openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Sampling /FiniteListSampling.lean
| import Mathlib.Algebra.BigOperators.Field | |
| import OAI.Computability.PerfectCompleteness.Foundations.OccurrenceStochasticLemmas | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.FiniteListSampling | |
| open scoped BigOperators | |
| open UniqueGamesTheorem.Foundations.Games | |
| noncomputable section | |
| variable {A B : Type*} [Fintype A] [Fintype B] [DecidableEq A] [DecidableEq B] | |
| def uniformList (L : Finset A) (nonempty : L.Nonempty) : FiniteDistribution A where | |
| weight a := if a ∈ L then 1 / (L.card : ℝ) else 0 | |
| nonnegative a := by | |
| split | |
| · exact one_div_nonneg.mpr (Nat.cast_nonneg _) | |
| · exact le_rfl | |
| normalized := by | |
| have hcard : (L.card : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr (ne_of_gt nonempty.card_pos) | |
| simp only [Finset.sum_ite_mem_eq, Finset.sum_const, nsmul_eq_mul, mul_one_div] | |
| exact div_self hcard | |
| @[simp] theorem uniformList_weight_mem (L : Finset A) (hne : L.Nonempty) | |
| (a : A) (ha : a ∈ L) : (uniformList L hne).weight a = 1 / (L.card : ℝ) := by | |
| simp [uniformList, ha] | |
| omit [DecidableEq A] in | |
| theorem probability_ge_weight (μ : FiniteDistribution A) (event : A → Bool) | |
| (a : A) (ha : event a = true) : μ.weight a ≤ μ.probability event := by | |
| change μ.weight a ≤ ∑ x, if event x then μ.weight x else 0 | |
| have h := Finset.single_le_sum | |
| (f := fun x : A => if event x then μ.weight x else 0) | |
| (fun x _ => by split; exact μ.nonnegative x; exact le_rfl) (Finset.mem_univ a) | |
| simpa only [ha, ite_true] using h | |
| def pairAcceptance (μ : FiniteDistribution A) (ν : FiniteDistribution B) | |
| (accepts : A → B → Bool) : ℝ := | |
| μ.expectation (fun a => ν.probability (accepts a)) | |
| omit [DecidableEq A] [DecidableEq B] in | |
| theorem pairAcceptance_nonnegative (μ : FiniteDistribution A) (ν : FiniteDistribution B) | |
| (accepts : A → B → Bool) : 0 ≤ pairAcceptance μ ν accepts := by | |
| apply Finset.sum_nonneg | |
| intro a _ | |
| exact mul_nonneg (μ.nonnegative a) (ν.probability_nonnegative _) | |
| omit [DecidableEq A] [DecidableEq B] in | |
| theorem pair_weight_le_acceptance (μ : FiniteDistribution A) (ν : FiniteDistribution B) | |
| (accepts : A → B → Bool) (a : A) (b : B) (accepted : accepts a b = true) : | |
| μ.weight a * ν.weight b ≤ pairAcceptance μ ν accepts := by | |
| have hinner := probability_ge_weight ν (accepts a) b accepted | |
| calc | |
| _ ≤ μ.weight a * ν.probability (accepts a) := | |
| mul_le_mul_of_nonneg_left hinner (μ.nonnegative a) | |
| _ ≤ _ := Finset.single_le_sum | |
| (fun x _ => mul_nonneg (μ.nonnegative x) (ν.probability_nonnegative _)) | |
| (Finset.mem_univ a) | |
| theorem uniform_pair_lower (L : Finset A) (R : Finset B) | |
| (hL : L.Nonempty) (hR : R.Nonempty) (accepts : A → B → Bool) | |
| (witness : ∃ a ∈ L, ∃ b ∈ R, accepts a b = true) : | |
| 1 / ((L.card : ℝ) * (R.card : ℝ)) ≤ | |
| pairAcceptance (uniformList L hL) (uniformList R hR) accepts := by | |
| obtain ⟨a, ha, b, hb, hab⟩ := witness | |
| have h := pair_weight_le_acceptance (uniformList L hL) (uniformList R hR) accepts a b hab | |
| simpa only [uniformList_weight_mem L hL a ha, uniformList_weight_mem R hR b hb, | |
| ← one_div_mul_one_div] using h | |
| theorem bounded_uniform_pair_lower (L : Finset A) (R : Finset B) | |
| (hL : L.Nonempty) (hR : R.Nonempty) (s : Nat) | |
| (hLs : L.card ≤ s) (hRs : R.card ≤ s) (accepts : A → B → Bool) | |
| (witness : ∃ a ∈ L, ∃ b ∈ R, accepts a b = true) : | |
| 1 / (s : ℝ) ^ 2 ≤ | |
| pairAcceptance (uniformList L hL) (uniformList R hR) accepts := by | |
| have hLc : 0 < (L.card : ℝ) := Nat.cast_pos.mpr hL.card_pos | |
| have hRc : 0 < (R.card : ℝ) := Nat.cast_pos.mpr hR.card_pos | |
| have hLs' : (L.card : ℝ) ≤ s := by exact_mod_cast hLs | |
| have hRs' : (R.card : ℝ) ≤ s := by exact_mod_cast hRs | |
| have hprod : (L.card : ℝ) * (R.card : ℝ) ≤ (s : ℝ) ^ 2 := by | |
| simpa only [pow_two] using | |
| mul_le_mul hLs' hRs' (Nat.cast_nonneg R.card) (Nat.cast_nonneg s) | |
| exact (one_div_le_one_div_of_le (mul_pos hLc hRc) hprod).trans | |
| (uniform_pair_lower L R hL hR accepts witness) | |
| omit [DecidableEq A] [DecidableEq B] in | |
| theorem pairAcceptance_eq_expectation (μ : FiniteDistribution A) (ν : FiniteDistribution B) | |
| (accepts : A → B → Bool) : | |
| pairAcceptance μ ν accepts = μ.expectation | |
| (fun a => ν.expectation (fun b => if accepts a b then 1 else 0)) := by | |
| unfold pairAcceptance | |
| apply FiniteDistribution.expectation_congr | |
| intro a | |
| simp [FiniteDistribution.probability, FiniteDistribution.expectation, mul_ite] | |
| end | |
| end PerfectCompleteness.FiniteListSampling | |
| end | |
| end OAI | |