| import Mathlib.Basic.Real.Basic |
| import Mathlib.Algebra.BigOperators.Fin |
| import Mathlib.Algebra.Order.BigOperators.Group.Finset |
| import Mathlib.Tactic.Linarith |
| import Mathlib.Tactic.NormNum |
| import Mathlib.Tactic.Positivity |
| import Mathlib.Tactic.Ring |
|
|
| namespace OAI |
|
|
| /-! |
| # Coefficients for the finite tuple estimate |
|
|
| Replacing `e` by `3` gives the coefficients `3 / 64^(i+1)`. Their finite |
| sum has an explicit positive remainder below `1 / 21`. A finite product |
| inequality then bounds the product of their complements from below. |
| -/ |
|
|
| namespace TalagrandThreshold |
|
|
| open scoped BigOperators |
|
|
| / |
| noncomputable def tupleCoefficient (i : ℕ) : ℝ := |
| 3 / 64 ^ (i + 1) |
|
|
| theorem tupleCoefficient_eq (i : ℕ) : |
| tupleCoefficient i = (3 / 64 : ℝ) * (1 / 64 : ℝ) ^ i := by |
| simp only [tupleCoefficient, pow_succ, div_eq_mul_inv, mul_inv_rev, one_mul, |
| inv_pow] |
| ring |
|
|
| theorem tupleCoefficient_pos (i : ℕ) : 0 < tupleCoefficient i := by |
| unfold tupleCoefficient |
| positivity |
|
|
| theorem tupleCoefficient_nonneg (i : ℕ) : 0 ≤ tupleCoefficient i := |
| (tupleCoefficient_pos i).le |
|
|
| theorem tupleCoefficient_le_first (i : ℕ) : tupleCoefficient i ≤ 3 / 64 := by |
| rw [tupleCoefficient_eq] |
| have hpow : (1 / 64 : ℝ) ^ i ≤ 1 := |
| pow_le_one₀ (by norm_num) (by norm_num) |
| nlinarith |
|
|
| theorem tupleCoefficient_lt_one (i : ℕ) : tupleCoefficient i < 1 := |
| lt_of_le_of_lt (tupleCoefficient_le_first i) (by norm_num) |
|
|
| / |
| theorem sum_tupleCoefficient_add_remainder (s : ℕ) : |
| (∑ i : Fin s, tupleCoefficient i.val) + |
| (1 / 21 : ℝ) * (1 / 64 : ℝ) ^ s = 1 / 21 := by |
| induction s with |
| | zero => norm_num |
| | succ s ih => |
| rw [Fin.sum_univ_castSucc] |
| simp only [Fin.val_castSucc, Fin.val_last] |
| rw [tupleCoefficient_eq s, pow_succ] |
| nlinarith [ih] |
|
|
| theorem sum_tupleCoefficient_lt (s : ℕ) : |
| (∑ i : Fin s, tupleCoefficient i.val) < 1 / 21 := by |
| have hrem : 0 < (1 / 21 : ℝ) * (1 / 64 : ℝ) ^ s := by positivity |
| linarith [sum_tupleCoefficient_add_remainder s] |
|
|
| theorem sum_tupleCoefficient_le (s : ℕ) : |
| (∑ i : Fin s, tupleCoefficient i.val) ≤ 1 / 21 := |
| (sum_tupleCoefficient_lt s).le |
|
|
| / |
| theorem one_sub_sum_le_prod_one_sub {ι : Type*} [Fintype ι] |
| (c : ι → ℝ) (h0 : ∀ i, 0 ≤ c i) (h1 : ∀ i, c i ≤ 1) : |
| 1 - ∑ i, c i ≤ ∏ i, (1 - c i) := by |
| classical |
| suffices h : ∀ s : Finset ι, |
| 1 - ∑ i ∈ s, c i ≤ ∏ i ∈ s, (1 - c i) by |
| exact h Finset.univ |
| intro s |
| induction s using Finset.induction_on with |
| | empty => simp |
| | @insert a s ha ih => |
| rw [Finset.sum_insert ha, Finset.prod_insert ha] |
| have hs : 0 ≤ ∑ i ∈ s, c i := |
| Finset.sum_nonneg (fun i _ => h0 i) |
| have hm := mul_le_mul_of_nonneg_left ih (sub_nonneg.mpr (h1 a)) |
| nlinarith [mul_nonneg (h0 a) hs] |
|
|
| theorem prod_one_sub_tupleCoefficient_gt (s : ℕ) : |
| (20 / 21 : ℝ) < ∏ i : Fin s, (1 - tupleCoefficient i.val) := by |
| have hprod := one_sub_sum_le_prod_one_sub (fun i : Fin s => tupleCoefficient i.val) |
| (fun i => tupleCoefficient_nonneg i.val) (fun i => (tupleCoefficient_lt_one i.val).le) |
| have hsum := sum_tupleCoefficient_lt s |
| linarith |
|
|
| end TalagrandThreshold |
|
|
| end OAI |
|
|
