏ i, ∑ a, (P i).weight a * f i a | |
| simp only [← Finset.prod_mul_distrib] | |
| exact (Fintype.prod_sum (fun i a => (P i).weight a * f i a)).symm | |
| def factor (i : I) (f : Ω i → ℝ) (j : I) (a : Ω j) : ℝ := | |
| if h : j = i then f (h ▸ a) else 1 | |
| omit [Fintype I] [∀ index, Fintype (Ω index)] in | |
| @[simp] theorem factor_self (i : I) (f : Ω i → ℝ) (a : Ω i) : | |
| factor i f i a = f a := by simp [factor] | |
| omit [Fintype I] [∀ index, Fintype (Ω index)] in | |
| @[simp] theorem factor_other (i j : I) (h : j ≠ i) (f : Ω i → ℝ) (a : Ω j) : | |
| factor i f j a = 1 := by simp [factor, h] | |
| omit [Fintype I] [∀ index, Fintype (Ω index)] in | |
| @[simp] theorem factor_function_self (i : I) (f : Ω i → ℝ) : | |
| factor i f i = f := by funext a; exact factor_self i f a | |
| omit [Fintype I] [∀ index, Fintype (Ω index)] in | |
| @[simp] theorem factor_function_other (i j : I) (h : j ≠ i) (f : Ω i → ℝ) : | |
| factor i f j = fun _ => 1 := by funext a; exact factor_other i j h f a | |
| omit [∀ index, Fintype (Ω index)] in | |
| theorem product_factor (i : I) (f : Ω i → ℝ) (x : (j : I) → Ω j) : | |
| (∏ j, factor i f j (x j)) = f (x i) := by | |
| classical | |
| rw [Finset.prod_eq_single i] | |
| · exact factor_self i f (x i) | |
| · intro j _ h | |
| exact factor_other i j h f (x j) | |
| · simp | |
| theorem expectation_factor_product (P : (i : I) → FiniteDistribution (Ω i)) | |
| (i : I) (f : Ω i → ℝ) : | |
| (∏ j, (P j).expectation (factor i f j)) = (P i).expectation f := by | |
| classical | |
| rw [Finset.prod_eq_single i] | |
| · congr 1 | |
| funext a | |
| exact factor_self i f a | |
| · intro j _ h | |
| rw [factor_function_other i j h, SmallBias.expectation_const] | |
| · simp | |
| theorem expectation_eval (P : (i : I) → FiniteDistribution (Ω i)) | |
| (i : I) (f : Ω i → ℝ) : | |
| (law P).expectation (fun x => f (x i)) = (P i).expectation f := by | |
| calc | |
| _ = (law P).expectation (fun x => ∏ j, factor i f j (x j)) := by | |
| apply FiniteDistribution.expectation_congr | |
| intro x | |
| exact (product_factor i f x).symm | |
| _ = ∏ j, (P j).expectation (factor i f j) := expectation_product P _ | |
| _ = _ := expectation_factor_product P i f | |
| theorem expectation_eval_mul (P : (i : I) → FiniteDistribution (Ω i)) | |
| (i j : I) (hij : i ≠ j) (f : Ω i → ℝ) (g : Ω j → ℝ) : | |
| (law P).expectation (fun x => f (x i) * g (x j)) = | |
| (P i).expectation f * (P j).expectation g := by | |
| have hfactor : ∀ k, | |
| (P k).expectation (fun a => factor i f k a * factor j g k a) = | |
| (P k).expectation (factor i f k) * (P k).expectation (factor j g k) := by | |
| intro k | |
| by_cases hki : k = i | |
| · subst k | |
| simp only [factor_other j i hij, factor_function_other j i hij, | |
| SmallBias.expectation_const, mul_one] | |
| · simp only [factor_other i k hki, factor_function_other i k hki, | |
| SmallBias.expectation_const, one_mul] | |
| calc | |
| _ = (law P).expectation | |
| (fun x => ∏ k, factor i f k (x k) * factor j g k (x k)) := by | |
| apply FiniteDistribution.expectation_congr | |
| intro x | |
| rw [Finset.prod_mul_distrib, product_factor i f x, product_factor j g x] | |
| _ = ∏ k, (P k).expectation (fun a => factor i f k a * factor j g k a) := | |
| expectation_product P (fun k a => factor i f k a * factor j g k a) | |
| _ = (∏ k, (P k).expectation (factor i f k)) * | |
| (∏ k, (P k).expectation (factor j g k)) := by | |
| simp only [hfactor, Finset.prod_mul_distrib] | |
| _ = _ := by rw [expectation_factor_product P i f, expectation_factor_product P j g] | |
| end | |
| end PerfectCompleteness.FiniteProduct | |
| end | |
| end OAI | |