| import Mathlib | |
| namespace OAI | |
| open scoped BigOperators | |
| namespace Problem367 | |
| theorem sum_sq_le_of_two_positive {ι : Type*} [DecidableEq ι] | |
| (s : Finset ι) (r : ι → ℕ) {i j : ι} | |
| (hi : i ∈ s) (hj : j ∈ s) (hij : i ≠ j) | |
| (hri : 0 < r i) (hrj : 0 < r j) : | |
| (∑ k ∈ s, r k ^ 2) ≤ (∑ k ∈ s, r k - 1) ^ 2 + 1 := by | |
| have hsum : r i + ∑ k ∈ s.erase i, r k = ∑ k ∈ s, r k := | |
| Finset.add_sum_erase s r hi | |
| have hsumSq : r i ^ 2 + ∑ k ∈ s.erase i, r k ^ 2 = ∑ k ∈ s, r k ^ 2 := | |
| Finset.add_sum_erase s (fun k => r k ^ 2) hi | |
| have hj' : j ∈ s.erase i := Finset.mem_erase.mpr ⟨Ne.symm hij, hj⟩ | |
| have hrest : 1 ≤ ∑ k ∈ s.erase i, r k := | |
| le_trans hrj (Finset.single_le_sum (fun k _ => Nat.zero_le (r k)) hj') | |
| have hsq : (∑ k ∈ s.erase i, r k ^ 2) ≤ (∑ k ∈ s.erase i, r k) ^ 2 := | |
| Finset.sum_sq_le_sq_sum_of_nonneg (fun k _ => Nat.zero_le (r k)) | |
| have htotal : 1 ≤ ∑ k ∈ s, r k := by omega | |
| have htotalSub := Nat.sub_add_cancel htotal | |
| have hiSub := Nat.sub_add_cancel hri | |
| have hrestSub := Nat.sub_add_cancel hrest | |
| nlinarith [Nat.zero_le ((r i - 1) * ((∑ k ∈ s.erase i, r k) - 1))] | |
| theorem critical_moment_absorption {b theta : ℝ} | |
| (hb : 0 ≤ b) (htheta : 0 ≤ theta) (hthetaOne : theta < 1) | |
| (hcritical : b * theta ^ 2 = 1) {p : ℕ} (hp : 3 ≤ p) : | |
| 0 < 1 - theta ∧ 1 - theta ≤ 1 - b * theta ^ p := by | |
| constructor | |
| · linarith | |
| · have hpow : theta ^ p ≤ theta ^ 3 := | |
| pow_le_pow_of_le_one htheta hthetaOne.le hp | |
| have hmul := mul_le_mul_of_nonneg_left hpow hb | |
| have hcube : b * theta ^ 3 = theta := by nlinarith [hcritical] | |
| linarith | |
| end Problem367 | |
| end OAI | |