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import Mathlib
set_option linter.unusedVariables.analyzeTactics true
open Real
lemma mylemma_1
(a b c : ℝ)
(x y z : ℝ)
(h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
(h₂: c ≤ b ∧ b ≤ a)
(h₃: z ≤ y ∧ y ≤ x) :
a * z + c * y + b * x ≤ c * z + b * y + a * x := by
suffices h₄: c * (y - z) + b * (x - y) ≤ a * (x - z)
. linarith
. have h₅: c * (y - z) + b * (x - y) ≤ b * (y - z) + b * (x - y) := by
simp
refine mul_le_mul h₂.1 ?_ ?_ ?_
. exact le_rfl
. exact sub_nonneg_of_le h₃.1
. exact le_of_lt h₀.2.1
refine le_trans h₅ ?_
rw [mul_sub, mul_sub, add_comm]
rw [← add_sub_assoc, sub_add_cancel]
rw [← mul_sub]
refine mul_le_mul h₂.2 ?_ ?_ ?_
. exact le_rfl
. refine sub_nonneg_of_le ?_
exact le_trans h₃.1 h₃.2
. exact le_of_lt h₀.1
lemma mylemma_2
(a b c : ℝ)
(x y z : ℝ)
(h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
(h₂: c ≤ b ∧ b ≤ a)
(h₃: z ≤ y ∧ y ≤ x) :
b * z + a * y + c * x ≤ c * z + b * y + a * x := by
suffices h₄: c * (x - z) + b * (z - y) ≤ a * (x - y)
. linarith
. have h₅: c * (x - z) + b * (z - y) ≤ b * (x - z) + b * (z - y) := by
simp
refine mul_le_mul h₂.1 ?_ ?_ ?_
. exact le_rfl
. refine sub_nonneg_of_le ?_
exact le_trans h₃.1 h₃.2
. exact le_of_lt h₀.2.1
refine le_trans h₅ ?_
rw [mul_sub, mul_sub]
rw [← add_sub_assoc, sub_add_cancel]
rw [← mul_sub]
refine mul_le_mul h₂.2 ?_ ?_ ?_
. exact le_rfl
. exact sub_nonneg_of_le h₃.2
. exact le_of_lt h₀.1
-- case #1
lemma mylemma_cba
(a b c : ℝ)
(hap : 0 < a )
(hbp : 0 < b )
(hcp : 0 < c )
(h₁ : c < a + b)
-- (h₂ : b < a + c)
(h₃ : a < b + c)
(hba: b ≤ a)
(hcb: c ≤ b) :
0 ≤ a^2 * b * (a - b) + b^2 * c * (b - c) + c^2 * a * (c - a) := by
have g₀: b * c ≤ a * c := by exact (mul_le_mul_iff_of_pos_right hcp).mpr hba
have g₁: a * c ≤ a * b := by exact (mul_le_mul_iff_of_pos_left hap).mpr hcb
have g₂: a * (b + c - a) ≤ b * (a + c - b) := by
have g₂₁: 0 ≤ (a-b) * (a+b-c) := by
refine mul_nonneg ?_ ?_
. exact sub_nonneg_of_le hba
. refine le_of_lt ?_
exact sub_pos.mpr h₁
linarith
have g₃: b * (a + c - b) ≤ c * (a + b - c) := by
have g₃₁: 0 ≤ (b - c) * (b + c - a) := by
refine mul_nonneg ?_ ?_
. exact sub_nonneg_of_le hcb
. refine le_of_lt ?_
exact sub_pos.mpr h₃
linarith
have g₄: (a * b) * (a * (b + c - a)) + (b * c) * (b * (a + c - b)) + (a * c) * (c * (a + b - c))
≤ (b * c) * (a * (b + c - a)) + (a * c) * (b * (a + c - b)) + (a * b) * (c * (a + b - c)) := by
refine mylemma_1 (a * b) (a * c) (b * c) (c * (a + b - c)) (b * (a + c - b)) (a * (b + c - a)) ?_ ?_ ?_
. constructor
. exact mul_pos hap hbp
. constructor
. exact mul_pos hap hcp
. exact mul_pos hbp hcp
. exact { left := g₀, right := g₁ }
. exact { left := g₂, right := g₃ }
linarith
-- tight version
lemma mylemma_cba_tight
(a b c : ℝ)
(hap : 0 < a )
(hbp : 0 < b )
(hcp : 0 < c )
(h₁ : c < a + b)
-- (h₂ : b < a + c)
(h₃ : a < b + c)
(hba: b ≤ a)
(hcb: c ≤ b) :
0 ≤ a^2 * c * (a - c) + c^2 * b * (c - b) + b^2 * a * (b - a) := by
have g₀: b * c ≤ a * c := by exact (mul_le_mul_iff_of_pos_right hcp).mpr hba
have g₁: a * c ≤ a * b := by exact (mul_le_mul_iff_of_pos_left hap).mpr hcb
have g₂: a * (b + c - a) ≤ b * (a + c - b) := by
have g₂₁: 0 ≤ (a-b) * (a+b-c) := by
refine mul_nonneg ?_ ?_
. exact sub_nonneg_of_le hba
. refine le_of_lt ?_
exact sub_pos.mpr h₁
linarith
have g₃: b * (a + c - b) ≤ c * (a + b - c) := by
have g₃₁: 0 ≤ (b - c) * (b + c - a) := by
refine mul_nonneg ?_ ?_
. exact sub_nonneg_of_le hcb
. refine le_of_lt ?_
exact sub_pos.mpr h₃
linarith
have g₄: (a * c) * (a * (b + c - a)) + (a * b) * (b * (a + c - b)) + (b * c) * (c * (a + b - c))
≤ (b * c) * (a * (b + c - a)) + (a * c) * (b * (a + c - b)) + (a * b) * (c * (a + b - c)) := by
refine mylemma_2 (a * b) (a * c) (b * c) (c * (a + b - c)) (b * (a + c - b)) (a * (b + c - a)) ?_ ?_ ?_
. constructor
. exact mul_pos hap hbp
. constructor
. exact mul_pos hap hcp
. exact mul_pos hbp hcp
. exact { left := g₀, right := g₁ }
. exact { left := g₂, right := g₃ }
linarith
theorem imo_1983_p6
(a b c : ℝ)
(h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
(h₁ : c < a + b)
(h₂ : b < a + c)
(h₃ : a < b + c) :
0 ≤ a^2 * b * (a - b) + b^2 * c * (b - c) + c^2 * a * (c - a) := by
wlog ho₀: b ≤ a generalizing a b c
. clear this
push_neg at ho₀
wlog ho₁: c ≤ b generalizing a b c
. clear this
push_neg at ho₁ -- a < b < c
rw [add_comm] at h₁ h₂ h₃
have g₀: 0 ≤ c ^ 2 * a * (c - a) + a ^ 2 * b * (a - b) + b ^ 2 * c * (b - c) := by
exact mylemma_cba_tight c b a h₀.2.2 h₀.2.1 h₀.1 h₃ h₁ (le_of_lt ho₁) (le_of_lt ho₀)
linarith
. wlog ho₂: c ≤ a generalizing a b c
. clear this -- a < c ≤ b
push_neg at ho₂
rw [add_comm] at h₁ h₂
have g₀: 0 ≤ b ^ 2 * c * (b - c) + c ^ 2 * a * (c - a) + a ^ 2 * b * (a - b) := by
exact mylemma_cba b c a h₀.2.1 h₀.2.2 h₀.1 h₃ h₂ ho₁ (le_of_lt ho₂)
linarith
. -- c ≤ a < b
rw [add_comm] at h₁
have g₀: 0 ≤ b ^ 2 * c * (b - c) + c ^ 2 * a * (c - a) + a ^ 2 * b * (a - b) := by
exact mylemma_cba_tight b a c h₀.2.1 h₀.1 h₀.2.2 h₁ h₂ (le_of_lt ho₀) ho₂
linarith
. wlog ho₁: c ≤ b generalizing a b c
. clear this
push_neg at ho₁
wlog ho₂: c ≤ a generalizing a b c
. clear this
push_neg at ho₂ -- b < a < c
rw [add_comm] at h₂ h₃
have g₀: 0 ≤ c ^ 2 * a * (c - a) + a ^ 2 * b * (a - b) + b ^ 2 * c * (b - c) := by
exact mylemma_cba c a b h₀.2.2 h₀.1 h₀.2.1 h₂ h₁ (le_of_lt ho₂) ho₀
linarith
. rw [add_comm] at h₃
exact mylemma_cba_tight a c b h₀.1 h₀.2.2 h₀.2.1 h₂ h₃ ho₂ (le_of_lt ho₁)
. exact mylemma_cba a b c h₀.1 h₀.2.1 h₀.2.2 h₁ h₃ ho₀ ho₁
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