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import Mathlib
set_option linter.unusedVariables.analyzeTactics true
open Real
lemma le_a_sq
(a b c : ℝ) :
(a + b - c) * (a + c - b) ≤ a ^ 2 := by
have h1: (a + b - c) * (a + c - b) = a ^ 2 - (b - c) ^ 2 := by
linarith
have h2: 0 ≤ (b - c) ^ 2 := by exact pow_two_nonneg (b - c)
rw [h1]
exact sub_le_self _ h2
theorem imo_1964_p2
(a b c : ℝ)
(h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
(h₁ : c < a + b)
(h₂ : b < a + c)
(h₃ : a < b + c) :
a ^ 2 * (b + c - a) + b ^ 2 * (c + a - b) + c ^ 2 * (a + b - c) ≤ 3 * a * b * c := by
have ha : 0 < b + c - a := by exact sub_pos.mpr h₃
have hb : 0 < a + c - b := by exact sub_pos.mpr h₂
have hc : 0 < a + b - c := by exact sub_pos.mpr h₁
have h₄: ((a + b - c) * (a + c - b) * (b + c - a)) ^ 2 ≤ (a * b * c) ^ 2 := by
have h₄₁: (a + b - c) * (a + c - b) ≤ a ^ 2 := by
exact le_a_sq a b c
have h₄₂: (a + b - c) * (b + c - a) ≤ b ^ 2 := by
rw [add_comm a b]
exact le_a_sq b a c
have h₄₃: (a + c - b) * (b + c - a) ≤ c ^ 2 := by
rw [add_comm a c, add_comm b c]
exact le_a_sq c a b
have h₄₄: ((a + b - c) * (a + c - b) * (b + c - a)) ^ 2 = ((a + b - c) * (a + c - b)) *
((a + b - c) * (b + c - a)) * ((a + c - b) * (b + c - a)) := by
linarith
rw [h₄₄]
repeat rw [mul_pow]
refine mul_le_mul ?_ h₄₃ ?_ ?_
. refine mul_le_mul h₄₁ h₄₂ ?_ ?_
. refine le_of_lt ?_
exact mul_pos hc ha
. exact sq_nonneg a
. refine le_of_lt ?_
exact mul_pos hb ha
. refine le_of_lt ?_
simp_all only [sub_pos, gt_iff_lt, pow_pos, mul_pos_iff_of_pos_left]
have h₅: (a + b - c) * (a + c - b) * (b + c - a) ≤ a * b * c := by
refine le_of_pow_le_pow_left₀ (by norm_num) ?_ h₄
refine le_of_lt ?_
refine mul_pos ?_ h₀.2.2
exact mul_pos h₀.1 h₀.2.1
linarith
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