| import Mathlib.Algebra.BigOperators.Ring.Finset |
| import Mathlib.Analysis.Complex.Exponential |
| import OAI.Computability.UniqueGames.Inverse.RowErasureLemmas |
|
|
| namespace OAI |
|
|
| section |
|
|
| namespace UniqueGamesTheorem.Inverse.RowErasure |
|
|
| open scoped BigOperators Classical |
|
|
| noncomputable section |
|
|
| theorem table_expect_prod {X Y : Type*} [Fintype X] [Fintype Y] |
| (f : X → Y → ℝ) : |
| Finset.univ.expect (fun r : X → Y => ∏ x, f x (r x)) = |
| ∏ x, Finset.univ.expect (f x) := by |
| classical |
| simp only [Fintype.expect_eq_sum_div_card, ← Fintype.prod_sum, |
| Finset.prod_div_distrib, Finset.prod_const, Finset.card_univ, |
| Fintype.card_fun, Nat.cast_pow] |
|
|
| private theorem exp_half_le_seven_quarters : Real.exp (1 / 2 : ℝ) ≤ 7 / 4 := by |
| have h := Real.exp_bound' (x := (1 / 2 : ℝ)) (by norm_num) (by norm_num) |
| (n := 3) (by norm_num) |
| norm_num [Finset.sum_range_succ] at h |
| linarith |
|
|
| private theorem exp_half_indicator (P : Prop) : |
| Real.exp (indicator P / 2) = 1 + (Real.exp (1 / 2 : ℝ) - 1) * indicator P := by |
| classical |
| by_cases h : P <;> simp [indicator, h] |
| |
| theorem table_coordinate_exp_le {Y : Type*} [Fintype Y] [Nonempty Y] |
| (P : Prop) (target : Y) : |
| Finset.univ.expect (fun y => Real.exp (indicator (P ∧ y = target) / 2)) ≤ |
| Real.exp (3 / (4 * (Fintype.card Y : ℝ))) := by |
| classical |
| have hcard : (0 : ℝ) < Fintype.card Y := Nat.cast_pos.mpr Fintype.card_pos |
| by_cases hP : P |
| · have hsingle : Finset.univ.expect (fun y : Y => indicator (y = target)) = |
| 1 / (Fintype.card Y : ℝ) := by |
| simp [indicator] |
| have heq : Finset.univ.expect |
| (fun y => Real.exp (indicator (P ∧ y = target) / 2)) = |
| 1 + (Real.exp (1 / 2 : ℝ) - 1) / (Fintype.card Y : ℝ) := by |
| simp_rw [hP, true_and, exp_half_indicator] |
| rw [Finset.expect_add_distrib, ← Finset.mul_expect, hsingle] |
| simp [div_eq_mul_inv] |
| rw [heq] |
| calc |
| 1 + (Real.exp (1 / 2 : ℝ) - 1) / (Fintype.card Y : ℝ) ≤ |
| 1 + 3 / (4 * (Fintype.card Y : ℝ)) := by |
| have h := div_le_div_of_nonneg_right |
| (show Real.exp (1 / 2 : ℝ) - 1 ≤ 3 / 4 by |
| linarith [exp_half_le_seven_quarters]) hcard.le |
| simpa only [div_div, add_comm] using add_le_add_left h 1 |
| _ ≤ Real.exp (3 / (4 * (Fintype.card Y : ℝ))) := by |
| simpa only [add_comm] using Real.add_one_le_exp (3 / (4 * (Fintype.card Y : ℝ))) |
| · simp only [hP, false_and, indicator_false, zero_div, Real.exp_zero, |
| Fintype.expect_const] |
| exact Real.one_le_exp (by positivity) |
|
|
| end |
| end UniqueGamesTheorem.Inverse.RowErasure |
|
|
| end |
|
|
| end OAI |
|
|
