theorem extracted_formal_statement_213 {a b c : ℝ} (h : a ^ 2 ≤ b * c) (hpos : 0 ≤ b + c) (this : 4 * b * c ≤ (b + c) ^ 2) :
  2 * a ≤ b + c := sorry

State before:
a b c : ℝ
h : a ^ 2 ≤ b * c
hpos : 0 ≤ b + c
this : 4 * b * c ≤ (b + c) ^ 2
⊢ 2 * a ≤ b + c

Tactic: exact (abs_le_of_sq_le_sq' (by linarith) hpos).2

State after:

Relevant lemmas:
abs_le_of_sq_le_sq' : ∀ {α : Type u_1} [inst : LinearOrderedRing α] {a b : α}, a ^ 2 ≤ b ^ 2 → 0 ≤ b → -b ≤ a ∧ a ≤ b