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import Mathlib
set_option linter.unusedVariables.analyzeTactics true
open Real
lemma imo_1964_p2_1
(a b c : ℝ)
(ha : 0 < -a + b + c)
(hb : 0 < a - b + c)
(hc : 0 < a + b - c)
(g1 : (a + b - c) * (a - b + c) * (-a + b + c) ≤ a * b * c) :
((a + b - c) * (a - b + c) * (-a + b + c)) ^ 2 ≤ (a * b * c) ^ 2 := by
refine pow_le_pow_left₀ (le_of_lt ?_) g1 2
exact mul_pos (mul_pos hc hb) ha
lemma imo_1964_p2_2
(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
rw [h₁]
refine sub_le_self _ ?_
exact sq_nonneg (b - c)
lemma imo_1964_p2_3
(a b c : ℝ) :
a ^ 2 - (b - c) ^ 2 ≤ a ^ 2 := by
simp
exact sq_nonneg (b - c)
lemma imo_1964_p2_4
(a b c : ℝ)
(h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
(h₁ : c < a + b)
(h₂ : b < a + c)
(h₃ : a < b + c) :
((a + b - c) * (a + c - b) * (b + c - a)) ^ 2 ≤ (a * b * c) ^ 2 := 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 h₄₁: (a + b - c) * (a + c - b) ≤ a ^ 2 := by
exact imo_1964_p2_2 a b c
have h₄₂: (a + b - c) * (b + c - a) ≤ b ^ 2 := by
rw [add_comm a b]
exact imo_1964_p2_2 b a c
have h₄₃: (a + c - b) * (b + c - a) ≤ c ^ 2 := by
rw [add_comm a c, add_comm b c]
exact imo_1964_p2_2 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]
lemma imo_1964_p2_5
(a b c : ℝ)
-- (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
-- (h₁ : c < a + b)
-- (h₂ : b < a + c)
-- (h₃ : a < b + c)
(ha : 0 < b + c - a)
(hb : 0 < a + c - b)
(hc : 0 < a + b - c)
(h₄₁ : (a + b - c) * (a + c - b) ≤ a ^ 2)
(h₄₂ : (a + b - c) * (b + c - a) ≤ b ^ 2)
(h₄₃ : (a + c - b) * (b + c - a) ≤ c ^ 2) :
((a + b - c) * (a + c - b) * (b + c - a)) ^ 2 ≤ (a * b * c) ^ 2 := by
repeat rw [mul_pow]
rw [pow_two, pow_two, pow_two]
have h₅: ((a + b - c) * (a + c - b)) * ((a + b - c) * (b + c - a)) ≤ a ^ 2 * b ^ 2 := by
refine mul_le_mul h₄₁ h₄₂ ?_ ?_
. refine le_of_lt ?_
exact mul_pos hc ha
. exact sq_nonneg a
have h₆: ((a + b - c) * (a + c - b)) * ((a + b - c) * (b + c - a)) * ((a + c - b) * (b + c - a))
≤ a ^ 2 * b ^ 2 * c ^ 2 := by
refine mul_le_mul h₅ h₄₃ ?_ ?_
. refine le_of_lt ?_
exact mul_pos hb ha
. refine mul_nonneg ?_ ?_
. exact sq_nonneg a
. exact sq_nonneg b
linarith
lemma imo_1964_p2_6
(a b c : ℝ)
-- (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
-- h₁ : c < a + b
-- h₂ : b < a + c
-- h₃ : a < b + c
(ha : 0 < b + c - a)
(hb : 0 < a + c - b)
(hc : 0 < a + b - c)
(h₄₁ : (a + b - c) * (a + c - b) ≤ a ^ 2)
(h₄₂ : (a + b - c) * (b + c - a) ≤ b ^ 2)
(h₄₃ : (a + c - b) * (b + c - a) ≤ c ^ 2)
(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))) :
((a + b - c) * (a + c - b) * (b + c - a)) ^ 2 ≤ a ^ 2 * b ^ 2 * c ^ 2 := by
rw [h₄₄]
have h₅: ((a + b - c) * (a + c - b)) * ((a + b - c) * (b + c - a)) ≤ a ^ 2 * b ^ 2 := by
refine mul_le_mul h₄₁ h₄₂ ?_ ?_
. refine le_of_lt ?_
exact mul_pos hc ha
. exact sq_nonneg a
have h₆: ((a + b - c) * (a + c - b)) * ((a + b - c) * (b + c - a)) * ((a + c - b) * (b + c - a))
≤ a ^ 2 * b ^ 2 * c ^ 2 := by
refine mul_le_mul h₅ h₄₃ ?_ ?_
. refine le_of_lt ?_
exact mul_pos hb ha
. refine mul_nonneg ?_ ?_
. exact sq_nonneg a
. exact sq_nonneg b
linarith
lemma imo_1964_p2_7
(a b c : ℝ)
-- (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
-- (h₁ : c < a + b)
-- (h₂ : b < a + c)
-- (h₃ : a < b + c)
(ha : 0 < b + c - a)
-- (hb : 0 < a + c - b)
(hc : 0 < a + b - c)
(h₄₁ : (a + b - c) * (a + c - b) ≤ a ^ 2)
(h₄₂ : (a + b - c) * (b + c - a) ≤ b ^ 2) :
-- (h₄₃ : (a + c - b) * (b + c - a) ≤ c ^ 2)
-- (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))) :
(a + b - c) * (a + c - b) * ((a + b - c) * (b + c - a)) ≤ a ^ 2 * b ^ 2 := by
refine mul_le_mul h₄₁ h₄₂ ?_ ?_
. refine le_of_lt ?_
exact mul_pos hc ha
. exact sq_nonneg a
lemma imo_1964_p2_8
(a b c : ℝ)
(h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
-- (h₁ : c < a + b)
-- (h₂ : b < a + c)
-- (h₃ : a < b + c)
-- (ha : 0 < b + c - a)
-- (hb : 0 < a + c - b)
-- (hc : 0 < a + b - c)
(h₄ : ((a + b - c) * (a + c - b) * (b + c - a)) ^ 2 ≤ (a * b * c) ^ 2) :
(a + b - c) * (a + c - b) * (b + c - a) ≤ a * b * c := by
refine le_of_pow_le_pow_left₀ ?_ ?_ h₄
. norm_num
. refine le_of_lt ?_
refine mul_pos ?_ h₀.2.2
exact mul_pos h₀.1 h₀.2.1
lemma imo_1964_p2_9
(a b c : ℝ)
-- (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
-- (h₁ : c < a + b)
-- (h₂ : b < a + c)
-- (h₃ : a < b + c)
-- (ha : 0 < b + c - a)
-- (hb : 0 < a + c - b)
-- (hc : 0 < a + b - c)
-- (h₄ : ((a + b - c) * (a + c - b) * (b + c - a)) ^ 2 ≤ (a * b * c) ^ 2)
(h₅ : (a + b - c) * (a + c - b) * (b + c - a) ≤ a * b * c) :
a ^ 2 * (b + c - a) + b ^ 2 * (c + a - b) + c ^ 2 * (a + b - c) ≤ 3 * a * b * c := by
repeat rw [mul_sub]
repeat rw [mul_add]
linarith
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