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f61e547 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 | 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
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